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Three essays in macroeconomics and international economics Tang, Yao 2009

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Three Essays in Macroeconomics and International Economics by Yao Tang B.A., Beijing Second Foreign Language Institute, 1998 M.A., Simon Fraser University, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Economics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2009 c⃝ Yao Tang 2009 Abstract This dissertation examines two issues in international economics and macroe- conomics. The first is to understand the response of productivity to major real exchange rate appreciations and the second concerns how to compare the fits of different calibrated macroeconomic models. In the first chapter, I construct a model to clarify how the increased competition due to an exchange rate appreciation provides incentive for firms to improve productivity. However, if a firm is in an industry shielded by a high trade cost, then the incentive is weaker. In industries with fewer firms, profits are more responsive to productivity improvements, therefore, firms are more likely to invest more heavily in productivity improvement. Empirical analysis of Canadian manufacturing data from 1997 to 2006 finds evidence consistent with the model predictions. The second chapter presents testing procedures for comparison of mis- specified calibrated models. The proposed tests are of the Vuong-type (Vuong, 1989; Rivers and Vuong, 2002). In the framework here, an econo- metrician selects values for the parameters in order to match some char- acteristics of the data with those implied by the theoretical model. We assume that all competing models are misspecified, and suggest a test for the null hypothesis that all considered models provide equal fit to the data ii Abstract characteristics, against the alternative that one of the models is a better approximation. The Carlstrom and Fuerst (1997) model and the Bernanke, Gertler and Gilchrist (1999) model are two leading models that study financial frictions in macroeconomic models. In particular, these models show that due to financial frictions, net worth plays an important role in obtaining external finance, and that at an aggregate level, net worth can propagate technology shocks and monetary shocks. However, neither paper examines whether the models can reproduce cyclical properties of net worth. The third chapter addresses this issue by applying the comparison method developed in the third chapter. Results indicate both models do reasonably well. In addition, price rigidity seems to play an important role in the latter model. However, both models can only partially capture the positive correlation between risk premium and net worth. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Does Productivity Respond to Exchange Rate Apprecia- tions? A Theoretical and Empirical Investigation . . . . . 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Basic Model Setup . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Exchange Appreciation and Investment Decision . . . . . . . 22 iv Table of Contents 2.4 Manufacturing Productivities in Canada . . . . . . . . . . . . 37 2.4.1 Specification and Data . . . . . . . . . . . . . . . . . 39 2.4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . 46 2.4.3 Robustness Checks . . . . . . . . . . . . . . . . . . . 50 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Comparison of Misspecified Calibrated Models: The Mini- mum Distance Approach . . . . . . . . . . . . . . . . . . . . . 62 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Properties of the CMD Estimators of Structural Parameters 77 3.4 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . 83 3.4.1 Nested Models . . . . . . . . . . . . . . . . . . . . . . 84 3.4.2 Strictly Non-nested Models . . . . . . . . . . . . . . . 87 3.4.3 Overlapping Models . . . . . . . . . . . . . . . . . . . 89 3.5 Model Comparison with Estimation and Evaluation . . . . . . 90 3.6 Averaged and Sup Tests for Model Comparison, . . . . . . . . 94 3.6.1 Averaged and Sup Tests . . . . . . . . . . . . . . . . 95 3.6.2 Confidence Sets for Weight Matrices . . . . . . . . . . 99 3.7 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.7.1 CIA Model . . . . . . . . . . . . . . . . . . . . . . . . 100 3.7.2 PAC Model . . . . . . . . . . . . . . . . . . . . . . . . 103 3.7.3 Model Estimation and Comparison Results . . . . . . 104 3.8 Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . 110 v Table of Contents Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4 An Exploration of the Role of Net Worth in Business Cycles 125 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.2 Cyclical Properties of Net Worth . . . . . . . . . . . . . . . . 128 4.3 Overviews of Two Competing Models . . . . . . . . . . . . . 136 4.3.1 The Carlstrom and Fuerst (1997) Model . . . . . . . 136 4.3.2 The Bernanke, Gertler and Gilchrist (1999) Model . . 143 4.4 Comparison of the Models . . . . . . . . . . . . . . . . . . . 149 4.4.1 Informal Comparison of the Models . . . . . . . . . . 149 4.4.2 Overview of the Formal Comparison Methodology . . 150 4.4.3 Formal Comparison of the Models . . . . . . . . . . . 154 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 vi List of Tables 2.1 Means of Key Variables between 1997 and 2006 . . . . . . . . 43 2.2 Benchmark Fixed Effect Estimations . . . . . . . . . . . . . . 48 2.3 Alternative Dependent Variables . . . . . . . . . . . . . . . . 52 2.4 Alternative Specification of Lags . . . . . . . . . . . . . . . . 53 2.5 Effects of Entry and Exit of Establishments . . . . . . . . . . 54 2.6 Other Robustness Checks . . . . . . . . . . . . . . . . . . . . 56 3.1 CIA and PAC Parameters’ Estimates and Their Standard Er- rors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.1 Net Worth, GDP, Consumption, and Investment . . . . . . . 133 4.2 Net Worth and Investment in Greater Details . . . . . . . . . 134 4.3 Moments of Calibrated Models . . . . . . . . . . . . . . . . . 151 4.4 Parameters Estimates of CF and BGG . . . . . . . . . . . . . 156 4.5 Moments of Estimated Models . . . . . . . . . . . . . . . . . 157 vii List of Figures 2.1 An Illustration of the Industrial Organization . . . . . . . . . 15 2.2 The Benefit and Cost of Adopting the Disruptive Technology 34 2.3 Illustration of The Relation between 푛푖 and Choice of 휎 . . . 35 2.4 Level of Technology Adoption and Number of Firms in the Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.5 Movements of Canadian Dollar Exchange Rate Since 1990 . . 42 3.1 Model Prediction Errors of the Inflation Impulse Responses with 95% Confidence Bands . . . . . . . . . . . . . . . . . . . 109 3.2 Model Prediction Errors of the Output Impulse Responses with 95% Confidence Bands . . . . . . . . . . . . . . . . . . . 109 4.1 Net Worth and GDP, Consumption, Hours Worked and In- vestment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2 Net Worth and Investment in Greater Details . . . . . . . . . 131 4.3 Net Worth and Interest Rates . . . . . . . . . . . . . . . . . . 132 viii Acknowledgements First thanks go to Paul Beaudry for his supervision and support. I am also grateful for help and advice from Michael Devereux, Amartya Lahiri, Vadim Marmer, Viktoria Hnatkovska, Anji Redish, Henry Siu, and colleagues and staff of the UBC economics department. ix Dedication To my parents and my wife. x Statement of Co-Authorship Chapter 3 of my dissertation is a joint work with Viktoria Hnatkovska and Vadim Marmer. I have been actively involved in the area of research pro- gram identification and design. In terms of execution, I have verified the applicability of the proposed econometric method to macroeconomic models and provide an application. I am also responsible for collection of data in the application section and provision of the manuscript draft of the section. Chapter 2 and 4 are my own independent research works. xi Chapter 1 Introduction 1 Chapter 1. Introduction This dissertation examines two issues in macroeconomics and interna- tional economics. The first issue is whether productivity responds to major real exchange rate appreciations and the second is about how to compare the fits of different calibrations in macroeconomics. The second chapter addresses the former issue. The third chapter develops a method to an- swer the latter and the fourth chapter applies the method to compare two macroeconomic models with financial frictions. The second chapter studies how exchange rate appreciations affect pro- ductivity growth. While there has been a large literature (see Obstfeld and Rogoff (1996)) on the casual effect of productivity on exchange rate, the reverse effect has not been studied, except for the study of Fung (2008). In this chapter, I address this question both theoretically and empirically. In a theoretical model, I adapt the assumption of disruptive technological change of Holmes, Levine and Schmitz (2008) to clarify the effect of increased com- petition due to an exchange rate appreciation. One of the costs for adopt- ing a cost-reducing technology, is the profit loss due to a temporarily high marginal cost of production during the transition. When the exchange rate appreciates, there is less profit to be made and the profit loss due to adopt- ing the new technology is also smaller. However if firms are in an industry shielded by high trade cost, then their profitability will be less influenced by an appreciation and the incentive to improve productivity provided by an ap- preciation will be smaller. The incentive also depends the the concentration of the industry. In industries with fewer firms, profits are more responsive to productivity improvements, therefore they are more likely to invest more in productivity improvement. Testing the predictions with Canadian man- 2 Chapter 1. Introduction ufacturing data from 1997 to 2006, I find that within the group of highly traded Canadian industries, the more concentrated ones experienced larger growth in labour productivity during the period of Canadian dollar appre- ciation. The empirical analysis controls for energy use growth, material use growth, R&D expenditure growth, productivity growth in corresponding US industries, industry fixed effects and year specific effects, and the results are robust to various specifications. The third chapter presents testing procedures for comparison of misspec- ified calibrated models. The proposed tests are of the Vuong-type (Vuong, 1989; Rivers and Vuong, 2002). In the framework here, an econometrician selects values for the parameters in order to match some characteristics of the data with those implied by the theoretical model. It is assumed that all competing models are misspecified, and suggest a test for the null hypothesis that all considered models provide equivalent fit to the data characteristics, against the alternative that one of the models is a better approximation. This chapter considers both nested and non-nested cases. The discussion includes the situation when parameters are estimated to match one set of moments and the model is evaluated by its ability to match another. This chapter also relaxes the dependence of ranking of the models on the choice of weight matrix by suggesting averaged and sup procedures. The proposed method is applied to comparison of cash-in-advance and portfolio adjust- ment cost models. The constructed test statistic indicates that compared to the cash-in-advance model, impulse responses generated by the portfolio adjustment cost model do not provide a better approximation for the output and price dynamics in the data. 3 Chapter 1. Introduction The fourth chapter applies the method developed in the third chapter to examine which of the Carlstrom and Fuerst (1997) model and the Bernanke, Gertler and Gilchrist (1999) model can capture the cyclical properties of net worth better. The models in the papers are two leading models that study fi- nancial frictions in macroeconomic models. In particular, these models show that due to financial frictions, net worth plays an important role in obtaining external finance, and that at an aggregate level, net worth can propagate technology shocks and monetary shocks. Intuitively, a higher level of net worth helps a firm to obtain external finance, as it allows the firm to post more collaterals and to have interests more in line with those of creditors. However, neither paper examines whether the models can reproduce cyclical properties of net worth. This chapter documents the cyclical properties of net worth and show that it is pro-cyclical, and its co-movements with GDP, investment and interest rates seemed to undergo changes around the 1980s. Then, I applies the econometric test developed in chapter 3 to compare the quantitative performances of Carlstrom and Fuerst (1997) model and the Bernanke et al. (1999) model in replicating the cyclical properties of net worth. The results indicate they both do reasonably well. In addition, price rigidity seems to play an important role in the Bernanke et al. (1999) model, as it improves the quantitative performance of the model significantly. How- ever, the models can only partially capture the positive correlation between risk premium and net worth. 4 Bibliography Bernanke, Ben S., Mark Gertler, and Simon Gilchrist, “The finan- cial accelerator in a quantitative business cycle framework,” in J. B. Taylor and M. Woodford, eds., Handbook of Macroeconomics, Vol. 1 of Handbook of Macroeconomics, Elsevier, 1999, chapter 21, pp. 1341–1393. Carlstrom, Charles T and Timothy S Fuerst, “Agency Costs, Net Worth, and Business Fluctuations: A Computable General Equilibrium Analysis,” American Economic Review, December 1997, 87 (5), 893–910. Fung, Loretta, “Large real exchange rate movements, firm dynamics, and productivity growth,” Canadian Journal of Economics, May 2008, 41 (2), 391–424. Holmes, Thomas J., David K. Levine, and James A. Schmitz, “Monopoly and the Incentive to Innovate When Adoption Involves Switchover Disruptions,” NBER Working Paper, 2008, No. W13864. Obstfeld, Maurice and Kenneth S. Rogoff, Foundations of Interna- tional Macroeconomics, Vol. 1 of MIT Press Books, The MIT Press, 1996. Rivers, D. and Q. Vuong, “Model Selection Tests For Nonlinear Dynamic Models,” Econometrics Journal, 2002, 5 (1), 1–39. 5 Chapter 1. Bibliography Vuong, Quang H., “Likelihood Ratio Tests For Model Selection and Non- Nested Hypotheses,” Econometrica, 1989, 57 (2), 307–333. 6 Chapter 2 Does Productivity Respond to Exchange Rate Appreciations? A Theoretical and Empirical Investigation1 1A version of this chapter will be submitted for publication. Tang, Yao, “Does Pro- ductivity Respond to Exchange Rate Appreciations? A Theoretical and Empirical Inves- tigation”. 7 2.1. Introduction 2.1 Introduction Substantial exchange rate movements over the last decade have raised an important question: What are the impacts of a major real exchange rate appreciation on firm performance? Conventional wisdom suggests that such appreciation worsens terms of trade and weakens the competitiveness of home firms. Meanwhile, the possibility remains that to maintain competi- tiveness, firms will be forced to raise productivity by reducing their costs. This chapter addresses theoretically and empirically the question of whether manufacturing productivity responds to real appreciations. First, I con- struct a model in which currency appreciations can provide incentives for firms to improve productivity if they are highly exposed to trade. The model also predicts that among highly traded industries the highly concentrated ones will invest more in productivity improvements since the marginal bene- fits of productivity gain will be greater for firms with a larger market share. Second, I test the predictions empirically by using Canadian manufacturing data from 1997 to 2006. The results confirm that manufacturing productiv- ity growth responded positively to the appreciation of the Canadian dollar between 2002 and 2006. Within industries exposed to a substantial amount of trade, the highly concentrated ones experienced a larger gain in labour productivity during the appreciation period. In addressing the research question, this chapter makes two contribu- tions. Theoretically, it studies whether firms will improve productivity by adopting new technologies to counter the effect of appreciations, and what type of firms will invest more in new technologies. Empirically, the esti- 8 2.1. Introduction mates in this chapter suggest the productivity responses of Canadian man- ufacturing industries to appreciations were positive and significant during the Canadian dollar appreciation between 2002 and 2006. In a neoclassical framework, profit maximization by firms automatically implies cost minimization. However, some economists have long argued that product market competition forces firms to lower costs and thus improve productivity. Nickell (1996) contains a review of earlier contributions along this line of thinking. Some of the theoretical models are based on contract theory, for example Hart (1983) and Raith (2003). Vives (forthcoming) ex- amines a wide variety of industrial organization models, and concludes that, in general, increased competition encourages product and process innova- tions. Holmes, Levine and Schmitz (2008) provide a simple setup to explain the positive relation between competition and adoption of new technology, based on the empirical observation that technology changes are often dis- ruptive in the sense that the transition to higher productivity often features initially higher marginal costs. This chapter adapts the Holmes et al. (2008) assumption of disruptive technological change to clarify the effect of increased competition due to real exchange rate appreciations on productivity. In the model, one of the costs of adopting a cost-reducing technology is profit loss due to a temporarily high marginal cost of production during the transition. When the exchange rate appreciates, there is less profit to be made, and so profit loss due to adopting new technology is also smaller. However, if firms in an industry are shielded by high trade costs, then their profitability is less influenced by appreciations, and the incentive to improve productivity provided by 9 2.1. Introduction appreciations is smaller. Unlike Holmes et al. (2008) and other previous papers which focus on when firms are likely to adopt new technologies to improve productivity, this chapter also studies what types of firms are likely to invest more in pro- ductivity improvement. The model predicts that within the set of industries subject to low trade costs, productivity improvement is positively correlated with the concentration level of the industry. In industries with fewer firms, since the marginal benefits of productivity improvements are greater, firms in these industries are likely to invest more in productivity improvements. There is a number of studies that provide evidence of a positive correla- tion between competition and productivity improvement, with competitive pressure measured as the number of competitors, concentration ratio, trade barriers, or the effect of competition policy. MacDonald (1994) finds that import competition improved productivity in highly concentrated US indus- tries. Nickell (1996) suggests that an increase in the number of competitors was associated with total factor productivity (TFP) gain in a sample of 700 firms in the UK. Symeonidis (2008) exploits the variation arising from the introduction of anti-cartel laws in UK industries, and finds that collusion reduced industry-level productivity growth. Galdon-Sanchez and Schmitz (2002) and Syverson (2004) are two papers that focus on individual indus- tries. The former paper investigates Canadian and American iron ore pro- ducers, who doubled labour productivity, and increased material efficiency by 50% in response to intense price competition from Brazilian firms. The latter paper examines ready-mixed concrete plants in the US, and finds that an increase in local competition led to higher average productivity and lower 10 2.1. Introduction productivity dispersion. As far as I am aware, Fung (2008) is the only previous paper that looks into the effect of major appreciations on productivity. The productivity gain in Fung (2008) came from exit of less efficient firms and bigger production scale of surviving firms after a major appreciation. Relative to her paper, this chapter focuses on the channel that firms’ effort to improve efficiency of production. Controlling for exits and production scale, this chapter finds em- pirical evidence that competitive pressure of appreciations encourage firms to improve productivity. To test the predictions of the theoretical model, I use data on 237 Cana- dian manufacturing industries between 1997 and 2006 to study how industry- level labour productivity growth interacts with exchange rate movements, concentration, and trade costs. Although the sample period is restricted by data availability, the Canadian dollar experienced such substantial move- ments in the period as to allow us to investigate the productivity response to a major appreciation. I find that growth rates of labour productivity, measured as value added per production worker, were on average higher during the Canadian dollar appreciation between 2002 and 2006, suggesting that the manufacturing industries responded to competitive pressure of the appreciation. Within the industries with a high trade-to-revenue ratio, the highly concentrated ones experienced greater growth in labour productivity. The empirical analysis controls for energy use growth, material use growth, R&D expenditure growth, productivity growth in corresponding US indus- tries, industry fixed effects, and GDP growth rates in Canada and the US. These findings also add to the stock of evidence demonstrating the relation 11 2.2. Basic Model Setup between competition and productivity. In economic policy circles, some (e.g. Porter (1990)) suggest that a “hard currency”, meaning a currency less prone to depreciation, can contribute to higher productivity growth. Harris (2001) argues that the Canadian dollar depreciation in the 1990s was partially responsible for the Canadian pro- ductivity decline. By providing both a model linking the exchange rate and productivity, and an empirical analysis, this chapter illustrates a channel through which exchange rate policy and competition policy can affect pro- ductivity. The empirics supply an assessment of the existence and magnitude of the effect of currency appreciations on labour productivity. The next section lays out the modeling environment. Section 3 intro- duces the technological opportunity for home firms to improve productivity, and examines how home firms’ choices interact with an appreciation. Section 4 tests the model predictions on Canadian manufacturing data and section 5 concludes. 2.2 Basic Model Setup There are two countries, the home (h) and the foreign (f), and each has a representative household. The two households have the same given wealth 푊 and consume a continuum of goods indexed by 푖 with 푖 ∈ [0, 1].2 Labour supplies in both countries are perfectly inelastic. The home household’s 2The model is a partial equilibrium one. In Tang (2008), I endogenize the income of the households. Since the purpose of the chapter is to explore the interaction between market concentration and productivity during exchange rate appreciations, assuming that firms take aggregate expenditure 푊 as given is a useful simplification. 12 2.2. Basic Model Setup problem is to maximize 2∑ 푡=1 훽푡−1 ∫ 1 0 푙표푔(퐶푖푡)푑푖 subject to the life-time budget constraint 2∑ 푡=1 훽푡−1 ∫ 1 0 푃푖푡퐶푖푡푑푖 ≤푊 (2.1) 퐶푖푡 denotes the quantity of good 푖 and 푃푖푡 is its price. Similarly the foreign household maximizes 2∑ 푡=1 훽푡−1 ∫ 1 0 푙표푔(퐶∗푖푡)푑푖 subject to the life-time budget constraint 2∑ 푡=1 훽푡−1 ∫ 1 0 푃 ∗푖푡퐶 ∗ 푖푡푑푖 ≤푊 ∗ (2.2) Following the convention in international economics, the superscript ∗ de- notes variables in the foreign country. The household preferences determine the demand functions for good 푖 in both countries 퐶푖푡 = 푊/(1 + 훽) 푃푖푡 (2.3) 퐶∗푖푡 = 푊/(1 + 훽) 푃 ∗푖푡 (2.4) where 푊/(1 + 훽) is normalized to be 1. For each good 푖, there are 푛푖 home firms and 푛푖 foreign firms who can produce it. I will refer to these firms as firms in industry 푖. In both periods, all home firms are endowed with a constant marginal cost of 푐푖ℎ푡 = 푐ℎ unit of labour and the foreign firms are endowed with a constant marginal cost of 푐푖푓푡 = 푐푓 . Thus in the model, home and foreign labour productivities 13 2.2. Basic Model Setup in any industry are 1푐ℎ and 1 푐푓 . Labour is the only input and is not mobile across countries. Every good is tradable, subject to an iceberg trade cost 휏푖 for good 푖, meaning that for each 휏푖 unit of good 푖 shipped to the other country only one unit will arrive. 휏푖 and 푛푖 are drawn from the joint CDF 퐹 (휏, 푛) with support [1,∞)× [1, 2, ⋅ ⋅ ⋅ , 푛]3. The market structure within each industry is similar to that found in Brander and Krugman (1983). The home firms and foreign firms of industry 푖 produce using labour in their respective countries. However, they are free to sell their production in both countries. For a given period, the home and foreign firms of industry 푖 play a Cournot game in the home market to determine the quantities of good 푖 produced by each firm for the home market. Simultaneously, the same firms also compete in a Cournot game in the foreign market. As mentioned before, in all periods both the home and foreign firm face an iceberg trade cost 휏푖 when they sell in the non-native market. Figure 2.1 illustrates the market structure. 3In this model, the number of firms in an industry is exogenously given. This treatment can be viewed as a simplification of the case where firms can enter and exit an industry freely and the number of firms in equilibrium is determined by the exogenous fixed cost of entry. 14 2.2. Basic Model Setup Home firm 1 in industry 푖 Home household 0 1 0 1 Foreign household 퐶푖푡 = 푥 1 푖ℎ푡 + 푥 2 푖ℎ푡 + 푥 1 푖푓푡 + 푥 2 푖푓푡 퐶∗푖푡 = 푥 1∗ 푖ℎ푡 + 푥 2∗ 푖ℎ푡 + 푥 1∗ 푖푓푡 + 푥 2∗ 푖푓푡 퐶푖푡 퐶∗푖푡 푥1푖ℎ푡 푥1∗푖ℎ푡 Home industry Foreign industry b b b b Foreign firm 1 in industry 푖 푥1푖푓푡 푥1∗푖푓푡 b b b b Three firms in industry 푖′ There are two countries and both produce the same continuum of consumption goods indexed by 푖 with 푖 ∈ [0, 1] at time periods 푡 = 1, 2. Every good is tradable subject to an iceberg trade cost 휏푖 for good 푖. In industry 푖 there are 푛푖 home firms and 푛푖 foreign firms who produce good 푖. The home firms and foreign firms of industry 푖 produce with labour in their respective country, and sell their production in both countries. In each period, the home and foreign firms of industry 푖 play a Cournot game in the home market to determine the quantities of good 푖 output. Similarly the same firms also compete in a Cournot game in the foreign market. Figure 2.1: An Illustration of the Industrial Organization 15 2.2. Basic Model Setup The problem4 of home Firm 푗 of industry 푖 is max 푥푗 푖ℎ1,푥 푗∗ 푖ℎ1,푥 푗 푖ℎ2,푥 푗∗ 푖ℎ2 Π푗푖ℎ = 휋 푗 푖ℎ1 + 푒1휋 푗∗ 푖ℎ1 + 훽(휋 푗 푖ℎ2 + 푒2휋 푗∗ 푖ℎ2) (2.5) where 푥푗푖ℎ1 and 푥 푗∗ 푖ℎ1 are the quantities it produces for home and foreign markets in period 1, and 푥푗푖ℎ2 and 푥 푗∗ 푖ℎ2 are the quantities for home and foreign markets in period 2. 휋푗푖ℎ1 and 휋 푗 푖ℎ2 are profits from the home market in periods 1 and 2. 휋푗∗푖ℎ1 and 휋 푗∗ 푖ℎ2 are profits from the foreign market, measured in the foreign currency. 푒1 and 푒2 are the exchange rates in the two periods. They are defined as the price of one unit of foreign currency in terms of home currency, so a decrease in 푒푡 is an appreciation of the home currency. Nominal money supplies are constant for both periods in both countries, so if we define a country’s real balance as the nominal money supply divided by the wage rate and normalize wages to 1, 푒푡 is also the real exchange rate. The exchange rates are determined exogenously and known to all firms at the beginning of period 1. At the beginning of period 1 all firms observe each other’s marginal costs for all times. Then all firms in industry 푖 play a game to determine quan- 4I assume firms will discount future at the rate of time preference of the household, who is also the owner of the firms. In reality, firms may differ in the discount factor. For firms who place little value on future, there is very little incentive for them to adopt a technology that will bring a future benefit, holding other factors constant. The objective function also features no expectation operator, as I assume firms have perfect foresight of future. While expectation plays an important role in decision, I choose to suppress it here so as to focus discussion on how exchange rate lowers opportunity cost of adopting new technology. On empirical section, it is argued that firms in Canada have a good idea about the path of exchange rate since appreciations tend to be persistent and commodity prices are a good forecaster of exchange rate of the Canadian dollar. 16 2.2. Basic Model Setup tities of output in the four markets (home and foreign markets in period 1 and 2). The strategy of home firm 푗 in industry 푖 is the set of quantities{ 푥푗푖ℎ1, 푥 푗∗ 푖ℎ1, 푥 푗 푖ℎ2, 푥 푗∗ 푖ℎ2 } , and the strategy of foreign firm 푗 in industry 푖 is the set of quantities { 푥푗푖푓1, 푥 푗∗ 푖푓1, 푥 푗 푖푓2, 푥 푗∗ 푖푓2 } . There are four subgames, one for each market in each period. I focus on the subgame perfect equilibrium, in which firms in industry 푖 of each country play symmetric strategies. Since firms have to determine simultaneously the quantities in both markets in a period, the two subgames in period 2 are independent. In period 2, firms have to play a Nash equilibrium in the subgames. By the standard back- ward induction principle, they will also have to play a Nash equilibrium in the subgames in period 1. Thus all four subgames are independent, so the subgame perfect equilibrium involves firms playing the symmetric Nash equilibrium in each subgame. The output quantities in each subgame are determined as the symmetric Nash equilibrium quantities in that subgame. We can calculate in the maximized total profit as the sum of maximized profits from each subgame. Normalizing home wage to be 1, the profit of the home firm 푗 of industry 푖 in the home market at time 푡 is 휋푗푖ℎ푡 = (푃푖푡 − 푐ℎ)푥푗푖ℎ푡 = ( 1∑푛푖 푘=1 푥 푘 푖ℎ푡 + ∑푛푖 푘=1 푥 푘 푖푓푡 − 푐ℎ)푥푗푖ℎ푡 (2.6) where 푥푘푖ℎ푡 and 푥 푘 푖푓푡 are the quantities of good 푖 produced by home firm 푘 and foreign firm 푘 for the home market. The last equality follows from (2.3) and the market clearing condition 퐶푖푡 = ∑푛푖 푘=1 푥 푘 푖ℎ푡 + ∑푛푖 푗=1 푥 푘 푖푓푡. When the home firm 푗 chooses 푥푗푖ℎ푡 to maximize (2.6), the first order condition is∑ 푘 ∕=푗 푥 푘 푖ℎ푡 + ∑푛푖 푘=1 푥 푘 푖푓푡 ( ∑푛푖 푘=1 푥 푘 푖ℎ푡 + ∑푛푖 푘=1 푥 푘 푖푓푡) 2 − 푐ℎ ≤ 0 (2.7) 17 2.2. Basic Model Setup Similarly the profit of foreign firm 푗 of industry 푖 in the home market at time 푡 is 휋푗푖푓푡 = (푃푖푡 − 푒푡휏푖푐푓 )푥푗푖푓푡 = ( 1∑푛푖 푘=1 푥 푘 푖ℎ푡 + ∑푛푖 푘=1 푥 푘 푖푓푡 − 푒푡휏푖푐푓 )푥푗푖푓푡 (2.8) When the foreign firm 푗 chooses 푥푗푖푓푡 to maximize (2.8), the first order con- dition is ∑푛푖 푘=1 푥 푘 푖ℎ푡 + ∑ 푘 ∕=푗 푥 푘 푖푓푡 ( ∑푛푖 푘=1 푥 푘 푖ℎ푡 + ∑푛푖 푘=1 푥 푘 푖푓푡) 2 − 푒푡휏푖푐푓 ≤ 0. (2.9) (2.7) and (2.9) implicitly define the best responses functions of the home 푗 and foreign firm 푗 to quantities produced by other firms. Combining (2.7) and (2.9) and imposing symmetry among all home firms and symmetry among all home firms, we have the equilibrium relation between outputs of home and foreign firms 푥푗푖푓푡 = 푛푖푐ℎ − (푛푖 − 1)푒푡휏푖푐푓 푛푖푒푡휏푖푐푓 − (푛푖 − 1)푐ℎ푥 푗 푖ℎ푡 = 훼1(푡, 푖)푥 푗 푖ℎ푡 (2.10) where 훼1(푡, 푖) = 푛푖푐ℎ−(푛푖−1)푒푡휏푖푐푓 푛푖푒푡휏푖푐푓−(푛푖−1)푐ℎ . A careful examination of (2.7) suggests that if 푐ℎ is large, then the home firms will produce zero quantities, and foreign firms will produce large quantities. This is because foreign firms know that, given the quantities they produced, home firms’ the marginal revenue in the home market (the first term in (2.7)) is always less than the marginal cost for all 푥푗푖ℎ푡 ≥ 0 and home firms will optimally choose zero. In this case, the denominator of 훼1 will be negative and (2.10) will no longer describe the relation between home and foreign quantities of output. Similarly when 푒푡휏푖푐푓 is large, foreign firms will produce zero quantities, and the numerator of 훼1(푡, 푖) will be negative. It can be shown that the necessary 18 2.2. Basic Model Setup conditions for both home and foreign firms to produce positive quantities in the home market is that both numerator and denominator of 훼1(푡, 푖) be positive. These conditions can be expressed as 휏푖 > 푛푖 − 1 푛푖 1 푒푡 푐ℎ 푐푓 휏푖 < 푛푖 푛푖 − 1 1 푒푡 푐ℎ 푐푓 (2.11) If (2.11) is satisfied, we can substitute the last expression into (2.7) and (2.9) and solve for 푥푗푖ℎ푡 and 푥 푗 푖푓푡 푥푗푖ℎ푡 = 푛푖 − 1 + 푛푖훼1(푡, 푖) (푛푖 + 푛푖훼1(푡, 푖))2푐ℎ (2.12) 푥푗푖푓푡 = 푛푖 − 1 + 푛푖/훼1(푡, 푖) (푛푖 + 푛푖/훼1(푡, 푖))2푒푡휏푖푐푓 (2.13) In particular if 푛푖 = 1 the solution is 푥푗푖ℎ푡 = 1 푒푡휏푖푐푓 (1 + 푐ℎ 푒푡휏푖푐푓 )2 (2.14) 푥푗푖푓푡 = 1 푐ℎ(1 + 푒푡휏푖푐푓 푐ℎ )2 (2.15) If we substitute (2.12) and (2.13) into (2.6) and (2.8), we have 휋푗푖ℎ푡 = 1 (푛푖 + 푛푖훼1(푡, 푖))2 (2.16) 휋푗푖푓푡 = 1 (푛푖 + 푛푖/훼1(푡, 푖))2 (2.17) Thus for industry 푖 we have a unique symmetric equilibrium in the home market under (2.11). Similarly the home firm’s and foreign firm’s profit functions in the foreign 19 2.2. Basic Model Setup market, denoted in foreign currency, are 휋∗푖ℎ푡 = (푃 ∗ 푖푡 − 휏푖푐ℎ 푒푡 )푥푗∗푖ℎ푡 = ( 1∑푛푖 푘=1 푥 푘∗ 푖ℎ푡 + ∑푛푖 푘=1 푥 푘∗ 푖푓푡 − 휏푖푐ℎ 푒푡 )푥푗∗푖ℎ푡 휋푗∗푖푓푡 = (푃 ∗ 푖푡 − 푐푓 )푥푗∗푖푓푡 = ( 1∑푛푖 푘=1 푥 푘∗ 푖ℎ푡 + ∑푛푖 푘=1 푥 푘∗ 푖푓푡 − 푐푓 )푥푗∗푖푓푡 In a symmetric equilibrium in which firms of both countries produce positive quantities, the equilibrium output and profits are given by 푥푗∗푖ℎ푡 = 푛푖 − 1 + 푛푖훼2(푡, 푖) (푛푖 + 푛푖훼2(푡, 푖))2푐ℎ (2.18) 푥∗푖푓푡 = 푛푖 − 1 + 푛푖/훼2(푡, 푖) (푛푖 + 푛푖/훼2(푡, 푖))2푒푡휏푖푐푓 (2.19) 휋∗푖ℎ푡 = 1 (푛푖 + 푛푖훼2(푡, 푖))2 (2.20) 휋∗푖푓푡 = 1 (푛푖 + 푛푖/훼2(푡, 푖))2 (2.21) where 훼2(푡, 푖) = 푛푖휏푖푐ℎ−(푛푖−1)푒푡푐푓 푛푖푒푡푐푓−(푛푖−1)휏푖푐ℎ . The necessary condition for both home and foreign firms to produce positive quantities in the home market is 휏푖 > 푛푖 − 1 푛푖 푒푡 푐푓 푐ℎ 휏푖 < 푛푖 푛푖 − 1푒푡 푐푓 푐ℎ (2.22) Given 푐ℎ, 푐푓 and 푒푡, (2.11) and (2.22) imply that in industries in the set Θ(푒푡) = {(푛푖, 휏푖) ∈ [1,∞)× [1, 2, ⋅ ⋅ ⋅ , 푛] : 휏푖 > 푛푖 − 1 푛푖 1 푒푡 푐ℎ 푐푓 , 휏푖 < 푛푖 푛푖 − 1 1 푒푡 푐ℎ 푐푓 , 휏푖 > 푛푖 − 1 푛푖 푒푡 푐푓 푐ℎ , 휏푖 < 푛푖 푛푖 − 1푒푡 푐푓 푐ℎ } , (2.23) both home and foreign firms will produce positive quantities in both markets at time 푡.5 For these industries, total profits for home and foreign firms are 5I use the notation Θ(푒푡) to emphasize the set depends on 푒1. 20 2.2. Basic Model Setup given by Π푗푖ℎ = 1 (푛푖 + 푛푖훼1(푡 = 1, 푖))2 + 푒1 (푛푖 + 푛푖훼2(푡 = 1, 푖))2 + 1 (푛푖 + 푛푖훼1(푡 = 2, 푖))2 + 푒2 (푛푖 + 푛푖훼2(푡 = 2, 푖))2 Π푗푖푓 = 1 푒1(푛푖 + 푛푖/훼1(푡 = 1, 푖))2 + 1 (푛푖 + 푛푖/훼2(푡 = 1, 푖))2 + 1 푒2(푛푖 + 푛푖/훼1(푡 = 2, 푖))2 + 1 (푛푖 + 푛푖/훼2(푡 = 2, 푖))2 . (2.24) Proposition 1. (a) For industries in the set Θ(푒푡), the period 푡 profit of home firm 푗 in industry 푖 is a decreasing function of 푐ℎ and an increasing function of exchange rate 푒푡. (b) For industries with the same 휏푖 in the set Θ(푒푡), the period 푡 profit for home firms 푗 is decreasing in 푛푖. (c) For industries with the same 휏푖 and in which only home firms are producing positive quantities, the period 푡 profit for home firms 푗 is decreasing in 푛푖. Proof: (a) From (2.16) and (2.21), we can see the period 푡 profit of home firm 푗 in industry 푖 is decreasing in 훼1 and 훼2. Since both 훼1 and 훼2 are increasing in 푐ℎ and decreasing in 푒푡, the conclusion follows. (b) For industries in Θ(푒푡), the period 푡 profit for home firm 푗 in the home market is given by 휋푗푖ℎ푡 = 1 (푛푖 + 푛푖훼1(푡, 푖))2 (2.16) If 푐ℎ > 푒푡휏푖푐푓 , then 훼1(푡, 푖) = 푛푖푐ℎ−(푛푖−1)푒푡휏푖푐푓 푛푖푒푡휏푖푐푓−(푛푖−1)푐ℎ = 푐ℎ−(푛푖−1)(푐ℎ−푒푡휏푖푐푓 ) 푒푡휏푖푐푓−(푛푖−1)(푒푡휏푖푐푓−푐ℎ) is increasing in 푛푖. Thus 휋 푗 푖ℎ푡 is decreasing in 푛푖. If 푐ℎ = 푒푡휏푖푐푓 , then 훼1(푡, 푖) = 1 so 휋푗푖ℎ푡 = 1 (푛푖+푛푖훼1(푡,푖))2 is decreasing in 푛푖. Lastly, when 푐ℎ < 푒푡휏푖푐푓 , we 21 2.3. Exchange Appreciation and Investment Decision can prove 휋푗푖ℎ푡 is decreasing in 푛푖 by showing the derivative of the numerator of (2.16) with respect to 푛푖 is positive. ∂ ∂푛푖 (푛푖 + 푛푖훼1(푡, 푖)) 2 = ∂ ∂푛푖 (푛푖 + 푛푖 푐ℎ 푒푡휏푖푐푓−푐ℎ − 푛2푖 + 푛푖 푒푡휏푖푐푓 푒푡휏푖푐푓−푐ℎ + 푛푖 − 1 )2 = 2 푐ℎ(푒푡휏푖푐푓+푐ℎ) (푒푡휏푖푐푓−푐ℎ)2 ( 푒푡휏푖푐푓 푒푡휏푖푐푓−푐ℎ + 푛푖 − 1)2 > 0 Therefore, we have shown that 휋푗푖ℎ푡 is always decreasing in 푛푖. Similarly, we can show the period 푡 profit of home firm 푗 in the foreign market is decreasing in 푛푖. (c) For industries in which only home firms are producing positive quan- tities for both markets, it is easy to verify that the period 푡 profit for firm 푗 is 푛푖 − 1 푛2푖 + 푒푡 푛푖 − 1 푛2푖 , (2.25) decreasing in 푛푖. ■ The proposition confirms the intuition that an appreciation of home currency erodes the profit of home firms and validates the usual Cournot competition result that profit dissipates with the number of firms. 2.3 Exchange Appreciation and Investment Decision In this section I introduce the possibility of cost-saving technology. The term technology is defined as in Jones (2001), as ways to transform factors into 22 2.3. Exchange Appreciation and Investment Decision output. In general, they can be product innovations, but in this chapter I refer to a cost-saving process innovation. For example the innovation could be an improvement in labour practice as emphasized in Baily, Gersbach, Scherer and Lichtenberg (1995), and Schmitz (2005). To simplify the problem, I assume all home and foreign firms in each industry are endowed with the same cost, 푐ℎ = 푐 = 푐푓 for both periods. All home firms have access to technology that reduces the second-period marginal cost from 푐ℎ to 1 휎 푐ℎ, where 휎 is the improvement in labour produc- tivity. However the technology is also disruptive in the sense that, if a firm chooses 휎 > 1, it raises the first period marginal cost from 푐ℎ to 훾푐ℎ, where 훾 is a constant greater than 1.6 Since adoption at time 푡 will raise the cost at that period, no firm would adopt the innovation at 푡 = 2. Proposition 1 implies the technology will bring higher profit in the second period but entail a loss of profit in the first. Firms can choose 휎 in the range [1, 휎) but will have to pay a fixed cost 퐼(휎). I assume 퐼(휎) is strictly convex in 휎 for all 1 < 휎 < 휎, 퐼(휎 = 1) = 0, lim휎→1 퐼(휎) > 0, and lim휎→휎 퐼(휎) =∞. 7 I assume that no foreign firms have the option to upgrade their technol- ogy. The assumption is made to simplify the interaction between home and foreign firms regarding the choice of 휎, which would vary across industries. 6It is possible that firms could improve productivity by adopting other new technologies that are not disruptive and are always profitable to implement. I choose not to model such technology opportunities as they would not interact with exchange rate movements. In the empirical section of the chapter, I will try to account for this possibility. 7In general 훾 can be increasing in 휎, however, since the assumptions regarding 퐼(휎) ensure that the first-period cost of adoption (which equals 퐼(휎) plus the profit loss due to a high marginal cost 훾푐ℎ) is increasing in 휎, I do not pursue this complication. 23 2.3. Exchange Appreciation and Investment Decision In Tang (2008), I show that if firms can only choose between the status quo (sq), i.e. 휎 = 1, and some fixed 휎 > 1, then the unique equilibrium is for the home firms to adopt and foreign firms to keep the status quo when there is a large appreciation. 8 My assumption regarding new technology follows that of Holmes et al. (2008), which suggests that technology change is disruptive in the sense that there is a costly transition to lower cost of production. Holmes et al. (2008) motivate this assumption by citing a large number of empirical observations. For illustrative purposes consider the following scenario. The implementa- tion of new technology requires a fixed investment in the training of employ- ees and during the transition, as a result workers are less productive as they are learning to master the new technology. As mentioned in the introduc- tion, Vives (forthcoming) studies a wide variety of industrial organization models and concludes that in general more competition induces a bigger effort to improve productivity. Holmes et al. (2008) obtain similar predic- tions with the empirically motivated assumption of disruptive technology changes. I follow their assumption to maintain model tractability. It is clear from the nature of the technology that the tradeoff between current costs and future gain is crucial for adoption choices. A two-period 8In the setting in which both home and foreign firms can choose 휎 from [1,∞), it is very difficult to predict the equilibrium outcome in an technology adoption game. It is possible to show home firms’ incentive to adopt increases with an appreciation given the choice of foreign firms, and foreign firms’ incentive decreases with an appreciation given the choice of home firms. Since it appears that foreign firms’ incentive to improve productivity is weaker with an appreciation, I assume the extreme case that foreign firms simply cannot upgrade and focus on how the choices of home firms vary with industry characteristics. 24 2.3. Exchange Appreciation and Investment Decision world is the minimum structure that allows us to study the tradeoff between the present and the future. Adding more periods simply requires one to re- place second-period profits in firms’ objective functions with value functions. Both a second-period profit function and a value function should be increas- ing in productivity and there will be a future gain. Since the focus of this chapter is on how first-period loss interacts with exchange rate movements, a two-period model is sufficient. Since the two countries are symmetric, it is reasonable to conjecture that at steady state exchange rate 푒푡 = 1, 9 will hold in both periods. The timing of the game in industry 푖 is the following: ∙ Stage 0, an exogenous shock to exchange rate is realized, firms have perfect foresight that 푒1 < 1 and 푒2 = 1 10; ∙ Stage 1, home firm 푗 determines its choices of 휎푗 and pay 퐼(휎푗), for 푗 = 1, 2, ⋅ ⋅ ⋅ , 푛푖; ∙ Stage 2, the choices of home firms in stage 1 are observed by all (so every firm knows the marginal cost of each firm in both periods), and firms play the Cournot game as described in section 2 to determine outputs in each of the four markets (home and foreign markets in period 1 and 2). 9In Tang (2008), I close the model and derive the equilibrium exchange rate as a function of firm productivities and shock to currency demand. In a steady state in which the productivities are equal across countries and currency demand shocks equal zero, the equilibrium exchange rate is 1. 10Or I can assume 푒2 equals any other constant value commonly expected or known. This change would only scale the second period profit gain of adopting 휎 > 1. 25 2.3. Exchange Appreciation and Investment Decision The game is solved by standard backward induction. In stage 2, given{ 휎1, 휎2, ⋅ ⋅ ⋅ , 휎푛푖} firms play the Cournot game described in section 2 and the payoffs are as derived in section 2. In stage 1, given how the equi- librium profit depends on { 휎1, 휎2, ⋅ ⋅ ⋅ , 휎푛푖}, home firm 푗 chooses 휎푗 , for 푗 = 1, 2, ⋅ ⋅ ⋅ , 푛푖. Again, I will focus on a symmetric equilibrium between home firms in stage 1. In stage 2, I focus on the choices of 휎 for industries in which firms of both countries produce positive quantities in all markets, except that home firms may be forced out of the foreign market during the period 1 appreciation. If all home firms in industry 푖 choose the same 휎 > 1, and if firms of both countries are producing positive quantities then the total profit of the home firm 푗 before paying 퐼(휎) is Π푗푖ℎ(휎) = 1 (푛푖 + 푛푖 푛푖훾−(푛푖−1)푒1휏푖 푛푖푒1휏푖−(푛푖−1)훾 )2 + 푒1 (푛푖 + 푛푖 푛푖휏푖훾−(푛푖−1)푒1 푛푖푒1−(푛푖−1)휏푖훾 )2 ⋅ 1(푒1 > 푛− 1 푛 휏푖훾) + 훽 (푛푖 + 푛푖 푛푖/휎−(푛푖−1)푒2휏푖 푛푖푒2휏푖−(푛푖−1)/휎 )2 + 훽푒2 (푛푖 + 푛푖 푛푖휏푖/휎−(푛푖−1)푒2 푛푖푒2−(푛푖−1)휏푖/휎 )2 (2.26) where 1(푒1 > 푛−1 푛 휏푖훾) is an indicator function. When 푒1 > 푛−1 푛 휏푖훾 fails, the home firms are driven out of the foreign market, and make zero profit. If all home firms choose status quo (sq), i.e. 휎 = 1, the total profit is Π푗푖ℎ(푠푞) = 1 (푛푖 + 푛푖 푛푖−(푛푖−1)푒1휏푖 푛푖푒1휏푖−(푛푖−1) )2 + 푒1 (푛푖 + 푛푖 푛푖휏푖−(푛푖−1)푒1 푛푖푒1−(푛푖−1)휏푖 )2 ⋅ 1(푒1 > 푛− 1 푛 휏푖) + 훽 (푛푖 + 푛푖 푛푖−(푛푖−1)푒2휏푖 푛푖푒2휏푖−(푛푖−1) )2 + 훽푒2 (푛푖 + 푛푖 푛푖휏푖−(푛푖−1)푒2 푛푖푒2−(푛푖−1)휏푖 )2 I refer to the difference Π푗푖ℎ(휎)−Π푗푖ℎ(푠푞) as the benefit of adopting the dis- ruptive technology. Choosing some 휎 > 1 dominates 휎 = 1, if the associated 26 2.3. Exchange Appreciation and Investment Decision benefit is greater than the cost 퐼(휎). The benefit has two components, the profit loss in the first period ∣퐿1∣ = 1 (푛푖 + 푛푖 푛푖−(푛푖−1)푒1휏푖 푛푖푒1휏푖−(푛푖−1) )2 + 푒1 (푛푖 + 푛푖 푛푖휏푖−(푛푖−1)푒1 푛푖푒1−(푛푖−1)휏푖 )2 − ( 1 (푛푖 + 푛푖 푛푖−(푛푖−1)푒1휏푖 푛푖푒1휏푖−(푛푖−1) )2 + 푒1 (푛푖 + 푛푖 푛푖휏푖−(푛푖−1)푒1 푛푖푒1−(푛푖−1)휏푖 )2 ) (2.27) and the profit gain in the second 퐺2 = 훽 (푛푖 + 푛푖 푛푖/휎−(푛푖−1)푒2휏푖 푛푖푒2휏푖−(푛푖−1)/휎 )2 + 훽푒2 (푛푖 + 푛푖 푛푖휏푖/휎−(푛푖−1)푒2 푛푖푒2−(푛푖−1)휏푖/휎 )2 − ( 훽 (푛푖 + 푛푖 푛푖−(푛푖−1)푒2휏푖 푛푖푒2휏푖−(푛푖−1) )2 + 훽푒2 (푛푖 + 푛푖 푛푖휏푖−(푛푖−1)푒2 푛푖푒2−(푛푖−1)휏푖 )2 ) (2.28) Similar to (2.23), given 푒1 < 1 and 푒2 = 1, we can formally define the set of industries with {푛푖, 휏푖, 휎푖} such that firms of both countries produce positive quantities in all markets, except that home firms may produce zero for the foreign market during the period 1 appreciation, as Θ휎(푒1) = {(푛푖, 휏푖, 휎푖) ∈ [1,∞)× [1, 2, ⋅ ⋅ ⋅ , 푛]× 휎푖 ∈ [1, 휎) : 휏푖 < 푛푖 (푛푖 − 1)휎푖 , 휏푖 > (푛푖 − 1)훾 푛푖푒1 , 휏푖 > (푛푖 − 1)휎푖 푛푖 } (2.29) To make it possible for the adoption decision problem to interact with the exchange rate, I assume ∙ (i) For industries in Θ휎(푒1 = 1), Π푗푖ℎ(휎) − Π푗푖ℎ(푠푞) < 퐼(휎) for all 휎 ∈ (1, 휎); ∙ (ii) If 휏푖 = 1, for all 푛푖 ∈ [1, 2, ⋅ ⋅ ⋅ , 푛] we can find an interval Σ푛푖 ⊂ (1, 휎) such that the second-period profit gain of firms in industry 푖 is strictly greater than the cost 퐼(휎) for all 휎 ∈ Σ푛푖 . 27 2.3. Exchange Appreciation and Investment Decision Assumption (i) implies it is not profitable to choose any 휎 > 1 with 푒1 = 1, and assumption (ii) says that if the first-period profit loss is zero, it will be profitable for home firms of industry 푖 to adopt 휎 ∈ Σ푛푖 . The following two propositions show how benefits in adopting disrup- tive new technologies are affected by 푒1 and 휏푖. Firstly given 휏푖 and 푛푖, an exchange appreciation lowers the first period profit loss, so choosing some 휎 > 1 can be profitable. Secondly, given 푒1 and 푛푖, a large trade cost 휏푖 insu- lates home firms from trade and the influence of exchange rate movements. Home firms will have no incentive to choose 휎 > 1, even if they experience an appreciation. Proposition 2. Consider industries in Θ휎(푒1). Given 푛푖, and 휏푖 close enough to 1, for all 휎 ∈ Σ푛푖 there exists an exchange rate threshold such that it is profitable to adopt 휎 for home firms for all 푒1 below the threshold. Proof: The absolute value of the first-period profit loss due to adoption (2.27) is bounded by the first-period profit in the status quo 1 (푛푖 + 푛푖 푛푖−(푛푖−1)푒1휏푖 푛푖푒1휏푖−(푛푖−1) )2 + 푒1 (푛푖 + 푛푖 푛푖휏푖−(푛푖−1)푒1 푛푖푒1−(푛푖−1)휏푖 )2 ⋅ 1(푒1 > 푛− 1 푛 휏푖). As 푒1 tends to 푛푖−1 휏푖푛푖 from above, the first-period profit will tend to zero and so will the first-period profit loss due to adoption. By assumption (ii), for industries with 휏푖, the benefit of adopting 휎 > 1 is greater than the cost for all 휎 ∈ Σ푛푖 . Since the profit functions are continuous in 휏푖, by assumption (ii) for 휏푖 close enough to 1, the second-period profit gain of firms in industry 푖 will be strictly greater than 퐼(휎) for all 휎 ∈ Σ푛푖 . Therefore for each 28 2.3. Exchange Appreciation and Investment Decision 휎 ∈ Σ푛푖 we can find an 푒1 such that for all 푒1 < 푒1, the first period loss ∣퐿1∣ < 퐺2 − 퐼(휎). Thus for all 푒1 < 푒1, adopting 휎 ∈ Σ푛푖 is profitable since Π푗푖 (휎)−Π푗푖 (푠푞) = 퐺2 − ∣퐿1∣ > 퐼(휎).■ Proposition 3. Given an 푒1 < 1, there exists a threshold 휏̂ such that adopt- ing the technology of any level 휎 will not not profitable for all firms in any industry with 휏푖 ≥ 휏̂ . Proof: Consider an industry with 푛 firms. There are two possibilities. Firstly, given 푒1 adopting any 휎 ∈ (1, 휎) will not be profitable for all 휏 ∈ [1,∞). In this case, set the threshold to be 휏̂푛 = 1. Secondly, given 푒1, adopting some 휎 ∈ (1, 휎) will be profitable for some 휏 ∈ [1,∞). If a new technology of level 휎 is not profitable for all 휏푖, set the threshold for the level 휎 to be 휏̂푛(휎) = 1. Otherwise, the new technology of level 휎 will be profitable for some level of 휏 . Note as 휏 → 푛푖휎푛푖−1 , home firms operate almost only in the home market. The limit of firm 푗’s gain (which equals benefit minus cost) from adopting the new technology of level 휎 is lim 휏→ 푛푖휎 푛푖−1 Π푗푖ℎ(휎)−Π푗푖ℎ(푠푞)− 퐼(휎) = −퐼(휎), Let 휏̂푛(휎) = 푛푖휎 푛푖−1 . Therefore, all firms in all 푛-firm industries with 휏푖 ≥ 휏̂푛(휎) will not adopt the technology of level 휎. The threshold for the 푛-firm industries is 휏̂푛 = sup {휏̂푛(휎) : 휎 ∈ (1, 휎)}. To find the trade cost threshold for all possible 푛, we take 휏̂ = max {휏̂푛 : 푛 = 1, 2, ⋅ ⋅ ⋅ , 푛} and the conclusion follows. ■ 29 2.3. Exchange Appreciation and Investment Decision The consequence of Proposition 3 is that given an appreciation of a certain magnitude, 휏̂ will partition firms into two sets. The first set of industries with low 휏푖 may choose a new technology of level 휎 > 1 and the second set of firms will not.11 The remaining part of the section examines how home firms choose 휎. We will see that if the first set contains industries with the same trade cost but the different 푛푖, then those with low 푛푖 are likely to choose a large 휎. In stage 2 of the game, the first-period profit is not dependent on the choice of 휎, and the equilibrium quantities and profits are similar to section 2. The second-period profits for home firm 푗 and foreign firm 푗 in the home market which depend on 휎 are 휋푗푖ℎ2 = ( 1∑푛푖 푘=1 푥 푘 푖ℎ2 + ∑푛푖 푘=1 푥 푘 푖푓2 − 푐ℎ 휎푗 ) 푥푗푖ℎ2 휋푗푖푓2 = ( 1∑푛푖 푘=1 푥 푘 푖ℎ2 + ∑푛푖 푘=1 푥 푘 푖푓2 − 휏푖푐푓 푒2 ) 푥푗푖푓2 and the first order conditions are∑ 푘 ∕=푗 푥 푘 푖ℎ2 + ∑푛푖 푘=1 푥 푘 푖푓2 ( ∑푛푖 푘=1 푥 푘 푖ℎ2 + ∑푛푖 푘=1 푥 푘 푖푓2) 2 − 푐ℎ 휎푗 ≤ 0∑푛푖 푘=1 푥 푘 푖ℎ2 + ∑ 푘 ∕=푗 푥 푘 푖푓2 ( ∑푛푖 푘=1 푥 푘 푖ℎ2 + ∑푛푖 푘=1 푥 푘 푖푓2) 2 − 휏푖푐푓 푒2 ≤ 0 (2.30) The first order conditions implicitly define the optimal output 푥푗푖ℎ2 as a function of 휎⃗ = [휎1, 휎2, ⋅ ⋅ ⋅ , 휎푛푖 ]. Denote it as 푥푗푖ℎ2(휎⃗). Similarly we define the optimal output function in the foreign market as 푥푗∗푖ℎ2(휎⃗) In stage 1, home firm 푗 foresees the equilibrium output functions in the 11The relation between 휏 and adoption choice is not monotonic, as some firms with 휏푖 < 휏̂ may choose not to adopt. 30 2.3. Exchange Appreciation and Investment Decision second stage and chooses 휎푗 to maximize total profit Π푗푖ℎ(휎 푗)− 퐼(휎푗) =휋푗푖ℎ1 + 푒1휋 푗∗ 푖ℎ1 + 훽휋 푗 푖ℎ2(휎 푗) + 훽푒2휋 푗∗ 푖ℎ2(휎 푗)− 퐼(휎푗) =휋푗푖ℎ1 + 푒1휋 푗∗ 푖ℎ1 − 퐼(휎푗) + 훽 ( 1∑푛푖 푘=1 푥 푘 푖ℎ2(휎⃗) + ∑푛푖 푘=1 푥 푘 푖푓2(휎⃗) − 푐ℎ 휎푗 ) 푥푗푖ℎ2(휎⃗) + 훽푒2 ( 1∑푛푖 푘=1 푥 푘∗ 푖ℎ2(휎⃗) + ∑푛푖 푘=1 푥 푘∗ 푖푓2(휎⃗) − 휏푖푐ℎ 푒2휎푗 ) 푥푗∗푖ℎ2(휎⃗) By the Envelop Theorem, the first order condition for an interior solution is ∂ ∂휎푗 Π푗푖ℎ(휎 푗) = 퐼 ′(휎푗) ⇒ 훽 푐ℎ (휎푗)2 [ 푥푗푖ℎ2(휎⃗) + 휏푖푥 푗∗ 푖ℎ2(휎⃗) ] = 퐼 ′(휎푗) (2.31) Imposing symmetry among home firms’ choices of 휎, we have 푥푗푖ℎ2(휎⃗) = 푥푘푖ℎ2(휎⃗) for all 푘. Using this knowledge to simplify the (2.30), we have 푥푗푖ℎ2 = ( 1 푛푖(1 + 훼1) − 1 푛2푖 (1 + 훼1) 2 ) 휎푗 푐ℎ 푥푗푖ℎ2 = ( 1 푛푖(1 + 훼2) − 1 푛2푖 (1 + 훼2) 2 ) 휎푗 푐ℎ 푥푗푖푓2 = ( 1 푛푖(1 + 1/훼1) − 1 푛2푖 (1 + 1/훼1) 2 ) 1 휏푖푐푓 푥푗푖푓2 = ( 1 푛푖(1 + 1/훼2) − 1 푛2푖 (1 + 1/훼2) 2 ) 1 휏푖푐푓 (2.32) where 훼1 = 푛푖−휎 푗휏푖(푛푖−1) 푛푖휎푗휏푖−푛푖+1 and 훼2 = 푛푖휏푖/휎 푗−푛푖+1 푛푖−(푛푖−1)휏푖/휎푗 . Substituting (2.32) into (2.31) we obtain 훽 휎푗 [( 1 푛푖(1 + 훼1) − 1 푛2푖 (1 + 훼1) 2 ) + 휏푖 ( 1 푛푖(1 + 훼2) − 1 푛2푖 (1 + 훼2) 2 )] = 퐼 ′(휎푗) (2.33) 31 2.3. Exchange Appreciation and Investment Decision which can be solved for the equilibrium 휎푗 . Proposition 4. Let 푒1 < 1 and consider industries with the same trade cost 휏 < 휏̂ in Θ휎(푒1). If all 휎 > 1 in some interval in (1, 휎) are profitable for firms in industries with different 푛푖, then the choice of 휎 is decreasing in 푛푖 for 2 ≤ 푛푖 ≤ 푛. Proof: Using the left-hand-side of (2.33) we have ∂ ∂푛푖 ( ∂ ∂휎푗 Π푗푖ℎ(휎 푗)) = 1 휎푗 [ (1푗휎휏푖)(2− 푛푖 − 푛푖훼1) (푛푖휏푖휎푗 − 푛푖 + 1)푛3푖 (1 + 훼1)3 + 휏푖(휏푖/휎 푗 + 1)(휏푖/휎 푗)(2− 푛푖 − 푛푖훼2) (푛푖 − (푛푖 − 1)휏푖/휎푗)2푛3푖 (1 + 훼2)2 ] which is negative if 푛푖 ≥ 2. This means the marginal benefit of 휎푗 is bigger for industries with a smaller 푛푖, provided 푛푖 ≥ 2. By the Envelop Theorem again we have ∂2 ∂휎푗∂휎푗 Π푗푖ℎ(휎 푗) = 훽 −2푐ℎ (휎푗)3 [ 푥푗푖ℎ2(휎⃗) + 휏푖푥 푗∗ 푖ℎ2(휎⃗) ] < 0 Thus Π푗푖ℎ(휎 푗) is a strictly concave function. Let 푛′ and 푛′′ be the number of firms in two industries with the same trade cost 휏 < 휏̃ and 2 ≤ 푛′ < 푛′′ ≤ 푛. Denote the firms’ optimal choices of technology levels as 휎푛′ and 휎푛′′ . Suppose 휎푛′′ ≥ 휎푛′ . Then we have ∂Π푗푖ℎ(휎 푗 , 푛푖 = 푛 ′) ∂휎푗 ∣∣ 휎푛′′ > ∂Π푗푖ℎ(휎 푗 , 푛푖 = 푛 ′′) ∂휎푗 ∣∣ 휎푛′′ = ∂ ∂휎푗 퐼(휎푗) ∣∣ 휎푛′′ which means the profit for firm 푗 in the 푛′-firm industry Π푗푖ℎ(휎 푗 , 푛푖 = 푛′)−퐼(휎푗) is increasing at some level no smaller than 휎푛′ . This increase con- tradicts that 휎푛′ is the optimal choice for firms in the industry with 푛푖 firms, 32 2.3. Exchange Appreciation and Investment Decision unless there is another local maximizer 휎∗ with 휎∗ > 휎푛′ . However, since Π푗푖ℎ(휎 푗) is strictly concave and 퐼(휎) is strictly convex, there are no other local maximizers. Thus we conclude that 휎푛′′ < 휎푛′ for all 2 ≤ 푛′ ≤ 푛′′ ≤ 푛. ■ Note when 휎 is greater (1 + 1푛푖−1) 1 휏푖 , all foreign firms in industry 푖 are forced out of the home market. Given a 휏푖 we can make 퐼(휎) rise fast enough so that it will exceed the benefit of adoption at 휎 = (1+ 1푛−1) 1 휏푖 . This ensures all home firms will have interior choices of 휎, i.e. the foreign firms will not be out of the home and foreign market. Figure 2.2 illustrates this point. 33 2.3. Exchange Appreciation and Investment Decision Π푗푖ℎ(휎)−Π푗푖ℎ(푠푞),퐼(휎) 휎 퐼(휎) (1 + 1푛−1) 1 휏푖 Π푗푖 (휎)−Π푗푖 (푠푞) (1, 0) Optimal 휎 Π푗푖 (휎)−Π 푗 푖 (푠푞) is the benefit of adopting technology of level 휎 and 퐼(휎) is the fixed cost. For a given 휏푖, when improvement in home productivity 휎 is larger than (1 + 1 푛−1 ) 1 휏푖 , foreign firms in industries with 푛 firms begin to drop out of the market and home firms has a jump in profit as they are competing only against each other. When 퐼(휎) rises fast enough, choosing 휎 > (1 + 1 푛−1 ) 1 휏푖 is not optimal and home firms will choose an interior 휎. For industries with 푛푖 < 푛, their jump points in profits are bigger than (1 + 1 푛−1 ) 1 휏푖 . Firms in these industries will choose interior 휎 as well as long as this is the case in the 푛-firm industry. Figure 2.2: The Benefit and Cost of Adopting the Disruptive Technology 34 2.3. Exchange Appreciation and Investment Decision The key for the proof is that among industries with the same 휏 , the profit of firms in industries with lower 푛푖 is more responsive to 휎. Thus the marginal profit with respect to 휎 is equal to the marginal cost 퐼 ′(휎) at a bigger value. Figure 2.3 demonstrates the argument graphically. Π푗푖ℎ(휎)−Π푗푖ℎ(푠푞),퐼(휎) 휎 퐼(휎) Π푗푖ℎ(휎)−Π푗푖ℎ(푠푞) for 푛푖 = 푛 (1, 0) Optimal 휎 for industry with 푛푖 = 푛 Π푗푖ℎ(휎)−Π푗푖ℎ(푠푞) for 푛푖 = 푛+ 1 Optimal 휎 for industry with 푛푖 = 푛+ 1 Π푗 푖ℎ (휎)−Π푗 푖ℎ (푠푞) is the benefit of adopting technology of level 휎 and 퐼(휎) is the fixed cost. Proposition 4 shows, the benefit of adoption for industries with 푛 firms is increasing faster in 휎 than industries with 푛+ 1 firms. Given the same fixed cost 퐼(휎), the optimal choice of 휎 for firms in industries with fewer firms is larger. Figure 2.3: Illustration of The Relation between 푛푖 and Choice of 휎 Putting Propositions 2, 3 and 4 together yields the following predictions. First, among industries with trade cost lower than the threshold 휏̂ there is negative correlation between the number of firms per industry and the choice of 휎 if 푛푖 ≥ 2. Since the concentration level of an industry is inversely related to the number of firms, if we regresses 휎 on concentration for the set 35 2.3. Exchange Appreciation and Investment Decision of industries with 휏̂ , OLS is predicted to find a positive relation. Second, industries with trade costs greater than 휏̂ will not adopt the disruptive technology. For these industries, a regression of 휎 on concentration will yield a zero slope coefficient. Figure 2.4 illustrate the adoption choices for firms in different industries. Overall, if we simply pool all industries together and regress 휎 on concentration, we are likely to find a positive relation. 푛푖 휎 1 2 3 4 5 6 7 8 1 b b b b b b b b Choices of 휎 for industries with trade cost above 휏̂ Choices of 휎 for industries with trade cost below 휏̂ b bb bb bb bb bb bb b The model suggests, 1) for industries with trade cost lower than 휏̂ the choice of productivity improvement 휎 is negatively correlated with the number of firms per industry 푛푖 , and 2) industries with trade cost greater than 휏̂ will not adopt the disruptive technology (denoted as choosing 휎 = 1 in the figure). Figure 2.4: Level of Technology Adoption and Number of Firms in the Industry Compared to Holmes et al. (2008) and other previous theoretical papers which focus on the question of whether firms will adopt a new technol- ogy when there is more competition, this chapter studies both the condi- tions for adoption and the intensity of adoption. The model presented here differentiates between two types of competition, the competitive pressure from appreciations, and market concentration. The competitive pressure 36 2.4. Manufacturing Productivities in Canada . . . from appreciations is predicted to provide an incentive for adopting new technologies, consistent with finding of previous papers. However, firms in highly-concentrated industries, i.e. those subject to less competition in this dimension, are likely to invest more to achieve bigger productivity improve- ments. Thus, in this model the effect of competition on adoption of new technologies is subtle. 2.4 Manufacturing Productivities in Canada Over the Last Decade When the home country experiences an appreciation, the model developed in sections 2 and 3 offers the following two key predictions. First, in general appreciations provide incentives for firms to improve productivity. Second, among industries with low trade costs, the highly concentrated ones will implement bigger improvements to productivity, as profits of firms with a bigger market share will be more responsive to change in productivity. In- dustries with high trade costs will have no incentive to improve productivity regardless of the concentration level, as the high trade cost will limit com- petition from foreign industries.12 12It should be recognized that an important alternative mechanism can potentially also give rise to similar predictions. That is, when exchange rate appreciates, foreign capital goods and intermediate goods that embody better technology will become cheaper. Such mechanism will predict increase in capital or intermediate goods purchase. Without access to detail data on the capital investment and intermediate good trade for Canadian manu- facturing industries, I am currently unable to differentiate between the two hypothesis in the empirical section. 37 2.4. Manufacturing Productivities in Canada . . . In the model I assume that in the country that experiences a depre- ciation, productivity will not respond to deprecation. The assumption is needed to simplify the analysis when industries in the other country are allowed to choose the level of productivity improvement. If this assumption is a reasonable approximation of firms’ behaviour during depreciation, we would see the firms’ productivity fall relative to their counterparts in the other countries as the latter group of firms have an incentive to improve productivity to counter the movement of the exchange rate. To test the predictions of the model, I analyze how the productivity of Canadian manufacturing industries responded between 1997 and 2006 to the interactions between exchange rate movements, trade costs, and concentra- tions. There are a few advantages to using Canadian manufacturing data. First, Canada is a highly open economy, and its manufacturing industries are exposed to a substantial amount of trade. In particular, because of the Free Trade Agreement with the US, Canada’s main trading partner, we may consider the trade costs of Canadian industries reflect mostly exogenous factors. Second, during the sample period the Canadian dollar experienced first a moderate depreciation then a major appreciation. Since there is evidence (see for instance Maier and DePratto (2007)) that the recent exchange move- ments are partly driven by movements in commodity prices, it is reason- able to suggest the movements are exogenous to manufacturing industries. Although productivity of manufacturing industries may contribute to the movements in exchange rates, such effects are likely to be dominated by the commodity factor. 38 2.4. Manufacturing Productivities in Canada . . . Third, since both Canada and the US have adopted the North Ameri- can Industry Classification System (NAICS), I am able to use productivity growth in the US manufacturing industries to control for some of the unob- served industry characteristics. Among others, this would capture techno- logical spillovers from US industries. 2.4.1 Specification and Data The sample used in this study involves the annual data of 237 6-digit NAICS Canadian manufacturing industries from 1997 to 200613. The sources of Canadian data are the Annual Survey of Manufacturers (ASM) published by Statistics Canada, the Canadian Socioeconomic Information Manage- ment (CANSIM) Database, the Bank of Canada, the Annual Survey of Manufactures (ASM) published by the US Census Bureau, and the Basic Economics database (DRI/McGraw-Hill). The specification is 푑푙푛(푝푟표푑푢푐푡푖푣푖푡푦)푖푡 = 훽0 + 훽1 ⋅ 푑푙푛(푒푥푐ℎ푎푛푔푒 푟푎푡푒)푡−1 + 훽2 ⋅ 푐표푛푐푒푛푡푟푎푡푖표푛푖푡−1 + 훽3 ⋅ 푑푙푛(푒푥푐ℎ푎푛푔푒 푟푎푡푒)푡−1 ⋅ 푐표푛푐푒푛푡푟푎푡푖표푛푖푡−1 + 훽4 ⋅ 푑푙푛(푒푥푐ℎ푎푛푔푒 푟푎푡푒)푡−1 ⋅ 푐표푛푐푒푛푡푟푎푡푖표푛푖푡−1 ⋅ 푇푟푎푑푒 퐷푢푚푚푦푖 + 훽5 ⋅ (표푡ℎ푒푟 푐표푛푡푟표푙푠) + 푢푖 + 휖푖푡 (2.34) where 푖 is the index for the industries and 푡 for year. 푢푖 is the industry specific effect and 휖푖푡 is the error term assumed to be i.i.d. across industries 13The total number of 6-digit NAICS manufacturing industries is 262. 25 industries are missing from the sample. 39 2.4. Manufacturing Productivities in Canada . . . and time. In the specification, I use last period exchange rate movement as a re- gressor. In the model, at the beginning of the first period firms decide whether to improve productivity conditioned on an expected appreciation. Since exchange rate movement is highly persistent, an appreciation in the last period is a good predictor that the current period exchange rate will stay at an appreciated level. The interaction between exchange rate and concentration corresponds to the model prediction, that in general, there is a positive relation between market concentration and productivity growth during appreciations. The triple interaction term reflects the model predic- tion that, during an appreciation, market concentration level is positively associated with productivity gain within the group of highly traded indus- tries. Since an appreciation is defined as an decrease in the exchange rate, a negative 훽4 supports the prediction. The traditional measure of productivity, total factor productivity (TFP), is not available for Canadian manufacturing industries as Statistics Canada does not provide data on capital stock or investment necessary for the com- putation of TFP. Thus I use labour productivity instead, and the main measure is value added per production worker. In robustness checks I also explore manufacturing revenue per production worker as an alternative mea- sure of labour productivity. Value added per production worker is often used to measure labour productivity in the international trade literature, for in- stance in Bernard and Jensen (1999). Trefler (2004) uses “value added in production activities per hour worked by production workers” as the mea- sure for productivity. While the analysis of Trefler (2004) is based on the 40 2.4. Manufacturing Productivities in Canada . . . 3-digit SIC manufacturing industries, this chapter is based on 6-digit NAICS classification of industries. As the hours worked are not reported by Statis- tics Canada for the 6-digit NAICS industries, it is not possible for this chapter to use the same measure. The measure of exchange rate is the Canadian-dollar effective exchange rate index (CERI) created by the Bank of Canada. It is defined by the Bank of Canada as “a weighted average of bilateral exchange rates for the Cana- dian dollar against the currencies of Canada’s major trading partners”14. Since the US dollar carries a weight of 0.7618, the movement of the CERI closely mimics the movement of the Canada/US exchange rate, as shown in Figure 2.5. I deflate the CERI by the inflation rate in Canada and the weighted inflation rate of the major trading partners to obtain movements in real exchange rate. 15 14These currencies are the US dollar, the European Union euro, the Japanese yen, the UK pound, the Chinese yuan, and the Mexican peso. Details can be found at http://www.bank-banque-canada.ca/en/rates/ceri.pdf. 15In unreported regressions, I use the Canada/US real exchange rate and find results are not sensitive to this treatment. 41 2.4. Manufacturing Productivities in Canada . . . 1990 1992 1994 1996 1998 2000 2002 2004 2006 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Year N om in al  e xc ha ng e ra te   Canada/US Trade weighted $C The solid line is the Canada/US nominal exchange rate and the dashed line is the Canadian-dollar effective exchange rate (CERI). Both are measured at annual frequency. Note the original CERI has a base value of 100 and is defined as the price of Canadian dollar in terms of the basket of foreign currencies. To make it compatible with the definition in the chapter, I divide the original CERI by 100 and take the inverse. The dashed line plots the edited CERI series. Figure 2.5: Movements of Canadian Dollar Exchange Rate Since 1990 42 2 .4 . M a n u fa ctu rin g P ro d u ctiv ities in C a n a d a ... Table 2.1: Means of Key Variables between 1997 and 2006 Whole period Depr(1997-2002) Appr(2002-2006) dln(value added per production worker), CND 1.4% (0.39%) 1.0% (0.53%) 2.0% (0.56%) dln(value added per production worker), US 3.0% (0.31%) 1.3% (0.42%) 5.2% (0.42%) dln(revenue per production worker), CND 1.5% (0.29%) 0.6% (0.02%) 2.5% (0.39%) dln(effective exchange rate) -1.4% (4.9%) 1.6% (2.6%) -5.6% (4.4%) trade to revenue ratio 1.39 (0.04) 1.36 (0.05) 1.43 (0.05) 4-firm concentration ratio 48.3% (0.52%) 48.3% (0.69%) 48.0% (0.80%) dln(manufacturing revenue) 0.8% (0.31%) 2.3% (0.45%) -1.1% (0.43%) dln(value of export) 0.9% (0.48%) 4.0% (0.73%) -3.0% (0.57%) dln(value of import) 2.8% (0.44%) 4.2% (0.68%) 1.1% (0.49%) dln(number of production workers) -0.7% (0.33%) 1.7% (0.43%) -3.7% (0.48%) dln(R&D expenditure) 7.9% (0.36%) 12.3% (0.54%) 2.5% (0.79%) dln(energy per production worker) 5.2% (0.37%) 2.7% (0.49%) 8.2% (0.55%) dln(material per production worker) 1.5% (0.34%) 1.5% (0.45%) 1.6% (0.51%) establishment size 60 (2.53) 66 (3.90) 52 (2.89) Notes: 1) The numbers are the means of 237 6-digit NAICS industries over the time period indicated, except for the case of R&D expenditure where the means are calculated from 4-digit NAICS industries. 2) “dln” denotes first differences in log, as approximations for growth rates. 3) The numbers in the parenthesis are standard errors.43 2.4. Manufacturing Productivities in Canada . . . The concentration of production in each industry is measured by the 4- firm concentration ratio (CR4) reported by Statistics Canada. In the model firms are symmetric, so CR4 has an inverse relation with the number of firms in the industry. In reality firms differ in size so CR4 might be a better measure of concentration compared to the number of firms 16. Since data on CR4 is not available beyond 2003, I use the 2003 values for the years 2004 and 200517. Trade costs of industries are not observed but in the model they have an inverse relation with the trade to sales ratio. I construct the ratio for an in- dustry as the value of total import plus export divided by the manufacturing revenue of the industries between 1997 and 200618. Other control variables included in the regressions are growth in energy per production worker, growth in material per production worker, growth in R&D expenditure, average establishment size, productivity growth in 16CR4 is used by MacDonald (1994) to study how the change in productivity varies with market power after an import surge. This chapter is similar in that it also studies how the effect of competition differs with cross section difference in CR4. 17Note CR4 enters the regression model with a one-period lag. Using 2003 values for the year of 2004 and 2005 does not have a major impact on the results, since CR4 is stable over time (see Table 2.1) and most of the variation in CR4 comes from the cross-section. In the robustness check subsection, I show the main results hold even if I use the 1990 CR4 values as the measure for concentration between 1997 and 2006. 18The construction calculates a constant trade-to-revenue ratio that does change over the years for a particular industry. I choose this ratio because I will estimate a threshold regression model based on the trade to revenue ratio where the ratio is used as a measure for trade cost. If one allows the trade to revenue ratio of an industry to vary across years, the industry can be classified as a high-trade-cost industry in one year and a low-trade-cost one in another, which is probably not desirable. 44 2.4. Manufacturing Productivities in Canada . . . corresponding US industry, and GDP growth in Canada and the US. Lastly, industry fixed effects and year effects are also used in most of the regressions. The inclusion of the year effects of course precludes GDP growth rates. As mentioned before, there are no direct measures of the capital stock, its utilization variation, and changes in hours worked per worker. Including energy and material use provides a limited remedy. The model in the chapter focuses on the adoption of a known technology, and the inclusion of R&D expenditure helps to control for the improvement to productivity due to firms’ search for new technologies. However, R&D is available only for 3- digit or 4-digit NAICS industries, at a higher level of aggregation than 6-digit NAICS industries. Average establishment size is computed as the number of production workers per establishment in an industry. It is included to control for return-to-scale effects. Since it is possible for Canadian industries to benefit from technological spillover from foreign industries, especially US industries, I include productivity growth in the corresponding US industry to capture such learning opportunities. Adding real GDP growth rates of Canada and US will control for the effects of macroeconomic productivity and demand shocks. Before turning to regression results, it is useful to have a brief look at a number of key variables during the depreciation sub-period (1997-2002) and the appreciation sub-period (2002-2006) in Table 2.1. We can see that during appreciation export growth and employment of production workers dropped. Meanwhile, Canadian manufacturing labour productivity, measured by both value added per production worker and manufacturing revenue per production worker increased, although it was 45 2.4. Manufacturing Productivities in Canada . . . outpaced by US labour productivity growth. Judging from the means re- ported, we cannot rule out the possibility that the higher labour produc- tivity growth in Canada had come from spillover from the 5.2% growth in US labour productivity. It could also be case that higher energy use per production worker contributed to the labour productivity growth. Growth in R&D expenditures and scale effects as measured by establishment sizes, on the other hand, appear to be poor explanations for the higher productiv- ity growth in the appreciation sub-period, as the two variables were lower during the appreciation. Lastly, it’s worth noting there was little change in the average concentration ratio. 2.4.2 Main Results I estimate all specifications with the linear model with industry fixed effects. The only complication comes from the threshold effect of trade. Conditioned on whether trade exceeds a threshold level, the model predicts different relations between concentration of production and productivity gain during appreciation. The trade threshold is unknown and has to be estimated. The estimation of the threshold follows Hansen (2000), and is based on least- square regressions. I first construct a grid of trade-ratios with the step size being 0.5 of a centile and then search the grid for a threshold at which the effect of concentration-exchange-rate interaction changes significantly. The estimated threshold is located at the 83.5th centile, translating to a trade to revenue ratio of 1.89. There are 39 industries with a trade ratio above the threshold. The 95% confidence interval for the threshold is between the 76th and 92.5th centiles, or [1.63, 2.80] in terms of trade-to-revenue ratio. Using 46 2.4. Manufacturing Productivities in Canada . . . the threshold estimate, I estimate the threshold regression model specified in (2.34). Standard errors are computed with methods suggested in Hansen (2000). Though not predicted by the theory, it is plausible that the effect of exchange rate on labour productivity growth may also change with a trade threshold. The application of the threshold estimation method on the in- teraction between trade and exchange rate movement indicates there is no statistically significant threshold effect19. In essence, I have used the method in Hansen (2006) to guide the empirical specification. In one of the robust- ness checks, the interaction between the trade dummy and exchange rate movement is included to show key results are insensitive to its inclusion. The first three columns of Table 2.2 report the benchmark regression results. The specification in column (1) includes year dummies, thus pre- cluding variables that are invariant across the cross-section, in particular the last-period exchange rate movement. Specification (2) and (3) estimate the same specification using only the subsamples. In column (1) of Table 2.2, the level of concentration ratio is not sig- nificant, consistent with the theory prediction that it should not matter independent of the exchange rate. The interaction between concentration and exchange rate is negative and significant, with a coefficient of -0.009. This estimate implies that during a 5% appreciation20 an industry with a 20% higher concentration ratio will experience labour productivity growth 19In unreported regressions, the interaction between trade-to-revenue ratio and other variables, such as the concentration ratio, are also included as regressors. Such interactions are always highly insignificant. 20Note again, appreciation is defined as decrease in the exchange rate. 47 2.4. Manufacturing Productivities in Canada . . . Table 2.2: Benchmark Fixed Effect Estimations Dependent variable Full sample Appr. Depr. dln(productivity) 1997-2006 2002-2006 1997-2002 퐶푅4 -0.001 -0.003 -0.001 (0.006) (0.003) (0.001) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 -0.009** -0.005 -0.013 (0.003) (0.004) (0.010) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 ⋅ 푇푟푎푑푒퐷 -0.005 -0.011** 0.001 (0.003) (0.004) (0.009) 푑푙푛(푅&퐷) 0.002 -0.013 -0.018 (0.027) (0.046) (0.044) 퐸푠푡푎푏 푠푖푧푒 0.004*** 0.001 0.001** (0.001) (0.001) (0.0002) 푑푙푛(퐸푛푒푟푔푦) 0.199*** 0.106** 0.326** (0.028) (0.040) (0.049) 푑푙푛(푀푎푡푒푟푖푎푙) 0.250** 0.207** 0.276** (0.028) (0.039) (0.050) 푑푙푛(푃푟표푑푢푐푡푖푣푖푡푦 푈푆) 0.159** 0.223* 0.165 (0.074) (0.098) (0.151) year dummies included included included industry fixed effects included included included 푅2 0.11 0.09 0.10 Observations 2068 906 1162 Industries 237 231 237 Notes: 1) ***, ** and * indicate significance levels of 1%, 5% and 10%. 2)“dln” denotes first differences in log, as approximations for growth rates. 3) The dependent variable is labour productivity, measured as value added per production worker. RER, CR4, TradeD, R&D, Estab size, Energy, Material, and Productivity US denote respectively real exchange rate, 4-firm concentration ratio, a dummy variable for highly-trade industries, R&D expenditure, average establishment size, energy used per production worker, material used per production worker, growth in value added per production worker in the corresponding US industry. 48 2.4. Manufacturing Productivities in Canada . . . that is 0.9% higher. Since the average labour productivity growth rate be- tween 1997 and 2006 is was 1.4%, and that the standard deviation of the concentration ratio was 24%, we can say this is an economically significant effect. Meanwhile, the coefficient on the triple interaction of exchange rate, concentration and trade dummy is -0.005, which is economically large but not statistically significant. The growth rate in R&D expenditure appears to have had no effect on labour productivity growth. While the establishment size did have a impact on labour productivity growth, the magnitude was not big as a coefficient of 0.004 meant that an increase of establishment size by 100 workers only raised labour productivity growth by 0.04%21. The coefficient on the energy and material variables suggest that the energy and material elasticity of productivity are 0.199 and 0.250 respectively. Both are highly significant. Lastly, the labour productivity growth in Canadian industries was positively correlated with the growth in US. A 1% increase in productivity in an US industry is associated with a 0.159% increase in the corresponding Canadian industry. Column (2) is estimated with the subsample between 2002 and 2006, i.e. the appreciation period, while column (3) is estimated with the subsample of the depreciation period. The discussion will be focused on the interaction terms, as estimates of other coefficients are similar to column (1). In column (2), the interaction between concentration and exchange rate becomes in- 21The unit of measurement for establishment is scaled up to 10 workers to facilitate the presentation of results, i.e. to avoid many fractions with four digits after the decimal point. 49 2.4. Manufacturing Productivities in Canada . . . significant while the triple interaction term becomes significant. A coefficient of -0.011 on the triple interaction term implies that during a 5% appreciation an industry with a 20% higher concentration ratio will experience a labour productivity growth that is 1.1% higher. The estimates are more in line with the predictions of the theory, i.e. we expect to see a positive correlation be- tween concentration and labour productivity growth only for the high-trade industries. On the other hand, the estimation on the depreciation subsample indicates no threshold effect and the effect of concentration-exchange-rate interaction is large but not statistically significant. It is worth noting that most of the variation in concentration ratio comes from the cross-section, rather than variation in the time dimension. Over the sample period, 98% of the variance in concentration is accounted for by the variance in the industry average concentration ratio. Namely, within most industries, the concentration levels had experienced very little changes. Therefore, in interpreting the results, we can roughly view the concentra- tion level as fixed over time and regard the regression coeffients on the concentration-exchange-rate interactions as reflection of the different effects of exchange rates movements on industries with different pre-determined concentration levels. 2.4.3 Robustness Checks In this subsection, I conduct several robustness checks. Table 2.3 reports the results with alternative dependent variables. The dependent variable in columns (1) through (3) is difference between Canadian and US labour productivity growth rates. Adopting this dependent variable is equivalent 50 2.4. Manufacturing Productivities in Canada . . . to imposing the restriction that the coefficient on US productivity growth is 1 in the regressions in Table 2.2. Careful comparison between the first three columns of Table 2.3 and Table 2.2 suggests they are very similar. In the last three columns, the dependent variable is manufacturing revenue per production worker, arguably a poorer measure for labour productivity not accounting for costs of other inputs. Although the overall fit of the three regressions are much better, we can only find a weak relation between concentration and labour productivity growth and there is no evidence of a threshold effect. In the baseline estimations, I look at the effect of exchange rate change between year 푡− 1 and 푡 on productivity growth between 푡 and 푡+ 1. Since in their decision-making, firms may look into exchange rate change over a longer period in the past, and the change in productivity may realize over a longer period too, I also estimate equations with alternative assumption about the length of periods. In Table 2.4, the first three columns present effects of exchange rate change between 푡−2 and 푡 on productivity between 푡 and 푡+2. The last three columns are effects of exchange rate change between 푡−3 and 푡 on productivity between 푡 and 푡+1. While there are some changes in parameter estimates, the coefficients on the triple interaction term for the appreciation period are very similar to the benchmarks in 2.2. After a major appreciation, productivity can improve due to firms up- grade their technologies, as suggested in this chapter. However, productiv- ity increase can also result from exits of less efficient firms. In Table 2.5, I present results from specifications augmented by change in the number of establishments. We can see the coefficients on the interaction terms are 51 2.4. Manufacturing Productivities in Canada . . . Table 2.3: Alternative Dependent Variables Dependent variable Full sample Appr. Depr. Full sample Appr. Depr. 1997-2006 2002-2006 1997-2002 1997-2006 2002-2006 1997-2002 (1) (2) (3) (4) (5) (6) 퐶푅4 -0.001 -0.002 -0.001 -0.001** -0.001 -0.001 (0.001) (0.003) (0.001) (0.0003) (0.001) (0.001) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 -0.006* -0.002 -0.014 -0.003* -0.002 -0.002 (0.004) (0.003) (0.011) (0.002) (0.003) (0.007) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 ⋅ 푇푟푎푑푒퐷 -0.006* -0.013*** -0.001 -0.001 -0.002 0.003 (0.004) (0.005) (0.010) (0.002) (0.002) (0.006) 푑푙푛(푅&퐷) 0.056** 0.073 0.020 0.002 0.040** -0.043 (0.027) (0.047) (0.044) (0.015) (0.019) (0.029) 퐸푠푡푎푏 푠푖푧푒 0.005*** 0.003 0.008*** 0.001* 0.002 0.002* (0.001) (0.003) (0.002) (0.0001) (0.002) (0.001) 푑푙푛(퐸푛푒푟푔푦) 0.207*** 0.092** 0.352*** 0.118*** 0.139*** 0.102*** (0.029) (0.042) (0.050) (0.015) (0.017) (0.032) 푑푙푛(푀푎푡푒푟푖푎푙) 0.238*** 0.211*** 0.252*** 0.584*** 0.536*** 0.637*** (0.029) (0.040) (0.050) (0.079) (0.016) (0.032) 푑푙푛(푃푟표푑푢푐푡푖푣푖푡푦 푈푆) - - - 0.079* 0.165*** -0.015 (0.041) (0.041) (0.099) year dummies included included included included included included industry fixed effects included included included included included included 푅2 0.09 0.06 0.09 0.05 0.06 0.04 Observations 2068 906 1162 2068 906 1162 Industries 237 231 237 237 231 237 Notes: 1) ***, ** and * indicate significance levels of 1%, 5% and 10%. 2)“dln” denotes first differences in log, as approximations for growth rates. 3) The dependent variable in the first three columns is the growth rates difference in Canada and US value added per production worker. 4) The dependent variable in column (4) through (6) is manufacturing revenue per production worker. 5) RER, CR4, TradeD, R&D, Estab size, Energy, Material, and Productivity US denote respectively real exchange rate, 4-firm concentration ratio, a dummy variable for highly-trade industries, R&D expenditure, average establishment size, energy used per production worker, material used per production worker, growth in value added per production worker in the corresponding US industry. 52 2.4. Manufacturing Productivities in Canada . . . Table 2.4: Alternative Specification of Lags Dependent variable Full sample Appr. Depr. Full sample Appr. Depr. dln(productivity) 1997-2006 2002-2006 1997-2002 1997-2006 2002-2006 1997-2002 (1) (2) (3) (4) (5) (6) 퐶푅4 -0.001 -0.007* -0.001 -0.001 -0.003 -0.001 (0.001) (0.004) (0.001) (0.001) (0.003) (0.001) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 -0.005 0.002 -0.011 0.001 0.002 -0.005 (0.003) (0.004) (0.009) (0.002) (0.002) (0.014) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 ⋅ 푇푟푎푑푒퐷 -0.005 -0.011*** 0.013 -0.004** -0.008*** 0.019* (0.003) (0.004) (0.008) (0.002) (0.002) (0.013) 푑푙푛(푅&퐷) 0.003 0.073 -0.038 0.008 -0.015 0.023 (0.024) (0.042) (0.043) (0.017) (0.028) (0.041) 퐸푠푡푎푏 푠푖푧푒 0.001*** 0.002** 0.001*** 0.003** -0.001 0.001*** (0.0002) (0.001) (0.0002) (0.0001) (0.001) (0.0002) 푑푙푛(퐸푛푒푟푔푦) -0.022 -0.014 -0.067 0.057*** 0.037 0.161*** (0.031) (0.046) (0.054) (0.022) (0.030) (0.049) 푑푙푛(푀푎푡푒푟푖푎푙) 0.103*** -0.033 0.191*** 0.089*** 0.010 0.197*** (0.031) (0.047) (0.053) (0.021) (0.031) (0.048) 푑푙푛(푃푟표푑푢푐푡푖푣푖푡푦 푈푆) 0.299*** 0.408*** 0.441** 0.188** 0.253*** 0.200 (0.097) (0.125) (0.076) (0.097) (0.160) year dummies included included included included included included industry fixed effects included included included included included included 푅2 0.05 0.06 0.09 0.06 0.09 0.04 Observations 1818 903 915 2048 906 1162 Industries 237 233 235 239 231 237 Notes: 1) ***, ** and * indicate significance levels of 1%, 5% and 10%. 2)“dln” denotes first differences in log, as approximations for growth rates. 3) The dependent variable in the first three columns is productivity growth rate in Canada between year 푡 and 푡 + 2. All independent variables are also measured between 푡 and 푡 + 2, except for that RER measures the exchange rate change between 푡− 2 and 푡. 4) In column (4) through (6) is manufacturing revenue per production worker, the dependent variable and all independent variables are measured between year 푡 and 푡 + 1, except for that RER measures the exchange rate change between 푡− 3 and 푡. 5) RER, CR4, TradeD, R&D, Estab size, Energy, Material, and Productivity US denote respectively real exchange rate, 4-firm concentration ratio, a dummy variable for highly-trade industries, R&D expenditure, average establishment size, energy used per production worker, material used per production worker, growth in value added per production worker in the corresponding US industry. 53 2.4. Manufacturing Productivities in Canada . . . similar to the benchmarks. However, adding change in the number of estab- lishments is a crude way to control for the effect of entries and exits. Ideally one should control for the size of entrants and exiting firms, but these data have not been publicly available. Table 2.5: Effects of Entry and Exit of Establishments Dependent variable Whole sample Appr. Depr. dln(labour productivity, Canada) 1997-2006 2002-2006 1997-2002 (1) (2) (3) 퐶푅4 -0.001 -0.003 -0.001 (0.001) (0.003) (0.001) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 0.005 -0.006 -0.012 (0.003) (0.004) (0.010) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 ⋅ 푇푟푎푑푒퐷 -0.005 -0.011*** -0.005 (0.003) (0.004) (0.009) 푑푙푛(푅&퐷) 0.011 -0.012 0.002 (0.027) (0.046) (0.043) 퐸푠푡푎푏 푠푖푧푒 0.001*** 0.001 0.001*** (0.0001) (0.001) (0.002) 푑푙푛(퐸푛푒푟푔푦) 0.197*** 0.100** 0.319*** (0.028) (0.040) (0.048) 푑푙푛(푀푎푡푒푟푖푎푙) 0.242*** 0.216*** 0.244*** (0.028)* (0.039) (0.049) 푑푙푛(푃푟표푑푢푐푡푖푣푖푡푦, 푈푆) -0.854*** -0.785*** -0.812*** (0.075) (0.093) (0.156) 푑푙푛(퐸푠푡푎푏푙푖푠ℎ푚푒푛푡푠) -0.005 0.062** -0.107*** (0.020) (0.025) (0.039) year dummies excluded excluded included industry fixed effects included included included 푅2 0.16 0.15 0.13 Observations 2068 906 1162 Industries 237 231 237 Notes: 1) ***, ** and * indicate significance levels of 1%, 5% and 10%. 2)“dln” denotes first differences in log, as approximations for growth rates. 3) The dependent variable is labour productivity, measured as value added per production worker. RER, CR4, TradeD, R&D, Estab size, Energy, Material, Productivity US, Establishments denote respectively real exchange rate, 4-firm concentration ratio, a dummy variable for highly-trade industries, R&D expenditure, average establishment size, energy used per production worker, material used per production worker, growth in value added per production worker in the corresponding US industry, and the number of establishments. In column (1) and (2) of Table 2.6, I allow for an interaction between 54 2.4. Manufacturing Productivities in Canada . . . the trade dummy and exchange rate movement, with the triple interaction absent in column (1). This interaction is not always significant. In column (1) we see a significant effect of the concentration-exchange-rate interaction, and in column (2) there is a threshold effect, significant at the 10% level. Lastly, it is reasonable to suspect the concentration ratio may affect labour productivity growth one period later via channels other than its in- teraction with the exchange rate, for example, the consolidation of firms in the current period can raise concentration and the resulting synergy can lead to productivity gains in the future periods. To show that this suspicion is unlikely, I use CR4 in 1990 to interact with exchange rate movements and trade between 1997 and 2006. In this case, only the lagged cross-section variation in CR4 is used in estimation. The results are reported in column (3) of Table 2.6. We can still see a positive relation between concentration and labour productivity growth, and a trade threshold effect, although the interaction terms are only significant at the 10% level. On the balance, the evidence suggest the appreciation provided incentive for Canadian manufacturing industries to improve productivity. In partic- ular, highly-concentrated industries experienced higher labour productivity growth during an appreciation. On the other hand, the theoretical model does not offer a direct prediction for periods of deprecation, and the evi- dence during the 1997-2002 sub-period is inconclusive. Lack of productivity responses during the depreciation sub-period could be due to that the de- preciation between 1997 and 2002 was too moderate to trigger responses from competitors of Canadian firms. 55 2.4. Manufacturing Productivities in Canada . . . Table 2.6: Other Robustness Checks Dependent variable Whole sample Whole sample Whole sample dln(labour productivity, Canada) 1997-2006 1997-2006 1997-2006 (1) (2) (3) 퐶푅4 -0.001 -0.001 - (0.001) (0.001) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 -0.010*** -0.006 -0.007* (0.003) (0.004) (0.004) 푑푙푛(푅퐸푅) ⋅ 푇푟푎푑푒퐷 -0.153 0.555 - (0.191) (0.444) 푑푙푛(푅퐸푅) ⋅ 퐶푅4 ⋅ 푇푟푎푑푒퐷 - -0.014* -0.006* (0.008) (0.003) 푑푙푛(푅&퐷) 0.002 0.002 -0.009 (0.027) (0.027) (0.028) 퐸푠푡푎푏 푠푖푧푒 0.004*** 0.004*** 0.004*** (0.001) (0.001) (0.001) 푑푙푛(퐸푛푒푟푔푦) 0.200*** 0.198*** 0.167*** (0.028) (0.028) (0.028) 푑푙푛(푀푎푡푒푟푖푎푙) 0.250*** 0.250*** 0.269*** (0.028)* (0.028) (0.030) 푑푙푛(푃푟표푑푢푐푡푖푣푖푡푦, 푈푆) 0.158** 0.157** 0.162** (0.074) (0.074) (0.076) year dummies excluded excluded included industry fixed effects included included included 푅2 0.11 0.11 0.10 Observations 2068 2068 1987 Industries 237 237 224 Notes: 1) ***, ** and * indicate significance levels of 1%, 5% and 10%. 2)“dln” denotes first differences in log, as approximations for growth rates. 3) The dependent variable is labour productivity, measured as value added per production worker. RER, CR4, TradeD, R&D, Estab size, Energy, Material, and Productivity US denote respectively real exchange rate, 4-firm concentration ratio, a dummy variable for highly-trade industries, R&D expenditure, average establishment size, energy used per production worker, material used per production worker, and growth in value added per production worker in the corresponding US industry. 56 2.5. Conclusion 2.5 Conclusion This chapter is motivated by the question of how productivity responds to major real exchange rate movements. Drawing on observations of disrup- tive technological changes documented in Holmes et al. (2008), I have built a partial equilibrium model to clarify how productivity responses of industries vary with trade costs and market concentration during an appreciation. Sim- ilar to results in previous literature, I find that competitive pressure resulting from appreciations increases incentives to improve productivity, as the ap- preciation lowers the profit loss during costly transitions. Meanwhile, higher trade costs reduce the incentives by diminishing the competitive pressure of appreciations. In addition, this chapter contributes to the theoretical litera- ture by studying the intensity of technology adoption, suggesting a positive relation between market concentration and the intensity of adoption. It is firms in highly concentrated industries that will invest more in productivity improvements, as their marginal benefits from adopting better technologies are greater. Empirical analysis of 237 6-digit Canadian manufacturing industries be- tween 1997 and 2006 supports the theoretical model’s predictions. During the appreciation period between 2002 and 2006, labour productivity growth was on average higher after controlling for industry fixed effects, and growth in all of energy use, material use, R&D expenditure, productivity in corre- sponding US industries, and GDP in Canada and the US. Highly concen- trated industries experienced high productivity growth, conditional on their exposure to a substantial amount of trade. The theoretical model does not 57 2.5. Conclusion offer predictions for productivity response to depreciations, and during the depreciation period between 1997 and 2002, there is little empirical evi- dence that labour productivity growth had been correlated with exchange rate movements or concentration. The empirical analysis is the first to study productivity response of surviving firms to real exchange rate movements, also adding to the evidence of a positive relationship between competitive pressure and productivity improvement. A logical next step would be to investigate firm level data. The theoreti- cal model of this chapter conjectures about firm behaviour, and the empirics test the implications at the industry level. Although industry-level evidence suggests adjustments have been made to counteract an appreciation, it is only natural to ask, what exactly these firms did. 58 Bibliography Baily, Martin Neil, Hans Gersbach, F. M. Scherer, and Frank R. Lichtenberg, “Efficiency in Manufacturing and the Need for Global Com- petition,” Brookings Papers on Economic Activity. Microeconomics, 1995, 1995, 307–358. Bernard, Andrew B. and Bradford J. Jensen, “Exceptional exporter performance: cause, effect, or both?,” Journal of International Eco- nomics, February 1999, 47 (1), 1–25. Brander, James and Paul Krugman, “A ’reciprocal dumping’ model of international trade,” Journal of International Economics, November 1983, 15 (3-4), 313–321. Fung, Loretta, “Large real exchange rate movements, firm dynamics, and productivity growth,” Canadian Journal of Economics, May 2008, 41 (2), 391–424. Galdon-Sanchez, Jose E. and James A. Jr. Schmitz, “Competitive Pressure and Labor Productivity: World Iron-Ore Markets in the 1980’s,” The American Economic Review, 2002, 92 (4), 1222–1235. 59 Chapter 2. Bibliography Hansen, Bruce E., “Sample Splitting and Threshold Estimation,” Econo- metrica, May 2000, 68 (3), 575–604. Harris, Richard G., “Is There a Case for Exchange-Rate-Induced Produc- tivity Changes,” Canadian Institute for Advanced Research, 2001, Work- ing Paper No. 164. Hart, Oliver D., “The Market Mechanism as an Incentive Scheme,” The Bell Journal of Economics, 1983, 14 (2), 366–382. Holmes, Thomas J., David K. Levine, and James A. Schmitz, “Monopoly and the Incentive to Innovate When Adoption Involves Switchover Disruptions,” NBER Working Paper, 2008, No. W13864. Jones, Charles I., Introduction to Economic Growth, W. W. Norton; 2 edition, 2001. MacDonald, James M., “Does Import Competition Force Efficient Pro- duction?,” The Review of Economics and Statistics, 1994, 76 (4), 721–727. Maier, Philipp and Brian DePratto, “The Canadian dollar and com- modity prices: Has the relationship changed over time?,” Bank of Canada Discussion Paper Series, November 2007. Nickell, Stephen J., “Competition and Corporate Performance,” The Journal of Political Economy, 1996, 104 (4), 724–746. Porter, Michael E., The Competitive Advantage of Nations, New York: New York: Free Press, 1990. 60 Chapter 2. Bibliography Raith, Michael, “Competition, Risk, and Managerial Incentives,” The American Economic Review, 2003, 93 (4), 1425–1436. Schmitz, James A., “What Determines Productivity? Lessons from the Dramatic Recovery of the U.S. and Canadian Iron Ore Industries Follow- ing Their Early 1980s Crisis,” Journal of Political Economy, June 2005, 113 (3), 582–625. Symeonidis, George, “The Effect of Competition on Wages and Produc- tivity: Evidence from the United Kingdom,” Review of Economics and Statistics, 2008, 90 (1), 134–146. Syverson, Chad, “Market Structure and Productivity: A Concrete Exam- ple.,” Journal of Political Economy, 2004, 112 (6), 1181 – 1222. Tang, Yao, “Exchange Rate Appreciation and Productivity,” Mimeo, May 2008. Trefler, Daniel, “The Long and Short of the Canada-U.S. Free Trade Agreement,” American Economic Review, September 2004, 94 (4), 870– 895. Vives, Xavier, “Innovation and Competitive Pressure,” Journal of Indus- trial Economics, forthcoming. 61 Chapter 3 Comparison of Misspecified Calibrated Models: The Minimum Distance Approach22 22A version of this chapter has been submitted for publication. Hnatkovska, V., Marmer, V. and Tang, Y., “Comparison of Misspecified Calibrated Models: The Minimum Distance Approach”. 62 3.1. Introduction 3.1 Introduction This chapter presents a method for the comparison of calibrated mod- els. While calibration is now an essential tool of quantitative analysis in macroeconomics, surprisingly, there is no generally accepted definition of calibration, and calibration is rather viewed as a research style character- ized by a certain attitude toward modelling, assigning parameters’ values, and model assessment (Kim and Pagan, 1995). A number of authors de- fine calibration as a sequence of steps allowing one to reduce the general theoretical framework to a quantitative relationship between variables. For instance, Cooley and Prescott (1995) outline three such steps: imposing parametric restrictions; constructing a set of measurements consistent with the parametric class of models; and assigning values to the model parame- ters. Canova and Ortega (1996) adopt a broader definition of calibration, by including model evaluation into the list of steps. The calibration approach takes an explicitly instrumental view of eco- nomic models: a calibrationist acknowledges that the model is false and will be rejected by the data (Canova, 1994). The objective of the calibrationist is not an assessment of whether the model of interest is true, but rather which features of the data it can be used to capture. Furthermore, a calibrationist may be interested in learning which of the competing but “false” models provides a better fit to the data. In a typical calibration exercise, the calibrationist selects values for the parameters in order to match some characteristics of the observed data with those implied by the theoretical model. For example, a model can be cal- 63 3.1. Introduction ibrated to match empirical moments, cross-correlations, impulse responses, and stylized facts. Such characteristics will be referred as the properties of a reduced-form model or the reduced-form parameters, since they can be consistently estimated from the data regardless of the true data generat- ing process (DGP). Calibrated parameters can be obtained using informal moment matching, the generalized method of moments (GMM), simulated method of moments (SMM), or maximum likelihood (ML) estimation (Kim and Pagan, 1995). Calibration was also formalized as an example of minimum distance es- timation in Gregory and Smith (1990, 1993). If the structural model is cor- rectly specified, the calibrated parameters are consistent and asymptotically normal estimators of the structural (or deep) parameters, and statistical in- ference can be performed using the standard asymptotic results (see, for example, Newey and McFadden (1994)). However, if the structural model is misspecified, the asymptotic distribution of calibrated parameters has to be corrected for misspecification. In this chapter, we explicitly consider the case of misspecified structural models. Our methodology uses a classical minimum distance (CMD) esti- mation procedure to calibrate model parameters. We then show that under some regularity conditions, the calibrated parameters converge in probabil- ity to the values of the structural parameters that minimize the distance between the population characteristics of the data and those implied by the structural model (pseudo-true values). Further, the CMD estimator is asymptotically normal, however, due to misspecification some adjustments to the asymptotic variance matrix are required. Gallant and White (1988) 64 3.1. Introduction and Hall and Inoue (2003), Hall and Inoue (2003) hereafter, established such results for GMM estimators. After choosing parameter values, the calibration exercise continues with evaluation of the structural model. This is usually done by comparing model-implied reduced-form characteristics with those of the actual data (Gregory and Smith, 1991, 1993; Cogley and Nason, 1995; Kim and Pagan, 1995). However, according to the calibrationist’s approach, while evaluating a model, one should keep in mind that it is only an approximation and there- fore should not be regarded as a null hypothesis to be statistically tested (Prescott, 1991). There is a large literature in econometrics that considers misspecified models. For example, Watson (1993) and Diebold et al. (1998) propose measures of fit for calibrated models that take into account pos- sible misspecification. Many papers also advocate evaluating a structural misspecified model against another misspecified benchmark model (Diebold and Mariano, 1995; West, 1996; Schorfheide, 2000; White, 2000; Corradi and Swanson, 2007). In this chapter, we compare misspecified models by the means of an asymptotic test. In this test, under the null hypothesis the two misspecified models provide an equivalent approximation to the data in terms of char- acteristics of the reduced-form model. Our approach is related most closely to Vuong (1989) and Rivers and Vuong (2002), Rivers and Vuong (2002) hereafter. Vuong (1989) proposed such a test in the maximum likelihood framework, and Rivers and Vuong (2002) discussed it in a more general set- ting allowing for a broad class of lack-of-fit criteria including that of GMM. The contribution of our chapter relative to Rivers and Vuong (2002) 65 3.1. Introduction is threefold. First, Rivers and Vuong (2002) focused solely on non-nested models. Our CMD framework allows us to analyze both non-nested and nested cases. The nested case is particularly important because, if the null of models equivalence is not rejected, one can replace the bigger model with a more parsimonious one. Furthermore, many hypotheses can be expressed in terms of parameter restrictions and thus fall into the nested category.23 The nested case also differs from non-nested in terms of the asymptotic null distribution. Rivers and Vuong (2002) show that in the non-nested case the difference between the sample lack-of-fit criteria of the two models is asymptotically normal. We derive a similar result for non-nested case, but also show that our test statistic has a mixed 휒2 distribution in the nested case, similarly to Vuong (1989).24 Second, we analyze the situation where the models are estimated using one set of reduced-form characteristics and are compared using another. This is a very common approach in the calibration literature. For instance, a structural model can be estimated to match the first moments of the data, and evaluated in terms of its ability to match the second moments. In this case, we show the asymptotic null distribution is always normal, regardless of whether the models are nested or non-nested. This fact substantially simplifies the testing procedure in such situations. The reason for asymptotic 23Note however that, in our framework, the nested case does not necessarily some re- strictions on some structural or deep parameters. 24Also, by considering the framework of MD estimation, we provide more specific as- sumptions and asymptotic results than in Rivers and Vuong (2002); and while, as a result, our treatment of the problem is less general than in Rivers and Vuong (2002), it covers the important case of calibration. 66 3.1. Introduction normality in the case of nested models is that there is no selection criteria minimization when the models are estimated and evaluated on different sets of reduced-form parameters. Third, we address the issue of choosing weights for reduced-form char- acteristics when comparing models. When models are misspecified, the pseudo-true values of their parameters and the ranking of the models de- pend on the choice of the weighting scheme. In particular, the null hypoth- esis changes when one applies different weights (see, for example, Hall and Inoue (2003) and Hall and Pelletier (2007)). In this chapter, we relax the dependence of models ranking on the choice of the weighting scheme by sug- gesting procedures that take into account the models’ relative performance for various choices of the weight matrix. We propose averaged and sup pro- cedures for model comparison. The averaged test corresponds to the null hypothesis that the two models have equal lack-of-fit on average. The null hypothesis of the sup test says that one model cannot outperform another for any choice of the weighting matrix. We also propose a simple procedure for constructing confidence sets for the weighting schemes favorable for one of the models. The problem of comparison of misspecified models should be discerned from non-nested hypothesis testing problems (Davidson and MacKinnon, 1981; MacKinnon, 1983; Smith, 1992). Suppose that the two alternative models are non-nested and therefore cannot be both true at the same time. According to our model comparison null hypothesis, the models have equal measures of fit and, consequently, the null hypothesis implies that they are both misspecified. However, in the literature on non-nested hypothesis 67 3.1. Introduction testing, the null hypothesis is that one of the models is true. Thus, the two approaches, the non-nested testing and the model comparison testing of misspecified models in the spirit of Vuong (1989), are not competing but rather complementary. The first approach can be used in a search for the true specification, while the later approach can be adopted when the econometrician believes that all alternative models are misspecified or when they all have been rejected by the overidentified restrictions or non-nested tests. Comparison of misspecified calibrated models has also been studied from the Bayesian perspective by Schorfheide (2000). Our method can be viewed as a frequentist counterpart of the Schorfheide (2000) procedure.25 Corradi and Swanson (2007) designed a Kolmogorov-type test for comparison of misspecified calibrated models. In their paper, the models are compared in terms of the distances between the historic empirical cumulative distribution function (CDF) and the CDFs implied by the model. Thus, their approach is similar to that of Vuong (1989) and Kitamura (2000) who use the Kullback- Leibler Information Criterion.26 We on the other hand focus on the ability of a model to approximate some reduced form characteristics that do not require knowledge of the CDF. Recall that when the models are misspecified, 25While in Schorfheide (2000) a structural model that achieves the lowest average pos- terior loss is selected, we follow the approach of Vuong (1989) and suggest a test for the null hypothesis that the two models have equal losses. 26According to Corradi and Swanson (2007) approach, the model description must in- clude the assumptions that allow one to simulate the data. Our approach allows us to compare and evaluate structural models that do not necessarily provide a complete dis- tribution for the data. 68 3.1. Introduction different measures can lead to different ranking of the models. The issue of misspecified calibrated models was also addressed by Dridi et al. (2007) using indirect inference.27 The main focus of their paper is consistent estimation of some deep parameters when the model is misspec- ified with respect to some nuisance parameters. They also emphasize the necessity of correcting asymptotic variances formulas when there is a possi- bility of misspecification; such corrections are discussed in our chapter for the CMD estimators. In a recent paper, Kan and Robotti (2008), use the Hansen-Jagannathan distance in a Vuong-type test to compare potentially misspecified asset pric- ing models.28 We apply our methodology to compare two standard monetary business cycle models. The first model is a cash-in-advance (CIA) model, while the second model is the Lucas (1990) and Fuerst (1992) model with portfolio adjustment costs (PAC). The two models have the same underlying struc- ture except in the information sets that agents possess when making their decisions. In particular, we assume that the portfolio decisions must be made before the current period shocks are realized. We judge the perfor- mance of the models based on their ability to replicate the dynamics of the business cycles in the US. As our comparison criteria or reduced-form characteristics, we use the response of inflation and the growth rate of out- put to an unanticipated monetary shock. A structural vector autoregression 27As a matter of fact, our method can be viewed as an example of indirect inference without simulations. 28The Hansen-Jagannathan distance uses the second moments of returns as weights for the vector of pricing errors derived from the model. 69 3.2. Definitions (SVAR) is employed to obtain model-free estimates of the impulse responses against which we judge the performance of the two structural models. The structural shocks are identified using the Blanchard and Quah (1989) de- composition under the restriction of the long-run monetary neutrality of output. According to our results, the null hypothesis that the two models have the same lack of fit cannot be rejected on the basis of equally-weighted twenty-periods output and inflation impulse responses. We conclude that the assumed rigidity in portfolio choice and adjustment costs of the PAC model do not play a significant role in approximating inflation and output impulse response dynamics. The chapter proceeds as follows. Section 3.2 introduces the framework. Section 3.3 describes the asymptotic properties of CMD estimators under misspecification. Section 3.4 suggests a QLR-type statistic for model com- parison. We discuss the distribution of the suggested statistic in the cases of nested, strictly non-nested and overlapping models. In Section 3.5, we con- sider the situation when a model is estimated using one set of reduced-form parameters and evaluated with respect to another. Section 3.6 discusses the averaged and sup tests, and confidence sets for weighting schemes. Section 3.7 illustrates the technique with an empirical application. All proofs are in the Appendix. 3.2 Definitions This section formally defines calibration as CMD estimation and intro- duces the framework for comparison of two calibrated models. The definition 70 3.2. Definitions of calibration is similar to that of Gregory and Smith (1990); it is viewed as an example of CMD estimation (for a discussion of CMD see Newey and McFadden (1994)). CMD estimation of optimization based models was considered recently in the econometrics literature by Moon and Schorfheide (2002). Let 푌푛 (휔) be a data matrix of the sample size 푛 defined on the proba- bility space (Ω,ℱ , 푃 ). All random quantities in this chapter are some func- tions of the data 푌푛. We use ℎ to denote an 푚-vector of parameters of some reduced-form model. Its true value, ℎ0 ∈ 푅푚, depends on the true unknown structural model of the economy and its parameters. For example, ℎ can be a vector of moments, cross-correlations, impulse responses, etc. While the true structural model is unknown, we will assume that reduced-form parameter ℎ0 can be estimated consistently from the data. Let ℎ̂푛 denote an estimator of ℎ. We assume that ℎ̂푛 has the following properties. Assumption 1. (a) ℎ̂푛 →푝 ℎ0 ∈ 푅푚. (b) 푛1/2 ( ℎ̂푛 − ℎ0 ) →푑 푁 (0,Λ0), where Λ0 is positive definite 푚 ×푚 ma- trix. (c) There is Λ̂푛 such that Λ̂푛 →푝 Λ0. According to Assumptions 1(a) and (b), ℎ̂푛 is a consistent and asymptot- ically normal estimator of ℎ0. Similarly to ℎ0, its asymptotic variance, Λ0, depends on the unknown true structural model and its parameters. Part (c) of the assumption requires that Λ0 also can be estimated consistently from the data. The above assumptions are of high level; they can be verified under more primitive conditions. For example, Assumption 1 holds when ℎ0 and ℎ̂푛 71 3.2. Definitions are functions of the first two population and sample moments of 푌푛 respec- tively, 푌푛 = (푦 ′ 1, . . . , 푦 ′ 푛) ′, such that {푦푡} is a stationary mixing sequence with 휙 of size −푟/ (푟 − 1), 푟 ≥ 2, or 훼 of size −2푟/ (푟 − 2), 푟 > 2, 퐸 ∥푦푡∥4푟+훿 <∞, where ∥⋅∥ denotes the Euclidean norm, and 푉 푎푟 (푛−1/2∑푛푡=1 푦푡) is uniformly positive definite (White, 2001); alternatively, Assumption 1 can be verified using a linear processes structure under the conditions of Phillips and Solo (1992). In our framework, 푚 is fixed by the calibrationist and independent of the data. We assume that the calibrationist chooses ℎ and 푚 according to the economic importance of the reduced-form characteristics that a model is used to explain; note, however, that there are recent methods allowing one to choose 푚 using data and a statistical information criterion (Hall et al., 2007). Let 휃 ∈ Θ ⊂ 푅푘 be a vector of deep parameters corresponding to a structural model specified by the calibrationist. We assume that one can compute analytically the value of the reduced-form parameters ℎ given the model and a value of 휃. The mapping from the space of 휃 to the space of reduced-form parameters is given by the function 푓 : Θ→ 푅푚, which we call the binding function using the terminology of indirect inference (Gouriéroux et al., 1993; Dridi et al., 2007). In the remainder of the chapter, structural models are referred by their binding functions. Vector 휃 denotes only the free parameters that are estimated using the sample information; any preset parameters are included as constants into the binding function 푓 . Such parameters usually are assigned values based on extra-sample information, and we assume that model comparison and 72 3.2. Definitions evaluation is performed conditional on the choice of preset parameters. This is a common practice in calibration literature (Gregory and Smith, 1990). A calibrationist distinguishes between free parameters that must be estimated, and other parameters with values chosen on the basis of what is considered to be reasonable values in the literature, or by other methods independent of the data used to calibrate the free parameters. The presence of preset parameters is only a problem if one wants to treat the structural model as a true DGP, since such parameters are likely to be set to wrong values (see, for example a Monte Carlo experiment in Gregory and Smith (1990)). On the other hand, when the model is treated as misspecified, the preset parameters do not pose an additional challenge. Vector 휃 is chosen to minimize the distance between the sample reduced- form characteristics of the data, ℎ̂푛, and those implied by the chosen struc- tural model, 푓(휃). Let 퐴푛 be a possibly random 푚 × 푚 weight matrix. The weight matrix can be nonrandom and chosen by the calibrationist to put more weight on the relatively more important reduced-form parameters. Alternatively, it can be data dependent and, therefore, random. For exam- ple, in the spirit of GMM estimation, 퐴푛 can be set such that 퐴 ′ 푛퐴푛 = Λ̂ −1 푛 , which exists with probability approaching one due to Assumption 1(c). Assumption 2. 퐴푛 →푝 퐴, where 퐴 is of full rank. The calibrated 휃, or the CMD estimator of 휃, is given by the value that minimizes the weighted distance function: 휃̂푛 (퐴푛) = argmin 휃∈Θ ∥∥∥퐴푛 (ℎ̂푛 − 푓 (휃))∥∥∥2 . (3.1) The structural model is said to be correctly specified if for some value 73 3.2. Definitions 휃0 ∈ Θ the binding function 푓 produces exactly the true value of the reduced- form parameter. The following definition is similar to Definitions 1 and 2 of Hall and Inoue (2003). Definition 1. The structural model 푓 is said to be correctly specified if there exists some 휃0 ∈ Θ such that 푓 (휃0) = ℎ0; 푓 is said to be misspecified if inf휃∈Θ ∥(ℎ0 − 푓 (휃))∥ > 0. Naturally, the structural model chosen by the calibrationist is correctly specified in the sense of Definition 1 in the unlikely situation that 푓 is the true data generating process. Also, the model 푓 is correctly specified according to Definition 1 in the case of exact identification, i.e. when 푚 = 푘, even if the structural model and its binding function describe an incorrect DGP. Thus, the structural model is misspecified if it is overidentified and, for no value of 휃, it can replicate the reduced-form characteristics. The requirement on overidentification is a crucial one since an exactly identified model is never misspecified according to Definition 1 (see Hall and Inoue (2003) for a dis- cussion of overidentification and misspecification). Typically, the number of reduced-form parameters available for calibration exceeds 푘, and, therefore, the calibrationist can always choose the binding function and reduced-form parameters so that the model is overidentified. We assume that the calibrationist considers two competing structural models. The second structural model is given by the binding function 푔 and the vector of deep parameters 훾 ∈ Γ ⊂ 푅푙. Let 훾̂푛 be the calibrated value of 훾, where 훾̂푛 is constructed similarly to 휃̂푛 in (3.1): 훾̂푛 (퐴푛) = argmin 훾∈Γ ∥∥∥퐴푛 (ℎ̂푛 − 푔 (훾))∥∥∥2 . 74 3.2. Definitions We assume that 푓 and 푔 are both overidentified and misspecified in the sense of Definition 1. Assumption 3. 푓 and 푔 are misspecified according to Definition 1. Next, we define the pseudo-true values of the structural parameters 휃 and 훾. The pseudo-true value minimizes the distance between ℎ0 and the binding functions for a given weight matrix 퐴. Assumption 4. (a) There exists a unique 휃0 (퐴) ∈ Θ such that for all 휃 ∈ Θ, ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥ ≤ ∥퐴 (ℎ0 − 푓 (휃))∥ . (b) There exists a unique 훾0 (퐴) ∈ Γ such that for all 훾 ∈ Γ, ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥ ≤ ∥퐴 (ℎ0 − 푔 (훾))∥ . The pseudo-true value is written as a function of 퐴 to emphasize that different choices of weight matrix may lead to different minimizers of ∥퐴 (ℎ0 − 푓 (휃))∥ (see Maasoumi and Phillips (1982) and Hall and Inoue (2003)). For notational brevity, we may suppress the dependence on 퐴 if there is no ambiguity regarding the choice of 퐴. The uniqueness of 휃0 and 훾0 is usually assumed in the literature on misspecified models (see Assumption 3 of Rivers and Vuong (2002) and Assumption 3 of Hall and Inoue (2003)). The uniqueness of the pseudo-true value can be verified with the probabil- ity approaching one since the binding functions are known, 퐴푛 →푝 퐴, and ℎ̂푛 is a consistent estimator of ℎ0 by Assumption 1(a). When the pseudo- true value lies in the interior of Θ, it uniquely solves the following equation, 75 3.2. Definitions provided that 푓 is differentiable: ∂푓 (휃0 (퐴)) ′ ∂휃 퐴′퐴 (ℎ0 − 푓 (휃0 (퐴))) = 0. (3.2) Due to Assumption 2 on 퐴, ∂푓 (휃0 (퐴)) /∂휃 ′ must have rank 푘 for 휃0 (퐴) to be unique. The calibrationist’s objective is to choose between the two wrong models the one that provides a better 퐴-weighted fit to the reduced-form parameters ℎ0. We suggest a testing procedure for the null hypothesis that the two models are equally wrong 퐻0 : ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥ = ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥ , (3.3) against the alternatives in which one of the models provides a better fit. The calibrationist prefers the model 푓 if the following alternative is true. 퐻푓 : ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥ < ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥ . (3.4) Similarly, the calibrationist prefers the model 푔 when 퐻푔 : ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥ > ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥ is true. The hypotheses are analogous to those of Vuong (1989) and Rivers and Vuong (2002). Note that, in the current framework, the decision de- pends on the choice of the weight matrix 퐴. Thus, under the null, the two structural models provide equivalent fit for the reduced-form characteristics for a given weighting scheme 퐴. Naturally, different weighting schemes may lead to different ranking of 푓 and 푔. In order to test the null hypothesis in (3.3), it is natural to consider a sample counterpart of the difference in fit between the two competing models 76 3.3. Properties of the CMD Estimators of Structural Parameters which is given by the following QLR statistic 푄퐿푅푛 ( 휃̂푛 (퐴푛) , 훾̂푛 (퐴푛) ) = − ∥∥∥퐴푛 (ℎ̂푛 − 푓 (휃̂푛))∥∥∥2+∥∥∥퐴푛 (ℎ̂푛 − 푔 (훾̂푛))∥∥∥2 . (3.5) Given our assumptions, 푄퐿푅푛 consistently estimates the difference in the population measures of fit −∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥+∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥, as implied by the results presented in the next section. 3.3 Properties of the CMD Estimators of Structural Parameters In this section, we discuss the asymptotic properties of the CMD esti- mators defined in the previous section. We make the following assumptions about the binding functions 푓 and 푔 and their parameters’ spaces Θ and Γ. Assumption 5. (a) Θ and Γ are compact. (b) 휃0 lies in the interior of Θ; 훾0 lies in the interior of Γ. (c) 푓 is continuous on Θ; 푔 is continuous on Γ. The following theorem gives consistency of the CMD estimators of 휃 and 훾. Theorem 5. Suppose that Assumptions 1, 2, 4, and 5 hold. Then, 휃̂푛 →푝 휃0 and 훾̂푛 →푝 훾0. As usual, the asymptotic distribution of the CMD estimators centered around their pseudo-true values can be derived from the mean value expan- 77 3.3. Properties of the CMD Estimators of Structural Parameters sion of the sample first-order conditions for the minimization problem in (3.1). We make the following assumption. Assumption 6. The binding function 푓 is twice continuously differentiable in the neighborhood of 휃0; the binding function 푔 is twice continuously dif- ferentiable in the neighborhood of 훾0. It follows from Theorem 5 and Assumption 6 that the binding functions evaluated at the corresponding CMD estimators are twice continuously dif- ferentiable with probability approaching one. Thus, the CMD estimator of 휃 must satisfy the first-order conditions: ∂푓 ( 휃̂푛 ) ∂휃 ′ 퐴′푛퐴푛 ( ℎ̂푛 − 푓 ( 휃̂푛 )) = 0. Using the mean value theorem twice to expand 푓 ( 휃̂푛 ) around 푓 (휃0) and ∂푓 ( 휃̂푛 ) /∂휃′ around ∂푓 (휃0) /∂휃 ′, and taking into account the population first-order conditions (3.2), we obtain the following equation determining the asymptotic distribution of the CMD estimators in the misspecified case: ( 휃̂푛 − 휃0 ) = 퐹−1푛 ∂푓 ( 휃̂푛 )′ ∂휃 ×( 퐴′푛퐴푛 ( ℎ̂푛 − ℎ0 ) + ( 퐴′푛퐴푛 −퐴′퐴 ) (ℎ0 − 푓 (휃0)) ) ,(3.1) where 퐹푛 = ∂푓 ( 휃̂푛 )′ ∂휃 퐴′푛퐴푛 ∂푓 ( 휃̃푛 ) ∂휃′ −푀푓,푛, 푀푓,푛 = ( 퐼푘 ⊗ (ℎ0 − 푓 (휃0))′퐴′퐴 ) ∂ ∂휃′ 푣푒푐 ( ∂푓 ( 휃푛 ) ∂휃′ ) . In the above equations, 휃̃푛 and 휃푛 denote the mean values between 휃0 and 휃̂푛, and 푣푒푐 (⋅) denotes column vectorization of a matrix. The term 푀푓,푛, 78 3.3. Properties of the CMD Estimators of Structural Parameters which involves the second derivatives of 푓 , reflects the fact that the model is misspecified; 푀푓,푛 and the second summand in (3.1) are zero if the model is correctly specified. The analogous result holds for 푔 and 훾̂푛. This expansion is similar to equation (9) of Hall and Inoue (2003), however, in the case of CMD it involves one less term than in the GMM case. This is due to the fact that, in the case of CMD, the data and the parameters are additively separated: the data enters through ℎ, and the parameters through the binding function. The result in (3.1) is also similar to Assumptions 10 and 13 of Rivers and Vuong (2002). The expansion in (3.1) shows that the convergence rate of the CMD of structural parameters in the misspecified case depends on that of the reduced-form parameters and the weight matrices. In many situations, it is natural to assume that 푛1/2 ( ℎ̂푛 − ℎ0 ) is asymptotically normal as we do in Assumption 1(b). In regards to the weight matrices, Hall and Inoue (2003) distinguish several cases: (i) fixed weight matrices, (ii) 푛1/2푣푒푐 (퐴′푛퐴푛 −퐴′퐴) being asymptotically normal, and (iii) 퐴′푛퐴푛 being the inverse of centered or uncentered HAC estimator. In the current framework, the case of fixed weight matrices plays an important role, since the weight matrix defines the relative importance of various reduced-form characteristics of the data. Consider the correctly specified case: ℎ0 − 푓 (휃0) = 0. In this case, Assumptions 1, 2, 4, 5, and 6 and Theorem 5 imply that 푛1/2 ( 휃̂푛 − 휃0 ) has asymptotically normal distribution with the variance matrix( ∂푓 (휃0) ′ ∂휃 퐴′퐴 ∂푓 (휃0) ∂휃′ )−1 ∂푓 (휃0) ′ ∂휃 퐴′퐴Λ0퐴 ′퐴 ∂푓 (휃0) ∂휃′ ( ∂푓 (휃0) ′ ∂휃 퐴′퐴 ∂푓 (휃0) ∂휃′ )−1 . As usual, in the correctly specified case, the efficient CMD estimator cor- 79 3.3. Properties of the CMD Estimators of Structural Parameters responds to 퐴′푛퐴푛 = Λ̂ −1 푛 . However, when the model is misspecified, such a choice no longer leads to statistical efficiency. Furthermore, when Λ̂푛 is a HAC estimator and the model is misspecified, 휃̂푛 has a convergence rate slower that 푛1/2 as shown in Hall and Inoue (2003). In this chapter, we focus on cases (i) and (ii). Case (i) corresponds, for example, to a situation where the calibrationist knows the relative impor- tance of different reduced-form characteristics of the data. In case (ii), the matrix 퐴′푛퐴푛 can be given, for example, by the matrix of second moments of the data, as in the case of Hansen-Jagannathan distance (Kan and Robotti, 2008). Define 퐹0 = ∂푓 (휃0) ′ ∂휃 퐴′퐴 ∂푓 (휃0) ∂휃′ −푀푓,0, where (3.2) 푀푓,0 = ( 퐼푘 ⊗ (ℎ0 − 푓 (휃0))′퐴′퐴 ) ∂ ∂휃′ 푣푒푐 ( ∂푓 (휃0) ∂휃′ ) , and 퐺0 = ∂푔 (훾0) ′ ∂훾 퐴′퐴 ∂푔 (훾0) ∂훾′ −푀푔,0, where (3.3) 푀푔,0 = ( 퐼푙 ⊗ (ℎ0 − 푔 (훾0))′퐴′퐴 ) ∂ ∂훾′ 푣푒푐 ( ∂푔 (훾0) ∂훾′ ) . Assumption 7. 퐹0 and 퐺0 are non-singular. The above assumption is similar to Assumption 5 of Hall and Inoue 80 3.3. Properties of the CMD Estimators of Structural Parameters (2003). Theorem 5 implies that 퐹푛 →푝 퐹0. We define further 푉푓푓,0 = 퐹 −1 0 ∂푓 (휃0) ′ ∂휃 퐴′퐴Λ0퐴 ′퐴 ∂푓 (휃0) ∂휃′ 퐹 ′−10 , 푉푓푔,0 = 퐹 −1 0 ∂푓 (휃0) ′ ∂휃 퐴′퐴Λ0퐴 ′퐴 ∂푔 (훾0) ∂훾′ 퐺′−10 , 푉푔푔,0 = 퐺 −1 0 ∂푔 (훾0) ′ ∂훾 퐴′퐴Λ0퐴 ′퐴 ∂푔 (훾0) ∂훾′ 퐺′−10 , and 푉0 = ⎛⎝ 푉푓푓,0 푉푓푔,0 푉 ′푓푔,0 푉푔푔,0 ⎞⎠ . (3.4) The following theorem describes the asymptotic distribution of the CMD estimators in the fixed weight matrix case. Theorem 6. Suppose that 퐴푛 = 퐴 for all 푛 ≥ 1. Under Assumptions 1, 2, 4, and 5-7, 푛1/2 ⎛⎝ 휃̂푛 − 휃0 훾̂푛 − 훾0 ⎞⎠→푑 푁 (0(푘+푙)×1, 푉0) . When the weight matrix depends on the data, we extend Assumption 1 with Assumption 8 below, which assumes that the elements of 퐴′푛퐴푛 are root-푛 consistent and asymptotically normal estimators of the elements of 퐴′퐴 and they can be correlated with ℎ̂푛. Assumption 8. (a) 푛1/2 (( ℎ̂푛 − ℎ0 )′ , 푣푒푐 (퐴′푛퐴푛 −퐴′퐴)′ )′ →푑 푁 ( 0,Λ퐴0 ) , where Λ퐴0 is a positive definite 푚(푚+ 1)×푚(푚+ 1) matrix Λ퐴0 = ⎛⎝ Λ0 Λ0퐴 Λ′0퐴 Λ퐴퐴 ⎞⎠ . (b) 퐴 has full rank. (c) There is Λ̂퐴0 such that Λ̂ 퐴 0 →푝 Λ퐴0 . 81 3.3. Properties of the CMD Estimators of Structural Parameters This assumption is similar to condition (12) of Theorem 2 in Hall and Inoue (2003). In particular, it allows 퐴′푛퐴푛 to depend on ℎ̂푛, 휃̂푛, and 훾̂푛. However, as discussed above, Assumption 8 rules out HAC based estimators of 퐴′퐴. Now, in view of expansion (3.1) and a similar expansion for 훾̂푛, the asymptotic distribution of 휃̂푛 and 훾̂푛 depends on that of 퐴 ′ 푛퐴푛. Define 푉 퐴푓푓,0, 푉 퐴 푔푔,0, and 푉 퐴 푓푔,0 to be the asymptotic variance of 휃̂푛, the asymptotic variance of 훾̂푛, and the asymptotic covariance of 휃̂푛 and 훾̂푛 respectively: 푉 퐴푓푓,0 = 퐹 −1 0 ∂푓 (휃0) ′ ∂휃 퐷퐴푓,0Λ 퐴 0퐷 퐴′ 푓,0 ∂푓 (휃0) ∂휃′ 퐹 ′−10 , where 퐷퐴푓,0 = ( 퐴′퐴 퐼푚 ⊗ (ℎ0 − 푓 (휃0))′ ) ; (3.5) 푉 퐴푔푔,0 = 퐺 −1 0 ∂푔 (훾0) ′ ∂훾 퐷퐴푔,0Λ 퐴 0퐷 퐴′ 푔,0 ∂푔 (훾0) ∂훾′ 퐺′−10 , where 퐷퐴푔,0 = ( 퐴′퐴 퐼푚 ⊗ (ℎ0 − 푔 (훾0))′ ) ; 푉 퐴푓푔,0 = 퐹 −1 0 ∂푓 (휃0) ′ ∂휃 퐷퐴푓,0Λ 퐴 0퐷 퐴′ 푔,0 ∂푔 (훾0) ∂훾′ 퐺′−10 , and 푉 퐴0 = ⎛⎝ 푉 퐴푓푓,0 푉 퐴푓푔,0 푉 퐴′푓푔,0 푉 퐴 푔푔,0 ⎞⎠ . The joint asymptotic distribution of 휃̂푛 and 훾̂푛 is given in the next theorem. Theorem 7. Under Assumptions 2, 4, and 5-8, 푛1/2 ⎛⎝ 휃̂푛 − 휃0 훾̂푛 − 훾0 ⎞⎠→푑 푁 (0(푘+푙)×1, 푉 퐴0 ) . The asymptotic variance of 휃̂푛 and 훾̂푛 can be consistently estimated by the plug-in method, i.e. by replacing ℎ0 − 푓 (휃0) with ℎ̂푛 − 푓 ( 휃̂푛 ) and so on. 82 3.4. Model Comparison 3.4 Model Comparison The distribution of the 푄퐿푅푛 statistic in (3.5) depends on the relation- ship between the two models. Similarly to Vuong (1989), we consider the following three cases: nested, strictly non-nested, and overlapping models 푓 and 푔. Define ℱ = {ℎ ∈ 푅푚 : ℎ = 푓 (휃) , 휃 ∈ Θ} , 풢 = {ℎ ∈ 푅푚 : ℎ = 푔 (훾) , 훾 ∈ Γ} . The subsets of 푅푚, ℱ and 풢, represent the spaces for the reduced-form parameter ℎ that are spanned by the structural models 푓 and 푔 respectively. The relationship between the two structural models can be defined in terms of ℱ and 풢. Definition 2. The two structural models 푓 and 푔 are said to be (a) nested if ℱ ⊂ 풢 or 풢 ⊂ ℱ , (b) strictly non-nested if ℱ ∩ 풢 = ∅, (c) overlapping if ℱ ∩ 풢 ∕= ∅, ℱ ∕⊂ 풢, and 풢 ∕⊂ ℱ . Note that the nested case does not necessarily correspond to zero re- strictions on the elements of structural parameters. The two models can be totally different in terms of their construction so that 휃 and 훾 are not directly comparable, and still be nested with respect to the spaces they span for ℎ. The asymptotic behavior of the QLR statistic and resulting inference procedure depend on whether 푓 and 푔 are nested, strictly non-nested, or overlapping. 83 3.4. Model Comparison Further, note that in the strictly non-nested case, the two models provide absolutely different predictions for the reduced form characteristics for any values of the structural parameters. It appears therefore that in the calibra- tion context, the non-nested case is less realistic then nested and overlapping cases. 3.4.1 Nested Models Suppose that 풢 ⊂ ℱ . In this case, model 푔 cannot provide a better fit than model 푓 . Thus, in this case the calibrationist is interested in testing 퐻0 against 퐻푓 , i.e. whether the approximation to the reduced-form character- istics of the data obtained from the smaller model is equivalent to that from the bigger model. Since the models are nested, and under Assumption 4 of unique pseudo-true values, the null hypothesis can be equivalently stated as 푓 (휃0) = 푔 (훾0). Indeed, let ℎ푓,0 = 푓 (휃0) and ℎ푔,0 = 푔 (훾0). Then under the null of models equivalence we have ∥퐴 (ℎ0 − ℎ푓,0)∥ = ∥퐴 (ℎ0 − ℎ푔,0)∥. However, since the models are nested, ℎ푔,0 ∈ ℱ , and there should be some 휃̃0 ∈ Θ such that ℎ푔,0 = 푓 ( 휃̃0 ) which violates Assumption 4 if ℎ푓,0 ∕= ℎ푔,0. We have the following result. Lemma 1. Suppose that Assumption 4 holds, and the models 푓 and 푔 are nested according to Definition 2. Then, under 퐻0 in (3.3), 푓 (휃0) = 푔 (훾0). The QLR statistic depends on the weight matrix explicitly and through the estimators 휃̂푛 and 훾̂푛. The following theorem establishes the distribution of the QLR statistic in the case of fixed weight matrices. 84 3.4. Model Comparison Theorem 8. Suppose that 퐴푛 = 퐴 for all 푛 ≥ 1, 퐴 is of full rank, Assump- tions 1, 3, 4, 5-7 hold, and 풢 ⊂ ℱ . (a) Under 퐻0, 푛푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) →푑 푍 ′Λ ′1/2 0 퐴 ′퐴 (푊푔,0 −푊푓,0)퐴′퐴Λ1/20 푍, where 푍 ∼ 푁 (0, 퐼푚) , 푊푓,0 = 푊푓,0(1)−푊푓,0(2)−푊푓,0(3), 푊푓,0(1) = ∂푓 (휃0) ∂휃′ 퐹 ′−1 0 ∂푓 (휃0) ∂휃 ′ 퐴′퐴 ∂푓 (휃0) ∂휃′ 퐹−10 ∂푓 (휃0) ′ ∂휃 , 푊푓,0(2) = ∂푓 (휃0) ∂휃′ ( 퐹 ′−1 0 + 퐹 −1 0 ) ∂푓 (휃0)′ ∂휃 , 푊푓,0(3) = ∂푓 (휃0) ∂휃′ 퐹 ′−1 0 ( 푀 ′푓,0 +푀푓,0 ) 퐹−10 ∂푓 (휃0) ′ ∂휃 , and 푊푔,0 is defined analogously to 푊푓,0 with 휃0, ∂푓/∂휃, 퐹0, and 푀푓,0 replaced by 훾0, ∂푔/∂훾, 퐺0, and 푀푔,0 respectively. (b) Under 퐻푓 , 푛푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) → +∞ with probability one. According to Theorem 8, the re-scaled QLR statistic has a mixed 휒2 distribution under the null. This result is similar to the one established by Vuong (1989) for MLE in the case of nested models. According to part (a) of the theorem, one should reject the null when 푛푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) > 푐1−훼, where 푐1−훼 is the critical value satisfying 푃 ( 푛푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) > 푐1−훼∣퐻0 ) → 훼 as 푛→∞. Under the null, the distribution of the statistic is nonstandard 85 3.4. Model Comparison and depends on the unknown parameters ℎ0, 휃0, 훾0, and Λ0. However, its asymptotic distribution can be approximated by simulations using the con- sistent estimators of the unknown parameters. First, let 푊̂푓,푛 and 푊̂푔,푛 be the plug-in estimators of 푊푓 and 푊푔 defined in part (a) of Theorem 8. To construct 푊̂푓,푛 and 푊̂푔,푛, one replaces ℎ0, 휃0, 훾0, and Λ0 by ℎ̂푛, 휃̂푛, 훾̂푛, and Λ̂푛 respectively. Next, simulate a vector of 푁 (0, 퐼푚) random variables, 푍푟, and calculate 푄퐿푅푛푟 = 푍 ′ 푟Λ̂ ′1/2 푛 퐴 ′퐴 ( 푊̂푔,푛 − 푊̂푓,푛 ) 퐴′퐴Λ̂1/2푛 푍푟. As 푛 → ∞, the asymptotic distribution of 푄퐿푅푛푟 is given in part (a) of Theorem 8. Repeating this for 푟 = 1, . . . , 푅 with 푍푟 being drawn indepen- dently across 푟’s, the simulated critical value 푐1−훼,푛,푅 is the 1 − 훼 quantile of {푄퐿푅푛푟 : 푟 = 1, . . . , 푅}. Hence, in practice, in the case of nested models, one rejects the null when 푛푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) > 푐1−훼,푛,푅. When the weight matrix is data dependent, the following result estab- lishes the null distribution of the QLR statistic. Theorem 9. Suppose that Assumptions 3, 4, and 5-8 hold, and 풢 ⊂ ℱ . Then, under 퐻0, 푛푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) →푑 푍 ′ ( Λ퐴0 )′1/2 ( 푊퐴푔,0 −푊퐴푓,0 ) ( Λ퐴0 )1/2 푍, 86 3.4. Model Comparison where 푍 ∼ 푁 (0, 퐼푚(푚+1)) , 푊퐴푓,0 = 푊 퐴 푓,0(1)−푊퐴푓,0(2)−푊퐴푓,0 (3)−푊퐴푓,0 (4) , 푊퐴푓,0(1) = 퐷 퐴′ 푓,0 ∂푓 (휃0) ∂휃′ 퐹 ′−10 ∂푓 (휃0) ∂휃 ′ 퐴′퐴 ∂푓 (휃0) ∂휃′ 퐹−10 ∂푓 (휃0) ′ ∂휃 퐷퐴푓,0, 푊퐴푓,0(2) = ( 퐴′퐴 0 )′ ∂푓 (휃0) ∂휃′ 퐹−10 ∂푓 (휃0) ′ ∂휃 퐷퐴푓,0 +퐷퐴′푓,0 ∂푓 (휃0) ∂휃′ 퐹 ′−10 ∂푓 (휃0) ′ ∂휃 ( 퐴′퐴 0 ) , 푊퐴푓,0 (3) = 퐷 퐴′ 푓,0 ∂푓 (휃0) ∂휃′ 퐹 ′−10 ( 푀 ′푓,0 +푀푓,0 ) 퐹−10 ∂푓 (휃0) ′ ∂휃 퐷퐴푓,0, 푊퐴푓,0 (4) = ( 0 퐼푚 ⊗ (ℎ0 − 푓 (휃0))′ )′ ∂푓 (휃0) ∂휃′ 퐹−10 ∂푓 (휃0) ′ ∂휃 퐷퐴푓,0 +퐷퐴′푓,0 ∂푓 (휃0) ∂휃′ 퐹 ′−10 ∂푓 (휃0) ∂휃 ′ ( 0 퐼푚 ⊗ (ℎ0 − 푓 (휃0))′ ) . Here 퐷퐴푓,0 is defined in (3.5) and 푊 퐴 푔,0 is defined similarly to 푊 퐴 푓,0. As in the fixed 퐴 case, the null asymptotic distribution is mixed 휒2. The mixing matrices 푊퐴푓,0 and 푊 퐴 푔,0 depend on the unknown parameters, however, they can be consistently estimated by the plug-in method, and the critical values can be obtained by simulations as outlined above. 3.4.2 Strictly Non-nested Models In the case of strictly non-nested models, the space of reduced-form parameters generated under 푓 and 푔 does not have any common points. Either one of the models can be chosen as providing a better fit to the reduced-form parameters. In this case, consistent with Vuong (1989) and Rivers and Vuong (2002), under the null, the asymptotic distribution of re-scaled QLR statistic is normal. 87 3.4. Model Comparison Theorem 10. Suppose that 퐴푛 = 퐴 for all 푛 ≥ 1, and 퐴 has full rank. Suppose that Assumptions 1, 3, 4, 5-7 hold, and ℱ ∩ 풢 = ∅. Then, (a) 푛1/2푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) →푑 푁 ( 0, 휔20 ) , where 휔0 = 2 ∥∥∥Λ1/20 퐴′퐴 (푓 (휃0)− 푔 (훾0))∥∥∥, under 퐻0. (b) 푛1/2푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) →∞ with probability one under 퐻푓 ; 푛1/2푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) → −∞ with probability one under 퐻푔. Asymptotic normality of the QLR statistic in the non-nested case is due to the fact that, in the asymptotic expansion, there appears a dominating term of order 푂푝 ( 푛−1/2 ) , (푓 (휃0)− 푔 (훾0))′퐴′퐴 ( ℎ̂푛 − ℎ0 ) , as we show in the proof of the theorem in the appendix. When the models are nested, this term disappears because 푓 (휃0) = 푔 (훾0) under the null. In practice, the null should be rejected in favor of 퐻푓 when 푛1/2푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) /휔̂푛 > 푧1−훼/2, where 푧훼 is the 훼 quantile of the standard normal distribution, and 휔̂푛 = 2 ∥∥∥Λ̂1/2푛 퐴′퐴(푓 (휃̂푛)− 푔 (훾̂푛))∥∥∥ . The null should be rejected in favor of 퐻푔 when 푛1/2푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) /휔̂푛 < −푧1−훼/2. One can see from part (a) of Theorem 10 that 휔20 > 0 whenever 푓 (휃0) ∕= 푔 (훾0). This condition always holds when the models are non-nested in the sense of Definition 2.29 When 휔20 = 0, 푄퐿푅푛 has a mixed 휒 2 distribution as described in the previous subsection. 29This agrees with the conclusions in Section 6 of Rivers and Vuong (2002). 88 3.4. Model Comparison In the case of data dependent weight matrix, the following result provides the null asymptotic distribution of the QLR statistic. Theorem 11. Suppose that Assumptions 3, 4, and 5-8 hold, and ℱ ∩ 풢 = ∅. Then, under 퐻0, 푛 1/2푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) →푑 푁 ( 0, 휔2퐴,0 ) , where 휔퐴,0 is given by∥∥∥∥∥∥(Λ퐴0 )1/2 ⎛⎝ 2퐴′퐴 (푓 (휃0)− 푔 (훾0)) (ℎ0 − 푔 (훾0))′ ( 퐼푚 ⊗ (ℎ0 − 푔 (훾0))′ )− (ℎ0 − 푓 (휃0))′ (퐼푚 ⊗ (ℎ0 − 푓 (휃0))′) ⎞⎠∥∥∥∥∥∥ . Again, as in the case of fixed weight matrices, 휔퐴,0 is strictly positive unless the models are nested (푓 (휃0) = 푔 (훾0)), and the asymptotic variance 휔2퐴,0 can be consistently estimated by the plug-in method. 3.4.3 Overlapping Models The models are overlapping when the intersection of ℱ and 풢 is non- empty, however, neither model nests the other. One has to consider two possibilities when the models are overlapping. First, if 푓 (휃0) = 푔 (훾0), then 휔20 = 0 and 푛푄퐿푅푛 has an asymptotic mixed 휒 2 distribution. Second, if 푓 (휃0) ∕= 푔 (훾0) but ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥ = ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥, then 푛1/2푄퐿푅푛 is asymptotically normal. In order to test the null hypothesis, one has to determine which of the two possibilities applies. Vuong (1989) proposed the following sequential procedure when the models are overlapping. In the first step, one tests whether 푓 (휃0) = 푔 (훾0). This hypothesis is rejected when 푛푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) exceeds a critical value from the mixed 휒2 distribution, say 푐1−훼1 , where 훼1 denotes the significance level used in step one. If not rejected, then one concludes that 푓 (휃0) = 푔 (훾0) 89 3.5. Model Comparison with Estimation and Evaluation . . . and the two models have the same lack-of-fit. The null can be rejected ei- ther because 푓 (휃0) ∕= 푔 (훾0), but the models have the same lack of fit (퐻0 is true); or because one of the models has a better fit (퐻푓 or 퐻푔 are true). If 퐻0 : 푓 (휃0) = 푔 (훾0) is rejected in the first step, one continues to the second step. In the second step, 퐻0 : ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥ = ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥ is rejected when 푛1/2푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) /휔̂푛 > 푧1−훼2/2, in which case 푓 is the preferred model, or 푛1/2푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) /휔̂푛 < −푧1−훼2/2, in which case 푔 is preferred. Here 훼2 denotes the significance level in step two. When the weight matrix is data dependent, one should use a consistent estimator of 휔퐴,0 in place of 휔̂푛. If 퐻0 is not rejected in the second step, one concludes that the two models are equivalent. Vuong (1989) shows that the asymptotic significance level of the sequential procedure is max (훼1, 훼2). 3.5 Model comparison with estimation and evaluation on different sets of reduced-form parameters In the calibration literature, model parameters are often estimated or calibrated using one set of reduced-form characteristics, while the model evaluation is conducted on another. For example, a structural model can be estimated to match first moments, and evaluated with respect to second moments. Such case is discussed in this section; it is analogous to out-of- 90 3.5. Model Comparison with Estimation and Evaluation . . . sample model evaluation in the forecasting literature30; it also corresponds to the case of model comparison without lack-of-fit minimization in Rivers and Vuong (2002). We find that when a model is estimated and evaluated on different sets of reduced-form parameters, the QLR statistic has asymptotically normal distribution regardless of whether 푓 and 푔 are nested or non-nested. The reason is that even when the models are nested a bigger model does not necessarily provides a better fit, since the deep parameters are not calibrated to minimize the distance between the truth and the part of the model used for evaluation. This conclusion is in agreement with the results in Section 6 of Rivers and Vuong (2002). Next, we introduce the notation and assumptions of this section. We partition ℎ0 = ( ℎ′1,0, ℎ ′ 2,0 )′ , where ℎ1,0 is an 푚1-vector, and ℎ2,0 is an 푚2- vector, 푚1 + 푚2 = 푚. Similarly, we partition ℎ̂푛 = ( ℎ̂′1,푛, ℎ̂ ′ 2,푛 )′ , 푓 (휃) =( 푓1 (휃) ′ , 푓2 (휃) ′)′, and 푔 (훾) = (푔1 (훾)′ , 푔2 (훾))′. Next, consider the weight matrices 퐴1 and 퐴2, where 퐴푖 is 푚푖 × 푚푖, 푖 = 1, 2. At the estimation stage, the parameters are calibrated using only the first 푚1 reduced-form characteristics and the weight matrix 퐴1: 휃̂푛 (퐴1,푛) = argmin 휃∈Θ ∥∥∥퐴1,푛 (ℎ̂1,푛 − 푓1 (휃))∥∥∥2 , and 훾̂푛 (퐴1,푛) = argmin 훾∈Γ ∥∥∥퐴1,푛 (ℎ̂1,푛 − 푔1 (훾))∥∥∥2 . At the evaluation stage, the models are compared using the remaining 푚2 30See, for example, West and McCracken (1998) 91 3.5. Model Comparison with Estimation and Evaluation . . . reduced-form characteristics and the weight matrix 퐴2: 퐻0 : ∥퐴2 (ℎ2,0 − 푓2 (휃0 (퐴1)))∥ = ∥퐴2 (ℎ2,0 − 푔2 (훾0 (퐴1)))∥ . (3.1) 퐻푓 : ∥퐴2 (ℎ2,0 − 푓2 (휃0 (퐴1)))∥ < ∥퐴2 (ℎ2,0 − 푔2 (훾0 (퐴1)))∥ . (3.2) 퐻푔 : ∥퐴2 (ℎ2,0 − 푓2 (휃0 (퐴1)))∥ > ∥퐴2 (ℎ2,0 − 푔2 (훾0 (퐴1)))∥ . (3.3) We make the following assumption. Assumption 9. (a) 푓2 and 푔2 are misspecified according to Definition 1. (b) 퐴1,푛 →푝 퐴1, 퐴2,푛 →푝 퐴2; 퐴1 and 퐴2 have full ranks. (c) Assumptions 4 and 7 hold for 퐴1, 푓1, and 푔1. (d) ∂푓2(휃0(퐴1)) ′ ∂휃 퐴 ′ 2퐴2 (ℎ2,0 − 푓2 (휃0 (퐴1))) ∕= 0; ∂푔2(훾0(퐴1)) ′ ∂훾 퐴 ′ 2퐴2 (ℎ2,0 − 푔2 (훾0 (퐴1))) ∕= 0. According to part (a) of the assumption, the models are misspecified with respect to the second set of reduced-form parameters ℎ2. Note that the pseudo-true values of the parameters are defined with respect to 퐴1 and the first 푚1 reduced-form characteristics. Consequently, the first-order condition (3.2) does not hold for 푓2, 푔2, ℎ2, and 퐴2, since 휃0 (퐴1) and 훾0 (퐴1) are not the minimizers of the CMD criterion for the remaining 푚2 reduced- form characteristics, as described in part (d). The QLR statistic is now defined as 푄퐿푅푛 ( 휃̂푛 (퐴1,푛) , 훾̂푛 (퐴1,푛) , 퐴2,푛 ) = − ∥∥∥퐴2,푛 (ℎ̂2,푛 − 푓2 (휃̂푛 (퐴1,푛)))∥∥∥2 ∥∥∥퐴2,푛 (ℎ̂2,푛 − 푔2 (훾̂푛 (+퐴1,푛)))∥∥∥2 . 92 3.5. Model Comparison with Estimation and Evaluation . . . Define further 퐽푓,0 = ( −∂푓2(휃0(퐴1))∂휃′ 퐹−11,0 ∂푓1(휃0(퐴1)) ′ ∂휃 퐴 ′ 1퐴1 퐼푚2 ) , 퐽푔,0 = ( −∂푔2(훾0(퐴1))훾′ 퐺−11,0 ∂푔1(훾0(퐴1)) ′ ∂훾 퐴 ′ 1퐴1 퐼푚2 ) , where 퐹1,0 and 퐺1,0 are defined similarly to 퐹0 and 퐺0 in (3.2) and (3.3) respectively, but using 퐴1, ℎ1,0, 푓1, and 푔1. In the case of fixed weight matrices, we have the following result. Theorem 12. Suppose that Assumptions 1 and 9 hold, and 퐴1,푛 = 퐴1, 퐴2,푛 = 퐴2 for all 푛. (a) Under 퐻0 in (3.1), 푛 1/2푄퐿푅푛 ( 휃̂푛 (퐴1) , 훾̂푛 (퐴1) , 퐴2 ) →푑 푁 ( 0, 휔221,0 ) , where 휔21,0 = 2 ∥∥∥Λ1/20 (퐽 ′푔,0퐴′2퐴2 (ℎ2,0 − 푔2 (훾0 (퐴1)))− 퐽 ′푓,0퐴′2퐴2 (ℎ2,0 − 푓2 (휃0 (퐴1))))∥∥∥ . (3.4) (b) Under 퐻푓 in (3.2), 푛 1/2푄퐿푅푛 ( 휃̂푛 (퐴1) , 훾̂푛 (퐴1) , 퐴2 ) → ∞ with prob- ability one; under the alternative 퐻푔 in (3.3), 푛1/2푄퐿푅푛 ( 휃̂푛 (퐴1) , 훾̂푛 (퐴1) , 퐴2 ) → −∞ with probability one. As before the QLR statistic is asymptotically normal when the models are non-nested. Now, however, it is asymptotically normal also in the nested case. This is because there is no minimization of the lack-of-fit functions in (3.4). Thus, when the models are estimated using one set of reduced-form parameters and evaluated using another, one follows the rule regardless of 93 3.6. Averaged and Sup Tests for Model Comparison, . . . whether the models are nested, non-nested, or overlapping. One should reject the null of equivalent models when 푛1/2 ∣∣∣푄퐿푅푛 (휃̂푛 (퐴1) , 훾̂푛 (퐴1) , 퐴2)∣∣∣ /휔̂21,푛 > 푧1−훼/2, where 휔̂21,푛 is a consistent estimator of 휔21,0. A consistent estimator of 휔21,0 can be obtained by the plug-in method, since all the elements of 휔21,0 can be consistently estimated. Note that, when 푓2 (휃0 (퐴1)) = 푔2 (훾0 (퐴1)), which can occur if the models are nested or overlapping, the columns corresponding to 퐼푚2 in 퐽푓,0 and 퐽푔,0 do not contribute to the asymptotic variance; however, this will be reflected automatically by any consistent estimator 휔̂21,푛. When the weight matrices are data dependent, one can adjust the asymp- totic variance of the QLR statistic in a manner similar to that in Theorem 11. 3.6 Averaged and Sup tests for model comparison, and confidence sets for weight matrices The choice of the weight matrix 퐴 plays a crucial role when the models are misspecified: the null hypothesis (3.3) changes with different weight matrices, and as a result different weighing schemes can lead to different ranking of the models. One way to relax this dependence is to consider a procedure that takes into account the models’ performance for various weighting schemes. 94 3.6. Averaged and Sup Tests for Model Comparison, . . . 3.6.1 Averaged and Sup Tests In this section, we propose averaged and sup procedures for model compar- ison. We assume that the models are estimated and evaluated on the same set of reduced-form parameters. Let 픸 be a sub-space of 푚 ×푚 full-rank matrices, ∥퐴∥ = 푡푟 (퐴′퐴)1/2, ℬ (픸) be a 휎-field generated by open subsets of 픸, and 휋 be a probability measure on ℬ (픸). We make the following assumption. Assumption 10. (a) 픸 is compact. (b) Assumption 4 holds for all 퐴 ∈ 픸. The null hypothesis of the averaged procedure is stated as 퐻푎0 : ∫ 픸 ( ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥2 − ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥2 ) 휋 (푑퐴) = 0. According to 퐻푎0 , the two models 푓 and 푔 provide equivalent approximations to the true ℎ0 on average, where the average is taken in the class 픸 with respect to the probability measure 휋. For example, 픸 may consist of a finite number of matrices 퐴, and 휋 assigns equal weights to all 퐴’s. Note that the pseudo-true values 휃0 (퐴) and 훾0 (퐴) continue to depend on 퐴. The null hypothesis 퐻푎0 will be tested against alternatives 퐻푎푓 : ∫ 픸 ( ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥2 − ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥2 ) 휋 (푑퐴) > 0, or 퐻푎푔 : ∫ 픸 ( ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥2 − ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥2 ) 휋 (푑퐴) < 0. The null hypothesis of the sup procedure is given by 퐻푠0 : sup 퐴∈픸 ( ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥2 − ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥2 ) ≤ 0. 95 3.6. Averaged and Sup Tests for Model Comparison, . . . According to 퐻푠0 , the model 푓 cannot outperform the model 푔 for any con- sidered weight matrix 퐴 ∈ 픸. Thus, 퐻푠0 imposes a much stronger restriction than 퐻푎0 . The null 퐻 푠 0 will be tested against the following alternative: 퐻푠푓 : sup 퐴∈픸 ( ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥2 − ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥2 ) > 0. According to 퐻푠푓 , there is a weight matrix 퐴 such that model 푓 outperforms model 푔. Again, we consider the QLR statistic defined in (3.5), however, it is now explicitly indexed by 퐴: 푄퐿푅푛 ( 휃̂푛 (퐴) , 훾̂푛 (퐴) , 퐴 ) = ∥∥∥퐴(ℎ̂푛 − 푔 (훾̂푛 (퐴)))∥∥∥2 − ∥∥∥퐴(ℎ̂푛 − 푓 (휃̂푛 (퐴)))∥∥∥2 . (3.1) The averaged and sup statistics are given by: 퐴푄퐿푅푛 = ∫ 픸 푄퐿푅푛 ( 휃̂푛 (퐴) , 훾̂푛 (퐴) , 퐴 ) 휋 (푑퐴) , 푆푄퐿푅푛 = sup 퐴∈픸 푄퐿푅푛 ( 휃̂푛 (퐴) , 훾̂푛 (퐴) , 퐴 ) . The asymptotic null distributions and ranking of the models according to 퐴푄퐿푅푛 or 푆푄퐿푅푛 depend on the choice of the measure 휋. The asymptotic null distributions of the averaged and sup statistics also depend on whether 푓 and 푔 are nested or non-nested. When the models are nested, 풢 ⊂ ℱ , the model 푔 cannot outperform the model 푓 , and the inequality in 퐻푠0 holds as an equality. We have the following result. Theorem 13. Suppose that Assumptions 1, 3, 5-7, 10 hold, and 풢 ⊂ ℱ . Let 푍 ∼ 푁 (0, 퐼푚), and, for a given 퐴 ∈ 픸, define the matrices 푊푓,0 (퐴), 푊푔,0 (퐴) as 푊푓,0, 푊푔,0 in Theorem 8. 96 3.6. Averaged and Sup Tests for Model Comparison, . . . (a) Under 퐻푎0 , 푛퐴푄퐿푅푛 →푑 푍 ′Λ ′1/2 0 (∫ 퐴′퐴 (푊푔,0 (퐴)−푊푓,0 (퐴))퐴′퐴휋 (푑퐴) ) Λ 1/2 0 푍 . Under 퐻푎푓 , 푛퐴푄퐿푅푛 →∞ with probability one; under 퐻푎푔 , 푛퐴푄퐿푅푛 → −∞ with probability one. (b) Under 퐻푠0 , 푛푆푄퐿푅푛 →푑 sup 퐴∈픸 ( 푍 ′Λ ′1/2 0 퐴 ′퐴 (푊푔,0 (퐴)−푊푓,0 (퐴))퐴′퐴Λ1/20 푍 ) . Under 퐻푠푓 , 푛푆푄퐿푅푛 →∞ with probability one. According to Theorem 13, when the models are nested, the asymptotic distribution of the averaged statistic is mixed 휒2. However, the weights are now given by the average of matrices 푊푓,0 and 푊푔,0. Note that 푊푓,0, 푊푔,0, 퐹0, 푀푓,0 depend on 퐴. Since 푊푓,0 (퐴) and 푊푔,0 (퐴) can be estimated consistently by the plug-in method, the critical values of the mixed 휒2 dis- tribution can be computed by simulations as described in Section 3.4.1. The asymptotic null distribution of the sup statistic depends on the sup trans- formation of the mixed 휒2 distribution. Its critical values can be obtained by simulations as well. In the case of non-nested models, the asymptotic null distribution is a functional of a Gaussian process. Note that when the models are non-nested, 퐻푠0 does not determine the null distribution uniquely. It is a composite hypothesis, and the null distribution depends on whether the restriction is 97 3.6. Averaged and Sup Tests for Model Comparison, . . . binding or not, and the least favorable alternative, as usual, corresponds to the case when the restriction is binding. Theorem 14. Suppose that Assumptions 1, 3, 5-7, 10 hold, and ℱ ∩ 풢 = ∅. Let {푋 (퐴) ∈ 푅 : 퐴 ∈ 픸} be a mean zero Gaussian process such that the covariance of 푋 (퐴1) and 푋 (퐴2), 퐴1, 퐴2 ∈ 픸, is 휔0 (퐴1, 퐴2), where 휔0 (퐴1, 퐴2) = 4 (푓 (휃0 (퐴1))− 푔 (훾0 (퐴1)))′퐴′1퐴1Λ0퐴′2퐴2 (푓 (휃0 (퐴2))− 푔 (훾0 (퐴2))) . (a) Under 퐻푎0 , 푛 1/2퐴푄퐿푅푛 →푑 푁 ( 0, ∫ 픸 ∫ 픸 휔0 (퐴1, 퐴2)휋 (푑퐴1)휋 (푑퐴2) ) . Under 퐻푎푓 , 푛1/2퐴푄퐿푅푛 →∞ with probability one; under 퐻푎푔 , 푛1/2퐴푄퐿푅푛 → −∞ with probability one. (b) Under 퐻푠0 , lim푛→∞ 푃 ( 푛1/2푆푄퐿푅푛 > 푐 ) ≤ 푃 (sup퐴∈픸푋 (퐴) > 푐). Under 퐻푠푓 , 푛1/2푆푄퐿푅푛 →∞ with probability one. According to Theorem 14, the averaged statistic has a normal distribu- tion. The variance is given by the weighted average of variances and covari- ances of the QLR statistics for different 퐴’s; it can be estimated consistently by the plug-in method. For the sup statistic, the asymptotic distribution is that of the sup of the Gaussian process, and the critical values for a test based on 푆푄퐿푅푛 can be obtained by simulations. In the case of overlapping models, one can apply a sequential procedure similar to the one discussed in Section 3.4.3. 98 3.6. Averaged and Sup Tests for Model Comparison, . . . 3.6.2 Confidence Sets for Weight Matrices When all the considered models are misspecified, it is possible that model 푔 provides a better approximation to one set of reduced-form characteristics, say ℎ1, and model 푓 performs better on another set of ℎ. In such a case, it might be of interest to see how large the weight of ℎ1 has to be for model 푔 to be preferred to 푓 overall. Let 풜0 be a collection of weighting schemes under which 푔 is preferred to 푓 : 풜0 = {퐴 ∈ 픸 : ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥ − ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥ ≤ 0} . In this section, we discuss construction of a confidence set (CS) for 풜0. The CS for 풜0, 퐶푆푛,1−훼 is defined as lim 푛→∞ 푃 (퐴 ∈ 퐶푆푛,1−훼) ≥ 1− 훼, for all 퐴 ∈ 풜0, and can be constructed by inversion of the basic QLR test discussed in Section 3.4. First, given 퐴 ∈ 픸, compute 푄퐿푅푛 (퐴). Next, test 퐻0 : 퐴 ∈ 풜0 as follows: reject 퐻0 when 푄퐿푅푛 (퐴) > 푧1−훼휔̂푛/ √ 푛, if the models are non-nested. If the models are nested, assuming that 풢 ⊂ ℱ , one can use the mixed 휒2 critical values as described in Section 3.4.1 to test 퐻0. If the models are overlapping, one can apply the sequential procedure of Section 3.4.3. The confidence set 퐶푆푛,1−훼 is given by the collection of all 퐴 for which 퐻0 : 퐴 ∈ 풜0 cannot be rejected. 99 3.7. Application 3.7 Application In this section we apply our proposed test to the two monetary macroe- conomic models, the cash-in-advance (CIA) model and the portfolio adjust- ment cost (PAC) model. Detailed discussions of these models can be found in Christiano (1991) and Christiano and Eichenbaum (1992). We compare the performance of the two models based on their ability to match the re- sponses of output and inflation to a monetary growth shock. Therefore, the latter impulse responses comprise ℎ0 - the vector of reduced form charac- teristics of interest in our application. We obtain a consistent estimate of ℎ0 from a structural vector autoregression (SVAR) model of GDP and infla- tion. The identification scheme employed for the SVAR follows Blanchard and Quah (1989). The particular restriction applied to identify the SVAR model is that money is neutral in the long run, which is satisfied by both CIA and PAC models. Since both CIA and PAC models are standard in the literature, we out- line them only briefly below. We also want to compare the results of our testing procedure with those obtained by Schorfheide (2000). For this pur- pose we follow his models specifications closely. 3.7.1 CIA Model The model economy is populated by a representative household, a firm, and a financial intermediary. At the beginning of period 푡 the household owns the economy’s entire money stock푀푡 and decides how to allocate it between purchases of consumption goods and deposits in the financial intermediary, 100 3.7. Application 푀푡 − 푄푡, where 푄푡 is money allocated to purchases of consumption goods. Consumption purchases must be financed with 푄푡 and wage earnings. Thus, the objective of the household is to choose real consumption, 퐶푡, working hours, 퐻푡, and nominal deposit, 푀푡 −푄푡, to solve the following problem: max {퐶푡,퐻푡,푀푡+1,푄푡} 피0 [ ∞∑ 푡=0 훽푡 [(1− 휙) ln퐶푡 + 휙 ln (1−퐻푡)] ] , subject to 푃푡퐶푡 ≤ 푄푡 +푊푡퐻푡, 푄푡 ≤ 푀푡, 푀푡+1 = (푄푡 +푊푡퐻푡 − 푃푡퐶푡) +푅퐻,푡 (푀푡 −푄푡) + 퐹푡 +퐵푡. Here 피푡 denotes conditional expectation at date 푡, 훽 is the subjective dis- count factor, and 휙 is the share of leisure in per period utility. 푃푡 denotes economy’s price level, while 푊푡 and 푅퐻,푡 denote nominal wage rate and re- turn on deposit. Household also receives nominal profits paid by the firm, 퐹푡, and the financial intermediary, 퐵푡. The production technology in the economy is 푌푡 = 퐾 훼 푡 (풜푡푁푡)1−훼, where 퐾푡, 푁푡, and 풜푡 are capital stock, labor input, and labor-augmenting tech- nology, respectively. Firms must pay the total wage bill up-front to workers, so they borrow 푊푡푁푡 from financial intermediary. Loans must be repaid at the end of period 푡. The representative firm’s problem is max 퐹푡,퐾푡+1,푁푡,퐿푡 피0 [ ∞∑ 푡=0 훽푡+1 퐹푡 퐶푡+1푃푡+1 ] , subject to 퐹푡 ≤ 퐿푡 + 푃푡 [푌푡 −퐾푡+1 + (1− 훿)퐾푡]−푊푡푁푡 − 퐿푡푅퐹,푡, 푊푡푁푡 ≤ 퐿푡. 101 3.7. Application The objective of the financial intermediary in this economy is simple. At the beginning of each period, it loans out the household’s deposit 푀푡 − 푄푡 and the money injection 푋푡 received from the central bank to the firm. At the end of period, it collects the loan plus interest 퐿푡푅퐹,푡 and pays the amount to the household. The household, firm and financial intermediary all take prices as given. Technology 풜푡 and money growth rate 푚푡 =푀푡+1/푀푡 follow stochastic processes ln풜푡 = 휓 + ln풜푡−1 + 휖풜,푡, with 휖풜,푡 ∼ 푁 ( 0, 휎2풜 ) , ln푚푡 = (1− 휌) ln푚푠푠 + 휌 ln푚푡−1 + 휖푀,푡, with 휖푀,푡 ∼ 푁 ( 0, 휎2푀 ) . Here 푚푠푠 is the steady state inflation rate, and 휓, 휌 are parameters. To solve the model, we first re-scale all real variables by technology level 풜푡, prices by 푀푡/풜푡, and nominal variables by 푀푡. Then we log-linearize the equilibrium conditions around the deterministic steady state and solve the resulting system of linear difference equations. The state space repre- sentation for the exogenous and endogenous state variables is 푘̂푡+1 = 휂1푘̂푡 + 휂2푎̂푡 + 휂3푚̂푡, 푎̂푡+1 = 휖풜,푡+1, 푚̂푡+1 = 휌푚̂푡 + 휖푀,푡+1, where ‘ˆ’ over a variable is used to denote a log deviation of that (re-scaled) variable from its steady state value. The coefficients 휂1, 휂2, and 휂3 are func- tions of model parameters 푚푠푠, 훼, 훽, 훿, 휓, 휙, and 휌. We use this state space 102 3.7. Application representation to obtain the theoretical impulse responses of the model. These impulse responses are conditional on the structural model parame- ters 휃 = [푚푠푠, 훼, 훽, 훿, 휓, 휙, 휌, 휎 2 풜, 휎 2 푀 ] ′. In the minimum distance estimation, we look for vector 휃̂ that minimizes the distance between the theoretical im- pulse responses for output growth and inflation and the impulse responses generated by the data. 3.7.2 PAC Model The production function and stochastic processes governing technology and money growth in the PAC model are the same as in the CIA model. The key difference between the two model is in the information sets that the household faces. In particular, in the PAC model the household’s contin- gency plan for deposit holdings is not a function of period-푡 realizations of shocks. This rigidity of 푄푡 implies that any positive money shock must be absorbed by firms. For firms to be willing to do so voluntarily, the interest rate must fall. To make this liquidity effect persistent, Christiano (1991) introduce the second distinct feature of the PAC model - the existence of an adjustment cost 푝̃푡 given by 푝̃푡 = 훼1 [ exp ( 훼2( 푄푡 푄푡−1 −푚푠푠) ) + exp ( −훼2( 푄푡 푄푡−1 −푚푠푠) ) − 2 ] . The household’s problem in the PAC model is max {퐶푡,퐻푡,푀푡+1,푄푡+1} 퐸0 [ ∞∑ 푡=0 훽푡[(1− 휙) ln퐶푡 + 휙 ln(1−퐻푡 − 푝̃푡)] ] , 103 3.7. Application subject to 푃푡퐶푡 ≤ 푄푡 +푊푡퐻푡, 푄푡 ≤ 푀푡, 푀푡+1 = (푄푡 +푊푡퐻푡 − 푃푡퐶푡) +푅퐻,푡(푀푡 −푄푡) + 퐹푡 +퐵푡. The firm’s problem and the financial intermediary’s problem are identical to those in the CIA model. We solve this model and calculate its theoret- ical impulse responses using the same procedure as for CIA model. These impulse responses are conditional on the set of structural model parameters 훾 = [푚푠푠, 훼, 훽, 훿, 휓, 휙, 휌, 휎 2 풜, 휎 2 푀 , 훼1, 훼2] ′. As before, we use the minimum distance estimation to find vector 훾̂ that minimizes the distance between the theoretical impulse responses for output growth and inflation and the impulse responses generated by the data. 3.7.3 Model Estimation and Comparison Results The CIA and PAC models both provide predictions for evolution of multiple time series. In this application we focus on the growth rates of GDP per capita and price level. Thus, vector ℎ0 consists of twenty-periods output growth impulse responses and 20 periods inflation impulse responses to a money growth shock. We use SVAR model on the GDP per capita growth and inflation series to obtain the consistent estimate of ℎ0, which we denote ℎ̂푛. The data used in the empirical analysis are the US GDP per capita growth rate and inflation rate available from the Basic Economics 104 3.7. Application database produced by DRI/McGraw-Hill.31 Our sample covers the 1947:Q2- 2003:Q3. To conduct our testing procedure, we search for (a) values of parameter vector 휃 = [푚푠푠, 훼, 훽, 훿, 휓, 휙, 휌, 휎 2 풜, 휎 2 푀 ] ′ that minimize the distance between the theoretical impulse responses in the CIA model, 푓(휃), and empirical impulse responses from SVAR model, ℎ̂푛; (b) values of the parameter vector 훾 = [푚푠푠, 훼, 훽, 훿, 휓, 휙, 휌, 휎 2 풜, 휎 2 푀 , 훼1, 훼2] ′, which minimize the corresponding distance for the PAC model, 푔(훾). To reduce the computation time, in our calibration exercise we fix the values of some parameters. Following Christiano and Eichenbaum (1992), we set 훼 = 0.36, 훽 = (1.03)−0.25, 휙 = 0.797, 훿 = 0.012. We borrow the value of 휎풜 = 0.014 from Christiano (1991). We calibrate the rest of the parameters. Because our procedure requires that the parameter vectors are defined on compact sets, we restrict the ranges of models parameters as follows. We assume that the steady state growth rate of money, 푚푠푠, belongs to [0, 0.05]; the steady state growth rate of productivity, 휓, belongs to [0, 0.1]. Persistence of money shock is between 0 and 1, 휌 ∈ [0, 1]. We assume that 휎푀 ∈ [0.0001, 0.004]. Finally, we restrict the range for parameters in the adjustment cost technology. Note that in the log-linearized PAC model parameters 훼1 and 훼2 only enter through a combination 훼1훼 2 2. Therefore, we can only identify 훼1훼 2 2. In the calibration procedure we draw on Christiano and Eichenbaum (1992) who set 훼1 = 0.00005 and 훼2 = 1000, and restrict 31The GDP per capita series are obtained as a ratio of GDP series (GDP215 in the DRI database) and population series (POP in the DRI database). For the price level, we choose the GDP deflator series (GDPD15 in the DRI database). 105 3.7. Application 훼1훼 2 2 ∈ [10, 90]. Table 3.1 summarizes the ranges for models parameters and their estimates with the weighting matrix being an identity matrix (퐴푛 = 퐴 = 퐼). 106 3 .7 . A p p lica tio n Table 3.1: CIA and PAC Parameters’ Estimates and Their Standard Errors Parameter Range CIA estimates PAC estimates 훼 capital share - 0.36 0.36 fixed fixed 훽 discount factor - 0.9926 0.9926 fixed fixed 훿 depreciation - 0.012 0.012 fixed fixed 휙 leisure share in utility - 0.797 0.797 fixed fixed 휎풜 std.dev. of productivity innovations - 0.014 0.014 fixed fixed 휓 steady state productivity growth [0,0.1] 0.0001 0.1 (3.71E-07) (2.58E-05) 휌 money shock persistence [0,1] 0.89 0.85 (2.35E-05) (4.84E-04) 휎푀 std.dev. of money growth innovations [0.0001,0.004] 0.0024 0.0032 (6.90E-03) (8.80E-03) 푚푠푠 steady state money growth [0,0.05] 0.001 0.05 (2.49E-05) (2.89E-04) 훼1훼22 adjustment cost parameter [10, 90] - 32.86 (130.20)107 3.7. Application Figures 3.7.3 and 3.7.3 plot the models’ prediction errors for the impulse responses of inflation and output, ℎ̂푛− 푓 ( 휃̂푛 ) and ℎ̂푛− 푔 (훾̂푛). Figure 3.7.3 suggests that both CIA and PAC models attain some success in replicating inflation dynamics, while Figure 3.7.3 indicates that both models lack in their ability to match the real-side dynamics. These results agree with the findings of Nason and Cogley (1994). At the same time, PAC model has a marginally better fit to the data than the CIA model in terms of the output impulse responses.32 The reason is that the CIA model generates virtually no output dynamics. In the CIA model the households can always rebalance their money holdings to nullify the real effect of money growth shock. In contrast, the PAC model generates a positive output response to money shock although the magnitude is much smaller than that of SVAR. The better fit provided by the PAC model is not surprising though, since it is richer and nests the CIA model. In order to determine whether the better performance of the PAC model in approximating the impulse responses is statistically significant, we com- pute the test statistic proposed in Section 3.4.1. The value of the 푄퐿푅푛 statistic is equal 0.0008. Since the distribution of the test statistic has a mixed 휒240 distribution, we simulate its critical values. The 5% and 10% critical values are 0.1192 and 0.1050, respectively, both bigger than 0.0008. The p-value of the test is 0.4905. Therefore we fail to reject that the CIA model fits the data as well as the PAC model. We conclude that both CIA 32This is in agreement with the findings of Schorfheide (2000) that the PAC impulse response dynamics provide a better approximation to the posterior mean impulse response function than the CIA model. 108 3.7. Application and PAC models provide equally poor fit to the output and inflation re- sponses to the money supply shocks in the data. Our findings indicate that the rigidities underlying the persistent liquidity effect in the PAC model do not play a significant role in approximating the inflation and output impulse response dynamics. 0 5 10 15 20 −4 −3 −2 −1 0 1 2 3 4 x 10−3 0 5 10 15 20 −4 −3 −2 −1 0 1 2 3 4 x 10−3 Left panel: PAC, right panel: CIA Figure 3.1: Model Prediction Errors of the Inflation Impulse Responses with 95% Confidence Bands 0 5 10 15 20 −4 −2 0 2 4 6 8 x 10−3 0 5 10 15 20 −4 −2 0 2 4 6 8 x 10−3 Left panel: PAC, right panel: CIA Figure 3.2: Model Prediction Errors of the Output Impulse Responses with 95% Confidence Bands 109 3.8. Proofs of Theorems 3.8 Proofs of Theorems Proof of Theorem 5. For consistency of 휃̂푛, it is sufficient to show uniform convergence of ∥∥∥퐴푛 (ℎ̂푛 − 푓 (휃))∥∥∥2 to ∥퐴 (ℎ0 − 푓 (휃))∥2 on Θ. The desired result will follow from Assumptions 4 and 5 by the usual argument for ex- tremum estimators (see, for example, Theorem 2.1 in Newey and McFadden (1994)).∥∥∥퐴푛 (ℎ̂푛 − 푓 (휃))∥∥∥2 − ∥퐴 (ℎ0 − 푓 (휃))∥2 = 푅1,푛 − 2푅2,푛 (휃) +푅3,푛 (휃) , where 푅1,푛 = ℎ̂푛퐴 ′ 푛퐴푛ℎ̂푛 − ℎ′0퐴′푛퐴푛ℎ0, 푅2,푛 (휃) = ( ℎ̂푛 − ℎ0 )′ 퐴′푛퐴푛푓 (휃) 푅3,푛 (휃) = (ℎ0 − 푓 (휃))′ ( 퐴′푛퐴푛 −퐴′퐴 ) (ℎ0 − 푓 (휃)) . By Assumption 1(a) and 2, ∣푅1,푛∣ →푝 0. Let ∥퐴∥ = 푡푟 (퐴′퐴)1/2. Due to Assumption 5 (a) and (c), 푓 is bounded on Θ (Davidson, 1994, Theorem 2.19), and, therefore, sup 휃∈Θ ∣푅2,푛 (휃)∣ ≤ ∥퐴푛∥2 ∥∥∥(ℎ̂푛 − ℎ0)∥∥∥ sup 휃∈Θ ∥푓 (휃)∥ →푝 0, by Assumptions 1(a) and 2. sup 휃∈Θ ∣푅3,푛 (휃)∣ ≤ ∥∥퐴′푛퐴푛 −퐴′퐴∥∥ sup 휃∈Θ ∥ℎ0 − 푓 (휃)∥2 ≤ ∥∥퐴′푛퐴푛 −퐴′퐴∥∥(∥ℎ0∥+ sup 휃∈Θ ∥푓 (휃)∥ )2 →푝 0. 110 3.8. Proofs of Theorems The proof of 훾̂푛 →푝 훾0 is identical with 푓 and 휃 replaced by 푔 and 훾. ■ Proof of (3.1). First, applying the mean value expansion to 푓 ( 휃̂푛 ) , 0 = ∂푓 ( 휃̂푛 ) ∂휃 ′ 퐴′푛퐴푛 ( ℎ̂푛 − 푓 ( 휃̂푛 )) = ∂푓 ( 휃̂푛 ) ∂휃 ′ 퐴′푛퐴푛 ⎛⎝ℎ̂푛 − 푓 (휃0)− ∂푓 ( 휃̃푛 ) ∂휃′ ( 휃̂푛 − 휃0 )⎞⎠ = ∂푓 ( 휃̂푛 ) ∂휃 ′ ( 퐴′푛퐴푛 ( ℎ̂푛 − ℎ0 ) + ( 퐴′푛퐴푛 −퐴′퐴 ) (ℎ0 − 푓 (휃0)) ) + ∂푓 ( 휃̂푛 ) ∂휃 ′ 퐴′퐴 (ℎ0 − 푓 (휃0))− ∂푓 ( 휃̂푛 ) ∂휃 ′ 퐴′푛퐴푛 ∂푓 ( 휃̃푛 ) ∂휃′ ( 휃̂푛 − 휃0 ) , where 휃̃푛 is the mean value. Next, ∂푓 ( 휃̂푛 ) ∂휃 ′ 퐴′퐴 (ℎ0 − 푓 (휃0)) = ( 퐼푘 ⊗ (ℎ0 − 푓 (휃0))′퐴′퐴 ) 푉 푒푐 ⎛⎝∂푓 ( 휃̂푛 ) ∂휃′ ⎞⎠ = ∂푓 (휃0) ∂휃 ′ 퐴′퐴 (ℎ0 − 푓 (휃0)) + ( 퐼푘 ⊗ (ℎ0 − 푓 (휃0))′퐴′퐴 ) ∂ ∂휃′ 푉 푒푐 ( ∂푓 ( 휃푛 ) ∂휃′ )( 휃̂푛 − 휃0 ) = 푀푓,푛 ( 휃̂푛 − 휃0 ) , (3.1) where 휃푛 is the mean value; note that the last equality follows from the population first-order condition (3.2). ■ Proof of Theorem 6. One can expand the first-order conditions for 훾̂푛 similarly to that of 휃̂푛, equation (3.1). Taking into account that 퐴푛 = 퐴 for 111 3.8. Proofs of Theorems all 푛, 푛1/2 ⎛⎝ 휃̂푛 − 휃0 훾̂푛 − 훾0 ⎞⎠ = ⎛⎝ 퐹−1푛 ∂푓(휃̂푛)′∂휃 퐺−1푛 ∂푔(훾̂푛) ′ ∂훾 ⎞⎠퐴′퐴푛1/2 (ℎ̂푛 − ℎ0) , where 퐺푛 = ∂푔 (훾̂푛) ′ ∂훾 퐴′푛퐴푛 ∂푔 (훾̃푛) ∂훾′ −푀푔,푛, 푀푔,푛 = ( 퐼푙 ⊗ (ℎ0 − 푔 (훾0))′퐴′퐴 ) ∂ ∂훾′ 푣푒푐 ( ∂푔 (훾푛) ∂훾′ ) , and 훾̃푛, 훾푛 are between 훾̂푛 and 훾0. The result follows from Theorem 5, As- sumptions 1(b) and 7. ■ Proof of Theorem 7. The result follows immediately from (3.1), a similar expansion for 훾̂푛, and the assumptions of the theorem by writing⎛⎝ 휃̂푛 − 휃0 훾̂푛 − 훾0 ⎞⎠ = ⎛⎜⎝ 퐹−1푛 ∂푓(휃̂푛) ′ ∂휃 ( 퐴′푛퐴푛 퐼푚 ⊗ (ℎ0 − 푓 (휃0))′ ) 퐺−1푛 ∂푔(훾̂푛) ′ ∂훾 ( 퐴′푛퐴푛 퐼푚 ⊗ (ℎ0 − 푔 (훾0))′ ) ⎞⎟⎠ ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠ . ■ Proof of Theorem 8. In the case of the fixed weight matrix, using (3.1) the following expansion is obtained.∥∥∥퐴(ℎ̂푛 − 푓 (휃̂푛))∥∥∥2 = ∥∥∥퐴(ℎ̂푛 − 푓 (휃0))∥∥∥2 + (ℎ̂푛 − ℎ0)′퐴′퐴푊푓,푛퐴′퐴(ℎ̂푛 − ℎ0) +표푝 ( 푛−1 ) , (3.2) 112 3.8. Proofs of Theorems where 푊푓,푛 = 푊푓,푛(1)−푊푓,푛(2)−푊푓,푛(3), 푊푓,푛(1) = ∂푓 ( 휃̂푛 ) ∂휃′ 퐹 ′−1 푛 ∂푓 ( 휃̃푛 ) ∂휃 ′ 퐴′퐴 ∂푓 ( 휃̃푛 ) ∂휃′ 퐹−1푛 ∂푓 ( 휃̂푛 )′ ∂휃 , 푊푓,푛(2) = ∂푓 ( 휃̂푛 ) ∂휃′ 퐹 ′−1 푛 ∂푓 ( 휃̃푛 ) ∂휃 ′ + ∂푓 ( 휃̃푛 ) ∂휃′ 퐹−1푛 ∂푓 ( 휃̂푛 )′ ∂휃 , 푊푓,푛(3) = ∂푓 ( 휃̂푛 ) ∂휃′ 퐹 ′−1 푛 ( 푀 ′푓,푛 +푀푓,푛 ) 퐹−1푛 ∂푓 ( 휃̂푛 )′ ∂휃 . To show (3.2), write∥∥∥퐴(ℎ̂푛 − 푓 (휃̂푛))∥∥∥2 = ∥∥∥퐴(ℎ̂푛 − 푓 (휃0))∥∥∥2 + 푆1,푛 + 푆2,푛 + 푆3,푛, (3.3) where 푆1,푛 = ( 푓 ( 휃̂푛 ) − 푓 (휃0) )′ 퐴′퐴 ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) , 푆2,푛 = −2 ( ℎ̂푛 − ℎ0 )′ 퐴′퐴 ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) , 푆3,푛 = −2 (ℎ0 − 푓 (휃0))′퐴′퐴 ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) . Now, one obtains (3.2) by expanding 푓 ( 휃̂푛 ) in 푆1,푛, 푆2,푛, and 푆3,푛 around 푓 (휃0) and using (3.1); in the case of 푆3,푛, after expanding 푓 ( 휃̂푛 ) , one can apply the result in (3.1) to (ℎ0 − 푓 (휃0))′퐴′퐴 ( ∂푓 ( 휃̃푛 ) /∂휃′ ) , which leads to 푀푓,푛 in the expression for 푊푓,푛(3). An expansion similar to (3.2) is available for ∥∥∥퐴(ℎ̂푛 − 푔 (훾̂푛))∥∥∥2 with 푓 , 휃, and 퐹 replaced by 푔, 훾, and 퐺. Hence, 푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) = − ∥∥∥퐴(ℎ̂푛 − 푓 (휃0))∥∥∥2 + ∥∥∥퐴(ℎ̂푛 − 푔 (훾0))∥∥∥2 + ( ℎ̂푛 − ℎ0 )′ 퐴′퐴 (푊푔,푛 −푊푓,푛)퐴′퐴 ( ℎ̂푛 − ℎ0 ) . (3.4) 113 3.8. Proofs of Theorems Under the null, the first summand on the right-hand side of (3.4) is zero by Lemma 1, and, due to Assumption 1(c) and Theorem 6, the second summand, when multiplied by 푛, converges in distribution to the random variable defined in part (a) of the theorem. Since under 퐻푓 , ∥퐴 (ℎ0 − 푓 (휃0))∥2 ≤ ∥퐴 (ℎ0 − 푔 (훾0))∥2, part (b) of the theorem follows. ■ Proof of Theorem 9. As in the proof of Theorem 8, write∥∥∥퐴푛 (ℎ̂푛 − 푓 (휃̂푛))∥∥∥2 = ∥∥∥퐴푛 (ℎ̂푛 − 푓 (휃0))∥∥∥2 + 푆퐴1,푛 + 푆퐴2,푛 + 푆퐴3,푛, (3.5) where 푆퐴1,푛 = ( 푓 ( 휃̂푛 ) − 푓 (휃0) )′ 퐴′푛퐴푛 ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) , 푆퐴2,푛 = −2 ( ℎ̂푛 − ℎ0 )′ 퐴′푛퐴푛 ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) , 푆퐴3,푛 = −2 (ℎ0 − 푓 (휃0))′퐴′푛퐴푛 ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) . Under the null in the nested case, 푓 (휃0) = 푔 (휃0), and therefore ∥∥∥퐴푛 (ℎ̂푛 − 푓 (휃0))∥∥∥2 = ∥∥∥퐴푛 (ℎ̂푛 − 푔 (훾0))∥∥∥2. Define 퐷퐴푓,푛 = ( 퐴′푛퐴푛 퐼푚 ⊗ (ℎ0 − 푓 (휃0))′ ) . By expanding 푓 ( 휃̂푛 ) around 푓 (휃0) and using (3.1), we obtain the following expression for 푆퐴1,푛:⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠′퐷퐴′푓,푛∂푓 ( 휃̂푛 ) ∂휃′ 퐹 ′−1푛 ∂푓 ( 휃̃푛 )′ ∂휃 퐴′푛퐴푛 × ∂푓 ( 휃̃푛 ) ∂휃′ 퐹−1푛 ∂푓 ( 휃̂푛 )′ ∂휃 퐷퐴푓,푛 ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠ , 114 3.8. Proofs of Theorems where 휃̃푛 is the mean value. Similarly, for 푆 퐴 2,푛 we obtain⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠′⎛⎝ 퐴′푛퐴푛 0 ⎞⎠ ∂푓 ( 휃̃푛 ) ∂휃′ 퐹−1푛 ∂푓 ( 휃̂푛 )′ ∂휃 퐷퐴푓,푛 ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠ + ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠′퐷퐴′푓,푛∂푓 ( 휃̂푛 ) ∂휃′ 퐹 ′−1푛 ∂푓 ( 휃̃푛 )′ ∂휃 ⎛⎝ 퐴′푛퐴푛 0 ⎞⎠′ ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠ . Next, for 푆퐴3,푛 write − 2푆퐴3,푛 = (ℎ0 − 푓 (휃0))′퐴′퐴 ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) +(ℎ0 − 푓 (휃0))′ ( 퐴′푛퐴푛 −퐴′퐴 ) ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) . (3.6) For the first summand on the right-hand side of (3.6), applying the mean- value expansion to 푓 ( 휃̂푛 ) around 푓 (휃0) and by (3.1), we obtain (ℎ0 − 푓 (휃0))′퐴′퐴 ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) = (ℎ0 − 푓 (휃0))′퐴′퐴 ∂푓 ( 휃̃푛 ) ∂휃′ ( 휃̂푛 − 휃0 ) = ( 휃̂푛 − 휃0 )′ 푀푓,푛 ( 휃̂푛 − 휃0 ) = ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠′퐷퐴′푓,푛∂푓 ( 휃̂푛 ) ∂휃′ 퐹 ′−1푛 푀푓,푛 ×퐹−1푛 ∂푓 ( 휃̂푛 )′ ∂휃 퐷퐴푓,푛 ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠ . 115 3.8. Proofs of Theorems For the second summand on the right-hand side of (3.6), write (ℎ0 − 푓 (휃0))′ ( 퐴′푛퐴푛 −퐴′퐴 ) ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) = ( 푣푒푐 ( 퐴′푛퐴푛 −퐴′퐴 ))′ (퐼푚 ⊗ (ℎ0 − 푓 (휃0))) ( 푓 ( 휃̂푛 ) − 푓 (휃0) ) = ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠′⎛⎝ 0 퐼푚 ⊗ (ℎ0 − 푓 (휃0)) ⎞⎠ ∂푓 ( 휃̃푛 ) ∂휃′ ×퐹−1푛 ∂푓 ( 휃̂푛 )′ ∂휃 퐷퐴푓,푛 ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠ . Now, collection the above expressions for 푆1,푛, 푆2,푛, 푆3,푛, and establishing similar expansions for ∥∥∥퐴푛 (ℎ̂푛 − 푔 (훾̂푛))∥∥∥2, the result follows by Assump- tion 8. ■ Proof of Theorem 10. From (3.2), by adding and subtracting ℎ0, we obtain∥∥∥퐴(ℎ̂푛 − 푓 (휃̂푛))∥∥∥2 = ∥퐴 (ℎ0 − 푓 (휃0))∥2 +2 (ℎ0 − 푓 (휃0))′퐴′퐴 ( ℎ̂푛 − ℎ0 ) +푂푝 ( 푛−1 ) , (3.7) with a similar expression for ∥∥∥퐴(ℎ̂푛 − 푔 (훾̂푛))∥∥∥2. Hence, 푄퐿푅푛 ( 휃̂푛, 훾̂푛 ) = −∥퐴 (ℎ0 − 푓 (휃0))∥2 + ∥퐴 (ℎ0 − 푔 (훾0))∥2 +2 (푓 (휃0)− 푔 (훾0))′퐴′퐴 ( ℎ̂푛 − ℎ0 ) +푂푝 ( 푛−1 ) . (3.8) Since ℱ ∩ 풢 = ∅, we have that 푓 (휃0) ∕= 푔 (훾0), and the result follows from Assumption 1(b). ■ 116 3.8. Proofs of Theorems Proof of Theorem 11. From (3.5) we have∥∥∥퐴푛 (ℎ̂푛 − 푓 (휃̂푛))∥∥∥2 − ∥퐴 (ℎ0 − 푓 (휃0))∥2 = (ℎ0 − 푓 (휃0))′ ( 퐴′푛퐴푛 −퐴′퐴 ) (ℎ0 − 푓 (휃0)) +2 (ℎ0 − 푓 (휃0))′퐴′푛퐴푛 ( ℎ̂푛 − ℎ0 ) +푂푝 ( 푛−1 ) = ( 2 (ℎ0 − 푓 (휃0))′퐴′푛퐴푛 (ℎ0 − 푓 (휃0))′ ( 퐼푚 ⊗ (ℎ0 − 푓 (휃0))′ ) ) × ⎛⎝ ℎ̂푛 − ℎ0 푣푒푐 (퐴′푛퐴푛 −퐴′퐴) ⎞⎠+푂푝 (푛−1) . Using a similar expansion for ∥∥∥퐴푛 (ℎ̂푛 − 푔 (훾̂푛))∥∥∥2, the result follows by Assumption 8. ■ Proof of Theorem 12. From (3.3), ∥∥∥퐴2 (ℎ̂2,푛 − 푓2 (휃̂푛 (퐴1)))∥∥∥2 can be expanded as ∥퐴2 (ℎ2,0 − 푓2 (휃0 (퐴1)))∥2 + 2 (ℎ2,0 − 푓2 (휃0 (퐴1)))′퐴′2퐴2 ( ℎ̂2,푛 − ℎ2,0 ) −2 (ℎ2,0 − 푓2 (휃0 (퐴1)))′퐴′2퐴2 ∂푓2 (휃0 (퐴1)) ∂휃′ ( 휃̂푛 (퐴1)− 휃0 (퐴1) ) +표푝 ( 푛−1/2 ) = ∥퐴2 (ℎ2,0 − 푓2 (휃0 (퐴1)))∥2 −2 (ℎ2,0 − 푓2 (휃0 (퐴1)))′퐴′2퐴2 ∂푓2 (휃0 (퐴1)) ∂휃′ 퐹−11,0 ∂푓1 (휃0 (퐴1)) ′ ∂휃 퐴′1퐴1 ⋅ ( ℎ̂1,푛 − ℎ1,0 ) +2 (ℎ2,0 − 푓2 (휃0 (퐴1)))′퐴′2퐴2 ( ℎ̂2,푛 − ℎ2,0 ) +표푝 ( 푛−1/2 ) = ∥퐴2 (ℎ2,0 − 푓2 (휃0 (퐴1)))∥2 + 2 (ℎ2,0 − 푓2 (휃0 (퐴1)))′퐴′2퐴2퐽푓,0 ( ℎ̂푛 − ℎ0 ) +표푝 ( 푛−1/2 ) . 117 3.8. Proofs of Theorems ■ Proof of Theorem 13. First, note that in the case of nested models for all 퐴 ∈ 픸, ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥2 ≥ ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥2, and thus, under 퐻푎0 , we have that for all 퐴 ∈ 픸, ∥퐴 (ℎ0 − 푔 (훾0 (퐴)))∥2 = ∥퐴 (ℎ0 − 푓 (휃0 (퐴)))∥2. We show next that under 퐻푎0 , 푛푄퐿푅푛 ( 휃̂푛 (퐴) , 훾̂푛 (퐴) , 퐴 ) converges weakly to a stochastic process indexed by 퐴. According to Theorem (10.2) of Pollard (1990), for weak convergence one needs to show finite dimensional convergence and stochastic equicontinuity of 푛푄퐿푅푛 ( 휃̂푛 (퐴) , 훾̂푛 (퐴) , 퐴 ) with respect to 퐴. Finite dimensional convergence follows by the same arguments as in the proof of Theorem 8. For stochastic equicontinuity, from (3.2) one can show that 푛 ∣∣∣푄퐿푅푛 (휃̂푛 (퐴1) , 훾̂푛 (퐴1) , 퐴1)−푄퐿푅푛 (휃̂푛 (퐴2) , 훾̂푛 (퐴2) , 퐴2)∣∣∣ ≤ 푛 ∥∥∥ℎ̂푛 − ℎ0∥∥∥2퐾푛 ∥퐴1 −퐴2∥훿 + 표푝 (1) , where 훿 > 0, 퐾푛 = 푂푝 (1) and independent of (퐴1 −퐴2), and 표푝 (1) term is uniform in 퐴; this is because 푊푓,푛 and 푊푔,푛 are continuous in 퐴, and 표푝 ( 푛−1 ) term is uniform in 퐴. Stochastic equicontinuity of 푛푄퐿푅푛 ( 휃̂푛 (퐴) , 훾̂푛 (퐴) , 퐴 ) follows from Lemma 2(a) of Andrews (1992). The results of the theorem follow now from weak convergence by the continuous mapping theorem (CMT). ■ Proof of Theorem 14. Convergence of finite dimensional distributions and stochastic equicontinuity can be established from (3.8). 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A firm with higher net worth can provide more internal finance for a project of a given size, thus can be subject to a lower risk premium, which is the excess return beyond return to risk-free assets requested by the cred- itor, as the firm has stronger interests in the project. Alternatively, a firm with higher net worth can post more collateral and apply to a bigger loan. Therefore it is plausible that net worth is closely related to investment ac- tivity and risk premium. The relationships between net worth, investment, and risk premium can manifest themselves at an aggregate level. In the ag- gregate economy, when the total factor productivity (TFP) growth is above its trend level, return to capital is higher, and so is the value of the net as- sets. The increase in net worth of the corporate sector can stimulate higher investment level and further expansion of aggregate output in the future. This chapter documents the cyclical properties of net worth of non- financial sectors and examine whether leading dynamic stochastic general equilibrium (DSGE) models that incorporate the role of net worth can repli- cate these features quantitatively. Since it is difficult to provide an appropriate survey of a large literature on net worth and financial frictions, I elect to mention three papers that examine the role of net worth in business cycle, Kiyotaki and Moore (1997), Carlstrom and Fuerst (1997) and Bernanke, Gertler and Gilchrist (1999). 126 4.1. Introduction Kiyotaki and Moore (1997) explores a partial equilibrium model with limited enforcement of debt contract. In their setup, a firm must post fixed assets as collaterals and the loan they obtain in equilibrium may not exceed the value of collaterals. Positive TFP shocks are shown to generate business- cycle-like fluctuations. Carlstrom and Fuerst (1997) and Bernanke et al. (1999) embed asymmetric information in entrepreneur return respectively into a real business cycle model and an new-Keynesian model, and evaluate the quantitative performance of the models. In equilibrium, the amount of loans obtained by entrepreneurs are increasing in their net worth. Carlstrom and Fuerst (1997) find that the financial friction helps to generate hump- shaped output impulse responses and autocorrelation in output growth, as observed in the data. In Bernanke et al. (1999), a similar financial friction strengthens the propagation of TFP and monetary shocks, and such an amplification effect is termed “financial accelerator”. While net worth plays a critical role in Carlstrom and Fuerst (1997) and Bernanke et al. (1999), they has not studied on whether such models can match the cyclical properties of aggregate net worth. Rather, they examines some quantitative aspects of DSGE models, such as whether the models with financial frictions can propagate productivity and monetary shocks. Relative these papers, this chapter contributes to the literature of financial frictions in aggregate economy by employing the econometric technique developed in chapter 3 to explicitly examine whether the models of Carlstrom and Fuerst (1997) and Bernanke et al. (1999) can capture the cyclical properties of net worth. In the chapter, I document that the cyclical properties of net worth have 127 4.2. Cyclical Properties of Net Worth three main features: 1) net worth is pro cyclical, 2) the volatility of net worth is higher in the later part of the sample starting at the first quarter of 1983, and 3) the co-movement between net worth and macroeconomic variables are strong in the later part of the sample. Applications of econometric procedures suggest that there is no significant difference in the performance of the two models in matching quantitatively the cyclical properties of net worth, and price rigidity plays an important role in Bernanke et al. (1999). However, both models only partially capture the positive correlation between net worth and risk premium. The next section describes the cyclical properties of net worth. Overviews of the models used in Carlstrom and Fuerst (1997) and Bernanke et al. (1999) are provided in section 3. In section 4, I carry out the econometric test in chapter 3 to compare the two models formally. Section 5 concludes. 4.2 Cyclical Properties of Net Worth In this section, I document the cross-correlations between net worth and GDP, consumption, hours worked, investment, T-Bill interest rate, prime interest rate, and risk premium in the quarterly U.S. data since the first quarter of 1952. Except for the interest rates which are simply HP-filtered series, all series are real per capita variables in log and HP-filtered. In this chapter, I focus on the net worth of non-financial sectors34 as it corresponds best to the net worth in the models of Carlstrom and Fuerst 34Bernanke and Gertler (1995) discuss in general how balance sheets of banks and non- financial companies affect the transmission of monetary policy. Chen (2001) and Meh and Moran (2008) are examples of models that examine the role net worth of banks. 128 4.2. Cyclical Properties of Net Worth (1997) and Bernanke et al. (1999). There are two measures of net worth. The first is net worth at market value, which is the difference between total assets at market value and total financial liabilities. The second is net worth at historical (purchase) cost, which is the difference between total assets at historical value and total financial liabilities. Among the total assets at market value, on average about 43% are financial assets, with the rest being mostly equipment, software, and real estate. Almost all liabilities are financial liabilities. To provide rough measures of the relative prices of net assets, I compute the ratio between total asset worth at market value and total asset worth at historical cost, and the ratio between net worth at market value and net worth at historical cost. The investment series used in the chapter is non-residential investment net of investment in inventory. The T-Bill interest rate is calculated from 3-month treasury bill secondary market rate, and the prime interest rate is the average prime rate charged by banks on short-term loans to business. Risk premium is defined as the difference between the two interest rates. The series on total asset values and total liabilities are extracted from the Federal Reserve Bank’s Flow of Funds Accounts database. Other variables are from Basic Economics database produced by DRI/McGraw-Hill. 129 4.2. Cyclical Properties of Net Worth -.12 -.08 -.04 .00 .04 .08 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at market value GDP -.12 -.08 -.04 .00 .04 .08 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at market value Consumption -.12 -.08 -.04 .00 .04 .08 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at market value Hours worked -.3 -.2 -.1 .0 .1 .2 .3 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at market value Investment Figure 4.1: Net Worth and GDP, Consumption, Hours Worked and Invest- ment In Figure 4.1, the HP-filtered series of net worth at market value are plotted against those of GDP, consumption, hours worked and investment. An “eyeball” examination suggests that net worth is pro-cyclical. In addi- tion, its volatility increased substantially in the latter part of the sample, getting close to the volatility of investment, the most volatile component of GDP. In particular, the net worth at market value series has tracked the investment series quite closely since late 1980’s. 130 4.2. Cyclical Properties of Net Worth -.3 -.2 -.1 .0 .1 .2 .3 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at market value Investment -.3 -.2 -.1 .0 .1 .2 .3 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at historical cost Investment -.3 -.2 -.1 .0 .1 .2 .3 50 55 60 65 70 75 80 85 90 95 00 05 Asset (market)/Asset (cost) Investment -.3 -.2 -.1 .0 .1 .2 .3 50 55 60 65 70 75 80 85 90 95 00 05 Net worth(market)/Net worth(cost) Investment Figure 4.2: Net Worth and Investment in Greater Details The change in net worth at market value, as plotted in upper left graph in Figure 4.2, can be roughly decomposed into two parts, change in quantities of net assets and change in the price of net assets. Since the measure of net worth at historical cost plotted in the upper right graph is net of change in prices, we can see that there has been pro cyclical change in the quantities of net assets. Meanwhile, if we look at the bottom two graphs which plot two rough measures of asset prices, we can see the prices are also pro cyclical. As in Figure 1, we can also see higher volatility of quantities and prices of net assets in later part of the sample. 131 4.2. Cyclical Properties of Net Worth -.3 -.2 -.1 .0 .1 .2 .3 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at market value Investment -.12 -.08 -.04 .00 .04 .08 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at market value T-Bill rate -.12 -.08 -.04 .00 .04 .08 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at market value Prime rate -.12 -.08 -.04 .00 .04 .08 50 55 60 65 70 75 80 85 90 95 00 05 Net worth at market value Risk premium Figure 4.3: Net Worth and Interest Rates Judging from Figure 4.3, we can argue the synchronizations between net worth and T-Bill rate, and between net worth and the prime rate, are more salient. Overall, visual inspection of the three figures above suggest 1) net worth is pro cyclical, 2) the volatility of net worth is higher in the later part of the sample, and 3) the co-movement between net worth and macroeconomic variables are strong in the later part of the sample. The next two tables provide a more formal description of these features. In tabulating the cross-correlation between net worth and other vari- ables, I split the sample at the end of the 1981/1982 recession, i.e. the last quarter of 1982, to see if the correlations differ in the two samples. Although 132 4.2. Cyclical Properties of Net Worth Table 4.1: Net Worth, GDP, Consumption, and Investment Corr(NW,X) 휎(푋) 휎(퐺퐷푃 ) 푋푡−4 푋푡−2 푋푡−1 푋푡 푋푡+1 푋푡+2 푋푡+4 1983Q1-2008Q4 NW(market) 2.96 0.45* 0.80* 0.92* 1 0.92* 0.80* 0.45* GDP 1 0.53* 0.50* 0.47* 0.51* 0.43* 0.32* 0.03 Investment 6.11 0.46* 0.45* 0.41* 0.39* 0.31* 0.22* -0.02 1952Q1-1982Q4 NW(market) 0.63 0.23* 0.58* 0.76* 1 0.76* 0.58* 0.23* GDP 1 0.50* 0.48* 0.35* 0.17 -0.04 -0.21* -0.39* Investment 4.90 0.48* 0.53* 0.44* 0.31* 0.05 -0.15 -0.39* Note: * denotes statistical significance at 5% level. the choice seems ad hoc, splitting the sample at the end of 1973/1975 reces- sion or the end of 1990/1991 recession produce similar results. Notably, the standard deviation of net worth relative to GDP increased from 0.63 in the first period to 2.96 in the second. From Table 4.1, we can see the autocorrelation of net worth is higher. More importantly, the positive correlation between net worth and GDP and investment are stronger. In particular, net worth only has positive and 133 4.2. Cyclical Properties of Net Worth statistically significant correlations with leads of GDP and investment after the last quarter of 1982. This feature present in the latter part of the sample is more consistent with the premise of Carlstrom and Fuerst (1997) and Bernanke et al. (1999) that net worth helps to propagate expansion of output. Table 4.2: Net Worth and Investment in Greater Details Corr(NW,X) 휎(푋) 푋푡−4 푋푡−2 푋푡−1 푋푡 푋푡+1 푋푡+2 푋푡+4 1983Q1-2008Q4 T-Bill rate 0.011 0.22* 0.54* 0.68* 0.68* 0.61* 0.52* 0.29* Prime rate 0.012 0.18 0.53* 0.69* 0.70* 0.64* 0.56* 0.33* Risk premium 0.003 -0.07 0.07 0.20* 0.26* 0.29* 0.30* 0.25* 1952Q1-1982Q4 T-Bill rate 0.017 0.01 -0.05 -0.06 -0.13 -0.02 -0.00 0.12 Prime rate 0.018 -0.09 -0.01 -0.01 0.12 0.10 0.17 0.26 Risk premium 0.007 -0.10 0.10 0.23* 0.27* 0.33* 0.34* 0.28* Note: * denotes statistical significance at 5% level. Judging from Table 4.2, there had been no significant correlations be- tween net worth and the T-Bill rate and the prime rate before the last 134 4.2. Cyclical Properties of Net Worth quarter of 1982. Afterward, positive and statistically significant correlations emerged. Meanwhile, the correlations between net worth and risk premium are stable over the two subsamples.35 The change in volatility and cyclical properties after 1982 is certainly intriguing, but exploring the cause would require a separate study. In the rest of chapter I will take the cyclical properties after 1982 as given and examine whether DSGE models can quantitatively replicate these features. 35The measure of risk premium presented in Table 4.2 follows that of Carlstrom and Fuerst (1997) and Bernanke et al. (1999). I also explore an alternative measure of risk premium defined as the difference between Moody’s yield on AAA seasoned corporate bonds of all industries and BAA seasoned corporate bonds of all industries. When this measure is used, net worth is negatively related to lags of risk premium and its correlations with future risk premiums are positive. With this measure, I obtain model comparison results similar to Section 4.4. 135 4.3. Overviews of Two Competing Models 4.3 Overviews of Two Competing Models In this section, I reproduce the setups of the two models to facilitate com- parison and discussion in section 4. In essence, Carlstrom and Fuerst (1997) adds to the benchmark RBC model a capital production sector where capital goods producers face stochastic return and have to rely partially on external finance. In comparison, Bernanke et al. (1999) is based on a new-Keynesian model and the financial friction occurs in the production of consumption goods. 4.3.1 The Carlstrom and Fuerst (1997) Model There are a continuum of agents of measure one, among which there are households of measure 1− 휂 and entrepreneurs of measure 휂. There are two goods, consumption goods and capital goods. A typical household supplies labour 푙푡 to consumption good firms and make investment decisions 푖푡 to maximize life-time utility 퐸0 ∞∑ 푡=0 훽푡푈(푐푡, 푙푡) = 퐸0 ∞∑ 푡=0 훽푡[푙푛(푐푡) + 휈(1− 푙푡)] 푠푢푏푗푒푐푡 푡표 푐푡 + 푖푡 ≤ 푞푡푘ℎ푡 (1− 훿) + 푟푡푘ℎ푡 + 푤푡푙푡 where 푞푡 is the end of period price of capital, 푘 ℎ 푡 is the beginning of period capital stock held by a household36, 훿 is the depreciation rate of capital, 푟푡 is the rental rate of capital, and 푤푡 is the wage rate. Note the households are 36For the quantity variables, upper cases denote the aggregates. For instance, 퐾ℎ푡 denotes the aggregate capital stock held by all households. 136 4.3. Overviews of Two Competing Models also owners of consumption good firms, who make zero profit in equilibrium since they have constant to return technologies and face perfect competition. Consumption goods producers are standard. They hire labour and rent capital from households and entrepreneurs. The production technology is 푦푡 = 퐴푡푘 훼1 푡 푙 훼2 푡 (푙 푒 푡 ) 1−훼1−훼2 where 퐴푡 is the stochastic TFP, which follows an AR(1) process with auto- correlation coefficient 휌퐴. The crucial difference of the model from RBC lies in the production of capital goods. In a benchmark RBC model, consumption goods can be converted one-to-one into investment goods, while in the current model, households must purchase capital goods from entrepreneurs via financial intermediaries. In the model, a continuum of financial intermediaries receive the savings from households and make loans to entrepreneurs, who are the producer of capital goods. The entrepreneurs have access to a stochastic technology that transfer 푖푒푡 unit of consumption goods into 휔푖 푒 푡 unit of capital goods, where 휔 is a random variable with pdf 휙(휔) and cdf Φ(휔). The entrepreneurs also supply labour inelastically 푙푒푡 = 1 to consumption good producers so they can have positive net wealth 푛푡. However, to finance an project of size 푖 푒 푡 , they could borrow 푖푒푡 − 푛푡 from the financial intermediaries to finance the production of capital goods. While entrepreneurs can observe the realization of 휔 for free but financial intermediaries can only observe it by paying a fee which is proportional to the size of the investment37. Contracts last for one period 37This is the costly state verification assumption introduced by Townsend (1979) 137 4.3. Overviews of Two Competing Models and both sides are anonymous, therefore there’s no reputation building from repeated games. In this circumstance, a debt contract is optimal38. The contract specifies an interest rate 푟푘푡 on the loans. When the realization of 휔 is above some threshold 휔, the entrepreneurs will repay the principal and interest; other- wise, they will default. In case of defaults, the financial intermediaries will pay the verification fee 휇푖 which can be interpreted as bankruptcy cost, and capture the residual value of the investment 휇푖휔. The threshold is given by 휔 = (1 + 푟푘푡 )(푖 푒 푡 − 푛푡) 푖푒푡 (4.1) since if 휔 is lower than the level, even using the whole return on the project to repay the financial intermediary will still fail to meet the specified interest rate, i.e. 휔푖푒푡 푖푒푡 − 푛푡 < 1 + 푟푘푡 Hence, the expected income to an entrepreneur 푞푡 [∫ ∞ 휔 휔푖푒푡푑Φ(휔)− (1− Φ(휔))(1 + 푟푘푡 )(푖푒푡 − 푛푡) ] which by using (4.1) is rewritten as 푞푡푖 푒 푡 (∫ ∞ 휔 휔푑Φ(휔)− (1− Φ(휔)(1 + 푟푘푡+1)(푖푒푡+1 − 푛푡+1) ) = 푞푡푖 푒 푡푓(휔) (4.2) The expected income to a creditor is 푞푡 [∫ 휔 0 휔푖푒푡푑Φ(휔)− Φ(휔)휇푖푒푡 + (1− Φ(휔))(1 + 푟푘푡 )(푖푒푡 − 푛푡) ] 38See Gale and Hellwig (1985) and Williamson (1987) 138 4.3. Overviews of Two Competing Models which by using (4.1) is rewritten as 푞푡푖 푒 푡 (∫ 휔 0 휔푑Φ(휔)− Φ(휔)휇+ (1− Φ(휔))휔 ) = 푞푡푖 푒 푡푔(휔) (4.3) The optimal contract is to maximize the entrepreneur’s income39 max 푖푡,휔 푞푡 [∫ ∞ 휔 휔푖푒푡푑Φ(휔)− (1− Φ(휔))(1 + 푟푘푡 )(푖푒푡 − 푛푡) ] subject to the participation constraint of the creditor 푞푡 [∫ 휔 0 휔푖푒푡푑Φ(휔)− Φ(휔)휇푖푒푡 + (1− Φ(휔))(1 + 푟푘푡 )(푖푒푡 − 푛푡) ] ≥ 푖푒푡 − 푛푡 where the alternative gross return to creditor’s funds is 1. The solution to the problem can be solved from the first order conditions: 푞푡(1− Φ(휔)휇) + 휙(휔)휇 ∫∞ 휔 휔푑Φ(휔)− (1− Φ(휔))휔 Φ(휔)− 1 = 1 (4.4) 푞푡 [∫ 휔 0 휔푖푒푡푑Φ(휔)− Φ(휔)휇푖푒푡 + (1− Φ(휔))(1 + 푟푘푡 )(푖푒푡 − 푛푡) ] ≥ 푖푒푡 − 푛푡 (4.5) The last two equations define entrepreneur’s equilibrium project size 푖푒푡 as a function of capital price 푞푡 and net worth 푛푡. Carlstrom and Fuerst (1997) shows that 푖푒푡 is increasing in both 푞푡 and 푛푡. The entrepreneurs are risk neutral and maximize 퐸0 ∞∑ 푛=0 (훽훾)푡푐푒푡 훾 is assume to be less than 1, so entrepreneurs are impatient and they will not accumulate a level of net worth large enough to self-finance their investment projects. The entrepreneur’s net worth is 푛푡 = 푤푡푙 푒 푡 + 푘 푒 푡 [푞푡(1− 훿)− 푟푡] 39This formulation assumes the entrepreneur captures the whole surplus generated by the contract. 139 4.3. Overviews of Two Competing Models where 푘푒푡 is the holding of capital by entrepreneurs at the beginning of the period. As return to net worth (see equation (4.6) below) is higher than 1, and entrepreneurs are risk neutral, they will invest all the net worth in the project. At the end of period, entrepreneurs who stay solvent make their consumption and savings decision subject to the budget constraint 푐푒푡 + 푞푡푘 푒 푡+1 ≤ 푞푡[푖푒푡휔 − (1 + 푟푘푡 )(푖푒푡 − 푛푡)] (4.6) Where 푖푒푡휔− (1+ 푟푘푡 )(푖푒푡 − 푛푡) is the realized return on the project of size 푖푒푡 . Since the expected return on capital in period 푡+ 1 is [푞푡+1(1− 훿) + 푟푡+1] [ 푞푡+1푓(휔푡+1) 1− 푞푡+1푔(휔푡+1) ] , where the fraction in the expression is the expected return to net worth in- vested in the time 푡+1 project derived from equation (4.5), the Euler’s equa- tion that governs the consumption and savings decision for the entrepreneur is 푞푡 = 퐸푡훽훾[푞푡+1(1− 훿) + 푟푡+1] [ 푞푡+1푓(휔푡+1) 1− 푞푡+1푔(휔푡+1) ] Note that in the last equation, net worth does not show up, indicating the decision rule of entrepreneurs are independent of their net worth. The ag- gregation of entrepreneurs’ budget constraints yields the equation describing evolution of 퐾푒푡 , the capital stock held by entrepreneurs 퐾푒푡+1 = (휂푤푡푙 푒 푡 +퐾 푒 푡 [푞푡(1− 훿) + 푟푡]) [ 푞푡+1푓(휔푡+1) 1− 푞푡+1푔(휔푡+1) ] − 휂푐푒푡/푞푡 = 휂푛푡 [ 푞푡+1푓(휔푡+1) 1− 푞푡+1푔(휔푡+1) ] − 휂푐푒푡/푞푡 The timing of events in a period is 140 4.3. Overviews of Two Competing Models 1. The TPF shock 퐴푡 is realized. 2. Consumption goods firm hire labour and capital for production. 3. Households make consumption and investment decisions. 4. The financial intermediaries use investment funds from households to make loans to entrepreneurs. 5. Entrepreneurs produce capital from consumption goods with the stochas- tic technology. 6. Entrepreneur-specific technology shock 휔 is realized. Entrepreneurs make decision about whether to repay the loan or default. 7. Solvent entrepreneurs decide on consumption and capital holding. Given the state variables (퐾푡,퐾 푒 푡 , 퐴푡), 40 a recursive competitive equilibrium are characterized by decision rules for 퐾푡+1,퐾 푒 푡+1, 퐿푡, 푞푡, 푛푡, 푖 푒 푡 , 휔푡, 푐 푒 푡 and 푐푡. 40Note 퐾푡 = 퐾 ℎ 푡 +퐾 푒 푡 . 141 4.3. Overviews of Two Competing Models They are implicitly defined by 푈퐿(푡) = 푈퐶(푡)퐴푡훼2퐾 훼1 푡 퐿 ( 푡1− 훼2)(퐿푒푡 )1−훼1−훼2 푞푡푈퐶(푡) = 훽퐸푡푈퐶(푡+ 1)[푞푡+1(1− 훿) +퐴푡+1훼1퐾훼1−1푡 퐿(푡훼2)(퐿푒푡 )1−훼1−훼2 ] 퐾푡+1 = (1− 훿)퐾푡 + 휂푖푒푡 [1− Φ(휔푡)휇] 푌푡 = (1− 휂)푐푡 + 휂푐푒푡 + 휂푖푒푡 푞푡 = [ 1− Φ(휔푡)휇+ 휙(휔푡)휇 푓(휔푡) 푓 ′(휔푡) ]−1 푖푒푡 = 푛푡 1− 푞푡푔(휔푡) 푛푡 = 퐴푡(1− 훼1 − 훼2)퐾훼1푡 퐿훼2푡 (퐿푒푡 )−훼1−훼2 + 퐾푒푡+1 휂 [ 푞푡(1− 훿) +퐴푡훼1퐾훼1푡 퐿훼2푡 (퐿푒푡 )1−훼1−훼2 ] 퐾푒푡+1 = 휂푛푡 [ 푞푡+1푓(휔푡+1) 1− 푞푡+1푔(휔푡+1) ] − 휂푐푒푡/푞푡 푞푡 = 퐸푡훽훾[푞푡+1(1− 훿) +퐴푡+1훼1퐾 훼1 푡+1퐿 훼2 푡+1(퐿 푒 푡+1) 1−훼1−훼2 ] [ 푞푡+1푓(휔푡+1) 1− 푞푡+1푔(휔푡+1) ] (4.7) In addition, markets clear for two types of labour, consumption goods, and capital goods: 퐿푡 = (1− 휂)푙푒푡 퐿푒푡 = 휂 푌푡 = (1− 휂)푐푡 + 휂푐푒푡 + 휂푖푒푡 퐾푡+1 = (1− 훿)퐾푡 + 휂푖푒푡 [1− Φ(휔푡)휇] The model is solved by linearizing the system of equations (4.7) near its steady state. 142 4.3. Overviews of Two Competing Models 4.3.2 The Bernanke, Gertler and Gilchrist (1999) Model To facilitate comparison, I modify the model of Bernanke et al. (1999) slightly and present it in a fashion similar to subsection 4.3.1. There are a continuum of agents of measure one, among which there are households of measure 1− 휂 and entrepreneurs of measure 휂. There are three types of goods, wholesale goods, differentiated goods and final goods. The represen- tative household supplies labour 퐿푡 to wholesale good producers and make decisions on consumption 퐶푡, deposit 퐷푡+1 and real balance 푀푡 푃푡 to maximize life-time utility 퐸0 ∞∑ 푡=0 훽푡[푙푛(퐶푡) + 휁푙푛( 푀푡 푃푡 ) + 휈푙푛(1− 퐿푡)] subject to the budget constraint 퐶푡 ≤푊푡퐿푡 − 푇푡 +Π푡 +푅푡퐷푡 −퐷푡+1 + 푀푡−1 −푀푡 푃푡 where 푇푡 is lump sum taxes, Π푡 is dividends from ownership of retail firms, and 푅푡 is the return on deposit. The solution is characterized by the stan- dard first order conditions 1 퐶푡 = 퐸푡훽푅푡+1 1 퐶푡+1 푊푡 1 퐶푡 = 휉 1 1− 퐿푡 푀푡 푃푡 = 휁퐶푡( 푅푛푡+1 − 1 푅푛푡+1 )−1 where 푅푛푡+1 is the gross nominal interest rate. A continuum of financial intermediaries receive the deposits from house- holds and make loans to entrepreneurs. Entrepreneur 푗 use a stochastic 143 4.3. Overviews of Two Competing Models technology to produce homogeneous wholesale goods 푌 푗푡 = 휔퐴푡(퐾 푗 푡 ) 훼1(퐿푗푡 ) 훼2(퐿푒푗푡 ) 1−훼−훼2 (4.8) where 퐴푡 is the stochastic TFP, which follows an AR(1) process 퐴푡 = 휌퐴퐴푡−1 + 휖 퐴 푡 (4.9) Since in equilibrium all entrepreneurs will choose the same capital-labour ratio, we can write the return to capital for entrepreneur 푗 as 휔푗훼1퐴푡(퐾 푗 푡 ) 훼1−1(퐿푗푡 ) 훼2(퐿푒푗푡 ) 1−훼−훼2 =휔푗훼1퐴푡(퐾푡) 훼1−1(퐿푡) 훼2(퐿푒푡 ) 1−훼−훼2 =휔푗푅푘푡 where 푅푘푡 is the aggregate return to capital. The price of wholesale good relative to final good is denoted as 1/푋푡. The entrepreneurs borrow from the financial intermediaries to finance the purchase of capital goods made from the final good. With the purchased cap- ital, a entrepreneur hires labour from both households and entrepreneurs to produce wholesale goods. They also supply labour 퐿푒푡 on the market so they can have positive net wealth. Again, entrepreneur 푗 can observe the realiza- tion of 휔푗 for free but financial intermediaries can only observe it by paying a fee which is proportional to the gross return to capital, 휔푗푅푘푡 푞푡퐾 푗 푡 . The optimal contract is very similar to subsection 4.3.1, so is the entrepreneurs’ maximization problem41. It can be shown that the desired capital level is 41In the original setup of Bernanke et al. (1999), they assume there are constant birth and death of entrepreneurs, with the survival probability in the next period being 훾. The entrepreneurs maximize wealth level and consume their wealth the period of death. Both setups will yield the same aggregate behaviour of the entrepreneurs. 144 4.3. Overviews of Two Competing Models given by 푞푡퐾 푗 푡+1 = 휄(퐸푡 푅푘푡+1 푅푡+1 )푁 푗푡+1 (4.10) where 휄(⋅) is an increasing function with 휄(1) = 1, 푞푡 is the price of capital, and 푁 푗푡 is the net worth the entrepreneur 푗. Note the equation implies the aggregate capital evolution is linear in aggregate net worth and independent of the distribution of net worth. The functional form of 휄(⋅) depends on the distribution function of 휔. The threshold value for return, 휔푗 is determined by 푅푡+1 = { [1− Φ(휔푗푡 )]휔푗푡 − (1− 휇) ∫ 휔푗푡 0 휔푑Φ(휔) } 푅푘푡+1푞푡퐾 푗 푡+1 푞푡퐾 푗 푡+1 −푁 푗푡+1 (4.11) Since (4.10) implies 퐾푗푡+1 is proportional to 푁 푗 푡+1, the fraction in the right hand side of (4.11) is a constant. Therefore the threshold value is determined only by the aggregate valuables 푅푡+1 and 푞푡, i.e. all entrepreneurs will choose the same 휔푗푡 . The entrepreneurs sell the wholesale goods to retailers, who differentiated the wholesale goods effortlessly. The differentiated goods are aggregated into a final good by a CES aggregator. The final good can be used as both consumption or capital. Let 푌푡(푧) denote the amount of output sold by retailer 푧 in terms of wholesale goods, then the total final good 푌 푓푡 is given by 푌 푓푡 = [ ∫ 1 0 푌푡(푧) (휖−1)/휖]휖/(휖−1) (4.12) with its nominal price defined as 푃푡 = [ ∫ 1 0 푃푡(푧) (휖−1)/휖]휖/(휖−1) (4.13) 145 4.3. Overviews of Two Competing Models Given the relation between wholesale goods and the final good specified in (4.12), the demand for goods of retailer 푧 is 푌푡(푧) = ( 푃푡(푧) 푃푡 )−휖푌 푓푡 The retailers take the demand and price of wholesale goods 푃푤푡 as given and set 푃푡(푧) to maximize profit. The retailers engage in monopolistic compe- tition and adjust nominal prices in a Calvo type price setting environment. In each period, a retailer can adjust its price with probability 1− 휃. Let 푃 ∗푡 and 푌 ∗푡 (푧) be the optimal price and quantity for a retailer who can reset its price. The retailer 푧 maximizes expected profit ∞∑ 푡=0 휃푡퐸0 [ 훽푡퐶0 퐶푡 푃 ∗0 − 푃푤푡 푃푡 푌 ∗푡 (푧) ] where 훽 푡퐶0 퐶푡 is the intertemporal marginal rate of substitution of the house- hold, who is the owner. The optimal price setting rule is given by the first order condition ∞∑ 푡=0 휃푘퐸푡−1 { 훽푡퐶0 퐶푡 ( 푃 ∗0 푃푡 )휖푌 ∗푡 (푧) [ 푃 ∗0 푃 ∗푡 − ( 휖 휖− 1) 푃푤푡 푃푡 ]} = 0 Then rewriting (4.13) yields the aggregate price evolution 푃푡 = [휃푃 1−휖 푡−1 + (1− 휃)(푃 ∗푡 )1−휖]1/(1−휖) Lastly, the government conducts fiscal and monetary policies. Its expen- ditures 퐺푡 are financed by lump sum taxes and revenue from money printing. The government’s budget constraint is 퐺푡 = 푀푡 −푀푡−1 푃푡 + 푇푡 146 4.3. Overviews of Two Competing Models The monetary policy is characterized by the policy rule 푅푛푡 = 휌푅 푛 푡−1 + 휁Π푡−1 + 휖 푟푛 푡 (4.14) where 푅푛푡 , Π푡−1, and 휖 푟푛 푡 are nominal interest rate, inflation and the i.i.d. monetary policy shock. Given the state variables (퐾푡, 퐴푡, 푁푡) and the government policies퐺푡, 푅 푛 푡 , a recursive equilibrium are characterized by decision rules for 퐼푡, 퐶푡, 퐶 푒 푡 , 푅 푘 푡 , 푅푡, 푋푡, 푞푡, 푌푡, 퐿푡, 퐿 푒 푡 ,Π푡,퐾푡+1, 푁푡+1 and 휔푡. They are implicitly 147 4.3. Overviews of Two Competing Models defined by 푌푡 = 퐶푡 + 퐶 푒 푡 + 퐼푡 +퐺푡 + 휇 ∫ 휔푡 0 휔푑Φ(휔)푅푘푡 푞푡−1퐾푡 1 퐶푡 = 훽퐸푡 1 퐶푡+1 푅푡+1 퐶푒푡 = (1− 훾) ⋅ [ 푅푘푡 푞푡−1퐾푡 − ( 푅푡 + 휇 ∫ 휔푡 0 휔푑Φ(휔)푅 푘 푡 푞푡−1퐾푡 푞푡−1퐾푡 −푁푡−1 ) (푄푡−1퐾푡 −푁푡−1) ] 푞푡퐾 푗 푡+1 = 휄(퐸푡 푅푘푡+1 푅푡+1 )푁 푗푡+1 푅푡 = 푅 푛 푡+1/Π푡 퐸푡(푅 푘 푡+1) = ( 1 푋푡+1 훼푌푡+1 퐾푡+1 +푄푡+1(1− 훼) 푄푡 ) 푞푡 = [Γ ′( 퐼푡 퐾푡 )]−1 푤ℎ푒푟푒 Γ(⋅) 푖푠 푐표푛푐푎푣푒 푌푡 = 퐴푡퐾 훼1 푡 퐿 훼2 푡 (퐿 푒 푡 ) 1−훼1−훼2 퐿푒푡 = 1 1 퐶푡 = 휈 1 1− 퐿푡 1 훼2퐴푡퐾 훼1 푡 퐿 훼2−1 푡 (퐿 푒 푡 ) 1−훼1−훼2 휋푡 = 퐸푡−1( 훽휋푡+1 휅푋푡 ) 퐾푡+1 = Γ( 퐼푡 퐾푡 )퐾푡 + (1− 훿)퐾푡 푁푡+1 = 훾 [ 푅푘푡 푞푡−1퐾푡 − ( 푅푡 + 휇 ∫ 휔푡 0 휔푑Φ(휔)푅 푘 푡 푞푡−1퐾푡 푞푡−1퐾푡 −푁푡−1 ) (푄푡−1퐾푡 −푁푡−1) ] + 1 푋푡 (1− 훼1)(1− 훼2 1− 훼1 − 훼2 )퐴푡퐾 훼1 푡 퐿 훼2 푡 푅푡+1 = { [1− Φ(휔푡)]휔푡 − (1− 휇) ∫ 휔푡 0 휔푑Φ(휔) } 푅푘푡+1푞푡퐾푡+1 푞푡퐾푡+1 −푁푡+1 (4.15) The solution of the model is obtained by linearizing the system of equations (4.15) around steady state. 148 4.4. Comparison of the Models 4.4 Comparison of the Models In this section, I address the question which of the two models capture quantitatively better the cyclical properties of net worth. I present a in- formal assessment, before presenting results from applying the econometric method developed in chapter 3, which provide a likelihood-ratio type test which gauge whether two or more competing models capture equally well the same set of statistics observed in the data. One crucial advantage of their framework is that the models are allowed to be misspecified, i.e. they need not be the true data-generating process. An econometrician would vary the parameters of each model and find the parameters values that minimize the weighted distance between the model moments and moments observed in the data. Then the econometrician can take as the test statistic the dif- ference in the two distances generated respectively by the two estimated models, and compared it to a desired critical value. If the test statistic is greater than the specified critical value, the econometrician rejects the null that both models provide equally good fits for the data. 4.4.1 Informal Comparison of the Models In table 4.3, I present the moments from the data (the first four rows), the calibrated Carlstrom and Fuerst (1997) model (row 5 to 8) and the calibrated Bernanke et al. (1999) model (the last four rows).The parameters used in the simulations are similar to the ones used in the papers. We can see both calibrated models predict overly strong correlation between net worth and GDP, and between net worth and investment. Meanwhile, they erroneously 149 4.4. Comparison of the Models predict negative correlation between net worth and risk premium. 4.4.2 Overview of the Formal Comparison Methodology In evaluating the quantitative performances of the models, both Carlstrom and Fuerst (1997) and Bernanke et al. (1999) rely on calibration. Calibra- tion is a technique often applied in macroeconomics and also other fields of economics. Typically, researchers calibrate (a subset of) the model param- eters such that certain features of the calibrated data can ‘match’ those in the observed data. For instance, researchers may want to match moments, correlations, impulse response or other stylized facts of interests. In appli- cations, different theoretical models are often calibrated to match features the same data set. Comparison among competing models naturally requires metrics that have desirable properties. If the theoretical (structural) model is correctly specified, calibration can often be viewed as an example of minimum distance estimation (MD) (Gregory and Smith (1993)). For such scenarios, one can apply the usual statistical tools to judge the goodness of fit. However, researchers are often aware that their models are very unlikely to represent the true data gener- ating process. Frequently the theoretical models contain a small number of parameters, while the stylized facts of the data involve many more parame- ters. In such cases, it is impossible for the theoretical to match all stylized facts as it is restricted. Chapter 3 proposes a formal test for comparison for comparison of two misspecified calibrated models. The test is of the likelihood ration type and based on the difference of the MD criterion functions corresponding 150 4.4. Comparison of the Models Table 4.3: Moments of Calibrated Models Corr(NW,X) 휎(푋) 휎(퐺퐷푃 ) 푋푡−4 푋푡−2 푋푡−1 푋푡 푋푡+1 푋푡+2 푋푡+4 1983Q1-2008Q4 data NW(market) 2.96 0.45* 0.80* 0.92* 1 0.92* 0.80* 0.45* GDP 1 0.53* 0.50* 0.47* 0.51* 0.43* 0.32* 0.03 Investment 6.11 0.46* 0.45* 0.41* 0.39* 0.31* 0.22* -0.02 Risk premium 0.003 -0.07 0.07 0.20* 0.26* 0.29* 0.30* 0.25* calibrated CF NW(market) 3.30 0.76 0.87 0.94 1 0.94 0.87 0.76 GDP 1 0.85 0.87 0.88 0.88 0.88 0.89 0.90 Investment 6.66 0.94 0.93 0.95 0.98 0.99 0.94 -0.05 Risk premium 0.05 -0.43 -0.97 -0.99 -0.99 -0.99 -0.99 -0.99 calibrated BGG NW(market) 1.35 0.98 0.99 0.99 1 0.99 0.98 GDP 1 0.97 0.99 0.99 0.99 0.97 0.96 0.94 Investment 1.14 0.96 0.94 0.91 0.84 0.54 -0.03 -0.82 Risk premium 0.08 -0.45 -0.94 -0.97 -0.95 -0.96 -0.95 -0.93 Note: * denotes significance at 5% level. 151 4.4. Comparison of the Models to the two competing models. They argue that among two misspecified models, the econometricians should prefer one that has a better match to the reduced form characteristics of the data. The procedure is an asymptotic test that under the null the two misspecified models provide an equivalent approximation to the data in terms of characteristics of the reduced form model. Vuong (1989) proposed such tests for misspecified models in the maximum likelihood framework. He showed that a preferred model has a smaller Kullback-Leiber distance from the true distribution. The definition of calibration here is similar to Gregory and Smith (1993) and it is viewed as classic minimum distance (CMD) estimation. 푌푛(휔) is a data matrix of sample size 푡 defined on probability space (Ω,ℱ , 푃 ). All random quantities are some functions of the data 푌푛. 휅 ∈ 풦 ⊂ ℝ푘 are a 푘−푣푒푐푡표푟 of parameters of a structural model. We use ℎ to denote a푚-vector of parameters of some reduced form model. Its true value ℎ0 depends on the true unknown structural model of the economy and its parameters. For example, ℎ can be moments, correlations, impulse response and etc. While the true structural model is unknown, we will assume that the reduced form parameters ℎ0 can be estimated consistently from the data. Let ℎ푛 denote an consistent estimator of ℎ. Given that model and a value of 휃, we assume that one can compute analytically the value of the reduced form parameters. Let 휓 ∈ Ψ ⊂ ℝ푗 be a vector of parameters that corresponds to the sec- ond structural model specified by the econometrician. Formally competing models 퐹 : 풦 → ℝ푚 and 퐺 : Ψ → ℝ푚 are mappings from the parameter space 풦 and Ψ into the space of reduced form parameters.We assume 푚 ≥ 푘 and 푚 ≥ 푗, i.e. the structural models are more restrictive than the reduced 152 4.4. Comparison of the Models form model. 퐹,퐺 are misspecified in the sense that inf 휅∈풦 ∥ℎ0 − 퐹 (휅)∥ > 0 (4.16) and inf 휓∈Ψ ∥ℎ0 − 푔(휓)∥ > 0. (4.17) The estimated 휅, i.e. the CMD estimator under 퐹 is given by the value that minimizes the weighted distance function: 휅̂푛(퐴푛) = argmin 휅∈풦 ∥퐴푛(ℎ푛 − 퐹 (휅))∥2 where ∥⋅∥ is the Euclidean norm and {퐴푛} is a sequence of positive definite weighting matrices such that converge in probability to some matrix 퐴 with full rank. Similarly the estimated 휓, i.e. the CMD estimator under 퐹 is given by the value that minimizes the weighted distance function: 휓̂푛(퐴푛) = argmin 휓∈Ψ ∥퐴푛(ℎ푛 −퐺(휅))∥2 Under suitable assumptions, a Quasi-Likelihood-Ratio (QLR) test is defined as 푄퐿푅푛(휅̂푛(퐴푛), 휓̂푛(퐴푛)) = −∥퐴푛(ℎ푛 − 퐹 (휅̂푛(퐴푛)))∥2 + ∥∥∥퐴푛(ℎ푛 −퐺(휓̂푛(퐴푛)))∥∥∥2 It is shown 푛푄퐿푅푛(휅̂푛(퐴푛), 휓̂푛(퐴푛)) has a mixed 휒 2 distribution. When 푛푄퐿푅푛 is greater than the critical value of a specified significance level, the econometrician will reject the null that both models provide equally good fits for the data. 153 4.4. Comparison of the Models 4.4.3 Formal Comparison of the Models In this application, I take the target moments ℎ0 as the cross-correlation coefficients documented in the first four rows of table 4.342. The distance metric is the Euclidean norm. To reduce computation time, I fix a number of parameters at values used by Bernanke et al. (1999) as the literature provides relatively good information about their values. To check whether sticky price is an important factor in replication of cyclical properties of net worth, I estimate two versions of the Bernanke et al. (1999) model. In the first one I fix the Calvo price setting parameter to be 0, meaning nominal prices are fully flexible. In the second one, I allow 휃 to be vary in [0, 1] to minimize distance from the target moments. Therefore the difference between the first version and Carlstrom and Fuerst (1997) mainly lies in the sector in which financial friction occurs. The second version adds the additional factor of price rigidity. Table 4.4 summarizes the parameter estimates and distances from target moments. Table 5 presents the moments from estimated models. Applica- tion of the econometric test of chapter 3 suggests that 1) difference in fits (measured as distances) of the Carlstrom and Fuerst (1997) model and the second version of the Bernanke et al. (1999) model is statistically insignifi- cant at 10% level; and 2) the fit of the first estimated version of Bernanke et al. (1999) are significantly worse than the other two estimated models, at 1% level. It appears that price rigidity plays an important role in the Bernanke et al. (1999) model, by this standard. 42In general it is possible to add the relative volatilities as targets, but it is not clear how to weight them against the correlations coefficients in the distance function. 154 4.4. Comparison of the Models From Table 5 we can see the first estimated version of the Bernanke et al. (1999) model fails to capture positive correlations between net worth and leads of GDP, and the positive correlations between net worth and risk premium. The other two models have reasonable performance in replicating the positive co-movement of net worth with GDP and investment. How- ever, they only partially reproduce the correlation between net worth and risk premium. This difficulty may stem from the built-in mechanism of both papers via which an increase in net worth lowers risk premium holding other factors constant. In a general equilibrium framework, a lower risk premium will face pressure of upward adjustment as firms have incentive to borrow more. However, for an increase in net worth to cause increase in risk pre- mium, it may require investment to go up substantially more than net worth which is not very plausible in the both models. 155 4.4. Comparison of the Models Table 4.4: Parameters Estimates of CF and BGG Parameter Estimates Name Meaning Range CF BGG 1 BGG 2 훼1 capital share - 0.360 0.360 0.360 fixed fixed fixed 훼2 share of non-entrepreneur labour - 0.630 0.630 0.630 fixed fixed fixed 훽 discount factor - 0.990 0.990 0.990 fixed fixed fixed 훿 depreciation rate - 0.020 0.020 0.020 fixed fixed fixed 휇 bankruptcy cost as - 0.120 0.120 0.120 a fraction of revenue 휃 fraction of firms can’t reset prices [0,1] - 0 0.792 to inflation in Taylor rule - fixed (0.194) 훾 extra discount factor [0,1] 0.982 0.710 0.798 of entrepreneurs (0.527) (0.131) (0.258) 휈 coefficient on leisure in preference [0,1] 0.601 0.590 0.590 (0.153) (0.189) (0.276) 휌퐴 persistence of TFP shock [0,1] 0.696 0.974 0.965 (0.073) (0.253) (0.337) 휎퐴 std of TFP shock - 0.010 0.010 0.010 fixed fixed fixed 휌 persistence of nominal interest rate - - 0.9 0.9 to inflation in Taylor rule - fixed fixed 휁 responsiveness of nominal interest rate - - 0.11 0.11 to inflation in Taylor rule - fixed fixed d Euclidean distance - 2.1375 3.2466 2.1871 between model and data moments Note: the numbers in parentheses are standard errors. 156 4.4. Comparison of the Models Table 4.5: Moments of Estimated Models Corr(NW,X) 휎(푋) 휎(퐺퐷푃 ) 푋푡−4 푋푡−2 푋푡−1 푋푡 푋푡+1 푋푡+2 푋푡+4 NW(market) 2.96 0.45* 0.80* 0.92* 1 0.92* 0.80* 0.45* GDP 1 0.53* 0.50* 0.47* 0.51* 0.43* 0.32* 0.03 Investment 6.11 0.46* 0.45* 0.41* 0.39* 0.31* 0.22* -0.02 Risk premium 0.003 -0.07 0.07 0.20* 0.26* 0.29* 0.30* 0.25* Estimated CF NW(market) 3.07 0.26 0.59 0.64 1 0.64 0.59 0.26 GDP 1 0.62 0.93 0.96 0.73 0.70 0.70 0.69 Investment 6.65 0.79 0.96 0.96 0.42 0.64 0.59 0.24 Risk premium 0.0001 -0.71 -0.73 -0.70 0.08 0.61 0.52 -0.07 Estimated BGG 1 NW(market) 1.35 0.22 0.34 0.33 1 0.33 0.34 0.22 GDP 1 -0.45 -0.43 -0.47 0.64 0.85 0.09 0.22 Investment 1.12 0.45 0.43 0.48 0.64 0.78 0.08 -0.22 Risk premium 0.56 -0.88 -0.90 -0.90 -0.44 0.05 -0.35 -0.22 Estimated BGG 2 NW(market) 0.98 0.78 0.89 0.63 1 0.63 0.89 0.78 GDP 1 0.99 0.99 0.87 0.86 0.84 0.82 0.75 Investment 9.53 0.64 0.67 0.12 0.60 0.81 0.83 0.75 Risk premium 0.39 0.51 0.50 -0.12 0.40 -0.28 0.25 -0.33 Note: * denotes significance at 5% level. 157 4.5. Conclusion 4.5 Conclusion This chapter is an exploration of the role of net worth in business cycles in the U.S. since 1952. It documents that 1) net worth is pro-cyclical; 2) net worth has had higher volatility since 1983; and 3) the synchronization of net worth with GDP, investment and interest rates has been stronger since 1983. Applications of formal econometric test developed in chapter 3 suggest both the Carlstrom and Fuerst (1997) model and Bernanke et al. (1999) model can capture reasonably well the cyclical properties of net worth. In addition, price rigidity seems to play an important role in the Bernanke et al. (1999) model, as it improves the quantitative performance of the model significantly. However, both models can only partially capture the positive correlation between risk premium and net worth. Given that both models have built in mechanism which tends to predict a negative or at best zero correlation, it remains to be explored whether other omitted factors contribute to such a positive relationship. 158 Bibliography Bernanke, Ben S and Mark Gertler, “Inside the Black Box: The Credit Channel of Monetary Policy Transmission,” Journal of Economic Perspec- tives, Fall 1995, 9 (4), 27–48. Bernanke, Ben S., Mark Gertler, and Simon Gilchrist, “The finan- cial accelerator in a quantitative business cycle framework,” in J. B. Taylor and M. Woodford, eds., Handbook of Macroeconomics, Vol. 1 of Handbook of Macroeconomics, Elsevier, 1999, chapter 21, pp. 1341–1393. Carlstrom, Charles T and Timothy S Fuerst, “Agency Costs, Net Worth, and Business Fluctuations: A Computable General Equilibrium Analysis,” American Economic Review, December 1997, 87 (5), 893–910. Chen, Nan-Kuang, “Bank net worth, asset prices and economic activity,” Journal of Monetary Economics, 2001, 48 (2), 415 – 436. Gale, Douglas and Martin Hellwig, “Incentive-Compatible Debt Con- tracts: The One-Period Problem,” Review of Economic Studies, October 1985, 52 (4), 647–63. Gregory, Allan W. and Gregor W. Smith, “Statistical Aspects of Cal- ibration in Macroeconomics,” in G. S. Maddala, C. R. Rao, and H. D. 159 Chapter 4. Bibliography Vinod, eds., Handbook of Statistics, Vol. 11, Amsterdam: North-Holland, 1993, chapter 25, pp. 703–719. Kiyotaki, Nobuhiro and John Moore, “Credit Cycles,” Journal of Po- litical Economy, April 1997, 105 (2), 211–48. Meh, Csaire and Kevin Moran, “The Role of Bank Capital in the Prop- agation of Shocks,” Working Papers 08-36, Bank of Canada October 2008. Townsend, Robert M., “Optimal contracts and competitive markets with costly state verification,” Journal of Economic Theory, October 1979, 21 (2), 265–293. Vuong, Quang H., “Likelihood Ratio Tests For Model Selection and Non- Nested Hypotheses,” Econometrica, 1989, 57 (2), 307–333. Williamson, Stephen D, “Costly Monitoring, Loan Contracts, and Equi- librium Credit Rationing,” The Quarterly Journal of Economics, February 1987, 102 (1), 135–45. 160 Chapter 5 Conclusion 161 Chapter 5. Conclusion The second chapter of the dissertation studies whether exchange rate appreciations would affect productivity growth. In the theoretical part, I introduce industry heterogeneity to the industrial organization environment of Dornbusch (1987) and adopt the assumption of disruptive technological change of Holmes, Levine and Schmitz (2008) to study the effect of increased competition due to an exchange rate appreciation. Exchange rate apprecia- tion can provide incentive to upgrade technology, since when the exchange rate appreciates, the profit loss due to adjustment to the new technology, which is part of the opportunity cost of technology upgrade, is low. In addition, industry heterogeneity plays an important role, in the sense that different industries are affected by the same exchange rate appreciation in different manners. For firms in industries shielded by high trade cost, they are less likely to respond. Meanwhile, for firms in highly traded industries, those in highly concentrated industries will invest more in technology up- grade, as their profit is more responsive to change in technology. The empirical section of the chapter adds to the limited empirical knowl- edge on the topic, by looking into the productivity performances of Canadian manufacturing industries between 1997 and 2006. In essence, this chapter regards the major Canadian dollar appreciation between 2002 and 2006 as driven mostly by commodity prices and exogenous to the manufacturing industries. After examining the path of productivity growth after the ap- preciation of Canadian dollar, I find evidence that compared to industries who trade less, highly traded industries had experienced faster productivity growth, and that among these industries, there is a positive relationship between concentration and productivity improvement. 162 Chapter 5. Conclusion However, when studying productivity growth, with the publicly available data, it is impossible to control perfectly for the effect of entry and exit of firms. In the regression analysis, I control for the change in the number of establishments, but have no information about the size of the entrants and exiting firms. In the future, the empirical evidence can be advanced in two directions. Firstly, applying for access to firm-level dataset in Canada can help to get a better picture of what firms did after appreciations, and to gain information on the size of establishments moving in and out of an industry. Secondly, it is interesting to look into evidence from other countries with similar appreciation experiences. To tackle the issue of comparing different calibrations, a popular quan- titative practice in economics, the third chapter proposes an econometric testing procedures for comparison of misspecified calibrated models. Rela- tive to the previous literature, we explicitly allow the models of interests to be misspecified in a frequentist framework. The test here is similar to that of Vuong (1989) and Rivers and Vuong (2002), and can be viewed as the fre- quentist counterpart of Schorfheide (2000). The null hypothesis of our test is that both models provide equal fit to some characteristics of data, against the alternative that one performs better. The fit of model can be interpreted as an in-sample forecast performance. We consider the cases where models are nested, non-nested, and overlapping. In addition, we also extend the test to the case where the model parameters are estimated to match one set of data characteristics while the model evaluation is based on another set of data characteristics. Since the model comparison is often dependent of the weights associated with the data characteristics, this chapter considers 163 Chapter 5. Conclusion the averaged and sup tests to alleviate the dependence on the choice weight- ing matrix. Overall, the method proposed is applicable to many calibration practices where researchers are interested in fits of different models. The fourth chapter applies the method developed in chapter 3 to examine fits of two leading macroeconomic models with financial frictions, Carlstrom and Fuerst (1997) model and the Bernanke, Gertler and Gilchrist (1999) model. In both models, due to financial friction, a higher level of net worth helps a firm to obtain external finance, as it allows the firm to post more collaterals and to better align its interests with those of the creditor. The two papers has shown that net worth can propagate technology shocks and monetary shocks, as the positive shocks will lead to a higher level of net worth which facilitate more borrowing and investment in the future. Since net worth places a critical role the both models, this chapter addresses the question which model replicates the cyclical properties of net worth better. The econometric test results indicate both perform reasonably well. Inter- estingly, it seems price rigidity play an important role in the Bernanke et al. (1999) model, as it improves the quantitative performance of the model. However, the models can only partially account for the positive correlation between risk premium and net worth. 164 Bibliography Bernanke, Ben S., Mark Gertler, and Simon Gilchrist, “The finan- cial accelerator in a quantitative business cycle framework,” in J. B. Taylor and M. Woodford, eds., Handbook of Macroeconomics, Vol. 1 of Handbook of Macroeconomics, Elsevier, 1999, chapter 21, pp. 1341–1393. Carlstrom, Charles T and Timothy S Fuerst, “Agency Costs, Net Worth, and Business Fluctuations: A Computable General Equilibrium Analysis,” American Economic Review, December 1997, 87 (5), 893–910. Dornbusch, Rudiger, “Exchange Rates and Prices,” The American Eco- nomic Review, 1987, 77 (1), 93–106. Holmes, Thomas J., David K. Levine, and James A. Schmitz, “Monopoly and the Incentive to Innovate When Adoption Involves Switchover Disruptions,” NBER Working Paper, 2008, No. W13864. Rivers, D. and Q. Vuong, “Model Selection Tests For Nonlinear Dynamic Models,” Econometrics Journal, 2002, 5 (1), 1–39. Schorfheide, Frank, “Loss Function-Based Evaluation of DSGE Models,” Journal of Applied Econometrics, 2000, 15, 645–670. 165 Chapter 5. Bibliography Vuong, Quang H., “Likelihood Ratio Tests For Model Selection and Non- Nested Hypotheses,” Econometrica, 1989, 57 (2), 307–333. 166

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