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Essays in housing and macroeconomy Huang, Haifang 2008

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ESSAYS IN HOUSING AND MACROECONOMY by Haifang Huang B.A., The People’s University of China, 1999 M.A., The University of New Brunswick, 2001 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Economics) The University Of British Columbia (Vancouver) November, 2008 c© Haifang Huang 2008 Abstract Compared to the previous twenty years, residential investments in the US appear more stable after the mid-1980s. Chapter 2 explores key hypothe- ses regarding the underlying causes. In particular, it uses estimated DSGE models to examine whether a more responsive interest rate policy stabilizes the housing market by keeping inflation in check. These estimations indeed found a policy that has become more responsive over time. Counter-factual analysis confirms that the change stabilizes inflation as well as nominal in- terest rate. It does not, however, find the change in policy to have stabilizing effect on real economic activity including housing investment. It finds that smaller TFP shocks make modest contributions, while the biggest contribut- ing factor to the fall in the housing volatility is a reduction in the sensitivity of the investment to demand variations. Chapter 3 constructs a richly specified model for the housing market to examine the empirical relevance of various costs and frictions, including the investment adjustment cost, sticky construction costs, search frictions, and sluggish adjustment of house prices. Using the US national-level quarterly data from 1985 and 2007, we find that the gradual adjustment of house prices is the most important and irreplaceable feature of the model. The key to developing an optimization-based empirical housing model, therefore, is to provide a structural interpretation for the slow adjustment in house prices. Chapter 4 uses US national-level time series of residential investment, price index of new houses, consumption and interest rate to explore whether the US, as a nation, experienced a drop in the price elasticity of supply of new housing. Maximum likelihood estimations with a simple stock-and- flow model found a statistically significant drop of the elasticity from 10 to 2.2, when the quarterly data between 1971 and 2007 are split at 1985. A richer model with mechanisms of gradual adjustment also indicates such a reduction, when existing knowledge about the adjustment parameters is incorporated in the analysis. For the Federal Reserve, an inelastic supply can be a source of concern, because policy-driven demand in housing market is more likely to trigger undesirable swings in prices. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Changes in Business Cycle Dynamics of Residential Invest- ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Residential Investment in Business Cycles: Changes . . . . . 6 2.3 The Model Economy . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 Households . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Producers of final goods . . . . . . . . . . . . . . . . 15 2.3.3 Retailers . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.4 Producers of generic goods . . . . . . . . . . . . . . . 17 2.3.5 Producers of capital goods . . . . . . . . . . . . . . . 19 2.3.6 Monetary authority . . . . . . . . . . . . . . . . . . . 21 2.3.7 Government . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.8 Aggregate resource constraint and the accumulation of capital stock . . . . . . . . . . . . . . . . . . . . . 22 2.3.9 Exogenous shocks . . . . . . . . . . . . . . . . . . . . 22 2.3.10 Steady state of the model, and the calibration strategy 24 2.3.11 The linearized system . . . . . . . . . . . . . . . . . . 26 2.3.12 Constructing a state-space econometric model . . . . 28 iii Table of Contents 2.4 Estimations and Counter-factual Analysis . . . . . . . . . . . 29 2.4.1 Overview of the strategy: Bayesian approach, split sample estimation, and counter-factual analysis . . . 29 2.4.2 Parameters to be estimated and their prior distribu- tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.3 Full-sample estimation and its in-sample fit . . . . . . 31 2.4.4 Out-of-sample forecast of the estimated model . . . . 33 2.4.5 Split-sample estimations . . . . . . . . . . . . . . . . 33 2.4.6 Breaking down the reduction in variance using counter- factual analysis . . . . . . . . . . . . . . . . . . . . . 35 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . 40 3 Costs and Frictions in the Housing Market. . . . . . . . . . 51 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1.1 The stock-flow model and its empirical performance . 54 3.2 The Augmented Model . . . . . . . . . . . . . . . . . . . . . 59 3.2.1 The asset market . . . . . . . . . . . . . . . . . . . . 60 3.2.2 Supplier of construction inputs . . . . . . . . . . . . . 61 3.2.3 Construction arm of residential builders . . . . . . . . 63 3.2.4 Retail arm of residential builders . . . . . . . . . . . . 64 3.2.5 Homebuyers . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.6 The linearized model . . . . . . . . . . . . . . . . . . 68 3.3 Compare the Estimated Models . . . . . . . . . . . . . . . . 70 3.3.1 Log-likelihood statistics . . . . . . . . . . . . . . . . . 70 3.3.2 One-step-ahead forecast . . . . . . . . . . . . . . . . . 71 3.3.3 Medium-run forecast . . . . . . . . . . . . . . . . . . 72 3.3.4 Second moments . . . . . . . . . . . . . . . . . . . . . 73 3.4 Some Discussions on the New Features of the Model . . . . . 74 3.4.1 Investment adjustment cost versus price rigidity . . . 74 3.4.2 Sticky construction costs, volatility of investments, and price rigidity . . . . . . . . . . . . . . . . . . . . 75 3.4.3 Inventory and changes in interest rate . . . . . . . . . 76 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.6 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Is there a Reduction in the Supply Elasticity of New Hous- ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 iv Table of Contents 4.2.1 Price of houses . . . . . . . . . . . . . . . . . . . . . . 90 4.2.2 Sluggish adjustment of house price . . . . . . . . . . . 91 4.2.3 Construction cost . . . . . . . . . . . . . . . . . . . . 91 4.2.4 Adjustment cost in construction industry . . . . . . . 92 4.2.5 Market clearance . . . . . . . . . . . . . . . . . . . . 94 4.2.6 Reduce the model to a simple stock-flow model . . . 94 4.2.7 The linearized system . . . . . . . . . . . . . . . . . . 94 4.2.8 Stochastic processes . . . . . . . . . . . . . . . . . . . 96 4.2.9 Observable variables . . . . . . . . . . . . . . . . . . . 96 4.3 The Estimations . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3.1 Estimating the basic model . . . . . . . . . . . . . . . 98 4.3.2 Estimating the richer model: a MLE approach . . . . 99 4.3.3 Estimating the richer model: a Bayesian approach . . 101 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.5 Table and figures . . . . . . . . . . . . . . . . . . . . . . . . 104 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Appendices A Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 121 A.1 Model Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.1.1 Households’ optimality conditions . . . . . . . . . . . 121 A.1.2 Retailers’ optimality conditions . . . . . . . . . . . . 123 A.1.3 Phillips curve . . . . . . . . . . . . . . . . . . . . . . 123 A.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.3 Kalman Filter and Likelihood Function . . . . . . . . . . . . 125 A.4 Bayesian Estimation and RWM Algorithm . . . . . . . . . . 127 A.4.1 Maximizing posterior density to estimate the mode . 127 A.4.2 Use RWM algorithm to explore the distribution around the estimated mode . . . . . . . . . . . . . . . . . . . 127 A.5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 B Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 129 B.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 B.2 Calibrated parameters . . . . . . . . . . . . . . . . . . . . . . 129 B.3 Deriving first order conditions for the inventory model . . . . 130 v Table of Contents C Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 133 C.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 C.2 Calibrated parameters . . . . . . . . . . . . . . . . . . . . . . 133 vi List of Tables 2.1 Descriptive statistics of observed variables between 1963-2006. . . 40 2.2 Parameters: prior and posterior distributions from the full- sample estimation. . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 The estimated DSGE model’s performance in out-of-sample forecast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Variance decomposition of the 8-quarter ahead forecast error. . . 43 2.5 Posterior distributions from the split-sample estimations. . . . 44 2.6 Breaking down the reduction in variance. . . . . . . . . . . . . 45 3.1 Testing the empirical importance of costs and frictions in the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.2 The performance of the estimated models in medium-run fore- cast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1 Estimating the basic model. . . . . . . . . . . . . . . . . . . . 104 4.2 Estimating the model with gradual adjustment in prices and constructions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 vii List of Figures 2.1 Observable variables: The data and the one-step-ahead forecast 46 2.2 Output gap and residential investments . . . . . . . . . . . . 47 2.3 Housing starts, change from same month last year . . . . . . 48 2.4 Output gap and mortgage flow . . . . . . . . . . . . . . . . . 49 2.5 Residential investments and the price index of residential struc- tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1 Compare one-step-ahead forecasts and realizations . . . . . . 80 3.2 Compare impulse responses from the stock-flow model and those from the data . . . . . . . . . . . . . . . . . . . . . . . 81 3.3 Compare different models for their accuracy in one-step-ahead forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.4 Impulse responses to an interest rate change : IAC V.S. rigid- ity in house price . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.5 Impulse response to an interest rate change: sticky construc- tion costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.6 Compare impulse responses from the full model and those from the data . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.7 Inventory and housing market’s response to an interest rate shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.1 Quarterly growth rate of housing price and investment in the US . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2 Maximized log-likelihood at different values of ψ: basic stock- flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.3 Maximized log-likelihood at different values of τ . . . . . . . 108 4.4 Maximized log-likelihood at different values of ϕhs . . . . . . 109 4.5 Maximized log-likelihood at different values of ψ: with grad- ual adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.6 Contour chart of log-likelihood in (ψ, τ) space: 1971Q1-1984Q4111 4.7 Contour chart of log-likelihood in (ψ, τ) space: 1985Q1-2007Q2112 4.8 Maximized log posterior function at different value of ψ . . . 113 viii Acknowledgements I would like to thank Dr. Paul Beaudry and the rest of my committee for their support and guidance. I also thank Mike Peters, Yoram Halevy, Ivar Ekeland, Vadim Marmer, and Laura Turner for their help in various forms. Sincere thanks also go to Dr. John Helliwell, for his very long-term support, academically and financially. ix Dedication To my mum, my brother, my father, my extended family. To D. Deng. x Chapter 1 Introduction This dissertation explores housing market and its relation with the broader economy. The second and the fourth chapters focus on changes in the dy- namic pattern of the US residential investments after the mid-1980s, and their policy implications. The third chapter augments a housing model with several features, and examines their empirical relevance. Chapter 3 bridges the gap between chapter 2, which uses a simple stock-flow model for housing investment, and chapter 4, which uses a relatively richer model for the same investment. Chapter 2 looks at the reduction in the volatility of residential invest- ments in the United States since the mid-1980s. The chapter provides a quantitative examination of underlying causes in the framework of a dy- namic stochastic general equilibrium (DSGE) model. The objective is two- fold: 1) to evaluate the hypotheses based on improvements in monetary policy; and 2) to detect and evaluate other contributing factors. The anal- ysis does not find evidence that the estimated change in monetary policy contributed to stabilizing housing investments. The reason is that the esti- mated DSGE model does not have a strong link between nominal volatility and real volatility. We find that smaller TFP shocks make modest contri- butions. The biggest contributing factor is a reduction in the sensitivity of the investments to demand variations. The reduced sensitivity may be an indication of structural changes in the housing market. Chapter 4 explores one particular hypothesis: a reduction in supply elasticity of new housing. Chapter 3 is motivated by the fact that the standard stock-flow model for housing investment, which chapter 2 uses, performs poorly in keeping track of the housing data and capturing their stylized patterns. This chap- ter augments the basic model with several costs and frictions, and examines the empirical relevance the newly added features. The new features come from both housing and macro literature. They include the investment and capital adjustment costs, sticky construction costs, and sluggish adjustment of house prices. Using quarterly US data between 1985 and 2007, and multi- ple empirical criterions, the chapter finds that the slow adjustment of house prices is the most important and irreplaceable feature of the model. The 1 Chapter 1. Introduction chapter concludes that the key to developing an optimization-based empir- ical housing model is to provide a structural interpretation for the gradual adjustment in the house price. Chapter 4 observes that the reduction in the volatility of housing invest- ment in US since the mid-1980s is accompanied by a relative increase in the variability of prices. The relative increase appears at the high-frequency quarter-to-quarter changes, as well as at the low-frequency swing over hous- ing cycles. The chapter sets out to examine if the supply of new houses in US, at the national level, has become less elastic. Using maximum likeli- hood estimations and time series of housing investment, price, consumption and interest rate, the chapter finds evidence to suggest a significant reduc- tion of the elasticity of new supply in a simple stock-flow model. A richer model with mechanisms of gradual adjustment also suggests such a reduc- tion, when knowledge about the parameter of gradual adjustments is incor- porated in the estimations. An inelastic supply calls for greater attention to the housing market from the Federal Reserve when conducting its interest rate policy. With inelastic supply, fluctuations in housing demand driven by policy changes are more likely to trigger undesirable swings in house prices. 2 Chapter 2 Changes in Business Cycle Dynamics of Residential Investment in the U.S.: an Investigation with Estimated DSGE Models 2.1 Introduction Since the mid-1980s, U.S. housing-related economic activity appears more stable than it had been in the preceding two decades. The standard devia- tion of the quarterly growth of residential fixed investment in the national income and product accounts (NIPA) was 6% between 1962 and 1985, and 2.4% between 1986 and 2006. The variability of other measures of housing investment, such as housing starts and issuance of housing permits, have also become smaller.1 Residential investment also appears less pro-cyclical. Historically, hous- ing investment tends to fall before recessions and picks up during recoveries. The tendency of positive co-movement has been weaker in recent years. In the 1990 economic recession, the housing retrenchment was modest by his- torical standards. In the 2001 recession, the growth of residential investment was largely undisturbed. What has caused these changes is not well understood. Many aspects of the US economy are different now than they were in the 1960s and 1970s. Some of these changes are likely to affect residential investments’ business cycle properties. One indication of the shifting economic landscape is the so-called “Great Moderation,” or widespread reduction in macroeconomic volatility, that be- 1The reduction in variance is also found at the state level in housing permit data. 3 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . gan in the early-1980s. This included a 50% drop in the standard deviation of the quarterly output growth, a 2/3 reduction in the standard deviation of inflation, and smaller measures for volatility in almost all major NIPA aggregates (Table 3, Stock and Watson (2002)). What produced the great moderation, and did it also affect residential investment? Three explanations have been suggested for the cause(s) of the great moderation. The first explanation is the smaller economic shocks, in- cluding less frequent disruptions of oil supply and smaller variance in Solow residuals, an empirical measure of productivity growth. The second explana- tion focuses on the apparent changes in Federal Reserve’s monetary policy. The Federal Reserve is believed to have taken a more aggressive stand in fighting inflation since the early-1980s, and have succeeded in bringing down inflation and its variability. Because inflation volatility and real volatility tend to move together in the US and in other industrialized countries (Blan- chard and Simon (2001)), the policy change is regarded as a likely factor behind the great moderation.2 Finally, the third explanation focuses on structural changes in the US economy.3 For residential investment, both smaller economic shocks and the change in monetary policy are likely to have impacts. Oil shocks and productivity shocks affect all sectors. A reduction in their frequency and magnitude can reduce variance in housing market as well as the correlation across sectors. An improvement in the monetary policy, according to a conjecture in Taylor (2007), keeps inflation low and steady, thereby reducing boom-bust cycles and large swings in interest rates that can cause fluctuations in housing market. The monetary policy improvement, therefore, not only accounts for, but also ties together, the great moderation and the housing modera- tion. Empirically, residential investment is very sensitive to monetary policy shocks (Figure 3 in Bernanke and Gertler (1995a)). Such sensitivity high- lights the importance to examine how the policy change have affected the housing sector. Sector-specific structural changes in the housing market may also have played an important role. McCarthy and Peach (2002) describe the transfor- mation of the US mortgage market in the past two decades with its banking 2See discussion in Bernanke (2004). Clarida et al. (2000) developed theoretical models that relate nominal volatility to real volatility. 3In the literature, structural changes that are suggested to be possible causes of the great moderation include improved management of business inventories, increased depth and sophistication of financial markets, deregulation, a shift away from manufacturing toward services, and increased openness to trade and international capital flows (see dis- cussion in Bernanke (2004)) 4 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . deregulation, mortgage securitization, and financial innovations, all of which have the potential to improve the resilience of housing financing system. Important changes also occurred in the construction industry. The fall of unionism is one such example. The share of union in the construction work- force has dropped from 70% in the 1970s to 20% in the 1990s (see Allmon et al. (2000), also Thieblot (2001)). Unions in the industry were once known for their strong “hiring hall” practice, which often meant fixed compensation and workers queueing for job opportunities (see Abowd and Farber (1982) for their characterization of unions’ hiring hall4). With fixed compensation, wages were unresponsive in the short-run, leaving employment and hours worked to do most of the adjustments. A reduction in the presence of union may lead to a more balanced response of wages and employments. Finally, residential builders’ business practices have also changed. The share of homes that are sold before construction starts has increased from 19% before 1981 to 36% in 1999 (Kahn (2000)). With a smaller share of speculative construction, builders can reduce inventory buildup in down- turns, and therefore avoid slowing down too much compared to what they otherwise would have to. This paper is an effort to quantitatively sort out the importance of com- peting explanations for the lower volatility of residential investment. It uses a structural macroeconomic model to explore the consequence of the change in monetary policy, and for other structural interpretations. Its choice of macro model belongs to the class of sticky-price DSGE models (or new neo- classical synthesis models.) This class of models combines Real Business Cycle models and New Keynesian features, thus admitting simultaneously the explanations based on economic shocks and those based on monetary policy. Because of the advantage, these models are popular in the literature that tries to understand the great moderation (for example Justiniano and Primiceri (2006), Leduc and Sill (2007), and Stock and Watson (2002)). To capture possible changes that are specific to the housing sector, this paper uses a model fashioned along the line of Christiano et al. (2005) and Smets and Wouters (2003). This particular string of models employs rich specifications of nominal and real rigidity, whose parameters are important determinants of the dynamics of endogenous variables. A change in these rigidity parameters, if detected, is indicative of potential structural changes in the relevant sector. For housing sector, the key parameters to watch are 4Abowd and Farber (1982) does not focus on construction union. It studies the practice of union firing hall in general. Construction industry is one of their examples for strong practice of hiring hall. 5 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . those that govern the adjustment mechanism of the investment. This paper follows a two-step approach for quantification analysis. In the first step, it estimates the model using split-samples to detect changes in parameters of the model. In the second step it performs counter-factual analysis based on the estimated changes. The estimation follows a Bayesian approach and uses Random Walk Metropolis algorithm to provide measures of statistical confidence. An and Schorfheide (2006) reviews the methodol- ogy in details. The appendix of this paper presents a brief description. Existing literature has seen few effort to explain the observed change in residential investments using a structural business cycle model. One notable exception is Campbell and Hercowitz (2005). They construct a borrower- lender model in which credit-constrained households adjust labor supply to satisfy downpayment requirement when purchasing durable goods. They show that an easing of the borrowing constraint in the 1980s can explain a substantial fraction of the decline in the volatility of hours worked, GDP, and households’ investments in durable goods. Their analysis is based on calibration and does not explore alternative explanations. Our paper is different in our use of estimation-based quantitative analysis to compare different hypotheses. This paper is related to a growing literature that studies the interaction between the housing market and the broader economy in a DSGE model. Examples include Aoki et al. (2004), Iacoviello (2005a), Gammoudi and Mendesy (2005), and Pescatori and Mendicino (2005). The organization of the paper is as follows: Section 2 documents the changes in the dynamic pattern of residential investment. It also presents evidence for smaller economic shocks and discusses the change in monetary policy and its implications for residential investment. Section 3 presents the linearized version of the model (The original model is in Appendix A). Section 4 provides an overview of the estimation strategy before presenting the findings, and Section 5 concludes. 2.2 Residential Investment in Business Cycles: Changes The residential investment spending, which the paper focuses on, is from BEA’s national income and product accounts (NIPAs). It consists mostly of homeowners’ investments in private residential structures and landlord’s investments in residential structures and equipment. It is about 5% of the total GDP. Even though the share of the investment is small, its volatility 6 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . is quite large, especially in the 1970s and the early 80s. The investment could swing, in the matter of a few quarters, by 2-3% of the GDP (Figure 2.2). This definition of housing investment does not include the purchase of consumer durables, which often comes with the purchase of residence units. When the value of consumer durable are included, the share of housing investment is larger than business sector’s fixed investment in structure and equipment. The fourth panel of Figure 2.1 plots the quarterly growth of residential fixed investment between 1963 and 2006. The period before the mid-1980s was significantly more volatile than the more recent period. The difference is visually apparent. An alternative measure of residential investment is the series of housing starts published in monthly frequency by the Census Bureau. Figure 2.3 plots the % change of housing starts from the same month one year ago. The observation is similar to those from Figures 2.1, that the series was more volatile before the mid-1980s. Figure 2.2 illustrates the weakened correlation between residential in- vestments and business cycles. It plots residential investments as a share of potential GDP, together with the output gap from 1963Q2 to 2007Q1. For most of the sample, between the late-1960s and early-1990s, residential in- vestment exhibits clear peaks and troughs, largely synchronized with those of the output gap, with residential investment usually leading the cycle. In the more recent period, movements of residential investment and output ap- pear more detached from each other. The 2001 recession sent the GDP into prolonged stagnation. Residential investment, on the other hand, continued to grow steadily. The lower panel of Table 2.1 shows the decline in the correlation coeffi- cients between the growth of residential investment and the growth of total output. The growths are defined as % change from the same quarter one year ago. The sample is split into two groups: one from 1963Q1 to 1985Q4, and the other from 1986Q1 to 2006Q2. Four correlation coefficients are pre- sented for each of the two sub-samples. These four coefficients measure the correlation between the growth of residential investment and the growth of GDP in the same-quarter, one-quarter-ahead, two-quarter-ahead and three- quarter-ahead, respectively. All four measures are lower in the more recent sample. For example, the correlation between housing investment growth and the three-quarter-ahead GDP growth is 0.71 in the pre-1986 sample, and 0.48 in the post-1986 sample. Another piece of evidence that suggests a weaker correlation comes from the mortgage market. Figure 2.4 plots the mortgage flow (defined as the net 7 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . increase of household home mortgage debt outstanding) as a share of poten- tial GDP, again with the output gap in the same panel. The observations are similar to those for the residential investments. The ups and downs of the mortgage flow largely match the output gap between the early-1970s and the late-1980s. Nevertheless, in 2001, they rose more quickly as the econ- omy fell into recession and stagnation. The level of mortgage flow reached an unprecedented level in 2006 before falling in the later part of 2006 and in 2007. Fisher and Quayyum (2006) discuss what they call the “great turn- of-the-century” housing boom, attributing part of it to the development of a sub-prime mortgage market. In the literature, Stock and Watson (2002) observed that the housing sector had the largest relative decline in volatility among major NIPA ag- gregates. Campbell and Hercowitz (2005) documented that household debt, including home mortgages, became less correlated with hours worked. Dy- nan and Sichel (2006) found that residential investment became less sensitive to changes in interest rates. McCarthy and Peach (2002) found that the in- vestment’s response to monetary policy shock became smoother over time. Great Moderation: widespread reduction in output volatility The reduction in the volatility of housing investment occurred at roughly the same period as the so-called great moderation. The great moderation is “one of the most striking features of the economic landscape over the past twenty years.” (Bernanke (2004)) Figure 2.1, besides presenting the time series of housing investment, also plots the time series of GDP, consumption, and business investment, together with the federal funds rate and inflation. All, except for the business-fixed investment series, experienced a large decline in their volatility after the mid-1980s. Regarding the timing, Kim and Nelson (1999) estimated that 1984Q1 was the most likely date for a structural break. Besides the decline in variance, Stock and Watson (2002) found that GDP growth was easier to forecast, and that the decline in volatility was widespread across sub-components of output. Among the main NIPA aggregates that they reviewed, 21 of 22 had lower standard deviations in the post-1984 period (Table 3, in Stock and Watson (2002)). The decline in standard deviation is not uniform across the series. The largest relative decline in volatility happened with residential investments. 8 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Smaller economic shocks Several papers attribute the great moderation primarily to smaller oil price and productivity shocks.5 The first evidence of smaller shocks is seen in the oils shocks that become less frequent. The following information is from Table 4 of Hamilton (2003). It documents the military conflicts and revolutions that have disrupted the worlds’ oil supply. Date Event Drop in world production Nov. 1973 Arab-Israeli War 7.8% Dec. 1978 Iranian Revolution 8.9% Oct. 1980 Iran-Iraq War 7.2% Aug. 1990 Persian Gulf War 8.8% In the seven years between 1973 and 1980, three major disruptions oc- curred, while after 1980, only one disruption took place. Smaller TFP shocks were also reported. Leduc and Sill (2007) used in- put, output, and oil price information to measure the historical productivity growth in the US economy. They fed the measured productivity growth into a macro model and found that the change associated with the series explains most of the reduction in the variance of output growth. Changes in monetary policy Explanations based on the change in monetary policy are motivated by the view that the Federal Reserve pursued a more aggressive monetary policy in fighting inflation after Volcker became the chairman of the Federal Reserve system in 1979, and the policy change succeeded in bringing down infla- tion and its variability. The change in policy probably came from a better understanding of the expectational nature of the unemployment-inflation trade-off. Clarida et al. (2000) find “striking difference” in the policy reaction func- tion across time. In particular, they find that the response to inflation in- creased significantly from the pre-Volcker period to the Volcker-Greenspan era. They theorize that the policy change had a stabilizing effect in real activity by mitigating the effect of fundamental shocks and preventing self- fulfilling “sun-spot” shocks. The hypothesis receives significant attention in the literature. Many researchers confirm that the monetary policy be- came more responsive to inflation (For example, Ireland (2004), Boivin and Giannoni (2006), and Lubik and Schorfheide (2004)). 5For example Stock and Watson (2002), and Leduc and Sill (2007). 9 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . More controversial, however, is whether or not the change in monetary policy reduces real volatility. Stock and Watson (2002) and Leduc and Sill (2007) compared the shocks-base explanation and the policy-based explana- tion in quantitative models and attributed most of the great moderation to smaller shocks. This paper will contribute to the discussion by including housing invest- ment in a DSGE model. The inclusion of residential investment in the macro monetary model expands the investigation on the change in monetary policy. Empirically speaking, residential investment is a spending component that is most sensitive to monetary policy shocks. The investment drops quickly and sharply following a monetary tightening. The sensitivity appears to be greater than the business investment, and the structure investment in the business sector.6 The latter observation is particularly interesting since resi- dential investment and business structure investment both involve long-lived assets with similar characteristics (Edge et al. (2004)). One explanation that was put forward by Edge et al. (2004) to explain the difference between residential investment and structure investment is that the former has a shorter project planning and completion time. Their calibrated business cycle model achieves some success in replicating the dif- ference. Based on this hypothesis, however, it is not clear why residential investment has now shown a smoother response to monetary policy shocks (a finding in McCarthy and Peach (2002)). Has the planning and completion horizon of the investment become longer? Another explanation for the high sensitivity of residential investment to monetary policy actions involves the possible existence of “banks-lending channel.” The banks-lending view hypothesizes that when a monetary au- thority uses open market operations to implement monetary policy, it with- draws or injects reserve funds into the banking system. Because a reserve fund is needed by banks to satisfy reserve requirements, the fund availability affects the banks’ ability to sell deposits and therefore their ability to make loans. An implication thus is that bank-dependent investors suffer more from monetary contraction than do those who have alternative sources of funds. As far as residential investment is concerned, households are the most bank-dependent, especially when the US mortgage market was domi- nated by the savings institutions in the 1960s and 1970s. The banks-lending channel may therefore explain the investment’s high policy sensitivity.7 Be- 6Housing investments’ initial response in % terms is larger than that of business in- vestments (Figure 3, Bernanke and Gertler (1995a)). 7Iacoviello and Minetti (2002) have discussions on banks-lending channel specific to the housing sector. 10 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . cause the strength of the channel depends on the tightness of constraints that are facing the banks, a relaxation of the reserve requirement reduces the channel’s significance. Since the Federal Reserve lowered or eliminated many reserve requirements, this might explain why investment now exhibits a smoother response to monetary policy shocks. Even if we take the policy sensitivity as given, how the change in mone- tary policy will affect the housing market is unclear. An aggressive monetary tightening, as a response to inflation, can send residential investment into a tailspin. An easing of monetary policy, for fear of deflation, can send housing market soaring. It is therefore possible that a more responsive policy makes the housing market more volatile. On the other hand, a more responsive policy can works through the channel of expectation, achieving stabilizing effect with better anchored inflation expectations. In other word, it can avoid boom-bust cycles in the economy and therefore the need for aggres- sive policy actions in the first place. It is the second channel that Taylor (2007) emphasizes when he hypothesizes the link between the change in policy and the volatility of the housing market. Beside the indirect effect described above, the stabilization of inflation may have direct benefits in the housing sector as well. The sector has much to gain from stable inflation because it often uses long-term fixed-rate mort- gages as a means of finance, which exposes it to inflation risks. Rudebusch and Wu (2004) construct a macro-finance model that interprets the latent factors of term structure as macroeconomic variables. One of the macro variables is the perception of investors about the inflation target, which is closely identified with the behavior of monetary policy. A policy that has a credible inflation target most likely will have a stabilizing effect on the long-maturity end of the yield curve, which, in turn, can make the decision on housing investment easier and smoother, since the incentive to time the cost will be reduced. 2.3 The Model Economy This paper uses an DSGE model to tie together macroeconomic variables, residential investments, economic shocks, and monetary policy. The model jointly specifies the structure of the economy, the monetary policy reaction function, and the property of exogenous shocks. The model involves a substantial number of parameters. For this reason, the estimations and statistical inferences are performed with the Bayesian approach and the Markov Chain Monte Carlo algorithms. The estimated 11 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . model in this paper achieves good empirical performance. For example, it substantially outperforms an unrestricted VAR in out-of-sample forecasts. The theoretical model abstracts from long-term growth by assuming zero growth rate for population and productivity in the long run. Reflecting this imposed stationarity in the theoretical model, we detrend the time series of macroeconomic variables before using them to estimate the model. These macroeconomic variables are the total output, consumption, business investment, housing investment, inflation, and the nominal interest rate. 2.3.1 Households Households’ preference is identical to that in Smets and Wouters (2007) with one exception, that the definition of consumption in this paper is expanded to include the consumption of housing services. Households maximize ex- pected life-time utility max (ct,ht+1,lt,Bt+1) Et ∞∑ s=0 Λt+s (cb,t+s − λbcb,t+s−1)1−σc 1− σc exp { σc − 1 1 + σl l1+σlt+s } In the utility function, lt indicates hours worked, and cb,t is the CES mixture of nondurable goods (ct) and housing service (ht). The consumption bundle is defined as cb,t = [ c1−γt + vhh 1−γ t ] 1 1−γ (2.1) The parameter γ (0 < γ <∞) is the inverse of the elasticity of substitution. When γ = 0, nondurable goods ct and housing service ht are perfectly substitutable. As γ approaches ∞, the substitutability becomes zero, and we end up with the Leontief preference. The parameter vh is one of the factors that determine the split between housing services and non-durable goods in households’ total consumption expenditure. This model defines the unit of housing service by normalizing that one unit of housing capital provides one unit of service. Therefore ht is also the stock of housing capital.8 The momentary utility function also features consumption habit forma- tion. The external habit is captured by the parameter λb, which is between zero and one. When λb is zero, utility depends on the level of consumption. 8Households’ utility function should have also included a preference for real monetary balance. In the equilibrium, however, the demand for money is met by monetary authority, who chooses an interest rate target and supply as much amount of money as necessary to achieve the target. As the result there is no need to model money in utility explicitly, as long as the preference for real monetary balance is separable to others arguments. 12 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . When λb is one, utility depends on changes in the consumption. When λb is between zero and one, both the level and the change affect utility. The term Λc,t indicates the weight that households assign to the utility in period t. As the result, the discount factor between two periods is Λc,t+sΛc,t . The dynamics of Λc,t are captured by Λc,t+1 Λc,t = βezχc,t A positive shock to zχc at time t raises households’ preference on current- period utility over that in the future. For this reason, the stochastic com- ponent in zχc,t is interpreted as consumption preference shocks. Households’ budget constraint is: wtlt + Bt−1 Pt + pih,t ≥ Tt Pt + ct + Bt 1 +Rt 1 Pt +qh,tezmh,t [ht+1 − (1− δh)ht]−mhqh,tezmh,tht The left hand side of the budget constraint sums together the sources of revenues in period t. The first term, or wtlt, is the employment income expressed in real terms. The second term, or Bt−1Pt , is the real value of bonds reaching maturity in period t (since the model has one-period bonds only, all bonds purchased in period t − 1 mature in period t). The last term, or pih,t, is the total amount of profit distributed to households by businesses in the economy. The right hand side of the budget constraint is the expenditure lay-out. The first term, TtPt , is the lump-sum tax collected by the government. The second term, ct, is the consumption expenditure. In the third term, Bt is the face value of discount bonds purchased in period t. The capital-cased Rt is the discount rate. Therefore Bt1+Rt 1 Pt is the real investment expenditure in bonds. The fourth term is the total cost incurred to the households for expanding their stock of housing capital from ht to ht+1. The total per unit cost of the capital is qh,tezmh,t , with qh,t being the unit price, and zmh,t (which has a mean of zero) indicating the extra costs. This extra cost may include transaction costs and mortgage costs (to the extent the mortgage rate is different from the cost of capital), among others. Since zmh,t affects demand for housing capital, it is interpreted as housing demand shocks. The last term is a property tax mh levied on the value of the housing capital. The tax rate affects the spread between the return from home ownership and the return from alternative investment in bonds. Technically, this spread gives us the degree of freedom that is necessary to calibrate the model according to 13 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . the observed data. More specifically, we can choose mh so that the steady- state solution to the model replicates the share of housing rents (including the market rent and the imputed rent) in total output, which is reported in the NIPA. The steady state solution is presented later in the paper. The consumption Euler is Ξt = βezχc,tEt [(1 + rt+1) Ξt+1] (2.2) In this Euler equation, the ex-post real interest rate is defined as the differ- ence between the pre-set discount rate and the realized inflation between t and t+ 1, i.e, 1 + rt+1 = (1 +Rt) Pt Pt+1 (2.3) The marginal utility of consumption, in the absence of preference shocks, is denoted as Ξt, Ξt = (cb,t − λbcb,t−1)−σc cγb,tc−γt exp { σc − 1 1 + σl l1+σlt } The optimal condition for hours worked, whose derivation is shown in sub-section (A.1.1), is captured by (cb,t − λbcb,t−1) c−γb,t cγt lσlt = wt (2.4) It is helpful, especially when solving the steady state of the model, to note that equ(2.4) has an alternative expression ( 1− λb cb,t−1cb,t ) etl σl t = wt, where et = c 1−γ b,t c γ t is the total consumption expenditure including rent. To see why et is the total consumption expenditure, we simply need to use the definition of cb,t (i.e., equ(2.1)) to expand c 1−γ b,t c γ t into ct+vh ( ct ht )γ ht, and realize that it is the sum of nondurable consumption and the value of housing services expressed in units of nondurable goods. The value of housing services is vh ( ct ht )γ ht, where ht is the amount of housing services provided by ht units of housing capital, and vh ( ct ht )γ is the intratemporal marginal rate of sub- stitution between consumption and housing services, or the relative price of housing services in units of nondurable goods. The investment Euler on housing capital is ezmh,tqh,tΞt = βezχc,tEt {[ vh ( ct+1 ht+1 )γ + (1− δh −mh) ezmh,t+1qh,t+1 ] Ξt+1 } (2.5) 14 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . 2.3.2 Producers of final goods To model nominal rigidity, this paper uses a framework that involves three parties in the production process of final goods: competitive producers of generic goods, monopolistically-competative retailers, and competitive pro- ducers of final goods. The setup is similar to that in Bernanke et al. (1999). The producers of generic goods sell their output to retailers, who package them into differentiated products and re-sell them at a markup. The pro- ducers of final goods combine differentiated products into final goods, with a technology that features imperfect substitutability. Let’s start with the producers of final goods. The profit maximization problem is max yi,t Ptyt − ∑ i Pi,tyi,t subject to yt = (∑ i (yi,t) 1 λf )λf In this problem, the variable yt is the output in term of final goods. The variable yi,t is the input purchased from retailer i. The parameter 1 < λf < ∞ indicates the difficulty of substitution. The price of product i is Pi,t. The aggregate level of price is P . The optimal demand for product i is yi,t yt = ( Pt Pi,t ) λf λf−1 The price of the final goods can be derived through the following relation Pt = ∑ i Pi,t yi,t yt = (∑ i P 1 1−λf i,t )1−λf 2.3.3 Retailers Retailer i can transform one unit of generic goods into one unit of inter- mediate goods, without incurring any cost. The retailer has monopolistic power over the supply of its product. Individual retailers face a Calvo-type “stickiness” in price setting. In each period, there is a chance ξp that a retailer will have to stick with its previously-set price. The chance of being stuck is random and history-independent. At the aggregate level, (1− ξp) 15 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . proportion of retailers can change their prices if they want to. The remain- ing proportion ξp index their prices according to lagged inflation. Inflation indexation automatically adjusts the price from pi,t−1 to pi,t−1pi ιp t−1pi 1−ιp∗ , where pi∗ is a steady-state inflation. Retailers who have the option to change their prices face a profit maxi- mization problem. Note that in the following representation of the problem, the subscript i is omitted, because all retailers will end up choosing the same new price. The profit maximization problem is max P̃t Et ∞∑ s=0 (ξp) s Λc,t+sΞt+s Λc,tΞt 1 Pt+s piRt+s where the profit in period t+ s is piRt+s = ( P̃tΠsl=1pi ιp t+l−1pi 1−ιp∗ − Pt+s xt+s ) ∗ ( Pt P̃tΠsl=1pi ιp t+l−1pi 1−ιp∗ ) λf λf−1 yt+s In the function that describe piRt+s, the first term is per-unit profit, and second term is the sales volume. The variable P̃t is the new price to be chosen. The variable xt+s is defined as the ratio of the aggregate price Pt+s over the price of generic goods Pg,t+s, or xt+s ≡ Pt+s Pg,t+s From the definition of x, it is clear that the price of input (i.e., generic goods) in period t + s is Pt+sxt+s . The price of output in the same period, assuming that the retailer can not re-adjust its price since, is P̃tΠsl=1pi ιp t+l−1pi 1−ιp∗ . The per-unit profit is the difference between the price of output and the price of input , or P̃tΠsl=1pi ιp t+l−1pi 1−ιp∗ − Pt+sxt+s . The sale volume is determined by the demand from the final goods sector, or ( Pt P̃tΠsl=1pi ιp t+l−1pi 1−ιp ∗ ) λf λf−1 yt+s. The total profit in the period is the product of per-unit profit and the sales volume. The profit is then divided by Pt+s so it is expressed in real terms. Real profit is then discounted into the present value using Λc,t+sΞt+sΛc,tΞt , which is the same as the discount factor used by households. The coincidence of discount factors comes from the assumption that households are the ultimate owner of retailers and their profit. Finally, (ξp) s is the probability that the retailer is stuck with P̃t after s periods. 16 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . The first order condition satisfies (see sub-section(A.1.2) for the deriva- tion) P̃t Pt = λf ∑∞ s=0 ξ s pEt Λt+sΞt+s ΛtΞt ( Pt+s PtGt+s ) λf λf−1 yt+s 1 xt+s∑∞ s=0 ξ s pEt Λt+sΞt+s ΛtΞt ( Pt+s PtGt+s ) λf λf−1 −1 yt+s (2.6) where Gt+s = Πsl=1pi ιp t+l−1pi 1−ιp∗ . The aggregate price Pt is the weighted average of newly-set prices and the past price. Pt = [ ξp ( Pt−1pi ιp t−1pi 1−ιp∗ ) 1 1−λf + (1− ξp) P̃ 1 1−λf t ]1−λf (2.7) The linearized version of equ(2.6) and equ(2.7) gives rise to a forward looking Phillips curve with back-ward looking indexation term, or equ(2.37). Sub-section(A.1.3) shows the linearization process. 2.3.4 Producers of generic goods Producers of generic goods use a Cobb-Douglas production technology with two inputs, capital service (kt) and labor service (lt). These producers own the capital and employ workers for their labor service. They take the stock of capital in the current period as given, but they have the option to increase or reduce capacity utilization (ut) of the existing stock. Higher utilization incurs a higher cost (such as faster depreciation). The output, in the form of generic goods, is sold to retailers at a relative price of 1xt (or Pt xt in nominal terms). Since these producers own the capital stock, they are responsible for investing in the stock. There is a market for new capital goods where addi- tional capital can be purchased. The installation of new capital is costless, but the production of these goods features increasing marginal cost in the short run. Finally, these generic-goods producers are assumed to operate in a competitive environment. The profit maximization problem is max (yt,ut,lt,kt+1) 1 Pt Et ∞∑ s=0 Pt+spi g t+s Πsl=0 (1 +Rt+l−1 +mk) where the variable pig is the dividend payout, which is defined as the total revenue after the deduction of wage, capacity utilization cost, and invest- 17 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . ment cost in the capital stock, or pit = 1 xt yt − wtlt − Φu (ut) kt −qk,t [kt+1 − (1− δ) kt] The production technology is Cobb-Douglas, yt = eza,t (utkt) α l1−αt (2.8) The random term zat in the production function is the total factor produc- tivity (TFP). Finally, in the objective function of the firms, there is a extra spread mk in the discount factor. Similar to the use of mh in households’ budget constraint, this extra parameter mk creates a necessary degree of freedom to calibrate the model’s steady state to the long term averages observed in the national account. More specifically, one can choose mk so that the model’s steady state replicates the observed averages of the investment to output ratio, for any given level of depreciation rate and interest rate in the steady state. The literature, such as Greenwood and Hercowitz (1991), some times uses capital-income tax for similar purpose. The optimality condition for labor demand is wt = 1 xt (1− α) yt lt (2.9) The optimality condition for capacity utilization is Φ′u (ut)ut = 1 xt αyt kt (2.10) which relates the extent of utilization to the productivity of capital ser- vice. A higher productivity gives the firms the incentive to intensify the use of their capital stock. For this reason, the function Φu (.) is chosen such that Φ′u (ut)ut is increasing with ut. The utilization in the steady state is normalized to be one. It is further assumed that Φu (1) = 0. The investment Euler is Ptqk,t (1 +Rt) = EtPt+1 [ 1 xt+1 α yt+1 kt+1 − Φu (ut+1) + (1− δk) qk,t+1 ] (2.11) 18 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . 2.3.5 Producers of capital goods Following the long tradition in DSGE models, it is assumed that the expan- sion of capital stock takes resources that are otherwise consumable. In this model, the production of new capital goods uses the consumable final goods as the input. This assumption means that the model abstracts from the likely heterogeneity, between sectors, in term of capital and labor intensity. There is adjustment cost in the sector. The transformation technology, from final goods to capital goods, is assumed to be one for one in the steady state. So the value of capital goods is one in the long run. In the short-run, however, the price of new capital goods can deviate from unity, because of adjustment costs. There are at least two types of adjustment costs in the existing literature. One type of the cost increases with the level of investment (often normalized by the stock of capital). This is the capital adjustment cost (or CAC), which gives rise to the Tobin’s Q in neo–classical models. Another type of adjustment costs increases with speed of changes in the level of investment. It is often called the investment adjustment cost (or IAC) in DSGE literature. IAC is popularized by Christiano et al. (2005), and is rapidly gaining popularity because of its ability to replicate the hump- shaped response of investment to economic shocks (see discussions in Smets and Wouters (2007)). The model in this paper uses a hybrid version of adjustment costs, which admits both CAC and IAC. There are two types of capital goods in the economy: business capital goods and housing capital goods. Their production is similarly modeled. We will start with business capital, and then move to housing capital. Producers of new business capital For the producers of new business capital, the adjustment cost function is Φk ( it i $k t−1 ) , where i is the amount of resource used in the production of the capital. The parameter $k has a value between zero and one. It indicates the weight between CAC and IAC. When $k = 0, the adjustment cost is increasing only in the level of current investment. It has nothing to do with the level of investment in the past. This is the case of pure CAC. If $k = 1, we have the case of pure IAC, when the adjustment cost is a determined by itit−1 , the level of current investment relative to the level of investment in the previous period. Finally, it is assumed that Φk ( (i∗)1−$k ) = 0 and Φ′′k ( (i∗)1−$k ) = ϕk > 0. 19 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Finally, there is a stochastic component in the sector’s productivity. For it amount of investment expenditure (in real terms) in the sector, the total amount of output (denoted by yKI,t), is yKI,t = ezi,tit [ 1− Φk ( it i$kt−1 )] where zi,t is the investment-specific productivity shocks. These producers’ objective is to maximize the present value of expected profit, or max it Et ∞∑ s=0 Pt+s Πsl=0 (1 +Rt+l−1) {qk,t+syKI,t+s − it+s} The optimality condition is Pt = Ptqk,tezi,t [ 1− Φk ( it i$kt−1 ) − Φ′k ( it i$kt−1 ) it i$kt−1 ] (2.12) +Et [ $k Pt+1qk,t+1e zi,t+1 1 +Rt+l−1 +mk Φ′k ( it+1 i$kt ) i2t+1 i$k+1t ] Producers of new housing capital Producers of new housing capital also face adjustment costs in the short- run. The adjustment cost function is Φh ( dt d $h t−1 ) , where the subscript h differentiates the function from the one in the production of business capital. The weighting parameter between CAC and IAC is $h, which also has a subscript of h for differentiation. It is assumed that Φh ( (d∗)1−$h ) = 0 and Φ′′h ( (d∗)1−$h ) = ϕh > 0. Finally, the amount of input in the housing sector is denoted as dt. This model assumes that supply shocks in the housing market is small, so that the productivity in the sector is not model as a stochastic variables. Instead the productivity is assumed constant. Empirical literature, such as Topel and Rosen (1988), suggests that majority of housing fluctuation is driven by demand factors. The profit maximization problem in the sector is max dt Et ∞∑ s=0 Pt+s Πsl=0 (1 +Rt+l−1) {qh,t+syHI,t+s − dt+s} 20 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . such that yHI,t = dt+s [ 1− Φh ( dt d$ht−1 )] The optimality condition is Pt = Ptqh,t [ 1− Φh ( dt d$ht−1 ) − Φ′h ( dt d$kt−1 ) dt d$ht−1 ] (2.13) +Et [ $h Pt+1qh,t+1 1 +Rt Φ′h ( dt+1 d$ht ) d2t+1 d$h+1t ] 2.3.6 Monetary authority The monetary authority uses the one-period discount rate as its policy tool. It sets an interest rate target and supply (or subtract) as much money as necessary until the interest rate is at the target. The demand for money comes from households, whose utility function should have included a real monetary balance. Under the assumption of separability in the preference for money and other inputs, it is not necessary to model the demand for money nor the supply, as long as the monetary authority respects the equality between money injections and transfers (see discussions in Bernanke et al. (1999) and Iacoviello (2005a)) The monetary authority is assumed to follow a forward-looking Taylor rule that responds to changes in output, and the expected inflation. In addition, the policy rule has a interest rate smoothing component. The policy rate in period t is Rt = (1− ρR)R∗ + ρRRt−1 + (1− ρR) ρpi (Etpit+1 − pi∗) (2.14) + (1− ρR) ρ∆y (yt − yt−1) + εR,t where the variables R, pi, and y denote interest rate, inflation and output, respectively. The R and pi that have “*” subscript are the steady state values. The non-systemic component of policy rate is denoted as εR,t. The interest-rate smoothing parameter is ρR, which is between zero and one. The parameter ρpi captures the response of the policy rate to inflation. The parameter ρ∆y captures the response to changes in output. 21 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . 2.3.7 Government Government collects taxes from households. It also issues discounted bonds. The government operates with the following budget constraint gt + Bt−1 Pt = Tt Pt +mhezmh,tqh,tht + 1 Pt Bt 1 +Rt The left hand side is the expenditure. The first term gt = g∗ezg,t is the government spending. The spending fluctuates around a steady state level g∗. The stochastic component is zg,t. The second term is the real value of bonds that the government has to honor in period t. The right hand side describes the sources of revenue. The first two terms are the tax revenue that include TtPt the net lump-sum tax, and mhe zmh,tqh,tht the property tax (or subsidies). The last term 1Pt Bt 1+Rt is the proceed from issuing new bonds. 2.3.8 Aggregate resource constraint and the accumulation of capital stock Aggregating the resource constraints across households and that of the gov- ernment, taking into account all the profit transfer from businesses to house- holds, and the net transfer between households and the government, there exists an economy-wide resource constraint yt = ct + it + dt + gt (2.15) In other words, total output is allocated between consumption ct, business investment it, residential investment dt, and government spending gt. The cost of capacity utilization is assumed to be small and therefore drops out from the constraint. The accumulation of capital stock follows kt = (1− δk) kt−1 + ezi,tit [ 1− Φk ( it i$kt−1 )] (2.16) and ht = (1− δh)ht−1 + dt [ 1− Φh ( dt d$ht−1 )] (2.17) 2.3.9 Exogenous shocks There are six exogenous shocks in this system. They all are approximated with AR(1) processes except for the consumption preference shock. Fol- lowing Smets and Wouters (2007), the preference shocks are modeled as a 22 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . ARMA(1,1) process. The moving average term picks up the high-frequency fluctuations in the quarterly consumption series. Let ρz be the auto-correlation coefficient of shock z, let εz,t be a random draw from the standard normal distributions at time t, let σε,z be the stan- dard deviation of innovations to z, we can describe the stochastic process of these shocks with the following set of equations: The TFP shocks za,t = ρaza,t−1 + σε,aεa,t (2.18) The investment-specific technology shock in the capital goods sector zi,t = ρizi,t−1 + σε,iεi,t (2.19) The consumption preference shocks zχc,t = ρχczχc,t−1 + σε,χcεχc,t − µ ∗ σε,χcεχc,t−1 (2.20) where µ is the moving average term. The government expenditure shocks zg,t = ρgzg,t−1 + σε,gεg,t (2.21) The monetary policy shocks zR,t = ρzR,zR,t−1 + σε,RεR,t (2.22) The housing demand shocks zmh,t = ρmhzmh,t−1 + σε,mhεmh,t (2.23) Among these shocks, TFP shocks and the investment specific technology shocks are regarded in the literature as being major sources of fluctuation. Several papers have found the moderation in the volatility of these shocks to be important for the great moderation. Leduc and Sill (2007) makes a case for the moderation in TFP shocks. Justiniano and Primiceri (2006) makes a case for investment shock. The government spending shocks and consump- tion preference shock belong to the minimum set of perturbations needed by this model to avoid singularity. Monetary policy shock is an intrinsic part of monetary models. The housing demand shock drives fluctuations in the housing sector. 23 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . 2.3.10 Steady state of the model, and the calibration strategy The model has no explicit solution. This paper relies on linear approxima- tion around steady state to capture the model’s short-run dynamics. The steady state is solved with the following steps. The first step is to solve for the steady-state real interest rate from the Euler equation, or equ(2.2), in its steady state. The result is r∗ = 1 β − 1 The second step is to calculate the investment to output ratio, or i∗y∗ , us- ing the conditions that describe the equilibrium in the market for business capital goods. These conditions include i∗k∗ = δk (which is the steady-state counterpart of equ(2.16)), qk,∗ = 1 (which is the steady-state counterpart of equ(2.12)), and αx∗ y∗ k∗ = qk,∗ (r +mk + δk) (which is the steady-state coun- terpart of equ(2.11). These three equations together give i∗y∗ as i∗ y∗ = α x∗ δk r∗ +mk + δk The next step is to solve for d∗y∗ and c∗ y∗ , which are the shares of hous- ing investment and consumption of non-durable goods in the total output. For this purpose we can use the resource constraint d∗y∗ + c∗ y∗ = 1− i∗y∗ − g∗ y∗ , and a linear relation between d∗y∗ and c∗ y∗ that is captured by d∗ y∗ / c∗ y∗ = d∗ c∗ = δh [ 1 vh (r +mh + δh) ]− 1 γ . The resource constraint is self explanatory. On its right hand side g∗y∗ is exogenously given, and i∗ y∗ has been solved in the previous steps. The linear relation between d∗ and c∗ can be derived by com- bining vh ( c∗ h∗ )γ = qh,∗ (r∗ +mh + δh) (which is equ(2.5) in the steady state), qh,∗ = 1 (equ(2.13) in the steady state), as well as d∗ = δhh∗ (equ(2.17) in the steady state). The solutions to d∗y∗ and c∗ y∗ are c∗ y∗ = 1− g∗y∗ − i∗y∗ 1 + δh [ 1 vh (r +mh + δh) ]− 1 γ and d∗ y∗ = 1− g∗ y∗ − i∗ y∗ − c∗ y∗ 24 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Knowing c∗y∗ and d∗ y∗ (the latter of the two gives us h∗ y∗ through d∗ = δhh∗), we can sum together the consumption of non-durable goods and the con- sumption of housing services as the total consumption expenditure, ex- pressed as a share of total output. The result is e∗ y∗ = c∗ y∗ + (r∗ +mh + δh)h∗ y∗ which corresponds to the item “private consumption expenditure” defined in the national account. To calculate the hours worked in the steady state, we need to use the clearance condition in the labor market. The clearance condition is (1− λb) e∗lσl∗ = (1− α) 1x∗ y∗ l∗ , from equ(2.9) and from the alternative expression for equ(2.4). Using this relation we can calculate l∗ as l∗ = ( 1− α 1− λb 1 x∗ ) 1 1+σl ( e∗ y∗ )− 1 1+σl Finally, using l∗ we can calculate y∗ using the production function, which gives us y∗ = ( k∗ y∗ ) α 1−α l∗ = ( 1 δk i∗ y∗ ) α 1−α l∗ This economy has a balance growth path, even if labor productivity has a positive growth rate in the long run. Productivity growth, in essence, increases the amount of effective labor in the economy, which pushes up the total output by the same proportion, but leaving i∗y∗ , c∗ y∗ , d∗ y∗ , e∗ y∗ , l∗ constant. The steady state solution clarifies the calibration strategy of the model. The objective of the calibration is to make sure the model’s steady state replicates long-term averages observed in the data. In this paper, the aver- ages I use are those for the real interest rate and various expenditure shares. More specifically, I first choose β to produce a desired r∗, then mk for i∗y∗ , then mh for e∗y∗ , and finally vh for c∗ y∗ . The calibration takes several parame- ters directly from the data, including δk, δh and g∗ y∗ . Two other parameters are taken from the literature. One of them is x∗, the steady-state markup charged by retailers. This parameter is taken from Iacoviello (2005a). The parameter σl, which determines the labor supply elasticity, is taken from Smets and Wouters (2007). The reason why this parameter is not estimated is because the estimations do not include wage and employment as observed variables. The data therefore have limited information about σl. Other parameters of the model come from estimations. The steady state of the model contains no information about parameters that have only short- run implication. Such parameters include those of the monetary policy 25 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . rule, those of nominal rigidity and real rigidity, and those that describe the properties of the exogenous shocks. These parameters have to come from estimation. The estimation is done using a linear state-space econometric system that approximates the model around its steady state. The following section presents the linearized system, and the appendix provides a brief description of the estimation method. 2.3.11 The linearized system Let’s first explain the notation in the linear system. All variables are mod- ified to have a hat over their original symbols. For majority of them, hat- ted variables means percentage deviation from the steady-state level, i.e., ŷt = ln ( yt y∗ ) , ît = ln ( it i∗ ) , etc. The exceptions are the interest rates R̂t and r̂, and the inflation pi. The hats for these three variables indicate percentage-point difference from their steady-state counterparts. In other words, R̂t = Rt −R∗, r̂t = rt − r∗, and pit = pit − pi∗. Starting from households’ utility maximization. The consumption Euler, or equ(2.2), is linearized into Etr̂t+1 = ( σc 1− λb − γ ) Etĉbt+1 − ( σc 1 + λb 1− λb − γ ) ĉbt (2.24) +σc λb 1− λb ĉbt−1 + γ (Etĉt+1 − ĉt) − (σc − 1) w∗l∗ e∗ 1 1− λb ( Et l̂t+1 − l̂t ) − zχc,t where w ∗l∗ e∗ is the ratio of total wage and total private consumption expen- diture, measured at the steady state. The variable ĉb comes from linearizing equ(2.1), ĉbt = κcĉt + (1− κc) ĥt (2.25) where the coefficient κc = c∗e∗ is the share of non-durable consumption in total private consumption expenditure. Equ(2.3), which defines the real interest rate, becomes R̂t − pit ≡ r̂t (2.26) The linear version of equ(2.4), which is the optimality condition for labor supply, becomes 1− γ (1− λb) 1− λb ĉbt + γĉt − λb 1− λb ĉbt−1 + σl l̂t = ŵt (2.27) 26 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . The linear version of equ(2.5), which is the investment Euler on housing capital, becomes q̂h,t = (1− εh,1)Et ( γĉt+1 − γĥt+1 ) (2.28) +εh,1 (Etq̂h,t+1 + Etzmh,t+1)− (Etr̂t+1 + zmh,t) where εh,1 ≡ 1−δh(r∗+δh+mh)+1−δh . The linear version of equ(2.8), which describes the total output as a function of inputs and productivity, becomes ŷt = (1− α) l̂t + αk̂t + αût + za,t (2.29) The linear version of equ(2.9), which governs the optimal demand for labor, is now ŷt − l̂t − x̂t = ŵt (2.30) The linear version of equ(2.10), which governs the optimal capacity uti- lization, is now ût = 1− ψ ψ ( ŷt − k̂t − x̂t ) (2.31) The coefficient 1−ψψ measures the response of utilization rate to rental price. The parameter ψ is between zero and one. When ψ = 1 the cost of vary- ing the utilization is extremely high, so that change in the productivity of capital does not affect the utilization rate. The case of ψ = 0 is the other extreme, when changes in the productivity of capital has infinite impact on the utilization rate. The linear version of equ(2.11), which is the investment Euler on business capital, is q̂k,t = (1− εk,1)Et ( ŷt+1 − k̂t+1 − x̂t+1 ) + εk,1Etq̂k,t+1 − Etr̂t+1 (2.32) where εk,1 ≡ 1−δk(r∗+δk+mk)+1−δk . The linear version of equ(2.12), which describes the supply of new busi- ness capital, is 1 ϕk q̂k,t = ( 1 + β$2k ) ît −$k ît−1 −$kβEt̂it+1 − ( 1 + β$2k ) zi,t (2.33) The linear version of equ(2.13), which describes the supply of new hous- ing capital, is 1 ϕh q̂ht = ( 1 + β$2h ) d̂t −$hd̂t−1 −$hβEtd̂t+1 (2.34) 27 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . The total stock of business capital evolves according to k̂t+1 = (1− δk) k̂t + δk ît + δkzi,t (2.35) The total stock of housing capital evolves according to ĥt+1 = (1− δh) ĥt + δhd̂t (2.36) Linearizing equ(2.6) and equ(2.7) gives rise to a Phillips curve, pit − ιppit−1 = β (Etpit+1 − ιppit)− (1− ξp) (1− ξpβ) ξp x̂t (2.37) The policy reaction function, or equ(2.14), is now R̂t+1 = (1− ρR) ρpiEtpit+1 + (1− ρR) ρ∆y (ŷt − ŷt−1) + ρRR̂t + zR,t (2.38) Finally, the economy-wide resource constraint, or equ(2.15), is now ŷt − c∗ y∗ ĉt − i∗ y∗ ît − d∗ y∗ d̂t − zg,t = 0 (2.39) 2.3.12 Constructing a state-space econometric model The linearized model is a system of linear expectational difference equations, or AEtXt+1 +G0EtZt+1 = BXt +G1Zt Zt = ΦZt−1 + Σεt The system can be solved numerically using the Blanchard-Khan type procedure modified by Klein (2000). The solution to the linearized model, barring indeterminacy, expresses jump variables in the system, or Xft , as functions of predetermined state variablesXbt and the current state of stochas- tic processes Zt, Xbt+1 = MxX b t +MzZt Xft = FxX b t + FzZt The solutions and the process of the shocks together define the law of motion. Once we choose a set of observed variables, we can define the observed variables from Xft and X b t and their lags to complete the state-space system. 28 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . The regression uses quarterly data from 1963Q1 to 2006Q2. Six observed variables are in the regression: the growth rate of output (with the value of housing service subtracted from GDP), private consumption expenditure (net of housing service), private non-residential investment, and residential investment; the level of effective federal funds rate; and the growth rate of GDP price index. The measurement equations of the state space system are, gr yt − γy = ŷt − ŷt−1 gr ct − γc = ĉt − ĉt−1 gr it − γi = ît − ît−1 gr dt − γd = d̂t − d̂t−1 Rt −R = R̂t pit − pi = pit Here γy = ∑2006Q2 t=1963Q2 gr yt 173 is the average growth rate of real GDP (net of housing output) over the sample. Other terms, γc, γi and γd are similarly defined. R = ∑2006Q2 t=1963Q2Rt 173 and pi = ∑2006Q2 t=1963Q2 pit 173 are the sample means of R and pi. 2.4 Estimations and Counter-factual Analysis 2.4.1 Overview of the strategy: Bayesian approach, split sample estimation, and counter-factual analysis Here we briefly discuss the key elements in the empirical strategy of the paper. The estimation and the statistical inference follow a Bayesian ap- proach. The prior distributions of the usual DSGE parameters follow those of Smets and Wouters (2007). The priors for housing-related parameters are loosely specified. The paper uses Sims’ maximization routine csminwel.m to estimate the mode of posterior distribution by maximizing the log of pos- terior function. It then uses a Random Walk Metropolis algorithm (RWM algorithm, described in detail in An and Schorfheide (2006)) to explore the posterior distribution around the mode.9 9This algorithm makes a large number of draws in the neighborhood of the mode in order to provide a complete picture of the posterior distribution. These draws, numbered at hundred of thousands for each of the multiple chains, provide the basis for researchers to approximate the percentiles of the distribution of parameters and of the transformation of parameters, such as the variance decomposition. 29 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Figure 2.1 presents the in-sample fit of the one-step-ahead forecast by comparing the data to the prediction. Table 2.3 presents the out-of-sample forecast performance of the estimated model over an unrestricted VAR(1). Table 2.4 presents the variance decomposition to explore the source of fluctu- ations in the housing market. Table 2.5 presents the posterior distributions from the split sample estimations. The break point is 1986Q1, the same as the one used in McCarthy and Peach (2002), who use split-sample VAR estimations to capture the difference in the transmission of monetary policy to residential investment. Finally, an identical set of priors is used for the split-sample estimations, so that the difference in the posterior distribution comes from the data as much as possible. After the estimations I use counter-factual analysis to pinpoint the most important changes. It is safe to guess that the split-sample estimations will detect more than one change in the parameters of the model. Changes can occur in the monetary policy rule, in the nominal and real rigidity parameters, and in the stochastic processes of disturbance. The question of interest is how these individual blocks of changes affect the variance of endogenous variables. To answer this question, this paper uses a sequence of counter-factual analysis. The sequential begins with a data-generating process (DGP), defined by the modes of the parameters estimated from the pre-86 data. It then replaces all parameters, block by block, until finally reaching a DGP defined by the post-86 estimates. In each step, the variance of the endogenous variables is recorded. A quantitative assessment of the contribution made by one particular change of parameters is the difference between the variance at the step when the change is made and those in the previous step. The result is recorded in Table 2.6. 2.4.2 Parameters to be estimated and their prior distributions Four sets of parameters need to be estimated. They are gathered by groups, their prior distribution is presented in the four panels of Table 2.2. The first is the “shock” parameters, including the autocorrelation coeffi- cients, the moving average coefficients, and the standard deviations of the six stochastic processes. Following Smets and Wouters (2007), the prior distri- butions are harmonized as much as possible. This way the difference in the posteriors reflects the information from the data. All the autocorrelation co- efficients are beta distributed with mean 0.5 and a standard error of 0.2. All standard deviations of innovations are assumed to follow an inverse gamma distribution with mean 0.3 and two degrees of freedom, which amounts to a 30 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . loose prior. The only moving average parameter for consumption preference shocks has a mean of 0.7 and a standard error of 0.15. The second set of parameters are those in the monetary policy reaction function. The smoothing parameter is beta with mean 0.75 and standard error of 0.1. The response to inflation is Gamma with mean 1.7 and a standard error of 0.3. The response to output growth is Gamma with mean 0.12 and standard error of 0.05. The third set of parameters contains all the nominal and real rigidity parameters. As for the prior distribution, the Calvo stickiness is Beta of 0.75[0.1], the adjustment cost parameter for capital investment is normal of 4[1.5], which is adopted from Smets and Wouters (2007).10 I use the same distribution for the adjustment cost parameter of the housing investment, since the prior is rather loose. Other parameters in the group have to lie between zero and one. I assume them all are beta distributed with mean 0.5 and standard error of 0.15. The last set has the preference parameters. The risk aversion parameter σ and the intratemporal elasticity of substitution between housing service and consumption, or γ, are assumed to be normally distributed with mean 2 and a large standard error of 0.4. The habit parameter λb is normal with 0.5[0.2]. 2.4.3 Full-sample estimation and its in-sample fit Here I use a full-sample estimation to provide a brief overview of the esti- mation before the split-samples. The full-sample estimation is also used to assess the in-sample fit of the model. Table 2.2 presents the mode, the mean, and the percentiles of the posterior distributions of parameters. The mode is from the maximization routine of Sims.11 The percentiles are the sample statistics of 300,000 draws from a RWM algorithm with three chains.12 The first observation is that the data is very informative for the stochas- 10who in turn chooses the prior based on Christiano et al. (2005). 11Chris Sims’ csminwel.m is for unconstrained maximization. Because some parame- ters in the model are clearly constrained, parameter transformation is used first so that csminwel.m can be applied. Transformation is used only in for estimation of the mode. For the MCMC draw, no transformation is used. An and Schorfheide (2006) describes the estimation steps in greater detail. 12Each of these chains is initiated with a random draw around the mode of the posterior distribution, using a variance covariance matrix that equals to 0.5 times of the inverse Hessian approximated at the mode. The jumping distribution uses a covariance matrix that is 0.17 of the inverse Hessian. This choice of scale generates a acceptance rate ranging from 22-27%. 31 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . tic processes of shocks. The posterior distributions of these parameters, presented in the top panel of Table 2.2 are quite different from the assumed priors. Furthermore, despite the harmonized prior distributions, the esti- mates vary significantly for different types of shocks. Among these different shocks, the productivity shocks are highly persistent. Persistence of the pro- ductivity shocks is a feature of DSGE models that assumes a deterministic trend of productivity (for example, see Smets and Wouters (2007)). Mone- tary policy shocks and consumption preference shocks, on the other hand, are far less persistent. The housing-specific shock is also persistent, though not as much as the TFP shocks. The data is quite informative for the adjustment cost parameters for housing and capital investment, but not for the Calvo stickiness and ca- pacity utilization. In the bottom panel, most of the preference parameters have posteriors that are similar to the assumed priors, except for the habit parameter. Figure 2.1 provides a visual diagnostic of the estimated model’s fit in one-step-ahead forecast. The figure plots the actual time-series of the six variables together with the forecast. First of all, the time series generally fit well, even for growth rates. This means that the model is quite capable of capturing the direction of movements, not just for levels. Another observa- tion is that the series of forecasts is smoother than the actual data, gliding through the middle of the data’s high frequency fluctuations. This suggests that the forecast achieves a satisfactory balance between keeping track of the trends and not over-reacting. Table 2.4 provides a variance decomposition for the eighth-quarter-ahead forecast error. The lower panel is the corresponding standard errors from the MCMC draws.13This table shows that fluctuations in GDP growth and consumption growth are mostly driven by TFP shocks (70% and 64%, re- spectively); investment fluctuations is mainly driven by investment-specific shocks (63%); and housing fluctuations is driven by housing-specific shocks (64%), and to a lesser extent TFP shocks (28%). The main contributor to the movements in the Federal funds rate and inflation are the TFP shocks, investment shocks, and consumption preference shocks. The non-systemic 13Variance decomposition breaks down the variance of J-step ahead forecasting errors, for each of the six endogenous variables, to components attributable to each of the six shocks. That is why this table has the 6-by-6 matrix, with endogenous variables in the column, and individual shocks in the rows. These decompositions are expressed in shares. Each column sums to 100 by construction. The matrix in the upper panel of the table is the mean of these shares from the every 500th draw in the MCMC chain. The lower panel is the corresponding standard errors. 32 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . monetary policy shocks account for only a small part of the variance. 2.4.4 Out-of-sample forecast of the estimated model Table 2.3 compares the performance of the DSGE model in out-of-sample forecast to that of an unrestricted VAR(1), as in Smets and Wouters (2007). The forecast target is an eight-quarter window within the period between 1990Q3 and 2006Q2. Both models are re-estimated yearly using the sample from the earliest date (1963Q2) up to the quarter immediately before the window of forecast. In other words, the data between 1963Q2 and 1990Q2 are used in the first round to estimate the model and make forecasts for 1990Q3-1992Q2. In the second round, the sample is expanded by four quar- ters, the model parameters are re-estimated, and projections are made for the next 8 quarters. The process continues until the end of the sample. The Root Mean Square Error (RMSE) is used as the measure of accuracy. For quantity variables, the growth rate is converted to the level before the calculation of RMSE. The accuracy gain is measured by the percentage reduction in RMSEs. For the four quantity series, the DSGE model out-performs the unre- stricted VAR(1) for the entire forecast window. The percentage improve- ment is the largest for the forecast of GDP, consumption, and investment, peaking at levels from 30 to 50%, similar to those reported in Smets and Wouters (2007). For residential investment, the best performance is found at the second and the third quarter, with a 10-15% improvement, which gets somewhat smaller over time. For nominal interest rate and inflation, a few early quarters show that the DSGE model is worse than the VAR. The loss is not large, however, with the biggest loss being -17% for federal funds rate forecast at the second quarter, followed by a -8% loss for inflation in the third quarter. Smets and Wouters (2007) found an almost identical loss of accuracy for the Federal funds rate, though not for inflation. The loss turns into a gain for both series from the fourth quarter on, reaching about a 20% gain by the eighth quarter for both nominal rate and for inflation. 2.4.5 Split-sample estimations The split-sample estimations in this paper allow all parameters to change between split-periods. The estimation and the RWM algorithm are identical 33 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . to those for the full sample exercise and uses the same set of priors.14 Table 2.5 presents the posterior distributions for the two sub-samples side by side. A majority of the parameters are quite stable over the samples. Some are different. One way to measure the statistical significance of the difference is to see if, for the same parameter but in difference periods, the 95% percentile of the lower estimate is still smaller than the 5% percentile of the higher estimate (in other words, the ”confidence intervals” do not overlap). With this criterion, there are only four parameter that experienced significant changes from pre-85 period to the post-85 period. They are the standard deviation of the innovations to TFP shocks (smaller in the more recent period); the standard deviation of the innovation to government spending shocks (smaller); standard deviations of monetary policy shocks (also smaller). These changes suggest that the economic shocks has become smaller over time. Interestingly, there is no evidence that housing-specific shocks are now smaller. The estimated auto-correlation of the shocks, as well as the standard deviation of the innovation to the shocks, are almost identical between the two samples. The fourth, and the last significant change in parameter is an increase in the responsiveness of interest rate policy to inflation, as suggested in Taylor (2007), Clarida et al. (2000), and many other papers. The first- period estimate of the parameter is 1.4. The second-period estimate is 2.1. Furthermore, albeit barely missing the bar for significance, the smooth- ing parameter of the monetary policy increases from 0.58 to 0.72. The smoother policy rule confirms the suggestion in Ireland (2004) that the Fed- eral reserves appears to have abandoned “a stylized pattern of ’stop-go’ pol- icy, according to which an initial tightening is quickly reversed, presumably in an attempt to partially offset the negative effects on output”. The adjustment cost parameter of housing investment, or ϕh, increases from 1.63 to 3.58, although it is short of outright significance. A higher adjustment cost parameter in the model smoothens out the response of residential investment to demand shocks. Higher adjustment cost means a steeper short-run supply curve. With steeper supply in the short run, the incentive to spread out investment over time is higher. The equilibrium housing investment therefore exhibits reduced sensitivity to demand shocks as ϕh becomes higher. A higher adjustment cost slows down changes in investments, and is likely 14The Hessian matrices are approximated separately for each sub-sample at their pos- terior modes. The jumping distribution uses a scale factor of 0.17 for both sub-samples, and it results in acceptance rate about 20-28% for individual chains. 34 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . to shift the adjustment mechanism toward house prices. Is there evidence that housing prices are now more responsive? According to Figure 2.5, the answer may be yes. The figure plots the level of investment with the log of price index. The latter is derived from Davis and Heathcote (2005). Between the late-1970s and the mid-1980s, the price index appears to be rather sluggish, with little movement despite the large swings in investments. From the mid-1980s, however, the price series and investment series move more closely with each other. For example, the price series follows the turns of the investment series in the mid-1980s, the early-1990s, and in 2006. 2.4.6 Breaking down the reduction in variance using counter-factual analysis Table 2.6 reports a sequence of counter-factual analysis that are used to break down the reduction in the standard deviation of endogenous vari- ables. The first column presents the unconditional standard deviation of the six variables calculated from the mode of the parameters from the pre- 1986 estimation. The last column is the same set of standard deviations from the mode of the post-1986 estimation. Between these columns is a sequence of hybrid data generating processes created by replacing the pa- rameters, block by block, in the following order. First, the monetary policy parameters are switched from the pre-1986 estimate to the post-1986 esti- mate. The resulting data generating process is called Hybrid-1. Next, all behavior parameters in Hybrid-1, except for housing adjustment cost pa- rameters, are replaced with post-1986 estimates, to reach Hybrid-2. The housing adjustment cost parameters are then substituted (Hybrid-3), fol- lowed by the substitution of monetary policy shock parameters (Hybrid-4), TFP shock parameters (Hybrid-5), and the housing-specific shock parame- ters (Hybrid-6). The last step is to replace all remaining shock parameters. With this last substitution, we reach the post-1986 data-generating process. The difference in the level of standard deviation measures the contribution from the changes in each block of parameters. Sources of moderation for residential investment Over the sequence of counter-factual analysis, there are only two significant reductions in the standard deviations of quarter-to-quarter growth of residential investment. The first one happens when moving from Hybrid 2 to Hybrid 3, namely replacing the adjustment cost parameters for housing investment from the pre-1986 estimates to the post-86 ones. The variability measure drops from 6% to 3.1%. The second largest reduction is is between Hybrid 4 and Hybrid 35 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . 5, when substituting the parameters of the TFP process. The standard deviation drops from 3.1% to 2.5%. Our estimate of higher adjustment cost is consistent with empirical find- ings in Dynan and Sichel (2006) regarding a reduced sensitivity of residential investments to changes in interest rates. They attributed the change to fi- nancial innovations or other developments in the mortgage market. Without excluding financial innovation as a cause for the reduced sensi- tivity, we believe there are other likely alternatives. One possibility is better management of inventory, as discussed in the introduction. Another possibil- ity is the fall in union’s presence in the construction industry. Construction unions are known for strong union hiring halls, which often fix compensation and queues up workers for job opportunities. Such labor market structure suppresses wage response, at the cost of more volatile building activity. The fall in union’s presence can switch the market towards more price adjust- ment, and less quantity adjustment. There is no escaping from one more obvious possibility, that the supply becomes sluggish because it is more difficult, or takes longer time, to build new houses. A drop in supply capacity, either due to the lack of land, or due to a greater regulatory burden on the use of land, can prevent supply from keeping up with the demand. A recent article by Glaeser et al. (2005) coins the term ”man-made scarcity” to describe the consequence of regulations in the US housing market. Regardless the cause, a reduced supply is likely to reduce the sensitivity of housing investments. On the other hand, lower supply elasticity increases the responsiveness of house prices, a prediction that seems to be supported by observations in Figure 2.5. Even though lower supply elasticity reduces investments’ response to demand shocks un- der normal situations, it could potentially be a source of instability, such as when speculations on price movements kick in (see discussions in Malpezzi and Wachter (2005)). It is difficult to evaluate alternative hypotheses for the reduced sensi- tivity of housing investment. There are two major hurdles in testing and comparing them. The first one is an overly simplified structure of the hous- ing model. The model, as it is right now, lacks inventory, union, and does not have a upward-sloped supply curve, which would have created diverging relative prices that his paper, and others (for example Smets and Wouters (2007)), avoid. The other hurdle is the absence of housing prices in the set of observable variables. The absence of capital price is common when estimating DSGE model, but it does creates difficulty in interpretating the change in the adjustment process. 36 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Source of moderation in other real volatility As for GDP and con- sumption, most of the moderation comes from smaller TFP shocks. From Hybrid-4 to Hybrid-5, when the TFP parameters are replaced, the standard deviation of output growth drops by 0.5%, which is large, considering that the total reduction is 0.61%. At the same time, the standard deviation of consumption drops by 0.57%, out of the total 0.6%. The standard deviation of business investment drops by 0.35%, out of a total reduction of 0.8%. To summarize, according to the estimated models, smaller TFP shocks are the most important driving force for the moderation in the volatility of output and consumption. Source of moderation in nominal volatility As for nominal variables, namely inflation and nominal interest rate, the drop in their standard de- viations is mostly driven by the systematic changes in the monetary policy rule. The standard deviation of inflation drops by 50% as we replace the parameters of the policy rule. The nominal rates experiences a 40% drops as well. In comparison, smaller non-systematic monetary policy shocks did not lead to much reduction in the standard deviations of these two nominal variables (see the difference between Hybrid-3 and Hybrid-4). This suggests that the non-systematic part of the monetary policy is a small component of the story. The change that really matters is in the policy rule, presumably due to its ability to anchor expectations. Changes in monetary policy: effective in stabilizing nominal vari- ables but not the real variables A more responsive monetary policy has no stabilizing effect on the four quantity variables. If anything, the standard deviations increase slightly under the new policy rule (see the dif- ference between Pre-86 and Hybrid-1 in Table 2.6). The finding that an more aggressive policy does not reduce real economic volatility is not un- usual. Stock and Watson (2002) and Leduc and Sill (2007) draw similar conclusions regarding the effect of the policy changes on total output. Since the policy change substantially reduces nominal volatility, and it is well-known that nominal volatility and real volatility tend to move together in the data (as in the case of US and other G-8 countries, see Blanchard and Simon (2001)), it is inevitable to ask if the estimated model fails to capture a link that should be there. If a link exists between nominal and real volatility, housing sector may be a place where such a link is important. The sector extensively uses long-term, nominally fixed-rate mortgages as means of financing. As a consequence, the cost of capital in the sector is directly 37 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . related to long-term interest rates. For households, it is the mortgage rates. For investors in mortgage debt, it is the ten-year treasury yield. Since these long-term rates are sensitive to medium and the long-term inflation risks, it is likely that a more aggressive interest rate policy, by keeping inflation expectation in check, can reduce cost fluctuations, thus having a stabilizing effect in housing market. Because this paper uses a first-order approximation that removes vari- ations in term premiums, it has very little to say to what extent the such variations can add to the link between nominal and real volatility. To admit time-varying term premium requires a higher order approximation. More importantly, the theory between premium and economic activity is not well understood. In fact, a theoretical model in Rudebusch et al. (2006) found that ”there is no structural relationship running from the term premium to economic activity”. Empirically speaking, however, there is evidence that is indicative of a link between monetary policy, economic activity, and changes in premi- ums. On the one hand, macro-finance literature, such as Rudebusch and Wu (2004), finds a relation between the price of risks and macroeconomic variables, including those closely identified with monetary policy, such as in- flation targets. One the other hand, the above-mentioned Rudebusch et al. (2006) finds evidence from their data that a decline in the term premium has typically been associated with stimulus to real economic activity,. To summarize, it is an interesting and worthy research question whether the change in monetary policy has an extra effect on real volatility through it impact on term premium. But to pursue the hypothesis is beyond the scope of the current paper. 2.5 Conclusion This paper estimates a sticky-price DSGE model that is expanded to include housing investments. The purpose is to investigate what could explain the fall in the volatility of US residential investment in the past two decades. It uses split-sample estimations, and counter-factual analysis, to compare three hypotheses: a monetary policy that becomes more responsive to infla- tion, smaller economic shocks, and structural changes in the housing sector. According to the conjecture in Taylor (2007), a more responsive monetary policy keeps inflation under control, thereby reduces boom-bust cycles and large swings in interest rates, which in turn, leads to a more stable housing market. 38 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . The analysis confirms only part of the hypothesis regarding monetary policy. We detect the policy changes, and confirm that the estimated change is capable of reducing the variance of nominal interest rates and inflation substantially. Nevertheless, we do not find the change to reduce the vari- ability of real economic activity, including that of residential investments. The estimated DSGE model appears to have only weak connections between nominal and real volatility. The paper touches on the interesting but diffi- cult question, on whether or not allowing for a time-invariant premium in term structure will enhance the link. The paper finds that smaller economic shocks make a modest contri- bution in reducing the variance of the housing investment, whereas, the biggest contributing factor is an increase in the parameters of adjustment costs, which indicates a reduced sensitivity of housing investments. This suggests the possibility of structural change in the housing sector. The pa- per briefly discusses some hypotheses that are not very often covered in the literature. These include the decline of union in the construction industry, an increase of housing starts with pre-committed buyers, and the possibility of a reduced capacity to supply new housing. 39 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . 2.6 Tables and Figures Table 2.1: Descriptive statistics of observed variables between 1963-2006 Std. Std. Dev. Std. Dev. Mean Dev. ’63Q2-85Q4 ’86Q1-’06Q2 GDP, Quarterly Growth 0.82 0.93 1.17 0.55 Private Consumption, Q. Growth 0.90 0.81 0.98 0.55 Business Investment, Q. Growth 1.24 2.27 2.50 1.98 Resid. Investment, Q. Growth 0.70 4.65 5.97 2.41 Federal Funds Rate 6.36 0.82 0.89 0.55 Inflation 3.87 0.60 0.65 0.23 1GDP, PCE and the Investment spendings are in real terms 2Total value of housing service is subtracted from GDP and PCE 3Data Source: GDP, PCE, and Investment series are from BEA Table 1.1.5. Total value of housing service is from BEA Table 2.3.5. Federal funds rate is from the Federal Reserve Correlation Coefficient 1963Q1-1985Q4 1986Q1-2006Q2 between dt/dt−4 and yt/yt−4 0.22 0.17 between dt/dt−4 and yt+1/yt−3 0.49 0.30 between dt/dt−4 and yt+2/yt−2 0.66 0.42 between dt/dt−4 and yt+3/yt−1 0.71 0.48 1yt is GDP net of housing service in quarter t; dt is residential investment 40 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Table 2.2: Parameters: prior and posterior distributions from the full-sample estimation Prior Posterior Type Mean Std.er Mode Mean 5% 95% ρ: TFP Shocks Beta 0.5 0.2 0.99 0.98 0.97 0.99 ρ: Investment Shocks Beta 0.5 0.2 0.95 0.94 0.91 0.97 ρ: Gov.Spending Shocks Beta 0.5 0.2 0.99 0.99 0.99 1.00 ρ: Monetary Shocks Beta 0.5 0.2 0.35 0.32 0.20 0.46 ρ: Consumption Shocks Beta 0.5 0.2 0.90 0.89 0.82 0.95 µ: Consumption Shocks Beta 0.7 0.15 0.80 0.72 0.43 0.93 ρ: Housing Shocks Beta 0.5 0.2 0.94 0.92 0.85 0.95 σ: TFP Shocks I.G.1 0.3 2 0.83 0.90 0.71 1.17 σ: Investment Shocks I.G. 0.3 2 0.65 0.72 0.54 0.88 σ: Gov.Spending Shocks I.G. 0.3 2 0.62 0.63 0.57 0.68 σ: Monetary Shocks I.G. 0.3 2 0.25 0.26 0.23 0.29 σ: Consumption Shocks I.G. 0.3 2 0.17 0.23 0.13 0.44 σ: Housing Shocks I.G. 0.3 2 0.42 0.70 0.36 1.46 Prior Posterior Type Mean Std.er Mode Mean 5% 95% Policy Rule: ρR Beta 0.75 0.1 0.59 0.61 0.49 0.70 Policy Rule: ρpi G.1 1.7 0.3 1.47 1.57 1.41 1.79 Policy Rule: ρ∆y G. 0.12 0.05 0.16 0.18 0.10 0.28 Prior Posterior Type Mean Std.er Mode Mean 5% 95% Calvo Stickiness ξp Beta 0.75 0.1 0.73 0.75 0.68 0.82 Adjustment Cost, ϕk N.m.1 4 1.5 2.11 2.46 1.60 3.68 Adjustment Cost, $k Beta 0.5 0.15 0.86 0.82 0.73 0.91 Adjustment Cost, ϕh N.m. 4 1.5 2.04 2.60 1.61 4.03 Adjustment Cost, $h Beta 0.5 0.15 0.69 0.69 0.57 0.81 Capacity Utilization ψ Beta 0.5 0.15 0.48 0.51 0.46 0.58 Inflation Indexation ιp Beta 0.5 0.2 0.04 0.07 0.02 0.13 Prior Posterior Type Mean Std.er Mode Mean 5% 95% Preference: σc N.m. 2 0.4 2.49 2.52 1.83 3.23 Preference: γ N.m. 2 0.4 2.21 2.28 1.80 2.76 Preference: λb Beta 0.5 0.2 0.31 0.31 0.21 0.43 1I.G. is the Inverse Gamma distribution; G. is the Gamma distribution; N.m. is the Normal distribution 2Preference parameter σl is set at 1.92. Since employment data is not in- cluded in the estimation, there is not much information for estimating pa- rameter related to labor supply elasticity. 1.92 is the estimate in Smet and Wouters (2007) 41 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Table 2.3: Compare the estimated DSGE model’s performance in out-of- sample forecast with a VAR model Panel a: Root Mean Square Error (RMSE) from a VAR(1) model Q-1 Q-2 Q-3 Q-4 Q-5 Q-6 Q-7 Q-8 y, Level 0.6 1.2 1.8 2.2 2.6 3.0 3.3 3.5 c, Level 0.6 1.2 1.7 2.0 2.2 2.6 2.8 3.0 i, Level 1.3 2.5 4.6 5.8 7.0 8.5 10.0 10.8 d, Level 2.7 5.8 8.0 9.1 8.8 9.2 9.3 9.3 R 0.1 0.2 0.3 0.3 0.4 0.5 0.5 0.6 pi 0.2 0.2 0.2 0.3 0.3 0.4 0.3 0.5 Panel b: Percentage improvement of DSGE model over the VAR(1) in reducing the size of RMSE Q-1 Q-2 Q-3 Q-4 Q-5 Q-6 Q-7 Q-8 y, Level 16.6 37.7 35.9 39.3 36.1 40.0 43.2 45.7 c, Level 22.0 27.6 38.2 39.1 38.5 45.9 50.0 47.4 i, Level 0.5 33.8 31.1 27.8 25.6 27.3 24.5 23.5 d, Level 3.7 15.3 14.5 10.3 7.8 7.1 9.6 5.6 R -6.2 -17.1 -4.5 3.8 9.9 13.8 17.2 19.9 pi 7.1 10.6 -8.7 16.8 16.7 20.1 13.3 21.8 1y is GDP, c is consumption, i is business investment, d is residential investment, R is federal funds rate, pi is inflation 42 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Table 2.4: Variance decomposition of the 8-quarter ahead forecast error Panel-a: Mean taken from the every 500th MCMC draws yt/yt−1 ct/ct−1 it/it−1 dt/dt−1 Rt pit TFP Shocks 70.98 64.38 29.21 27.66 36.92 35.66 Investment Shocks 0.74 2.73 62.78 1.66 23.97 22.81 Gov.Spending Shocks 14.88 11.85 0.19 5.24 5.42 4.55 Monetary Shocks 7.61 6.91 5.46 1.05 6.58 5.27 Consumption Shocks 4.95 12.92 1.42 0.62 17.61 22.43 Housing Shocks 0.84 1.21 0.93 63.78 9.50 9.28 Panel-b: Standard error taken from the every 500th MCMC draws yt/yt−1 ct/ct−1 it/it−1 dt/dt−1 Rt pit TFP Shocks 4.99 5.39 7.90 6.19 11.65 11.98 Investment Shocks 0.68 1.18 9.58 0.86 7.10 5.95 Gov.Spending Shocks 2.30 2.45 0.18 1.62 2.84 2.47 Monetary Shocks 1.83 1.75 1.90 0.45 1.61 2.53 Consumption Shocks 2.84 5.19 1.00 0.45 7.05 6.81 Housing Shocks 0.73 0.65 0.52 8.43 4.12 3.68 1y is GDP, c is consumption, i is business investment, d is residential investment, R is federal funds rate, pi is inflation 43 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Table 2.5: Posterior distributions from the split-sample estimations Sample: 1963Q2-1985Q4 Sample: 1986Q1-2006Q2 Mode 5% 95% Mode 5% 95% ρ: TFP Shocks 0.98 0.96 0.99 0.98 0.96 0.99 ρ: Investment Shocks 0.95 0.91 0.97 0.95 0.92 0.98 ρ: Gov.Spending Shocks 0.99 0.98 1.00 0.99 0.97 1.00 ρ: Monetary Shocks 0.29 0.13 0.47 0.51 0.38 0.59 ρ: Consumption Shocks 0.89 0.75 0.92 0.94 0.89 0.97 µ: Consumption Shocks 0.81 0.52 0.93 0.60 0.31 0.85 ρ: Housing Shocks 0.93 0.88 0.95 0.93 0.80 0.94 σ: TFP Shocks 1.09 0.90 1.56 0.54 0.46 0.71 σ: Investment Shocks 0.61 0.51 0.95 0.79 0.63 1.19 σ: Gov.Spending Shocks 0.73 0.66 0.83 0.45 0.40 0.51 σ: Monetary Shocks 0.32 0.29 0.38 0.13 0.11 0.17 σ: Consumption Shocks 0.20 0.16 0.50 0.10 0.08 0.22 σ: Housing Shocks 0.50 0.36 1.19 0.48 0.38 1.28 Sample: 1963Q2-1985Q4 Sample: 1986Q1-2006Q2 Mode 5% 95% Mode 5% 95% Policy Rule: ρR 0.58 0.45 0.69 0.72 0.66 0.81 Policy Rule: rhopi 1.38 1.30 1.72 2.12 1.89 2.80 Policy Rule: ρ∆y 0.14 0.08 0.25 0.09 0.05 0.19 Calvo Stickiness ξp 0.77 0.69 0.86 0.69 0.63 0.77 Adjustment Cost, ϕk 2.44 1.73 4.70 2.61 1.58 5.40 Adjustment Cost, $k 0.88 0.72 0.92 0.73 0.50 0.83 Adjustment Cost, ϕh 1.63 1.25 3.51 3.58 2.52 5.67 Adjustment Cost, $h 0.67 0.51 0.79 0.64 0.51 0.83 Capacity Utilization ψ 0.51 0.46 0.65 0.55 0.52 0.66 Inflation Indexation ιp 0.06 0.03 0.20 0.09 0.04 0.30 Preference: σc 2.30 1.48 2.92 2.14 1.53 2.72 Preference: γ 2.11 1.69 2.68 2.08 1.67 2.57 Preference: λb 0.31 0.21 0.48 0.34 0.20 0.43 44 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Table 2.6: Breaking down the reduction in variance Standard deviations of endogenous variables under the estimated and hypothetical data generating process Pre-86 Hybrid-1 H-2 H-3 H-4 H-5 H-6 Post-86 yt/yt−1 1.28 1.33 1.26 1.26 1.22 0.76 0.76 0.65 ct/ct−1 1.44 1.51 1.49 1.61 1.57 1.00 1.00 0.85 it/it−1 2.68 2.73 1.94 1.95 1.90 1.55 1.55 1.88 dt/dt−1 6.01 6.08 6.02 3.12 3.10 2.53 2.48 2.39 Rt 1.19 0.74 0.74 0.78 0.73 0.62 0.62 0.56 pit 0.95 0.47 0.61 0.61 0.54 0.48 0.48 0.40 1y is GDP, c is consumption, i is business investment, d is residential investment, R is federal funds rate, pi is inflation 2Pre-86 DGP describes the estimated data generating process before 1986; Post-86 DGP, in the last column, describes the estimated data generating process after 1986 3Hybrid-1 replaces monetary policy parameters in Pre-86 with those from Post-86; Hybrid-2 replaces non-monetary, non-housing behavior parameters in Hybrid-1 with those from Post-86; Hybrid-3 replaces the housing adjustment cost parameters in Hybrid-2 with those from Post-86; Hybrid-4 replaces the monetary policy shocks parameters in Hybrid-3 with those from Post-86; Hybrid-5 replaces the TFP shocks parameters in Hybrid-4 with those from Post-86; Hybrid-6 replaces the housing-specific shocks parameters in Hybrid-5 with those from Post-86 45 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Figure 2.1: Observable variables: The data and the one-step-ahead forecast 1963Q2 1986Q1 2006Q2 -5 0 5 GDP, Quarterly Growth Rate 1963Q2 1986Q1 2006Q2 -5 0 5 Private Consumption Expenditure, Q. Growth Rate 1963Q2 1986Q1 2006Q2 -10 0 10 Non Residential Investment, Q. Growth Rate 1963Q2 1986Q1 2006Q2 -40 -20 0 20 Residential Investment, Q. Growth Rate 1963Q2 1986Q1 2006Q2 -2 0 2 4 Federal Funds Rate 1963Q2 1986Q1 2006Q2 -2 0 2 4 Growth Rate of GDP Price Index Figure 2.1: Observable variables: the data and the one-step ahead forecast The thicker line is the one-step ahead forecast 46 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Figure 2.2: Output gap and residential investments Data Source: The real GDP and Investment is from BEA's GDP quantity index adjusted to dollar value. Real Potential GDP is the Congressional Budget Office's Estimates. The series used in this figure is from Table 2-2 in "The Budget and Economic Outlook: Fiscal Years 2008 to 2017" Figure 2.2: Output gap and residential investments Correlation coefficients: 0.64 before 1986, 0.29 after -12 -8 -4 0 4 8 12 Q1 19 63 Q1 19 67 Q1 19 71 Q1 19 75 Q1 19 79 Q1 19 83 Q1 19 87 Q1 19 91 Q1 19 95 Q1 19 99 Q1 20 03 Q1 20 07 %  o f R ea l P ot en tia l G D P Output Gap Residential Investment Spending as % of Potential GDP 47 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Figure 2.3: Housing starts, change from same month last year Data Source: U.S. Department of Commerce: Census Bureau Assessed on June 27, 2007 at Economic Data - FRED® Figure 2.3:  Housing starts, change from same month last year (%) -80 -40 0 40 80 120 Ma y-5 9 Ma y-6 1 Ma y-6 3 Ma y-6 5 Ma y-6 7 Ma y-6 9 Ma y-7 1 Ma y-7 3 Ma y-7 5 Ma y-7 7 Ma y-7 9 Ma y-8 1 Ma y-8 3 Ma y-8 5 Ma y-8 7 Ma y-8 9 Ma y-9 1 Ma y-9 3 Ma y-9 5 Ma y-9 7 Ma y-9 9 Ma y-0 1 Ma y-0 3 Ma y-0 5 Ma y-0 7 %  C hg . f ro m  1  Y r. A go Total New Privately Owned Housing Units Started In The U.S., Year-to-Year Growth 48 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Figure 2.4: Output gap and mortgage flow Data Source The real GDP and Investment is from BEA's GDP quantity index adjusted to dollar value. Real Potential GDP is the Congressional Budget Office's Estimates. The series used in this figure is from Table 2-2 in "The Budget and Economic Outlook: Fiscal Years 2008 to 2017" The flow of home mortgages is the net increase in liabilities of households and nonprofit organizations, or FA153165105 of the Federal Flow of Funds Account, that is deflated by GDP price index Figure 2. 4: Output gap and mortgage flow -10 -5 0 5 10 15 Q1 19 63 Q4 19 65 Q3 19 68 Q2 19 71 Q1 19 74 Q4 19 76 Q3 19 79 Q2 19 82 Q1 19 85 Q4 19 87 Q3 19 90 Q2 19 93 Q1 19 96 Q4 19 98 Q3 20 01 Q2 20 04 Q1 20 07 %  o f P ot en tia l R ea l G D P Real Output Gap Net Increase in Home Mortgage as % of Potential Real GDP 49 Chapter 2. Changes in Business Cycle Dynamics of Residential Investment. . . Figure 2.5: Residential investments and the price index of residential struc- tures Data Source: The real residential Investment is from BEA's GDP quantity index adjusted to dollar value. The residential structure price index is from Davis and Heathcote (2005). Real series is derived by dividing the nominal series by the price index of GDP. Figure 2.5: Residential investments and the price index of residential structures -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Q1  19 75 Q2  19 76 Q3  19 77 Q4  19 78 Q1  19 80 Q2  19 81 Q3  19 82 Q4  19 83 Q1  19 85 Q2  19 86 Q3  19 87 Q4  19 88 Q1  19 90 Q2  19 91 Q3  19 92 Q4  19 93 Q1  19 95 Q2  19 96 Q3  19 97 Q4  19 98 Q1  20 00 Q2  20 01 Q3  20 02 Q4  20 03 Q1  20 05 Q2  20 06 Log of real price index of residential structures Log of residential invesment multiplied by .2 and minus a constant 50 Chapter 3 Costs and Frictions in the Housing Market: a Macro and Quantitative Perspective 3.1 Introduction Several recent papers have expanded DSGE (dynamic stochastic general equilibrium) macroeconomic models to include a housing sector (e.g., Aoki et al. (2004), Iacoviello (2005a), and Gammoudi and Mendesy (2005)). This development reflects the consensus that housing investments and house prices play prominent roles in macroeconomy.15 Most of these papers de- scribe the market with a standard stock-flow model that is unable to generate sufficient persistence in the growth of house prices.16 There also exists am- biguity in terms of motivating the slow adjustment of capital stock.17 This class of models performs poorly when applied to the data, especially when the house price is included in the estimation. The empirical weakness of the models presents a hurdle for developing DSGE-based macroeconometric models (i.e., Christiano et al. (2005) and Smet and Wouters(2003, 2007)) to cover the housing-macro interaction. This paper constructs a richly specified stock-flow model for housing prices and investments. The model has detailed descriptions of the cost structure in the construction industry, differentiating the short-term and the 15At least four reasons explain why the housing sector is important for the macroecon- omy. Housing investment is a big contributor to economic fluctuations (Leamer (2007)). The investment is also sensitive to monetary policy shocks (Chart 3 of Bernanke and Gertler (1995b)). House prices influence consumer spending (Greenspan (2001)), as well as financial stability. Trichet (2005) indicates the bust of a house price bubble can often bring more damage to the economy that the bust of a equity price bubble. 16For example, Gammoudi and Mendesy (2005) estimates a housing-DSGE model and find that the house prices respond much more rapidly to shocks in the model than in the data. 17For example, Aoki et al. (2004) motivates the slow adjustment through capital ad- justment cost, while Gammoudi and Mendesy (2005) uses investment adjustment cost. 51 Chapter 3. Costs and Frictions in the Housing Market. . . long-term supply elasticity, and admitting possible stickiness in the prices of construction inputs. It also features search frictions and gradual adjustment of house prices in part of the market. The paper estimates the model using maximum likelihood estimations and the US quarterly data between 1985 and 2007, for the purpose of testing the empirical relevance of different features in the model. The goal of the paper is not to generate precise estimates of the numerous parameters, which is difficult given the complex interaction between various features of the model. Instead, the paper focuses on the loss of empirical capability when individual features is turned off from the full model. The empirical capability is measured by the likelihood function and other mea- sures. The objective of the paper is to use the data to help determine the structure of the model, when the theory is ambiguous. This is one step in de- veloping an optimization-based, yet empirically competent, housing model that may one day be built into a full-fledge DSGE model for macroeconomic analysis in places like central banks. The full model in the paper includes several well-known costs and fric- tions. It has a long-run supply curve that is not necessarily flat, which brings into the model the complication of diverging relative prices.18 The model also features short-term costs of adjusting construction activity, and adjusting the stock of housing capital. In addition, it admits possible rigid- ity of house price in the asset market. Most of these features are frequently found in empirical housing literature. For example, Topel and Rosen (1988) estimates a model of ”internal adjustment costs” in construction industry, while DiPasquale and Wheaton (1994) estimates a mechanism of gradual price adjustment in house prices. The model also include other features of interest. One of the new fea- tures is the demand for inventory, motivated by search frictions between homebuyers and residential builders. To deal with such frictions, builders use inventory to facilitate sales of new homes, which explains why, in the US, a majority of housing construction are started without committed buy- ers (see data in Kahn (2000)). The stock of inventory of new homes in the market is substantial, equivalent to half a year’s sales on an ordinary day. The model also features sticky construction cost, generated in the model by sticky movements in the prices of construction inputs. The modeling as- sumption is motivated by the heavy presence of unions in the construction industry, especially in the 60s and 70s, when unions’ share of construction 18DSGE models, such as those in Smet and Wouters (2007), often abstract from upward sloped long run supply of investment to avoid diverging prices. 52 Chapter 3. Costs and Frictions in the Housing Market. . . workforce was more than 70%. The share has subsequently fallen to about 20% in the 1990s (Allmon et al. (2000)). The presence of union affects how the industry responds to changes in market demand. This is due to con- struction unions’ practice of “hiring hall”, which often fixes workers’ com- pensation, while at the same time queuing up workers for job opportunities. Such labor market structure makes construction activity more sensitive than it otherwise would be. If the cost is not responsive, output is likely to be volatile, especially given unions’ mechanism of placing workers in and out of the queue. The final model is rich and is likely over-parameterized relative to the information in the data. We therefore do not seek precise estimate for in- dividual parameters. Instead, we test the importance of individual features by removing them from the full model. Smets and Wouters (2007) uses a similar method to test the importance of the features in their large scale DSGE macroeconomic model. We use maximum likelihood estimation on a linearized versions of the models. The observable variables include the real residential investments, the real price of new houses, the real consumption, and the real interest rates. The data are in quarterly frequency between 1985 and 2007Q2. Besides the log-likelihood statistics, we also compare dif- ferent models’ accuracy in forecasting individual time series in the short and in the medium run, their ability to replicate second moments of the data, and occasionally their ability to match empirical impulse responses. The findings in the paper provide helpful information for modeling hous- ing market. We find that the most important and irreplaceable feature of the model is the gradual adjustment of house price, a mechanism that is emphasized in DiPasquale and Wheaton (1994). In addition, the findings can be interpreted as cautious notes about the interpretation of investment adjustment cost (or IAC for short) that has become a staple feature of the latest DSGE models.19 Groth and Khan (2006) also finds empirical evidence that questions the IAC cost structure. They are using the U.S. two-digit industry data while we are using aggregate U.S. housing data. The paper is organized as follows. The remainder of this introduc- tion presents the standard version of the stock-flow model and estimates it with MLE. The model’s empirical performance is then discussed. Section 2 presents the full model. Section 3 estimates various versions of the model and discussed their empirical performance. Section 4 has more discussion about features in the model. Section 5 concludes. 19Investment adjustment cost is present in Christiano et al. (2005), Smet and Wouters (2003, 2007) and many others 53 Chapter 3. Costs and Frictions in the Housing Market. . . 3.1.1 The stock-flow model and its empirical performance Macroeconomic models typically regard housing capital as durable goods and describe its pricing, investment and accumulation with a stock-flow model. In this type of model, the value of a house is determined in the asset market. It is the expected discounted value of the stream of services that it provides over its lifetime, minus maintenance and other costs. The discount factor is the cost of capital, adjusted for risk premiums. The level of investment is determined by intersecting the house price with an upward sloped supply schedule in construction industry (sometimes the increasing marginal cost is modeled as the result of adjustment costs). The flow of investment in each period expands the housing stock. The more recent application of stock-flow model in macro-housing models include Aoki et al. (2004), Gammoudi and Mendesy (2005), and Davis and Heathcote (2005). The theoretical model The first equation of a stock-flow model is the “stock” equation. It describes the clearance of the market for existing housing stock. The clearance takes the form of an non-arbitrage condition, in which the price of housing services (i.e., the dividend from housing asset) is determined in the rental market. If we use qh as the notation for asset price, R (.) for rental price, rt+1 for a risk-less rate of return, and m as the asset-specific risk premium, we can express the non-arbitrage condition as (1 + rt+1 +m) qh,t = Et [R (ct+1, ht+1, zd,t+1) + (1− δh) qh,t+1] (3.1) Rental price R (.) is determined by the clearance of the rental market. It is increasing in the total consumption c, which proxies wealth, but decreasing in the size of the existing housing stock h. The rental price is also affected by a demand shifter zd. This paper assumes a specific functional form for R (), that R (ct, ht, zd,t) = zd,t ct ht This expression of rental price embeds the assumption of a Cobb-Douglas type utility function. The assumption of Cobb-Douglas utility guarantees a balance growth path. The data does support the hypothesis of balance growth, because the share of spending on residential investment tends to revert back to the average around 5-6% over the past 40 years, despite its volatile fluctuations in the short run. 54 Chapter 3. Costs and Frictions in the Housing Market. . . The second equation is the “flow” equation. It describes residential builders’ response to changes in house prices. These builders are assumed to operate with an upward-sloped marginal cost curve. In the equilibrium, builders optimally choose to produce at a level that equates the marginal cost to the price of their output. We can therefore back out the level of construction from the following equation qh,t = 1 zh,t (1 + γx) t (dt) ψ (3.2) In this equilibrium condition, the variable dt is the equilibrium output of new houses. The parameter ψ determines the response of marginal cost to changes in the output. In the denominator, (1 + γx) t is the housing-sector- specific (relative) level of productivity. The productivity will have a time trend if γx 6= 0.20. The marginal cost is also affected by a temporary cost shifter zh,t The stock of housing stock evolves according to (1 + γp)ht+1 = (1− δh)ht + (dt + δhht) (3.3) The parameter γp is the growth rate of the population. The total output from the construction industry is (dt + δhht), within which δhht replenishes depreciations, and dt is the new construction. Similar versions of the stock-flow model underly most DSGE literature’s modeling approach for investments in durable goods. In housing literature, the formalization of stock-flow model is often attributed to Poterba (1984). Estimating the basic model This section describes the estimation and the estimation results from the basic model. We use maximum likelihood estimation, for which we need to specify the stochastic processes of forcing variables.21 In the current system, the forcing variables are the demand and cost shifters, as well as interest rate and consumption. For the demand and cost shifters, the underlying processes are ẑd,t = ρzdẑd,t−1 + εzd,t (3.4) 20Since γx is relative to economy-wide technology, it can be either positive or negative. 21In fact, we use MLE with boundary constraints on the parameters being estimated. Imposing a boundary, but no prior within the range, our approach can be viewed as a naive version of the Bayesian approach. 55 Chapter 3. Costs and Frictions in the Housing Market. . . ẑs,t = ρzsẑs,t−1 + εzs,t (3.5) where the hatted variables are expressed as percentage deviations. The ran- dom shocks εzd,t and εzs,t are draws from normal distributionsN ( 0, diag ( σ2ε,zd, σ 2 ε,zs )) . The aggregate private consumption and interest rates are assumed to be exogenous to the housing market. The real interest rate is assumed to be stationary. Consumption moves around a deterministic time trend with positive growth γy. The cyclical components of consumption and real interest rate are not independent from each other, but follow auto-regressive processes with recursive timing assumptions. The recursive assumption is that consumption innovations affect interest rate in the current period, but interest rate shocks affect consumption with one lag. This assumption in timing is to identify the impulse responses to interest rate shocks from a recursive VAR, thus enabling the comparison between the model and the data in term of impulse responses. Using ĉt to denote the percentage deviation of consumption from its trend level, and r̂t to denote the difference of (real) interest rate from its normal value, we can capture their stochastic processes with ĉt = 4∑ j=1 ρcc,j ĉt−j + 4∑ j=1 ρcr,j r̂t−j + εc,t (3.6) r̂t = ρrc,0ĉt + 2∑ j=1 ρrc,j ĉt−j + 2∑ j=1 ρrr,j r̂t−j + εr,t (3.7) where the random shocks εc,t and εr,t are drawn from normal distributions N ( 0, diag ( σ2ε,c, σ 2 ε,r )) . We now proceed to construct a linear state-space econometric model for the purpose of estimations. First we divide variables in equ(3.1), equ(3.2),and equ(3.3) by their respective growth rates in a balance-growth path, thus turning the system into a stationary form.22 We then linearize the model around the steady state. The resulted linear system is combined with the exogenous stochastic processes to be solved using the solution method de- scribed in Klein (2000). The solution to the model describes the transition rule of the state vector in the system, and expresses the jump variables as functions of state variables. The next step is to construct a state-space sys- tem by appending to the solution an observation system, which expresses 22Of course, the growth rates for qh,t, dt, and ht are endogenous. We start with assumed growth rates, and the solve the model to confirm that the conjecture is correct. 56 Chapter 3. Costs and Frictions in the Housing Market. . . the observed data as functions of variables and parameters in the theoret- ical model. With the resulted state-space system and Kalman filter, we can evaluate the likelihoods of the observed time series, based on the as- sumed distributions. Estimating the model is to search for the combination of parameters that maximizes the likelihood.23 The observation equations are ln ( qnewt qnewt−1 ) = q̂h,t − q̂h,t−1 + γ2 ln ( ih,t ih,t−1 ) = ( 1− δh γp + γ1 + δh )( d̂t − d̂t−1 ) + δh γp + γ1 + δh ( ĥt − ĥt−1 ) + γ1 ln ( ct ct−1 ) = ĉt − ĉt−1 + γy Rt − pit = r̂t + r The left hand side of the system are the observed data. The Fisher price index of new houses from the Census Bureau is denoted with qnewt . The quantity index of total residential investment (including maintenance, im- provement and new construction) from the NIPA account is denoted with ih,t. The real consumption from NIPA is ct. The last equation expresses the real interest rate as the difference between the effective federal funds rate and the growth of the GDP price index, or Rt − pit.24 The right hand side of the observation system are variables and parame- ters in the theoretical model. At the end of the first equation is the balance- path growth rate of house prices, γ2 = ψγy−γx 1+ψ . At the end of the second equation is the balance-path growth rates of the quantity of investment, γ1 = γy+γx 1+ψ . Exogenous parameters in the system include γy, γp, and γx. They are the exogenous growth rate of consumption, population, and relative productivity in the construction sector, respectively. The second equation looks more complicated than others. The reason is that the investment data in NIPA includes both new construction and maintenance/improvements. As the result, the growth rate in the data corresponds to, in the model, a weighted average of the growth rate of new investments and the growth rate of the maintenance. The parameters of estimation interest are (ψ, ρzd, ρzs, σ2zd, σ 2 zs, σ 2 y , σ 2 r ). There are restrictions on the ranges of individual parameters during the 23Hansen and Sargent (2005) provides a detailed description of the method. 24All nominal series are turned into real series with the GDP price index as the deflator. 57 Chapter 3. Costs and Frictions in the Housing Market. . . estimations, but there’s no prior distributions within those ranges.25 The parameters that describe the stochastic processes of consumption and inter- est rate are fixed during the estimations. They are fixed to the averages of estimates that are obtained from a set of trial estimations.26 The empirical performance of the model The only behavior parameter in the basic model is the long-run supply elasticity, which is estimated to be about 2. Figures 3.1 and 3.2 assess the empirical performance of the estimated model. Figure 3.1 compares the model’s one-step-ahead forecast with actual realizations. Figure 3.2 presents the comparison between theoretical and empirical impulse responses to in- terest rate shocks. The theoretical impulse response is from the estimated model, while the empirical responses are from a recursive VAR. The order of variables in the VAR is: consumption, interest, house price, and residential investment. The comparison between one-step ahead forecast and the subsequent re- alization is shown in Figure 3.1. The comparison suggests that the model has a difficulty in keeping track of the time series of house prices and in- vestments. The projected changes from the model have little persistence compared to what is observed from the data. The model fails to follow the direction of housing busts and booms, and does not reflect the deep and long decline of house price in the late-1980s. For example, the model con- tinuously predicts house price deflation in the early-2000s, while in fact the prices grew significantly. Figure 3.2 compares the theoretical and empirical impulse responses of house price and investment to interest rate shocks. The two plots on top are the impulse responses of consumption and interest rate to a 30-bp interest rate shock. The two plots at the bottom show the response of house price and investment. Each subplot has two lines. The thin line is the empirical impulse response and their 95% confidence bounds. The thick lines are IRF from the estimated model. For house price, the model describes a rapid response. The price drops immediately to the bottom and then increases 25The range restricts the supply elasticity to be non-negative, and auto-correlation to lie between zero and one. Finally standard deviations are strictly positive. 26More specifically, we first perform an initial round of estimations of the model, in- cluding the shock parameters. We run the estimations for all different specifications of the model that is presented later in the text. To fix the parameters that describe the shock processes, we take a simple average of all estimates of these parameters and use them for all estimations in the formal analysis. The reason for keeping these parameter unchanged is to focus the comparison to the difference between specifications of the model. 58 Chapter 3. Costs and Frictions in the Housing Market. . . gradually. In the data, however, the price’s trajectory is very sticky. It barely moves at all in the quarter of impact. Instead it moves slowly down- ward, and does not begin to rise until after several quarters. The response of investment to interest rate shock show similar failure of the model. In the data the investment drops gradually. In the model the investment drops immediately to the bottom. To summarize, the model has difficulty in generating the degree of auto- correlation for house prices and residential investments observed in the data. The model’s impulse responses to interest rate shocks are too fast compared to the sluggish movements found in a VAR. Finally, the estimated model significantly over-predicts the volatility of house prices. The growth rate of house price in the model has a standard deviation that is three times as large as that in the data. In the data the standard deviation is 0.74%. The same measure is 2.24% in the simulate series from the model. 3.2 The Augmented Model This section discusses a model of housing investment with several new fea- tures. It is the ”full model” of this paper. In the model, the structure of the economy is as follows. Four active players are in the market: homebuy- ers, the retail arm of residential builders, the construction arm of residential builders, and various contractors. Contractors provide necessary construc- tion inputs (think labor, materials, components, and service of heavy equip- ment). The construction arm of residential builders works with contractors to create physical residential units. The retail arm purchases the completed units and is responsible for the sales. Households are buyers in the market. They have the alternative option of purchasing existing homes instead of newly constructed ones. Five costs and frictions are added to the basic stock-flow model. Two of these are the sluggishness in the adjustment of prices, two others are costs in adjusting construction-related activity. The last one is search friction that creates demands for inventory. We discuss, in greater detail, each of these new features in the following sections. A brief list of descriptions is provided here. One of the price ad- justments is the rigidity in the asset market, reflecting the possibility that the houses adjust only slowly to changes in fundamentals. We point out that the asset value of a house can be different from the price of a newly constructed unit. A new house has to be purchased by households to gen- erate rents. Before that, it stays in the inventory of residential builders. Its 59 Chapter 3. Costs and Frictions in the Housing Market. . . price can be influenced by the stock of inventory. Holding all else constant, an increase in inventory reduces the price of new homes. The second price rigidity is on the price of construction inputs, reflecting our concerns about the flexibility of the cost structure in the construction industry. Historically, the industry has had a high share of unionized workers in its workforce. The industry was well known for its strong practice of ”union hiring hall”. In these hiring halls, the wage was fixed but workers may have to wait in queue for job opportunities (see Abowd and Farber (1982)27). Such labor structures present an form of sticky wage that forces quantity to adjust more than it would if the price was fully flexible. There is an investment adjustment costs in the housing market, which penalizes rapid changes in construction activity. Topel and Rosen (1988) calls it the “internal adjustment cost” of construction industry. Investment adjustment cost (or IAC) is the more familiar term in DSGE literature. There is another adjustment cost that is the capital adjustment cost (or CAC), which says it is costly for the economy to adjust the stock of housing (think building roads to connect housing and provide water and electricity). These two adjustment costs have different implications on the dynamics of investments. For example, IAC produces hump-shape response of invest- ment, while IAC does not. The last friction is search. Search friction makes holding inventory mandatory for residential builders, despite the cost of carrying it. Builders can however decide how much inventory they want to carry. The use of inventory in this model is similar to that in the model of sale-facilitating in- ventory in Bils and Kahn (2000). Furthermore, the option of using inventory as a storage tool give residential builders the tool to arbitrage, dynamically, for profit or to avoid loss. 3.2.1 The asset market There is a fundamental price of houses. It is the solution to the difference equation describe in 3.1. Using q∗,t to denote the fundamental price, we can rewrite the non-arbitrage condition into (1 + rt+1 +m) q∗,t = EtR (ct+1, ht+1, zd,t+1) + Et (1− δh) q∗,t+1 The price of existing homes in the asset market can deviate from the q∗,t in the short run. Similar to DiPasquale and Wheaton (1994), the market 27Abowd and Farber (1982) does not focus on construction union. It studies the practice of union firing hall in general. Construction union is their example for strong practice of hiring hall. 60 Chapter 3. Costs and Frictions in the Housing Market. . . price qg,t adjusts slowly according to qg,t qg,t−1 = (1 + γ2) 1−τ ( q∗,t qg,t−1 )τ where the balance path growth rate of house price equals to γ2 . The lin- earized version of this price adjustment equation is q̂g,t = (1− τ) q̂g,t−1 + τ q̂∗,t, which is identical to that used in DiPasquale and Wheaton (1994). These authors find strong support for gradual adjustment of the house price from the statistical superiority of the model with lagged house prices. The consensus from the literature on house price dynamics suggests that house prices exhibit a positive serial correlation in the short-run and a mean reversion in the long run (see Cho (1996) for a survey of the literature). Several mechanisms may create gradual adjustments, which can be classi- fied into three classes. The first two classes are described in DiPasquale and Wheaton (1994) and in their references. One is learning under imperfect in- formation, for example, time-consuming search that may result in stochastic sale time that would delay the seller’s recognition that fundamentals have changed. The second is the existence of rule of thumb households that form expectations by looking at the past. The third class is related to credit and liquidity constraints. Krainer et al. (2005) describe an interesting model for the downward stickiness of house price. Their model involves search friction and a heterogenous willingness to pay for the same house. Hoping to find someone who is willing to pay a higher price, home owners may hold up the sales of their houses even if they recognize the downturn of the mar- ket. Past prices enter the current and future prices because they determines household’s debt, or the extent of ”debt overhang”. In DiPasquale and Wheaton (1994), the estimated speed of adjustment is 29% in the first year following a shock. This translates into a quarterly adjustment rate of slightly more than 7%. McCarthy and Peach (2002) use a similar adjustment mechanism in their vector error correction model, and estimate the adjustment rate to be 5% per quarter. 3.2.2 Supplier of construction inputs This section describes the pricing decision of contractors and the equilib- rium dynamics of construction cost. Contractors are suppliers of various construction inputs (think labor, construction materials and components, and service of heavy equipment). These inputs are not perfect substitu- tions. The optimal proportion of combination depends on the degree of 61 Chapter 3. Costs and Frictions in the Housing Market. . . substitutability and relative price. As the result, individual suppliers face a demand that is decreasing in the relative price of their product. We use monopolistic competition and infrequent price adjustment to motivate the possible stickiness in construction costs. We assume that the suppliers of different inputs have monopolistic rights to their own prod- ucts. They preset a price before production (think labor unions entering a contract). Following the Calvo-type probabilistic approach, we assume individual suppliers have a random opportunity to re-negotiate a price con- tract in each period. The chance for re-optimizing is (1- ξ). Under these assumptions, the equilibrium features slowly adjusting construction costs. In the equilibrium, there exists a mark-up between what is paid by res- idential builders and what would be called the “true” cost of building a house. The cost facing the builders is the actual cost plus a monopolistic markup charged by contractors, cc,t = dψt zh,t (1 + γx) t + xcc,t where the total cost incurred to the builder is cc,t, the “true” marginal cost of production is d ψ t zh,t(1+γx) t . The markup over the cost is xcc,t. In the marginal cost function, the variable d is the level of new construction in period t. The marginal cost is increase in the output if ψ > 0. The marginal cost function has a trend component captured by (1 + γx) t, where the trend growth rate is γx. The marginal cost also has a cost shifter zh,t. Secondly, the markup responds to changes and expected changes in cc,t. The following is the linearized version of the relation, − (1− ξ) ( 1− 11+rξ ) ξ xcc,t = picc,t − 11 + rEtpicc,t+1 The variable picc,t = cc,t − cc,t−1 is the price change. Since no nominal price is involved here, the change is in real terms. The parameter ξ is the Calvo stickiness parameter, which, in the theoretical model, literally means the chance that an individual supplier cannot change the price. Higher ξ means more stickiness. When ξ becomes zero, the coefficient (1−ξ)(1− 11+r ξ) ξ becomes infinity, which means xcc,t has to be zero at all time. In other words, the con- struction cost reflects exactly the true construction cost. In this case, the two equations are reduced into one, which is cc,t = dψt zh,t(1+γx) t . 62 Chapter 3. Costs and Frictions in the Housing Market. . . 3.2.3 Construction arm of residential builders Besides the construction cost, the construction arm of residential builders also face a cost in adjusting construction activity. With the adjustment cost, the short-run supply elasticity is smaller than the long-run elasticity. Topel and Rosen (1988) finds that the response of housing starts to a given price change is gradual. Instead of jumping to a level where the new price intersects with the long term supply, housing starts move gradually to reach the new level. They attribute the difference to what they call “internal adjustment cost,” a cost that penalizes rapid change in output. This internal cost can be interpreted as planning cost, or bottlenecks in supply chains. These costs may disappear in the long run, but will still have an impact on production and prices. The builder’s profit maximization problem is max dt+s ∞∑ s 1 Πs (1 + rt+1+s) [ qc,t+s ( 1− Φ1 ( dt+s dt−1+s )) dt+s − cc,t+sdt+s ] where the level of output is d, the price of output received by the builder is qc, the cost of capital is r. The adjustment cost function is Φ1 ( dt+s dt−1+s ) . It is assumed that in the balance-growth path Φ1 (.) = Φ′1 (.) = 0 and Φ′′1 (.) = ϕhs. The construction industry faces a convex cost if ϕhs > 0, in which case the industry has the incentive to smoothen out its construction plan over time. The optimal condition for the profit maximization problem is 1− cc,t qc,t = Φ1 ( dt dt−1 ) + Φ′1 ( dt dt−1 ) dt dt−1 (3.8) −Et 11 + rt+1 qc,t+1 qc,t Φ′1 ( dt+1 dt )( dt+1 dt )2 In DSGE literature, what Topel and Rosen (1988) call the “internal ad- justment cost” is known as “Investment adjustment cost (IAC)”. IAC is used in DSGE literature primarily for its ability to keep track of invest- ment series and generate hump-shaped response to shocks.(Christiano et al. (2005), Smets and Wouters (2007), Justiniano and Primiceri (2006)). In housing models, Gammoudi and Mendesy (2005) use the adjustment cost to motivate slow stock adjustment of housing capital. From now on, I will fol- low the convention in DSGE literature and refer to such cost as investment adjustment cost. 63 Chapter 3. Costs and Frictions in the Housing Market. . . Groth and Khan (2006) find that the evidence from US two-digit industry (1) does not support the IAC cost structure, (2) implies a very high elasticity of investment with respect to the shadow price of capital relative to aggregate estimates. 3.2.4 Retail arm of residential builders This section focuses on the retail arm of a residential builder. One salient feature of the housing market is the large stock of inventory of new homes. According to the US Census Bureau, a typical day in the US will have 5.5 months’ sales of houses looking for buyers. The stock of inventory is pro- cyclical with a lag, as it can takes a long time for inventory to clear after a housing slump. Maisel (1963) found that the inventory of unsold new homes has the most rapid impact on new starts, which he attributes to the carrying costs. The author also expressed concern about the possibility of a large backlog of unsold units discouraging lenders from lending to builders when needed. The current housing slump sees an unprecedented level of inventory buildup of unsold new homes.28 Many now view it as a major concerns.29 Housing inventory has not caught much attention in the macro-modeling of housing. Kahn (2000) provides a narrative description of the dynamics and composition of inventory in the housing market. Krainer (2006) found supply-to-sale ratio to be a better indicator than house price for predicting future investments. McCarthy and Peach (2002) included inventory stock in their estimated short-run housing supply equation. Given the large cost of holding inventory, what can explain the need for such a large stock of inventory in the market? Production lag is one explanation, production smoothing is another. Residential construction, however, is more volatile than sales. It is therefore questionable whether production smoothing is the main justification for inventory in this market. Inventory is also useful for satisfying the diversity in demand. The housing market features heterogenous buyers and heterogenous houses. A successful trade requires a seller’s goods to match reasonably well with the buyer’s demands. Maintaining an inventory provides households with options. If search involves cost, then an inventory is valuable for its ability to increase the chance of a successful search. 28As the aftermath of the unprecedented housing boom, the inventory buildup has also exceeded all previous records. In terms of inventory to sales ratio, August 2007 had 7.5 months sales in stock; the average is 5.5. The ratio is still slightly lower than that seen in the early 1980s. 29For example by Bernanke in his Jacksonhole remark of August 2007 (Bernanke (2007)). 64 Chapter 3. Costs and Frictions in the Housing Market. . . The paper captures the search friction with a “matching” function. Let the number of successful match be st, let the stock of inventory be vt, and let households’ search effort be bt, the matching function is st = vt ( 1− e− bt vt ) The matching function has the following features: 1. st < vt, meaning that the number of sales will be strictly smaller than the stock; 2. Sales are increasing in both vt and bt ; 3. the matching function has a constant return to scale. From the builders’ perspective, the probability that an individual house gets sold in a period t is stvt = 1− e − bt vt . Before modeling the investment decision in inventory, it may be helpful to describe in detail the flow of funds and goods: In period t , the retail arm of residential builders pays the construction arm for constructing new houses dt. The price they pay is qc,t per unit. The houses are constructed during the period and becomes part of vt+1, or the retailer’s inventory in the next period. In period t + 1, part of vt+1 will be sold. The remainder goes into period t+ 2 as inventory, together with new productions in t+ 1. The retail arm of residential builders uses two instruments for profit max- imization. One is qh,t, the market price of new houses. The other instrument is dt, the order of new constructions. The following is the maximization problem max (dt,qh,t) Et ∞∑ s=0 qh,t+svt+s [ 1− e− bt+s vt+s ] − qc,t+sdt+s Πs (1 + rt+1+s)  subject to vt+1 = vte − bt vt + dt (3.9) and bt = arg max bt (qb,t − qh,t) vt ( 1− e− bt vt ) − bt (ωb ∗ qc,t) The first constraint describes the accumulation of the stock of inventory. The second constraint is the reaction function from households that have the alternative option to buy from the market of existing homes. A buyer arbitrages the price difference between new homes and existing homes. The variable qb,t is the willingness of a buyer to pay for a new house before taking into account search cost. For the buyer, the per-unit gain from a successful 65 Chapter 3. Costs and Frictions in the Housing Market. . . match is the difference between the willingness to pay and the price, or qb,t− qh,t. The total gain is unit gain multiplied by the number of successful matches, or vt ( 1− e− bt vt ) . The total gain minus total search cost is the net gain, which is the objective function of the buyer. The total search cost is bt (ωb ∗ qt). In this cost function, the search effort is bt, and the cost per unit of effort is ωb fraction of the price of the house. With θt denoting e − bt vt , the buyer’s optimality condition is (qb,t − qh,t) θt = ωb ∗ qc,t This optimality condition equates the expected marginal return to search effort (the left hand side of the equation) to the marginal cost of the effort. The return is increasing with the difference between qb,t and qh,t. Meaning that the return is high when the market price is low relative to the buyer’s valuation. The return is also high if θt is high, which indicates a “buyer’s market”: slow sale and large inventory. Sellers take into account the relation between the price and the speed of sales when setting their price. In the equilibrium, they choose a level of price and sales ratio so that the allocation of total gain between seller and buyer satisfies: qh,t − qc,t qb,t − qh,t = (1− θt) θt The left hand side of the equation is the ratio of “seller’s markup” over “buyer’s net gain”. Seller’s markup is the numerator qh,t − qc,t, or the difference between the sales price of a house and it acquisition cost. Buyer’s net gain is the denominator, or qb,t− qh,t, the difference between the buyer’s valuation and the sales price. From the left hand side we can see that the relative allocation is determined by θt. Recall that θt = e − bt vt , the chance that a house fails to sell. A high θt indicates a buyer’s market, which increases the buyer’s net gain relative to seller’s markup. A low θt indicates a seller’s market, which increase the seller’s markup relative to the buyer’s net gain. The two preceding equations can be put together to express qh,t and qb,t in term of qc,t and θt. They are qh,t − qc,t = qc,t (1− θt) θ2t ωb (3.10) and qb,t − qc,t = qc,t 1 θ2t ωb (3.11) 66 Chapter 3. Costs and Frictions in the Housing Market. . . The above two equations are in fact arbitrage conditions. Consider the case when there is no search fiction, i.e., ωb = 0 (zero search cost means infinite search effort and therefore immediate sales), we have qh,t = qc,t = qb,t: the production cost equals the market price and the buyer’s willingness to pay. The last optimal condition is the Euler equation of investing in inventory (1 + rt+1) = Et [( 1 θt+1 − 1 )2 ωb + 1 ] qc,t+1 qc,t (3.12) Again, if we take out search friction from the model, we have (1 + rt+1) = Et qc,t+1 qc,t , which indicates a cost-less storage technology. The Euler equation has two notable properties. One is that ∂Etθt+1 ∂rt+1 < 0 which states that higher interest rate increase the cost of carrying inventory, thus reducing θt, meaning that the sales to inventory ratio has to increase to reduce the average holding time. Secondly ∂θt ∂Et qc,t+1 qc,t > 0 which states that expected increases in replacement cost increase the θt. It other word, it reduces the speed of sales. This happens because sellers have the incentive to arbitrage over time: if the product’s replacement cost is expected to increase, the incentive to sell the product today is lower, since expected appreciation remedies the cost of carrying the inventory. 3.2.5 Homebuyers We now discuss qb,t , the willingness of buyers to pay for a new house. It is defined as qb,t = qg,t 1 + Φ2 ( dt ht ) The willingness to pay equals to the price of houses in the market for exist- ing homes, or qg,t, adjusted by an adjustment cost Φ2 (.). The adjustment cost, assumed to be paid by homebuyers, is increasing in the speed of devel- opment. This captures the short-term cost of connecting new development with roads and various types of services such as electricity, water, and ca- ble. This cost, in essence, is the capital adjustment cost (or CAC) that is 67 Chapter 3. Costs and Frictions in the Housing Market. . . commonly used in neo-classical models to generate Tobin’s Q. In housing lit- erature, Jud and Winkler (2003) define qg,tqh,t as the Tobin’s Q in the housing market and find it positively correlated with residential investments. Tech- nically, this paper assumes that Φ′′2 (.) = ϕq ≥ 0 in the balance growth path. The findings in Jud and Winkler (2003) will be replicated with a positive ϕq. 3.2.6 The linearized model Before applying maximum likelihood estimations to evaluate the maximized likelihood statistics associated with different versions of the model, the paper needs to construct a linear state-space econometric model. As the first step of the linearization, we stabilize the system by dividing variables that have a trend component with the corresponding trend. We then solve the stationarized system for its steady-state, and linearize the model around the solution. The resulted linear system is a system of difference equations that can be solved numerically. We then append a observational equation to relate the model with the observed data. We use the same set of observed variables as the estimation of the basic model, i.e., the consumption growth, the (real) interest rate, the growth rate of (real) house price, and the growth rate of the quantity index of residential investment, taking into account the allocation between maintenance and new construction. Here we present the linearized system only. A technical appendix can be requested from the author. From the asset market q̂∗,t = (r + δh − γr)Et ( zd,t+1 + ŷt+1 − ĥt+1 ) + [1− (r + δh − γr)]Etq̂∗,t+1 − rt+1 − zd,t q̂g,t = (1− τ) q̂g,t−1 + τ q̂∗,t If asset price adjusts immediately, τ = 1 and therefore q̂g,t = q̂∗,t. From the suppliers of construction inputs, ĉc,t − x̂cc,t = ψd̂t − zs,t βEtpicc,t+1 = picc,t + (1− ξ) ( 1− 11+rξ ) ξ xcc,t From the construction arm of residential builders we have 68 Chapter 3. Costs and Frictions in the Housing Market. . . ϕhsd̂t = 1 (1 + γ1) 2 (1 + β) (q̂c,t − ĉc,t) + ϕhs 1 + β d̂t−1 + ϕhs ( 1− 1 1 + β ) Etd̂t+1 where β = 11+r is the steady-state discount factor. In the absence of invest- ment adjustment cost (i.e., ϕhs = 0), the equation is reduced into q̂c,t = ĉc,t. From the retail arm of residential builders v̂t+1 = θ 1 + γ1 ( v̂t + θ̂t ) + ( 1− θ 1 + γ1 ) d̂t r̂t+1 = Et (q̂c,t+1 − q̂c,t)− 2 ( r − γ2 1− θ ) Etθ̂t+1 q̂h,t − q̂c,t = − ( 1− 1r−γ2 1−θ + 1 )( θ 1− θ + 2 ) θ̂t q̂b,t − q̂c,t = − 1− 1r−γ2 (1−θ)2 + 1  2θ̂t where θ solves ωbr−γ2 = ( θ 1−θ )2 and is interpreted as the chance that a house fails to sell in the steady state. Without search friction, the system becomes v̂t+1 = d̂t, r̂t+1 = Et (q̂c,t+1 − q̂c,t), and q̂h,t = q̂c,t = q̂b,t replace the system above, which would have been identical as setting the search cost parameter ωb as zero. If storage technology is not allowed, then r̂t+1 = Et (q̂c,t+1 − q̂c,t) drops out of the equation completely. From a households’s arbitrage condition q̂b,t = q̂g,t − ϕq ( d̂t − ĥt ) If the system has no capital adjustment cost, we will have ϕq = 0 and q̂b,t = q̂g,t. The accumulation of housing asset is ĥt+1 = 1 1 + γp + γ1 ĥt + ( 1− 1 1 + γp + γ1 )( v̂t − θ1− θ θ̂t ) when there is no search friction, θ = 0 and ĥt+1 = 11+γp+γ1 ĥt+ ( 1− 11+γp+γ1 ) v̂t 69 Chapter 3. Costs and Frictions in the Housing Market. . . and v̂t = d̂t−1. The system is completed by adding the four equations that describe the stochastic processes of exogenous shocks. 3.3 Compare the Estimated Models This section discusses the empirical relevance of the new features added to the standard stock-flow model. The strategy is to compare the empirical performance of the full model to a set of smaller models. Each of these smaller models has a single feature that is turned off from the bigger one. Since the only difference between the full model and a smaller model is the elimination of one feature, the change in empirical performance reflects the empirical relevance of the feature that is turned off. We examine ”empirical performance” with four criterions, one for each of the following four sub-sections. The first sub-section looks at the log- likelihood statistics, the second sub-section at the accuracy of one-step- ahead forecasts for prices and for investments, the third one at the accuracy of medium-run forecast, and the last one at the model’s ability to replicate second moments in the data 3.3.1 Log-likelihood statistics This sub-section uses the fall in log-likelihood from the full model to the smaller model as a metric to measure the empirical importance of the cost/friction that is turned off. For statistical inference, we first use the log-likelihood ratio test (LRT) between the parental model and the smaller model. With LRT, twice the difference in log-likelihoods is compared to the 5% critical value of a Chi-squared distribution with one degree of freedom. An alterna- tive statistical inference is the SIC (Schwarz Information Criterion). SIC will select the full model over the smaller one if the reduction in log likelihood is greater than ln(N)2 . Table 3.1 presents the results from the experiments. The first column of the table is from the full model. The estimate paints a picture of the housing market that has: a) a high degree of price rigidity in the asset market (the speed of adjustment is 12% per quarter); b) elastic long-term supply (about 6% increase in supply for 1% increase in price); c) short-run adjustment costs in both the supply side and the demand side; and d) construction costs that appear to be very flexible, 80% of the suppliers can adjust their price in one quarter. 70 Chapter 3. Costs and Frictions in the Housing Market. . . Then, we start to turn off individual features from the full model. First, we look at the two price sluggishness. The first column assumes immediate adjustment in the asset market. The log-likelihood drops by more than 60. Twice this reduction is 120, which is well beyond the critical value for either the LRT or the criterion of the SIC, which are 3.8 and 4.9, respectively.30 By turning off the sticky prices for construction inputs, the log-likelihood is reduced by 3, which is barely significant. We proceed to the two quantity adjustment costs. By eliminating the IAC (or investment adjustment cost), the log-likelihood is reduced by 13. Eliminating the CAC (capital adjustment cost) leads to a reduction of 8. In the estimations, the search friction parameter ωb has been calibrated to the level so that, in the steady-state, the model generates a quarterly sales-to-inventory ratio of 0.55, the average ratio in the US market for new, single-family residential units. Slightly varying the value of this particular parameter does not change the likelihood statistics much. Of more impor- tance, however, is the removal of the residential builders’ option to invest in inventory. The removal forces builders to clear their productions in each period. Under this assumption, the estimation finds a radically different economy that fits the data equally well (the reduction in likelihood is al- most zero). The estimates describe a rigid economy. More than 80% of construction suppliers do not have the option to reset their prices, and a large adjustment cost facing households. We believe that investment in in- ventory is a matter of fact, so we choose to work on the full model that allows this option. The fit between the model’s impulse response and the empirical response is shown in Figure 3.6. The fits improve compared to the simpler model (Figure 3.2). 3.3.2 One-step-ahead forecast In Figure 3.3, we present a set figures to illustrate the match between the one-step-ahead forecast and the actual realization for house prices and for residential investment. The first column is for house price and the second column is for residential investment. The figures have five rows, each corre- sponding to one model. The ordering, at which the models are presented, is the same as that in Table 3.1. The full model is the top row. The second 30Schwarz information criterion (SIC) has the formula SIC = −2 ln (L) + k ln (N) where ln (L) is log-likelihood an, k is number of parameter and N is number of obervation. 71 Chapter 3. Costs and Frictions in the Housing Market. . . row assumes immediate adjustment of house price in the asset market. The third row removes sticky construction cost from the full model. The fourth row turns off IAC. The last row turns off CAC. Figure 3.3 is helpful for understanding how individual frictions or costs affect the model’s ability to keep track of house prices and residential invest- ments. It demonstrates that the price rigidity in the asset market is, by far, the most important feature. Comparing row 2 to row 1, we see that taking away the price rigidity worsens the fit for both prices and investments. For example, in the housing boom during the early 2000s, the model without price rigidity forecasts a price drop instead of an increase. It also misses the investment boom during the same period. Again during the late 2006 and early 2007, the removal of price rigidity cause the model to miss the magnitude of price and investment drop. The figure also suggests that the feature of sticky construction cost does not improve the fit for either prices or for investments. On the other hand, the two adjustment costs, IAC and CAC, both improve the fit of model in term of residential investment. 3.3.3 Medium-run forecast This section examines the accuracy of different models in term of medium- run forecasts. In the exercise, the estimated models are made to stop at a given quarter and forecast the window of the next eight quarters, using only the current and the past data. For example, when the model stands at 2005Q2, it forecasts the house price and investment for every quarter between 2005Q3 and 2007Q2. We then compare the forecast to the actual realization, to derive a measure of forecast error. We repeat the exercise using the second quarter of every year between 1998 and 2005 as the stop- and-forecast point. The measure of forecast error is the Root Mean Square Error (RMSE), which is square root of the mean of squared errors. The RMSE from an estimated unrestricted VAR(2) is used as the benchmark for comparison. Table 3.2 presents the results from the exercise, and shows that the estimated full model has the forecast ability that is comparable to an unre- stricted VAR(2). In averaging over the time horizon of eight quarters, the estimated model has a RMSE that is 8% higher than that from a VAR(2) in the case of house price, and 1% higher in the case of residential investment. This does not sound impressive unless we consider that the standard stock- flow model has an RMSE that is 124% higher than VAR(2) for the price, and 52% higher for the investment. 72 Chapter 3. Costs and Frictions in the Housing Market. . . Now we proceed to compare different specification of the housing models. The comparison shows that the gradual price adjustment in the asset market is the key to improving forecast accuracy. Removing rigidity in house price increases the RMSE substantially for both price and for investment (compare row 4 to row 3). Turing off IAC (investment adjustment cost),on the other hand, has little impact as long as CAC (capital adjustment cost) is present (compare row 5 to row3). Further turning off CAC, presented in row 6, has a greater impact for investments, but not for price. An interesting comparison is between row 7 and row 2. Row 7 is from a model in which IAC is the only extra feature. This is similar to how Christiano et al. (2005) and Smet and Wouters (2007) model investments in their DSGE model. Row 2 corresponds to the full model. The comparison between row 7 and row 2 shows that, despite IAC’s ability to improve the forecast for investments, it does virtually nothing to improve the forecast for house prices. This raises the interesting question about why investment adjustment cost fails to generate a better forecast for prices. Neither Chris- tiano et al. (2005) nor Smet and Wouters (2007) have capital prices in their estimation. That is probably why the failure of investment adjustment cost on the price front is not reported. Further exploration raises more questions about IAC. When the con- struction industry features IAC, the level of investment tends to move slowly. Similar gradual movements in the investment, however, can also be gener- ated by a slowly adjusting house price, along an positive-sloped marginal- cost curve. Is it possible that the price rigidity would be mistaken as IAC when the price is assumed to adjust immediately? 3.3.4 Second moments This subsection compares the second moments predicted by the models to those found in the data. We look at both the standard deviations and the correlation coefficients between price and investment. The middle panel of Table 3.1 shows the standard deviation of the quar- terly growth rate of the new house price and residential investment. The models are able to produce realistic-looking standard deviations as long as the price rigidity remains in the model. Once price rigidity is removed, the models predicts a volatility for house price that is far too high. This suggests that price rigidity is irreplaceable when it comes matching the volatility for house prices. The bottom panel of Table 3.1 presents the correlation coefficients be- tween the four-quarter growth in the price and the four-quarter growth in 73 Chapter 3. Costs and Frictions in the Housing Market. . . the investment. From the data, the correlation coefficient is 0.5. In the simulated data from the estimated models, the correlation coefficients reach a height of 0.5 only when there is no rigidity in house prices. When there is rigidity, however, the coefficients drop to 0.15-0.27. We perform further ex- periments by stripping the model of all other features except for the rigidity in the asset price. The exercise confirms price rigidity by itself is the main source for the reduction in the price-investment correlation. Why does rigidity in house price reduce the price-investment correlation? One explanation is that rigidity in house price slows down the response of in- vestment to demand changes, thus reducing the volatility of the investment. To compensate the loss in investment’s volatility, the model assigns more volatile supply shocks. Since demand shocks are the main source of posi- tive co-movement, while supply shocks create negative co-movements, the correlation coefficients drops as the result of greater supply shocks. Vari- ance decomposition provides supportive evidence. Without price rigidity, the contributions from demand shocks to the variance of the investment is equal to those from supply shocks. With rigidity, however, the contributions from demand shocks becomes smaller, while supply shocks dominate. 3.4 Some Discussions on the New Features of the Model We now use simulation exercises to explore the properties of the models. A set of experiments were conducted to study the role of individual features in the model. The following is a summary of the findings: 3.4.1 Investment adjustment cost versus price rigidity DSGE literature that emphasizes empirical performance often uses produc- tion inertia in the capital goods sector, i.e., IAC (investment adjustment cost), to keep track of the time series of investment. The existence of IAC makes it costly to change the level of production, thus giving producers the incentive to smoothen out their production plan. The result is greater ex- tent of auto-correlation in the models’ prediction. Such prediction matches data better than those from an alternative model without IAC. The analy- sis in this paper, however, raises questions on the use of IAC. The previous sections have shown that the IAC is not the only mechanism that can help the model to achieve better predictions on investment. Rigidity in house price also does. 74 Chapter 3. Costs and Frictions in the Housing Market. . . Furthermore, rigidity in house price generates better forecast for house price as well. IAC, on the other hand, improves the forecast only for invest- ment, but not for prices. The failure of IAC in providing better forecast of prices is rather unsettling. We might think that since a model with IAC gives better prediction about the future movements of investments, it must also has a better prediction about future prices. That does not appear to be the case. Another often-cited advantage of using IAC in the model of investments is its ability to generate a hump-shape response of investment to shocks (Smets and Wouters (2007)). Price rigidity can replace IAC in this aspect to certain extent. Figure 3.4 presents the impulse response of the housing market to an interest rate change. One of the models has IAC as the only extra feature. The other model replaces IAC with price rigidity. Both mod- els are seen to generate a slowly adjusting trajectory for housing investment. Price rigidity generates a smooth trajectory for price as well. In the case of IAC, however, price drops immediately to the bottom, contrary to the data. This again highlights IAC’s inability to keep tract of house price, de- spite its apparent success in tracking investment. In comparison, the model with price rigidity succeeds in giving both a smoother movement in house prices, and a hump-shape investment. It is easy to see why this is the case. With price moving slowly, the construction industry responds gradually, in adjusting their construction activity. 3.4.2 Sticky construction costs, volatility of investments, and price rigidity This subsection asks two questions: a). how does the stickiness in construc- tion costs affect housing market’s response to shocks? b). Is sticky cost responsible for price rigidity in the market? We find that sticky cost makes investment more sensitive, but it has little impact on the dynamics of price. The observations are from Figure 3.5, which compares a simple stock-flow model’s impulse responses, to an interest rate shock, with those from a model with sticky costs. The latter model has a Calvo stickiness at 0.65, which literally means the prices of 65% of construction input can not be re-adjusted in any a quarter. The figure shows that the stickiness in construction cost increases the immediate reaction of the investment by three times. It does not, however, affect the trajectory of price. In particular, it does not slow down the movement of house prices. The price still falls to the bottom at the period when the shock occurs. The inability for sticky cost to create rigidity in house price shall not 75 Chapter 3. Costs and Frictions in the Housing Market. . . be surprising. The quarterly flow of new homes into housing stock is small in relative size, accounting for only 0.6% of existing stock. As for house price in the theoretical model, it is the total unit of stock that counts. The rigidity in house price can not come from the construction industry, it must have more substantial sources. 3.4.3 Inventory and changes in interest rate The option for residential builders to hold inventory increase the impact of an interest rate change on housing investment, especially when house price adjusts slowly. With gradually adjusting house price, an increase in interest rate has two effects on residential builders. First, higher interest rate increases the cost of carrying the inventory. Second, the price is expected to decline further. The expected depreciation imposes another layer of cost for holding the inventory. A residential builder therefore has the extra incentive to draw down inventories. The drawing down can be accomplished by two means: reducing price, or reducing orders for new investment. Builders in the model use both tactics. Figure 3.7 shows the impulse response, to an interest rate change, from two models. One of the model has sticky price and allows builder to invest/disinvest in inventory. The other model removes inventory from the model all together. The comparison shows that, when given the opportunity to use inventory to optimize inter-temporarily, residential builders react more strongly in their orders of new housing units. House price also drops, but only by a small amount. 3.5 Conclusion This paper presents a richly specified stock-flow model for the housing mar- ket and empirically examines the importance of the different costs and fric- tions in the model. The model has detailed descriptions of the cost structure in the construction industry, differentiating the short-term and the long-term supply elasticity, and admitting possible stickiness in construction cost. It also features search frictions thus demand for inventories. Finally, it admits possible price rigidity in the asset market. We use several criteria to exam- ine the empirical relevance, including the likelihood statistics, the model’s forecast performance in the short run and in the medium-run, the model’s ability to replicate second moments in the data, and the ability to match empirical impulses. Using the US data between 1985Q1 and 2007Q2, we find that the grad- ual adjustment of house price in the asset market is by far the most impor- 76 Chapter 3. Costs and Frictions in the Housing Market. . . tant friction. We conclude that the key for constructing an optimization- based housing model is to structurally interpret the slow adjustment of house prices. In addition, the findings in the paper raise questions about the use of in- vestment adjustment cost (IAC) in the model of investment. IAC is a staple feature of the some recent DSGE models. Its popularity primarily comes from its ability to keep tract of the time series of investment. The analysis in the paper, however, demonstrates that IAC is not capable of improving the models’ fit and forecast of capital prices in the housing market, despite its apparent success in tracking housing investments. The paradox is rather unsettling and probably needs to be explained. In comparison, the sluggish adjustment of house prices improves the model’s fit and forecasts of both prices and investments. The price rigidity is also able to generate hump- shape response of the investments even without IAC. It will be interesting to see how much of the perceived evidence for IAC is in fact the result of price rigidity. 77 Chapter 3. Costs and Frictions in the Housing Market. . . 3.6 Tables and Figures Table 3.1: Testing the empirical importance of costs and frictions in the model Panel -a: reductions in log-likelihood when features are turned off Log-likelihood 316.65 376.89 319.71 329.68 324.73 Search friction* Yes Yes Yes Yes Yes Slope of the long run supply curve 0.170 0.157 0.145 0.231 0.740 Price rigidity in the asset mar- ket(Quarterly speed of adjustment) 0.123 off 0.123 0.110 0.049 Sticky input prices (Calvo stickiness) 0.208 0.174 off 0.001 0.104 Investment adjustment cost 0.107 0.064 0.104 off 4.722 Capital adjustment cost 0.441 0.388 0.460 0.278 off (Reduction in maximum likelihood)*2 120.49 6.13 26.06 16.17 Critical value (5%) from Chi-square dist. of 1 degree of freedom: 3.84 SIC (Schwarz information criterion): 4.95 Panel -b: models’ ability to match std dev. of prices and investments Stdev.: house price (0.74 in data) 0.80 1.40 0.75 0.81 0.74 Stdev.: investments (2.5 in data) 2.37 2.81 2.43 2.47 2.37 Panel -c: testing whether the estimated models can match correlation coefficients found in the data between the four-quarter growth of house price and the four-quarter growth of investment The price-investment corrcoef. 0.14 0.49 0.14 0.13 0.28 (0.5 in data) [0.14] [0.10] [0.14] [0.16] [0.16] 1Search cost is calibrated so that the sales to inventory ratio is 0.55 in the steady state 2The correlation coefficients of the estimated models are from simula- tions. Each simulation involves creating artificial data of comparable sample size(with early simulated data discarded). The table shows the mean and standard deviation from 200 simulations for each model. 78 Chapter 3. Costs and Frictions in the Housing Market. . . Table 3.2: The performance of the estimated models in medium-run forecast compared to VAR(2) RMSE (root mean square error) of a VAR(2) Q-1 Q-2 Q-3 Q-4 Q-5 Q-6 Q-7 Q-8 House price 0.7 1.2 1.3 1.4 1.5 2.1 2.6 3.2 Resd. investment 1.4 2.7 3.5 3.6 5.0 6.9 8.8 10.3 Percentage improvement over VAR(2), Avg. Negative value indicates worse forecast Impv. Row 2: Basic Stock-flow model House price -42 -61 -131 -186 -188 -141 -136 -104 -124 Resd. investment -89 -69 -63 -76 -61 -35 -16 -6 -52 Row 3: Full model House price 12 8 -8 -28 -14 -5 -16 -11 -8 Resd. investment -6 -12 -8 -12 -8 4 12 19 -1 Row 4: Full model minus price rigidity in housing market House price -23 -32 -77 -113 -105 -73 -74 -54 -69 Resd. investment -56 -59 -56 -67 -54 -31 -14 -4 -43 Row 5: Full model minus IAC (investment adjustment cost) House price 19 12 2 -15 -2 3 -8 -3 1 Resd. investment -20 -17 -16 -18 -9 1 9 17 -7 Row 6: Full model minus IAC and CAC (capital adjustment costs) House price 10 13 -2 -26 -13 -1 -9 -4 -4 Resd. investment -41 -32 -27 -30 -21 -5 6 13 -17 Row 7: Model with IAC only House price -41 -59 -129 -183 -184 -136 -132 -100 -121 Resd. investment -33 -31 -31 -46 -41 -20 -6 2 -26 1How RMSE of forecast is calculated: The model stops at the second quar- ter of each year between 1998-2005, and uses current and past information to forecast the accumulated growth of price and investment in each of the future eight quarters. After that the model moves to the second quarter of the next year and repeat the exercise. We calculate the Root Mean Square Error (RMSE) for each of the eight quarters from the ten forecasts. 2Percentage improvement is defined as the reduction in RMSE relative to RMSE of the VAR(2) 79 Chapter 3. Costs and Frictions in the Housing Market. . . Figure 3.1: Compare one-step-ahead forecasts and realizations 1985Q1 2000Q1 2007Q2 −2 0 2 4 Growth rate of house price 1985Q1 2000Q1 2007Q2 −10 −5 0 5 10 Growth rate of residential inv. 1985Q1 2000Q1 2007Q2 −2 −1 0 1 2 Growth rate of consumption 1985Q1 2000Q1 2007Q2 −1 0 1 2 Real interest rate Footnote: 1. All units are in percentage points. 2. The growth rates are in quarterly frequency, yearly interest rate has been divided by four.  Figure 3.1: Compare one−step−ahead forecasts and realizations The model: basic stock−flow Model; The thick line is the forecast 80 Chapter 3. Costs and Frictions in the Housing Market. . . Figure 3.2: Compare impulse responses from the stock-flow model and those from the data 0 5 10 15 20 −1.5 −1 −0.5 0 0.5 Resonse: House price 0 5 10 15 20 −0.4 −0.2 0 0.2 0.4 0.6 Shocks: Interest rate Resonse: Consumption 0 5 10 15 20 −0.1 0 0.1 0.2 0.3 Shocks: Interest rate Resonse: Interest rate 0 5 10 15 20 −3 −2 −1 0 1 Resonse: Resid. Investment Figure 3.2: Compare impulse responses from the stock−flow model                      to those from the data  Footnote: 1. All units are in percentage points. 2. The empirical impulse responses is from a recursive VAR. The order of variables is the same     as the clock−wise order of plots. 3. Tme frame is in quarters                  The thick lines are the IFR from the estimated model,                  The thin lines are IRF from the data. Bounds indicate 95% confidence interval 81 Chapter 3. Costs and Frictions in the Housing Market. . . Figure 3.3: Compare different models for their accuracy in one-step-ahead forecasting 1985Q1 2000Q1 2007Q2 −2 0 2 Growth rate of house price : Full model 1985Q1 2000Q1 2007Q2 −10 0 10 Growth rate of residential inv. Full model 1985Q1 2000Q1 2007Q2 −2 0 2 Price rigidity is turned off 1985Q1 2000Q1 2007Q2 −10 0 10 Price rigidity is turned off 1985Q1 2000Q1 2007Q2 −2 0 2 Sticky cost is turned off 1985Q1 2000Q1 2007Q2 −10 0 10 Sticky cost is turned off 1985Q1 2000Q1 2007Q2 −2 0 2 CAC (capital adjustment cost) is turned off 1985Q1 2000Q1 2007Q2 −10 0 10 CAC (capital adjustment cost) is turned off 1985Q1 2000Q1 2007Q2 −2 0 2 IAC (investment adjustment cost) is turned off 1985Q1 2000Q1 2007Q2 −10 0 10 IAC (investment adjustment cost) is turned off Footnote: 1. All units are in percentage points. 2. The growth rates are in quarterly frequency, yearly interest rate has been divided by four.  Figure 3.3: Different models’ accurary in one−step−ahead forecasting The thick line is the forecast, the thin line is the data 82 Chapter 3. Costs and Frictions in the Housing Market. . . Figure 3.4: Impulse responses to an interest rate change : IAC V.S. rigidity in house price 0 5 10 15 20 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 Resonse: House price 0 5 10 15 20 −0.8 −0.6 −0.4 −0.2 0 0.2 Resonse: Resid. Investment Figure 3.4: Impulse responses to an interest rate change                        Rigidity in house price  V.S.  IAC The thin line is from the model with IAC( investment adjustment cost) The thick line is from a model with slowly adjusting house price  83 Chapter 3. Costs and Frictions in the Housing Market. . . Figure 3.5: Impulse response to an interest rate change: sticky construction costs 0 5 10 15 20 −1.5 −1 −0.5 0 0.5 Resonse: House price 0 5 10 15 20 −8 −6 −4 −2 0 2 Resonse: Resid. Investment Figure 3.5: Impulse response to an interest rate change: sticky construction costs The thin line is from a simple stock−flow model The thick line is from a model in which 65% of the suppliers in a quarter can not change their prices 84 Chapter 3. Costs and Frictions in the Housing Market. . . Figure 3.6: Compare impulse responses from the full model and those from the data 0 5 10 15 20 −0.4 −0.2 0 0.2 0.4 0.6 Shocks: Interest rate Resonse: Consumption 0 5 10 15 20 −0.1 0 0.1 0.2 0.3 Shocks: Interest rate Resonse: Interest rate 0 5 10 15 20 −1 −0.5 0 0.5 Shocks: Interest rate Resonse: House price 0 5 10 15 20 −3 −2 −1 0 1 Shocks: Interest rate Resonse: Resid. Investment Figure 3.6: Compare impulse responses from the full model and those from the data  Footnote: 1. All units are in percentage points. 2. The empirical impulse responses is from a recursive VAR. The order of variables is the same     as the clock−wise order of plots. 3. Tme frame is in quarters                   The thick line is IFR from the estimated model,                  The thin lines are IRF from the data. Bounds indicate 95% confidence interval 85 Chapter 3. Costs and Frictions in the Housing Market. . . Figure 3.7: Inventory and housing market’s response to an interest rate shock 0 5 10 15 20 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 Resonse: House price 0 5 10 15 20 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 Resonse: Resid. Investment Figure 3.7: Inventory and market’s responses to an interest rate shock The thick line is a model in which builders can invest in the inventory The thin line is the model that prohibits investment in the inventory Footnote:     Price adjustment speed is 20% per quarter     When inventoty investment is allowed, the steady state sales to inventory ratio is about 0.55 a quarter     The remaining part of the model is from the estimated standard stock flow model 86 Chapter 4 Is there a Reduction in the Price Elasticity of the Supply of New Housing in the US? 4.1 Introduction The US housing investments since the mid-1980s have become markedly less volatile and less sensitive compared to the earlier two decades. Figure 4.1 plots the time series of the quarterly growth rates of the investment from 1971 to the second quarter of 2007. The standard deviation of the percentage growth drops from 6% before 1985 to 2.5% after. Several pa- pers, including McCarthy and Peach (2002), Dynan and Sichel (2006), and Bernanke (2007) find that housing investments have become less responsive to economic shocks, or exhibit smoother responses to them. Chapter 2 of this dissertation also finds housing investment to have a reduced sensitivity in an estimated Dynamic Stochastic General Equilibrium (DSGE) model. During the same period, the US has seen a relative increase in the volatil- ity of house prices. New home prices, more specifically the quarterly growth of inflation-adjusted prices of newly constructed houses, also show sign of moderation. The fall in their standard deviation, however, is smaller in proportion than that of the investments. As the result, the ratio of the standard deviations increases from 0.16 before 1985 to 0.3 after. Figure 4.1 presents the time series of prices and investments. The relative increase in the variability of the price series is visually notable. Observations at a lower frequency also indicate a larger movement of house prices in the more recent period. During the latest upswing of housing cycle, the real price of new homes increased by 34% from the bottom in the early 1990s to the peak in 2005. Over the same upswing, the level of housing investments increased by 68% from the bottom to the peak. The ratio of price movement over investment movement is, therefore, 0.5. The identically defined ratio for the preceding two cycles are 0.08 (between the early and 87 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . the late 1980s) and 0.35 (between the mid and the late 1970s), respectively. Krainer (2005), when describing the 1990s experience, suggests “...the link between house prices and new residential construction shifted dramatically. Even though house prices shot up at the end of the decade, the response from the construction sector was unusually subdued.” These observations suggest that the supply of new houses may have be- come less elastic. Holding all else constant, a reduced capacity, or slower speed of new supply, suppresses the response of investments with greater re- sponses of prices. In the extreme case when the supply is perfectly inelastic, all the adjustments in the market go through prices, and there will be no response from investments, as in the theoretical model of Iacoviello (2005a). The possible change in supply conditions received little attention in the literature that involves interpreting the appearance of more stable housing investments. The reduced volatility and sensitivity of the investments are well documented. As possible explanations, the literature often refers to the perceived improvement in monetary policy or in the housing finance system. For example, Taylor (2007) suggests that the decline in volatility is “largely due to an improved monetary policy” that keeps inflation under control, which reduces the chances of boom-bust cycles and large swings in interest rates. Peek and Wilcox (2003) and Peek and Wilcox (2006) credit the improved resilience of the housing market to the expansion of a secondary market for mortgage debts, which they suggest has resulted in a more stable flow of mortgage credit. Dynan and Sichel (2006) credits financial innovations, and Bernanke (2007) suggest that “the easing of some traditional institutional and regulatory frictions seems to have reduced the sensitivity of residential construction to monetary policy”. It is important to take into account changes in supply conditions when trying to interpret the observed changes in the US housing market. Reduced supply capacity or speed has different implications than the explanations based on structural improvements. Explanations based on improvements have the implication that the housing sector has now become a smaller source of economic fluctuations. A lower supply elasticity, on the other hand, means that the prices are now doing more of the adjustments. In the case of extreme elasticity, housing market affects the real economy through residential construction. In the case of extreme inelasticity, housing market does so through property prices. One way for price to affect the real economy is through the so-called “wealth effect” on consumer spending, that higher prices increase aggregate consumption. This may happen because property values affect the perception of wealth. It is also likely that an increase in home value raises the collateral value of the asset, thus allowing credit 88 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . constrained home owners to borrow more and spend more. Iacoviello (2005a) provides a macro model that formalizes the collateral effect of housing prices. In his model housing market affects the economy only through prices. Unlike explanations based on improvements in the housing finance sys- tem, a lower supply elasticity calls for greater attention to housing market when conducting monetary policy. Bernanke (2007) suggested that housing market may be “no longer so central to monetary transmission as it was” because housing construction is now less responsive to policy actions. He assumes that the reduced sensitivity is the result of easing frictions in the financing system. If the reduced sensitivity is the consequence of tighter sup- ply conditions, however, the implication will be drastically different. Tighter supply leads to larger movement of house prices. Large movements of prop- erty value, in turn, can potentially trigger speculations and therefore ex- cessive buildings. Malpezzi and Wachter (2005) makes a case that inelastic supply is the necessary condition for speculation to generate large real-estate cycles. In their paper speculation is defined as demand that is “...a positive function of recent price changes.” To the extent that such speculation exists, the appearance of reduced sensitivity of construction could be a concern if it is accompanied by more responsive housing prices. A recent IMF report (World Economic Outlook, April 2008) urges central banks to pay greater attention to movements in house price, partly because “Recent innovations in housing finance markets have generally increased the impact of monetary policy on house prices.” A reduction in supply elasticity has similar effect. It, too, translates policy-induced demand into movements in prices. This paper uses US national-level data to estimate the supply elasticity of new housing. It estimates two versions of a housing model. The simpler version is a stock-flow model that is commonly used to study investments in durable goods. The second version is the same model augmented with mechanisms of gradual adjustments in housing prices and in construction activity. The paper then applies maximum likelihood estimations to the time series of residential investment from NIPA, and the price index of new, one-family houses from the Survey of Construction. The time series of con- sumption and interest rate are also included as forcing variables. The use of price index for new homes, instead of existing homes, reflects the paper’s focus on the supply of new housing. The estimations uses split samples to detect changes in the elasticity parameters. The earlier sample covers the quarters between 1971 and 1984. The more recent sample is between 1985Q1 to 2007Q2. The paper is organized as follows. Section 2 discusses the model and the estimation strategy. Section 3 presents the estimation results. Section 89 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . 4 concludes. 4.2 The Model This section presents the models used to estimate the supply elasticity. The model encompasses the stock-flow model laid out in Poterba (1984). It has two new features that Chapter 3 identifies as being the most empirically im- portant. Chapter 3 augments the basic stock-flow model with several new features, including the sluggish adjustment of house prices, costs of adjust- ing the level of construction activity and housing stock, as well as sticky construction costs, among others. It then examines the drop in maximized log-likelihoods as individual features are turned off from the full model, so as to measure empirical relevance. It is found that the sluggish adjustment of house prices (as in DiPasquale and Wheaton (1994)) is the most impor- tant feature. Following behind is the cost of adjusting construction activity, which is also called ”internal adjustment cost” in Topel and Rosen (1988). Reflecting these findings, these two features are built into the model in the current chapter. 4.2.1 Price of houses The value of a house comes from its ability to provide housing service and possible capital gains. House price is the solution to a non-arbitrage condi- tion in the housing market. (1 + rt+1 +m) q∗,t = EtR (ct+1, ht+1, zd,t+1) + Et (1− δh) q∗,t+1 (4.1) In this equation, the price of a housing asset is q∗. The housing unit is normalized to produce one unit of service. The value of the unit of service is R (.). The risk-free rate of return is rt+1. The asset-specific risk premium is m. Further regarding the value of housing service, the (relative) price is determined by the clearance of the rental market. The price is increasing in the total consumption c, which proxies wealth and demand for housing, but decreasing in the size of the existing housing stock h, which is the total supply of housing. The rental price is also affected by a demand shifter zd. This paper assumes a specific functional form for R (), that R (ct, ht, zd,t) = zd,t ct ht In the background is a utility function, on the part of households, that has Cobb-Douglas feature. With such a utility function, the intra-temporal 90 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . marginal rate of substitution between consumption and housing services is just zd,t ctht . The assumption of Cobb-Douglas utility guarantees a balance growth path, which does appear to be supported by the data. The share of spending on residential investment, over the past forty years, fluctuates significantly, but tend to revert back to a long term average that is about 5-6% of total GDP. The demand shifter is assumed to follow exogenous stochastic process. The exact specification will be described later in the text. The accumulation of housing stock follows ht+1 = ht + dt (4.2) where the variable dt denotes the net flow of new addition into existing housing stock. 4.2.2 Sluggish adjustment of house price The price q∗,t is the fundamental value of housing asset. In the short run, however, market price may deviate from the fundamental value. The ev- idence of sluggish adjustment has long been documented. For example, DiPasquale and Wheaton (1994) find that a model with lagged house price is statistically superior than a model without. The following equation describes the adjustment process of house prices, where qg,t denotes the market prices, qg,t qg,t−1 = (1 + γ2) 1−τ ( q∗,t qg,t−1 )τ (4.3) This equation uses τ , which is between zero and one, to denote the speed of adjustment. The equation basically says that the growth rate of house price is a weighted average of γ2 , which is the growth rate of house price in the balance-growth path, and the distance by which the fundamental value of housing asset exceeds the market price. The linearized version of the equation is q̂g,t = (1− τ) q̂g,t−1 + τ q̂∗,t. When there is immediately adjustment of house prices, we have τ = 1, which means q̂g,t = q̂∗,t for all t. The same adjustment mechanism is used in DiPasquale and Wheaton (1994). 4.2.3 Construction cost This section describes the cost structure in the construction industry. Let dt be the level of construction activity at the aggregate level, and let cc,t be 91 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . the cost of constructing a house, the relation between cost and construction is cc,t = dψt zh,t (1 + γx) t (4.4) The cost structure has three elements. The first is an upward slope captured by dψt with ψ > 0, in which case the cost of construction is increasing with the level of construction. The second element is a trend in cost captured by (1 + γx) t, with γx being the growth rate of exogenous costs. γx < 0 indicates a downward trend in cost while γx > 0 indicates a upward trend. The last element is mean-reverting cost shifter captured by variable zh,t, which follows an exogenous stochastic process. The process will be described later in the text. This paper takes the view that all three elements of the cost structure are necessary. The cost shifter zh,t captures exogenous and random ele- ments such as the cyclical variations in commercial building and/or high way constructions, which compete for construction inputs. The parameter ψ, if positive, gives rise to a long-term supply elasticity that is less than infinity. The supply elasticity is an actively researched topic in empirical housing literature. Some early papers suggests that the supply of new hous- ing is flat, i.e., ψ = 0 (See DiPasquale (1999) for a survey of the literature). The more recent papers, however, tend to estimate a positive ψ. Among them Topel and Rosen (1988) even find a modest supply elasticity of 3. Fi- nally, the time trend (1 + γx) t captures long term technological progress (or relative regress). As mentioned above, the real price of new homes, even after adjusting for quality, has an upward trend in the long run. Part of the trend may come from expansion of investment along an upward slope supply curve. But it may also come from trends in unspecified costs, such as the cost of land, and the cost of obtaining regulatory approval for development. The model’s inclusion of the exogenous time trend γx acknowledges the sec- ond possibility. The otherwise would be to fix the position of the supply curve over the course of four decades. 4.2.4 Adjustment cost in construction industry According to Topel and Rosen (1988), the supply of new housing in the US is more elastic in the long run than it is in the short run. They attribute the difference to what they call the “internal adjustment cost” in the con- struction industry. Such a cost penalizes rapid adjustment in construction activity. Let’s consider a hypothetical permanent increase of house price. 92 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Without the “internal adjustment cost”, construction activity jumps imme- diately to the level where the marginal cost equals to the new price. With the adjustment cost, construction activity approaches the higher level in a gradual manner. This explains why the long run supply elasticity is higher. With internal adjustment cost, the price of houses does not always equal to the marginal cost of construction, even if the market is competitive. For one, a builder needs to be compensated for the adjustment costs. Secondly, the optimal level of construction is determined in a dynamic optimizing manner: If a builder expects an increase of house price in the future, it may increase current construction to the point where the cost is higher than the price. This happens because a higher construction activity today reduces the cost of construction in the future. Mathematically, the dynamic optimization problem is described by max dt Et ∞∑ s 1 Πs (1 + rt+1+s) [ qc,t+s ( 1− Φ1 ( dt+s dt−1+s )) dt+s − cc,t+sdt+s ] The objective function is the expected discounted value of current and future profits. The discount factor is 1Πs(1+rt+1+s) . The periodical profit is inside the square bracket. In the profit function, the level of construction output is dt+s, the price of output is qc,t+s, the cost of construction cost per unit is cc,t+s. The adjustment cost function is Φ1 ( dt+s dt−1+s ) . It is assumed that in the balance-growth path Φ1 (.) = Φ′1 (.) = 0 and Φ′′1 (.) = ϕhs. The construction industry faces a convex cost if ϕhs > 0, in which case the industry has the incentive to smoothen out its construction plan over time. The optimal condition for profit maximization is 1− cc,t qc,t = Φ1 ( dt dt−1 ) + Φ′1 ( dt dt−1 ) dt dt−1 (4.5) − 1 1 + rt+1 Et qc,t+1 qc,t Φ′1 ( dt+1 dt )( dt+1 dt )2 In the DSGE literature, what Topel and Rosen (1988) call the “internal adjustment cost” is known as “Investment adjustment cost (IAC)”. IAC is used in DSGE literature primarily for its ability to keep track of investment series and generate hump-shaped response of investments to shocks. IAC is used in Christiano et al. (2005), Smets and Wouters (2007), and Justiniano and Primiceri (2006). 93 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . 4.2.5 Market clearance The market for new houses is cleared when the willingness to pay equals to the market price of a new house. The market price is denoted with qc,t. The willingness to pay, on the other hand, equals to the price of existing homes under certain assumptions. These assumptions include the absence of preferential differentials, and the absence of frictions and additional costs associated with the purchase of new houses. The price of an existing house is denoted with qgt as in equ(4.3), so we can describe the market clearance by qc,t = qg,t (4.6) 4.2.6 Reduce the model to a simple stock-flow model The system, consisting of equ(4.1) to equ(4.6), describe the behavior part of the model. The system can be reduced into a simpler version if we impose ϕhs = 0, and τ = 1. The former assumption excludes adjustment cost in the construction industry. The latter assumption excludes deviation of the price of a house from its fundamental value. The simplified model is, (1 + rt+1) qg,t = Etzd,t+1 ct+1 ht+1 + (1− δh)Etqg,t+1 qg,t = dψt zh,t (1 + γx) t ht+1 = ht + dt which is just a three equation stock-flow model. The first equation is the stock equation, describing the formation of house prices as a discounted sum of the value of housing services. The second equation is the flow equa- tion, describing the flow (or new investments) as a function of house price intersecting with an upward-sloped supply. The last equation is the accumu- lation equation, describing tomorrow’s stock as a function of today’s stock and flow. 4.2.7 The linearized system Some variables in the model, namely the consumption, the housing prices and the investments, may have long-term trends in time. For this reason, the model, as it is presented in the previous sections, does not have a steady state. To achieve a steady state, we stabilize the system by dividing the 94 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . variables that have a theoretical trend with the corresponding level of the trend. This way we rewrite the system in a stationary form. We then solve the stationarized system for its steady-state. After that, we log-linearize the system around the steady-state. During the estimation, the trend compo- nents is added back to the model. Here we present the stationarized and linearized system, In the balance growth path, the growth rate of residential investment in the balance-growth path is γ1 = γy − γx 1 + ψ The growth rate of house price in the balance-growth path is γ2 = ψγy + γx 1 + ψ he linearized version of equ(4.1) is q̂∗,t = (r + δh − γ2) ( Etẑd,t+1 + Etŷt+1 − Etĥt+1 ) + [1− (r + δh − γ2)]Etq̂∗,t+1 − r̂t+1 Equ(4.3) becomes q̂g,t = (1− τ) q̂g,t−1 + τ q̂∗,t Equ(4.2) becomes ĥt+1 = 1 1 + γ1 ĥt + ( 1− 1 1 + γ1 ) d̂t Equ(4.4) is now ĉc,t = ψd̂t − ẑh,t Equ(4.5) is now ϕhsd̂t = 1 (1 + γ1) 2 ( 1 + 11+r ) (q̂c,t − ĉc,t) + ϕhs 1 + 11+r d̂t−1 + ϕhs ( 1− 1 1 + 11+r ) Etd̂t+1 and finally the condition for market clearance, or equ(4.6), is q̂c,t = q̂g,t 95 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . 4.2.8 Stochastic processes The following describes the stochastic processes that are the drivers of fluc- tuations in the housing market. The processes, in the order that they are presented, are housing demand shocks, housing supply shocks, and the pro- cesses of consumption and (real) interest rates: ẑd,t = ρzdẑd,t−1 + εzd,t (4.7) ẑs,t = ρzsẑs,t−1 + εzs,t (4.8) ĉt = 4∑ j=1 ρcc,j ĉt−j + 4∑ j=1 ρcr,j r̂t−j + εc,t (4.9) r̂t = ρrc,0ĉt + 2∑ j=1 ρrc,j ĉt−j + 2∑ j=1 ρrr,j r̂t−j + εr,t (4.10) 4.2.9 Observable variables The system of linear difference equations can be solved numerically. The solution can then be used to construct a state-space system of econometric model. This paper uses four observed variables in the estimations. They are the quarterly growth rate of consumption, the level of real interest rate, the growth rate of inflation-adjusted house price, and the growth rate of residential investment from the national account. The relation between the observable data and the theoretical model is described by the following observation equations, ln ( qnewt qnewt−1 ) = q̂h,t − q̂h,t−1 + ψγy + γx1 + ψ ln ( ih,t ih,t−1 ) = ( 1− δh γ1 + δh )( d̂t − d̂t−1 ) + δh γ1 + δh ( ĥt − ĥt−1 ) + γy − γx 1 + ψ ln ( ct ct−1 ) = ĉt − ĉt−1 + γy Rt − pit = r̂t + r The left hand side of the system explains the use of the data. The inflation- adjusted Fisher price index of new houses from the Census Bureau is qnewt in the first equation. The estimations use the growth rate of prices, or 96 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . ln ( qnewt qnewt−1 ) . The quantity index of total residential investment (including maintenance, improvement and new construction) from the NIPA account is ih,t in the second equation. Again, the growth rate of the investment is used for estimations. The inflation-adjusted Private Consumption Expenditure from the NIPA is the consumption series. Its growth rate is ln ( ct ct−1 ) in the third equation. Finally, the real interest rate is defined as the difference between the effective federal funds rate and the growth of the GDP price index, or Rt − pit in the last equation. The right hand side of the observation system are variables and parame- ters in the theoretical model. In the first equation, the growth rate of house price is expressed as the change in q̂h between t and t-1, plus a trend growth rate ψγy+γx1+ψ . The trend growth rate is added because q̂h is defined as the percentage deviation from trend. The right hand side of the second equations is for the growth rate of the quantity of investment. It is constructed in a manner similar to that in the first equation, but with slight complication, because the investment in d is defined as new investment, while the data ih,t includes spending on mainte- nance/renovations. For this reason, the growth rate in the data is expressed as a weighted average of the growth rate of dt (new investments) and the growth rate of δhht (the maintenance under the assumption of constant rate of depreciation δh). The share of maintenance is δhγ1+δh . On the right hand side of the third equation, the growth rate of con- sumption is simply the change in ĉ plus the trend growth rate γy. On the RHS of the fourth equation, the the steady-state level of real interest rate equals to r. During the estimations, the paper calibrates γy and r to the average lev- els in the sample. Allowing these two parameters to be part of the estimation does not change the result in noticeable way. 4.3 The Estimations This section reports the estimations, focusing on the parameter of supply, or ψ. The estimations are performed in a basic stock-and-flow model as well as in a model with mechanisms of gradual adjustments. The basic model is the dominant way of modeling investments in durable goods in many macroeconomic models, so it is interesting to see what the model sees from the data. The model with gradual adjustment in price and construction is much more capable of tracking the dynamics of the time series of prices and investments. The additional features however makes estimating the 97 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . elasticity less precise, as we will see in the later discussion. 4.3.1 Estimating the basic model The basic model has only two parameters to be estimated, not including those that describe the stochastic processes of exogenous shocks. One of these two parameters is ψ, the slope of supply curve. The other is γx, which is the growth rate of the exogenous component in construction cost. Table 4.1 presents the estimation results. Its first two columns have the estimated parameters from a routine of maximizing likelihood. The first column is for the sample between 1971Q1 and 1984 Q4. The second column is for the sample after 1984 until 2007Q2. The slope from the pre-85 estimation is 0.10, implying a supply elasticity of ten. The same parameter from the second estimation is 0.46, implying a supply elasticity slightly greater than 2. The growth rate of exogenous cost is 0.27% per quarter in the earlier sample, and 0.07% is for the more recent sample. The average growth rate of house price in both sample is about 0.33% per quarter. The estimation therefore suggests that most of the growth in house price before 1985 came from the growth in exogenous cost, while after 1985 the growth is coming from the increase of residential investment along a steeper supply curve. The third and the fourth column of the table test the robustness of these findings by setting the exogenous growth of costs to be zero. The findings are robust. The slope is higher in the second sample, increasing from 0.12 to 0.48, which of course means a movement from a higher to lower supply elasticity. So far, the point estimates of supply elasticity fall within the range found in the literature. DiPasquale (1999) surveys the literature, and finds estimates ranging from infinity (Muth (1960), Follain (1979), and Stover (1986)), to between 4 to 14 in Malpezzi and Maclennan (2001) (published later after DiPasquale’s review), to about 3 in Topel and Rosen (1988), to be- tween 0.5 and 2.3 in Poterba (1984). The estimated elasticity in DiPasquale and Wheaton (1994), which takes into account gradual stock adjustment, is between 1.0 and 1.2. Figure 4.2 provides a plot of maximized likelihoods along the dimension of ψ. This gives us an idea whether the estimated difference between the samples is significant or not. The range of ψ in the figure is from 0.01 to 1.4. Because ψ is the slope of the supply, a higher ψ indicates lower elasticity. The range of ψ in Figure 4.2 corresponds to a range of elasticity from 0.7 to 100, a reasonable range given the findings in the literature. The likelihood is re-maximized at each point of ψ, allowing other parameters to re-adjust 98 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . in search of a new maximum. The figure shows that, the pre-85 log-likelihood peaks at about ψ = 0.1, and declines as ψ moves both to higher and lower level. When ψ increases to the level found in the second sample, the log-likelihood drops of the first sample by more than 20, a change that is well above 3.84, the 5% critical value of a Chi-squared distribution with one degree of freedom used in a likelihood ratio test. Similarly, the likelihood curve from the second sample assigns a significantly lower likelihood for the point estimate found in the earlier sample. To summarize, the data before 1985 produces a point estimate of supply elasticity about 10. The data after 1985 produces an point estimate of 2.15. The fall in the elasticity appears statistically significant judging by likelihood curves. 4.3.2 Estimating the richer model: a MLE approach The full model has two mechanisms of gradual adjustment. One of them is in housing prices. The other one is in construction activity. The adjustment mechanism in house prices brings into the model the parameter τ , which is the speed of adjustment. Higher τ indicates faster adjustment. Price adjusts immediately if τ = 1. The mechanism of adjustment in construction activity brings the parameter ϕhs. Higher level of ϕhs indicates greater cost penalty to rapid changes in construction. There is large improvement of fit from introducing the two adjustment mechanisms into the model. This is demonstrated by Figure 4.3 and 4.4, which plot the likelihood curves along the dimension of τ and ϕhs, respec- tively. The likelihood is re-maximized at each point on the horizontal axis, allowing other parameters to re-adjust in search of a new maximum. Figure 4.3 is for τ . Its horizontal axis starts from 0.01 and ends at 1. Moving from τ = 1 to τ = 0.1 increases the log-likelihood by 60 for the data of the earlier period, and by almost 100 for the data after 1985. Figure 4.4 presents the likelihood curve along ϕhs. To choose a range for ϕhs , we turn to Smets and Wouters (2007), which has an identically defined parameter of adjustment cost, but for investment defined more broadly to include both business and housing investment. Their prior of this parameter is a normal distribution with a mean of 4, and a standard error of 1.5. We use the range between zero and 12 for ϕhs. It turns out that the likelihood curve plateaus after 1.5. Increasing ϕhs to beyond 1.5 does not change the likelihood in substantial way. Moving from ϕhs = 0 to ϕhs = 1.5, however, improves the likelihood by 10 for the data before 1985, and by 20 for the 99 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . data after. Now moving on to ψ, the slope of supply. Figure 4.5 presents the like- lihood curves along different values of ψ between 0.01 to 1.4. The pattern here is quite different from Figure 4.2, which is the earlier findings under the simpler model. Most importantly, the log-likelihood curves, for both samples, are quite flat over the entire range. The post-1985 sample slightly favors steeper supply, because the likelihood is increasing with ψ. But the minimum at ψ =0.01 and the maximum at ψ =1.4 is within 4 units of log-likelihood. The lack of difference in likelihood is even worse for the sam- ple before 1985. Not only that the difference is small (smaller than one), but that there is no clear pattern whether the likelihood is increasing or decreasing in ψ. The difficulty to estimate ψ is a result from short samples, and the inclusion of mechanisms for gradual adjustment. For the data before 1985, the main problem comes from the interaction between ψ and τ . Higher (lower) ψ indicates a steeper (flatter) supply, which boosts (suppresses) the response of house price for a given change in residential investments. A lower (higher) τ , on the other hand, indicates a slower (faster) adjustment speed of house price, which reduces (boosts) the response of price. As the result, a model with a high ψ and a low τ may look a lot like a model with a low ψ and a high τ . Figure 4.6 presents a contour chart of likelihood over the discretized (ψ, τ) two-dimensional space. It shows that there is a series of combination of (ψ, τ) that produces likelihood very close to the maximum. There exists a ridge on the surface of likelihood moving along from the combination of high ψ & low τ , to the combination of low ψ & high τ . It follows that one way to achieve a more precise estimate of ψ is to impose an prior belief about τ . In fact, if we limit the estimation to τ ≥ 5% (i.e., the gap between the market price for new houses and its fundamental value disappears at a rate greater or equal to 5% per quarter), most of the high ψ region will disappear, leaving ψ staying in the region that is less than 0.2, which indicates a high elasticity of supply. For the data after 1986, the substitution between ψ and τ is less a problem. Instead the substitution between ψ and ϕhs matters more. ϕhs is the adjustment cost in construction industry. An extremely low ψ can be accommodated by a very high ϕhs. If we fix ϕhs at 4, the extremely low ψ become much less likely. To summarize, with gradual adjustment in prices and construction ac- tivity, there is little hope to generate a meaningful estimate for the supply elasticity. On a brighter side, these experiments suggest that if there exists a set of prior regarding the likely range or distributions of τ and ϕhs, the 100 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . information will be very useful for estimating the parameter of interest, in this case the elasticity of supply. It is noteworthy to point out that the simpler stock-flow model is just a extreme example, in which an econometrician sets τ = 1 and ϕhs = 0 before running the regression. 4.3.3 Estimating the richer model: a Bayesian approach As we find out from the preceding section, a MLE with minimum restriction makes it impossible to reach a meaningful estimate of the supply elasticity under the richer model. The simple stock-flow model gives a more pre- cise estimate, but it arbitrarily sets τ and ϕhs to extreme values despite their empirical importance demonstrated in Figure 4.3 and Figure 4.4. This section seeks the middle road by adopting a Bayesian approach. Bayesian approach allows econometricians to incorporate prior knowledge for some or all parameters during the estimations. In this case, we try to incorporate existing knowledge about τ and ϕhs into the analysis. Instead of setting these parameters to extreme values as the simpler model does, a Bayesian approach summarizes existing knowledge about the parameter in the form of prior distributions. As for τ , the speed of adjustment for house price, the estimate in Di- Pasquale and Wheaton (1994) is 29% in the first year following a shock, which translates into a quarterly adjustment rate slightly greater than 7%. Another paper, McCarthy and Peach (2002), estimates the speed to be 5%. We therefore assign this parameter a prior distribution that is Beta with a mean of 6% and a standard error of 1%. The prior distribution for ϕhs follows the prior on a similar parameter in Smets and Wouters (2007), being normal with a mean of 4 and a standard error of 2 (larger than the 1.5 in the cited paper). It has to be acknowledged that the adjustment cost parameter in Smets and Wouters (2007) is for investment broadly defined, including both housing and business investment. The choice rather reflects the lack of understanding about this particular parameter. For this reason, the standard error is intentionally set at a higher value. The same set of prior is used for both samples. Table 4.2 presents the estimation results. It shows that the supply elasticity in the sample before 1985 is closed to infinity (since ψ is approaching zero). Only the adjustment cost prevents an infinite jump of investment in response to changes in house prices. The elasticity estimated with data after 1985 is much lower, at about 1.25 (as an inverse to ψ = 0.8). There is no noticeable difference in 101 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . the adjustment cost parameter ϕhs between the two samples. Figure 4.8 plots the log posterior function along ψ for the two samples. The posterior function clearly favors high elasticity in the earlier sample, and low elasticity in the sample after 1985. 4.4 Conclusion This paper asks whether the US experienced a drop in the price elasticity of the supply of new housing. The comparison using the US national-level data, across time, appears to support the hypothesis of a fall in elasticity. Maximum likelihood estimations with the most basic form of the stock- and-flow model find an elasticity of 10 from the data between 1971Q1 and 1984Q4, and an elasticity of 2.15 between 1985Q1 and 2007Q2. In a richer model that incorporates mechanisms of gradual adjustment in house price and in construction activity, a MLE has difficulty in estimating the elasticity parameter. Nevertheless, when prior knowledge about the adjustment pa- rameters is incorporated, likelihood-based estimations again favor a higher elasticity in the earlier sample, and a low elasticity in the more recent sam- ple. The lower supply elasticity may come from limited availability of land. An indication of land scarcity is that the price of land has grown much faster than the price of houses. According to Figures 1 in Davis and Heath- cote (2005), the increase in the log of land’s price is close to 3 between 1975 and 2006, while the log of home prices increased only by one. Why would a country with low population density like the US experience a land shortage? Glaeser et al. (2005) attribute the trend to what they call the “man-made scarcity”: the “increasing difficulty of obtaining regulatory ap- proval for building new homes” and “significant increase in the ability of local residents to block new projects and a change of cities from urban growth machines to homeowners’ cooperatives.” Whether regulations reduces elas- ticity is beyond the reach of the paper. The question may be better answered using more disaggregate data that also need to have information about the existence and types of regulations. Regardless the cause for the reduction in supply elasticity, an inelastic supply is a source of concern for policy makers. Price movement can af- fect the broader economic by influencing consumer spending. To illustrate the quantitative impact of house price on consumption, I use the estimated model for the post-1986 data for an out-of-sample forecast. The forecast suggests that the next eight-quarters, following 2007Q2, will see a further 102 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . reduction of house price by 5% in real terms. This is probably at the conser- vative end of the predictions. Nevertheless, it still carries a sizable consump- tion effect. According to data compiled by Davis Morris and Heathcote, the aggregate market value of homes is $26.9 trillion dollars in the third quar- ter of 2007.31 A 5% reduction in house price means a $1.35 trillion loss in housing wealth. If we adopt the wealth effect, estimated by Carroll et al. (2006), which is 9 cents per dollar in the long run, the loss in housing wealth will reduce households’ spending by $120 billion, slightly less than 1% of the total GDP.32 To put this into context, the housing slump contributes only a 1.2% reduction in GDP over the course of the 6 quarters, from 2006Q1 to 2007Q2. The indirect effect of house price on consumption, therefore, carries a size that is almost comparable to the direct effect. Furthermore, Malpezzi and Wachter (2005) uses simulated models to show that an inelastic supply of new housing exacerbates the undesirable consequence from speculation, and is able to create a much larger price cycle than the alternative scenario with elastic supply. Large price cycle, in turn, can be destabilizing for the financial market and the economy. A reduced supply elasticity adds to the complexity facing the Federal Reserve. As they ease monetary policy, the resulted demand for housing investment is more likely to result in movement in house price, which has the potential of triggering an unwanted swings. 31The most up-to-the-date data is available from http://morris.marginalq.com/landdata.html. The cited number here is accessed on December 2007. 32The concept of “long-run” effect in Carroll et al. (2006) takes into account the serial correlation of consumption after the wealth shock. The authors’ interpretation of their ”long-run” is that ”it really reflects the medium-run dynamics of consumption (over the course of a few years)” before endogeneity kicks in. Some often cited estimates, such as those in Case et al. (2005), is before taking into account the propagations. Case et al. (2005) find a 10% increase in housing wealth has the effect to increase households’ consumption by 0.4%. 103 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . 4.5 Table and figures Table 4.1: Estimating the basic model Samples 1971Q1- 1985Q1- 1971Q1- 1985Q1- -1984Q4 -2007Q2 -1984Q4 -2007Q2 Log-likelihood 455.23 428.83 459.15 429.12 Parameters that describe the housing model and shocks ψ 0.10 0.46 0.12 0.48 γx 0.27 0.07 ρε,zd 0.99 0.96 0.97 0.97 σε,zd 0.14 0.10 0.19 0.09 ρε,zs 0.99 0.97 1.00 0.96 σε,zs 1.39 1.93 1.45 1.95 Parameters that describe the joint stochastic process of consumption and interest rates ρcr,1 -0.16 0.46 ρcr,2 0.02 -0.78 ρcr,3 0.07 0.25 ρcr,4 0.08 0.30 Same as Same as ρcc,1 1.24 0.94 the first the second ρcc,2 0.10 0.20 column1 column1 ρcc,3 -0.28 0.10 ρcc,4 -0.06 -0.33 ρrr,1 0.68 0.72 ρrr,2 0.10 0.16 ρrc,0 0.09 0.13 ρrc,1 -0.07 -0.12 ρrc,2 -0.04 -0.02 σε,c 0.80 0.43 σε,r 0.50 0.25 1The stochastic process for consumption and interest rate are estimated separately, the same set of point estimates are then used in all estimations. 104 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Table 4.2: Estimating the model with gradual adjustment in prices and constructions Samples 1971Q1- 1985Q1- -1984Q4 -2007Q2 Log-posterior function 352.76 321.84 Parameter that describe the housing model and shocks Prior distribution ψ None 0.01 0.80 ϕhs Normal(4,2) 5.33 4.70 τ Beta(0.06,0.01) 0.06 0.06 γx None 0.32 -0.13 ρε,zd None 0.38 0.49 σε,zd None 1.41 0.67 ρε,zs None 0.78 0.89 σε,zs None 2.59 2.04 1The stochastic process for consumption and interest rate uses the same point estimates as those in Table 4.1. 105 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Figure 4.1: Quarterly growth rate of housing price and investment in the US * The investment is the quantity index of residential investment in the US NIPA * The house price is the Price Deflator (Fisher) Index of New, One-Family Houses Under Construction, from the Survey of Construction Figure 4.1: Quarterly growth rate of housing price and investment in the US -30 -20 -10 0 10 20 Q1 97 10 1 Q1 97 50 1 Q1 98 00 1 Q1 98 50 1 Q1 99 00 1 Q1 99 50 1 Q2 00 00 1 Q2 00 50 1 Q2 00 70 2 C ha ng es  in  in ve st m en t -2 -1 0 1 2 3 C ha ng es  in  p ric e Residential Investment House price 106 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Figure 4.2: Maximized log-likelihood at different values of ψ from the basic stock-flow model Figure 4.2: maximized log-likelihood at different value of ȥ from the basic stock-flow model -640 -600 -560 -520 -480 -440 -400 ȥ= 0.0 11 ȥ= 0.0 5 ȥ= 0.1 ȥ= 0.2 ȥ= 0.4 ȥ= 0.6 ȥ= 0.8 ȥ= 1 ȥ= 1.2 ȥ= 1.4 1971Q1-1984Q4 1985Q1-2007Q2 107 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Figure 4.3: Maximized log-likelihood at different values of τ * IJ indicates the speed of adjustment for house price toward its fundamental value. When IJ=1, the market price of houses equals to fundamental value at all time Figure 4.3: Maximized log-likelihood at different values of IJ, from the model with gradually adjusting prices and construction activity -440 -420 -400 -380 -360 IJ=0.015 IJ=0.03 IJ=0.05 IJ=0.075 IJ=0.1 IJ=0.15 IJ=0.35 IJ=0.5 IJ=0.75 IJ=1 1971Q1-1984Q4 1985Q1-2007Q2, shifted downward by 30 units 108 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Figure 4.4: Maximized log-likelihood at different values of ϕhs * phi indicates adjustment cost in constructions. Higher phi indicates greater cost  penalty associated with rapid changes in construction activity Figure 4.4: Maximized log-likelihood at different value of ij_hs, from the model with gradually adjusting prices and construction activity -380 -375 -370 -365 -360 -355 -350 ij_ hs =0 ij_ hs =0 .5 ij_ hs =1 ij_ hs =1 .5 ij_ hs =2 ij_ hs =3 ij_ hs =4 ij_ hs =6 ij_ hs =8 ij_ hs =1 2 1971Q1-1984Q4 1985Q1-2007Q2, shifted downward by 30 units 109 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Figure 4.5: Maximized log-likelihood at different values of ψ Figure 4.5: maximized log-likelihood at different value of ȥ from the model with gradually adjusting prices and construction activity -358 -356 -354 -352 ȥ=0.011 ȥ=0.05 ȥ=0.1 ȥ=0.2 ȥ=0.4 ȥ=0.6 ȥ=0.8 ȥ=1 ȥ=1.2 ȥ=1.4 1971Q1-1984Q4 1985Q1-2007Q2, shifted downward by 30 units 110 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Figure 4.6: Contour chart of log-likelihood in (ψ, τ) space: 1971Q1-1984Q4 ȥ =0 .0 11 ȥ =0 .0 5 ȥ =0 .1 ȥ =0 .2 ȥ =0 .4 ȥ =0 .6 ȥ =0 .8 ȥ =1 ȥ =1 .4 IJ=0.015 IJ=0.03 IJ=0.05 IJ=0.1 IJ=0.15 Figure 4.6: Contour chart of log-likelihood in (ȥ, IJ)  space; Sample 1971Q1-1984Q4 -362.5--360 -360--357.5 -357.5--355 -355--352.5 111 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Figure 4.7: Contour chart of log-likelihood in (ψ, τ) space: 1985Q1-2007Q2 ȥ =0 .0 11 ȥ =0 .0 5 ȥ =0 .1 ȥ =0 .2 ȥ =0 .4 ȥ =0 .6 ȥ =0 .8 ȥ =1 ȥ =1 .4 IJ=0.015 IJ=0.03 IJ=0.05 IJ=0.1 IJ=0.15 Figure 4.7: Contour chart of log-likelihood in (ȥ, IJ)  space; Sample 1985Q1-2007Q2 -333--330.5 -330.5--328 -328--325.5 -325.5--323 112 Chapter 4. Is there a Reduction in the Supply Elasticity of New Housing. . . Figure 4.8: Maximized log posterior function at different value of ψ Figure 4.8: maximized log posterior function at different value of ȥ -365 -362.5 -360 -357.5 -355 -352.5 -350 ȥ=0.011 ȥ=0.05 ȥ=0.1 ȥ=0.2 ȥ=0.4 ȥ=0.6 ȥ=0.8 ȥ=1 ȥ=1.4 1971Q1-1984Q4 1985Q1-2007Q2, shifted downward by 30 units 113 Bibliography Abowd, J. M. and Farber, H. S. (1982). Job queues and the union status of workers. Industrial and Labor Relations Review, 35 (3), 354–367, available at http://ideas.repec.org/a/ilr/articl/v35y1982i3p354-367.html. Allmon, E., Haas, C. T., Borcherding, J. D. and Goodrum, P. M. (2000). 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Housing, Credit Constraints, and Macro Stabil- ity: The Secondary Mortgage Market and Reduced Cyclicality of Res- idential Investment. Tech. Rep. 298, Fisher Center Working Papers, http://repositories.cdlib.org/iber/fcreue/fcwp/298. Pescatori, A. and Mendicino, C. (2005). Credit Fric- tions, Housing Prices and Optimal Monetary Policy Rules. Money Macro and Finance (MMF) Research Group Conference 2005 67, Money Macro and Finance Research Group, available at http://ideas.repec.org/p/mmf/mmfc05/67.html. Poterba, J. M. (1984). Tax subsidies to owner-occupied housing: An asset-market approach. The Quarterly Journal of Economics, 99 (4), 729–52, available at http://ideas.repec.org/a/tpr/qjecon/v99y1984i4p729- 52.html. Rudebusch, G. and Wu, T. (2004). A macro-finance model of the term structure, monetary policy, and the economy. Proceedings, (Mar), available at http://ideas.repec.org/a/fip/fedfpr/y2004imarx8.html. Rudebusch, G. D., Sack, B. P. and Swanson, E. T. (2006). 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Tech. rep. 119 Bibliography Thieblot, A. (2001). The fall and future of unionism in con- struction. Journal of Labor Research, 22 (2), 287–306, available at http://ideas.repec.org/a/tra/jlabre/v22y2001i2p287-306.html. Topel, R. H. and Rosen, S. (1988). Housing investment in the united states. Journal of Political Economy, 96 (4), 718–40, available at http://ideas.repec.org/a/ucp/jpolec/v96y1988i4p718-40.html. Trichet, J.-C. (2005). Asset price bubbles and monetary policy. Speech by Jean-Claude Trichet, Mas lecture, Singapore. 120 Appendix A Appendix for Chapter 2 A.1 Model Appendix A.1.1 Households’ optimality conditions Labor supply The marginal rate of substitution between consumption and hours worked can be defined as the ratio of the partial derivatives of the momentary util- ity function with respect to lt over the derivative with respect to ct. The consumption shock Λt drops out when taking the ratio, wt = (cb,t − λbcb,t−1)1−σc exp ( σc−1 1+σl l1+σlt ) lσlt (cb,t − λbcb,t−1)−σc cγb,tc−γt exp { σc−1 1+σl l1+σlt } = (cb,t − λbcb,t−1) c−γb,t cγt lσlt To reach the alternative expression for the labor market equilibrium in term of the Private Consumption Expenditure (E), we first need to define E et ≡ ct + vh ( ht ct )−γ ht where vh ( ht ct )−γ is the rental price of housing asset. Intuitively one should be able to express et as the quantity of the mixture goods, cb,t (defined in equ(2.1)), divided by the relative price of cb,t, or et = cb,t( cb,t ct )γ = c1−γb,t cγt It is straight forward to verify that the intuition is right because c1−γb,t c γ t = ct + vh ( ht ct )−γ ht once we expand cb,t according to equ(2.1) 121 Appendix A. Appendix for Chapter 2 Plugging et = c 1−γ b,t c γ t into the labor market equilibrium, or equ(2.4), gives us wt = ( 1− λb cb,t−1 cb,t ) etl σl t or wtltet = ( 1− λb cb,t−1cb,t ) l1+σlt , which in steady state becomes w∗l∗ e∗ 1 1− λb = l 1+σl∗ Linearizing the optimality conditions The linearized form of equ(2.1) is ĉbt = κcĉt + (1− κc) ĥt where κc is the steady state share of consumption in private consumption expenditure. The linearized labor supply equation, or equ(2.4), is ŵt = 1− γ (1− λb) 1− λb ĉbt + γĉt − λb 1− λb ĉbt−1 + σl l̂t The linearized Euler equation, or equ(2.2), is 1 = Et  (rt+1 + zχc,t)− σc ( 11−λb cb,t+1 − 1+λb1−λb cb,t + λb1−λb cb,t−1)+ γ (cb,t+1 − cb,t) −γ (ct+1 − ct) + (σc − 1) w∗l∗e∗ 11−λb ( l̂t+1 − l̂t )  The term (σc − 1) w∗l∗e∗ 11−λb ( l̂t+1 − l̂t ) approximates σc−11+σl ( l1+σlt+1 − l1+σlt ) once we take into account the steady state solution of the labor supply equation, or w∗l∗e∗ 1 1−λb = l 1+σl∗ .33. With terms collected, the linearized Euler equation is (rt+1 + zχc,t) = ( σc 1− λb − γ ) ĉbt+1 − ( σc 1 + λb 1− λb − γ ) ĉbt + σc λb 1− λb ĉbt−1 +γ (ĉt+1 − ĉt)− (σc − 1) w∗l∗ e∗ 1 1− λb ( l̂t+1 − l̂t ) 33More about the approximation: σc−1 1+σl ( l 1+σl t+1 − l1+σlt ) ' σc−1 1+σl l 1+σl∗ (1 + σl) ( l̂t+1 − l̂t ) . The RHS can be written as (σc − 1) w∗l∗e∗ 11−λb ( l̂t+1 − l̂t ) after canceling out 1 + σl and making use of the steady state solution w∗l∗ e∗ 1 1−λb = l 1+σl∗ of the labor supply equaiton. 122 Appendix A. Appendix for Chapter 2 A.1.2 Retailers’ optimality conditions For simplicity in notation, let’s use Gt+s to denote Πsl=1pi ιp t+l−1pi 1−ιp∗ . The derivative of the objective function with respect to P̃t is 0 = ∑ ξsp Λt+sΞt+s ΛtΞt 1 Pt+s [ Gt+syi,t+s + ( P̃tGt+s − Pt+s xt+s ) ∂yi,t+s ∂P̃t ] 0 = ∑ ξsp Λt+sΞt+s ΛtΞt 1 Pt+s [ Gt+syi,t+s − λf λf − 1 ( P̃tGt+s − Pt+s xt+s ) yi,t+s P̃t ] The square bracket can be rewritten as yi,t+s [ Gt+s − λfλf−1 ( P̃tGt+s − Pt+sxt+s ) 1 P̃t ] , and then yi,t+s 1λf−1 [ −Gt+s + λfxt+s Pt+s P̃t ] . If we plug this term back into the FOC, the equation becomes 0 = 1 λf − 1 ∑ ξsp Λt+sΞt+s ΛtΞt yi,t+s 1 Pt+s [ −Gt+s + λf xt+s Pt+s P̃t ] Now we can move 1Pt+s into the bracket and multiply both sides of the equation by (λ− 1) P̃t, to have 0 = ∑ ξspEt Λt+sΞt+s ΛtΞt yi,t+s [ P̃tGt+s Pt+s − λf xt+s ] or P̃t = λf ∑∞ s=0 ξ s pEt Λt+sΞt+s ΛtΞt yi,t+s 1 xt+s∑∞ s=0 ξ s pEt Λt+sΞt+s ΛtΞt yi,t+s Gt+s Pt+s We now can substitute yi,t+s with ( Pt+s P̃tGt+s ) λf λf−1 yt+s to get P̃t = λf ∑∞ s=0 ξ s pEt Λt+sΞt+s ΛtΞt ( Pt+s Gt+s ) λf λf−1 yt+s 1 xt+s∑∞ s=0 ξ s pEt Λt+sΞt+s ΛtΞt ( Pt+s Gt+s ) λf λf−1 −1 yt+s The only remaining thing to do is to divide both sides by Pt to get equ(2.6). A.1.3 Phillips curve First linearize the optimal price setting condition, or equ(2.6). In the process we can cancel out Pt from the RHS of the equation, and cancel out terms 123 Appendix A. Appendix for Chapter 2 that exist in both the numerator and the denominator of the RHS. The result is ̂̃ P t = (1− ξpβ)Et ∞∑ s=0 ξspβ s ( P̂t+s − Ĝt+s − xt+s ) Moving this equation one period forward gives us Et ̂̃ P t+1 = (1− ξpβ)Et ∞∑ s=1 ξs−1p β s−1 [ P̂t+s − ( Ĝt+s − Ĝt+1 ) − xt+s ] = (1− ξpβ)Et ∞∑ s=1 ξs−1p β s−1Ĝt+1 + (1− ξpβ)Et ∞∑ s=1 ξs−1p β s−1 ( P̂t+s − Ĝt+s − xt+s ) = Ĝt+1 + (1− ξpβ)Et ∞∑ s=1 ξs−1p β s−1 ( P̂t+s − Ĝt+s − xt+s ) = ιppit + (1− ξpβ)Et ∞∑ s=1 ξs−1p β s−1 ( P̂t+s − Ĝt+s − xt+s ) in the last equality ιppit substitute Ĝt+1 from G’s definition. Now we can combine the expression for ̂̃P t and that for Et ̂̃P t+1 to derivễ P t − ξpβEt ̂̃P t+1, ̂̃ P t − ξpβEt ̂̃P t+1 = (1− ξpβ)(P̂t − xt)− ξpβιppit The result of canceling out common term and setting Ĝt = 0 at time t. We now move on to linearize equ(2.7). First we divide the equation by Pt on both sides and substitute Pt−1 Pt with 1pit . The linearized version of the resulting equation is ̂̃ P t − P̂t = ξp1− ξp (pit − ιppit−1) Finally, we need to combine the linearized equ(2.6) and the linearized equ(2.7). It takes some tedious rearrangement, but the result is just equ(2.37) in the text. 124 Appendix A. Appendix for Chapter 2 A.2 Calibration 1). β is set at 0.982, implying a real net interest rate of 7% per year. 2). δk is set at 10% per year. 3). δh is set at 1.6% per year used in Davis and Heathcote (2005). 4). α the share of capital is set at 0.3 following usual calibration conven- tion; x, the monopolistic markup is set at 1.05 following Iacoviello (2005b). 5). mh and mk are chosen so that in the steady state, the model fea- tures 61% consumption to output ratio, 12% investment to output ratio, 5% residential investment to output ratio (The remainder 22% of the out- put is treated as government consumption), and 15% of housing rent in total private consumption expenditure. A.3 Kalman Filter and Likelihood Function This section starts from a linear state space model that approximates the solution of the DSGE model around a unique stable rational expectations equilibrium. The objective is to illustrate how to construct a likelihood function when given a sample of observed endogenous variables and a set of parameters of the DSGE model. Readers can find a more detailed de- scription of the process from Chapter 9 of Hansen and Sargent (2005). This section provides a compact treatment. The state-space representation expresses the state vector xt and the vec- tor of observable yt as linear functions of initial state and histories of or- thogonal shocks: xt+1 = Axt + Cwt+1 yt = Gxt + vt where wt+1 ∼ iidN (0, In) and vt ∼ iidN (0, R). w represents structural shocks, and v is interpreted as the measurement errors. It is assumed that Ewt+1v ′ s = 0 for all t and s. The matrix A, C, G and R are constructed out of the parameters of the model, let’s call it θ. The measurement error vt is assumed to be zero is this paper. For the purpose of estimation, the representation is not useful because these orthogonal shocks are not observable. We shall use the Kalman filter to obtain the “whitening filter” that is cast in terms of the observed variables and the parameters of the model. It is by using this whitening filter that we can calculate recursively a Gaussian likelihood function. 125 Appendix A. Appendix for Chapter 2 The whitening filter is described with at = yt −GEt−1xt Etxt+1 = AEt−1xt +AΣtG′(GΣtG′ +R)−1at Σt+1 = AΣtA′ + CC ′ −AΣtG′(GΣtG′ +R)−1GΣtA′ Vt = GΣtG′ +R where Kt = AΣtG′(GΣtG′ + R)−1 is the Kalman gain matrix, constructed as the parameters of the state space model and the matrix Σt. The latter is initiated from some matrix Σ0 (more discussion follows later), and is updated according to the third equation.34 From Σt we can calculate Vt, the covariance matrix of prediction error. The filter accepts the data {yt} as the input and produces {at} as the output. {at} is the the one-step ahead prediction error of yt. The process from {yt} to {at} is called a “whitening” process, because {yt} is serially correlated and the {at} is serially un-correlated, or “white”. {at} can be said to be a fundamental white noise for the {yt} process. To construct the likelihood function for a sample of observations {yt}Tt=1, we need to feed {yt}Tt=1 into the filter described above with some initial E0x1 and Σ0. The output is a series of prediction error {at}Tt=1, and a series of matrix {Vt}Tt=1 that describe the conditional variance covariance of {at}Tt=1. The log-likelihood function of {yt}Tt=1 given model’s parameter is then lnL ( θ|yT ) = −n 2 ln (2pi)− 1 2 T∑ t=1 ln (|Vt|)− 12 T∑ t=1 atV −1 t a ′ t To initiate the filter I use the unconditional variance of xt as Σ0 and the unconditional mean of xt as E0x1. 34The matrix Σt+1 can be defined as the second moment of xt+1 − Etxt+1 xt+1 − Etxt+1 = Aoxt + Cwt+1 − [AoEt−1xt +Kt (Gxt + vt −GEt−1xt)] = (Ao −KtG) (xt − Et−1xt) + Cwt+1 −Ktvt the second moment of xt+1 − Etxt+1 is Σt+1 = (A o −KtG) Σt (Ao −KtG)′ + CC′ +KtRK′t = AoΣtA o′ +KtGΣt (KtG) ′ −AoΣt (KtG)′ −KtGΣtAo′ +CC′ +KtRK ′ t which can be rearranged into the updating rule after substituting in the Kalman gain matrix. 126 Appendix A. Appendix for Chapter 2 A.4 Bayesian Estimation and RWM Algorithm Bayesian estimation maximizes an objective function that is defined as the sum of the log of the data likelihood and the log prior density. The set of the parameter that maximizes the objective function is the mode of the posterior distribution of the parameters. To explore the posterior distribution around the neighborhood of the mode, I use the Random Walk Metropolis (RWM) algorithm. More details can be found in An and Schorfheide (2006). A.4.1 Maximizing posterior density to estimate the mode Denoting the prior density by p (θ), the likelihood function by L ( θ|yT ), the posterior density by p ( θ|yT ), we have p ( θ|yT ) = L (θ|yT ) p (θ) p (yT ) where p ( yT ) = ∫ θ L ( θ|yT ) p (θ) dθ is the marginal data density. Because p ( yT ) is independent of θ, it is sufficient, for the purpose of maximizing posterior density, to just maximize the log of the non-normalized posterior density L ( θ|yT ) p (θ), or max θ ln ( L ( θ|yT ))+ ln (p (θ)) I use the maximization routine “cswell.m” written by Chris Sims for the maximum likelihood estimation of a DSGE model. The algorithm uses a fairly simple line search and randomly perturbs the search direction if it reaches a cliff caused by non-existence or non-uniqueness of the DSGE solutions. The latest version is available for download in Sims’ website. Following An and Schorfheide (2006), I use parameter transformation to turn constraint maximization into unconstrained maximization in the search for mode. The transformation is NOT used in the RWM algorithm that will be described shortly, where the out-of-range draws are discarded instead. A.4.2 Use RWM algorithm to explore the distribution around the estimated mode RWM algorithm explores the posterior distribution of the model parameters in the neighborhood of the mode θ̃. Step 1: Numerically maximize the non-normalized posterior density with respect to parameter vector θ to obtain the posterior mode θ̃. 127 Appendix A. Appendix for Chapter 2 Step 2: Compute Σ̃, the negative of the inverse of the Hessian computed at the mode. Step 3: Draw an initial parameter vector θ0 from N ( θ̃, c20Σ̃ ) . Step 4: For s = 1, ...., Nsim, draw ϑ from the distributionN ( θ(s−1), c2Σ̃ ) , and jump from θ(s−1) to θ(s) with the following rule: θ(s) = ϑ with proba- bility min { 1, r ( θ(s−1), ϑ|yT )}, and θ(s) = θ(s−1) otherwise. The acceptance probability r ( θ(s−1), ϑ ) is r ( θ(s−1), ϑ ) = L ( ϑ|yT ) p (ϑ) L ( θ(s−1)|yT ) p (θ(s−1)) Step 5: Use the sample statistics from { θ(s) } s to approximate the poste- rior distribution such as the mean and the percentile. One can also approx- imate the posterior expected value of a parameter transformation h(θ) by 1 Nsim ∑Nsim s=1 h(θ s). Examples of h(θs) include the variance decomposition of prediction errors. In this paper the step sizes c0 = 0.3, and c is chosen to produce an acceptance rate of about 25%. A.5 Data Real output and real expenditures are defined as the nominal dollar value divided by a single deflator that is the implicit price index of GDP. All aggregate values are divided by US population over age of 16. Inflation is defined as the quarterly growth rate of the implicit GDP price. Interest rate is the effective federal funds rate. Total output is defined as nominal GDP net of the value of housing service. The value of housing service is available in BEA’s Table 2.3.5. Non-housing consumption expenditure is defined as the total Private Consumption Expenditure net of the value of housing service. 128 Appendix B Appendix for Chapter 3 B.1 Data The price index of new homes is the Price Deflator (Fisher) Index of New, One-Family Houses Under Construction, from the Survey of Construction, The survey collects information on the physical characteristics and prices of new, one-family houses. The value of land is subtracted from the sales price at a fixed proportion of about 11%. The price index is the major component of the price index of residential investment of the NIPA. In fact, the two series (price for residential invest- ment in the NIPA, and the Price Deflator of the Census Bureau for single- family houses) have a correlation of 0.94, in terms of quarter-to-quarter growth rate. Thus, they are hardly distinguishable. The quantity index of residential investment is the chain type quantity index of residential investment from the NIPA of the BEA. The consumption series is the total Private Consumption Expenditure from the NIPA. The real interest rate is the difference between the effective federal funds rate and the growth rate of the GDP deflator. B.2 Calibrated parameters The depreciation rate of houses is calibrated from the BEA fixed asset ta- bles (9.1, 9.2, and 9.3). The three tables show the Real Net Stock, Real Depreciation, and Real Investment of private residential fixed assets. These figures are available from 1990 on. The average yearly depreciation rate is calculated as a ratio of depreciation over stock, and is 1.59% for residential fixed asset. The steady-state real interest rate (2.4% yearly rate) and the steady-state growth rate of consumption per capita (2% yearly) are averages from the data. The population growth rate is set at 1.2% per year; and the average growth rate of the population (aged 16 and over) is for the US between 1984 129 Appendix B. Appendix for Chapter 3 and 2007. The exogenous growth rate of sector-specific technology (γx) is calibrated to produce a 1.2% annual growth rate in the real house price for the estimated slope of the marginal cost curve ψ. The alternative of setting the exogenous growth rate to zero does not change the result in a noticeably different way. B.3 Deriving first order conditions for the inventory model Let qb,t = qg,t 1+Φ2 ( dt ht ) , households’ optimal level of search effort satisfies the following condition (qb,t − qh,t) e− bt vt = cqc,t (B.1) The optimal condition, and equ(3.9) that describes the accumulation of inventory, are the two constraints in the profit maximization of the retail arm of residential builder. The Lagrangian of residential builder’s optimization problem is £ = max (dt,vt+1,qh,t,bt) Et ∞∑ s=0 qh,t+svt+s [ 1− e− bt+s vt+s ] − qc,t+sdt+s Πs (1 + rt+1+s)  + ∞∑ s=0 λ1,t+s vt+se− bt+svt+s + dt+s − vt+1+s Πs (1 + rt+1+s)  + ∞∑ s=0 λ2,t+s (qb,t+s − qh,t+s) e− bt+svt+s − cqc,t+s Πs (1 + rt+1+s)  130 Appendix B. Appendix for Chapter 3 The first order conditions are ∂£ ∂dt = 0 = −qt + λ1,t = 0 ∂£ ∂vt+1 = 0 = qh,t+1 1 + rt+1 [ 1− e− bt+1 vt+1 − e− bt+1 vt+1 bt+1 vt+1 ] − λ1,t + λ1,t+1 1 + rt+1 ( e − bt+1 vt+1 + e − bt+1 vt+1 bt+1 vt+1 ) + λ2,t+1 1 + rt+1 (qb,t+1 − qh,t+1) e− bt+1 vt+1 vt+1 bt+1 vt+1  ∂£ ∂qh,t = 0 = vt ( 1− e− bt vt ) − λ2,te− bt vt ∂£ ∂bt = 0 = (qh,t − λ1,t) e− bt vt − (qb,t − qh,t)λ2,t e − bt vt vt From ∂£∂dt = 0, λ1,t = qc,t From ∂£∂qh,t = 0, λ2,t e − bt vt vt = ( 1− e− bt vt ) which together with ∂£∂bt = 0 determines that (qh,t − qc,t) e− bt vt = ( 1− e− bt vt ) (qb,t − qh,t) (B.2) The equation has an alternative expression of qb,t−qh,t = e− bt vt (qb,t − qct). It states that the net gain from the trade, defined as qb,t−qh,t , is split between developer and households. The share that goes to household, or qb,t − qh,t, is e− bt vt . The share is larger when inventory-to-sales is high. The expression for ∂£∂vt+1 = 0 can be simplified substantially, if we notice that the last term of the expression can be written as 11+rt+1 (qh,t+1 − qc,t+1) e − bt+1 vt+1 bt+1 vt+1 using the expression for ∂£∂qh,t = 0 as well as equ(B.2). The simpler expres- sion can then be plugged into ∂£∂vt+1 = 0 to cancel out all terms that contains 131 Appendix B. Appendix for Chapter 3 e − bt+1 vt+1 bt+1 vt+1 . The resulted equation is qc,t (1 + rt+1) = qh,t+1 ( 1− e− bt+1 vt+1 ) + qc,t+1 ( e − bt+1 vt+1 ) (B.3) which is after substituting λ1,t away with qc,t. qc,t (1 + rt+1) = (qh,t+1 − qc,t+1) ( 1− e− bt+1 vt+1 ) + qc,t+1 = mt+1qc,t+1 ( 1− e− bt+1 vt+1 ) + qc,t+1, where mt+1 = qh,t+1 − qc,t+1 qc,t+1 (1 + rt+1) = [ mt+1 ( 1− e− bt+1 vt+1 ) + 1 ] qc,t+1 qc,t After solving Equ(B.2) and Equ (B.1) for qh,t and qb,t as a function of qc,t and e − bt vt , we can derive the equations in the text including equ(3.12). 132 Appendix C Appendix for Chapter 4 C.1 Data The price index of new homes is the Price Deflator (Fisher) Index of New, One-Family Houses Under Construction, from the Survey of Construction, The survey collects information on the physical characteristics and prices of new, one-family houses. The value of land is subtracted from the sales price at a fixed proportion of about 11%. The price index is the major component of the price index of residential investment of the NIPA. In fact, the two series (price for residential invest- ment in the NIPA, and the Price Deflator of the Census Bureau for single- family houses) have a correlation of 0.94, in terms of quarter-to-quarter growth rate. Thus, they are hardly distinguishable. The quantity index of residential investment is the chain type quantity index of residential investment from the NIPA of the BEA. The consumption series is the total Private Consumption Expenditure from the NIPA. The real interest rate is the difference between the effective federal funds rate and the growth rate of the GDP deflator. All nominal series are turned into real series with the GDP price index as the deflator. C.2 Calibrated parameters The depreciation rate of houses is calibrated from the BEA fixed asset tables (9.1, 9.2, and 9.3). The three tables show the Real Net Stock, Real Depre- ciation, and Real Investment of private residential fixed assets. The average yearly depreciation rate is calculated as a ratio of depreciation over stock. The result is 1.59% for residential fixed asset. The steady-state real interest rate (2.4% yearly rate) and the steady-state growth rate of consumption (3.6% yearly) are averages from the data. 133

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