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Purchasing power parity puzzle as a trade phenomenon Berka, Martin 2005

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Purchasing Power Parity Puzzle as a Trade Phenomenon by Martin Berka B.A., Management., Comenius University, Bratislava, 1994 M.B.A. , Comenius University, Bratislava, 1996 M.A. Econ., Central European University, Budapest, 1998 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in The Faculty of Graduate Studies (Economics) T H E UNIVERSITY OF BRITISH COLUMBIA May 2005 © Martin Berka, 2005 11 Abstract The purchasing power parity puzzle is among the central issues of international macroeconomics. In my thesis, I document and explain the persistence and volatility of its empirical counterpart -the real exchange rate - as a trade phenomenon arising from heterogeneous physical characteristics of products and geography. In the first chapter, a general equilibrium model with shipping costs that depend on physical characteristics of goods and distance leads to endogenous tradability of goods. Deviations of prices from parity are sustained as long as they do not exceed the heteroge-neous trade frictions. The real exchange rate exhibits deviations whose persistence matches the data and, when quadratic adjustment costs in change of trade volume are added to the model, also the volatility of the real exchange rate deviations matches the data as well. The second chapter studies monthly deviations from the law of one price for a group of 63 goods and services in Canada and USA between 1970 and 2000 and relates them to a separate dataset of price-to-weight and price-to-volume ratios. Threshold estimates are significantly negatively related to the estimates of the price-to-weight ratios and price-to-volume ratios, respectively. Physical charac-teristics of goods are important empirical determinants of heterogeneous non-linear behavior of deviations of their prices from parity. The third chapter studies the implications for monetary policy in a general equilibrium model where credit crunches occur due to shifts in the distribution of assets among heterogeneous households. Martin Berka. mberka@interchange.ubc.ca iii Contents Abstract ii Contents iii List of Tables vi List of Figures viii Acknowledgements x Summary xi 1 General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Per-sistence 1 1.1 Introduction 1 1.2 General equilibrium model of arbitrage trade 6 1.2.1 Households 7 1.2.2 Arbitrage trading firms 8 1.2.3 Market clearing 10 1.2.4 Equilibrium 10 1.3 Arbitrage trade model with adjustment costs to trade 15 1.3.1 Equilibrium 17 1.3.2 Solution method 22 1.4 Real exchange rate 24 1.4.1 Real exchange rate in a model with quadratic adjustment costs 27 1.4.2 Country-specific shocks 30 1.4.3 Sector-specific shocks 30 Contents iv 1.5 Calibration 32 1.6 Simulation results 34 1.6.1 Simulation results in a linear shipping cost model 34 1.6.2 Simulation results in a quadratic adjustment cost model 37 1.7 Conclusion 38 2 Non-linear Adjustment in International Prices and Physical Characteristics of Goods 40 2.1 Motivation and the literature 42 2.2 Data 48 2.2.1 Limitations of studying index-based RER 48 2.2.2 Dataset description 49 2.3 Empirical analysis and results 51 2.3.1 Model selection 51 2.3.2 Methodology 52 2.3.3 Non-linear estimation results 54 2.3.4 Testing 57 2.3.5 Determinants of no-arbitrage bounds 58 2.4 Conclusion 59 3 Basle Accord and Financial Intermediation: The Impact of Policy 61 3.1 Introduction 61 3.2 Model 63 3.2.1 Overview 63 3.2.2 Households 63 3.2.3 Financial Sector 67 3.2.4 Central bank 68 3.2.5 Market clearing 69 3.2.6 Equilibrium 71 3.3 Parametrization 71 3.4 Capital requirements, bank lending and monetary policy 73 Contents v 3.4.1 Countercyclical monetary policy 74 3.4.2 Procyclical monetary policy 75 3.4.3 Countercyclical capital requirements 76 3.4.4 Procyclical capital requirements 77 3.4.5 Credit crunch? What exactly happens in the model? 77 3.4.6 Does the equity market worsen or soften the credit decline? 78 3.5 Conclusion 79 Bibliography 80 A 89 A. l Approximating the absolute value function 89 Figure: linear model, autarky solution 91 Figure: linear model, trade volume 91 Figure: linear model, mean and volatility of price deviations 92 B 103 B. l Tsay's F-test for non-linearity 103 B.2 Constructing the price and weight, and volume datasets 104 Artifact: Sample of N Y C prices in Feb 2001 109 B. 3 Tables and Figures 112 Figure: Price-to-weight ratios 117 Figure: Price-to-weight ratios, no alcohol and energies 118 Figure: Price-to-weight ratios, no alcohol and energies 2 121 C 126 C. l Solving the banks' problem 126 C.2 On the assumption of a single portfolio optimization 127 C.3 The solution procedure 128 C.4 Figures 129 vi List of Tables 1.1 RER half lives (in quarters) and the shock process 25 1.2 Standard deviation of the RER and the shock process 26 1.3 RER half-lives and the relative trade friction 27 1.4 Mean half-life of convergence estimate of log(RER) when c=0.1 29 1.5 Mean half-life of convergence estimate of log(RER) when c=0.01 29 1.6 [Mean std(log(RER))]/[Mean std(log(GDP))] when c=0.1 29 1.7 [Median std(log(RER))]/[Median std(log(GDP))] when c=0.1 • • 30 1.8 Properties of the US-EU model simulation 35 B. l Coverage of CPI by the data 105 B.2 Part 2 of the CPI coverage table. Bold series are not included in the dataset. . . 106 B.3 Data sources on weights and prices - part 1 107 B.4 Data sources on weights and prices - part 2 108 B.5 Data sources on volume - part 1 110 B.6 Data sources on volume - part 2 I l l B.7 Long run properties of exchange rates 113 B.8 Long run properties of linearly detrended exchange rates 113 B.9 Long run properties of HP-detrended exchange rates 113 B.10 Long run properties:ADF and half-life convergence 114 B . l l EQ-TAR Summary 115 B.12 BAND-TAR Summary 115 B.13 AR(1) Results - Part I 116 B.14 AR(1) Results - part II 118 B.15 EQ-TAR(2,1,1) results 119 B.16 EQ-TAR(2,p,d) results - part I 120 List of Tables vii B.17 EQ-TAR(2,p,d) results - part II 121 B.18 BAND-TAR(2,p,d) results - part I 122 B.19 BAND-TAR(2,p,d) results - part II 123 B.20 Price-to-weight ratios, Price-to-volume ratios, and thresholds 123 B.21 Nonlinear relationship between price-to-weights and thresholds 124 B.22 Prices, weights and thresholds 124 B.23 Price-to-weight ratios, Price-to-volume ratios, and thresholds - robustness 125 viii List of Figures 1.1 Model solution: thresholds of price deviations in linear model 13 1.2 Simplified profit, functions in a quadratic and linear model when N-i = 0 20 1.3 Optimal trade volume in partial equilibrium in QAC model for various price deviations and N-i. c=0.01, t=0.2 22 1.4 RER persistence and volatility and the shock process in linear model 25 1.5 Properties of the real exchange rate and the trade friction in linear model 26 1.6 RER properties with country - and sector - specific endowment shocks in linear model 31 1.7 Calibration of per-kg-per-km shipping cost for cooled sea transport 32 A . l Approximating functions g(N) and G(N) for A = 105 89 A.2 Linear model: Autarky solution 91 A.3 Linear model: International trade 91 A.4 Linear model: means and standard deviations of the price deviations as functions of shock volatility 92 A.5 Linear model: average thresholds of the law of one price deviation as function of shock volatility 92 A.6 Linear model robustness: mean price deviations as a function of shock volatility and relative friction 93 A.7 Linear model robustness: volatility of price deviations as a function of shock volatil-ity and relative friction 93 A.8 Linear model robustness: price deviations thresholds as a function of shock volatil-ity and relative friction 94 A.9 Distribution of trade. US-EU simulation of the linear model 94 A. 10 Distribution of international prices. US-EU simulation of the linear model 95 List of Figures ix A.11 Distribution of the real exchange rate and the shock process. US-EU simulation of the linear model 95 A. 12 Approximating the indicator function in QAC model 96 A. 13 Thresholds in Q A C model when c=0.000001 97 A. 14 Thresholds in Q A C model when c=0.001 97 A. 15 Thresholds in Q A C model when c=0.1 98 A.16 Thresholds in Q A C model when c=l 98 A.17 Profits in partial equilibrium, for various price deviations and JV_i. c=0.01, t=0.2 . . . . 99 A. 18 Distribution of standard deviation estimates in a QAC model, c=0.1 99 A.19 Price deviations in a QAC model, c=0.1 100 A.20 Endowment shocks and RER in a QAC model, c=0.1 101 A.21 Trade in a QAC model, c=0.1 101 A. 22 Consumption co-movement in a QAC model, c=0.1 102 B. l Some prices in N Y C , February 2001 109 B.2 Nonlinear fit of price-weight ratios and thresholds 117 B.3 Nonlinear lit of price-weight ratios and thresholds 118 B. 4 Nonlinear fit of price-weight ratios and thresholds 121 C. l Benchmark economy as it cycles through all possible aggregate states 129 C.2 Benchmark and policy with interest rate reduction in worst case only 130 C.3 Benchmark and policy with gradual interest rate reduction in bad return situations 131 C.4 Benchmark and policy with interest rate increase in worst case only 132 C.5 Benchmark and policy with gradual interest rate increase as aggregate states worsenl33 C.6 Benchmark and policy with relaxing of capital requirements in worst case only . . 134 C.7 Benchmark and policy with gradual relaxing of capital requirements as aggregate states worsen 135 C.8 Benchmark and policy with tightening of capital requirements in worst case only . 136 C.9 Benchmark and policy with gradual tightening of capital requirements as aggregate states worsen 137 Acknowledgements I am greatly indebted to my supervisory committee, Professor Michael B. Devereux and Pro-fessor John Helliwell for their excellent guidance and support. I would like to thank Professor Christian Zimmermann, our common work is the foundation of my last essay. I would also like to thank Professor James Nason, Professor Paul Beaudry, Professor Brian Copeland, Professor Patrick Francois, Professor Francisco Gonzalez, and Professor Henry Siu for helpful suggestions and discussions at various stages of my dissertation. I am also grateful to my classmates, col-leagues and friends, Lilia Karnizova, Kristin L0seth, Christoph Schleicher, Genevieve Verdier, Jenny Xu, Jacob Wong and many others for helpful suggestions, discussions and support. I am solely responsible for any errors and misinterpretations. xi Summary The first chapter of my thesis shows how modelling of physical characteristics of goods and geography can explain the puzzling persistence and volatility in the deviations of the prices from parity. Consequently, it can also explain the purchasing power parity puzzle. In a two-country, three-good general equilibrium model, arbitrage firms trade goods across borders. Shipping costs depend on distance and physical characteristics of the product (weight or volume). Tradability of goods is endogenous - goods are traded only when the difference in their price from parity exceeds the shipping cost. The adjustment of prices across borders is non-linear, with heterogeneous thresholds that depend positively on the physical characteristic of a product and the distance an empirical regularity. Aggregation of the law of one price deviations implies a smooth threshold non-linearity in the real exchange rate, justifying a reoccurring finding in the recent empirical literature. When the stochastic endowments process is calibrated to match the quarterly HP-filtered US and E U GDPs, and trade costs equal on average less than 1% of good's value, the half-life of deviation in the real exchange rate matches the persistence found in the data. A model with quadratic adjustment costs in change of trade volume also matches real exchange rate volatility. The PPP puzzle can thus be explained entirely as a trade phenomenon. The second paper analyzes the law of one price deviations across 63 groups of goods and services between the US and Canada and combines them with a separate dataset of physical char-acteristics of products (weight and volume). The estimation and testing of linear- and threshold-autoregressive models on monthly data (1970-2000) shows that the adjustment of prices of goods and services is often non-linear: there are heterogeneous no-trade bands within which deviations of prices from parity do not revert towards the mean. The estimates of the band size vary from 0.5% for footwear to 30% for tobacco. First, in terms of the speed of convergence, services have an average half-life of 200 months while goods have an average half-life of 47 months. Second, one half of the persistence in the real exchange rate deviations can be accounted for by a proper mod-eling of the prices of tradable goods, in that it is contained within the aforementioned thresholds. Summary xii Third, there is a large heterogeneity across goods and services in the dynamics of the law of one price deviations. Threshold estimates of tradable goods are significantly negatively related their price-to-weight ratios, as well as to price-to-volume ratios. Less bulky goods (relative to price) goods have narrower no-trade bounds which is why their price adjusts faster to a shock to the law of one price deviation. A similar relationship is found between the heterogeneous half lives and price-to-weight ratios. The paper establishes stylized facts about the heterogeneity in the price adjustment across goods and finds a significant empirical support for the model from the first chapter of my dissertation. The third paper studies the loan activity in a context where banks have to follow Basle Accord type rules and need to find financing from households. Loan activity typically decreases when investment returns of entrepreneurs decline, and we study policies that may address this event in a general equilibrium model with heterogeneous agents. Active capital requirement policy can be effective as well if it implies tightening of regulation in bad times. Basle Accord rules are more powerful in this respect. We also show that an identification of a credit crunch may be difficult even when banks are clearly tightening credit, due to somewhat counterintuitive behaviour of some indicators. The paper also develops new solution methods for transition paths in dynamic heterogeneous agent models with aggregate shocks. Chapter 1 i General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 1.1 In t roduc t ion Why do prices of tradable goods between countries diverge in such a persistent manner? How do physical characteristics of products and the geographical distances influence the deviations of good prices from the law of one price? Can this explain the persistent purchasing power parity puzzle? This chapter aims to answer these questions. The concept of purchasing power parity (PPP) maintains that national price levels should be equal when expressed in the units of a common currency (Cassel (1918)). Translated into observ-ables, it states that the real exchange rate (a ratio of price indexes in two countries expressed in terms of a single currency) should be constant. The central puzzle in the international business cycle literature is that fluctuations in the real exchange rate are very large and very persistent. Traditionally, attempts to address this puzzle were based on the Harrod-Balassa-Samuelson ob-jection to PPP, utilizing the distinction between traded and non-traded goods (Balassa (1961)). The real exchange rate then equals the relative price of non-traded goods to traded goods. How-ever, these models were shown to be empirically unwarranted for developed economies1. Most notably, Engel (1999) shows that in the U.S. data, no more than 2% of the variation in the real exchange rate can be attributed to the fluctuations in the relative price of non-traded to traded goods. Dozens of empirical studies document large, volatile and persistent deviations in the prices of traded goods across countries. Therefore, deviations in the prices of traded goods are the empirically relevant cornerstone of 1 Harrod-Balassa-Samuelson proposition holds holds better for emerging and developing economies, and at lower frequencies. See, i.a., Choudhri & Khan (2004) Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 2 the current theoretical approaches2. Several avenues have been explored to motivate the deviations of prices of traded goods from the law of one price. Pricing to market combined with nominal rigidities has been used widely in creating volatile deviations in the real exchange rate (Betts & Devereux (2000), Bergin & Feenstra (2001)). In particular, a year-long price stickiness combined with a low degree of intertemporal elasticity of substitution and consumption - leisure separable preferences generates sufficient volatility but not sufficient persistence in the real exchange rate (Devereux (1997), Chari, Kehoe & McGrattan (2002)). A distribution costs approach (e.g., Corsetti & Dedola (2002), Burstein, Neves & Rebello (2001)) justifies wedges between the prices of tradable goods but has to rely on very large costs to product distribution to match the volatility of the real exchange rate. Differences in preferences across countries have also been used to rationalize deviations from the law of one price (e.g., Lapham & Vigneault (2001)) but must resort to volatile and highly persistent shocks to the preference substitution parameters in order to match the observed fluctuations in the prices of traded goods. Finally, models of the costs of arbitrage trade were relatively unsuccessful in generating deviations from the law of one price (e.g., Obstfeld & Rogoff (2000), Dumas (1992), Ohanian & Stockman (1997), Prakash & Taylor (1997), Sercu, Uppal & van Hulle (1995)). This paper fits into the last strand of literature. Recent empirical literature stresses the finding that the law of one price deviations behave in a non-linear and heterogeneous way (e.g., O'Connel & Wei (2002), Crucini, Telmer & Zachariadis (2001)). The causal relationship of this heterogeneity has not been understood so far. In the second paper of my dissertation I use post-Bretton Woods monthly price series for 63 traded goods in the U.S. and Canada and a separate dataset of product weights and volumes to show that the thresholds and half-lives in the deviations of the law of one price are significantly positively related to product weights and volumes and are also negatively related to the price-to-weight and price-to-volume ratios. Understanding of the thresholds is important for the understanding of the real exchange rate because they alone account for about 50% of the persistence in the real exchange rate. The two general equilibrium models presented in this chapter show how geography and physical characteristics of goods explain the persistence in the deviations of good prices - and the real exchange rate - from parity. In a world inhabited by identical consumers, three consumption 2 Real exchange rate differences can also result from differences in consumption expenditure weights. Given that the RER is very volatile and highly persistent, this modeling strategy alone may not be ideal. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 3 goods are traded across borders for arbitrage purposes. The goods only differ by their weight (a physical mass). The trade is carried out by arbitrage trading firms who decide on the timing and magnitude of trade in order to maximize their profits. Arbitrage revenue depends on law of one price deviations, and the volume of trade. Arbitrage costs in the first model depend on the distance between the countries and some physical characteristic of the shipment (e.g., weight of the product). When a profitable arbitrage opportunity arises, firms engage in trade until prices adjust, so that no viable arbitrage opportunities remain. All trade takes place within the period. Arbitrage costs in the second model depend on all of the above characteristics and also include quadratic adjustment costs in the volume of trade. Arbitrage trading firms find large changes in the volume of trade costly due to needed adjustment to legal contracts, infrastructure, etc. Consequently, firms choose to adjust the amount of goods more gradually following changes in their environment, which leads to more volatile behaviour of price differences. Prices of goods at Home and Abroad can deviate from parity as long as their difference does not exceed marginal costs of trade. In this sense, arbitrage costs create symmetric thresholds: for law of one price deviations below the thresholds, no trade occurs and prices deviate freely. Prices differentials in excess of the thresholds are immediately arbitraged away. The first implication of this is that the tradability of goods is determined endogenously. The second implication is that prices of tradable goods behave in a non-linear manner. Third, the non-linearity depends on product weights and distance. Non-linearity is a result of the thresholds, which delimit a maximal deviation in the law of one price for every traded good. Thresholds are heterogeneous across goods and increase in the weight of a good and the distance of shipment. Heavier and more distant goods require a larger deviation in the law of one price in order to become traded. In a stochastic environment, there is a concave relationship between the volatility of the law of one price deviations and the volatility of the endowment shock process. Large endowment shocks push the law of one price deviations above the thresholds, triggering immediate arbitrage. Physical characteristics of goods are not the only candidate source of heterogeneity in estimates of no-arbitrage thresholds. As has been highlighted by other studies (Obstfeld & Taylor (1997), Zussinan (2002), etc.), such thresholds tend to be significantly related to import taxes, distance between countries and nominal exchange rate volatility. In the spirit of the border effect literature (Helliwell (1998)), other potential candidates for heterogeneity in threshold estimates may include Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 4 home bias in preferences, some degree of domestic networking for certain types of goods (primarily non-homogeneous), as well as distribution costs. The logarithm of a real exchange rate in this model is approximately a weighted average of the logs of the law of one price deviations. It therefore inherits the volatility and persistence of its components and some of their non-linear behaviour. The non-linearity in the real exchange rate is not of an on-off switch type present in its components. Instead, it is akin to a string: for large deviations of RER from parity, thresholds of many goods are crossed, creating a relatively strong mean-reverting tendency. This leads to a smooth threshold non-linearity in the real exchange rate, justifying a reoccurring finding in the recent empirical literature (e.g., Taylor, Peel & Sarno (2001), Kilian & Taylor (2003)). When the stochastic endowments follow an AR(1) process, a simple arbitrage trade model without nominal variables and real rigidities can generate arbitrarily large half lives of the real exchange rate deviations. Real exchange rate persistence depends positively on the size of the trade friction, negatively on the volatility of the endowment shocks and positively on the persistence of the endowment process. Volatility of the real exchange rate depends positively on all three of the above factors. When both foreign and home countries are subject to endowment shocks, the correlation of the two shock processes influences the stochastic properties of the real exchange rate. Estimates of the half lives of deviations in the real exchange rate increase in the shock correlation parameter3 while the standard deviation of the real exchange rate decreases monotonically. When the shocks are sector-specific, their negative correlation has a minimal impact on volatility and persistence of the real exchange rate. For positive values of the shock correlation, the half life of deviations in the real exchange rate is strongly positively related to the correlation coefficient. The stochastic endowment process at Home is calibrated to match the autoregressive root in the post-Bretton Woods quarterly U.S. GDP series and the endowment process Abroad to match the correlation between U.S. and E U GDP data over that period4. The physical weights of the goods are calibrated using the dataset from the second chapter of my dissertation. The parameters of the symmetric preferences are calibrated according to literature standards. The model simulation matches exactly the persistence of the real exchange rate found in the data. It 3 This relationship is not monotonic, it reverses for correlation values near 1. See section 1.4.2. 4 T h e calibration is taken from Chari, Kehoe & McGrattan (2002) who use Prance, Italy, U K and Germany in their E U data. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 5 also generates a quantitatively close match to the persistence and co-movements of various price-and quantity- constructs. However, the model fails to create sufficiently volatile prices. The difficulty in generating high real exchange rate volatility in the model with linear trade costs is that, when the model is realistically calibrated, arbitrage is triggered for relatively small movements in the real exchange rate. In response to this, the model is extended to incorporate quadratic adjustment costs in the volume of trade. The second model differs from the first only in the structure of the costs faced by the arbitrage trading firms. In addition to the shipping costs which depend on the distance and physical characteristics of goods, firms also have to face quadratic adjustment costs in the volume of trade. Sudden changes in the volume of trade are costly because they require additional legal and infrastructure costs, such as costs of establishing new (or changing existing) business relationships and distribution networks. Introducing a quadratic adjustment cost into the decision of the firm leads to a dynamic and highly non-linear model but improves the results. An additional friction preserves the core feature of the linear shipping cost model - the heterogeneous trade frictions - while creating a dynamic environment from the point of view of the firm. Quadratic adjustment costs lead to larger and more long-lived deviations of law of one price for traded goods in equilibrium, because firms are more careful in reacting to changes in endowments by large adjustments in the volume of trade. Consequently, the arbitrage trade between countries and co-movement between their consumptions is limited. A simulation of the model matches both the persistence and volatility of the real exchange rate while giving qualitatively meaningful results along other dimensions. Although a more extensive quantitative investigation should be done in the future, it appears that both the persistence and volatility sides of the purchasing power parity puzzle can be explained by careful modeling of trade costs. We know that prices, like trade volumes, exhibit an apparent "border effect" anomaly when comparing equidistant locations. When border exists between locations, price volatility and per-sistence appears to be much higher than for equidistant locations within the same country (see, i.a., Engel & Rogers (1996) and Jenkins & Rogers (1995)). Although the model in this chapter does not distinguish transport costs between locations within the same country and across coun-tries, it would be easy to create such environment by including more than two locations. A tariff could be introduced between the countries that would apply to shipments of all 3 goods. As well, Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 6 adjustment costs could differ between countries due to additional language, legal, culture and other barriers. Such addition could potentially explain the Engel & Rogers (1996) findings. The empirical framework of the second chapter (threshold non-linear AR estimations) could be used on city data to establish whether (by how much) the border affects the threshold estimates. More generally, an extension of the model into a multi location (country) framework could improve our understanding of trade flows, bilateral exchange rate persistence and volatility and perhaps other multilateral economic phenomena. It is in particular possible to envision an exten-sion of the model in which trade technology exhibits increasing returns to scale which could, in multi-location framework, help explain the " missing trade" puzzle. It may also be feasible to link the trade costs to development: should country A be producing goods that are more costly to transport than goods produced by country B, subset of prices in country A will be more isolated from some "world shocks" leading to lower integration into the world economy. If we assume that price response is desirable and that country A is poorer than country B (both of which are not unlikely), such a version of the model could be understood as a trade-related source of a development trap. On the empirical side, a multi country estimation framework could be used to analyze the persistence and volatility of price deviations for bilateral pairs of goods across many countries (in the same way Engel and Rogers (1996) do for city data using linear models). The paper is structured as follows. Section 1.2 exposes the general equilibrium model of ar-bitrage trade with linear heterogeneous shipping costs and analyzes its equilibrium properties. Section 1.3 analyzes the properties of the arbitrage trade model that also has quadratic adjust-ment costs in trade volume. Section 1.4 constructs and analyzes stochastic properties of the real exchange rate in a variety of environments. Section 1.5 discusses parameter calibration. Section 1.6 analyzes persistence, comovement and volatility of the real exchange rate and other variables. Section 1.7 concludes. 1.2 Gene ra l equ i l i b r i um mode l of arb i t rage t rade The two-country world consists of households and arbitrage trading firms. Each country is en-dowed with positive amounts of three tradable goods. Goods differ in their physical characteristics, proxied here by their weight. The trade is costly because the arbitrage firm has to use resources to ship goods across borders. As shown in the transportation literature, physical characteristics Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 7 of goods are an important determinant of the shipment costs (e.g., Hummels (1999)). One of the goods is used by the firms as an input in the transportation technology (in addition to being consumed by households). This numeraire good has a zero trade friction5. I assume that the cost of shipment is linear in the weight of the product: shipment X tons of any good between any two equidistant locations will cost X times the shipping of one ton of a good6. 1.2.1 Households A representative household at Home chooses its consumption path to maximize an instantaneous CES utility function subject to a resource budget constraint: oo 1_A i l _ i A 1 x ft s-l2- fl . ft ss*-max ) i C'lt ,C2t ,C '3 t 7 i S C 1 V * + 7 | C ^ + 7 | C 3 t - * s.t. pltCu + P2tC2t + C3t = p « Fu + V2tYit + Y3t + ^APt (1.1) given APt and Yit, i = {1,2}, where Yu is an endowment of good i at time t, Yli=i 7t = 1> & > 1> prices pu,i — 1,2 are relative prices denoted in the units of the local currency and APt is the amount of current-period arbitrage profits transferred to the household from a firm, assuming an equal splitting rule between households at home and abroad. The first order conditions for this problem imply the usual demand functions: Cu = 7 W T J ^ (1-2) •ft C-tt = J2P2t^e (1-3) Czt = l3-~j (1.4) rt where Yt is home country's real GDP measured in the units of good 3 (Yt = puYit +P2tY2t + Y3t + \APt) and Pt is a composite price index Pt = {l\p\7e + l2p\t° The problem for the representative household abroad is identical, with prices and quantities denoted with an asterix. Preferences of households at Home and Abroad are identical. 5The zero trade friction is an innocuous assumption. A positive friction for good 3 would make the computation more complicated but would not change the nature of the results. insurance costs, costs of setting up distribution networks, and other costs are ignored in this specification of the transportation technology. This simplification is for the purpose of expositional clarity and mathematical simplicity. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 8 1.2.2 Arbitrage trading firms There is a representative arbitrage trading firm in each country. It chooses the time and the amount traded in each good, taking into account the transportation costs. Transportation costs are introduced in a way of a transportation technology, which uses good 3 as an input. oo max AIlt = max (3tAPt Ni,N2,N3 N1,N2,N3'-J oo = max NuN2,N3t=1 3 T,(rit-Pit)Nit-T(NluN2t) =i (1.5) where Nu is the amount of trade in good i (N > 0 implies exports from Home to Abroad) and T(Nu, N2t) is the c ° s t function of the arbitrage trading firm. An arbitrage firm has to purchase T(./Vi,./V2) units of good 3 to trade {A^i,^}. Because the trade in good 3 is costless, does not enter T(.) 7 . For a start, the cost function takes a simple linear form, accounting only for the shipment cost of Ni units of a good. A more realistic technology would also include fixed costs of arbitrage: legal costs, insurance costs, labor costs, etc. A second model in section 1.3 proxies for these costs of trade by introducing a quadratic adjustment cost in the volume of trade. For now, — ( a w x | ^ l t | + a w 2 | -^2t | ) — a ( ' t i ' l | ^ l t | + W2|^2t|) where is the weight of a good i and a is a constant8. A ratio of the shipment cost of two goods between the same locations then equals the ratio of their weights. The first order conditions for the arbitrage trading firm yield: I(N)(pJ - pi) = atui iff |7Yi| > 0 (1.6) I ( N ) ( p J - p i ) < a t « i i ff /Vi=0 I ( N ) (P2 - P2) = o.W2 iff |7V2| > 0 (1.7) I ( N ) ( P 2 - p 2 ) <aw2 iff A 2^ = 0 P*3 ~ P3 = 0 7This is an innocuous assumption, one could calculate a more complicated version of the model with a positive trade friction in all three goods. The parameters ti and £2 can be thought of as trade frictions of goods 1 and 2 relative to the trade friction of good 3. intuitively, a is a constant, per-kilogram fraction of good 3 which disappears when a good is transported between these two particular locations. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 9 where I(N) is an indicator function, such that I(N)= 1 when N > 0, I(N)= —1 otherwise. The left hand side of the first order conditions is the marginal revenue of arbitrage trade and the right hand side the marginal cost. Trade occurs when the marginal revenue of arbitrage exceeds the marginal cost. The trade brings about price convergence, and stops when all profit opportunities are eliminated. Hence, following the trade, law of one price deviation equals marginal trade friction and the FOC holds with equality. The first order conditions for goods 1 and 2 hold with inequality in an autarky solution when the law of one price deviation does not exceed the marginal costs of trade. The first two pricing equations can be rewritten to show the first-order effect of good heterogeneity: I ( N ) ^ ^ < p 3 a , i = l,2 (1.8) Wi or, more intuitively, M R A per kg ^~T^ - M C A per kg ^ Vi ~Vi ^ • -j 0 -p3a < < p3a i = l,2 Wi The left-hand side of equation (1.8) captures the marginal arbitrage revenue per kilogram of good i (MRi) and the right-hand side the marginal arbitrage cost per kilogram of good i (MCi). While the right-hand side is identical across goods, the left-hand side is not. Goods that are relatively heavy will need a larger deviation in the law of one price in order for MRA to exceed M C A and trigger arbitrage trade. Denoting t\ = aw\ and t2 = aw?, the first order conditions for the firm can be expressed as: LOPD bJ~^ ~Pi] < *i (1.9) IP2-P2I < * 2 (1.10) V3-V*3 = 0 (1.11) The law of one price ensues for good 3 because the trade in it is costless. Equations (1.9) and (1.10) imply there exists a maximum law of one price deviation for goods 1 and 2: max.(LOPDi) = ti for i = 1,2. Maximum LOPD is proportional to the weight of a product. This leads to heterogeneous filtering. Consider a shock x that affects the law of one price deviations for goods 1 and 2 identically (for example, an identical increase in home endowments of both goods). It follows that the value of such shock can be divided into three subsets. First, for a positive law of one price deviation, x G [0, x\) leads to autarky because the law of one price Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 10 deviations for goods 1 and 2 are in a no-trade region (\LOPDi\ < t{ MRi < MCi i = 1,2). Second, for 2), only the lighter good (thereafter good 1) is traded because while the law of one price deviation for good 1 is initially outside the no-trade region, this is not true for good 2: \LOPD-L] > h <S=^> MR\ > MCU \LOPD2\ < t2 MR2 < MC2. Third, for x G [a;2*,co), all goods are traded as their initial law of one price deviations are outside the no-trade region (\LOPDi\ > ti MRi > MCi * = 1,2). Identical shocks to the law of one price deviation can have differential effects on the price deviations9. Some goods will become traded while others will not. Consequently, prices will adjust in a heterogeneous fashion. 1.2.3 Market clearing There are no asset markets in this model. All three goods markets clear at home as well as abroad. The direction of trade in goods 1 and 2 depends on the size and sign of the initial deviation from a law of one price, as determined by the endowments of these goods. Because there are only two countries in this model, Ni = EXPi = IMP? — -EXP? = -N*. The market clearing conditions can then be written as follows. C i + /Vi = Ylt C{ - N 1 = Y{ (1.12) C2 + N2 = Y2, C*2 - N2 = Y2* (1.13) C3 + N3 + lT(NuN2) = Y3, CI - N3 + \T{N1,N2) = Y3* (1.14) 1.2.4 Equilibrium The equilibrium is a set of prices and quantities {pi,pl,p2,p2, Ci,C{,C2,C2,C3, C3, iV*i, N2, N3} such that the households maximize their utility (equations (1.1)-(1.4)), arbitrage trading firms maximize their profits (equations (1.9) to (1.11)) and markets clear (equations (1.12) - (1.14)). Frictionless trade Here I analyze the model with zero transportation costs (o = 0), which implies T(N\,N2) = 0 V Ni,N2. The profit maximization problem faced by the arbitrage trading firm (eq. (1.5)) 9In the case 2, P2 of the heavier good will adjust due to substitution and welfare effects of a change in pi . Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 11 yields a law of one price equality for all goods: Pi =Pi »e{i,2}. (1.15) The equilibrium relative prices then depend on the world endowments and the preference param-eters: Pi Pj PL Pi yW 7j (1.16) where = Yi + Y?. The equilibrium consumption levels are (S) Ci = Yi-yW + Yx Yi Yw c2 Yi Y,w te)5 yW yW I i - Y l yW ft)' yW ^ 0 + similarly for C\ and C | when all Yi in equations (1.17) and (1.17) are changed for Y?. Ci YU C* = Y* iff % = XL Y' • The usual comparative advantage argument ensues: the country that is endowed with a relatively larger amount of good i will export good i and import good j. Equilibrium with positive trade frictions When borders open to trade, three cases can arise. First, if endowments are such that the autarky prices do not bring the law of one price deviation above the marginal costs (i.e., (1.9) and (1.10) hold with inequality), neither of the goods 1 and 2 is traded. Second, if the endowments lead to autarky prices which exceed the marginal cost of arbitrage for one good but not the other, trade occurs in the good with a relatively smaller friction ((1.9) holds with equality) while the good with a relatively larger friction is not traded ((1.10) holds with inequality). Finally, for endowments which lead to autarky prices such that the law of one price exceeds MCi Vi G {1,2}, both goods are traded. In this subsection I study the properties of the equilibrium in all three of these cases. Case 1: No trade in goods 1 & 2 When the autarky prices {pia,plai P2a>P2a} a r e s u c n that the law of one price deviation stays inside of the no-trade region for both good 1 and 2, pricing Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 12 expressions (1.9) and (1.10) hold with inequalities. N± = N2 = 0 in this equilibrium. Using the law of one price equation (1.11) the system can be written as: pi-e Y pl-0 = Y2, p*l-8 Y Y* = Y{ = Y: 73 + = Y3 + Y3* pl-0 ' p*l-0 where Y = PlYx + p2Y2 + Y3, Y* = p\Y{ + p2Y2* + Y3*, P = (llP\-e + l2p\-0 + 7 3 ) 1 / ( 1 ' 9 ) and P* = (jip^1-0 +J2p21~° + 73) 1^ 1 -^- Due to Walras' law, this system can be uniquely solved for relative prices {pi,p2,pl,p2}, which recursively define all other equilibrium values. Case 2: No trade in good 2 When {pia,Pia>P2a,P2a} a r e such that the law of one price deviation stays inside of the no-trade region for good 2 but exceeds the marginal costs of arbitrage for good 1, pricing expression (1.9) holds with an equality while (1.10) hold with an inequality. The relative price of good 2 is determined in the local markets while prices for goods 1 and 3 are determined internationally. The equilibrium is then constructed as defined above with N2 = 0: Y o *-o Y 73 Y Pl Ifl+73 Y* p*i-e Yi ~ 71 (Pl ~ /(JVi)ii .-e Y pi-e = Yx + Y{ = Y2 = Y2* = Y3 + Y3* where Y = (p*1-I(N1)t1)Y1+P2Y2+Y3) and P = (liipl-IiN^)1'6 +l2{p2)1-6 +l3)l'^-d). Due to Walras' law, this system uniquely determines {p\,p2,p2} and consequently all other equilibrium values as functions of preferences, endowments, and the trade friction t\. Case 3: A l l goods traded Finally, when the law of one price deviations for both goods under autarky prices exceed their respective marginal costs of arbitrage, both goods are traded in equilibrium. The equilibrium prices then solve a reduced system 1 ( p * - j ( A 0 t i ) - f l i ^ + P i - 8 Y {p*2 - I{N)t2) Pi--o Y p*i-e , Y* P*i-e 7i (Yi + Y{ = ~(Y2 + Y2*) 72 Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 13 7 3 Y + 7 3 y * p*l-6 Yi - 7 1 (Pi - /(JVi)ti)" y P i - ( F 3 + F 3 * where Y = (p\ - I(Ni)ti)Yi + (p*2 - I(N2)t2)Y2 + Y3) and P = (7i(pJ - HNM1'6 + 7 2 ( P 2 -7(A^2)t2)1 ° + T a ) 1 ^ 1 - ^ . Walras' law reduces the above system into two equations that solve uniquely for {pl,p2} and implicit ly all other variables as functions of endowments, preferences, and the trade frictions t\, t2. Properties of the equilibrium The trade frictions affect the equilibrium prices and allocations in all three cases described above. The equilibrium values in case 3 depend directly on t\ and t2, and all equilibrium values in case 2 depend directly on t\. However, t\ and t2 also define the set of endowments for which the autarky (case 1 applies). Likewise, t2 defines the set of endowments for which only good 1 is traded (case 2)-Figure 1.1: Model solution: thresholds of price deviations in linear model Law of one price deviations and Real Exchange Rate (11=0.02, t2=0.l) -h L O P D 1 O L O P D 2 * R E R Thresholds for good 1 Region 3 IBHBBHi«Hn»»l»i«»ll!(!t»l-+»-tt-(hlHHH«-4~i- 4- -H-i-Region 3 Thresholds for goodi 2 : l i Reg ion 1 -40 -20 0 20 40 % difference in Home and Abroad relative endowments To understand the properties of the equilibrium, I solve the model numerically and analyze its properties across a range of values for relative endowments. Keeping the endowments Abroad fixed, I vary the size of the Home endowments of goods 1 and 2 by the same amount. This is the simplest way to model an environmental change in this model and it is chosen to clearly illustrate its properties. Figure 1.1 plots the law of one price deviations from the solution of the model Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 14 and Figure A. 3 plots the volume of trade. Depending on the level of the relative endowment (shown as a percentage change from the initial endowment), deviations from the law of one price fall into one of the three regions. In region 1 with deviations in the law of one price below the trade friction, no trade takes place. This corresponds to the case when both expressions (1.9) and (1.10) hold with inequalities. Marginal costs of trade are higher than marginal revenues for any volume of trade, therefore no trade takes place. When endowments exceed the first set of thresholds, trade occurs in good 1 but good 2 remains non-traded (region 2). Trade brings about reversion in the price differential: in the case of exports, price at Home rises while price Abroad falls and vice versa in the case of imports. After all arbitrage trade opportunities are extinguished, the law of one price deviation for good 1 reverses back to the threshold. Good 2 remains non-traded and its LOPD increases. Notice that the increase in the deviation from the law of one price for good 2 is smaller in region 2 than in region 1, although good 2 is non-traded in both cases. There are two reasons why the slope of the LOPD2 is less in region 2 than in region 1. First, as good 3 vanishes in transportation, its price increases (hence a small relative decrease in p2). Second, households substitute from the more expensive into a cheaper good 1 0. Finally, after the shocks exceed the second set of thresholds, both goods become traded, and the law of one price deviation for good 2 reverses towards its threshold. It is clear that the model generates heterogeneity in the thresholds of the law of one price deviations. The effects of thresholds on the persistence and volatility of the law of price deviations will be explored in the subsection 1.4 below. Next, I run a Monte Carlo simulation to illustrate stochastic properties of the model. Starting with identical endowments of all 3 goods in both countries, I introduce a country-specific shock which alters the balance of endowments of good 1 and 2 at home relative to abroad. The shock affects goods 1 and 2 symmetrically. The relative endowment shock is normally distributed (e ~ N(0, cr 2)) and alters the endowments in the following way: Y\ = Y2 — Y + e, Y* = Y2* = Y3* — Y3 = Y, and changes the autarky relative prices of goods 1 and 2 proportionally at home while keeping them unchanged abroad. Depending on the size of the shock, one of cases 1, 2 or 3 1 0 I f good 1 is exported, Home buyers substitute from it into good 2 which is now relatively cheaper. Buyers abroad substitute from good 2 into 1. Both imply a lower LOPD2 relative to its autarky value. If good 1 is imported an inverse of the argument applies. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 15 ensues. I simulate the model 1000 times and record equilibrium allocations after shocks. I repeat this process for a range of values of the standard deviation of the endowment shock <x. Figure A.4 shows standard deviations and mean price deviations as a function of the variability of the endowment shocks. Price deviations of heavier goods are more larger and more volatile than those of lighter good. Volatility of price deviations is concave in shock volatility. For a range of values of cr e [l,c>j] where c>2 > <xi, price volatility rises with shock volatility. For a > <7j, volatility remains constant as excessive price differences get arbitraged away. Figure A.5 plots the average thresholds of price deviations for different values of standard deviation of the shock volatility. The thresholds are symmetric, and depend positively on the trade friction (here t2 = 5r-i). As a robustness check, figures A.6 - A.8 show the behavior of the means, standard deviations and thresholds of price deviations across a range of values of the trade frictions. Because the ship-ping costs are linear, the properties of the model described above remain qualitatively unchanged for any t2/t\lx. As t2/t\ rises so does the maximum of LOPD2/LOPD\ (figure A.6). This is independent of the value of standard deviation of the endowment shock, as long as it sufficiently large to allow good 2 to cross its threshold. Secondly, STDLOPD2I'STDLOPD-I is increasing in t2jt\. It appears that this relationship is concave: LOPD variability increases less than the trade friction (panel 2 in figure A.7). Third, the thresholds are proportional to the levels of t2/t\ (figure A.8). 1.3 A r b i t r a g e t rade mode l w i t h ad justment costs to t rade The endowment and preference setting of the second model is identical to the first. However, trade costs consist of shipping costs as well as quadratic adjustment costs in the change of trade volume. Changes in the volume of trade from previous to current time period require hiring of labour resources, adjustment in the distribution system and possibly investment in (or a changes in the existing) trade infrastructure. Adjustment of trade to endowment shocks also requires time. Quadratic adjustment costs get at this idea in a smoother way than assuming pre-determined 1 1 Please refer to the earlier version of the paper with a partial equilibrium model in which a friction also consists of a fixed cost and a value-related cost (e.g., an insurance cost). It is shown how fixed costs of arbitrage brings about non-linearities to the relationships discussed here. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 16 shipment volume. Consequently, large swings in trade volume are costly. The arbitrage firms' problem can be written as follows: oo max Alit — max TP^APt AT- . AT. . AT- . AT. . —^ Nlt,N2t Nlt'N2tt=i oo = max V / 3 ( t _ 1 ) Nu,N2t fz{ Y(Pit-Pit)Nit-T(NluN2t) i = l (1.17) s.t. T(Nlt,N2t) = t1\Nlt\+t2\N2t\+c1AN^t + c2AN2zt (1.18) where Nn is the amount of trade in good i at time t (N > 0 implies exports from Home to Abroad) and T(N\t, N2t) represents the cost function of the arbitrage trading firm, and pu is the price of good i relative to good 3. An arbitrage firm has to purchase T(N\t, N2t) units of good 3 to trade {N\t, N2t}. Because the trade in good 3 is assumed to be costless, N3t does not enter T(.) 1 2 and its trade volume is determined according to the market clearing conditions and the preferences of households. The arbitrage trade cost function consists of a shipment cost and adjustment cost component. The shipping cost between two locations is again assumed linear in weight of the good, leading to identical trade frictions U = awi, i = 1,2 where Wi is the weight of good i and a is a constant. The adjustment cost component is quadratic in the change of volume of trade from the previous period to the current period. The quadratic adjustment cost parameters ci and c2 are not related to the physical characteristics of goods. Summarizing the behaviour of the firm by the first order conditions of its profit maximization decision is challenging because of the non-differentiability of the absolute value function at 0. I use a smooth approximation G(.) to the absolute value function to allow a continuous mapping between the first order conditions and the objective function. Function g(.) = dG(.) denotes the first order derivative of the " smooth" absolute value function, and is itself a smooth approximation to a step function. See the appendix for details. The first order optimality conditions with respect to goods i — 1,2 yield: 0 = Pi,t ~ Pi,t ~ [til(Nitt) + 2ci{Nitt - Nijt-x)] - /3Et[-2ci(Nijt+1 - Niit)] 1 2 This is an innocuous assumption, one could calculate a more complicated version of the model with a positive trade friction for trade in each of the three goods. The parameters ti and ti can be thought of as trade frictions of goods 1 and 2 relative to the trade friction of good 3. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 17 Rearranging, we get T~ \PU ~ Pi,t ~ tiI(Ni,t)} = - / 3 E t / V M + i + (1 + P)Niit - 7Yi,t-i = (-(3B~2 + (1 + ^ B " 1 - l)EtJV i > t -i = [-0 + (1 + P)B - B 2 ] E t A T i ) t + i Vi where B is a backshift operator and l(Nitt) is an approximation to the indicator function, such that I(iV i it) = 1 when Ni<t > 0, l(Niit) = -1 when JV i ) t < 0 and I(Niit) = 0 when Niit = 0. The quadratic form on the right hand side of the last equality has one stationary and one non-stationary root and can be re-written as — (B — 1)(B — (3). The first order conditions for the firm can then be written as ~hi " Pi'1 ~ t i l ^ N i A = ( 1 " B _ 1 ) ( 1 _ ^B _ 1 )Et^ i ,* - i V i After expanding the stable eigenvalue forward and the unstable one backward, the first order condition for the arbitrage trading firm can be re-written as i oo Ni,t = Ni,t-! + ^"Et £ Pj (plt+j - Pi,t+j - Ul(Nitt+j)) Vi (1.19) The optimal amount of trade in good i in period t depends positively on the volume of trade in the past period and on the expected future path of deviations of prices from parity in excess of the trade friction. Firms care about the future path of price differentials so that their trade patterns are smooth over time. An expectation of a deviation from parity in excess of the trade friction in the future period t + j increases trade in all future periods from t onwards. Note that the expectation of the direction of the trade flow in the future periods EtI(Nitt+j) enters a firm's decision rule. If a firm expects a future price process in which a set of time periods with expected export regime is followed by a set of time periods with an expected import regime13, it optimally lowers the trade volume relative to a scenario in which the firm expects only one regime to prevail. 1.3.1 Equilibrium The equilibrium is a set of prices and quantities {pi,t,P\tt,P2,t,P2p Ci,t,C{t,C21t, C | 4 , 6 3 , 4 , C$t,Nitt,N2,t, N3}t}t^0 such that the representative household maximizes its utility (equations 1 3 A process in which v*i,t+j+i > Pi,t+j+i+U =>• I(Nilt+j) = 1 is followed by p ' i t t + j + k + t i < p i i t + j + k => l(Nitt+j+k) = - 1 for k > I. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 18 (1.1)-(1.4)), arbitrage trading firms maximize their profits (equation (1.19) for both goods) and all markets clear (equations (1.12) - (1.14)). Household optimality conditions at home and abroad (equations (1.2), (1.3), (1.4), (1.2*), (1.3*), (1.4*)), firm's two optimality conditions described by (1.19), and 6 market clearing con-ditions ((1.12), (1.13), (1.14), (1.12*), (1.13*), (1.14*)) can be simplified into a 4-by-4 system in {PU,P2,t,P*iluP2,t}: Ay, l,t - 7 1 pu pi-. -e Yt-! • P i , t - i p i-e •Q- i - E t £ P [P*i,t+j ~ PU+i ~ hI(Nlit+j)] 2ci (1.20) 3=0 Ay2,t - 7 2 -0 Yt _ _0 Y t - i P2,t D l _ 0 P2,t-\ jjl-0 OO -EtY/P3[p*2,t+3-P2,t+j-t2m 2c2 3=0 t Yt t 7 2 P 2 , t ^ p e + 7 2 P ; / - ^ f : Yi,t + Ylt Y2,t + Y2*t (1.21) (1.22) (1.23) where Yt = pi,tYi,t+P2,tY2,t + Y3>t + \APU Pt is a composite price index (Pt = ( 7 1 ^ 1 ^ + l2p\~t6 + 7 3 ) 1 - " ) , l(N) is a smooth approximation to the indicator function such that I(-/V,]t) — 1 •<==> Nitt > 0 and 1(^4) — — 1 =^>- iV t^ < 0 and APt are the current-period arbitrage profits. Equations (1.20) and (1.21) are the two intertemporal equilibrium conditions. Equations (1.22) and (1.23) are the two infratemporal equilibrium conditions. Intuition Before solving the infinite version of the model, I illustrate some of its main features using a simpler one-period version. The first obvious improvement is the lack of inteterminacy of trade volume in partial equilibrium setting. If a price deviation exceeds marginal cost of shipment in a model without quadratic adjustment costs, firms prefer to ship an infinite amount of goods in a partial equilibrium. Market clearing imposed in the general equilibrium forces prices to adjust until price difference reaches the threshold and trading stops (therefore the volume traded is finite). Trade volume is effectively determined outside the firm. On the other hand, in a model Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 19 with quadratic adjustment costs, the optimal volume of trade chosen by the firm is finite in a partial equilibrium. The quadratic adjustment cost model encompasses the linear shipping cost model. If we multiply the inter-temporal Euler equations (1.20) and (1.21) by c\ and c2, respectively, so as to avoid having adjustment cost parameters in the denominator of a ratio, we can study the properties of a system as C j approaches 0. Smaller c limits the linkage between today's and yesterday's volumes of trade N^t and Nitt-i, respectively (note that the expressions on the left-hand side of equations (1.20) and (1.21) are just A7V i t i = 1,2). When c = 0, the two inter-temporal equations become identical to equations (1.6) and (1.7) from the linear shipping cost model. The inter-temporal linkage ceases to exist, and the system collapses onto system described in cases 1, 2, or 3 of the linear shipping cost model equilibrium (section 1.2.4). Consider a one-period version of the model, with only one good with a positive trade friction. Then, the firm's first order condition implies: p* - p - l(N)t = 2c(N — N-\) where W_i is the last period's trade volume. Without the quadratic adjustment cost function (c = 0), as long as the law of one price deviation exceeds the trade friction t, the volume of trade is indeterminate in a partial equilibrium. From the point of view of the firm, an infinite volume of trade is optimal because that maximizes its profits. With c > 0 and \p* — p\ > t, the firm chooses a finite volume of trade that depends positively on the law of one price deviation (p* — p) and last period's trade volume N-i, and negatively on the shipping cost t and quadratic adjustment cost parameter c. The general equilibrium conditions imposed by market clearing and price adjustment further limit the volume of trade because adjustment of prices narrows the profit margin of the firm. Note that with \p* — p\ < t, and N-i = 0, the firm optimally chooses not to trade. When \p* — p\ < t and iV_i ^ 0, balance has to be struck between the loss-making trade and the costs of trade deceleration (note that profits are negative in this situation). Figure (1.2) plots the revenue, cost and profit functions for linear and quadratic models in two situations, assuming N-i = 0. In the upper segment, price abroad is 30% higher than price at home. In the lower segment, price abroad is 15% lower than the price at home. Trade friction t = 0.2 in both cases, i.e., 20% of good 3 is consumed by shipment. Note that the revenue functions are identical in both models, while the cost function in the quadratic adjustment cost model lies always above the cost function in the linear model (due to the additional cost). It Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 20 Figure 1.2: Simplified profit functions in a quadratic and linear model when ./V_i = 0 Trade with arbitrage trade opportunity (t=0.2. A p=0.3) - S O S volume of trade No trade despite arbitrage trade opportunity (t=0.2. A p=-0.15) 3 r Linear model -5 0 5 volume of trade Linear model -5 0 5 volume of trade -5 0 5 volume of trade follows that the profit function in a quadratic adjustment cost model is a parabola with a kink and lies below the profit function of the linear cost model in all cases other than N = 0, when the two profit functions are identical. Consequently, trade will occur in the Q A C model for law of one price deviations only when it also occurs in the linear model. We know that trade occurs in a linear model if and only if \p* — p\ > t. Therefore \LOPD\ > t is a necessary condition for trade to occur in the QAC model. Also note that the no-trade bands in a quadratic adjustment cost model are identical to the no-trade bands in a linear cost model. As has been shown above, if \p* — p\ < t while N-\ = 0, firms will not trade. If \p* — p\ > t, trade does occur, but in a smaller (finite) volume than it would in a linear cost model. In particular, when c > 0 and N-i = 0, - l dNj dYi 2cB _ i Y ~°' Y 0 et < I (1.24) where ir = prob(\p* — p\ > t). The volume traded will be smaller than the increase in endowment (see the Appendix for proof). Because the arbitrage trade is the only source of price adjustment, a smaller trade volume leads to smaller price adjustment. Consumption closer to the autarky level requires that the equilibrium prices are closer to the autarky prices than they were in a linear model. Therefore, Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 21 equilibrium law of one price deviations in excess of the shipment thresholds t can be sustained. While the no-trade region in a model with quadratic adjustment costs is identical to the one in a model with linear shipment costs, law of one price deviation can exceed the shipping threshold in equilibrium. In particular, it can be shown that, in the simple one-period model, for any N-%, where D> 0 and A < — 1 (see the Appendix for derivation). Therefore, an increase in endowment will decrease the home price, i.e., increase the law of one price deviation p* — p when c > 0. Moreover, the deviation from the law of one price is increasing in c. Intuitively, larger quadratic adjustment costs lead to smaller adjustment in volumes, lessening the pressure on prices to con-verge. Figures (A.13)-(A.16) show initial responses of deviations from the law of one price for a range of dYij, i = 1,2, as well as for a range of cost adjustment parameters c.. As in the linear shipping cost model, three regions exist for any level of c: region 1 in which none of the goods are traded, region 2 in which only the lighter good is traded and region 3 with both goods traded. Price of good 2 is affected by the fact that good 1 is traded even when good 2 is not (region 2): as home price of good 1 rises, home consumers substitute to good 2. This expenditure switching effect from traded to non-traded goods at home increases home demand for the non-traded good 2 and increases its prices above what it would have been otherwise. Consequently, law of one price deviation for good 2 is smaller than it would have been if good 1 was not traded. The figures also show how total volume of trade changes with the endowment. Trade volume depends negatively on the quadratic adjustment cost c as well as on the trade friction t. Also, cumulative trade volume expressed as a percentage of the change in endowment is non-monotonic. Finally, costly adjustment leads firms to spread out trade in more steps of smaller magnitude: time period of adjustment depends positively on c and dY,. The intuition presented can change slightly when N-i ^ 0. Figures (A.13) - (A.16) showed that when N-\ — 0, volume of trade N increases in p* — p when \p* — p\ > t and is zero otherwise. N = N-i + ^ip* -p-I(N)t) and so as long as the good remains traded {I(N) does not change), qualitative relationship between the optimal trade volume and price deviation stays the same. But the range of autarkic values of deviations of prices from parity decreases in N-i (see figure 1.3). If -/V-i > 0, positive trade volume may be optimal even when p* — p < t due to costs of (1.25) Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 22 trade deceleration. The reverse can also hold. Therefore, optimality does not guarantee positive profits when c > 0 (see figure A.17). Large values of | i V _ i | require larger \p* — p\ for arbitrage profits to be positive. Figure 1.3: Optimal trade volume in partial equilibrium in QAC model for various price deviations and N-i. c=0.01, t=0.2 P'-P 1.3.2 Solution method Due to a high degree of non-linearity, I solve the model numerically. A differentiable approx-imation to the absolute value function G(.) allows us to summarize the system by its first or-der conditions. The derivative of G(.) is a smooth approximation to a step function I(Nij) = I({pitt+i,Pi t+i,P2,t+i,P2t+i}i=o)' resulting in a highly non-linear, although differentiable, first order system. First, the time span of the model needs to be limited. I assume the existence of a steady state equilibrium to which countries converge following a shock. A maximum number of time periods T available for an adjustment is necessary to limit the size of the system. Conditional on T , the model can in principle be solved recursively. However, efforts to apply recursive solution methods have failed to find the equilibrium paths of the variables, primarily due to the non-linearity in the indicator function. Therefore, a method of relaxation (see Boucekkine (1995)) is applied in which a finite-period approximation f{-)t=1.T = 0 to the system f(.)t=i-.oo = 0 is solved by stacking all Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 23 equations for all time periods into one large system F(.) = [f(-)t=1 ••• f(.)t=T]i = 0 which is then solved numerically. The four unknown price variables in the equilibrium system /(.) above imply that the size of the Jacobian is 4T x 4T. Second, in order to compute the Jacobian of the stacked system F(.) in one step, a smooth I(Ni,t) approximation to the indicator function I(Nif) is necessary. Some inventive work leads to 2 I(Ni,t) = - arctan(A7V;,) 7T where A is a choice parameter which governs the approximation error. An inverse of a trigono-metric function tan(x), arctan(a;) has a range of [—7r/2,7r/2] for x 6 R and is monotonically increasing, continuously differentiable, and has a convenient property that arctan(x) < 0 when x < 0, arctan(:r) > 0 when x > 0 and arctan(a;) = 0 when x = 0. Because of the bounded range, and because arctan(Ax) x = 0 = 0 VA, arctan(A.T) can reach the bounds very quickly. Finally, premultiplying the function by 2/-7T, changes the range to [-1,1] - very convenient for a step func-tion. High A lowers the approximation error, as illustrated in figure (A. 12). A choice of A = 10 4 0 makes the approximation error indistinguishable from zero for any feasible stopping criterion of the numerical solver. I(Nitt) is a derivative of G(Nijt). Third, to facilitate the numerical solver in finding an equilibrium, the inter-temporal Euler equations (1-20) and (1.21) are replaced with their simpler forms which do not include an infinite forward-looking sum. An error in pu by the numerical solver then affects only the 4(4— 1) : 4(i+1) partition of the Jacobian (= 122 = 144 values), whereas before it would have influenced all 4T 2 (= 1600 if T — 10, 6400 if T = 20) values of the Jacobian, making convergence exceedingly difficult for relatively small error levels. 1 (ptt-Pi,t-hI(7htt)) = (l + P)Yu-(3Yu+1-Y1^1 2ci (1 + PhipJ-£o + PKPWI^O +liPu-i-£re (1-26) •Pt-i 1 (p*2,t-P2,t-t2l(N2,t)) - (1 + P)Y2,t ~ (3Y2,t+i ~ Y2,t-i 2c2 "(I + P)l7P2°-zh> + -Y20P2,?+1-§S + 7 2 P 2 l - i - § ^ (1-27) " t - i Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 24 A system f(.)tz=i, P a r t of the large stacked system F(.), consists of equations (1.26), (1.27), (1.22) and (1.23). Period T + 1 values which are found in the inter-temporal Euler equations of f(.)T are set to steady-state equilibrium values associated with a full adjustment to the shock. Finally, a two-step method is used in the calculation to ensure that the layers of approximation do not lead us to a wrong solution. First, a system F(.) above is solved, and time paths of all variables are computed. Because this is merely an approximation to the true system (for smaller levels of A, I(Nitt) can be distinguished from 1, —1, or 0), a second step involves replacing the I(Ni,t)) with 1, —1, or 0 using the first step estimates, and solving the system again. 1.4 R e a l exchange rate This section explains the behaviour of the real exchange rate in the model. Because the model is solved numerically, properties of the real exchange rate are explained using simulations across a range of ad-hoc parameter values. Section 1.5 calibrates most realistic parameter values for a two-country simulation of the model which is then compared to the moments of the data. A model-specific logarithm of the real exchange rate is a weighted average of the three law of one price deviations: log(RER) = 7 1 \og(LOPDi) + 7 2 \og(LOPD2) + 7 3 \og(LOPD3). To under-stand the model's implications for the persistence, an endowment process at Home is introduced to the model: Yitt = 0^,4-1 + (1 - ct)Y + ut i = 1,2 where ut ~ N(0, cr2). Lack of price reversion to parity for tradable goods creates persistence in their deviations from the law of one price and consequently in the real exchange rate. As long as the deviation from parity stays within the thresholds, reversion does not occur. A sufficiently small endowment shock can generate an infinite half-life of deviation in the real exchange rate from PPP. Conditional on the trade friction, the degree of persistence in the real exchange rate is positively related to the persistence of the endowment shock process (see Figure 1.4 and Table 1.1). For a < 0.9, half lives of the real exchange rate deviations do not exceed 7 time periods. Half life increases sharply in a for values near 1, to about 12 time periods when a — 0.95, 30 time periods for a = 0.975 up to 525 time periods when the endowments have a unit root (a = 1). Variability of the shocks inversely affects the half life in this region: for a = 6.8% of GDP, the maximum half does not exceed 19 periods. For given thresholds, high variability of the endowment shocks negatively affects the real exchange rate persistence because it increases the likelihood that the price deviation will exceed Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 25 the threshold, thus leading to price equalization. Figure 1.4: RER persistence and volatility and the shock process in linear model Half-lifes of RER and the shock process (2000 reps.) 600 r 0.024 0.022 0.02 0.018 STD a : S T DC T> STD Persistency of the endowment shock process O.OOB ' ' ' ' 0.6 0.7 0.8 0.9 1 Persistency of the endowment shock process Table 1.1: RER half lives (in quarters) and the shock process a = 0.65 a = 0.7 a = 0.75 a = 0.8 a = 0.85 a = 0.9 a = 0.95 a = 1 (7 = 0.008 1.66 1.99 2.44 3.11 4.16 6.13 11.52 525.12 a = 0.019 1.42 1.67 2.00 2.48 3.18 4.55 7.93 263.46 a = 0.034 1.21 1.41 1.65 1.98 2.50 3.50 5.86 59.39 cr = 0.068 1.02 1.16 1.34 1.59 2.03 2.82 4.99 18.56 Each result is based on 2000 Monte Carlo simulations of the linear shipping cost model (c, = 0 i = 1,2) when i 1 = 0 .02 and r-2=0.04. a is the AR(1) coefficient of the shock process, a is the standard deviation as a proportion of the mean GDP. The standard deviation of the real exchange rate is increasing in both persistence a and volatility cr of the shock process (see figure 1.4 and table 1.2). High volatility of the endowment shocks leads to high volatility in the law of one price deviations, as long as the shocks do not exceed the thresholds. Once a shock exceeds the threshold, additional volatility is neutral. More persistent shocks lead to longer-lived deviations from the law of one price (the shocks effectively last longer) and so increases their volatility. This effect is particularly visible when a is small enough so that most shocks leave LOPD below the thresholds. In such cases, std(RER) can exceed cr (see table 1.2). As o rises, price deviations are more frequently outside of their thresholds, and get arbitraged away. Therefore, higher shock persistence a has minimal influence on real exchange rate volatility when a is relatively high. Size of the trade (shipping) friction is a major determinant of the persistence of the real exchange rate. For any given a, a larger shipping friction t requires a larger endowment shock Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 26 Figure 1.5: Properties of the real exchange rate and the trade friction in linear model Half—lifes of R E R and the trade friction (2000 reps., a=1.9% of mean GCST^ndard deviation of R E R and the trade friction b.5 0.6 0.7 0.8 0.9 1 Pers is tency of the endowment shock p rocess 0.6 0.7 O.a 0.9 1 Pers is tency of the endowment shock p rocess Table 1.2: Standard deviation of the RER and the shock process a = 0.65 a = 0.7 a = 0.75 a = 0.8 a = 0.85 a = 0.9 a = 0.95 a = 1 o = 0.008 0.008 0.009 0.009 0.010 0.011 0.012 0.014 0.017 a = 0.019 0.013 0.013 0.014 0.014 0.015 0.016 0.017 0.021 a = 0.034 0.015 0.015 0.016 0.016 0.017 0.017 0.018 0.024 a = 0.068 0.016 0.016 0.017 0.017 0.018 0.019 0.019 0.027 Each result is based on 2000 Monte Carlo simulations of the linear shipping cost model (c; = 0 i = 1,2) when t 1 = 0.02 and £2=0.04. a is the AR(1) coefficient of the shock process, a is the standard deviation as a proportion of the mean GDP. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 27 in order for arbitrage trade to occur (see figure 1.5). For any given shock process, a larger trade friction t will increase RER volatility by making a larger subset of the endowment shocks non-neutral. Shipping costs generate persistence in the real exchange rate by creating a wedge between the prices of goods in two locations. The size of the wedge varies: it increases for shocks that are small relatively to the weight of a product, and stays constant for relatively large shocks above the threshold. The half life then depends negatively on the volatility of the endowment shocks because small (and persistent) shocks will need more time to cross a threshold. On the other hand, the half life of deviation in the RER depends positively on the weights of the products, as these determine the location of a threshold. Table 1.3: RER half-lives and the relative trade friction a = 0.6 a = 0.7 a = 0.8 a = 0.9 a = 1 = 2 1.4 1.9 2.9 5.0 263.5 « 2 / * l = 4 1.4 2.0 3.1 6.1 653.4 = 6 1.4 2.0 3.1 6.2 1060.2 V * i = 8 1.4 2.0 3.1 6.2 1641.0 Each result is based on 2000 Monte Carlo simulations of the linear shipping cost model (d = 0 i = 1,2) when ti=0.02 and t2=0.04. a is the AR(1) coefficient of the shock process, a is the standard deviation as a proportion of the mean GDP. 1.4.1 Real exchange rate in a model with quadratic adjustment costs The real exchange rate in a quadratic adjustment cost model is more volatile and more persistent than in the linear shipping cost model. The additional friction of quadratic adjustment costs in changes in trade volume narrows the profitability of the arbitrage opportunity to the firm after an endowment shock. As we have seen in section (1.3.1), the optimal choice set is narrowed in a non-linear fashion. The no-trade zones in the QAC model are identical to those in a linear shipping cost model (these are determined only by the cost-of-shipping parameters t\ and t2). Thresholds of the law of one price deviations are now upward sloping in the size of the endowment shock (price deviations for good i can exceed friction i j ) . Additional trading costs lower the trade volume because consumption sharing is now more costly. Equilibrium consumption sets are closer to the endowments, and prices are closer to autarky prices. This implies that the law of one price deviations are larger and reverse more slowly than in the case without additional costs14. 1 4Recalling that the L O P D ; = U in the linear model, it follows that L O P D ; > U in the Q A C model. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 28 Firms choose to adjust in more than one time period to a profitable shock in order to minimize the losses from trade deceleration. Negative contemporaneous arbitrage profits in the post-shock periods (there is no arbitrage opportunity that would warrant trade in post-shock periods) are smaller in discounted net present value than the costs of shutting down the trade channel. Both of the above differences lower the responses of firms to changes in endowments and make the real exchange rate more volatile and sticky in a model with quadratic adjustment costs. A word of caution needs to be said about the comparability of results between the two models. The linear shipping cost model is static, and the dynamics are introduced by an autoregressive endowment process. The computation of the equilibrium is not restricted by the time horizon and long time spans can be used to increase the precision persistence and volatility estimates. On the other hand, the quadratic adjustment cost model is inherently dynamic. The numerical solution method has to limit the maximum number of time periods allowed for adjustment (in this case, T = 20 - compare to T = 2000 in the linear shipping cost model) which makes the A R estimations less precise. Moreover, a shock to the autoregressive endowment process only occurs in the first period, and then gradually dies off. This condition is necessary for the convergence of the numerical solver15. A simulation of a length T is repeated M times. Half-life and volatility of the real exchange rate deviations are computed at the end of each simulation, and a mean of that distribution of statistics is reported in the tables below. This contrasts with the result tables for the linear shipping cost model which report one statistic based on a simulation of a considerably longer length. The endowment process is Y^t — aYi,t-i + (1 — ct)Y + ut i — 1,2 where ut ~ N(0,cr2). Moreover, I assume that ut = 0 for t > 1. I then simulate the model for a variety of adjustment cost parameters c. Table 1.4 shows that the half life of deviation of the real exchange rate behaves in a qualita-tively similar manner when quadratic adjustment costs are present - it is decreasing in the shock volatility and increasing in the shock persistence. A more volatile endowment process lowers the half-life of convergence because profitable arbitrage opportunities arise more frequently. A more persistent endowment process makes an arbitrage opportunity last longer and so increases the 1 5 If the AR(1) endowment process is shocked in every period, numerical solver may not converge because the assumption of full adjustment to the equilibrium by t = T may require very fast adjustment, especially in the latter periods. Shocking the system in later periods leaves less time for adjustment. Moreover, shocking the system in each period makes the adjustment more complicated due to the changes in the state variable (JVj,t-i). Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 29 Table 1.4: Mean half-life of convergence estimate of log(RER) when c=0.1 a = 0.7 a = 0.8 a = 0.9 a = 0.99 a = 0.8% 1.9444 3.12 7 a = 1.9% 1.9438 3.11 6.8 CT = 3.4% 1.9436 3.11 6.7 a = 6.8% 1.9433 3.1 6.6 Each result is based on 1000 simulations of the model when T = 20, i i = 0.0054 and t2 = 0.0174 are the shipping trade frictions (see Calibration section 1.5). a is the AR(1) coefficient of the shock process. adjustment time. Compared with Table 1.1, half lives of deviation are longer with quadratic ad-justment costs for any combination of a and a. Larger adjustment costs magnify these differences (tables 1.4), 1.5 and 1.1). The intuition from the one-period model carries through: quadratic ad-justment costs lower the speed of adjustment of trade volume to shocks, leading to more sluggish adjustment in prices and consequently the real exchange rate. Table 1.5: Mean half-life of convergence estimate of log(RER) when c=0.01 a = 0.7 a = 0.8 a = 0.9 Q = 0.99 <7 = 0.8% 1.943 3.105 6.567 68 a = 1.9% 1.9429 3.1048 6.567 66 a = 3.4% 1.9428 3.1047 6.515 65 a = 6.8% 1.9428 3.09 6.522 63 Each result is based on 100 simulations of the model when T = 20, ti = 0.0054 and t2 = 0.0174 are the shipping trade frictions (see Calibration section 1.5). a is the AR(1) coefficient of the shock process. Law of one price deviations are more volatile than in the linear shipping cost model (see Table 1.6). Unlike in the linear shipping cost model, estimates of real exchange rate volatility relative to that of the GDP are not sensitive to endowment volatility, as more volatile endowments lead to higher volatility in prices (therefore their ratio stays unchanged). The standard deviation estimates decline in a because of a smoother adjustment in prices imposed by the quadratic adjustment costs. Table 1.6: [Mean std(log(RER))]/[Mean std(log(GDP))] when c=0.1 a = 0.7 a = 0.8 a = 0.9 a = 0.99 (7 = 0.8% 27.62 14.39 12.59 8.49 a = 1.9% 17.83 14.88 12.35 8.42 a = 3.4% 17.68 14.52 12.24 8.54 a = 6.8% 17.07 14.25 12.14 8.75 Each result is based on 1000 simulations of the model when T = 20, t\ = 0.0054 and t2 = 0.0174 are the shipping trade frictions (see Calibration section 1.5). a is the AR(1) coefficient of the shock process. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 30 Table 1.7: [Median std(log(RER))]/[Median std(log(GDP))] when c=0.1 a = 0.7 a = 0.8 a = 0.9 a = 0.99 cr = 0.8% 1.951 1.952 1.953 2.03 cr = 1.9% 1.950 1.951 1.952 2.08 cr = 3.4% 1.950 1.950 1.951 2.36 0" = 6.8% 1.949 1.949 1.950 2.71 Each result is based on 1000 simulations of the model when T = 20, t\ = 0.0054 and *2 = 0.0174 are the shipping trade frictions (see Calibration section 1.5). a is the AR(1) coefficient of the shock process. 1.4.2 Country-specific shocks This section studies the addition to the linear shipping cost model of endowment shocks Abroad. To isolate the effect of a second shock process, the persistence of shocks at Home and Abroad is constrained to be the same across sectors as well as across countries: where Y\,t — Yi,t ut , ut = Ii,t * Yilt = a%-i + (1 - a)Y + ut for i = 1, 2 and ut ~ N(0, Cl) where Cl = ( cr* 7 . The correlation 7 CT" parameter r/ = ^ of the shocks at Home and Abroad affects the properties of the real exchange rate. Persistence of the real exchange rate deviations is lowest when the correlation of the shocks is close to zero (figure 1.6). Positively correlated shocks tend to produce relatively small average price deviations and lengthen the time needed to cross a threshold. Highly negatively correlated shocks lead to large price deviations which lower the persistence of the real exchange rate. There-fore, the real exchange rate persistence increases in On the other hand, volatility of the real exchange rate is a monotonically decreasing function of n. Negatively correlated shocks increase the real exchange rate volatility because they virtually act like a single unilateral shock of a larger magnitude. Positively correlated shocks lead to a minimal average deviation in the law of one price. Standard deviation of the real exchange rate approaches 0 as i] approaches 1. 1.4.3 Sector-specific shocks Here, endowment shocks differ across goods, possibly to proxy for sector-specific productivity shocks. Sectoral endowment shocks at Home are correlated, with no endowment shocks Abroad: Yt = aYt-i + (1 - a)Y + ut for i = 1,2 Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 31 Figure 1.6: RER properties with country - and sector - specific endowment shocks in linear model Shocks to Home and Abroad endowments (same across sectors, 1000 draws, shock persistency: a=0.95) Shocks to endowments ol goods 1 and 2 (1000 reps., shock persistency: ct=0.95) - H L R E F U o W .. 100-STD RER . -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 correlation coefficient of Home and Abroad shocks to endowments 1 In both simulations, shock volatility as a proportion of G D P is set to cr — 0.034. where Yt = Yi,t •Y2,t U2,t and ut ~ N(0, u) where Q, = a2 7 7 a' . Both persistence and standard deviation of the real exchange rate are increasing in fj (right panel of figure 1.6). When fj < 0 (shocks to goods 1 and 2 are negatively correlated), a positive law of one price deviation to good one tends to be offset by a negative law of one price deviation for another good, resulting in a small deviation in the real exchange rate. Even small real exchange rate deviations will be arbitraged away because they originate in two law of one price deviations of opposite signs but larger magnitudes. This lowers the half life of persistence in the real exchange rate (such offsetting effect disappears as n approaches zero). For fj > 0, shocks to goods 1 and 2 grow more similar and, as fj = 1, persistence of the real exchange rate approaches the value when good endowments are subject to a single shock (see table 1.1 above). The standard deviation of the real exchange rate is an increasing function of fj. Negatively correlated shocks tend to offset each other and lower the volatility of the real exchange rate. Positive correlation makes the two shocks resemble a single country-specific shock which moves law of one price deviations of both goods in unison, leading to a larger real exchange rate deviation. Note again that as fj approaches 1, the real exchange rate volatility approaches the value for a single country-specific shock. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 32 1.5 C a l i b r a t i o n In this section I describe the parameter choices for the calibration of the model. I choose the utility function weights symmetrically: 7 j = ^ V i The inverse of the elasticity of substitution 9 assumes the standard value in the literature 1.5 (see Chari, Christiano & Kehoe (1994) and McGrattan (1994)). Figure 1.7: Calibration of per-kg-per-km shipping cost for cooled sea transport Estimating the per—km-pot—ton shipping cost us function of distance 1000 2000 3000 4000 5000 6000 7000 8000 shipping Oislooca (km) Many authors calibrate the transportation costs using indirect estimates (e.g., Ravn & Mazzegna (2002)). I calibrate the shipping costs directly as a tax (or heterogeneous iceberg cost) that disap-pears in the course of good's shipment. I assume that, for any particular route, this tax depends multiplicatively on the distance and the weight of a good. Although unit shipping costs tend to decline with distance (possibly due to increasing returns to scale in transportation), this is not a problem here because the model has only two locations (one distance). Weight-dependance of shipping costs has been established by already (see Table 7 in Hummels (1999)). US and E U are chosen as locations because of their similar size. The distance is set equal to the New York -Hamburg shipping distance of 6000km. Most goods are shipped by sea between Europe and the US. I find two sources for shipping cost estimates. In a large survey of all modes of transportation, Runhaar et. al (2001) quote an average price in 2001 a standard 40' container on a route Rotterdam - Singapore (9650km) of N L G 3060 including a NLG 360 fuel surcharge (equal to USD 1220 in 2001). They estimate the average load of a 40' container is 16.25 ton. This yields an average rate of USD 0.0077 per ton per km in 2001. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 33 Perishable items such as most of foodstuffs are shipped in chilled containers. In their survey of shipping costs for fish containers Brox et. al. (1984) survey costs across a range of distances. The implied per ton per km shipping costs can be well approximated by a hyperbolic function (see figure 1.7). At the 6000km distance these imply a unit cost of USD 0.11 per ton per km between US and Europe. In order to compute the share of the good that is used up, I use the the dataset of physical weights and average prices in 2001 collected for chapter two. Approximately 24% of the goods in the dataset require refrigerating for transport. Using this as a weight, the average shipping cost per ton per km is USD 0.033. An average weight of a good in the dataset is 43kg, and the average price USD 745 (2001 prices). Because I need to calibrate two fricitions, I arbitrarily pick two weights (20kg and 66kg) which equal the average weight of a good in my dataset. Consequently, the per-kg-per-km fractions U of good i = 1,2 equal 0.0054 and 0.0174, respectively. That is, about 0.54% of good 1 and 1.74% of good 2 get used in transportation. These cost estimates are conservative compared to those found in the literature: Harrigan (1993) finds transportation barriers of 20%. Hummels (1999) uses 2-digit SITC data to estimate a transportation costs of 9%. Using 4-digit SITC data, Ravn & Mazzegna (2002) find that the weighted average of transportation costs declined from 6.31% in 1974 to 3.49% in 1994. IMF frequently uses 11% as a rule of thumb for transportation costs. All these are greater than the 1.14% average in my calibration. The stochastic endowment process at Home is calibrated to match the logged and H-P-filtered quarterly U.S. GDP series from 1973:1 to 1994:4 (see Chari, Kehoe & McGrattan (2002)). That implies the AR(1) coefficient a — 0.88 and the standard deviation of the residuals of this process is 0.8% of the mean GDP. Because the two countries in the model are approximately of the same size, the output process Abroad is calibrated to the total GDP of the main E U members: France, Italy, UK and Germany. From Chari, Kehoe & McGrattan (2002), this implies a correlation between the two output processes corr(Y, Y*) = fj — ^ — 0.6. The mean endowment of each good is normalized to 100. Calibration of the quadratic adjustment cost model is identical to that of a linear shipping cost model. No values for the quadratic adjustment cost parameter c were found in the literature. I therefore use an ad-hoc value c — 0.1. We saw in the discussion of the equilibrium properties Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 34 of the real exchange rate (section 1.4.1) that persistence will increase in c, as will the standard deviation of RER relative to GDP. At the same time, at c = 0.1, the thresholds of law of one price deviations are still distinctly different between the two goods. 1.6 S imu la t i on resul ts 1.6.1 Simulation results in a linear shipping cost model A bivariate vector of 10,000 normally distributed shocks ut and ul are used to generate the stochastic endowment vectors at Home and Abroad: Yitt = aV^t-i + (1 — a)Y + ut for i = 1,2 where Yi t Yi,t , ut = V " * * a2 7 and ut ~ N(0, fl) where fl = . The model is then 7 a2 solved for equilibrium price levels and allocations. The solution can be described qualitatively in the same way as in section 1.2.4. Due to trade frictions, tradability in goods is determined endogenously. Good 1 is traded more frequently (86% of the time periods) than good 2 (32%). Histogram in Figure A.9 illustrates the differential trade volume in the two goods. Consequently, prices of goods 1 and 2 deviate from parity in a persistent but heterogeneous manner. Figure A. 10 shows the bimodal distributions of the law of one price deviations. This bimodal nature is a direct result of the thresholds. Shocks that bring the law of one price deviations in autarky above a threshold also trigger arbitrage, equilibrium price adjustment and reversion of the the deviation from the law of one price back to the threshold. In a linear shipping cost model, this happens immediately which is why we observe bunching of equilibrium price deviations around the thresholds - lighter goods have narrower thresholds than heavier goods, and their law of one price deviations reverse more frequently. I analyze the persistence, comovements and volatility of the simulated series. Table 1.8 shows selected statistics. Persistence The model-generated logarithm of the real exchange rate matches the persistence in the data, as in Chari, Kehoe & McGrattan (2002) (see section 1.5 above). Model's AR(1) estimate in RERt = & RERt-i is a = 0.8286 with a standard error 0.005616. The persistence of the real exchange rate found in the data is 0.83. Deviations from parity for good 1 are more persistent than for good 2 (AR(1) estimates of 0.7379 vs. 0.847, respectively). 6This implies a half-life of convergence of about 3.7 quarters. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 35 Table 1.8: Properties of the US-EU model simulation data linear model 1 Q A C c = 0.05 model 2 c = 0.1 C K M c G 3 A u t o c o r r e l a t i o n s Ex. rates & prices R E R 0.83 0.8286 0.868 0.92 0.62 Business cycle stat G D P 0.88 0.88* 0.88* 0.88* 0.62 Consumption 0.89 0.88 0.854 0.69 0.61 Net Exports 0.82 0.87 0.700 0.49 0.72 S T D r e l . t o G D P Ex. rates & prices R E R 4.36 0.002 6.41 (1.65) 7.66 (2.17) 4.27 Business cycle stat Consumption 0.83 0.75 1 1 0.83 Net Exports 0.11 0.19 0.001 0.001 (0.49) 0.09 C r o s s - C o r r e l a t . GDPs 0.6 0.6* 0.6* 0.6* 0.49 Consumptions 0.38 0.62 0.283 0.27 0.49 N X & G D P -0.41 -0.03 0.050 0.04 (0.87) 0.04 R E R & G D P 0.08 0.69 -0.02 0.13 0.51 R E R & N X 0.14 0.88 (-0.02) 0.027 0.024 -0.04 R E R & Relat. C -0.35 0.96 0.956 0.95 1.00 based on 10,000 simulations of the linear shipping model with parameter calibration as described in section 1.5. 2 based on 5,000 simulations (each 20 periods long) of the quadratic adjustment cost model with c = 0.1. 3 results of the model simulation in C h a r i , Kehoe & M c G r a t t a n (2002). " denotes a calibrated value. The persistence of consumption and net exports in the model is driven by the persistence of the stochastic endowment process. Because of the risk-sharing role played by the international trade, consumption is less volatile than the endowments in the two countries (STDc/STDy=0.72 in model and 0.83 in the data.). On the other hand, zero trade volume in the linear shipping cost model lowers the volatility of exports relative to that of output (STDjv/STDy=0.19 in model and 0.11 in the data.). The existence of thresholds increases the volatility of consumption and decrease the volatility of net exports. Comovements Consumption levels at Home and Abroad have the same cross-country correlation as the GDPs. This is a result of the calibration of Y and Y* and of the fact that consumption dynamics is qualitatively similar to the shock dynamics, although smaller in magnitude. Because this is a real model, the good with the lowest friction sometimes flows in the opposite Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 36 direction than goods 1 and 2 for the aggregate constraint to be satisfied17. While the exports of goods 1 and 2 are positively correlated with the domestic GDP, corr(N3,Y) is negative. Conse-quently total net exports are slightly negatively correlated with the GDP (-0.03 in the model and -0.41 in the data). The thresholds provide only a partial disconnect of the real exchange rate in from the real economy. A sufficiently large endowment change lowers the price at Home relative to Abroad - a depreciation (increase) in the real exchange rate that leads to a positive corr(RER, Y) 1 8 . Backhus-Smith puzzle remains due to the expenditure-switching effect of the changes in the real exchange rate. Partial expenditure switching brings model's correlation of RER and relative consumptions away from unity and closer to the data (corr(RER, C/C*)=0.91 in the model and -0.35 in the data). Therefore, consumption correlation across countries is negatively related to the trade friction of a particular product: it changes from -1 (good 3) to 0.834 (good 1) to 0.686 (good 2). Volatility The linear shipping cost model fails to generate sufficient volatility in the real exchange rate, for three reasons. First, all adjustment takes place within the time period when the shock arrives. Second, maximal deviation is a weighted average of the the trade friction of the real exchange rate's components - all of them goods. A more realistic model would see the real exchange rate adjust over a longer time horizon (during which RER could exceed the thresholds), and would also model the real exchange rate components more realistically by including services with potentially large thresholds19. Third, in the absence of nominal and real rigidities, volatility of the price aggregates is limited by the volatility of the driving force of the model (see Figure A.11). The standard deviation of shocks is calibrated to 0.8% of GDP. Therefore, irrespective of other parameters, it must be that S T D ^ ^ ' < a = 0.8% of mean GDP. In this calibration, S T D m o d d = 0 . 2 % of mean GDP. This is inconsistent with the observation of S T D ^ E H ~ 436% of GDP in the data. 1 7 When goods 1 and 2 do not flow in the same directions, net flow of good 3 depends on the prices of goods 1 and 2 and the volume of trade. 18corr(RER, Y)=0.69 in the model but only 0.08 in the data 1 9 Nominal rigidities could also serve to increase the volatility of price aggregates over that of the endowment shocks. Unlike quadratic adjustment costs, these nominal rigidities would be orthogonal to the current workings of the model. Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 37 The volatility of consumption and net exports is roughly consistent with the data ( S T D ™ o d e Z =0.72, STDc?ta=0.83). Trade creates a quantitatively relevant consumption insurance. While the ratio of marginal utilities at home and abroad is not constant, it is less volatile than in the data. This is a necessary implication of a marginally higher volatility of the arbitrage trade in the model than in data (STD$ o d e 7STDgg^=0.19, STD^ t a /STDg^ P =0.11). In turn, high volatility in trade comes from the the fact that the arbitrage trade is the only adjustment mechanism in this model. Trade frequently changes from autarky (N=0) to full trade equilibria. A model that would also include non-arbitrage trade motive would generate a smoother {TV} process and so increase the volatility of consumption. 1.6.2 Simulation results in a quadratic adjustment cost model The addition of the quadratic adjustment costs (QAC) in change of volume of trade improves the qualitative results of the model. The additional trade friction brings countries closer to their autarkic consumption sets20 and leaves the equilibrium prices further away from parity for a longer time. Therefore, both persistence and volatility of the real exchange rate increase compared to the results of a linear shipping cost model. The trade volume between countries declines and becomes smoother due to the quadratic nature of the additional friction. Consumption behaviour therefore mimics the endowment dynamics. Persistence On average, the real exchange rate is more persistent in the Q A C model than in the data, with an AR(1) coefficient estimate of 0.92. A result of the upward-sloping thresholds in the law of one price deviations, the real exchange rate is more persistent than in the linear shipping cost model. Consumptions and trade are persistent to a lesser degree than the endowments, due to a limited pass-through of endowment shocks to prices and trade. Although the volatility of consumption aggregate is as high as that of the endowment shocks, the relatively small trade volume can not match the volatility of net exports from the data. As has been mentioned above, the adjustment costs further disconnect the two economies. Therefore, consumption co-moves less between countries than it did in the shipping cost model, although its value of 0.27 is closer to 2 0This happens in a more complicated way than by "increasing the distance" as would be the case in a linear shipping model (see section 1.3.1). Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 38 that of 0.38 found in the data. Similar results hold qualitatively for co-movement of RER and GDP, RER and NX, and RER and the relative consumption. The results are further improved when c = 0.05. Volatility The calibration of the quadratic adjustment cost model is significantly more successful in creating volatility of prices relative to GDP (see table 1.8). The average standard deviation estimate of 7.66 is well above the 4.35 found in the data (when c = 0.05, volatility is 6.4 times higher than that of GDP). Histogram of all standard deviation estimates can be found in Figure A. 18. Law of one price deviations for good 1 are more volatile than that of the real exchange rate. As can be seen in Figure A. 15, volatility of the law of one price deviation for good 2 is only marginally higher than for good 1 at c = 0.1 (see Figure A. 19). The distribution of mean law of one price deviations appears to be bimodal with higher mass near the thresholds (figure A. 19). The bimodality is not as pronounced as in the linear model due to the existence of quadratic adjustment costs. 1.7 C o n c l u s i o n Geography and physics delimit our lives but are somewhat marginalized in economic analysis. In a vast majority of cases, this is a justifiable omission due to a static nature of these constraints and a uniform way in which they are thought to affect economic variables. However, there are cases when geography and physics affect the marginal decisions in an economically relevant way. This chapter analyzes a particular example - the persistence and volatility dimensions of the purchasing power parity puzzle. The persistence in deviations of the real exchange rate - and of prices of traded goods - from parity is not puzzling. It can be explained as a byproduct of heterogeneous trade costs. This idea is intuitive and, as the following chapter shows, supported by the data. Nominal rigidities may not be necessary in explaining persistence and volatility in price differences across countries21. The price stickiness is a byproduct of the shipment costs which depend on distance and physical characteristics of goods. Because weight and volume of products matter in shipment (in a way which is unrelated to their price) differences in physical characteristics leads to a heterogeneity in the unit shipping 2 1 Al ternat ively , the model can be thought of as jus t i fy ing an op t ima l level of nomina l price stickyness Chapter 1. General Equilibrium Model of Arbitrage Trade and Real Exchange Rate Persistence 39 costs. Only price deviations in excess of the shipping costs are arbitraged away in the model. Consequently, tradability is endogenous and adjustment of prices over distance is non-linear. Goods with a larger trade friction need a larger deviation from parity to become traded - they are traded relatively less frequently, and can sustain larger deviations in prices from parity. This chapter studies two real models: a simple linear shipping cost model and a model with both linear shipping costs and quadratic adjustment costs in the changes of trade volume. In both models, the real exchange rate exhibits persistent deviations from parity due to trade frictions. Calibration exercise shows that half life in real exchange rate deviations can match the estimates observed in the data. The linear model also generates a quantitatively close match to the persis-tence and co-movements of various price- and quantity- constructs. However, volatility of price deviations is insufficient. Arbitrage firms in the second model also pay quadratic adjustment cost in the change of trade volume, due to additional legal and infrastructure expenses and to capture the time dimension of shipping. The core feature of the linear shipping cost model stay unchanged. However, a dynamic and highly non-linear environment emerges. Firms' aversion to react strongly to endowment shocks by large trade volume adjustments creates more pronounced and long-lived real exchange rate deviations. This further limits the arbitrage trade between countries and the co-movement between their consumptions. Simulation results of the second model can easily match both per-sistence and volatility of the real exchange rate while giving qualitatively meaningful results along other dimensions. Persistence and volatility of the real exchange rate need not be puzzling, they can be explained as a result of trade costs. 40 Chapter 2 Non-linear Adjustment in International Prices and Physical Characteristics of Goods This chapter analyzes the heterogeneous adjustment of international relative prices across a range of goods. It examines the relationship between the physical characteristics of goods and the heterogeneity in the adjustment of their prices to parity. The main result of this chapter is the significant negative relationship between price-to-weight and price-to-volume ratios and non-linear threshold estimates across goods in US and Canada1. Post-Bretton Woods CPI data on 64 groups of products and services in Canada and the US are used to study the dynamics of the law of one price deviations. It is found that changes in the LOPD behave in a non-linear way which can be described well as a threshold-type non-linearity. The setup of this estimating model is rooted in the theoretical model presented in the previ-ous chapter: deviations of prices from parity do not adjust inside of thresholds (observationally equivalent to the no-trade bounds in the linear shipping cost model), but exhibit relatively high speeds of adjustment outside of thresholds. It is found that the no-adjustment zones account for approximately one half of the persistence in the deviations of the real exchange rate. Further-more, these zones of no adjustment and half-lives of deviation are heterogeneous across goods. The most interesting result emerges when the no-adjustment zones are related to a separately collected dataset of product weights and prices. The threshold width is significantly negatively related to the price-to-weight ratios, and also positively to prices and negatively to weights when these are considered individually. The purchasing power parity puzzle is a special case of a broader phenomenon labelled "ex-change rate disconnect" for unexplainably low short-term links between real macroeconomic vari-ables and the nominal exchange rate. A fully encompassing explanation of high volatility and persistence in nominal exchange rates, it is sometimes argued, requires a model that would gen-X A similarly significant relationship is present when prices and weights are considered individually. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 41 erate similar behavior in all asset markets (see, i.a., Obstfeld & Rogoff 1996, chapter 9) 2 . One way to approach this problem is from " the bottom up". If the frictions in the goods markets are modelled explicitly, what is there to be learned about the adjustment of prices across borders? Do physical characteristics of goods matter for the behaviour of law of one price deviations? This chapter attempts to contribute to answering these questions. The general equilibrium model introduced in the previous chapter lays out the framework for the empirical investigation carried out here. In that model, heterogeneous goods in two countries are traded by profit-maximizing arbitrage firms who take prices as given and decide on the volume of trade for each good. The goods only differ in their physical characteristics which in part determine the shipping costs3. These differences lead to different marginal costs of arbitrage trade across goods. Consequently, an identical increase in an endowment for all goods may lead to profitable trade opportunities (hence mean reversion in prices) for some goods while other goods remain non-traded (and their prices in the two countries will continue to deviate). Because arbitrage trade is the only source of price adjustment in the model, endogenous tradability of goods leads to heterogeneous but symmetric thresholds: for law of one price deviations below the thresholds, no trade occurs and prices deviate freely. Prices differentials in excess of the thresholds are eventually arbitraged away. The linear shipping cost model has two predictions relevant for my empirical exercise: that prices of tradable goods behave in a non-linear manner as a consequence of the thresholds, and that the non-linearity thresholds depend on product weights and distance4. 2It is not difficult to generate arguments for why the nominal exchange rate - and asset prices in general - is more volatile than the prices of other goods. First, the utility of holding assets, as opposed to consumption goods, is derived from a stream of future dividends which have much longer time horizon (a few days up to maximum of near 20 years for durables, versus potentially hundreds of years for assets). Second, once a good is purchased, utility derived from it's consumption is, up to the first-order approximation, independent of any future (non-preference) shocks while the utility derived from holding an asset varies with the income stream the asset is expected to generate. Expectations of future shocks can then magnify the variance in subjective asset valuations relative to consumption products leading to larger asset price volatility (which is consistent with large observed trading volume). A third reason for high price variance specific to the foreign exchange market is that a high volatility in the foreign exchange market can additionally be caused by the role of nominal exchange rate as a medium of exchange. 3Distance is the other determinant. Although physical weight is an important determinant of the shipping costs for air freight, other physical characteristics such as volume are essential in container transport. 4 This also has the potential to explain why smooth threshold autoregressive models (STARs) are a very good fit for real exchange rates: the model predicts that the real exchange rate is more persistent around its equilibrium value when none of the components is far enough from the parity to trigger price reversion. The persistence decreases in the distance from the equilibrium as more of the components' thresholds are crossed. Consequently, the non-linearity of such construct is not an "on-off switch" non-linearity but a smooth nonlinearity akin to a string - just the idea behind a S T A R econometric model Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 42 The remainder of this section reviews the empirical and theoretical literature to motivate the topic. Section 2.2 discusses the data, section 2.3 presents an empirical analysis of the two datasets and section 2.4 concludes. 2.1 M o t i v a t i o n and the l i te ra ture Purchasing power parity is a very old topic in economics. While the term was coined by Cassel (1918), a question of whether national price levels are equal when expressed in terms of a common currency had been analyzed for a very long time before that5. No doubt, the appeal lies in part in the topic's accessability6, especially when much of the "puzzle" remains to be explained. The puzzle also reflects the strength of the priors most economists have about one variant of PPP or another as an anchor for the long-run real exchange rate (Rogoff (1996)). This idea is imbedded in most of the "old" (e.g., Dornbush (1980)) and many of the "new" (e.g., Obstfeld & Rogoff (1995, 1996)) open macroeconomic models. Typical general equilibrium models of exchange rates (e.g., Chari, Kehoe & McGrattan (2002)) contain a first order condition which relates the real exchange rate to the marginal rate of substitution between two countries' relative consumption baskets. In these models, properties of the real exchange rate are limited by the dynamics of relative consumptions. The theory of PPP implies that the real exchange rate (as measured by the price indexes) should be constant. Yet in the data, real exchange rates (RER) exhibit large degrees of volatility, and their deviations from the mean are highly persistent, particularly in the post Bretton-Woods era of flexible nominal exchange rates. This is why the Euler equation described above, combined with sticky prices, fares poorly. Empirical literature The empirical literature has largely followed two strands: studies of the law of one price deviations for the individual commodities and studies of aggregate indexes such as the real exchange rate. Using 7-digit ISTC categories, Isard (1977) finds large and persistent deviations from the law of one price. This is a recurring finding in many later studies. Giovannini (1988) reaches similar conclusion for extremely homogeneous products such as screws. Parsley & Wei (2001) find large 6 T h e issue has first been articulated in the sixteenth century by scholars of Salamanca school. For a survey see, e.g., Officer (1976a). 6 " A decade ago, PPP seemed like a fairly dull research topic. [...] Fortunately, the past decade has witnessed a tremendous degree of progress in the area [...] and several important results have emerged." Froot & Rogoff (1995). Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 43 deviations in the price levels of mainly food product across cities in US and Japan and lend further weight to the presence of a " border effect" (see below). Similarly, Crucini, Telmer & Zachariadis (2000, 2001) find that prices of consumer products vary greatly across the E U member states, but that overall they tend to cancel each other out so that some variant of PPP holds. Froot, Kim &; Rogoff (2000) construct data of grains and dairy products over the past 700 years to find that although law of one price holds on average, there are low frequency trends of about 200 years that tend to reverse themselves. Studies of aggregated indexes (either the real exchange rates or some of its aggregated com-ponents) tend to conclude that the Purchasing Power Parity holds over long horizons. While the earlier work (Lee (1976), Officer (1982)) found strong evidence in favor of PPP, it later became clear that real exchange rates do not revert or revert very slowly in post-Bretton Woods data7. In an important paper in terms of its theoretical implications, Engel (1999) found that nearly all of the deviations of bilateral US real exchange rates from mean (at all time horizons from one month to 30 years, and for five available definition of non-traded goods) can be attributed to the movements in the international relative price of the traded goods, rather than non-traded goods. Similarly, Chari, Kehoe & McGrattan (2000) showed that less than 2% of the variance of RER between the US and the E U is clue to fluctuations in the relative price of nontradables. These findings imply that, in short and medium run, deviations of prices of traded goods from parity are at the heart of the nonstationarity of the real exchange rate. The quest for more precise estimates of the real exchange rates persistence has often been addressed either by longer time series or by panel datasets8. It was found that purchasing power parity holds fairly well over longer horizons (e.g., Cheung & Lai (1994) and Lothian & Taylor (1996)). Due to low power of augmented Dickey-Fuller univariate unit root tests, the use of panel datasets has increased. These reject Ho of nonstationarity for groups of real exchange rates (e.g., Frankel & Rose (1996), Flood & Taylor (1996), Papell (1997), Taylor & Sarno (1998)). Imbs et al. (2004) caution about interpreting the RER persistence estimates due to sectoral aggregation 7A popular estimate range for half-lifes of convergence of detrended series is 3-5 years. This measure varies significantly across base countries. Chari, Kehoe & McGrattan (2002) find the post-Bretton Woods half life for U S - E U real exchange rate to be approximately 4 quarters. However, Murray & Papell (2002) show that the half-life estimates are highly misspecified and contain virtually no information regarding the true half-lifes. Yet, the correctly estimated half-lifes still range in the 3-5 year range. 8Distinguishing long memory processes from various alternatives is difficult. Diebold & Inoue (2001) show that even small amounts of a stochastic regime switching is very easily confused (even asymptotically) with long memory process. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 44 bias9. RER half-life estimates do not imply that the half-lifes of RER components are as persistent - they are not. It remains unclear how much weight should one place on the results of the early panel unit root tests. Taylor & Sarno (1998) show that such tests tend to reject non-stationarity if just one of the series in the panel does not have a unit root. Lyhagen (2000), Bornhorst (2003) and Banerjee, Marcellino and Osbat (2001) show that common stochastic trends - largely unaccounted for and introduced by the base currency's nominal exchange rate - yield large sizes of most of the panel tests. Nonlinear econometric models also became popular in analyzing the dynamics of the real ex-change rate. Dumas (1992), Ohanian & Stockman (1997) are a few examples of theoretical models that generate bands where the real exchange rate does not revert due existence of transportation costs. Empirical studies of this mold fit some type of a self-exciting threshold autoregressive (SETAR) model on log-deviations of real exchange rate or its major components10. It is known that an evidence of non-linear mean reversion invalidates the standard long-term tests of RER based on linear models (see, e.g., Obstfeld & Taylor (1997), Kilian & Taylor (2003)). Obstfeld & Taylor (1997) estimate a TAR model for CPI and its four major components across cities in Europe and North America and show that A R estimations of half life in the literature have to be biased upward if a data-generating process is nonlinear. Zussman (2002) estimates TAR(2,1,1) models on annual, quarterly and monthly bilateral real exchange rates for 108 countries and finds that the threshold estimates are significantly positively related to distance, import taxes, and nominal exchange rate volatility between countries11. Imbs et. al. (2003) find non-linear adjust-ment at sectoral level and show that the heterogeneity relates to distance and nominal exchange rate volatility. Studies that use smooth threshold autoregressive models (STAR - see Granger & Terasvirta (1993)) with smooth transition between stationary and non-stationary states find quicker estimated speeds of convergence than the linear A R models (e.g., Taylor, Peel & Sarno (2001), Kilian & Taylor (2003)). Theoretical justification of the STAR models remains the stum-bling block of this strand of research. Taylor (2000) uses multivariate as well as univariate tests 9 More volatile components of R E R receive a larger weight in the estimation, leading to upward bias. 1 0 A S E T A R model implies that the process is more stationary the bigger the deviation from the mean. A most basic model is the Threshold autoregressive (TAR) model with two symmetric bounds within which the process is a unit root while it reverts outside of the bounds. 1 1 The use of White heteroskedasticity-consistent standard errors as well as the ad-hoc choice - and lack of testing - of TAR(2,1,1) model is questionable Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 45 of higher power over long time horizons to show that while volatility of the RER varies across different exchange-rate regimes, persistence of the RER deviations is not significantly different from one exchange rate regime to another. A related " border effect" literature shows how the strength of dependance of prices and trade volume on distance changes dramatically for within- and cross- border comparisons. The cross-border price differences are much larger (and trade densities much lower) than what can be accounted for by standard gravity models. Engel (1993) finds that prices of similar goods across borders are a lot more volatile than relative prices of dissimilar goods within the US. Engel & Rogers (1996) show such effect in terms of price volatility and Jenkins & Rogers (1995) in terms of persistence. Engel & Rogers (2001) use monthly CPI data for 55 European cities and find that at monthly frequency, variance of changes in nominal exchange rates is the most important explanatory variable of the variance of changes in prices across locations. Yet at 48-month horizon, impact of the border dummy on the variance of prices across borders is 40-times larger (yet equally significant) than the impact of changes in nominal exchange rates. These results suggest that the sticky-prices and pricing-to-market hypothesis may well be present at frequencies below 48 months. O'Connel & Wei (2002) show how various bands of no-trade may arise from trade frictions in an optimal control framework of Dixit (1993). The authors analyze quarterly cost-of-living price level survey data covering 48 goods (mainly foods) and services in 24 US cities12 and find stationarity in either an AR, TAR, or ESTAR specifications. Theoretical literature Although the real exchange rates are aggregate objects consisting of nearly all goods and services sold in the economy, their theoretical analysis has traditionally been approached in a binary framework of tradable versus non-tradable goods. Such framework suggests two sources of real exchange rate deviations: from the deviations of the law of one price for tradable goods or from the movements in the relative prices of non-tradable to tradable goods. In the evidence of the importance of deviations of prices from parity for traded goods, the latter approach has been addressed by modeling endogeneous tradability. 1 2 This is an identical dataset to Parsley & Wei (1996). Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 46 Pricing to market Models of pricing to market (a term coined by Krugman (1997)) assume market segmentation for a subset of the tradable goods, and have a positive share of goods prices quoted in a different currency (local currency pricing). When preferences do not differ across countries market segmentation is non-vacuous only in the presence of real frictions (frequently imperfect competition). The former allows firms to choose their prices as a markup over their marginal costs and warrants a demand-determined and inefficiently low level of output. If prices are sticky, an excessive response of the exchange rate to a shock is necessary to clear the market for real balances, increasing the volatility. If prices are quoted in the currency of a producer and preferences are identical, an exchange rate change will leave unaffected home and foreign prices of home goods, and change proportionately home and foreign prices of foreign goods. Thus, nominal exchange rate movements will not induce deviations from the law of one price for traded goods. With different yet constant preference parameters, the ratio of prices in two countries is different from 1 but constant (relative PPP holds). Only in the presence of preference shocks is producer pricing capable of generating LOP deviations (e.g., in Lapham & Vigneault (2001)). Betts & Devereux (1996) build a model of pricing to market and show how it limits the exchange rate pass-through to foreign prices. In a dynamic model, Betts & Devereux (2000) show that pricing to market leads to overshooting in the exchange rate and, because of sticky prices, increases the volatility and persistence of the RER relative to a model with producer pricing. In models of this kind (Chari, Kehoe & McGrattan (2002), Bergin & Feenstra (2001), etc.), behavior of the real exchange rate around a stationary equilibrium q\ is a function of marginal utilities of consumption and a risk-aversion parameter13: q = A(c- c*) + B(m - m) + D(l - 1*) where c,m,l are logarithms of consumption, money and labor, "*" denotes a value of a variable abroad, " is a deviation from a steady state and A,B,D are constants of which only A is significantly different from zero. The variance of the real exchange rate is then determined by the variance of the consumption growth differential between countries, while the persistence of real 1 3 Char i , Kehoe & McGrattan (2000) show that preferences need to be additively separable in leisure and a consumption-money aggregate to generate this result. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 47 exchange rate deviations comes from autocorrelation in such consumption growth differentials: std{q) = astd(c - c*) corr(qt, qt-i) = corr(ct - c*t, c t_i - tt_x) Models of this type tend to generate sufficient variance but insufficient persistence in the RER movements, even with 12-month price rigidity. In Ravn (2001) endogenous (optimal) pricing-to-market leads to persistent and volatile RER fluctuations. Ghironi & Melitz (2004) model changes in the consumption basket composition due to monopolistically competitive firms facing heterogeneous productivity shocks and trade barriers. Nontraded goods Even when the law of one price holds for traded goods, price of non-traded goods relative to the price of traded goods depends on the local conditions, which can create a wedge in the real exchange rate. Therefore, purchasing power parity need not hold in the models with non-traded goods. Models of this sort tend to be nested in the Redux model of Obstfeld & Rogoff (1995) (see Obstfeld & Rogoff (1996, 2000a), Hau (2000)). Specifically, non-traded goods change the transmission of a positive monetary shock in three ways. First, the "relief effect" of a international price adjustment of tradables is curbed, which in equilibrium requires a larger nominal exchange rate adjustment (an exchange rate magnification effect). Second, domestic demand expands following a depreciation in home currency because the non-tradables become cheaper relative to the tradables. This home consumption bias effect limits the demand transmission of the shock and lowers the international comovements of C, Y. Third, a price differential drives a wedge between the real returns on foreign assets to home and foreign consumers which affects the intertemporal consumption smoothing effect. Bergin & Glick (2004) introduce endogeneous non-tradability by the way of heterogeneous iceberg costs. Good on the margin provides the linkage between prices of traded and non-traded goods. Ravn & Mazzenga (2002) generate persistence and volatility in RER deviations in a model where foreign imports and domestic factors of production are complements. The remainder of this chapter is structured as follows: section 2.2 describes the data, section 2.3 studies the dynamics of the deviations from parity for 63 goods and services, by estimat-ing appropriate non-linear models and analyzes potential sources of heterogeneity in threshold estimates. Section 2.4 concludes. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 48 2.2 D a t a This section aims to assess the evidence in support (or against) of the model presented in the first chapter. It first describes why data limitations do not constrain analysis of the dynamics of the law of one price deviations, and then describes the datasets. 2.2.1 Limitations of studying index-based R E R Study of disaggregated real exchange rates based on price indexes can only answer a subset of question related to the PPP puzzle. A level of an index does not have an economic meaning on its own, only when compared to other observations of the same index. The same applies to any construct that uses at least one index, for example the index-based real exchange rate. An observation of log(RER) = 3 does not tell us anything about the deviation from PPP: one goods basket could be overvalued, undervalued or at parity with the other one1 4. Therefore, questions related to the magnitudes of the law of one price deviations and PPP deviations can only be addressed directly with price level data. Consequently, one can not test for speed of adjustment to parity using index data because such data does not specify where the parity is. Index data can only be used for the analysis of deviations from a mean, interpretation of which is open to discussion15. If there are bands where prices do not adjust and if the deviations from parity within these bands are tied to nominal exchange rates, then the width of such bounds may be negatively related to the likelihood that the mean of the real exchange rate coincides with parity. In models which generate no-trade bounds, the exact location of the mean of the real exchange rate within no-arbitrage bounds is pinned down by the mean of the nominal exchange rate. Although most theories predict that the mean of log(RER) is zero, this is not true in the data. Demeaning of the series is then essential for two reasons. First, a constant has no meaning in terms of an index, and second, there is no theoretical justification for a mean log-deviation to have a non-zero value16. Detrending of the index-based series may be harmful because it makes the data more stationary and so increases the likelihood of rejecting nonstationarity (see Tables 1 4 I f one knew the size of the deviation from parity at any point of time, one could construct the series on actual deviations from law of one price deviation. 1 5See Crucini, Telmer & Zachariadis (2001) for a price level analysis that documents widespread law of one price violations (hence mean does not equal parity) across the E U . 1 6 A less convenient alternative to demeaning is to remove the starting value of the deviation from a law of one price (or a real exchange rate) for all series and study the dynamics of the relative prices starting from this assumed parity Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 49 B.7, B.8 and B.9). Although some argue that detrending removes a long-run trend and allows to capture better the short-run dynamics (Obstfeld & Taylor (1997)), this does not withstand closer scrutiny. The dynamics of an index series carries the same meaning as the dynamics of level series. Consequently, if there is a trend in the index series, that same trend is also present in the price level series. Such trend needs to be explained, not removed. I do not detrend the data. 2.2.2 Dataset description Tables B.7, B.8 and B.9 illustrate basic properties of the data and the issues raised above. First, the non-stationarity of the demeaned real exchange rate differs across the sample periods (see Table B.7). Augmented Dickey Fuller tests show that in the 1947-1970 period, levels of nominal as well as real exchange rates were more stationary than in the 1970-2000 period. This holds irrespective of whether and how the data is detrended (either linearly in Table B.8 or by Hodrick-Prescott filter in Table B.9). Second, detrending makes data more stationary and so lowers their half lives of convergence. Between 1947 and 1970, estimates of the half life of convergence of the real exchange rate decline from 27 to 15 to 4.3 months as we move from raw to linearly detrended and then to HP-detrended series, respectively. If we consider the full sample period (1970-2000), this measure drops from infinity to 228 to 4.6 months, respectively. Properties of the nominal exchange rates are similar. Price index dataset The price index dataset contains disaggregated price series of 63 groups of goods and services in the United States and Canada, as well as the aggregate consumer price index and the nominal exchange rate. The maximal time coverage is from 1970:1 to 2000:8 (some series start after 1970). The series were obtained from Bureau of Labor Statistics and Statistics Canada, respectively, and are all demeaned. Of the 63 groups included in the dataset, 49 cover goods and 14 services. The goods in the dataset cover 24.1% of the CPI while the services and nontradable goods cover 46.7% of the CPI 1 7 . That is, my dataset covers approximately 73.5% of the CPI. Using the taxonomy of Lebow & Rudd (2001), 77% of durable goods, 70% of nondurable goods and 39% of services are included in the data (see tables tables B . l and B.2 in Appendix B.l). 1 TSource: CPI all urban consumers, Bureau of Labor Statistics, December 2001. Some of the groups are a subset of other groups - all such double accounts are accounted for in this reported measure. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 50 Physical weights dataset I construct the dataset of physical weights and individual prices for each good (or group). These data can not be obtained from one source, leading me to the following data-collection algorithm. First, when available, weights are obtained from statistical agencies or government bodies. When unavailable, documents of manufacturers' associations are searched for average weights of partic-ular goods or product groups. In a minority of cases when neither of the the first two techniques is available, the weights are estimated as an average of the market's large manufacturer's product range (e.g., for watches, an average weight is set equal to a current average weight of a Timex watch). Average prices are obtained in a similar manner, US data sources being searched first. When a US price is unavailable, similar search is performed across the Canadian data sources. The price corresponds to an average price in year 2000 in US dollars. When the describing a group of products rather an individual product (e.g., womens' apparel), weight and price are computed as weighted averages of weights and prices of components using the expenditure shares from US urban average CPI in December 2001 as weights. When the price is not available for year 2000, the last available price is inflated by the CPI inflation rate of the relevant country. Composition of all groups, data sources, as well as price and weight estimates are documented in tables B.3 and B.4 in the Appendix B.2. Volume dataset The dataset of physical volumes of is calculated indirectly using data on stowage factors from the German Transportation Information Service database18. A stowage factor of a cargo is the ratio of weight to stowage space (the unit is ton/m3) required under normal conditions, including all required packaging. Consequently, the volume of a unit of the good can be calculated using the stowage factor and weight of the good. Because stowage factors for goods can vary depending on packaging, water contents, and compression, I use the average of all quoted stowage factors in calculating the volume of a good. I find stowage ratios for products that are not included in the German database from other sources (see the Appendix B.2 for a detailed list). 1 8 A website run by the German Insurance Association http: / /www.t is-gdv.de/t is e / inhal t .html Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 51 2.3 E m p i r i c a l analys is and resul ts This section studies the dynamics of the law of one price deviations across goods and services, and attempts to account for its sources. It has two goals: first, to establish stylized facts about the heterogeneity in international price adjustment across goods and second, to account for relevant features of the observed behaviour using the lens of the theoretical model from the first chapter. The second task tests the theory from chapter one. The variable of interest zf is a logarithm of law of one price deviation: zf = p\ — pf* + St, where t is a time index and g is a good (service) index, p and p* denote logarithm price indexes in US and Canada, respectively, and St is the logarithm of the nominal exchange rate. 2.3.1 Model selection A general discrete form of a threshold autoregressive model is chosen for analyzing the time series. The theoretical model from chapter one implies good-specific thresholds of no arbitrage which can be well captured by a group of self-exciting threshold autoregressive models19. A choice has to be between two kinds of models: "smooth" and not "smooth". Frequently used smooth transition autoregressive models (STAR) are, for the purposes of RER analysis, a fluid combination of a non-stationary and a stationary regime akin to a string2 0. However, good-specific no-arbitrage bounds from the linear shipping cost model imply discrete thresholds determined by the physical characteristics, distance, unit shipping costs, insurance costs, trade barriers, etc. With two locations, and assuming that the unit shipping costs do not change over time, such a model implies a 'sharp switch' non-linearity with two separate autoregressive regimes: one within and another outside the thresholds. Other choices need to be made about the exact form of a TAR model before estimation: selection of the number of thresholds, selection of a number of autoregressive lags p and of an optimal delay parameter d. Because the theoretical model predicts two thresholds for each good 19Self-exciting threshold autoregressive (SETAR) models can be thought of as a combination of several (typically two) regimes which differ in the degree of stationarity they impose on the series. The decision on which regime shall the variable observe depends on a position of a control variable - in "self-exciting" models this is just a lagged value of the examined series. 2 0 A larger deviation of the R E R rises with the weight placed on the stationary regime relative to the non-stationary one. Therefore, reversion occurs for any deviation and its strength rises in the size of the deviation (see Tong (1990), Granger & Terasvirta (1993) for reference on non-linear time series analysis) Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 52 a TAR(2,p,d) is selected21. Moreover, I assume that deviations from the law of one price behave symmetrically and therefore the two thresholds 7 1 and 7 2 are symmetric around zero, i.e., "ff — - 7 I = 7 S for each good g. Letting the optimal delay lag parameter be dp, the vector of the appropriate lagged values zt be zt, the symmetric threshold be 7 , and g again index the goods, an "equilibrium threshold autoregressive model" (EQ-TAR) can be written as: pg,outE9 + eout i f £ > 7 g f39<inz? + ef if 7 S > z°_dp > - 7 9 pg,out29 + eout i f _ 7 S > z9_d where e£"* ~N(0, ug4* ) and ef ~N(0, crl£ ). As an alternative econometric model, band-threshold autoregressive model (BAND-TAR) is estimated as well. BAND-TAR forces data to converge towards the bound, rather than towards its mean as in EQ-TAR: pg,out^9 _ 7 s) + eout i f z 9 > 7 s <39>inz? + ef if 7 5 > zl_dp > - 7 S p9,out^9 + 7 S ) + eout if _ 7 9 > £ where e^ "* ~N(0, cr^ ) and ef ~N(0, (T%g ). Because large deviations from the mean in a BAND-TAR model gravitate more strongly towards the band than the smaller ones, this model necessarily produces faster conditional speeds of convergence. Vectors of estimates j3g'm and /?s>ou' determine the conditional convergence speed of the law of one price deviation Azf outside the no-arbitrage bound 7 9 . It is assumed that a law of one price deviation that lies within these bounds does not exhibit mean reversion due to transportation-related costs, good characteristics, trade barriers and other impediments preventing arbitrage: a restriction of / 3 9 > M = 0. 2.3.2 Methodology Specification and estimation of a nonlinear model (in this case TAR(2,p,d)) proceeds in several steps22 that are repeated for all 63 product groups and for the CPI-based real exchange rate. First, an appropriate lag-structure p of the linear model is determined. Secondly, the delay parameter d is chosen. Finally, the non-linear model is estimated and tested against its linear alternative. Specifically: 2 1 One threshold following sufficient appreciation, another one after depreciation. 2 2See Granger and Terasvirta (1993), Terasvirta(1994), Tsay (1986), Tsay(1989). Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 53 1. With monthly data, up to 12 lags are considered. Examination of the partial autocorrelo-gram (Granger and Terasvirta (1993)) narrows the potential candidates for lags. Out of the potential candidates, I choose a combination with the lowest Akaike information criterion (or Schwarz Bayesian information criterion) as long as the residuals are not serially correlated and are normally distributed23. An appropriate model specification is important at this stage because omitted autocorrelation may lead to a rejection of the null hypothesis of a linear model and in general make if difficult to interpret test results (Kilian and Taylor (2003)). 2. For a given set S of feasible values of the delay parameter d (here, S = {1 ,2 ,12}) and for a given lag structure of the A R model determined in the previous step, the optimal dp is selected by a procedure suggested in Tsay (1989): F(p,dp) = max. F(p,v) where F is the F-statistic described in the Appendix B . l . This procedure selects the value of the delay parameter which gives the most significant result in testing for a non-linearity. P-values of the optimal dp can then also be used as a general nonparametric test of non-linearity. 3. For a given lag structure and a delay parameter dp, parametric estimation uses maximum likelihood estimation in a procedure described in Obstfeld and Taylor (1997) (who in turn follow Fanizza (1990), Balke and Fomby (1997) and Prakash (1996)). The procedure is a best-fit grid search for a threshold parameter 7 that maximizes the log-likelihood ratio L L R = 2(La - LQ) where LaWimPouuo-imO-ouui) = -^n i n [ log(27r)+ log[^ J i n t '" 1 l + ?^—?1 2 L L nin - 2 J nin J 1 f, ,„ > , , r5Z/ o l U et,out] , n<mt — 2] -^n o u t \ log 2TT -flog — ' + , 2 L L nout - 2 J nout J 1 . . r , - _ / « _ x , , . J ^ e ( l , » - 2 Lo(P,<r) = --n[log(27r)+log 2 L ' ° L n - 2 J n where et,in = Azttin - f3inzt}in 2 3Selected combination is tested for residual serial correlation using Breusch-Goodfrey L M test and by exam-ining the Q-statistic, and for the residual normality using Lomnicki-Jarque-Berra statistic. Residual normality is frequently rejected which can be result of the sample size. Most of the time, these criteria select the same model. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 54 e t ,ou£ — ^zt,out Poutzt,out and et = &zt - fizt Choices of 7 with less than 1 0 observation above the threshold are not considered in order to lower the sample bias. The half-life of adjustment is often used as a way of assessing the speed with which deviations from LOP return to their respective bands. When the regression includes more than one lag, the half-life of convergence to a shock is computed numerically24. 4. Two tests are used to assess the non-linear TAR against the linear alternative: likelihood ratio test and Tsay's general nonparametric F-test. First, the likelihood ratio test uses a statistic obtained during the grid-search. The likelihood ratio statistic does not follow the asymptotic x2 distribution in a non-linear model. Because the threshold parameter 7 is not identified under Ho of linearity25, Monte Carlo simulation is used to obtain p-values of the LLR statistic for this case Second, Tsay's general nonparametric F-test uses the minimal p-value of the two F-statistic described in Appendix B. l : one from an arranged regression using ascending ordering of the case data, another with descending orderings of the case data (see Obstfeld and Taylor (1997)). So far, I have only run this test on a TAR(2,1,1), not the general TAR(2,p,d) model. 2.3.3 Non-linear estimation results This subsection documents the findings of the time series estimations. In summary, the non-linear models perform significantly better than linear autoregressive models in characterizing the law of one price deviations. Most of the time series are non-stationary and significantly non-linear, well suited for analysis by threshold autoregressive models26. Application of augmented Dickey Fuller tests in this paper takes into account the appropriate lag structure chosen by analyzing the 2 4 F o r AR(1), half-life = - l o g '° 1 ' [ ff i . a ) where $*g is Kendall's bias-adjusted slope coefficient: $"9 = 2 5 T h i s is another way of saying that there exist other sets of restrictions that make the T A R model linear. 2 6 T h e precision with which we can conclude non-linearity or non-stationarity depends on the length and breadth of the sample as well as on whether the test statistic controls for the serial correlation of the error terms. O'Connel (1998) shows how failure to account for serial correlation leads to serious size distortions. Papell (1997) shows that various panel datasets provide stronger rejection of the unit root hypothesis than a similar time-series analyses. While panels improve the power of unit root tests, they suffer from series of other problems (see, e.g., Lyhagen (2000), Bornhorst (2003), Banerjee, Marcellino & Osbat (2001)). It should also be noted that, in addition to all problems mentioned above, the power of the unit root tests further drops when the underlying D G P is not linear. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 55 partial autocorrelation function. Table B.10 reports the results of the Augmented Dickey Fuller tests for all series. In the majority of cases, ADF can not reject Ho of a unit root. Unit roots appear to be rejected for the more valuable series with the notable exception of foods. Tables B.I1 and B.12 summarize the results for two versions of the threshold autoregressive model. AR(1) Although the slope coefficients on all stationary AR(1) regressions are significant, the majority of the regressions are misspecified because the optimal lag structure is rarely AR(1) (misspecification results are not reported). Unit root hypothesis can be rejected for just 16 of the initial 47 goods (see Table B.10). The half-life estimates show a large degree of heterogeneity in mean reversion across goods: they are 8 years on average with a standard deviation of 17 years. Pooling products into groups (see Tables B . l l and B.12), we see that vehicles and car parts, clothing and footwear have the lowest half-life of 20 months. Following them are toys (31 months), foods (46), fuels (51), furniture (60), vice goods and jewelry (72), laundry appliances (93), educational books and supplies (119), technological goods and equipment (156) and medical products and house chemicals (235 months). Two sources of model misspecification complicate interpretation of half-lives: a potential nonlinearity of the underlying series27, and an inappropriate lag structure even when the series is linear. These problems are to some extent eliminated by threshold autoregressive models. TAR(2,1,1) A TAR(2,1,1) eliminates one of the two sources of misspecification. However, it does so in an imprecise way by assuming that the optimal delay parameter d = 1 which need not be the case. Nevertheless, the threshold heterogeneity is clearly visible in the results (table B.15). The conditional half-lifes decline relative to the AR(1) estimates - a necessity with any SETAR model. The average estimate of a half-life declines from 96 months in AR(1) to 49 months in EQ-TAR (2,1,1) and 70 months in Band-TAR(2,1,1). Half-lives in BAND-TAR range from 2 months for watches to 630 months for medical care products, and in EQ-TAR from 2 months for potatoes to 346 months for car maintenance. The average size of a threshold is 11.7%, and varies from 0.8% for watches to 40% (31% in BAND-TAR) for potatoes. 2 70bstfeld & Taylor (1997) and Kilian & Taylor (2003) show how nonlinearity leads to an upward bias of the half-life estimates. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 56 I then estimate an appropriate lag-structure p (see table B.10) and the optimal delay parameter d for each of the 63 series. The appropriate lag structure is AR(1) in less than half of the situations. The optimal delay parameter equals one in only 10 cases and equals on average 6.8 months. This suggests that prices react to shocks with an average delay of more than half a year. The non-linearity of the series is also clearer when the autocovariance structure of the series is taken into account. Tests reject linearity for 66% of all stationary series. TAR(2,p,d) Due to treatment of miss-specification, the number of non-stationary series drops from 7 in AR(p) to one in a BAND-TAR and 5 in EQ-TAR. Thresholds Threshold estimates 7 9 denote the width of a symmetric no-arbitrage band: nom-inal exchange rate deviation has to cross the threshold before, d months later, reversion towards a mean (towards threshold in case of a BAND-TAR) occurs. The average threshold of price de-viation is ±7.9% (a threshold of 0.0785) from the mean. The thresholds vary from ±0.6% for footwear to ±30% for tobacco28. A weighted average of thresholds where weights follow the R E R weights equals ±4%, which is in between the TAR(2,1,1) and TAR(2,p,d) thresholds of ± 7 . 1 % and ±1.2%. Because the thresholds are symmetric, their estimates are obviously bounded from above by the degree of variability of the series. The correlation of std(zt) and 7 is 0.55. The relationship between the thresholds and physical characteristics of goods that has been suggested by the model of chapter one is examined in subsection 2.3.5 below. Persistence The half lives of convergence reported in the tables are conditional on a shock that brings zt outside the no-arbitrage bounds. Moreover, the unconditional half lives vary greatly across goods as they combine the effects of heterogenous threshold with that of a conditional half-life. A simple average of the half-life of deviation of a CPI component contained in the dataset is 94 months according to a BAND-TAR model. This amounts to 48% of the persistence of the CPI-based RER as measured by a half-life of convergence in a BAND-TAR model. A weighted average of the components in the dataset is 188 months, 97% of the persistence of a CPI-based 2 S T h e standard deviation of thresholds is 0.074. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 57 RER according to a BAND-TAR model2 9. This excercise illustrates that the persistence of the real exchange rate can be decomposed into persistence of its components30. Consequently, our understanding of the adjustment of individual goods' prices across borders is essential. 2.3.4 Testing The estimation results are tested in two ways. First, I test the appropriateness of the thresh-old non-linearity assumption in all threshold-autoregressive models using Tsay's F-test for non-linearity. Second, I test the likelihood that the non-linear model describes the data better than the linear autoregressive model using a log-likelihood ratio test. Both tests are meant to assess the robustness of the findings. As another robustness check, note that Equilibrium TAR and Band TAR give identical estimates of threshold in 67% of cases, while their estimates of speeds conditional on the thresholds coincide in only 15% of all cases. Tsay's F-test for non-linearity Tsay's test for non-linearity is described in detail in Appendix B . l . When applied under the assumption that AR(1) is the data generating process, Tsay's test rejects the Ho of linearity for 16 out of the 42 stationary series. The non-linearity of the series is clearer when the autocovariance structure of the series is taken into account. Under AR(p), linearity is rejected for 66% of the stationary series. Log-likelihood ratio test Log likelihood ratio (LLR) test statistic has been conveniently computed during the TAR(2,p,d) estimation. It measures the difference between log-likelihoods of an optimal TAR(2,p,d) and a corresponding AR(p) model. However, LLR statistic of a non-linear model does not follow the usual x2 distribution because the parameters of the nonlinear alternative are not identified under HQ of linearity (Obstfeld & Taylor (1997), Granger & Terasvirta (1993)), i.e., there exists more than one set of restrictions which makes a TAR(2,p,d) model linear. I use Monte Carlo simulation to obtain an empirical distribution of LLR for all goods and from it compute the empirical p-values of LLR statistics Tables B.16 and B.18 provide empirical p-values of the hypothesis that 2 9Conditional T A R half-life is used when L L R test shows that TAR(2,p,d) model is preferred to AR(p). Otherwise, AR(p) half-life is used. Also, categories of goods that overlap are only used once. 3 0Imbs et. al (2004) show how inferences can not be made in opposite direction: R E R persistence measure does not imply much for the persistence of the component series. Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 5 8 a TAR(2,p,d) is better than AR(p) for EQ-TAR and B A N D - T A R models, respectively. At 10% significance level, L L R test results imply that TAR(2,p,d) is a better model than the linear AR(p) for 40% of series (30% in Equilibrium TAR). 2.3.5 Determinants of no-arbitrage bounds I analyze the determinants of thresholds through a lens of the linear shipping cost model of chapter one. Estimates of the no-adjustment bands are related to the estimates of weights (and price-to-weight ratios) as well as to estimates of volume (and price-to-volume ratios) across goods. This amounts to an indirect test of the theory. Product weight and no-arbitrage bounds When a full dataset of goods is used, price-to-weight ratios are negatively but insignificantly correlated with the thresholds estimates (see the first estimation in table B.20). After removing tobacco from the dataset (tobacco data exhibits a structural break due to a differential tax change in 1984 - see figure B.2), the estimation results are significant at a 10% level (estimation 2 in table B.20). An increase in the price-to-weight ratio of a good by one $/kg narrows its no-adjustment threshold by 2.4 percentage points on average. This is a major finding. It shows that certain characteristic of a time-series estimation across a range of goods is significantly related to a completely different dataset of physical characteristics of those goods. Moreover, this is also a prediction of the model in chapter one. There are groups of products for which shipping costs present only a small fraction of arbitrage trade costs. Trade in natural gas or gasoline is controlled, and significant distribution costs are necessary for trade in these goods. Licensing requirements present a barrier to trade in alcoholic drinks. For these groups of products, shipping costs constitute a small fraction of the arbitrage costs in these products. Indeed, estimations that exclude these goods are more significant. Table B.23 presents findings of such regressions. If both alcohol and energy are excluded from the dataset, price-to-weight ratios are significant at a level of less than 1%. Figures B.2 and B.3 plot the relationship between thresholds and price-to-weight estimates. A non-linear least squares estimation which assumes hyperbolic relationship between thresholds and price-to-weight ratios delivers a better fit: 23% of variability in thresholds can be accounted Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 59 for by the variability in price-to-weight ratios (see table B.21) 3 1 . Relaxing the ad-hoc restriction about identical magnitude and opposite directions in which prices and weights affect the threshold improves the results3 2. The R 2 of the regression doubles and the significance of the regressors increases (see table B.22). In the absolute value, price elasticity of the threshold is 20 times lower than the weight elasticity. The effect of product weight is now 8-times larger than when price-to-weight ratio was considered. Standard deviation of the law of one price deviation is significantly negatively correlated with the weights of the products and their prices. A l l else constant, an increase in the weight of a product by 10kg is predicted to increase its no-trade bounds by 0.93 percentage points (on each side). Similarly, an increase in the price of a product by USD 100 will lower the no-arbitrage bound by 0.5 percentage points, ceteris paribus. The constant term (7.1%) absorbs all effects homogeneous across goods: unit shipping costs, identical trade barriers, etc. Product volume and no-arbitrage bound For many transportation modes, volume of the product is more important than weight for deter-mining the cost of shipping. This is in particular true of sea shipping. Shipping costs are quoted per container (with various sizes available). Therefore goods that take less space (relative to their price) will have a relatively smaller per-dollar shipping cost. At times, Leontief-type combination of weight and volume is used to determine shipping costs. I estimate regressions identical to those above with price-to-volume data. Table B.20 shows that price-to-volume ratios are negatively correlated to thresholds at 5% level of significance: more voluminous products have wider thresh-olds in price deviations. The coefficient is smaller than for price-to-weight ratios. This finding is present more significantly when goods with high non-shipping cost component of arbitrage trade costs are excluded (see table B.23). 2.4 Conc lus ion Physical characteristics of goods are important in explaining the dynamics of deviations of inter-national prices of goods from parity. In particular, they are important in explaining the threshold 3 1 A source of such nonlinearity could lie in an additional effect of fixed costs, different degrees of tradability or in any other factor I do not control for. 32Such assumption is only supported in a model without fixed costs to arbitrage (which in this model can be thought of as wage bill, legal, or other trade-related expenses homogeneous across goods). Chapter 2. Non-linear Adjustment in International Prices and Physical Characteristics of Goods 60 nonlinearity across goods. In this chapter, an econometric model with a threshold nonlinearity shows that about one half of the persistence of the RER is accounted for by the existence of good-specific thresholds. Moreover, good-specific thresholds are in turn significantly related to the price-to-weight ratios as well as price-to-volume ratios obtained from a different dataset in a direction predicted by the arbitrage trade models from chapter one. This chapter analyzes the law of one price deviations across 63 groups of goods and services between the US and Canada and combines them with a separate dataset on product weights. The estimation and testing of linear- and threshold-autoregressive models on monthly data (1970-2000) shows that the adjustment of prices of goods and services is often non-linear: there are bands of no adjustment where deviations of prices from parity do not revert towards the mean. First, in terms of the speed of convergence, services have an average half-life of 200 months while goods have an average half-life of 47 months. Second, one half of the persistence in the real exchange rate deviations can be accounted for by a proper econometric modeling of the prices of tradable goods, in that it is contained within the aforementioned thresholds that are left out as part of misspecification error in the AR models. Third, there is a large heterogeneity across goods and services in the dynamics of the law of one price deviations. Heterogeneity across thresholds is significantly negatively related to the estimates of the price-to-weight ratios of the tradable goods (the size of the no-adjustment bands varies from 0.5% for footwear to 30% for tobacco). More "valuable" goods have narrower no-trade bounds which is why their price adjusts faster to a shock to the law of one price deviation. A similar relationship is found between the heterogeneous half lives and price-to-volume ratios. These findings explain the heterogeneity of behaviour of price deviations from parity as a result of heterogeneity in physical characteristics of goods. They also yield support to the theoretical models outlined in chapter one which explain PPP puzzle as a trade-based phenomenon. 61 Chapter 3 Basle Accord and Financial Intermediation: The Impact of Policy 3.1 In t roduc t ion 1 Traditionally, the literature on financial intermediation and credit channels, especially credit crunches, emphasized the relationship between banks and entrepreneurs requiring credit and neglected the funding of banks. With this paper, we want to be more precise in this respect and study the impact of funding on credit. Indeed, regulation that has become world wide with the Basel Accord limits the amount of bank loans with minimal levels of bank equity. How much equity the banks can issue depends in particular on the demand for equity by the households. In our model economy, households have heterogenous asset holdings because they have different labor histories and because only some of them get credit as entrepreneurs (among those, the return on investment is stochastic). Non-entrepreneur households invest in bank deposits and bank equity, and banks maximize profits while following regulations. A central bank conducts the monetary policy and regulates the banks. When banks need to reduce their loan portfolio, the displaced entrepreneurs become new equity holders, thereby acting as "automatic stabilizers". However, banks typically cut loans as a consequence of their loan portfolio becoming too risky, and households may then want to hold less equity in banks that are now more risky. Whether and how much banks have to tighten credit depends primarily on the distribution of assets across households and their equity decisions. We solve the model using numerical methods, in particular for the transitional dynamics that may lead an economy into a possible credit crunch. We then analyze policies that may help the economy out of a trough. We find that the endogenous distribution of assets has strong implications that should not be neglected in future research. Also, monetary policy can only have positive real effects if the central bank is able to commit. We find some evidence in our model that a credit crunch can arise in the presence of capital 1 This chapter is based on work with Christian Zimmermann. Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 62 requirements. Numerical simulations show that the crunch is not very large. It is natural to look at whether flexible capital requirements can change the loan volume dynamics. Although one may think that loosening those requirements in a trough will expand the loan mass, the contrary is the case. As tighter capital requirements increase the demand for equity, they facilitate the financing of banks sufficiently to offset the reduction of allowable loans for a given level of equity. Again, this highlights the importance of household sayings decisions. This result is particularly important as the new Basel Accord with its more flexible requirements would essentially tighten requirements when the economy undergoes a recession. We are not the first to highlight the real impact of monetary policy through bank lending. Bernanke and Gertler (1995) highlight two channels. In the balance sheet channel, central bank's policy affects the financial position of borrowers and hence there ability to post collateral or self-finance. In the bank lending channel, central bank's policy shifts the supply of bank credit, in particular loans. They argue the importance of the latter channel has declined with deregulation, as this channel relies on reserves. Van de Heuvel (2001) identifies another channel stemming specifically from Basel Accord like rules. The "bank capital channel" arises from maturity trans-formation through banks: higher short term interest rates depress profits, thus equity and capital adequacy. Their model has a very detailed banking structure, but neglects the problems of house-holds and firms. Our model has a simpler banking structure but emphasizes the source of financing (households) and the demand for loans (entrepreneurs) by modeling occupational choice, savings and bankruptcy.2 Chami and Cosimano (2001) identify a similar channel, called "bank-balance sheet channel", using the concept of increasing marginal cost of external financing. As Van den Heuvel, they need market power in the banking industry to obtain the result. Our model has fully competitive banks. Furthermore, they summarize the demands for loans with a reduced form while we try to come closer to a general equilibrium framework. Bolton and Freixas (2001) find that, capital requirements can be the origin of a credit crunch. Their model is very detailed on the lending market and asymmetric information. Our model puts more emphasis on the financing side and does not require asymmetric information. The structure of this chapter is as follows: section 3.2.2 analyzes the heterogenous behavior 2 T h e heterogeneity of firms we obtain is then endogenous. Bernanke, Gertler and Gilchrist (1998) also have heterogeous firms, but they exogenously fix a share of firms to have easy access to credit. Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 63 of households, sections 3.2.3 and 3.2.4 analyze the (homogeneous) financial sector and the central bank, section 3.2.6 defines and analyzes the equilibrium and section 3.3 presents the calibration, of the model. Section 3.4 analyses bank lending and optimal monetary policy behavior following negative shocks. Section 3.5 concludes. The appendices give additional details about various aspects of the model and the solution strategy. 3.2 M o d e l 3.2.1 Overview There are three types of agents in the economy: households, banks, and a central bank. House-holds in a productive stage of their lives want to become entrepreneurs, but a shortage of internal financing forces them to apply for external funds. Successful applicants become entrepreneurs and the others remain workers. Each worker faces an idiosyncratic shock of becoming unemployed while entrepreneurs have risky returns on investment. All households in a productive stage of life (entrepreneurs, employed and unemployed workers) face a risk of becoming permanently retired, and all retirees face a risk of dying. New households are born to replace the deceased ones. When the households make their consumption-savings decision, savings are invested in bank deposits and bank equity. Banks collect deposits and equity, provide loans to entrepreneurs and purchase risk-free government bonds in order to maximize their profits. Banks screen loan applications and accept them according to the level of household's net worth. Banks have to purchase deposit insurance and are subject to a capital adequacy requirement imposed by the central bank. The central bank also controls the government bond rate. We now go through the model in more detail. The economy is subject to aggregate shocks and thus can be represented by an aggregate state vector including the current shock and the current distribution of assets and occupations that we ignore in the following to simplify notation. 3.2.2 Households In the model economy, there is a continuum of measure one of households, each maximizing their expected discounted lifetime utility by choosing an optimal consumption-savings path. A household can either be productive or retired, and the probability of a productive household retiring r is exogenous3. 3 Once retired, household cannot become productive again. Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 64 A productive household i is endowed with one investment project of size xl, which is always greater than the household's net worth m 1. We assume that the total investment is a fixed multiple of household's net worth: xl = <f>m% where <j) > 1. The project is indivisible, and so (<f> - l)ml has to be funded by the bank in order for a project to be undertaken4. If a household receives a loan it becomes an entrepreneur and invests into a project, receiving a return r% drawn from a trinomial distribution. In equilibrium, distribution of returns is such that households always prefer investing into projects and becoming entrepreneurs to becoming workers5. The returns are drawn independently across households (i.e. projects) and time. The lowest of the returns is sufficiently negative with a positive probability to lead to bankruptcy, in which case a household is guaranteed a minimal amount of consumption c m j n and starts next period with no assets. When the bank rejects a loan application, the household enters the work force and faces exogenous probabilities 1 — u of becoming employed and u of becoming unemployed. Workers inelastically supply their labor and receive an after tax wage income y. Unemployed workers receive unemployment benefits By where 9 is the replacement ratio. Labor supply is inelastic at an individual level. At the aggregate level, labor supply is de-termined by moves between the pools of workers, entrepreneurs, unemployed and retirees. This assumption increases the role asset accumulation plays in the economy. We use aggregate labor input data on the average hours per worker to calibrate the labor demand. After retirement, the household earns income from its savings and pension (which equals unemployment benefit payments). Retirees face a, probability 8 of dying. They are then replaced by agents with no assets and any remaining assets are lost (no bequests). The households make their consumption-savings decision to maximize their expected lifetime utility. The contemporaneous utility function is a CRRA type: 3 1 - p where j G {W, U, E, R}, I denotes leisure, c consumption and p is a risk-aversion parameter. As mentioned above, the labor supply is inelastic and the values lj represent market-clearing values for leisure. 4Therefore at a household level, demand for loans is uniquely determined by the net worth and so by the history of consumption-savings decisions and luck. 5 For entrepreneurs that receive loans, the participation constraint is always satisfied in our simulations. Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 65 Let Vj denote the value functions and m* be the minimum net worth necessary for external financing. A worker with a net worth m (< m*) faces probability (1 - u) of being employed, following which he receives labor income y = (1 - lw)w and interest income Rdm, pays a banking fee £ 6 , consumes a desired level and invests his remaining net worth m'7 in a bank. If unemployed, he receives unemployment benefit payment 9y and makes a similar consumption-savings decision. In the next period, depending on the level of m', a worker may either become an entrepreneur (borrower) or remain a worker (depositor). For an employed worker, the Bellman equation is: Vw(rrJ) = m^{Uw(lw, cl) + 0[(1 - r)[(l - u)VwW') + uVuinJ') + Er,VE{rrJ'',/)] + TVR^')]} (3.1) s.t. ci + mi' = (1 + rd)mi + y-£ For an unemployed worker: VuirrJ) = max^Uuih,^) + f3[{l - r)[(l - uJV^m*') + uVuim1') + Er-VEim1', r1')} + TV^m*')]} (3.2) s.t. c* + m*' = (1 + rd)m* + Oy - £ An entrepreneur i invests in a project of size xl, earns a stochastic net return rl and labor income y — (1 - l£)w and pays the borrowing cost rl(xl - ml), while making a consumption-savings decision to maximize her expected utility. Because the net wealth is constrained to be non-negative, significant project losses may drive the entrepreneur into bankruptcy. When bankrupt, an entrepreneur defaults on the portion of the debt he can not repay less a minimal consumption allowance Cmin which has to be granted by the bank. Upon default, entrepreneur starts the next period as a household with no assets and no liabilities. The returns on project r, are drawn independently across time and individuals and follow a trinomial distribution. The lowest 6 W e will justify in the calibration the use of £. 7 A prime ' denotes variable values in the next period. Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 66 of the returns is sufficiently negative to lead the entrepreneur to bankruptcy. For an entrepreneur: V k K . r ' ) = m a x ^ t e . c * ) + /3[(1 - r)[(l - t i ^ m * ' ) + uVuinJ') + E^VEirrJ', r1')} + rl^m*')]} (3.3) s.t. cl = maxjcmin, m i + y + (1 + rl)a;1 - rl(xl - m%) - £ - m1'} a;1 = cfim1 Note that we assumed the size of the project is proportional to the entrepreneur's asset hold-ings. This can be justified by collateral requirements typically observed in the credit markets. In addition, <f> can easily be quantified in the data. To stress the effects of the supply of credit, we assume that households ex ante always prefer to apply for a loan. This implies a participation constraint for households in a productive stage of their lives that needs to be satisfied for all households that effectively obtain a loan: ErVE{m, r) > (1 - u)Vw{m) + uVrj(m), Vm > m* (3.4) Every household faces an exogenous probability of retirement r. Once retired, the household collects retirement income yu — 8w and manages its assets subject to the risk of death S. VR(m) = max{.7R(l, c') + /3[(1 - ^(m*')]} (3.5) c',m* S.t. c* + m*' = (1 + rd)m + yR-£. Because of their risk aversion, the agents smooth their consumption over time. The presence of heterogeneous risks of unemployment and retirement as well as the heterogeneity in project returns lead to a non-degenerate distribution of assets in the economy. Intuitively, the individual risks along these dimensions substitute for the uncertainty of income which is modeled as fixed. Without these risks, there would be no reason to save other than to invest in a project, and the asset distribution would unrealistically collapse along m — 0 and m = m*. This would not allow for financial intermediation because of lack of funds (no depositors). Any equilibrium in this bimodal distribution is very unstable because all entrepreneurs can drift to zero assets following a Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 67 shock. The distribution of assets plays a crucial role in determining the dynamics of the aggregate variables. The decision to allocate savings between bank equity and bank deposits is obtained by maxi-mizing a risk-adjusted return on protfolio (rport): m a x r ^ - ha2port where rport = rejjj + rd(j^J = wrre + (1 - ujr)rd, UJt = E/M is a weight on the risky (equity) investment, A is a risk-aversion parameter and c 2 0 r t is a variance of the portfolio return. Because bank deposits carry no risk (cr2, = 0), the household maximizes: maxw r r e + ( 1 - u>r)rd - ^ A w 2 ! ? 2 e _ d which yields the optimal share of equity UJ* = r , £ . This in turn defines the demand for equity (and implicitly for deposits) given savings M: M = ~XoT ( 3 - 6 ) Note that we have separated this portfolio problem from the intertemporal utility maximiza-tion of the household. This is for computational reasons: given that with aggregate shocks we need to include the entire asset distribution in the state space, we need to avoid having to track for each household two separate assets to keep the dimension of the state space within reason-able bounds. This means also that the share of equity is independent of the asset level. Such assumption is not necessarily innocuous. As the Appendix C.2 shows, as long as the households have the same labor income, their optimal splitting rule between equity and deposits is constant and identical for all households due to C R R A preferences. With labor income varying between workers and unemployed/retired, the optimal splitting rule may change. 3.2.3 Financial Sector Bank The representative bank maximizes its expected profits, taking the asset distribution in the econ-omy as given. Profits equal asset returns less the funding costs, deposit insurance payments and the expected loan losses and liquidation costs. The bank's choice variables are loans L, bonds B, equity E and deposits D. Because the bank takes the distribution of assets as well as all returns Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 68 as given, the choice of loan volume is identical to choice of a threshold level of net worth m*. Formally, the problem can be stated as: max rlL + rbB ~ rDD - reE - S(^YD ~ (l + lc)eL + t (3.7) L,B,D,E \E' subject to B + L = D + E = M (3.8) | > a (3.9) D + E > L (3.10) where M is the total amount of loanable funds that are exogenous from the point of view of the bank8, 6 is a per-unit deposit insurance cost parameter, e is an expected share of loan losses, e determines the loans facing bankruptcy losses, and Ic is a liquidation cost parameter. Equation (3.8) is the usual balance sheet constraint, (3.9) is the regulatory requirement on capital adequacy and (3.10) is a non-negativity constraint on bond holdings. The profit function (3.7) is non-linear due to the inclusion of deposit insurance costs which are an increasing function of the deposit/equity ratio. Because profits increase in loans for any given asset distribution, one and only one of the constraints (3.9) and (3.10) will bind at any time9. The solution of the profit maximization is described in the appendix. 3.2.4 Central bank The central bank in the model sets the bond interest rate rb and elastically supplies (government) bonds at this rate. It also determines the capital asset ratio parameter a. a and rb are the only monetary policy instruments. In the simulation section 3.4 we show how different monetary policy actions, as represented by mean preserving changes in rb across the aggregate states, influence the behavior of the different types of households and of the representative bank. 8The total amount of assets flowing through the financial sector is determined by households' decisions. Half of the total "financial" assets (note that the self-financed part of entrepreneur's project does not enter financial sector) has to equal total bank liabilities=assets (see equation 3.14). 9The chances that both of them bind at the same time can be dismissed as arbitrarily low. Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 6 9 3.2.5 Market clearing On the financial side, markets for loans, bonds, equity and deposits must clear. The bond market clears automatically because of an infinitely elastic supply of bonds10. The remaining market clearing conditions are: DS = DD= Y <*>.•) (3-n) m*<m* Es = ED = Y m^r (3.12) L= Y (0-1)™* (3-13) m*>m* M= Y rni = D + E = B + L= Y (0 - (3.14) m'<m* m*>m* Moreover, expected losses of the bank must in equilibrium equal the realized loan losses: e = Y maxjo, (1 + fi)[rl(<t> - l)m* - (pm^l + r')] + c m i „} m'>m* where p are auditing costs. The market clearing equations (3.11) - (3.14) connect the new homogeneous part with the heterogenous part of the model. The sum of individual demands for deposits, equity and loans on the right-hand sides must equal the supply levels decided on an aggregate level. Equity market clearing implicitly defines the return on equity re as a function of all other returns. In the case of an interior solution, equations (C.2) and (3.6) imply: -5{re - rdf - (V - rd - (1 + Jc)e) + l] (r e - rd)2 + 2\a2e(re - rd) - X2aAe = 0 (3.15) In the case of a corner solution, equations (C.5) and (3.6) imply: r e 3 - r e 2 [2rd + r' - (1 + lc)e + l] - re \rd2 + 2rd{r' - (1 + lc)e + 1) + 2Aa2] - [rdV " (1 + Ich + 1) + 2A<JeV + <5A2cr4j = 0 (3.16) To illustrate the functioning of the equity market, it is useful to undergo a following thought experiment. Consider a case of an increase in the lending interest rate rl, possibly because of an increase in the demand for loans. As long as the ratio of expected losses as a proportion of loans e rises less than rl, the bank's profit margin on each new loan goes up, which prompts the 1 0 O n e can think of banks depositing their non-loanable investments at the central bank which also sets the deposit rate in this model. Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 70 bank to lend more. To do so, bank has to raise more equity (it starts without any excess of it: E = aL), which is why the equity supply equation (C.2) is increasing in the loan profit margin. The demand for equity (3.6) is unaffected by the return on loans, and so to raise more of equity, the bank's offered re has to increase. Note that because the government bond rate is exogenous and it determines the deposit rate in an interior solution (which is the norm), and because the bank can not choose the size of its balance sheet M, re plays an important role in the bank's liability management. Its increase will lead to a rise in the total amount of equity raised and to a more-than-proportional increase in the E/D ratio for any size of the balance sheet M11. It is therefore easy to see that when the bank increases the share of loans in its portfolio, it has to fund the higher equity holdings at an ever-increasing price. Eventually, the original profit margin disappears and a new optimal loan level is achieved. Two cases can occur. First, the total amount of new loans is less than the new balance sheet level, loan market clearing conditions are satisfied and constitute a potential equilibrium. Secondly, the total amount of new loans may exceed the new balance sheet volume M, which is what we defined earlier as a corner solution. In the latter case, the loan market does not clear and the banks ration some of the eligible loan applicants. Because there is no asymmetric information problem in this model (hence no adverse selection), an increase in the price of loans does not affect their quality and a higher rl is needed to clear the market. Therefore we have a choice of focusing on market-clearing equilibria which rule out corner solutions and equity "hoarding", or allowing credit rationing when multiple equilibria may arise and excess equity is kept as a backup in case the total amount of loanable funds M increases. For simplicity, we only focus on the market-clearing equilibria, and only equation (3.15) becomes relevant. One of the implications of this is that we will never observe banks hold excess equity in equilibrium, and so regulatory changes in capital adequacy ratio p will have a direct effect on the loan volume. The above market clearing condition (3.15) defines a return on equity as function of all other returns and some parameters: re=re(rl, rd, a2, A, a). The above cubic equation can be solved analytically but does not determine the re uniquely. Depending on the parameter values, two of the three roots are complex and disregarded in the following calculations. We now have a recursive system. Conditional on M, equation (C.2) determines the optimal 1 1 This follows from the fact that j | = and uir increases in r e . Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 71 level of equity E, equation (C.4) determines the optimal level of deposits D, equation (3.14) determines the optimal level of bonds B and equation (C.3) determines the optimal level of loans L. We therefore have {re, rd, E, D, L, B} as a function of {rl,M} and exogenous variables. 3.2.6 Equilibrium A recursive equilibrium in this model economy are four value functions Vj(m,s), where s repre-sents the aggregate state (current shock, distribution of m), for j £ {E, W, U, R}, decision rules {gfim, s),gdM{s),geM{s),g7§\s),grB{s)}, government policies {a(s), rb(s)}, prices {rd(s),rport(s),re(s)}, aggregate asset levels {L, D,B,E}, and a function ^{p) such that: 1. decision rules 5™(m, s) solve each household's problem with the associated value functions Vj(m,s). 2. decision rules gfjis) and geM(s) solve portfolio problem of the household. 3. decision rules g% (s) and grB (s) solve the banks' problems. 4. loan, equity and deposit markets clear: L(s) = J2 (<f> ~ l)mp(m, s) (3.17) m>m* e d E(s) = - — V mp(m,s) (3.18) e m<m re _ yd D(s) = l l 2 ~ YI TOMm.s) (3-19) 7 ( 7 5. the distribution of households is the fixed point of the law of motion $: p!(m, s) = ^(m, s) 3.3 Parameter izat ion To simulate the economy and obtain numerical results, we parametrize the model to the Canadian economy in the years of 1988 to 1992, in accordance with the available data on project return distributions. Indeed, these are the only years for which Statistics Canada published such data. First we calibrate the household sector. Several parameters are set in accordance with the literature: p = 2.5, (3 = 0.96 and a = 0.67. In accordance with the models that include explicit Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 72 leisure specification, lE = lyy — lu = 0.55 while IR=1, as a result of which the labor input of entrepreneurs and workers, and the search effort of unemployed are set to 0.45. Wages are exogenous and while they completely characterize the labor income of entrepreneurs and workers, the incomes of unemployed and retired are determined by the ratio of unemployment insurance benefits to wages 6 = 0.292912. The probability of unemployment is set equal to the average Canadian unemployment rate for the considered period: u = 0.0924. The probability of retirement T and the mortality rate S are set at 0.05 and 0.1, so that the number of expected periods while worker and retiree are 20 and 10, respectively. Longer expected lifetime horizon allows us to utilize the effect of savings over time more fully than in the usual 2-period models (e.g. Williamson (1987) and Bernanke and Gertler (1989). Now we turn to the financial side. Following the calibration in Yuan and Zimmermann (1999), we set the real bond rate rb at 1%, such that the deposit rate rd is about 0.9%, which corresponds to an average of savings rates and guaranteed investment certificate rates. The parameter a of the capital adequacy constraint is taken to represent the tier-1 capital requirements imposed by the Basle Accord (1988) and set to a = 0.08. The deposit insurance parameter 5 is calibrated using the premium rates of the Canadian Deposit Insurance Corporation for banks in 2000/2001 (0.0417% of insured deposits). This per-unit rate corresponds to S = 0.0000417 for an average D / E ratio of 10. The loan administration cost lc is assumed to equal 0. The account flat fee £ is set at 0.0003 by trial and error in order to get the banks to break even. The parameters of the equity market that need to be calibrated are A and a\. The variance of returns on equity of the banks is calculated from the T S E monthly series on financial enterprises' returns on equity from September 1978 until December 2000, which are deflated by the CPI. Therefore, a\ — 0.24. The risk-aversion parameter of the portfolio optimization problem A is calibrated from the market clearing condition (3.15) using the observed average real deposit, lending and ROE rates. This implies A = 16. The distribution of returns follows a two-state Markov process calibrated such that the high state occurs 75% of the time. Specifically, a high state has a 75% chance of reoccuring the next period, while a low state can repeat itself with a 25% chance. 1 2 This measure is based on the replacement rate of Hornstein and Yuan (1999). Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 73 The distributions of project returns in both aggregate states are calibrated from firms' re-turn on equity data. Statistics Canada (1994) reports the distribution of return on equity by non-financial enterprises from the fourth quarter of 1988 until the fourth quarter of 1992. Aver-age returns in each quarter are reported for the top, middle and bottom fertile. Assuming the underlying distribution is normal, we find the returns and associated probabilities for trinomial distributions such that a) average returns are replicated, b) we have have two extreme returns, one implying bankruptcy. We compute two such distributions, one for the high aggregate state, corresponding to the average of the 75% best quarters in the sample period, and the other for the low state. The returns and the associated distributions are the following: ( -50% 5.2% 60% \ / -50% 2.57% 60% ^ High: Low: \^  0.71% 98.48% 0.81% J ^ 1.79% 97.42% 0.79% J The ratio of investment to net worth (<j> — 1) is calibrated to equal the average debt-equity ratio during the reference period, and so <p — 2.2. With a minimum return on investment of -50%, we have occasional bankruptcies. The auditing costs v are assumed to equal 0.03. 3.4 C a p i t a l r e q u i r e m e n t s , b a n k l e n d i n g a n d m o n e t a r y p o l i c y We now want to understand the behavior of the model economy. This task is made more difficult by the complexity of the model and richness of the aggregate state space. Many histories can be imagined in this environment, and discretion needs to be used in selecting those we want to focus on. In this section we consider a history which is empirically relevant from the business cycle perspective. Starting from a steady state, the experiment begins with a succession of five Low shocks followed by five High shocks. Thus, the model economy goes through the whole cycle, bottoming out in the middle. Note that this a particular history of shocks among many others, and that this history is not anticipated. In Figure C l , we show the behavior of various indicators in a benchmark economy, that is with no policy intervention from the central bank on bond rates or capital requirements. When the initial bad shock hits the economy, the lending rate jumps up, essentially to cover higher than expected loan losses. As more bad news accumulate, the lending rate decreases as m* reacts and the households adapt their asset levels. Indeed, banks ration more and more as bad shocks accumulate, but revert to "normal" behavior as soon as good news come in. From Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 74 peak to trough, the amount of loans decreases by 3.0%, and 3.6% of all entrepreneurs are driven out. The consequence is that the size of an average loan increases by 0.6%, corresponding to the empirically documented phenomenon that small businesses are hurt more when credit conditions worsen. Do we have evidence of a credit crunch in this benchmark economy? Despite the fact that banks can increase the loan rate to compensate for higher rates, they have to decrease the total loan mass. The reason is the following. Facing increased risk, more entrepreneurs are forced to become workers, as the bankruptcy rate is higher. With more agents that save, the volume of assets increases. However, a smaller share of those assets are channeled to bank equity because its return is too low given its risk. The banks are then squeezed by the capital requirement and have to ration credit and invest more into "unproductive" government bonds. Without the capital requirement, banks could give more loans, in principle, by charging even higher loan rates, and entrepreneurs would still be ready to pay these rates. Although all agents behave optimally, we have a situation that can be described as a credit crunch, where marginal return and marginal costs of loans are not equal. Capital requirements imply that changes in the compostition of banks' liabilities affect the amount of credit in the economy. An adverse productivity shock increases the number of depositors and lowers the number of borrowers. Yet risk averse depositors shy away from the highly risky bank equity which leads to a further credit decline (due to the capital requirements). However, the movements described above are relatively small. 3.4.1 Countercyclical monetary policy The following experiments will help us understand what are the consequences of various policy actions. The first policy experiment, described in Figure C.2, involves a 25 basis point reduction of the bond rate in the worst aggregate state (current shock Low, long history of Low shocks).13 Thus, the central bank reacts only after a prolonged decline in the economy. Note that the deci-sions of the banks are changed only in this specific state: m* and the lending rate are unaffected when the central bank does not move, but when it does banks reduce the lending rate by the same margin and, more importantly, significantly relax their loan threshold m*. Thus the situation for entrepreneurs should improve noticeably: easier access to credit at better conditions. Loan ac-1 3Note that all experiments are designed such that the average rb or m* stay at the same level. Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 75 tivity is negatively affected, however, and equity is reduced compared to the benchmark. This is because workers' savings decline (interest rates are lower) and with them the total amount of equity (lower return). Note that household decisions are affected even when the central bank has left the bond rate untouched, in anticipation of possible changes. Ultimately, the same number of entrepreneurs gets loans and the average loan is now smaller. A one-time drop in the interest rate therefore does not appear to be an effective policy. What now if the interest rate is gradually reduced by 5 points after each bad shock, and goes back to normal whenever a good shock comes by? This policy takes better into account the anticipations the households formulate. On Figure C.3, we see that the outcome is quite different. Banks become much more generous to entrepreneurs in bad times, both in terms of lower lending rates in bad times (but higher in good ones) and quite significantly in terms of m*. In all states, there are more entrepreneurs, loans, deposits and equity. While the average loan is larger in normal times compared to the benchmark, it is smaller in almost any other. This means that asset accumulation has increased for households: entrepreneurship is more interesting as monetary policy counterbalances the increased risk in bad times. Indeed, while firms face lower average returns and higher bankruptcy rates, monetary policy forces banks to offer better conditions. This has an impact on asset accumulation even in good times. We conclude that an active countercyclical monetary policy can have a significant positive impact. Note, however, that it cannot remove the cyclical nature of loans. 3.4.2 Procyclical monetary policy If some policy of interest rate reduction may have negative consequences, one may naturally ask whether an interest rate increase can do some good. Indeed, higher bond rates mean higher returns on savings, and potentially more equity to satisfy the loan needs in the presence of capital requirements. In Figure C.4. we find that the model economy does not behave in a symmetric way, as compared to Figure C.2. While the lending rate increases as expected, m* stays essentially put rather then shoot up. Consequently, loan activity does not change much as households barely change their decisons compared to the benchmark. The sum of all tiny changes results, however, in a noticeably decrease in the average loan size, but not as strong as in the opposite policy. Comparing Figures C.3 and C.5, it appears that the same kind of asymmetry exists for a Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 76 gradual policy. A gradual increase of the bond rate has a negative, but much smaller impact on the various assets. An explanation of this asymmetry is as follows. Procyclical monetary policy induces a drop in ra*, leading to an increase in the loan volume as more smaller agents can become entrepreneurs. Moreover, a lower m* induces workers to save more (consumption drops) at any given deposit rate because the entrepreneurship is more likely to be attained (this move is slightly offset by the distributional movements as there are fewer workers and more entrepreneurs). Because of such boom in banks' liabilities, asset sides of banks' balance sheets expand which reinforces the initial loan volume increse. On the other hand, a countercyclical monetary policy induces a small rise in m*. This is a strong disincentive for saving for workers who want to become entrepreneurs, and leads to a drop in the volume of deposits and equity. Such drop is partly offset by an increase of the pool of depositors and a rise in the deposit interest rate. These offsetting moves are behind the relatively small changes in the volume and the composition of banks' balance sheets. Banks' decision to change ra* in an asymmetric way is just a reflection of the equilibrium nature of the problem. With procyclical monetary policy, banks' desire to give more loans requires a rise in their equity funding (capital requirements bind). Yet equity is more risky in bad states and households channel their savings away from equity and into deposits. Therefore, in order to expand their loans, banks must make the vision of entrepreneurship (a motivation for saving) highly desirable to get sufficient equity - hence a sharp drop in ra*. On the contrary, a countercyclical monetary policy motivates a loan volume drop which is achieved by an increase in ra*. Such increase can be small because for any amount of savings, risk-averse households prefer deposits in bad times anyway. The heterogenous agent setup of this model highlights the effects of the changes in distribution of assets and bank financing on loan activity. In particular, it shows the asymmetric propagation of the monetary policy. 3.4.3 Countercyclical capital requirements The interest rate is one of two instruments the central bank can use. The other is to modify the capital requirements, which in the benchmark economy are set at a 8% equity/loan ratio, as in the Basle Accord. As it appears capital requirements have an impact on the model economy, one Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 77 may want to establish whether it can be used for cyclical purposes as well. In the first experiment, Figure C.6, the equity/loan ratio is allowed to be reduced to 7% in the worst aggregate state only. While the banks can now offer more generous conditions, in this state only, households observe higher bank risk and shift from equity to deposits sufficiently to counterbalance and decrease the loan mass. As for a bond rate reduction, the average loan size decreases as the number of entrepreurs barely changes compared to the benchmark economy. The next experiment involves a gradual decline of the capital requirements during the bad shocks, Figure C.7. One would expect that the regulator allowing the banks to take more risks during a downturn may generate more loans. To the contrary, equity declines even more, resulting in a smaller loan mass. Interestingly, loans are lower even when the regulator does not intervene and has in fact slightly more stringent capital requirements to maintain the same average as in the benchmark. The reasons are the same as previously: households shy away from banks when they take on more risk. 3.4.4 Procyclical capital requirements If countercyclical capital requirements have adverse effects, maybe procyclical ones have a positive impact on lending ability. Figure C.8 looks at the punctual policy, Figure C.9 at the gradual one. Both policies have positive effects, locally and small for the first one, globally and massively for the second one. Thus it appears that tightening capital requirements is good for loan activity because it improves the financing of the banks. In this case the arguments are symmetric to the countercyclical policies. Note that we have no informational problem in the model economy that would actually require the imposition of capital requirements. One can easily imagine that if the model would include this it would only reinforce the result: the presence of more entrepreneurial risk leads to a higher impact of asymmetric information and risk, thus furthering the need for regulation. 3.4.5 Credit crunch? What exactly happens in the model? A negative aggregate shock lowers the expected project returns and increases their volatility. This affects the loan volume and the lending rate in four ways. First, both these effects decrease the ex-pected value of risk-averse entrepreneurs (ETVE) while the value functions of non-entrepreneurial Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 78 households do not change.14 Therefore the incentive to accumulate assets in order to be eligible for a loan declines. This lowers the demand for credit because fewer agents save enough to pass the m* cutoff. Second, the risk-neutral banks only care about the expected return of projects. The relative net payoff of bonds versus loans rises and induces a substitution from loans to bonds. The loan supply drops and the lending rate rL increases to compensate for higher loan losses. This is the credit supply effect (i.e. the "crunch"). Third, an increase in rL further discourages loan applicants because their net return on investment declines, and the equilibrium credit level drops further. Therefore the post-shock equilibrium exhibits a higher lending rate and a lower level of loans which further propagates the shock. Note that the decline in the market-clearing volume of credit is partly demand-driven, and cannot be only attributed to the the credit crunch behavior of the banks. Fourth, the household perceives more risk in the bank when entrepreneurial risk increases. It then shifts, on behalf of households, from equity to insured deposits, thus making it harder for banks to meet the capital requirements. 3.4.6 Does the equity market worsen or soften the credit decline? The existence of equity market can either amplify or reduce the impact of a negative shock on a volume of credit. Only the second and fourth of the above mentioned four effects is directly affected by the existence of an equity market. The equilibrium condition (3.15) shows that only changes in rL and e affect credit behavior through the equity channel, and they do so in an offsetting manner. An increase in e (higher loan losses) increases the return on equity rE, while an increase in rL lowers it. We therefore distinguish two cases. (A) If d (rL — (1 + lc^j e) < 0, then a rise in rE increases the cost of funds to the bank which squeezes the profit margin further and leads to an additional substitution from loans to bonds (L drops) as well as an increase in rL. At the same time rPORT increases, making borrowing relatively less attractive (demand for credit drops). In this case, the presence of the equity market worsens the credit decline: a higher rE is only compatible with a lower amount of equity E on the market15, which in turn requires an additional drop in loans due to a binding capital adequacy constraint (see equation (C.3)). Case (B) when d(rL — (1 + lc)t) > 0 has the opposite implication - it softens the effects of financial accelerator. According to the simulations (comparing peak and trough states), d(rL — (1 + Z c ) £ ) — 0.0002 and 1 4 There is only a second order effect coming from expectations to be an entrepeneur in the future. 1 5 T h i s is because households are risk averse while banks are risk neutral. Chapter 3. Basle Accord and Financial Intermediation: The Impact of Policy 79 we can conclude that the presence of the equity market softens the credit crunch. 3.5 C o n c l u s i o n We study the interaction of household saving decisions, project returns, Basel Accord type banking regulation and credit activity. We find that the Basel Accord has a noticeable impact on loans when project returns decline through the cycle. Active monetary policy through interest rate reductions in bad times increases the loan activity but does not remove its cyclical nature. A relaxation of the Basel Accord capital requirements in bad times obtains negative results, as households shy away from the equity banks need to make loans. As in models with informational problems, of which there are none here, a monetary policy tightening is needed. This calls for an active regulatory policy through the business cycle, not fixed policy rules currently in place. Our results also emphasize the importance of taking into account the financing of banks. Given capital requirements, banks are limited in their lending by the amount of bank equity that households are willing to hold. As this decision is influenced by interest rates, another channel of inonetary policy arises. 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An inverse of a trigonometric function tan(x), arctan(x) has a range of [—7r/2,7r/2] for x 6 R and is monotonically increasing, continuously differentiable, and has a convenient property that arctan(rr) < 0 when x < 0, arctan(x) > 0 when x > 0 and arctan(a;) = 0 when x = 0. Because of the bounded range, and because arctan(Aa;)x=0 = 0 VA, arctan(Ax) can reach the bounds very quickly. Finally, if we premultiply the function by 2/7T, the range becomes [-1,1] - very convenient for a step function. The choice parameter A is inversely related to the approximation error, a relationship that can be seen in figure (A. 12). However, it is misleading to use this approximation to describe the first order conditions of a system with \NiA because the absolute value function is not differentiable at 0. Therefore, a smooth approximation G(Nij) to \Nitt\ needs to be constructed first, and then differentiated. Conveniently, function G(Ni:t) = / g(Nit) XNiit [ ^ arctan(AJVi)t) - 0.51og(l + (AiV i ] t) 2) can be used to arbitrarily closely approximate |iV^ t| by a choice of A (see figure (A.l)). Figure A . l : Approximating functions g(N) and G(N) for A = 105. -1 - 0 . 8 - 0 . 6 -0 .4 -0 .2 0 0.2 0.4 0.6 0.G -0 .2 0 0.2 trade volume N Appendix A. 90 Derivation of equation (1.24) N = thereforedNi/dYi Noting that paut and that jV_ 1 + - ( p * - p - I ( J V ) t ) (dp*/dYi - dp/dYi - t)ir + 0(1 - TT) where TT = pro6(|p* - p\ > t) 2c dp*/8Yi = 0 dN/dYi = ± Yin Y3n dp/dYi = - i - i ^73 *3 7i "° 1 1 73 3^ 7* ii 1 1 73 — £0 , identical to equation (1.24) Y3 7i ' Assuming that the shares of 3 goods in the utility function are identical (7, = jj Vi,j), dNi/dYi < 1 when c is sufficiently large. 7T I Y-c> c = max < 0, — | — te v B which is automatically satisfied (c = 0) if r Y3 Yi>Yi [(toy 1 1+0 when exporting When Y3 = 100, 0 = 1.5 and t = 0.0174 (as in the model's calibration), Yi = 51.7. Therefore, this condition always holds when exporting. An analogous derivation can be made for the case of imports. Appendix A. 91 F i g u r e s F i g u r e A . 2 : L i n e a r m o d e l : A u t a r k y s o l u t i o n A u t a r k y s o l u t i o n ( e n d o w m e n t s of g o o d 2 & 3 f ixed et 1 OO) 1 .-4 : 1 .3 1 .2 1.1 • P n / P 3 1 2 0 1 3 0 140 1 5 0 160 170 190 2 0 0 1 1 O 1 2 0 F i g u r e A . 3 : L i n e a r m o d e l : I n t e r n a t i o n a l t r a d e I n t e r n a t i o n a l t r a d e O t r a d e 1 x t r a d e 2 <> t r a d e 3 - 2 0 0 2 0 r e l a t i v e e n d o w m e n t s h o c k Appendix A. 92 F i g u r e A . 4 : L i n e a r m o d e l : m e a n s a n d s t a n d a r d d e v i a t i o n s o f t h e p r i c e d e v i a t i o n s a s f u n c t i o n s o f s h o c k v o l a t i l i t y . 0.07 0.06 0.05 0.04 0.03 0.02 0.01 S t a n d a r d dev ia t ion of the L O P dev ia t ion ( e a c h point f rom a distrib. of 100 d r a w s . t2/t1=5) n S T D ( L O P D ) S T D ( L O P D ) 9 * • •••••••(^ ••••••••DCiO E>CJ ••DQGDDDQODBGDQDD D G i a o G j a o o a c i a o s a D i z i t j a o i M e a n of the L O P dev ia t ion ( e a c h point f rom a distrib. of 100 draws) 0.02 0 .015 0.01 0.005 0 Hi- M e a n ( L O P D ) M e a n ( L O P D ) 9 ' : , E a r i t - j n a C i n i r i c n a n a G ] D S T D of the relative e n d o w m e n t s h o c k a s a % of initial e n d o w m e n t F i g u r e A .5 : L i n e a r m o d e l : a v e r a g e t h r e s h o l d s o f t h e l a w o f o n e p r i c e d e v i a t i o n a s f u n c t i o n o f s h o c k v o l a t i l i t y T h r e s h o l d s of L O P D . F r i c t i o n . _ = 5 * F r i c t i o n M e a n ( T h r 1 ) M e a n ( T h r 2 A) S T D of the re la t i ve e n d o w m e n t s h o c k a s a % o f initial e n d o w m e n t Appendix A. 9 3 F i g u r e A . 6 : L i n e a r m o d e l r o b u s t n e s s : m e a n p r i c e d e v i a t i o n s a s a f u n c t i o n o f s h o c k v o l a t i l i t y a n d r e l a t i v e f r i c t i o n M e a n L O P D / L O P D a s a function of t IX a n d endowment shock S T D ( e n d o w m e n t shock) F i g u r e A . 7 : L i n e a r m o d e l r o b u s t n e s s : v o l a t i l i t y o f p r i c e d e v i a t i o n s a s a f u n c t i o n o f s h o c k v o l a t i l i t y a n d r e l a t i v e f r i c t i o n Appendix A. 94 Figure A.8: Linear model robustness: price deviations thresholds as a function of shock volatility and relative friction A v e r a g e T H R ( L O P D 2 ) as a function of t 2 /t 1 and endowment shock Figure A.9: Distribution of trade. US-EU simulation of the linear model Dist r ibut ion of the t rade in g o o d g o o d 2 a n d g o o d 1 (red) f rom the U S - E U m o d e l . 1 0 0 0 0 s i m u l a t i o n s . 5 0 0 0 h 4 0 0 0 -1 0 1 % of the m e a n e n d o w m e n t Appendix A. 95 F i g u r e A . 10: D i s t r i b u t i o n o f i n t e r n a t i o n a l p r i c e s . U S - E U s i m u l a t i o n o f t h e l i n e a r m o d e l Distr ibut ion of the L O P D for g o o d 2 and L O P D for good 1 (red) f rom the U S - E U mode l . 10OOO s imu la t ions 1400 | 1 1 , 1 , , . llllalllllllllllllllllllllllli §iii lilllllllllllHlllMllllMHlllllli. -0 .5 O 0.5 % from the parity F i g u r e A . l l : D i s t r i b u t i o n o f t h e r e a l e x c h a n g e r a t e a n d t h e s h o c k p r o c e s s . U S - E U s i m u l a t i o n o f t h e l i n e a r m o d e l Distr ibut ion of the L o g ( R E R ) f rom the U S - E U mode l and of the s h o c k p r o c e s s (red). 1O00O s imu la t i ons Appendix A. 9 6 E r r o r of t he a p p r o x i m a t i o n to a s t e p f u n c t i o n , X = 1 0 0 0 0 E r r o r of t he a p p r o x i m a t i o n to a s t e p f u n c t i o n , A,= 10 1 I -o.os 0 . 0 5 0.1 Appendix A. Figure A. 13: Thresholds in QAC model when c=0.000001 Initial r e s p o n s e s for c=0.000001 al (period 1) t rade in g o o d s c h a n g e in e n d o w m e n t (in %) # of time per iods g o o d s c h a n g e in e n d o w m e n t (in %) T o t a l trade in g o o d s , a s a p e r c e n t a g e of total c h a n g e in the e n d o w m e n t 20 | ;-y . ' - — <> g o o d 2 | g o o d 1 g o o d 2 | c h a n g e in e n d o w m e n t (in %) c h a n g e in e n d o w m e n t (in %) Figure A.14: Thresholds in QAC model when c=0.001 for c=0.001 Initial (period 1) trade in g o o d s c h a n g e in e n d o w m e n t (in %) # of time periods g o o d s are traded good 1 good 2 | Total trade in good: c h a n g e in e n d o w m e n t (in %) , as a percentage of total c h a n g e in the e c h a n g e in e n d o w m e n t (in %) c h a n g e in endi Appendix A. 98 Figure A. 15: Thresholds in QAC model when c=0.1 litiai r e s p o n s e s for c=0.1 (period 1) trade in g o o d s 0.3 0.25 0.2 0.15 0.1 0.05 / l o p d l I 1 oh, c h a n g e in e n d o w m e n t (in %) U of time per iods g o o d s are traded 0.2 0 2 c h a n g e in e n d o w m e n t (in %) Tota l trade in g o o d s , as a p e r c e n t a g e of total c h a n g e in the e n d o w m e n t c h a n g e in e n d o w m e n t (in %) Figure A. 16: Thresholds in QAC model when c=l Initial r e s p o n s e s for c=1 Initial (period 1) trade in g o o d s c h a n g e in e n d o w m e n t (in %) # of time per iods g o o d s are traded O g o o d 2 | 20 c h a n g e in e n d o w m e n t (in %) Tota l trade in g o o d s , as a p e r c e n t a g e of total c h a n g e in the e n d o • g o o d 1 g o o d 2 | c h a n g e in e n d o Appendix A. 9 9 Figure A.17: Profits in partial equilibrium, for various price deviations and 7V_i. c=0.01, t=0.2 Figure A . 18: Distribution of standard deviation estimates in a QAC model, c=0.1 Distribution of standard deviations of R E R (red) and L0PD1 (white) relative to std. of world GDP in a model with QAC . 5000 smulations, benchmark calibration. Cutout of a distribution of standard deviations of R E R (red) and LOPD1 (white) relative to std. of world GDP in a model with QAC . 5000 simulations, benchmark calibrate Appendix A. 100 Figure A.19: Price deviations in a QAC model, c=0.1 All LOPD 1 and LOPD 2 (red) in a model with QAC. 5000 simulations, benchmark calibration. 2000 h 1500h 1000h 500 h 0^ -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 Means (one per iteration) of LOPD 1 and LOPD 2 (red) in a model with QAC. 5000 simulations, benchmark calibration. Appendix A. Appendix B 103 B . l Tsay ' s F-test for non- l inear i ty The F-test for non-linearity consists of several steps. First, data is arranged into cases of (Azt,l, zt-i,zt-k) such that fc G M where M is a set of all relevant lags as determined in stage 1 of the model-specification (see section(2.3.2) above). Second, these cases are arranged in ascending order according to zt-d where d is the threshold delay parameter1 (see Tsay (1989), Obstfeld and Taylor (1997)). Third, an arranged autoregression is run on the ordered case data using recursive least squares: femax Azt= Y akZt-k + ut (B.l) fceM Recursive least square technique provides us with predictive residuals which are then used in Tsay's nonlinearity test. The recursive estimates are updated as follows (see Tsay (1989), Tong (1990), Barlet(????), Ertel and Fowlkes (1976)): Pm+1 = Pm + Km+i[Azm+i — PmZm] Km+1 — PmZm+l/Dm+i -Dm+1 = 1 + z'rn+lPm+lZm+1 p _ (T-P ^rn+iz'm+1 *m+l — y1 rm ) r m where m denotes a case, zm is a vector of all RHS variables in equation (B.l) (hence P = (di ... afcmax)') and I is an identity matrix. The recursion is initiated by a regular OLS on the first b cases where b = n/10 + p, n is the total number of observations and p is the number of elements in M2. The first b cases are then scrapped. The predictive (om) and standardized predictive (em) residuals are obtained as follows: 0"rn+\ = Azm+\ — PmZm em+l = O-m+ll V Dm+1 Fourth, standardized predictive residuals are regressed on the RHS variables: k m a x &m = ^ ^ 'YkZm—k ~T" £ m k&M F = (Ei2t-Ee2t)/(P+i) Ee2t/(n-d-b-p-h) { ' ; 1 Cases are analyzed in an ordered fashion because of the lack of knowledge of the position of a threshold ex-ante, therefore Pb = {zb'zb'1). Appendix B. 104 T h e a s s o c i a t e d F - s t a t i s t i c f o l l o w s a n F d i s t r i b u t i o n w i t h p + 1 a n d n — d — b — p — h d e g r e e s o f f r e e d o m w h e r e h = m a x { l , p + 1 — d}. I n t u i t i v e l y , a t h r e s h o l d i m p l i e s a p a r a m e t e r c h a n g e i n t h e a r r a n g e d a u t o r e g r e s s i o n ( B . l ) a t t h e t h r e s h o l d l e v e l . T h e r e f o r e , w h i l e p r e d i c t i v e r e s i d u a l s w i l l b e o r t h o g o n a l t o t h e r e g r e s s o r s f o r t h e c a s e s t h a t f a l l b e l o w t h e t h r e s h o l d , t h e y w i l l b e c o m e b i a s e d a b o v e t h e t h r e s h o l d , d e s t r o y i n g t h e o r t h o g o n a l i t y w i t h t h e r e g r e s s o r s . S u c h r e g i m e c h a n g e t h e n l e a d s t o a r e j e c t i o n o f o r t h o g o n a l i t y a n d c a n b e t e s t e d b y t h e F - s t a t i s t i c (B . 2 ) . R e j e c t i o n o f o r t h o g o n a l i t y i m p l i e s a r e j e c t i o n o f a l i n e a r A R m o d e l f o r a n o n - l i n e a r T A R a l t e r n a t i v e . B . 2 C o n s t r u c t i n g t h e p r i c e a n d w e i g h t , a n d v o l u m e d a t a s e t s Appendix B. 105 Table B . l : Coverage of CPI by the data. Relative importance of components in the Consumer Price Indexes (1999-2000 weights): U.S. city average, December 2001. Bold series are not included in the dataset. Item C P I - U C P I - W Item C P I - U C P I - W A l l items 100.000 100.000 Gas (piped) and electricity 3.466 3.778 Food and beverages 15.719 17.229 Electricity 2.521 2.762 Food 14.688 16.228 Uti l i ty natural gas service .945 1.017 Food at home 8.468 9.798 Water and sewer and trash collection services .857 .873 Cereals and bakery products 1.298 1.468 Water and sewerage maintenance .633 .660 Cereals and cereal products .444 .525 Garbage and trash collection .224 .213 Flour and prepared flour mixes .058 .070 Household furnishings and operations 4.840 4.101 Breakfast cereal .249 .278 W i n d o w & f loor c o v e r i n g s & o t h e r l inens .289 .254 Rice, pasta, cornmeal .137 .177 Furniture and bedding 1.051 .955 Bakery products .854 .944 Bedroom furniture .306 . .284 Meats, poultry, fish, and eggs 2.271 2.831 L iv ing room, kitchen, & dining room furniture .555 .495 Meats, poultry, and fish 2.178 2.712 Other furniture .181 .154 Meats 1.450 1.832 Unsampled furniture .010 .021 Beef and veal .693 .868 Appliances .364 .416 Uncooked ground beef .255 .334 Major appliances .199 .226 Uncooked beef roasts .115 .132 Other appliances .151 .176 Uncooked beef steaks .278 .351 Unsampled appliances .013 .015 Uncooked other beef and veal .045 .051 Other household equip. & furnishings .806 .565 Pork .468 .610 Tools, hardware, outdoor eq. & supplies .649 .595 Bacon, breakfast sausage, & rel. products .148 .190 Housekeeping supplies .862 .959 Ham .104 .132 Household cleaning products .392 .459 Pork chops .112 .156 Household paper products .200 .221 Other pork including roasts and picnics .105 .132 Miscellaneous household products .270 .279 Other meats .289 .355 Household operations .820 .357 Poultry .414 .518 Apparel 4.399 4.831 Chicken .329 .423 Men's and boys' apparel 1.122 1.243 Other poultry including turkey .085 .095 Men's apparel .880 .927 Fish and seafood .314 .362 Men's suits, sport coats, and outerwear .201 .189 Fresh fish and seafood .187 .219 Men's furnishings .191 .196 Processed fish and seafood .126 .143 Men's shirts and sweaters .263 .279 Eggs .093 .119 Men's pants and shorts .203 .241 Dairy and related products .916 1.021 Unsampled men's apparel .021 .023 Fruits and vegetables 1.204 1.307 Boys' apparel .242 .316 Fresh fruits and vegetables .928 .995 Women's and girls' apparel 1.807 1.864 Fresh fruits .466 .490 Women's apparel 1.515 1.504 Apples .084 .095 Women's outerwear .108 .111 Bananas .088 .100 Women's dresses .214 .247 Citrus fruits .079 .085 Women's suits and separates .762 .712 Other fresh fruits .215 .210 Women's underwear, nightwear, sportswear .400 .399 Fresh vegetables .462 .505 Unsampled women's apparel .032 .036 Potatoes .080 .092 Girls' apparel .292 .360 Lettuce .059 .066 Footwear .874 1.165 Tomatoes .094 .109 Men's footwear .290 .416 Other fresh vegetables .230 .238 Boys' and girls' footwear .177 .269 Processed fruits and vegetables .276 .312 Women's footwear .407 .480 Nonalc. beverages and bev. materials .967 1.132 Infants' and toddlers' apparel .203 .256 Juices and nonalcoholic drinks .710 .853 Jewelry and watches .394 .303 Carbonated drinks .364 .468 Watches .058 .057 Frozen noncarbonated juices and drinks .036 .039 Jewelry .336 .245 Nonfrozen noncarbonated juices and drinks .310 .346 Transportation 17.055 19.393 Beverage materials including coffee and tea .257 .280 Private transportation 15.845 18.452 Coffee .100 .105 New and used motor vehicles 8.614 10.145 Other beverage materials including tea .157 .174 New vehicles 5.083 4.897 Other food at home 1.811 2.038 Used cars and trucks 2.195 4.099 Sugar and sweets .315 .339 Leased cars and trucks 1.061 .925 Sugar and artificial sweeteners .056 .069 Car and truck rental .120 .085 Candy and chewing gum .200 .207 Unsampled new & used motor veh. .155 .140 Other sweets .059 .063 Motor fuel 2.564 3.153 Fats and oils .265 .316 Gasoline (all types) 2.536 3.120 Butter and margarine .090 .103 Other motor fuels .028 .033 Salad dressing .076 .089 Motor vehicle parts and equipment .421 .530 Other fats and oils including peanut butter .098 .125 Tires .234 .262 Other foods 1.232 1.383 Vehicle accessories other than tires .187 .268 Food away from home 6.220 6.430 Motor vehicle maintenance and repair 1.400 1.438 Ful l service meals and snacks 2.649 2.198 Motor vehicle body work .082 .077 Limited service meals and snacks 2.741 3.354 Motor vehicle maintenance and servicing .478 .475 Food at employee sites and schools .296 .375 Motor vehicle repair .821 .868 Food - vending mach. &c mobile vendors .151 .229 Unsampled service policies .020 .019 Other food away from home .383 .275 Motor vehicle insurance 2.288 2.679 Alcoholic beverages 1.031 1.001 Motor vehicle fees .558 .506 Alcoholic beverages at home .682 .709 Public transportation 1.211 .941 Beer, ale, and other malt beverages at home .352 .469 Airl ine fare .761 .508 Distilled spirits at home .109 .102 Other intercity transportation .187 .124 Wine at home .221 .139 Intracity transportation .256 .300 Alcoholic beverages away from home .348 .292 Unsampled public transportation .006 .008 Housing 40.873 38.141 Medical care 5.810 4.620 Shelter 31.522 29.212 Medical care commodities 1.377 1.006 Rent of primary residence 6.421 8.395 Prescription drugs and medical supplies .959 .680 Lodging away from home 2.702 1.523 Nonprescription drugs and medical supplies .418 .326 Housing at school, excluding board .241 .176 Internal &c respiratory over-the-counter drugs .304 .250 Other lodg. away from home incl. hotels 2.461 1.347 Nonprescription medical equip. & supplies .114 .076 Owners' equivalent rent of primary residence 22.046 18.980 Medical care services 4.434 3.614 Tenants' and household insurance .353 .314 Professional services 2.784 2.245 Fuels and utilities 4.511 4.829 Physicians' services 1.503 1.280 Fuels 3.654 3.955 Dental services .747 .584 Fuel oil and other fuels .188 .177 Eyeglasses and eye care .288 .240 Fuel oil .121 .105 Services by other medical professionals .247 .142 Other household fuels .068 .072 Hospital and related services 1.353 1.092 Appendix B. 106 T a b l e B . 2 : P a r t 2 o f t h e C P I c o v e r a g e t a b l e . Bold s e r i e s a r e not included i n t h e d a t a s e t . Item C P I - U C P I - W Item C P I - U C P I - W Hospital services 1.271 1.075 Cigarettes .864 1.360 Nursing homes and adult daycare .082 .017 Tobacco products other than cigarettes .057 .073 Health Insurance .297 .276 Unsampled tobacco and smoking prods .007 .008 Recreation 6.019 5.649 Personal care 3.384 3.059 Video and audio 1.645 1.803 Personal care products .706 .815 Televisions .150 .157 Hair , dental, shaving, & pers. care .374 .434 Cable television .928 1.034 Cosmetics, perfume, bath, nail preps .327 .374 Other video equipment .055 .064 Unsampled personal care products .005 .006 Video casset., discs, &L other media incl. rental .148 .182 P e r s o n a l c a r e s erv ices .901 .900 Audio equipment .117 .131 M i s c e l l a n e o u s p e r s o n a l serv ices 1.562 1.181 A u d i o discs , t apes and o t h e r m e d i a .147 .159 Miscellaneous personal goods .215 .183 U n s a m p l e d v i d e o and a u d i o .009 .076 S p e c i a l aggrega te indexes Pe t s , pet p r o d u c t s and serv ices .711 .703 A l l items 100.000 100.000 Sporting goods .628 .728 Commodities 41.300 45.559 Sports vehicles including bicycles .286 .413 Commodities less food and beverages 25.582 28.330 Sports equipment .333 .309 Nondurables less food and beverages 13.493 14.685 U n s a m p l e d s p o r t i n g g o o d s .009 .006 Nondurables less food, bev. &: apparel 9.094 9.854 Photography .241 .215 Durables 12.089 13.645 Photographic equipment and supplies .110 .092 Services 58.700 54.441 P h o t o g r a p h e r s a n d f i lm p r o c e s s i n g .129 .122 Rent of shelter 31.169 28.898 U n s a m p l e d p h o t o g r a p h y .001 .001 Transportation services 6.638 6.573 Other recreational goods .497 .512 Other services 10.963 10.033 Toys .360 .399 A l l items less food 85.312 83.772 S e w i n g m a c h i n e s , f a b r i c and s u p p l i e s .068 .062 A l l items less shelter 68.478 70.788 M u s i c Ins t ruments a n d accessor ies .062 .049 A l l items less medical care 94.190 95.380 U n s a m p l e d r e c r e a t i o n c o m m o d i t i e s .016 .012 Commodities less food 26.612 29.331 • R e c r e a t i o n serv ices 1.861 1.364 Nondurables less food 14.524 15.687 Recreational reading materials .436 .324 Nondurables less food and apparel 10.125 10.855 Newspapers and magazines .265 .210 Nondurables 29.212 31.915 Recreational books .170 .114 Nondurables less food 14.524 15.687 Unsampled recreational reading materials .001 .000 Nondurables less food and apparel 10.125 10.855 Education and communication 5.813 5.637 Nondurables 29.212 31.915 Education 2.726 2.382 Appare l less footwear 3.525 3.666 Educational books and supplies .220 .203 Services less rent of shelter 27.531 25.543 Tuit ion, other school fees, and childcare 2.506 2.178 Services less medical care services 54.266 50.827 College tuition and fees 1.162 .877 Energy 6.218 7.109 E l e m e n t a r y & h i g h s c h o o l t u i t i o n & fees .338 .258 A l l items less energy 93.782 92.891 C h i l d care and n u r s e r y s c h o o l .840 .896 A l l items less food and energy 79.094 76.663 T e c h n i c a l & bus iness sch . t u i t i o n & fees .084 .077 Commodities less food & energy 23.860 26.001 U n s a m p l e d t u i t i o n , fees, & c h i l d c a r e ' » .083 .071 Energy commodities 2.752 3.330 C o m m u n i c a t i o n 3.087 3.255 Services less energy services 55.234 50.663 Other goods and services 4.312 4.499 Domestically produced farm food 7.099 8.204 Tobacco and smoking products .928 1.441 Utilities and public transportation 8.785 9.217 Table B.3: Da ta sources on weights and prices - part 1 item unit price curr. weight (kg) p / w ( U S D / k g ) note Tota l R E R - C P I Apples kg 2.57 C N D 1 1.7 05/00-05/01 average, Statcan Table 326-0012 A u d i o equipment stereo unit 150 U S D 6 25 www.jandr.com (the largest retailer in US) , includes packaging Beef ground, 1kg 4.63 C N D 1 3.06 05/00-05/01 average, Statcan Table 326-0012 Beer six pack 5.40 U S D 2.30 2.35 Sec Grossmann & Markowitz (1999) C a r purchase car 24,923 U S D 1326.13 18.79 1996 avg. extrapolated to 2000, American Automobi le Manufacturers' Association 1996 C a r parts tire 100 U S D 10 10 Cheese kg 8.69 U S D 1 8.69 A v g . of American processed cheese (Series APU0000710211) and Cheddar cheese (Series APU0000710212) B L S , 2001 average monthly Clothes Clothes (men) basket* U S D 50.52 U.S . Department of Commerce, 2000 Clothes (women) basket* U S D 52.93 U.S . Department of Commerce, 2000 Coffee roast, 300g 3.27 C N D 0.3 7.20 05/00-05/01 average, Statcan Table 326-0012 E d u c . books &; supplies Eggs dozen 1.91 C N D 0.73 1.74 05/00-05/01 average, Statcan Table 326-0012 weight: a 30-dozen egg container weighs 471b. Electr ic i ty 500 k W h 48.55 U S D - - B L S , average 2001 price (Series APU000072621) Fats and oils basket* 1.81 U S D 0.598 3.68 StatCan, A v g price in Calgary in Nov 2001 for Salad dressing, avg. price in N Y C , Feb 2001 F i s h and seafood basket-*" 2.85 U S D 1 2.85 Fish processing industry data, wholesale prices. F l o u r 2.5kg 3.37 C N D 2.50 0.89 05/00-05/01 average, Statcan Table 326-0012 Footwear Footwear (men) pair, avg of casual and athletic 46.50 U S D 0.73 63.70 Footwear (women) pair, athletic 43.88 U S D 0.56 81.00 Fuel oil liter 0.34 U S D 0.86 0.39 A v g price, B L S 2001, Series APU000072511 Furniture bed 200 C N D 46.7 4.3 I K E A Gas 1000 f t 3 7.45 U S D 18.16 0.41 A v g price for year 2000, Energy Information Administrat ion, Natural Gas Monthly, Jan 2002 Gasoline liter 0.38 U S D 0.70 0.54 A v g . price, B L S , 2001, Series APU000074714 *Men's basket contains: coats, blazers, trousers, suits, women's basket contains: coats, dresses, blazers, trousers, suits, and skirts. *In accordance with the CPI definition of the category, the basket contains: Margarine (Canola, 1.36kg), Butter (Parchment, 454g), Shortening (454g), Oil (Canola, 11), Lard (454g), Peanut butter (500g), and Salad dressing (8oz). Weights are equal to the CPI weights. + Canned fish composition matches the composition of the fish processing industry data. Canned: Tuna (48%), Salmon (12%), Clams (8%), Sardines, Shrimp, Fillets: Cod (4.7%), Flounder (1.7%), Haddock, Rockfish, Pollock (11%) and Other (11%), Fresh fish approximated by 50% tuna and 50% salmon. Table B.4: Data sources on weights and prices - part 2 item unit price curr. weight p /w note (kg) ( U S D / k g ) House chemicals 75oz pack of 2.30 U S D 2.13 1.16 1997 N Y C price extrapolated into 2001 laundry deterg. Jewelry -L a u n d r y appliances washer 887 U S D 158.9 5.58 2002 avg. price for Maytag Liquor 750ml whiskey 11.74 U S D 0.75 15.65 B L S avg. price for 1986, adjusted by C P I inflation (scries APU0000720211) Medica l care products Non-prescription med. Parsley & Wei (2001) and U S Department of Commerce, avg. price 01/00-07/00 Pants pair, jeans, avg. 50.18 U S D 1.36 36.86 P C unit 1000 U S D 20 50 Dell .com average price in 2002. Personal care products a basket 1 12.58 C N D 8.31 2.77 05/00-05/01 average, Statcan Table 326-0012 Photo equipment -Pork kg, chops 9.29 C N D 1 6.14 05/00-05/01 average, Statcan Table 326-0012 Potatoes 4.54kg 3.83 C N D 4.54 0.56 05/00-05/01 average, Statcan Table 326-0012 Poultry kg 4.45 C N D 1 2.94 05/00-05/01 average, Statcan Table 326-0012 Prescription medicine -Sport equipment basket* 99.67 U S D 2.10 65.00 ht tp: / /www.usolympicteam.com/sports2 / ih /az equip.html Sport vehicles bicycle 225 U S D 15 15.00 Sugar l i b 0.43 U S D 0.45 0.95 B L S avg. price for 2001 (Series APU0000715212) Tobacco 200 cigs 37.78 C N D 0.25 99.80 05/00-05/01 average, Statcan Table 326-0012 Toys basket 31.33 U S D 2.55 13.19 average of 5 age-group categories from Toys 'R'Us 2001. Video equipment basket* 226.67 U S D 8.73 25.96 from J & R website, the largest US retailer, includes packaging. Watches piece 50 U S D 0.2 250 T i m e x website avg. price, weight approximated W i n e liter 5.96 U S D 1.3 4.58 B L S avg. price, 2001 (series APU0000720311) Fresh fruits basket 19.36 U S D 8 2.42 B L S avg. price, 2001 Reading materials book 30 U S D 0.5 60 Tomatoes kg 2.90 U S D 1 2.9 B L S avg. price, 2001 (series APU0000712311) *Sports basket contains ski boots, skis and bindings, tennis racquet, basketball, golf set (llpc), dozen golf balls, hockey stick, hockey skates, inline skates and hockey helmet * Average of a TV set, a VCR, and a camcorder. CD to to o 00 Appendix B. 109 Figure B . l : Some prices in NYC, February 2001 i Table B.5: Data sources on volume - part 1 item unit price stowage factor volume ( m 3 ) p / v ( U S D / m 3 ) note Total R E R - C P I Apples kg 2.57 2.622 0.003 647.4 boxes, http: / /www.t is-gdv.de/t is e /ware/obst /apfel /apfel .htm Audio equipment stereo unit 150 5.495 0.055 2730 http / /www. jr . com/JRProductPage.process?Product=3967701 Beef ground, 1kg 4.63 1 0.001 3057.8 http / /www.t is-gdv.de/t is e/ware/fleisch/gekuehlt/gekuehlt.htm Beer six pack 5.40 1.556 0.004 1508.9 http / /www.t is -gdv.de/ t i s e /ware/ lebensmi/bier /bier .htm Car purchase car 24,923 8.399 11.138 2237.7 http / /www.fordvehicles .com/Cars/focus/features/specdimensions/ C a r parts tire 100 4.041 0.04 2474.6 http / /amchouston.home.att .net/stowage factors.htm Cheese kg 8.69 1.397 0.001 6222 http / /www.t is -gdv.de/ t i s e/ware/milchpro/kaese/kaese.htm Clothes Clothes (men) basket* 4.728 10686.4 http:/ /www.tis-gdv.de/t is e /ware/text i l /konfektion/konfektion.htm Clothes (women) basket* 4.728 11208.1 http: / /www.t is-gdv.de/t is e /ware/text i l /konfektion/konfcktion.htm Coffee roast, 300g 3.27 1.961 0.001 3671.3 Rodrigues et. al. (2003) Educ . books & supplies Eggs dozen 1.91 2.755 0.002 630.7 measure Electric i ty 500 k W h 48.55 - -Fats and oils basket* 1.81 1.25 - 2944 German transportation database source for each component Fish and seafood basket**" 2.85 1.85 - 1537.8 German transportation database source for most components Flour 2.5kg 3.37 1.33 0.003 669.4 http:/ /amchouston.home.att .net/stowage factors.htm Footwear Footwear (men) pair, avg of casual 46.50 21.918 0.016 2906.3 Mens shoo box 14-3/4" x 10-1/8" x 5-5/8" and athletic Footwear (women) pair, athletic 43.88 28.351 0.014 2857.1 Fuel oil liter 0.34 1.163 0.001 338 Furniture bed 200 4.73 0.22 909.1 http:/ /www.ikea-usa.com/webapp/wcs/stores/servlet / . . . . . .ProductDisplay?catalogId=10101&storeId=12&productId = 32145&... . . . langId=-l&parentCats=10103*10144 Gas 1000 f t 3 7.45 1559.298 28.317 0.3 Gasoline liter 0.38 1.434 0.001 337 *Men's basket contains: coats, blazers, trousers, suits, women's basket contains: coats, dresses, blazers, trousers, suits, and skirts. *In accordance with the C P I definition of the category, the basket contains: Margarine (Canola, 1.36kg), Butter (Parchment, 454g), Shortening (454g), Oil (Canola, 11), Lard (454g), Peanut butter (500g), and Salad dressing (8oz). Weights are equal to the CPI weights. + C a n n e d fish composition matches the composition of the fish processing industry data. Canned: Tuna (48%), Salmon (12%), Clams (8%), Sardines, Shrimp, Fillets: Cod (4.7%), Flounder (1.7%), Haddock, Rockfish, Pollock (11%) and Other (11%), Fresh fish approximated by 50% tuna and 50% salmon. Table B.6: Data sources on volume - part 2 House chemicals Jewelry Laundry appliances Liquor Medica l care products Non-prescription med. Pants P C Personal care products Photo equipment Pork Potatoes Poul try Prescription medicine Sport equipment Sport vehicles Sugar Tobacco Toys Video equipment Watches W i n e Fresh fruits Reading materials Tomatoes 75oz pack of laundry deterg. washer 750ml whiskey pair, jeans, avg. unit a basket 1 kg, chops 4.54kg kg basket* bicycle l i b 200 cigs basket basket* piece liter basket book kg 887 11.74 50.18 1000 12.58 9.29 3.83 4.45 99.67 225 0.43 37.78 31.33 226.67 50 5.96 19.36 30 2.90 stowage factor 4.506 1.75 3.57 25 8.664 1.7 1 23.61 17.864 1.354 0.002 0.044 1.175 2.95 1.78 2.373 volume 0.716 0.001 0005 0.5 0.024 1 0.002 0.005 0.036 0.268 0.001 6 0.2 5 0.0012 0.0015 0.024 0.001 0.002 p / v | note ( U S D / m 3 ) 1238.8 8944.8 10328 2000 346.2 6.14 3609.1 557.1 2753.3 839.7 699.5 13861 156.7 5191.4 41667 3973.3 820.3 33707.9 1221.9 measurement ht tp: / /www.maytag.com/products / images /products /dmsearcywash.pdf http: / /www.t is-gdv.de/t is e /ware /genuss /rum/rum.htm http:/ /www.t is-gdv.de/t is e /ware/texti l /konfektion/konfektion.htm http:/ /www.shipit .co.uk/Overseas Removals Companies Volumes.htm measurement of basket items http: / /www.t is-gdv.de/t is e/ware/gemuese/kartoffe/kartoffe.htm assume same volume as beef various sources for i t ems* http: http: http: / /www.crateworks.com/frameset. html? page=features / /www.t is-gdv.de/t is e/ware/zucker/weiszuck/weiszuck.htm / /www. discount-cigarettes-online, biz / templates/ faq.php guess http: / /www.t is-gdv.de/t is e /ware/maschinen/unterhaltung/unterhaltung.htm dims: 20x10x5cm, volume direct same stowage factor as liquor German transportation database source for each component http: / /www.t is-gdv.de/t is e /ware/papier/zei tung/zei tung.htm http:/ /www.t is-gdv.de/t is e /ware/gemuese/tomaten/tomaten.htm *Sports basket contains ski boots (http://www.snowshack.com/head-boot-bag.html), skis and bindings (http://www.snowshack.com/salomon-equipe-2pr-skibag.html), tennis racquet, basketball (http://experts.about.eom/q/2551/1184149.htm), golf set ( l lpc , length 44in = 111cm), dozen golf balls (http:/ /www.overstock.com/cgi-bin/d2.cgi?PAGE=PROFRAME&PROD ID=676397), hockey stick (http://www.unleash.com/picks/sportinggoods/topsportinggoodshockeysticks.asp), hockey skates (15-in x 9-in x 15-in bag), and inline skates and hockey helmet http://securel.esportspartners.com/store-redskins/main detail.cfm?nCategoryID=4&nObjGroupID=134&nProductID=56453 'Average of a T V set, a V C R , and a camcorder. 13 13 Ct> ts a . X ' to 0 Appendix B. 112 B . 3 T a b l e s a n d F i g u r e s Appendix B. 113 Table B.7: Long run properties of exchange rates Sample: 1947:1-2000:8 Sample: 1947:1--1969:12 Sample: 1970:1-2000:8 N E R R E R Rel. CPI N E R R E R Rel. CPI N E R R E R Rel. CPI Lags 0 1 11 13 6 12 13 1 11 13 A D F stat.1 0.057 -0.923 -1.923* -0.667 -1.195 -1.444 0.471 -0.614 -1.495 A D F stat.2 0.067 -0.884 -1.931 -1.545 -2.083 -3.19** -0.225 -0.881 -1.882 A D F stat.3 -16.6*** -5.4*** -5.5*** -9.9*** -3.35** -4.2*** -12.6*** -4.2"* -4.0*** Half life4 - (1.001) - (1.0014) 146.4 74.2 27.3 10.5 14145 - (1.0015) 140 Table B.8: Long run properties of linearly detrended exchange rates Sample: 1947:1-2000:8 N E R R E R Rel. CPI Sample: 1947:1-1969:12 N E R R E R Rel. CPI Sample: 1970:1-N E R R E R -2000:8 Rel. CPI Lags 0 11 1 13 13 12 12 11 13 13 A D F stat.1 -2.93"* -2.09** -2.26*** -2.57*** -3.22*** -2.11** -1.87* -1.74* -1.98** A D F stat.2 -2.93** -2.09 -2.25 -2.74* 3.37** -2.1 -1.86 -1.72 -1.99 A D F stat.3 -6.01*** -5.3*** -5.35*** -4.39*** -3.35" -3.95*** -4.48*** -4.16"* -4.3*** Half life4 98.7 64.1 128 66 15.4 13 154.3 228.3 321.9 Table B.9: Long run properties of HP-detrended exchange rates Sampl( v. 1947:1--2000:8 Sampl 3: 1947:1--1969:12 Sample: 1970:1-2000:8 N E R R E R Rel. CPI N E R R E R Rel. CPI N E R R E R Rel. CPI Lags 0 11 11 13 15 13 13 12 11 17 A D F stat.1 -8 .2*" -7.4*** -7.8*** -4*** -4.6*** -5.3*** -5.4*** -5.9*** -5.6*** A D F stat.2 -8.2*** -7.4"* -7.9*** . 4 * * * -4.6*** -5.3*** -5.4*** -5.9*** -5.6*** A D F stat.3 -9.2*** -8.6*** 7.5*** -6*" -5.7*** -5.1*** -5.9*** -5.8*** -5.8*** Half life4 5 4.5 4.9 5 4.3 3.2 4.9 4.6 5.9 Al l series are demeaned and in logarithms. °Lags may vary for A D F tests in differences, depending on the P A C F criterions. 1Augmented Dickey-Fuller test in levels, no intercept, no trend. 2Augmented Dickey-Fuller test in levels, with intercept, no trend. 3Augmented Dickey-Fuller test in differences, with intercept, no trend. 4 AR(1) half-life is calculated without a constant. Appendix B. 114 Table B.10: Long run properties:ADF and half-life convergence G o o d t y p e o b s # l a g s 1 h a l f - l i f e ^ h a l f - l i f e ^ Apples 334 -1.49 13 5.6 6.2 Audio equipment 189 -0.005 4 187 -Beef 334 -1.197 4 20 31 Beer 334 -1.359 2 120 -Car maintenance 265 -0.917 9 1391 -Car parts 117 -2.233" 8 22 -Car purchase 334 -2.58*** 0 20 30 Cheese 189 -.151 3 - -Clothes 223 -1.563 6 26 84 Clothes (men) 223 -1.51 6 34 274 Clothes (women) 223 -1.581 4 15 26 Coffee 334 -0.842 5 78 -Educational books and supplies 70 -0.095 2 140 -Eggs 334 -4.716*** 2 5.2 6 Electricity 334 -1.182 1 74 -Fats and oils 141 -1.333 0 35 Fish and seafood 189 -0.315 2 180 -Flour 273 -2.098** 1 40 262 Footwear 334 -2.437** 7 22 35 Footwear (men) 189 -1.826* 4 15 27 Footwear (women) 273 -2.46** 8 19 32 Fuel oil 333 -1.418 5 39 118 Furniture 334 -1.319 0 60 -Gas 334 -2.69*** 2 16 22 Gasoline 334 -1.554 2 45 202 House chemicals 334 -0.146 4 859 -Jewelry 117 -2.704*** 2 21 -Laundry appliances 189 -0.589 4 94 -Liquor 273 -1.271 1 103 -Medical care products 265 -1.027 11 - -Non-prescription medicine 166 -0.616 1 133 -Pants 273 -2.147** 3 11 14 P C 34 -0.485 8 6 -Personal care products 334 -0.624 2 151 -Photo equipment 189 -0.137 0 639 -Pork 265 0.917 10 23 47 Potatoes 344 -2.374** 13 3.6 3.8 Poultry 334 -2.589*** 5 11 14 Prescribtion medicine 189 0.6 3 - -Sport equipment 265 -1.519 1 37 203 Sport vehicles 265 -1.589 2 48 -Sugar 134 -1.904* 7 39 -Tobacco 177 0.053 0 - -Toys 189 -2.09** 12 31 707 Video equipment 57 -1.459 2 10 -Watches 93 -2.710*** 5 2.4 3 Wine 330 -1.758* 2 69 -1 The number of lags L in AT/ £ __ L where Ayt is the L H S variable of A D F test is one which minimizes Akaike Information Criterion. 2 Half-life is calculated using sample-adjusted AR(1) coefficient from Kendall's formula. In cases denoted "+", adjustment for bias leads to non-stationary AR(1) process in which case there is no half-life. There, I compute half-life without adjustment for sample-size bias. Appendix B. 115 Table B . l l : E Q - T A R Summary STD AR(1) TAR(2,1,1) TAR(2,1,1) A R ( P ) TAR(2,p,d) TAR(2,p,d) half life threshold half life half life threshold half life Foods 0.147 45 0.144 34 41 0.083 58 Vice goods 0.188 72 0.115 79 91 0.134 105 Clothing and footwear 0.075 20 0.035 21 26 0.041 19 Tech stuff 0.085 156 0.077 45 540 0.063 38 Fuels 0.149 51 0.109 45 50 0.070 48 Medical and chemical 0.146 235 0.194 100 244 0.131 90 Cars and car parts 0.074 20 0.039 19 27 0.046 23 Laundry appliances 0.099 94 0.134 26 98 0.154 25 Furniture 0.092 59 0.125 30 67 0.145 34 Services 0.133 224 0.054 241 CPI-RER 0.111 0.071 162 0.012 213 Table B.12: B A N D - T A R Summary STD AR(1) TAR(2,1,1) TAR(2,1,1) A R ( P ) TAR(2,p,d) TAR(2,p,d) half life threshold half life half life threshold half life Foods 0.147 45 0.146 22 41 0.083 29 Vice goods 0.188 72 0.149 70 55 0.144 149 Clothing and footwear 0.075 20 0.027 23 26 0.022 31 Tech stuff 0.085 156 0.079 33 540 0.063 27 Fuels 0.149 51 0.097 43 50 0.069 50 Medical and chemical 0.146 235 0.193 332 244 0.105 527 Cars and car parts 0.074 20 0.039 22 27 0.035 26 Laundry appliances 0.099 94 0.074 111 98 0.154 45 Furniture 0.092 59 0.127 43 67 0.145 60 Services 0.133 224 0.065 160 CPI-RER 0.111 0.071 1733 162 0.012 193 Appendix B. 116 Table B.13: AR(1) Results - Part I AR(1) p-value AR(1) Tsay's lambda half life nonlinearity test Total RER-CPI 0.354 Apples -0.120 0.025 5 0.000 Audio equipment -0.004 0.011 182 0.019 Beef -0.035 0.015 20 0.169 Beer -0.006 0.005 114 0.087 Car purchase -0.036 0.014 19 0.218 Car maintenance -0.001 0.008 1216 0.104 Car parts -0.031 0.013 22 0.490 Cheese 0.001 0.010 0.106 Clothes -0.026 0.015 26 0.504 Clothes (men) -0.020 0.014 34 0.779 Clothes (women) -0.044 0.019 15 0.634 Coffee -0.005 0.007 130 0.022 Educational books and supplies -0.006 0.022 119 0.011 Eggs -0.127 0.025 5 0.003 Electricity -0.006 0.006 107 0.243 Fats and oils -0.020 0.015 35 0.316 Fish and seafood -0.004 0.011 175 0.203 Flour -0.017 0.010 40 0.572 Footwear -0.033 0.013 21 0.076 Footwear (men) -0.046 0.021 15 0.457 Footwear (women) -0.036 0.015 19 0.838 Fuel oil -0.018 0.010 39 0.704 Furniture -0.012 0.009 59 0.563 Gas -0.043 0.017 16 0.020 Gasoline -0.016 0.009 43 0.671 House chemicals 0.001 0.005 0.805 Jewelry -0.033 0.018 21 0.008 Laundry appliances -0.007 0.011 94 0.246 Liquor -0.007 0.008 103 0.327 Medical care products 0.003 0.006 0.030 Non-prescription medicine -0.005 0.009 132 0.159 Pants -0.062 0.021 11 0.118 P C -0.116 0.065 6 0.644 Personal care products -0.002 0.004 338 0.299 Photo equipment -0.001 0.009 579 0.056 Pork -0.030 0.017 23 0.489 Potatoes -0.171 0.029 4 0.029 Poultry -0.050 0.015 14 0.030 Prescription medicine 0.004 0.005 0.138 Sport equipment -0.018 0.011 37 0.419 Sport vehicles -0.015 0.009 48 0.000 Sugar -0.018 0.015 39 0.041 Tobacco 0.001 0.011 0.019 Toys -0.022 0.015 31 0.407 Video equipment -0.066 0.049 10 0.135 Watches -0.249 0.061 2 0.566 Wine -0.010 0.006 69 0.384 Appendix B. 117 Figure B.2: Nonlinear fit of price-weight ratios and thresholds 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 j V 70 75 80 85 90 95 00 L R X T O B A C C O 2 htonlinear estimation (excludes,tobacco) T H R = i y ( 9 . 1 4 5 + P W M K ) F f ^ O . I S (0.000) (0.0003) 150 200 250 3 0 0 3 5 0 400 450 500 Pr ice-weight ta tb ( U S t * g ) Appendix B. 118 Table B.14: AR(1) Results - part II AR(1) p-value AR(1) Tsay's lambda half life nonlinearity test Airfare 11 Cable T V 60 Car insurance 51 Child care Dental services Fresh fruits 11 Intra-city transport 43 Margarine 31 Medical services Reading materials 48 Rent 1175 Restaurant meals 65 Shelter 150 Tomatoes 6 Tuition 17 Water and sewerage 69 Figure B.3: Nonlinear fit of price-weight ratios and thresholds Nonlinear estimation (exc ludes tobacco , a lcohol , gasol ine a n d natural gas) o 0.2-° b o 0 50 100 150 200 250 500 350 400 450 500 Price-u'eight rate (U St/kg) Appendix B. 119 Table B.15: EQ-TAR(2,1,1) results L L R TAR(2,1,1) TAR(2,1,1) T A R (2,1,1) p-value threshold lambda half life Total RER-CPI -0.8 0.071 0.003 0.129 Apples 34.7 0.109 -0.124 5.2 0.200 Audio equipment 7.3 0.098 -0.004 160.9 0.134 Beef 5.8 0.024 -0.036 19.1 0.295 Beer 6.7 0.045 -0.006 115.2 0.130 Car purchase -1.9 0.034 -0.037 18.5 0.266 Car maintenance 1.6 0.041 -0.002 346.2 0.136 Car parts -0.8 0.043 -0.034 20.3 0.344 Cheese -3.5 0.042 0.002 0.129 Clothes -3.4 0.019 -0.026 26.3 0.205 Clothes (men) -0.7 0.010 -0.020 34.5 0.176 Clothes (women) -2.9 0.062 -0.037 18.6 0.313 Coffee 24.2 0.223 -0.005 144.1 0.113 Educational books and supplies 3.6 0.126 -0.102 6.4 0.134 Eggs 19.8 0.138 -0.174 3.6 0.335 Electricity 1.9 0.183 -0.009 81.2 0.151 Fats and oils 1.8 0.074 -0.029 23.4 0.213 Fish and seafood -4 0.141 -0.025 27.6 0.141 Flour 17.6 0.176 -0.033 20.9 0.152 Footwear -1.1 0.009 -0.033 20.6 0.249 Footwear (men) -5.2 0.026 -0.042 16 0.301 Footwear (women) 12.4 0.059 -0.029 23.7 0.237 Fuel oil 11.2 0.101 -0.016 43.8 0.198 Furniture 5.6 0.125 -0.023 29.9 0.163 Gas 24.8 0.085 -0.072 9.2 0.211 Gasoline 26.6 0.066 -0.016 43.8 0.129 House chemicals 1.3 0.164 0.004 0.145 Jewelry -0.9 0.214 -0.023 29.5 0.244 Laundry appliances -3.6 0.134 -0.026 25.8 0.153 Liquor -1 0.012 -0.007 103.1 0.153 Medical care products 4 0.072 0.003 0.115 Non-prescription medicine -2.1 0.156 -0.019 35.4 0.114 Pants 0.3 0.058 -0.076 8.8 0.314 PC -2 0.059 -0.163 3.9 0.390 Personal care products 3.6 0.288 -0.005 144.1 0.207 Photo equipment 1 0.194 -0.014 48.1 0.144 Pork -2.4 0.102 -0.009 80.3 0.245 Potatoes 15.2 0.399 -0.298 2 0.414 Poultry 47.1 0.180 -0.063 10.6 0.083 Prescription medicine -4.8 0.288 -0.006 121.3 0.162 Sport equipment -1.8 0.139 -0.021 32 0.199 Sport vehicles 30.3 0.189 -0.046 14.8 0.079 Sugar -3 0.048 -0.023 30.2 0.193 Tobacco 100.2 0.299 -0.005 150.3 0.000 Toys 8.9 0.148 -0.059 11.3 0.186 Video equipment 0.8 0.028 -0.078 8.6 0.203 Watches 7.9 0.008 -0.249 2.4 0.361 Wine 38.2 0.252 -0.028 24.8 0.055 Appendix B. 120 Table B.16: EQ-TAR(2,p,d) results - part I L L R TAR(2,p,d) TAR(2,p,d) p-value threshold halflife (old, 600) Total RER-CPI -9.1 0.012 213 0.056 Apples 20.9 0.073 6 0.126 Audio equipment 12.6 0.078 115 0.103 Beef 13.4 0.039 23 0.235 Beer 1.8 0.041 113 0.260 Car purchase 2.9 0.022 25 0.278 Car maintenance -1.9 0.086 470 0.114 Car parts 1 0.070 20 0.425 Cheese -2.7 0.081 563 0.113 Clothes 1 0.084 21 0.201 Clothes (men) -11.8 0.132 8 0.189 Clothes (women) -7.1 0.009 23 0.270 Coffee 1.3 0.223 0.070 Educational books and supplies 1.4 0.116 9 0.131 Eggs -1.9 0.034 7 0.414 Electricity 1.4 0.059 79 0.218 Fats and oils 2.7 0.033 53 0.211 Fish and seafood -2.1 0.152 27 0.107 Flour 20.5 0.157 27 0.124 Footwear -6.3 0.006 ' 34 0.192 Footwear (men) -2.4 0.019 22 0.367 Footwear (women) -13.4 0.020 13 0.214 Fuel oil 4.1 0.112 45 0.204 Furniture 10.4 0.145 34 0.194 Gas 44.5 0.043 18 0.029 Gasoline 28.3 0.064 48 0.190 House chemicals -1.3 0.166 0.152 Jewelry -2.1 0.080 18 0.386 Laundry appliances -2.2 0.154 25 0.185 Liquor -4.1 0.012 86 0.075 Medical care products -4.9 0.027 0.091 Non-prescription medicine 2 0.150 42 0.143 Pants -5.1 0.016 11 0.327 PC -1.4 0.018 18 0.000 Personal care products -3.2 0.263 138 0.211 Photo equipment -1.1 0.202 37 0.114 Pork -3.6 0.013 15 0.288 Potatoes 1.3 0.065 14 0.356 Poultry 20.5 0.086 15 0.103 Prescription medicine -3.7 0.047 0.077 Sport equipment -4.3 0.055 49 0.183 Sport vehicles 43.1 0.183 11 0.066 Sugar 1.6 0.030 23 0.410 Tobacco 106.8 0.301 172 0 Toys 10.8 0.145 18 0.201 Video equipment 5.9 0.012 18 0.177 Watches -5.2 0.007 3 0.332 Wine 30.1 0.230 47 0.029 Appendix B. 121 Table B.17: EQ-TAR(2,p,d) results - part II L L R TAR(2,p,d) threshold TAR(2,p,d) halflife Airfare -2.3 0.015 11 Cable T V 22.1 0.042 51 Car insurance 23.2 0.025 48 Child care 10.1 0.018 Dental services 9.4 0.046 Fresh fruits -12.5 0.051 11 Intra-city transport 4.9 0.036 51 Margarine 0.9 0.117 15 Medical services 0.3 0.161 Reading materials -2.2 0.017 48 Rent -3.3 0.016 1066 Restaurant meals 5 0.040 63 Shelter -2.2 0.033 146 Tomatoes -28.5 0.042 6 Tuition -0.05 0.017 17 Water and sewerage -2.7 0.114 75 Appendix B. 122 Table B.18: BAND-TAR(2,p,d) results - part I L L R TAR(2,p,d) threshold TAR(2,p,d) halflife p-value Total RER-CPI -9 0.012 193 0.057 Apples 23 0.073 6 0.086 Audio equipment 13 0.076 86 0.096 Beef 15.1 0.039 23 0.253 Beer 1.3 0.041 144 0.172 Car purchase 1.2 0.022 30 0.273 Car maintenance -1.9 0.167 58 0.106 Car parts 6.5 0.048 21 0.396 Cheese -1 0.163 24 0.099 Clothes -2.2 0.087 36 0.209 Clothes (men) -11.8 0.010 46 0.204 Clothes (women) -7.3 0.009 24 0.238 Coffee 1.4 0.225 107 0.067 Educational books and supplies -1.7 0.116 22 0.086 Eggs -4 0.033 8 0.412 Electricity 2.5 0.055 71 0.188 Fats and oils 3.1 0.033 49 0.191 Fish and seafood -2.5 0.126 50 0.116 Flour 17.3 0.157 48 0.182 Footwear -6.6 0.006 35 0.175 Footwear (men) -3 0.006 22 0.355 Footwear (women) -14.1 0.020 39 0.221 Fuel oil 5.9 0.112 48 0.219 Furniture 10.4 0.145 60 0.161 Gas 41.9 0.043 28 0.075 Gasoline 28.7 0.064 51 0.126 House chemicals -1.3 0.125 188 0.180 Jewelry -1.3 0.029 17 0.407 Laundry appliances -2.3 0.154 45 0.188 Liquor -4.1 0.012 86 0.095 Medical care products -4.9 0.027 9634 0.084 Non-prescription medicine 0.9 0.150 101 0.166 Pants -5.6 0.016 13 0.272 PC -1.6 0.018 22 0.000 Personal care products -2.2 0.177 165 0.242 Photo equipment 1.2 0.202 5 0.099 Pork -3.6 0.013 15 0.271 Potatoes -0.6 0.065 14 0.209 Poultry 18.6 0.086 34 0.291 Prescription medicine -4 0.047 1655 0.053 Sport equipment -5.6 0.009 50 0.241 Sport vehicles 40.5 0.183 13 0.075 Sugar 2.7 0.030 21 0.327 Tobacco 107.5 0.303 106 0.000 Toys 8.7 0.120 161 0.247 Video equipment 5.6 0.012 20 0.179 Watches -3.4 0.008 3 0.282 Wine 27 0.230 409 0.068 Appendix B. 123 Table B.19: BAND-TAR(2,p,d) results - part II LLR TAR(2,p,d) threshold TAR(2,p,d) halflife p-value Airfare -2.4 0.015 11 0.266 Cable TV 22.4 0.042 50 0.068 Car insurance 23.2 0.025 51 0.157 Child care 10 0.019 461 0.159 Dental services 8.4 0.046 0.161 Fresh fruits -13 0.051 11 0.311 Intra-city transport 5.2 0.061 60 0.218 Margarine 1 . 0.054 32 0.245 Medical services 0.9 0.166 165 0.178 Reading materials -2.4 0.017 51 0.106 Rent -3.1 0.016 542 0.145 Restaurant meals '5.1 0.040 68 0.061 Shelter -1.7 0.033 119 0.036 Tomatoes -28.5 0.042 6 0.460 Tuition 1.3 0.028 18 0.463 Water and sewerage -1.6 0.114 76 0.205 Table B.20: Price-to-weight ratios, Price-to-volume ratios, and thresholds Cnst P/W P/Wjvr P / V i V T R 2 p-value (F) LogL DW Thr B 8.4** -0.017 (0.000) (0.245) 0.03 0.245 -135 2.31 ThrB,NT 8.68*** -0.059*** (0.000) (0.012) 0.16 0.067 -126 2.36 ThrB,NT 8.42*** -0.0002** (0.000) (0.052) 0.099 0.059 -118 2.48 ThrB:NT 8.68*** -0.06 0.00001 (0.000) (0.123) (0.96) 0.157 -135 2.37 p-values in parentheses. A * denotes a significance at 10% level, ** at 5% level and *** at 1% level. 1) "NT" is a regression excluding tobacco due to a large discrete jump in its relative price in 1984/85 following a tax change. Appendix B. 124 Table B.21: Nonlinear relationship between price-to-weights and thresholds Hyperbola: ThrB,NT = 1/(C(1) + (P/WNT)C{2)) C(l) C(2) R 2 LogL DW T h r s . i v r 9.144** 0.697** (0.000) (0.000) 0.18 55.6 2.57 T h r s . j v T A E 8.697** 0.71** (0.000) (0.000) 0.23 50.9 2.59 1) " J V T " is a regression excluding tobacco due to a large discrete jump in its relative price in 1984/85 following a tax change. 2) " NTAE" is a regression excluding tobacco, alcohol and energies (gasoline, natural gas) due to poor tradability. Table B.22: Prices, weights and thresholds Cnst P w R 2 p-value (F) LogL DW T h r B -0.0046 0.082 (0.000) (0.161) (0.181) 0.06 0.3 -135 2.4 T h r ^ j v T 7.12** -0.0049* 0.088* (0.000) (0.090) (0.10) 0.08 0.19 -126 2.6 7.256** -0.0051* 0.092* (0.000) (0.086) (0.098) 0.095 0.18 -120 2.5 100*STDiVT 12.2 -0.005 (0.000) (0.221) 0.041 0.23 -113 1.72 100*STDivr 12.1 -0.0003 (0.000) (0.232) 0.04 0.23 -113 1.73 in parentheses. A * denotes a significance at 10% level, ** at 5% level and *** at 1% level 1) RSW,NT is a residual from estimation of S T D on W and a constant due to endogeneity of S T D (not shown). 2) " NT" is a regression excluding tobacco due to a large discrete jump in its relative price in 1984/85 following a tax change. " N T E " also excludes energy (gasoline, natural gas) due to poor tradability. Appendix B. 125 Table B.23: Price-to-weight ratios, Price-to-volume ratios, and thresholds - robustness Cnst P / W P / V R 2 DW 8.6*** -0.06*** (0.000) (0.012) 0.17 2.29 Thr NTE 8.6*** -0.0003** (0.000) (0.049) 0.11 2.42 ThrNTE 8.9*** -0.06 0.0001 (0.000) (0.123) (0.99) 0.17 2.30 Thr NTW 9.8*** -0.14*** (0.000) (0.016) 0.24 2.53 ThiNTW 8 Q*** -0.0003* (0.000) (0.082) 0.08 2.66 Thrjvrw 9.9*** -0.13* -0.0001 (0.000) (0.095) (0.80) 0.24 2.56 ThTNTAE 8.8*** -0.06*** (0.000) (0.009) 0.19 2.57 Thr^TAE 8.4*** -0.0002** (0.000) (0.043) 0.12 2.57 ThlNTAE 8.78*** -0.06 0.0001 (0.000) (0.11) (0.96) 0.19 2.57 p-values in parentheses. A * denotes a significance at 10% level, ** at 5% level and *** at 1% level. " NTE" excludes tobacco and energies (gasoline, natural gas) "NTAE" excludes tobacco, alcohol (liquor, beer, wine) and energies (gasoline, natural gas) "NTW" excludes tobacco and watches Appendix C 126 C l S o l v i n g t h e b a n k s ' p r o b l e m Due to the inequality constraints, we have to use a Kuhn-Tucker approach and be careful about the corner solutions. The Lagrangean for this problem is: £ = rlL + r b B - r r D - r e E - 5 ^ y D - (l + lc)eL +Xi(D + E-B-L) + X2(E/L - a) + X3(D + E - L) Then the first order conditions are: rl - Ai - X2E/L2 - X3 - e(l + lc) = 0 r b - A i = 0 - r d - c 5 ( 7 ~ l ) ( | ) 7 + A 1 + A 3 = 0 / £)\T+1 -re + Sj{~) + Ai + X2/L + A 3 = 0 As noted above, there are two possibilities: either constraint ( 3 . 9 ) or constraint ( 3 . 1 0 ) bind. In terms of the Lagrangean we therefore need to consider two cases. The one where A2 > 0 and A 3 = 0 (i.e. ( 3 . 9 ) binds while ( 3 . 1 0 ) does not) will be referred to as an "interior solution" because not all loanable funds are invested into loans. The opposite case where A 3 > 0 and A2 = 0 will be referred to as a "corner solution". For simplicity, in what follows we assume 7 = 1 . Interior solution This is the case when bank holds just enough equity to satisfy the capital adequacy requirement (E/L = a and therefore D + E > L). The above first order conditions can be combined into: ( C l ) ^ ( r ' - r " - ( l + - c ) £ ] * (C2) (C.3) (C.4) where (C.2) is an equity (or implicitly deposit) supply equation. Conditional on particular values of M and all levels of prices, equations ( C l ) to (C .4) form a recursive system which uniquely determines all quantities. Corner solution r M E L D = 1 E -E a M-E Appendix C. 127 M = 1 + — E L L = M D = M - E (1 + lc> = r b - rd I n a c o r n e r s o l u t i o n , b a n k h o l d s m o r e e q u i t y t h a n r e q u i r e d b y t h e c a p i t a l a d e q u a c y r e q u i r e m e n t (D + E = L a n d t h e r e f o r e E/L > a). N o w , rb > rd 1, a n d t h e a b o v e f i r s t o r d e r c o n d i t i o n s c a n b e c o m b i n e d i n t o : r' + d + y . i i ( a 5 ) (C.6) (C.7) rl - rb - (   lc)e  r   r d (C.8) w h e r e (C.5) i s a g a i n a n e q u i t y s u p p l y e q u a t i o n . N o t e t h a t n o w l o a n s a n d e q u i t y s u p p l y d e c i s i o n s a r e d i s c o n n e c t e d . E q u a t i o n (C.8) s h o w s a w e d g e b e t w e e n t h e b o n d a n d d e p o s i t r a t e s . T h e b o n d " p r e m i u m " o n t h e r i g h t h a n d s i d e e q u a l s t h e p r o f i t d i f f e r e n t i a l b e t w e e n n e t r e t u r n s o n l o a n s a n d b o n d s t h a t w o u l d e q u a l z e r o i n a n i n t e r i o r s o l u t i o n . C.2 O n t h e a s s u m p t i o n o f a s i n g l e p o r t f o l i o o p t i m i z a t i o n I t m a y s e e m p r o b l e m a t i c t o a s s u m e t h a t t h e a s s e t p o r t f o l i o i s a l l o c a t e d i n a n i d e n t i c a l m a n n e r f o r a l l h o u s e h o l d s . H e r e w e s h o w t h a t a s l o n g a s t h e l a b o r i n c o m e r e m a i n s t h e s a m e a c r o s s a l l d e p o s i t o r s , t h e o p t i m a l s p l i t t i n g r u l e s d e r i v e d f r o m t h e i r p r e f e r e n c e s w i l l b e i d e n t i c a l a c r o s s a l l o f t h e h o u s e h o l d s . T o p r o v e t h i s p o i n t , w e u s e a s i m p l i f i e d v e r s i o n o f t h e p r o b l e m . H o u s e h o l d s m a x i m i z e r -i [la c1~a)1~p — l m a x { c M , m i , t + i , d M , e t ] i } E o p S o / 3 u(ct,i)\ s . t . Ctj+dtj+etj = mt,i+yi, w h e r e U(ct,i) = ° c ' ' i _ p , mi,t+l — dtti(l + rf) + etti(l + rf) a n d ett%, dtj d e n o t e i n d i v i d u a l e q u i t y a n d d e p o s i t h o l d i n g s , r e -s p e c t i v e l y . T h e E u l e r e q u a t i o n s f o r t h i s p r o b l e m a r e : < i = / 3 E t [ ^ + l i i ( l + r t d + 1 ) ] = / 3 [ E t ( l + r t d + 1 ) E t [ c ^ j ] c * = (JEt [ < ^ + l i < ( l + r ? + 1 ) ] = 0 [ E t ( l + r t e + 1 ) E t + cov[(l + r f + 1 ) , w h e r e x = —(o~ + p ( l — cr)). S o l v i n g w i t h a m e t h o d o f u n d e t e r m i n e d c o e f f i c i e n t s , w e m a k e a n e d u c a t e d g u e s s t h a t e t , j = 7 e " i i , i a n d dtti = Jd.mt,i a n d r e w r i t e t h e E u l e r e q u a t i o n s a s : [(1 - 7e , i - ld,i)mt,i + yi\X = P(l + r d ) E t [(1 - 7e , i - ld,i)mt+i,i + Vi]* [(1 ~ 7e , i - ld,i)mt,i + yi\X = PEt [(1 + r f + 1 ) 1 / x [ ( l - 7e , i - ld,i)mt+i,i + J/t]]^  T h e a b o v e t w o e q u a t i o n s g i v e d e t e r m i n e t h e s h a r e s o f e q u i t y 7 e , i a n d d e p o s i t s ~fd,i m mt+i,i a s f u n c t i o n s o f i n d i v i d u a l a s w e l l a s a g g r e g a t e v a r i a b l e s . T h e i m p o r t a n t p o i n t f o r o u r a r g u m e n t i s t h a t i f a l l a g e n t s h a v e t h e s a m e l a b o r i n c o m e = y V i, t h e n w e c a n h a r m l e s s l y a s s u m e t h a t y = 0 a n d t h e s e t w o e q u a t i o n s c o l l a p s e i n t o : 1 = /3(l + r d ) E t [ [ 7 ( i ( l + r d ) - t - 7 e ( l + r- t e + 1 )]j (C.9) 1 = / 3 E t [ ( l + r e ) 1 ^ [ 7 d ( l + r ' i ) - r - 7 e ( l + r t e + 1 ) ] ] X (C.IO) J A lower demand for bank's financing by deposits (relative to equity) depresses their price. Appendix C. 128 Note that in equations (C.9) and (CIO), j e and Jd are independent of any individual variables. They are only functions of the rates of return and the parameters of the utility function. This way we have shown that as long as the agents have an identical labor income (and as long as their deposit and equity demands are linear in their asset holdings which can be proved for the case of y = 0), the portfolio-splitting decisions can be assumed to be made uniformly. Because in this model we work with two types of depositors (workers and retirees / unem-ployed), the assumption of an identical labor income is only justified within these two groups. In the future research, we should therefore model the number of distinct portfolio optimizations equal to 2. The optimality conditions will then present additional identification restrictions on the parameter values of the household's risk-aversion parameter A. We ignore this in the current version of the paper due to the computational difficulty. C.3 The solution procedure Heterogeneous agents models with aggregate shocks are difficult to solve because the distribution of agents is not invariant and becomes a highly dimensional state variable. The two main strategies to solve this problem is to either find a good way to summarize the distribution with very few variables, as Krusell and Smith (1998) demonstrate, or to work with linearization, as Cooley and Quadrini (1999) do. Unfortunately, neither is possible here due to some highly non-linear phenomena that are crucial in our model. For example, decision rules change abruptly in the vicinity of m*. Finally, second degree effects appear to be quite important, and they are likely to vanish with linearization. Our strategy uses the realization that aggregate shocks in a two-state Markov process lead to transitional states somewhere between two steady-states corresponding to repeated identical shocks. We therefore choose a sufficient number of aggregate states to represent a large proportion of actual aggregate states. The aggregate state space is assumed two dimensional: one dimension is the current shock, High or Low, the other is a counter of how far from the the High steady-state the economy is. Specifically, this counter is incremented by one each time a Low shocks occurred in the previous period, or decreased by one if a High shock occurred. The minimum counter value is one, the maximum is chosen such that this state occurs infrequently. We choose a maximum of 5, implying with the transition probabilities of the Markov process that the economy will in any of the aggregate states Ssc% of the time, where We then solve this model economy with the standard tools for heterogeneous agent economies, that is value function iterations followed by iterations on the invariant distribution (defined over the aggregate states as well). The equilibrium is reached by finding the set of lending rates rl and loan eligibility rules m* that balance all markets and satisfy all constraints. 50.2 16.7 5.6 1.9 0.6 16.7 5.6 1.9 0.6 0.2 Appendix C. 129 C.4 Figures Appendix C. 130 Figure C.2: Benchmark and policy with interest rate reduction in worst case only lending rate (%) bond rate (%) Appendix C. 131 Figure C.3: Benchmark and policy with gradual interest rate reduction in bad return situations lending rate (%) bond rate (%) 8 10 8 10 Appendix C. 132 Figure C.4: Benchmark and policy with interest rate increase in worst case only lending rate (%) bond rate (%) Appendix C. 133 Figure C.5: Benchmark and policy with gradual interest rate increase as aggregate states worsen lending rate (%) bond rate (%) Appendix C. 134 Figure C.6: Benchmark and policy with relaxing of capital requirements in worst case only lending rate (%) bond rate (%) Appendix C. 135 Figure C.7: Benchmark and policy with gradual relaxing of capital requirements as aggregate states worsen lending rate (%) bond rate (%) m* total deposits v 2 4 6 8 10 2 4 6 8 10 8 7 6 8 7.5 7 2 4 6 8 10 ^ / s / equity/loans (%) 2 4 6 8 10 entrepreneurs (%) \^—— t " 2 4 6 8 10 0.0535 0.05 2 4 6 8 10 Appendix C. 136 Figure C.8: Benchmark and policy with tightening of capital requirements in worst case only lending rate (%) bond rate (%) Appendix C. 137 Figure C.9: Benchmark and policy with gradual tightening of capital requirements as aggregate states worsen bond rate (%) lending rate (%) m* 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 

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