Essays on Household Preference andDynamic DecisionsbyTsenguun EnkhbaatarB.A., Osaka University, 2012M.P.P., The University of Tokyo, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2020© Tsenguun Enkhbaatar 2020The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation titled:Essays on Household Preference and Dynamic Decisionssubmitted by Tsenguun Enkhbaatarin partial fulfillment of the requirements for the degree of Doctor of Philosophy in Economics.Examining Committee:Hiroyuki Kasahara, Professor, Economics, UBCSupervisorPaul Schrimpf, Associate Professor, Economics, UBCSupervisory Committee MemberFlorian Hoffmann, Associate Professor, Economics, UBCSupervisory Committee MemberGiovanni Gallipoli, Professor, Economics, UBCUniversity ExaminerThomas Davidoff, Associate Professor, Business Administration, UBCUniversity ExaminerArvind Magesan, University of CalgaryExternal ExamineriiAbstractThis study examines the identification of household preference from a micro-panel dataset andthe effects of the stock market collapse and the Zero-Lower Bound monetary policy on householdconsumption decisions during the 2007-2008 global financial crisis (GFC).In the first chapter, I propose a procedure for estimating the household utility function from amicro-panel dataset using the inter-temporal Euler equation. Here, I demonstrate that the house-hold utility function can be estimated accurately from a micro-panel dataset using the intertemporalEuler equation by taking into account the differences in the portfolio compositions of householdsavings across households. In addition, I construct a new household dynamic model in which thehousehold stock-holding behavior is modeled by explicitly taking into account the hidden stockmarket participation cost pointed out in recent empirical studies as a potential explanation for thelow stock market participation rate among households in the United States.In the second chapter, I explore the effects of the stock market collapse and the Zero-LowerBound monetary policy on household consumption decisions during the GFC. The huge drop in thestock market return and the decline in the risk-free rate due to the Zero-Lower Bound monetarypolicy triggered a large percentage decrease in the consumption of wealthy households in this periodin the United States. Meanwhile, the consumption of households at the bottom part of the wealthdistribution, which generally do not participate in the stock market, increased slightly in percentagedue to the decline in the risk-free rate during the GFC in the United States. As a result, the stockmarket collapse and the Zero-Lower Bound monetary policy generated a substantial decrease inconsumption inequality among households.In the third chapter, I estimate the heterogeneity in household preference by employing a com-bination of the extremum and nonparametric estimation methods and find significant heterogeneityin the preferences across households in the United States and Italy. This estimated heterogeneityin household preference can be used to explain the observed wealth distribution and the differencesin career and investment choices across households in future research.iiiLay SummaryMy dissertation consists of three chapters, where I examine the identification of household preferencefrom a micro-panel dataset and the effects of the stock market collapse and the Zero-Lower Boundmonetary policy during the 2007-2008 global financial crisis (GFC) on household consumptiondecisions in the United States. In the first chapter, I show that the household utility functioncan be estimated accurately from a micro-panel dataset using the inter-temporal Euler equationby explicitly taking into account the portfolio composition of household savings as an endogenoushousehold choice in the household dynamic model. In the second chapter, I argue that the stockmarket collapse and the Zero-Lower Bound monetary policy were significant driving forces thattriggered a substantial decrease in consumption inequality among households in the United Statesduring the GFC. In the third chapter, I discuss the heterogeneity in household preference byemploying a combination of the extremum and nonparametric estimation methods and find thesignificant heterogeneity in preferences across households in the United States and Italy.ivPrefaceChapters 2, 3, and 4 of this thesis are my original, unpublished, and independent work.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Estimating Household Preference and Stock Market Participation Cost . . . . 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Stock Market Return Process . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 The Household Income Process . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 The Household Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . . . 112.2.5 The Quasi-Hyperbolic Discounting Preference . . . . . . . . . . . . . . . . . 112.3 Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Nonparametric Estimation of Household Consumption and Conditional ChoiceProbabilities of the Stock-Holding Share . . . . . . . . . . . . . . . . . . . . 132.3.2 Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . . . 142.3.3 The Quasi-Hyperbolic Discounting Preference . . . . . . . . . . . . . . . . . 152.3.4 Stock Market Participation Cost and Stock-Holding Share Adjustment Cost 152.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . . . 162.4.2 The Quasi-Hyperbolic Discounting Preference . . . . . . . . . . . . . . . . . 172.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19vi2.6 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 The Effect of the Stock Market Collapse on Household Consumption during the2007-2008 Global Financial Crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Stock Market Collapse and the Zero-Lower Bound Monetary Policy . . . . . . . . . 233.2.1 Stock Market Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Zero Lower Bound Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Household Stock-Holding Behavior . . . . . . . . . . . . . . . . . . . . . . . 243.3 Estimation of the Household Dynamic Model . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Stock Market Return Process and Household Income Process . . . . . . . . . 263.3.2 Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . . . 263.3.3 The Quasi-Hyperbolic Discounting Preference . . . . . . . . . . . . . . . . . 273.3.4 The Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 The Effects of the Stock Market Collapse and the Zero-Lower Bound Monetary Policyon Household Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.1 The Effect of the Stock Market Collapse on Household Consumption . . . . 303.4.2 The Effect of the Increase in the Expected Future Volatility of Stock MarketReturns on Household Consumption . . . . . . . . . . . . . . . . . . . . . . . 313.4.3 The Effect of the Increase in the Expected Future Volatility of Stock MarketReturns on Households’ Stock Holding . . . . . . . . . . . . . . . . . . . . . 333.4.4 The Effect of the Drop in the Risk-Free Rate on Household Consumption andStock Holding Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4.5 The Effects of the Stock Market Collapse and the Zero-Lower Bound Mone-tary Policy with the Quasi-Hyperbolic Discounting Preference . . . . . . . . 363.4.6 The Impact of the Stock Market Collapse and the Zero-Lower Bound Mone-tary Policy on the Household Consumption Distribution . . . . . . . . . . . 383.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.6 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.7 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Heterogeneous Household Preference . . . . . . . . . . . . . . . . . . . . . . . . . . 584.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Model and Estimation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.2 Estimation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Data Description and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.1 Data description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2 Data Summary and Parameter Specification . . . . . . . . . . . . . . . . . . 644.4 The Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65vii4.4.1 Estimation Result for the Survey on Household Income and Wealth Dataset 664.4.2 Estimation Result for the Panel Study of Income Dynamics Dataset . . . . . 674.4.3 Comparison to the Existing Literature . . . . . . . . . . . . . . . . . . . . . 684.4.4 The Interpretation of the Result . . . . . . . . . . . . . . . . . . . . . . . . . 694.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.6 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.7 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91AppendicesA Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.1 The Definitions of the Household Stock Holding Share and the Household PortfolioReturn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.2 The Derivations of the Inter-Temporal Euler Equations . . . . . . . . . . . . . . . . 96A.2.1 The Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . 96A.2.2 The Quasi-Hyperbolic Discounting Preference . . . . . . . . . . . . . . . . . 97A.3 The Detail of the Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 98A.3.1 The Estimation of the Utility Function . . . . . . . . . . . . . . . . . . . . . 98A.3.2 The Estimation of the Full Model . . . . . . . . . . . . . . . . . . . . . . . . 100A.3.3 The Bootstrapping for the Standard Error and the Bias Correction . . . . . 101B Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103B.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103B.1.1 Household Consumption Data . . . . . . . . . . . . . . . . . . . . . . . . . . 103B.1.2 Household Wealth Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104B.1.3 The Distribution of the Stock-Holding Share . . . . . . . . . . . . . . . . . . 105B.1.4 The Stock Market Return and the Risk-Free Rate Data . . . . . . . . . . . . 106B.2 The Estimation Result for the Constant Absolute Risk Aversion Utility Function . 106viiiList of Tables2.1 The Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . 202.2 The Quasi-Hyperbolic Discounting Preference . . . . . . . . . . . . . . . . . 202.3 The Estimated Curvature Parameter of the Constant Relative Risk Aver-sion Utility Function without the Portfolio Composition . . . . . . . . . . . 202.4 The Estimated Curvature Parameter of the Constant Relative Risk Aver-sion Utility Function with the Portfolio Composition . . . . . . . . . . . . . 213.1 The Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . 403.2 The Quasi-Hyperbolic Discounting Preference . . . . . . . . . . . . . . . . . 413.3 The Relative Change in Average Consumption of Households between2006 and 2008 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 The Relative Change in Average Consumption of Households between2006 and 2008 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.5 The Distribution of Stock Holding Share among Stock Holding House-holds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.6 The Stock Market Participation Rate: 2006 versus 2008 . . . . . . . . . . 423.7 The Stock Market Participation Rate: 2006 versus 2008 . . . . . . . . . . 423.8 The Average Effect of the Stock Market Collapse on Consumption . . . 423.9 The Median Effect of the Stock Market Collapse on Stock Holding House-hold’s Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.10 The Median Effect of the Stock Market Collapse on Consumption . . . 433.11 The Median Contributions of the Stock Market Return Drop and the Ex-pected Future Volatility Increase to the Change in Consumption between2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.12 The Median Contributions of the Stock Market Return Drop and the Ex-pected Future Volatility Increase to the Change in Consumption between2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.13 The Median Contributions of the Stock Market Return Drop and the Ex-pected Future Volatility Increase to the Change in Consumption between2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44ix3.14 The Median Contributions of the Stock Market Return Drop and theExpected Future Volatility Increase to the Change in Stock Market Par-ticipation Rate between 2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . 443.15 The Median Contributions of the Stock Market Return Drop and the Ex-pected Future Volatility Increase to the Change in Distribution of StockHolding Share between 2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . 453.16 The Average Effect of the Drop in the Risk-Free Rate on HouseholdConsumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.17 The Median Effect of the Drop in the Risk-Free Rate on the Stock MarketParticipation Rate between 2006 and 2008 . . . . . . . . . . . . . . . . . . . . 453.18 The Median Effect of the Drop in the Risk-Free Rate between 2006 and2008 on Household Consumption . . . . . . . . . . . . . . . . . . . . . . . . . 463.19 The Average Effect of the Drop in the Risk-Free Rate between 2006 and2008 on Household Consumption . . . . . . . . . . . . . . . . . . . . . . . . . 463.20 The Average Effect of the Drop in the Risk-Free Rate on Consumptionof Household with Low Wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.21 The Average Effect of the Stock Market Collapse on Household Con-sumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.22 The Median Effect of the Stock Market Collapse on Stock Holding House-hold’s Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.23 The Median Contributions of the Stock Market Return Drop and the Ex-pected Future Volatility Increase to the Change in Consumption between2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.24 The Median Contributions of the Stock Market Return Drop and the Ex-pected Future Volatility Increase to the Change in Consumption between2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.25 The Median Contributions of the Stock Market Return Drop and the Ex-pected Future Volatility Increase to the Change in Consumption between2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.26 The Median Contributions of the Stock Market Return Drop and the Ex-pected Future Volatility Increase to the Change in Consumption between2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.27 The Median Contributions of the Stock Market Return Drop and theExpected Future Volatility Increase to the Change in Stock Market Par-ticipation Rate between 2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . 493.28 The Median Contributions of the Stock Market Return Drop and the Ex-pected Future Volatility Increase to the Change in Distribution of StockHolding Share between 2006 and 2008 . . . . . . . . . . . . . . . . . . . . . . 49x3.29 The Median Effect of the Drop in the Risk-Free Rate on the Stock MarketParticipation Rate between 2006 and 2008 . . . . . . . . . . . . . . . . . . . . 493.30 The Median Effect of the Drop in the Risk-Free Rate on the Stock MarketParticipation Rate between 2006 and 2008 . . . . . . . . . . . . . . . . . . . . 503.31 The Average Effect of the Drop in the Risk-Free Rate on HouseholdConsumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.32 The Median Effect of the Drop in the Risk-Free Rate on HouseholdConsumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.33 The Median Effect of the Drop in the Risk-Free Rate on Consumptionof Household with Low Wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.34 The Relative Change of Average Consumption in Different PercentileIntervals of the Household Consumption Distribution . . . . . . . . . . . . 513.35 The Relative Change of Average Consumption in Different PercentileIntervals of the Household Consumption Distribution . . . . . . . . . . . . 524.1 The Survey on Household Income and Wealth Dataset and the PanelStudy of Income Dynamics Dataset: The Summary of the ConsumptionRatio Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 The Survey on Household Income and Wealth Dataset and the PanelStudy of Income Dynamics Dataset: The Summary of the Wealth Data . 73B.1 The Distribution of the Stock Holding Share among Households from thePanel Study of Income Dynamics Dataset . . . . . . . . . . . . . . . . . . . . . 105B.2 The Estimation Result: The Constant Absolute Risk Aversion UtilityFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107xiList of Figures3.1 The Annualized Stock Market Return . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 The Expected Stock Market Return . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 The Expected Stock Market Return Volatility . . . . . . . . . . . . . . . . . . . . . . 543.4 The Daily S&P500 Stock Price Index . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 The Nominal Federal Funds and 3-Month Treasury Bill Rates . . . . . . . . . . . . . 553.6 The 3-Month Treasury Bill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.7 The Relative Change in Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.8 The Relative Change in Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.9 The Drop in the Stock Market Return versus the Drop in the Risk-Free Rate . . . . . . . . 574.1 The Empirical Distribution of the Consumption Ratio from the Survey on HouseholdIncome and Wealth Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 The Empirical Distribution of the Consumption Ratio from the Panel Study of In-come Dynamics Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 The Empirical Distribution of the Household Wealth from the Survey on HouseholdIncome and Wealth Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 The Empirical Distribution of the Household Wealth from the Panel Study of IncomeDynamics Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.5 The Annual Real Interest Rate Data for the U.S. . . . . . . . . . . . . . . . . . . . . 774.6 The Annual Real Interest Rate Data for Italy . . . . . . . . . . . . . . . . . . . . . . 784.7 The Annual Inflation Rate Data for the U.S. . . . . . . . . . . . . . . . . . . . . . . 794.8 The Annual Inflation Rate Data for Italy . . . . . . . . . . . . . . . . . . . . . . . . 804.9 The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . 814.10 The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . 824.11 The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . 834.12 The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . 844.13 The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . 85xii4.14 The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . 864.15 The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . 874.16 The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility Function . . . . . . . . . . . . . 88B.1 Histogram for The Household Consumption Data from the PSID Dataset (CPI-adjusted to the 1998 USD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104B.2 Histogram for The Household Wealth Data from the SCF Dataset (2008 Wave) . . . 105xiiiAcknowledgementsFirst and foremost, I want to thank my supervisor, professor Hiroyuki Kasahara. He taught me alot and guided me through the hardship of the doctoral study, from the day I took his PhD-leveleconometrics course. This dissertation would not be possible without his advice and support. Hegenuinely cared about my future and helped me significantly to determine my career. Therefore, Iwould like to express my deepest gratitude to professor Hiroyuki Kasahara in this acknowledgement.I am also grateful for my committee members, professor Paul Schrimpf and professor FlorianHoffmann, for their continuous guidance and support. Their feedback and advice helped me tocomplete this thesis to a great extent. Furthermore, I want to thank my second year paper ad-visor, professor Vadim Marmer, for his great supervision. Finally, I would like to thank professorKyungchul Song, professor Paul Beaudry, professor Michael Devereux, professor Giovanni Gallipoli,and the other professors at the Vancouver School of Economics, the University of British Columbia,who attended my presentations and provided me with their feedback and advice.I would also like to thank my parents, Tseveen Enkhbaatar and Otgon Tserennadmid, forraising me with full love and caring for my well-being most. Moreover, I want to thank my twokids, Oyungerel Tsenguun and Orchlon Tsenguun, for their emotional supports and being the lightof my life. Finally, I would like to express my sincerest gratitude to my wife, MyagmarsurenBudaasuren, from the bottom of my heart. She backed me up all the way through my doctoralstudy. Without her strength and support, this thesis would have been impossible.xivChapter 1IntroductionThis thesis discusses the identification of household preferences from a micro-panel dataset andthe effects of the stock market collapse and the Zero-Lower Bound monetary policy on householdconsumption decisions during the 2007-2008 global financial crisis (GFC). In particular, in Chapter2, I propose a new procedure for estimating the household utility function from a micro-paneldataset using the intertemporal Euler equation and construct a novel household dynamic model inwhich the household stock-holding behavior is captured by explicitly taking into account the hiddenstock market participation cost. In Chapter 3, I explore the effects of the stock market collapse andthe Zero-Lower Bound monetary policy on household consumption decisions during the GFC, usingthe estimation procedure developed in Chapter 2. Finally, in Chapter 4, I estimate the heterogeneityin household preference by employing a combination of the extremum and nonparametric estimationmethods and find significant heterogeneity in preferences across households in the United Statesand Italy.An accurate estimation of a household utility function is vital for properly conducting coun-terfactual analyses and evaluating the effects of macroeconomic shocks and policy interventionson household well-being. A household utility function is typically estimated using the calibrationmethod in macroeconomics literature. However, the estimation result of the calibration methodis heavily influenced by the calibration target statistics selected by researchers. Furthermore, themajority of studies on household utility function estimation do not provide simulation analyses todemonstrate how well the estimation procedures based on the calibration method capture the trueparameter of a household utility function. Researchers have attempted to estimate a householdutility function from a micro-panel dataset using the intertemporal Euler equation as the momentcondition. The log-linearized version of the intertemporal Euler equation is often employed for es-timating the household utility from a micro-panel dataset, owing to the simplicity of its estimationprocedure. However, Dixon (2001) demonstrates that the approximation bias presents a substantialissue in the estimation of the constant relative risk aversion (CRRA) utility function from a micro-panel dataset using the log-linearized version of the intertemporal Euler equation. In addition,Dixon (2001) shows that the use of instrumental variables and the second-order approximation ofthe intertemporal Euler equation do not eliminate this approximation bias problem. Moreover,Dixon (2001) notes that the generalized method of moments methodology does not work well forestimating a household utility function using the intertemporal Euler equation as the moment con-dition when the household consumption data have significant measurement errors. In addition,researchers point out that the time dimensions of the existing household micro-panel datasets in1the United States are not sufficient for identifying a household utility function using the intertem-poral Euler equation estimation methodology. In fact, Attanasio and Low (2004) states that atime dimension of 40 or more years in a household micro-panel dataset is necessary for accuratelyestimating a household utility function from a micro-panel dataset using the said methodology.In Chapter 2 of this thesis, I avoid this approximation bias problem by not log-linearizing theintertemporal Euler equation. This imposes a more computationally intensive non-linear estima-tion procedure compared with one that is based on a log-linearized version of the intertemporalEuler equation. Furthermore, I solve the measurement error problem in the household consumptiondata by estimating a household consumption function in the first step of the estimation procedureusing the nonparametric estimation method while assuming that measurement errors in the house-hold consumption data are independent of the other observed household variables in a householdmicro-panel dataset. However, the main issue with the intertemporal Euler equation estimationmethodology arises from the lack of variation in the return on household savings caused by theshortness of time dimensions of available household micro-panel datasets in the United States. Itackle this issue by explicitly taking into account the portfolio composition of household savings.The heterogeneity in a portfolio composition of household savings across households creates anextra variation in the return on household savings. This extra variation enables researchers to es-timate accurately a household utility function from a micro-panel dataset using the intertemporalEuler equation. Furthermore, as one of the first in the household finance literature to study thistopic, this chapter succeeds in estimating the quasi-hyperbolic discounting (QHD) preference froma household micro-panel dataset using the intertemporal Euler equation as the moment condition.Empirical papers have pointed out the low stock market participation rate among households inthe United States. In fact, the Survey of Consumer Finances (hereinafter, SCF) and the Panel Studyof Income Dynamics (hereinafter, PSID) datasets indicate that approximately 30% of householdsparticipate in the stock market. Haliassos and Bertaut (1995) argues that this low level of thestock market participation rate among households cannot be explained by the heterogeneity ofbeliefs, habit persistence, and borrowing constraints. Furthermore, Haliassos and Bertaut (1995)states that households need to have an unrealistically high degree of risk aversion to be reluctantto participate in the stock market at this low level. The second interesting feature of householdstock-holding behavior is that a high level of persistence exists in the stock-holding shares of stock-holding households between the current and previous periods. In Chapter 2, I introduce the hiddenstock market participation and stock-holding share adjustment costs as the household disutilitiesin the household dynamic model to capture these features of the observed household stock-holdingbehavior, and estimate these costs using the estimated household utility function and the iterativemaximum likelihood estimation method from a micro-panel dataset.Hotz and Miller (1993) explores the estimation procedure for the dynamic discrete choice modelswith the finite dependence property (the dynamic discrete choice models with the terminal action) orthe optimal stopping problems. The estimation procedure for the models with the finite dependenceproperty is simpler because researchers do not need to obtain the full model solution to estimate2the model parameters. On the other hand, the finite dependence property does not hold for thehousehold dynamic model, developed in Chapter 2. Hence, the full model solution is neededto estimate the hidden stock market participation and stock-holding share adjustment costs inChapters 2 and 3 of this thesis. Moreover, Hotz, Miller, Sanders, and Smith (1994) developsthe simulation based estimation procedure for the dynamic discrete choice models. However, thisestimation procedure is relatively cumbersome to apply for the discrete-continuous dynamic choicemodels. The household dynamic model, constructed in Chapter 2, belongs to the family of thediscrete-continuous dynamic choice models. Furthermore, in Chapter 2, the household dynamicmodel is solved with the method similar to the method proposed in Iskhakov, Jørgensen, Rust, andSchjerning (2017). Nonetheless, the full model solution method for the household dynamic model,developed in Chapter 2, is more computationally intensive than the full model solution method,discussed in Iskhakov, Jørgensen, Rust, and Schjerning (2017), because the model, explored inIskhakov, Jørgensen, Rust, and Schjerning (2017), has fewer state variables, exogenous shocks, anddiscrete choices, compared with the household dynamic model, constructed in Chapter 2 of thisthesis.In Chapter 3, I evaluate the effects of the stock market collapse and the Zero-Lower Boundmonetary policy on household consumption decisions during the GFC using the methodology de-veloped in the first chapter from the PSID dataset. Existing literature, in general, has linked thechange in consumption distribution to the change in income distribution, even though householdincome inequality did not change much during the GFC. However, in this chapter, I show that thehuge decline in the stock market return and the drop in the risk-free rate due to the Zero-LowerBound monetary policy have triggered a substantial decrease in consumption inequality amonghouseholds in the United States. Furthermore, the second chapter demonstrates that the increasein the expected future volatility of the stock market return during the GFC almost did not affectthe consumption decisions of wealthy households. Meanwhile, the consumption decisions of house-holds at the bottom part of the wealth distribution were significantly affected by the increase inthe expected future volatility of the stock market return during the GFC.Furthermore, I acknowledge that the empirical analyses, conducted in Chapter 3, are largelyconstrained by the availability and completeness of the data I used. The PSID dataset, used inthe estimation of the household dynamic model, covers the time period between 1998 and 2014.Moreover, the household wealth data from the PSID dataset do not include the tax-deferred retire-ment savings account (401(K)). Since households can invest in the stock through this tax-deferredretirement savings account (401(K)), the absence of this tax-deferred retirement savings account(401(K)) is a serious issue. Meanwhile, the Survey of Consumer Finances (SCF) dataset, used inevaluating the effects of the stock market collapse and the Zero-Lower Bound monetary policy onhousehold decisions in Chapter 3, includes this tax-deferred retirement savings account (401(K)).Another issue, pointed out by researchers, is the inclusion of the credit card balance in the householdwealth from the PSID dataset. The household wealth data from the PSID dataset accommodatethe credit card balance explicitly. Hence, this issue is not huge regarding the PSID dataset. In ad-3dition, the household wealth data from the PSID dataset include the individual retirement account(IRA). The detailed description for the data, utilized in Chapter 3, can be found in the appendixof this thesis.In Chapter 4, I examine the heterogeneity in preferences across households by employing acombination of the extremum and nonparametric estimation methods. Furthermore, the thirdchapter shows that there is significant heterogeneity in household preference across householdsin the United States and Italy. This estimated heterogeneity in household preferences can beimportant for explaining the observed wealth distribution and the heterogeneity in household careerand investment choices in the United States. It can be useful in future research for exploringthe implications of the heterogeneity in household preferences (using the heterogeneous agentsmodels) to explain the observed wealth distribution and the heterogeneity in household career andinvestment choices in the United States.4Chapter 2Estimating Household Preference andStock Market Participation Cost fromPanel Data2.1 IntroductionIn this chapter, I propose a household dynamic model and explain the estimation procedure of thismodel. I assume the CRRA utility function and the QHD preference as the household preferences.The household dynamic model constructed in this chapter differs from the conventional dynamichousehold models in the household finance literature in two ways. First, it explicitly models house-holds’ stock-holding decisions as endogenous choices. Second, it imposes stock market participationand stock-holding share adjustment costs as the cost structure regarding households’ stock-holdingchoices. The estimation procedure of this household dynamic model is divided into two parts. Inthe first part, I estimate household preference using the intertemporal Euler equation. In the secondpart, I estimate the stock market participation cost and stock-holding share adjustment cost usingthe iterative maximum likelihood estimation method and the estimated household utility functionfrom the first part.In macroeconomics literature, the calibration method is frequently employed to estimate house-hold preferences. However, this method is heavily influenced by choices of the target statistics,and the confidence interval and the standard error of the household preference parameter are usu-ally not provided as a result. Meanwhile, estimation of the household utility function using theintertemporal Euler equation as the moment condition is often attempted in the household financeliterature. According to existing literature, identifying household preferences using the intertem-poral Euler equation from micro-panel data is problematic because of the following issues. First,the log-linearized version of the intertemporal Euler equation is frequently employed to estimatethe household utility function in the literature, owing to the simplicity of its estimation procedure.However, Dixon (2001) demonstrates that the approximation bias presents a substantial issue inthe estimation of the CRRA utility function using the log-linearized Euler equation. Furthermore,Dixon (2001) illustrates that the use of instrumental variables and the second-order approximationdoes not eliminate the approximation bias problem using the simulation analysis of the householddynamic model. Second, as noted by Dixon (2001), the generalized method of moments methodol-ogy does not work well for estimating a household utility function using the intertemporal Euler5equation as the moment condition when the consumption data have significant measurement errors.Third, the time dimensions of the existing micro-panel datasets are not sufficient for identifying thehousehold utility function using the Euler equation estimation methodology. In fact, Attanasio andLow (2004) states that a time dimension of 40 or more years in a micro-panel dataset is necessaryto estimate the household utility function when using the Euler equation estimation method.I tackle the first issue mentioned above by not log-linearizing the intertemporal Euler equation.This imposes a more computationally complicated non-linear estimation procedure compared withthe log-linearized Euler equation estimation method. However, the approximation bias problemcan be avoided. Furthermore, I solve the second issue of the measurement error by estimating thehousehold consumption function in the first step using the nonparametric estimation methodology.This eliminates the measurement error problem by assuming that the measurement error is inde-pendent of the observed variables as the standard assumption. Finally, the third issue arises due tothe lack of variation in the return on household savings caused by the shortness of time dimensionsof micro-panel datasets. The existing literature in household finance does not take into account theportfolio composition of household savings when estimating the household utility function usingthe intertemporal Euler equation. As a result, the return on household savings varies only overtime and does not change across households at a fixed time point. This implies that the shortnessof the time dimension of micro-panel datasets causes a lack of variation in the return on householdsavings for estimating accurately the household utility function from a micro-panel dataset. I solvethis issue by explicitly taking into account the portfolio composition of household savings. Thiscreates an extra variation in the return on household savings due to the differences in portfoliocompositions across households. With this extra variation in a return on household savings, I ac-curately estimate the household utility function using the intertemporal Euler equation from themicro-panel dataset. The estimation result of the household utility function from the intertempo-ral Euler equation estimation method improves greatly by including the portfolio composition ofhousehold savings in the household dynamic model as Tables 2.3 and 2.4 demonstrate.Empirical studies have frequently pointed out the low stock market participation rate amonghouseholds. In fact, the SCF and the PSID datasets indicate that approximately 30% of householdsparticipate in the stock market. Haliassos and Bertaut (1995) discusses that this low stock marketparticipation rate among households cannot be explained by the heterogeneity of beliefs, habitpersistence, and borrowing constraints. Furthermore, Haliassos and Bertaut (1995) states thathouseholds need to have an unrealistically high degree of risk aversion to be reluctant to participatein the stock market at this low level. The second interesting feature of the household stock-holding data is that a high persistence exists in the stock-holding shares of stock-holding householdsbetween the current and previous periods. The stock market participation cost and stock-holdingshare adjustment cost are estimated by using the household utility function estimated from theintertemporal Euler equation method discussed above. The households’ observed stock marketparticipation choices identify the stock market participation cost. Furthermore, the persistence inhouseholds’ stock-holding shares between the current and previous periods identifies the adjustment6cost of the household stock-holding share. This study succeeds in being the first in the householdfinance literature to estimate the CRRA utility function from a micro-panel dataset using the Eulerequation estimation method. Furthermore, in the literature, the QHD preference has never beenestimated from a micro-panel dataset using the intertemporal Euler equation estimation method.Moreover, this study is the first in financial economics literature to estimate explicitly the hiddenstock market participation cost using the household dynamic model. In the later sections of thischapter, I present the estimation results and discuss the importances and implications of suchresults. Finally, Section 2.5 summarizes the key conclusions of the chapter.2.2 ModelI will consider the household dynamic model with discrete and continuous choices. The continuouschoice of a household is the choice of optimal consumption. The discrete choices of a householdare stock market participation and stock-holding share choices. I discretize the stock-holding sharechoices as evenly distributed grid points on the interval [0, 1]. When the stock-holding share iszero, a household does not participate in the stock market or does not hold any stock in its savings.I consider two types of household preferences. The first type is the CRRA utility function. Thesecond type is the QHD preference. I explain the household dynamic model in detail in the followingsections.2.2.1 Stock Market Return ProcessI model the stock market return process following the autoregressive conditional heteroskedasticity(ARCH) model asrt+1 = (1− ρr)µr + ρrrt + rt+1, rt+1 ∼ N(0, σrt+1) (2.1)σrt+12 = α0 + α1rt2 (2.2)where I denote the stock market return at the period t by rt, the exogenous shock to the stockmarket return by rt , and the variance of the exogenous shock by σrt . Furthermore, the mean ofthe stock market return is µr, and the persistence parameter is denoted by ρr. As Equation (2.2)demonstrates, the variance of the exogenous shock to the stock market return varies over time.Furthermore, I assume that the exogenous shock to the stock market return, rt , follows the normaldistribution. Given α0, α1, and rt , the household formulates the variance of the exogenous shock tothe stock market return in the next period. With this, the household formulates its expectation ofthe stock market return, rt+1, for the next period according to the law of motion equation describedin Equation (2.1).72.2.2 The Household Income ProcessI model the household income process aslog(yit+1) = ρy log(yit) + yit+1, yit+1 ∼ N(0, σy) (2.3)where I denote the household income in the current period by yit and the exogenous shock to thehousehold income in the current period by yit. Here, the persistence parameter is ρy, and thevariance of the exogenous shock to household income is σy. Moreover, I assume that the exogenousshock to household income, yit, follows the normal distribution. This formulation of the householdincome process is very common in household finance literature.2.2.3 The Household Dynamic ModelGiven the exogenous household income and stock market return processes, a household seeks tomaximize the present value of expected lifetime utility, discounted at rate β ∈ (0, 1),E0∞∑t=0βtU¯(cit, dit, dit−1) (2.4)subject to the intertemporal budget constraint,xit+1 = (xit − cit)(1 + rf + dit(rt+1 − rf )) + yit+1 (2.5)where I denote the household’s extended utility function by U¯(·), its consumption by cit, its stock-holding share in the household savings by dit, its stock-holding share in the previous period bydit−1, its cash-in-hand variable by xit, the risk-free rate by rf , and the discount factor by β. Thehousehold’s extended utility function is defined as follows:U¯(cit, dit, dit−1) = U(cit)−AC · (dit − dit−1)2 − PC · 1{dit 6= 0} (2.6)where U(·) denotes the conventional household utility function. The household’s infinite horizonproblem defined in Equation (2.4) can be recursively recast asV (dit, dit−1, xit, sit) = maxcit{U(cit)−AC · (dit − dit−1)2 − PC · 1{dit 6= 0}+ βEV¯ (dit, xit+1, sit+1)}.(2.7)subject to the intertemporal budget constraint,xit+1 = (xit − cit)(1 + rf + dit(rt+1 − rf )) + yit+1sit = {rt, rt , yit} (2.8)8where V (·) is the household’s value function; c(·) is the household’s consumption function; V¯ (·)is the household’s ex ante value function; sit is a set of exogenous state variables at the currentperiod; AC is the adjustment cost of the household’s stock-holding share between the current andlast periods; and PC is the stock market participation cost for the current period. A householdinvests its current period savings, xt − ct, into the risk-free asset and the stock holding. The shareof the stock holding in household savings is denoted by dit. The return on household savings,(1 + rf + dit(rt+1− rf )), can be expressed as (1 + (1− dit)rf + ditrt+1). In this household dynamicmodel, the return on household savings depends on the risk-free rate, rf , the stock market return,rt+1, and the portfolio composition of household savings, dit. Furthermore, the discount factor, β,is exogenously given and set as 0.9 in the Chapters 2 and 3. The reason is that the historical averageof the real annualized stock market return is approximately 8.5% in the U.S. Huggett (1993) statesthat β(1 + r) has to be strictly lower than one for the asset space to be bounded in the householddynamic model. Here, r expresses the return on household savings and β denotes the discountfactor in Huggett (1993). If we set the discount factor, β, too high such as 0.95, β(1 + r) can gethigher than one and we may not be able to solve the model.Stock Market Participation CostI model the stock market participation cost as PC · 1{dit 6= 0}. When a household is not partici-pating in the stock market in the current period, the household’s choice of the stock-holding share,dit, is equal to 0; otherwise, it is not equal to 0. As Equation (2.7) indicates, I model the stockmarket participation cost as part of the household utility in the current period.Adjustment Cost of the Stock Holding ShareI model the adjustment cost of a stock-holding share in household savings as AC · (dit − dit−1)2.The adjustment cost increases when the difference between the current period stock-holding share,dit, and the previous period stock-holding share, dit−1, increases. This creates an incentive for ahousehold to not change its optimal stock-holding share frequently. When a household chooses itsoptimal-stock holding share, it tends to choose the stock-holding share closer to that of the previousperiod, as observed in the household micro-panel datasets. As Equation (2.7) illustrates, I modelthe adjustment cost of the stock-holding share in household savings as part of the household utilityin the current period.Household Choice of the Stock-Holding ShareA household chooses its optimal share of stock-holding in the household savings asdit = arg max{V (d1, dit−1, xit, sit) + ε1, ..., V (dK , dit−1, xit, sit) + εK}(2.9)ε = {ε1, .., εK}9where the choice specific preference shock, εk, is the independent and identically distributed randomvariable and follows the type-1 extreme value distribution and K is the number of choices fora household’s stock-holding share. I acknowledge that the assumption of the independent andidentically distributed random variable for the choice specific preference shock, εk, is a strongassumption. There may exist a cross correlation between the choice specific preference shocks andthey may not follow the type-1 extreme value distribution. However, it is hard to estimate thecross correlation between the choice specific preference shocks since we do not observe householdvalue functions directly from the data. Moreover, by assuming that the choice specific preferenceshock, εk, follows the type-1 extreme value distribution, we obtain the closed form solutions for theconditional choice probability of the stock-holding share and the household’s ex ante value function.Due to these advantages, it is commonly assumed that the choice specific preference shock, εk, isthe independent and identically distributed random variable and follows the type-1 extreme valuedistribution in the dynamic logit model literature. Finally, in Chapters 2 and 3, I assume thatthe choice specific preference shock, εk, is independent of the exogenous shock to the householdincome, yit, and the exogenous shock to the stock market return, rt . The reason is that it is hardto estimate the correlations between the choice specific preference shock and the other householdexogenous shocks since we do not observe household value functions directly from the data, andthis assumption simplifies the complexity of the model solution to a large extent.I define the household’s ex ante value function asV¯ (dit−1, xit, sit) = Eε max{V (d1, dit−1, xit, sit) + ε1, ..., V (dK , dit−1, xit, sit) + εK}. (2.10)Following Rust (1987), the conditional choice probability function of the stock-holding share isexpressed as a function of the value functions, {V (dk, dit−1, xit, sit)}Kk=1, as follows:Pr{dit|dit−1, xit, sit} = exp(V (dit, dit−1, xit, sit))∑d′it∈D exp(V (d′it, dit−1, xit, sit))(2.11)where Pr{dit|dit−1, xit, sit} is the conditional choice probability function corresponding to the stock-holding share choice, dit, in the current period, and D is the set of choices for the stock-holdingshare. Moreover, as noted by Rust (1987), the household’s ex ante value function is expressed as afunction of the value functions, {V (dk, dit−1, xit, sit)}Kk=1, asV¯ (dit−1, xit, sit) = ln( ∑d′t∈Dexp(V (d′t, dit−1, xit, sit)))+ γ (2.12)where γ is the Euler-Mascheroni constant.102.2.4 Constant Relative Risk Aversion Utility FunctionAs the first type of household preference, I consider the CRRA utilify function as the householdpreference, defined asU(cit) =c1−σit1− σ (2.13)where σ is the curvature parameter of the CRRA utility function.The Intertemporal Euler EquationI derive the intertemporal Euler equation from the model defined above using the Lagrangian methodand the envelope theorem when the CRRA utility function is assumed as the household preference.The intertemporal Euler equation 1 is derived asc(dit, dit−1, xit, sit)−σ = βEsit+1{ ∑d′it+1∈DPr{d′it+1|dit, xit+1, sit+1}(1 + rf + dit(rt+1 − rf ))c(d′it+1, dit, xit+1, sit+1)−σ}(2.14)where c(·) is the household consumption function. This intertemporal Euler equation has twodistinct features compared with the conventional Euler equations derived in the household financeliterature. First, the return on household savings, (1 + rf + dit(rt+1 − rf )), is a function of thestock holding share, dit. Second, the household’s expected marginal utility for the next period iscalculated as the weighted sum of the household marginal utilities in the next period. Here, theweights are the conditional choice probabilities of the stock-holding share for the next period.2.2.5 The Quasi-Hyperbolic Discounting PreferenceAs the second type of household preference, I consider the QHD preference as the household pref-erence, defined asW (dit, dit−1, xit, sit) = maxcit{c1−σit1− σ −AC · (dit − dit−1)2 − PC · 1{dit 6= 0}+ δβEV¯ (dit, xit+1, sit+1)}(2.15)V (dit, dit−1, xit, sit) =c1−σit1− σ −AC · (dit − dit−1)2 − PC · 1{dit 6= 0}+ βEV¯ (dit, xit+1, sit+1)(2.16)1The detailed derivation of the intertemporal Euler equation for the CRRA utility function is explained in theAppendix.11where W (·) is the household value function for the current period, and V (·) is the household valuefunction for the future periods. The value function, W (dit, dit−1, xit, sit), can be expressed asW (dit, dit−1, xit, sit) = δV (dit, dit−1, xit, sit) + (1− δ)U(c(dit, dit−1, xit, sit)). (2.17)The main distinction between the QHD preference and the CRRA utility function is that a house-hold with the QHD preference uses two types of discount factors as follows: the short-run discountfactor, δ, and the long-run discount factor, β. Meanwhile, a household with the CRRA utilityfunction uses only the long-run discount factor, β, as the discount factor. The short-run discountfactor, δ, imlies that a household values its current period utility more than its future utilities, asit discounts its future value function with an extra discount factor compared with a household withthe CRRA utility function. However, a household uses the long-run discount factor, β, when it com-pares the future period utilities. This creates a dynamic inconsistency regarding the household’sdecisions over time.The Intertemporal Euler EquationI derive the intertemporal Euler equation using the Lagrangian method and the envelope theoremwhen the QHD preference is assumed as the household preference. Here, the envelope theorem isapplied to W (dit, dit−1, xit, sit), instead of V (dit, dit−1, xit, sit). Meanwhile, the envelope theorem isapplied to V (dit, dit−1, xit, sit), in a case of the CRRA utility function. Due to this difference, ahousehold’s marginal propensity to consume, ∂c(dt+1,dit,xit+1,sit+1)∂xit+1 , enters the intertemporal Eulerequation as Equation (2.17) indicates. The intertemporal Euler equation2 for the QHD preferenceis derived as:c(dit, dit−1, xit, sit)−σ = E{ ∑d′it+1∈DP{d′it+1|dit, xit+1, sit+1}(1 + rf + dit(rt+1 − rf ))β[1− (1− δ)∂c(d′it+1, dit, xit+1, sit+1)∂xit+1]c(d′it+1, dit, xit+1, sit+1)−σ}(2.18)The intertemporal Euler equation described in Equation (2.18) has one important differencefrom the intertemporal Euler equation, derived when the CRRA utility function is assumed as thehousehold preference. That difference is that the intertemporal Euler equation includes the term,called as the household’s effective discount factor, defined asβ[1− (1− δ)∂c(d′it+1, dit, xit+1, sit+1)∂xit+1](2.19)In the existing literature, the short-run discount factor, δ, is assumed to be lower than or equal toone. The QHD preference collapses into the CRRA utility function when the short-run discount2The detailed derivation of the intertemporal Euler equation for the QHD preference is explained in the Appendix.12factor, δ, is exactly equal to one. The 95% confidence interval estimation result for the short-rundiscount factor, δ, from this chapter confirms that δ is statistically significantly lower than one.The household’s effective discount factor, defined in Equation (2.19), depends on a household’smarginal propensity to consume (MPC), ∂c(dt+1,dit,xit+1,sit+1)∂xit+1 . This implies that households withdifferent MPCs discount their next period’s marginal utilities by different magnitudes. In detail, ahousehold puts more weight toward the current period’s utility when its MPC is larger because itseffective discount factor becomes smaller. Therefore, the QHD preference can generate significantlydifferent consumption decisions among households with relatively greater MPC, compared with theconsumption decisions arising from the CRRA utility function. Equation (2.18) is similar to theintertemporal Euler equation, derived in Laibson (1998). The first main difference from Laibson(1998) is that the return on household savings, (1 + rf + dit(rt+1 − rf )), is a function of the stockholding share, dit. Moreover, the second main difference is that the conditional choice probabilitiesof the stock-holding share for the next period enter the intertemporal Euler equation (2.18).2.3 Estimation ProcedureIn this section, I explain in detail the estimation procedure of the dynamic household model. Asa preparation step, the stock market return and household income processes are estimated be-cause these processes are the exogenous processes in the household dynamic model. In the firststep of the main estimation procedure, the household consumption function and the conditionalchoice probabilities of the stock-holding share are estimated by the nonparametric kernel estimationmethod. In the second step, the household preference is estimated using the intertemporal Eulerequation as the moment condition and the nonparametrically estimated household consumptionfunction and conditional choice probabilities of the stock-holding share. In the third step, the stockmarket participation cost and stock-holding share adjustment cost are estimated using the esti-mated household preference from the second step and the iterative maximum likelihood estimationmethodology.2.3.1 Nonparametric Estimation of Household Consumption and ConditionalChoice Probabilities of the Stock-Holding ShareIn the first step of the estimation procedure, the household consumption function and condi-tional choice probabilities of the stock-holding share are estimated by the nonparametric estimationmethod. I employ the Nadarya-Watson kernel regression method for this purpose. The estimatedhousehold consumption function and conditional choice probabilities of the stock-holding share areused in the second step for estimating the household preference using the intertemporal Euler equa-tion estimation method. The main advantage of the nonparametric estimation method is that wedo not need to know the household preference parameters and the cost structure for the householdstock-holding decision to estimate the consumption function and conditional choice probabilities ofthe stock-holding share. The detail of the nonparametric estimation procedure is available in the13Appendix.2.3.2 Constant Relative Risk Aversion Utility FunctionMacroeconomics literature states that β(1+r) < 1 is necessary for the state space of the householdasset holding to be bounded when the household income process is stochastic 3. In addition, thehistorical average of the annualized real return of the S&P 500 Index is approximately 8.5%. Thisimplies that the time discount factor, β, cannot be set too high. In this chapter, the time discountfactor, β, is set to 0.9, to ensure that the state space of the household asset holding is bounded.The empirical approximation of the intertemporal Euler equation (2.14) is defined asQˆit(σ) = β1NsNs∑j=1∑d′it+1∈DPˆ r{d′it+1|dit, xjit+1, sjit+1}(1 + rf + dit(rjit+1 − rf ))·· cˆ(d′it+1, dit, xjit+1, sjit+1)−σcˆ(dit, dit−1, xit, sit)−σ− 1 (2.20)where Ns is the number of random draws of the exogenous state variables; Pˆ r{·} is the nonpara-metrically estimated conditional choice probability function of the stock-holding share from thefirst step; cˆ is the nonparametrically estimated consumption function from the first step; and sjit+1is the set {rjit+1, rjit+1, yjit+1}. I define the objective function using Equation (2.20) asQT (σ) =1NN∑i=11TT∑t=1Qˆit(σ) (2.21)where N is the number of households in the micro-panel dataset, and T is the time dimensionof the micro-panel dataset. I estimate the curvature parameter, σ, of the CRRA utility functionby minimizing the Euclidean norm of Equation (2.21). I estimate only one parameter in the casewhere the CRRA utility function is the household preference, the estimated parameter σˆ satisfiesthe following condition:QT (σˆ) = 0.3The detail is available at Chamberlain and Wilson (2000) and Huggett (1993).142.3.3 The Quasi-Hyperbolic Discounting PreferenceThe empirical approximation of the intertemporal Euler equation (2.18) is defined asQˆit(ζ) = β1NsNs∑j=1∑d′it+1∈DPˆ r{d′it+1|dit, xjit+1, sjit+1}(1 + rf + dit(rjit+1 − rf ))[δ∂cˆ(d′it+1, dit, xjit+1, sjit+1)∂xjit+1+(1− ∂cˆ(d′it+1, dit, xjit+1, sjit+1)∂xjit+1)]cˆ(d′it+1, dit, xjit+1, sjit+1)−σcˆ(dit, dit−1, xit, sit)−σ− 1(2.22)where ζ is the set of parameters σ and δ; ∂cˆ(dt+1,dt,xt+1,st+1)∂xit+1 is the nonparametric estimator of theMPC ; δ is the short-run discount factor; and β is the long-run discount factor. The main differencebetween this intertemporal Euler equation and the intertemporal Euler equation derived from theCRRA utility function is that the MPC enters the intertemporal Euler equation. I estimate theMPC using the nonparametric estimation method in the first step 4. The objective function in thecase of the QHD preference is defined asQT (ζ) =1NN∑i=11TT∑t=1Qˆit(ζ) (2.23)where ζ is {σ, δ}. The parameters σ and δ are estimated by minimizing the Euclidean norm of thisobjective function asζˆ = arg minζ‖QT (ζ)‖. (2.24)2.3.4 Stock Market Participation Cost and Stock-Holding Share AdjustmentCostIn the third step of the estimation procedure, the stock market participation cost and stock holdingshare adjustment cost are estimated by maximizing the log-likelihood function:Lˆ(θ; θg,X) =N∑i=1T∑t=1log[exp(Uˆ(c(dit, dit−1, xit, sit))−AC · (dit − dit−1)2 − PC · 1{dit 6= 0}+ βEV¯ (dit, xit+1, sit+1; θg))∑d′it∈D exp(Uˆ(c(d′it, dit−1, xit, sit))−AC · (d′it − dit−1)2 − PC · 1{d′it 6= 0}+ βEV¯ (d′it, x′it+1, sit+1; θg))](2.25)where Uˆ(·) is the household utility function estimated from the second step; V¯ (·) is the ex antevalue function with the parameter set, θg; X is the notation for the observed data; θ is the setof the parameters, AC and PC; and θg is the set of the parameters, ACg and PCg. Moreover,the consumption function, c(dit, dit−1, xit, sit), used in Equation (2.25), is the consumption func-tion from the model solution with the candidate parameter set θg and not the nonparametrically4The detail is available in the Appendix.15estimated consumption function from the first step of the estimation procedure. The log-likelihoodfunction Lˆ(θ; θg,X) is derived from the conditional choice probability function defined in Equation(2.11). I follow the common approach in the conditional choice probability estimation literature.Given the candidate parameter set θg, the ex ante value function V¯ (·) is obtained as the modelsolution. Using this ex ante value function and the household utility function estimated from thesecond step, the log-likelihood function Lˆ(θ; θg,X) is maximized with respect to the parameter setθ. Here, θ is the set of parameters AC and PC. The parameter set θg is updated with the newlyobtained parameter set θ. This procedure is repeated until the difference between θ and θg, |θ−θg|,is sufficiently close to 0. According to the conditional choice probability estimation literature, thisapproach is more efficient than the other approaches.2.4 Results and DiscussionIn this section, I apply the estimation procedure described in the previous section to the datasimulated from the true data generating process. I simulate the data mimicking the householdconsumption, income, and wealth data from the PSID dataset. A total of eight survey waves of thehousehold data were simulated from the true data generating process. Each survey wave containedapproximately 3,200 households. The survey waves were collected two years apart from each otheras the survey waves of the PSID dataset are collected in the same way. I set the parameters ofthe household dynamic model defined in this chapter based on the estimated parameters fromthe PSID dataset in Chapter 3. I present the result of the estimation procedure for the CRRAutility function as the household preference in Section 2.4.1. In addition, I present the result of theestimation procedure for the QHD preference as the household preference in Section 2.4.2.2.4.1 Constant Relative Risk Aversion Utility FunctionI present the result of the estimation procedure for the CRRA utility function as the householdpreference in Table 2.1. From Table 2.1, we observe that the estimated curvature parameter, σ,of the CRRA utility function is 3.29 when the true curvature parameter, σ, of the CRRA utilityfunction is 3.40. The bootstrap standard error for the estimated curvature parameter, σ, of theCRRA utility function is 0.15, while the bias-corrected estimator is 3.36 when the true curvatureparameter, σ, of the CRRA utility function is 3.40. The difference between the bias-correctedestimator and the true parameter is 1.19% according to Table 2.1. In macroeconomics literature,the curvature parameter, σ, of the CRRA utility function is estimated by the calibration method tobe between 2 and 4. However, the estimation procedure I employ in this chapter does not dependon the choices of the target statistics, as the calibration method usually does. The CRRA utilityfunction is estimated in the second step of the estimation procedure using the nonparametricallyestimated household consumption and conditional choice probabilities functions from the first step.Therefore, the estimation bias from the first step affects the estimation of the CRRA utility functionin the second step.16The stock market participation and stock-holding share adjustment costs are estimated in thethird step of the estimation procedure using the estimated household utility function from thesecond step of the estimation procedure. I model the stock market participation and stock-holdingshare adjustment costs as parts of the household utility or disutility. The stock market participationcost is estimated at 8.42, as shown in Table 2.1. The bootstrap standard error for the estimatedstock market participation cost is 0.54, as shown in Table 2.1. The bias-corrected estimator forthe stock market participation cost is 8.66 when the true stock market participation cost is 9.33.The difference between the true stock market participation cost and the bias-corrected estimatorfor the stock market participation cost is 7.74%, as shown in Table 2.1. Moreover, the estimatedstock-holding share adjustment cost is 2.07, as shown in Table 2.1. The bootstrap standard errorfor the stock-holding share adjustment cost is 0.03, while the bias-corrected estimator is 2.06 whenthe true stock-holding share adjustment cost is 1.87. The difference between the bias-correctedestimator and the true stock-holding share adjustment cost is −9.22%, as shown in Table 2.1. Theestimation biases from the first and second steps of the estimation procedure affect the estimationsof the stock market participation and stock-holding share adjustment costs in the third step of theestimation procedure. Therefore, the percentage differences between the estimated stock marketparticipation and stock-holding share adjustment costs and the true stock market participationand stock-holding share adjustment costs are larger than the percentage difference between theestimated curvature and the true curvature parameters of the CRRA utility function in absolutevalues.2.4.2 The Quasi-Hyperbolic Discounting PreferenceI present in Table 2.2 the result of the estimation procedure where the QHD preference is thehousehold preference. The estimated curvature parameter, σ, of the QHD preference is 3.30 whenthe true curvature parameter of the QHD preference is 3.41, as shown in Table 2.2. The bootstrapstandard error for the estimated curvature parameter of the QHD preference is 0.04. The bias-corrected estimator for the curvature parameter, σ, of the QHD preference is 3.37 when the truecurvature parameter of the QHD preference is 3.41. The percentage difference between the biascorrected estimator for the curvature parameter, σ, of the QHD preference and the true curvatureparameter is 1.19% according to Table 2.2. From Tables 2.2 and 2.1, we can observe that the bias-corrected estimators for the curvature parameters of the QHD preference and the CRRA utilityfunction are very similar.The estimated short-run discount factor, δ, of the QHD preference is 0.94, as shown in Table 2.1.The bootstrap standard error for the estimated short-run discount factor, δ, of the QHD preferenceis 0.02, while the bias-corrected estimator is 0.92 when the true short-run discount factor, δ, of theQHD preference is 0.94. The percentage difference between the bias-corrected estimator for theshort-run discount factor, δ, of the QHD preference and the true short-run discount factor, δ, ofthe QHD preference is 2.17% according to Table 2.2. We can see that the difference between thebias-corrected estimator for the short-run discount factor, δ, of the QHD preference and the true17short-run discount factor, δ, of the QHD preference is very small. Therefore, the QHD preferenceis estimated accurately in the second step of the estimation procedure.The stock market participation cost when the QHD preference is assumed as the household pref-erence is estimated at 9.77, as shown in Table 2.2. The bootstrap standard error for the estimatedstock market participation cost is 0.24, as shown in Table 2.2. The bias-corrected estimator for thestock market participation cost is 10.04 when the true stock market participation cost is 9.38. Thepercentage difference between the true stock market participation cost and the bias-corrected esti-mator for the stock market participation cost is −6.57%, as shown in Table 2.2. Furthermore, theestimated stock-holding share adjustment cost is 1.76, as shown in Table 2.2. The bootstrap stan-dard error for the stock-holding share adjustment cost is 0.02, while the bias-corrected estimator is1.77 when the true stock-holding share adjustment cost is 1.87. The percentage difference betweenthe bias-corrected estimator for the stock-holding share adjustment cost and the true stock-holdingshare adjustment cost is 5.65%, as shown in Table 2.2. The estimation biases from the first andsecond steps of the estimation procedure affect the estimations of the stock market participationand stock-holding share adjustment costs in the third step of the estimation procedure. Hence, thepercentage differences between the estimated stock market participation and stock-holding shareadjustment costs and the true stock market participation and stock-holding share adjustment costsare larger than the percentage differences between the estimated curvature parameter and short-rundiscount factor of the QHD preference and the true curvature parameter and short-run discountfactor of the QHD preference in absolute values.The QHD preference collapses into the CRRA utility function when the short-run discountfactor, δ, is equal to 1, as observed in Equations (2.14) and (2.18). Furthermore, the effectivediscount factor for the QHD preference gets closer to the long-run discount factor, β, as the MPC,∂c(dt+1,dit,xit+1,sit+1)∂xit+1, gets closer to zero. Meanwhile, the effective discount factor for the QHDpreference approaches βδ as the MPC, ∂c(dt+1,dit,xit+1,sit+1)∂xit+1 , gets closer to one, as shown in Equation(2.19). The MPC is larger for poorer households. In fact, the difference between the QHD preferenceand the CRRA utility function is very small for most households, as the results from Chapter 3demonstrate. The non-negligible difference between the QHD preference and the CRRA utilityfunction is observed only for households at the bottom part of the wealth distribution. The reasonis that the MPCs are very large for these households. Another interesting observation is that thebootstrap standard errors of the estimated parameters for the QHD preference are smaller thanthe bootstrap standard errors of the estimated parameters for the CRRA utility function, as shownin Tables 2.1 and 2.2. The reason is that the CRRA utility function is the subset of the QHDpreference since the QHD preference collapses into the CRRA utility function when the short-rundiscount factor, δ, is equal to one. Therefore, the QHD preference may fit the data better than theCRRA utility function in the estimation procedure.182.5 ConclusionThis chapter shows that the household utility function can be estimated accurately with the in-tertemporal Euler equation estimation method by explicitly taking into account the portfolio com-position of household savings as an endogenous household choice in the household dynamic model.The extra variation in the return on household savings caused by the differences in portfolio com-positions across households improves, to a great extent, the identification of the household utilityfunction with the intertemporal Euler equation estimation method. From the existing literature,the return on household savings varied only over time in household dynamic models. Since house-hold micro-panel datasets do not cover many periods, the intertemporal Euler equation estimationmethod did not work well for estimating the household utility function from household micro-paneldatasets because of the lack of variation in the return on household savings. Meanwhile, in thischapter, I estimated successfully the parameter of the household utility function from the householdmicro-panel dataset using the intertemporal Euler equation estimation method by explicitly mod-eling the portfolio composition of household savings as an endogenous household choice variable inthe household dynamic model. Furthermore, as the first to do so in the household finance literature,I estimated successfully the QHD preference as the household preference from the household micro-panel dataset using the intertemporal Euler equation estimation method. An accurate estimationof the household utility function is vital for precisely evaluating the effects of macroeconomic shocksand policy interventions on household well-being.Moreover, this chapter argues that the puzzle of the low stock market participation rate amonghouseholds might be explained well by modeling the hidden stock participation cost as a cost to thehousehold utility in a household dynamic model. This hidden stock market participation cost canbe estimated from household micro-panel datasets, such as the PSID dataset, using the estimationprocedure developed in this chapter. Another important contribution of this chapter to the existingliterature is that it finds that changes in stock market returns affect household decisions throughtwo channels. First, it affects a household’s expectation of the stock market return for the nextperiod. Second, it affects a household’s expectation of the future volatility of the stock marketreturn. The second channel can be important for evaluating the effect of the financial crisis onhousehold decisions, since the volatility of the stock market return usually increases significantlyduring financial crises. With the methodology developed in this chapter, researchers can assessmore realistically the impact of stock market crashes on household well-being.192.6 TablesTable 2.1: The Constant Relative Risk Aversion Utility FunctionParameter EstimatorBootstrapstandard errorBias correctedestimatorTrue parameterDifference fromtrue parameterσ 3.29 0.15 3.36 3.40 1.19%AC 8.42 0.54 8.66 9.33 7.74%PC 2.07 0.03 2.06 1.87 -9.22%Note: σ denotes the curvature parameter of the Constant Relative Risk Aversion utility function,AC denotes the stock holding share adjustment cost, and PC denotes the stock market participationcost.Table 2.2: The Quasi-Hyperbolic Discounting PreferenceParameter EstimatorBootstrapstandard errorBias correctedestimatorTrue parameterDifference fromtrue parameterσ 3.30 0.04 3.37 3.41 1.19%δ 0.94 0.02 0.92 0.94 2.17%AC 9.77 0.24 10.04 9.38 -6.57%PC 1.76 0.02 1.77 1.87 5.65%Note: σ denotes the curvature parameter of the Quasi-Hyperbolic Discounting preference, δ denotesthe short-run discount factor, AC denotes the stock holding share adjustment cost, and PC denotesthe stock market participation cost.Table 2.3: The Estimated CurvatureParameter of the Constant Relative RiskAversion Utility Function without thePortfolio CompositionMinimum Maximum Average True valueσ 1.23 2.56 1.70 3.40Note: Here, I simulated the household data from thehousehold dynamic model without the portfolio com-position of household savings 100 times. Then, I esti-mated the curvature parameter of the Constant Rela-tive Risk Aversion utility function from each of these100 simulated datasets using the Euler equation esti-mation method.20Table 2.4: The Estimated Curvature Parameter of the Constant RelativeRisk Aversion Utility Function with the Portfolio Composition10thpercentile30thpercentile50thpercentile70thpercentile90thpercentileAverage True valueσ 2.77 3.03 3.16 3.27 3.53 3.15 3.40Note: Here, I simulated the household data from the household dynamic model with the port-folio composition of household savings 100 times. Then, I estimated the curvature parameter ofthe Constant Relative Risk Aversion utility function from each of these 100 simulated datasetsusing the Euler equation estimation method.21Chapter 3The Effect of the Stock MarketCollapse on Household Consumptionduring the 2007-2008 Global FinancialCrisis3.1 IntroductionOne of the major events during the 2007-2008 global financial crisis (GFC) was the collapse ofthe stock market. Even though stock holdings are an important investment option for households,the effect of the stock market collapse during the GFC on household savings and consumption hasnot been explored much compared to the housing market collapse. The S&P 500 Stock MarketIndex declined from 1897.75 to 1112.80 between September 2007 and December 2008. As a resultof this huge decline in the S&P 500 Index, its real annualized return dropped to −37.28% in 2008as Figure 3.1 illustrates. This caused an enormous loss in the savings of stock-holding households.In this chapter, I evaluate the effect of the stock market collapse during the GFC on householdconsumption and savings using the household dynamic model developed in Chapter 2. In responseto the crisis, the Federal Reserve switched to the Zero-Lower Bound monetary policy during theGFC, which pushed the policy rate (the federal funds rate) to the zero lower bound as shown inFigure 3.5. Moreover, Figure 3.5 illustrates that the 3-Month Treasury Bill rate also dropped tothe zero lower bound as a result of the said policy. The 3-Month Treasury Bill rate is often used asthe risk-free rate in the household finance literature. The huge decline in the stock market returnand the drop in the risk-free rate both affected household savings and consumption.Consumption inequality in the U.S. declined significantly at the end of the GFC compared withhow it was at the beginning of the crisis as the PSID dataset indicates. Existing literature connectsthe change in consumption inequality to the change in income inequality. In addition, a major partof income inequality arises from labor income inequality. However, evidence from the PSID datasetreveals that income inequality and labor income inequality did not change much compared with thechange in consumption inequality during the GFC. Therefore, it is hard to claim that the change inincome inequality has caused a substantial decrease in consumption inequality among householdsin the U.S. during the 2007-2008 financial crisis. In this chapter, I identify that the huge decline inthe stock market return and the drop in the risk-free rate have caused the consumption inequality22to decrease significantly during the GFC.I employ the household dynamic model developed in Chapter 2 to investigate the effects ofthe stock market collapse and the monetary policy change during the GFC on household con-sumption. First, I estimate the stock market return process using the autoregressive conditionalheteroskedasticity (ARCH) model from the annualized return time-series data of the S&P 500 In-dex. Furthermore, I use two different risk-free rates. First is the risk-free rate before the GFC andsecond is the risk-free rate after the GFC. I obtain these risk-free rates from the 3-Month TreasuryBill rate data. Next, I estimate the household dynamic model developed in Chapter 2 using thePSID dataset as the household micro-panel dataset. Finally, I evaluate and discuss the effects ofthe stock market collapse and monetary policy change during the GFC on household consumption.3.2 Stock Market Collapse and the Zero-Lower Bound MonetaryPolicy3.2.1 Stock Market CollapseThe huge drop in the stock market return during the GFC created a substantial loss to house-hold wealth as a direct income effect. Furthermore, stock market crashes can trigger two changesin household expectations regarding the stock market return process in future periods. First,household expectation of the stock market return might drop as a decrease in the expectationof the first moment of the stock market return process. Second change is that a household ex-pectation of the stock market return volatility migh increase as an increase in the expectationof the second moment of the stock market return process occurs. A decrease in the householdexpectation of the stock market return generates a substitution effect and an income effect. Asthe substitution effect, households will favor risk-free investment options more than stock holdings.As the income effect, households’ expected income in the future will decrease. This decrease inhouseholds’ expectation of future income diminishes households’ demand for both stock-holdingand risk-free investment options. An increase in households’ expectation of stock market returnvolatility triggers the substitution effect as the main effect, which leads households to substitutemore toward risk-free investment options when households are risk-averse.To capture all effects discussed above, I estimate the stock market return process using theARCH model from the time-series annualized return data of the S&P 500 Index. The ARCHmodel captures two important features of a stock market collapse. First, it captures a decrease inthe household expectation of the stock market return caused by the stock market collapse. Second,it captures an increase in the household expectation of the stock market return volatility triggeredby the stock market crash. I model the stock market return process using the ARCH model asrt+1 = (1− ρr)µr + ρrrt + rt+1, rt+1 ∼ N(0, σrt+1) (3.1)σrt+12 = α0 + α1rt2 (3.2)23where I denote the stock market return at the period t by rt, the exogenous shock to the stockmarket return by rt , and the variance of the exogenous shock by σrt . The huge negative shock, rt ,means a substantially low value realization of the stock market return, rt. A low value of the stockmarket return, rt, means that the expected stock market return, E[rt+1], is also low, as shown inEquation (3.1). Furthermore, equation (3.2) demonstrates that the huge negative shock, rt , willtrigger an increase in household expectation of future stock market return volatility, σrt+1. In thehousehold dynamic model developed in Chapter 2, a household forms its expectations over thestock market return and the stock market return volatility for the next period. The ARCH modelprovides the structure for forming these expectations. The estimation results of the ARCH modelare displayed in Figures 3.2 and 3.3.3.2.2 Zero Lower Bound Monetary PolicyThe Federal Reserve reduced its policy rate, the federal funds rate, to a level close to zero as anexpansionary monetary policy during the GFC. As Figure 3.5 illustrates, the federal funds rate andthe 3-Month Treasury Bill rate both declined to near zero during the GFC. The 3-Month TreasuryBill rate is commonly used as a proxy for the risk-free rate in the household finance literature. Thisindicates that the nominal return on risk-free investment options (savings bonds, savings accounts,treasury bills, etc.) declined as a result of the Zero-Lower Bound monetary policy. Moreover, Figure3.6 shows that the real annualized return of the 3-Month Treasury Bill rate declined during the GFCdue to the Zero-Lower Bound monetary policy. The drop in the risk-free rate triggers a substitutioneffect and an income effect. As the substitution effect, it causes households to increase their demandfor stocks and decrease their demand for risk-free investment options. As the income effect, itdecreases households’ expected future income, which reduces household demand for both risky andrisk-free assets. The decline in the risk-free rate triggered by the Zero-Lower Bound monetarypolicy during the GFC affected household consumption differently depending on a household’sposition in the wealth distribution. A decline in the risk-free rate created a lower expected incomefor households with positive net savings. This means that households with positive net savingsdecreased their consumption and savings as the income effect. They decreased their savings andincreased their consumption as the substitution effect. Existing studies demonstrate that the incomeeffect dominates the substitution effect in most cases. Meanwhile, households at the bottom of thewealth distribution had negative net savings or debt. They financed their consumption by borrowingand did not hold stocks, as empirical evidence from the SCF and the PSID datasets suggests. As aresult, the decline in the risk-free rate triggered a drop in their borrowing costs. During the GFC,this enabled such households to increase their consumption. In fact, I observe this phenomenonfrom the PSID dataset.3.2.3 Household Stock-Holding BehaviorResearchers need to construct a model that accurately captures the observed household stock-holding behavior to evaluate precisely the effect of the stock market collapse on household well-24being. Evidence from the PSID and the SCF demonstrates that approximately 30% of householdsparticipate in the stock market in a given time period even though the average stock market returnis much larger than the risk-free rate. In fact, the historical average of the real annualized return ofthe S&P 500 Index is approximately 8.5%. Meanwhile, the average real annual risk-free rate, repre-sented by the 3-Month Treasury Bill rate, has been lower than 2% for the past 20 years. Haliassosand Bertaut (1995) points out that this low rate of participation in the stock market among house-holds cannot be explained by the standard expected utility maximization theories. Furthermore,they discuss that the heterogeneity of beliefs, habit persistence, and borrowing constraints do nothelp explain this phenomenon, and households need to have an unrealistically high degree of riskaversion to participate in the stock market at this low rate. Haliassos and Bertaut (1995) suggeststhat one possible inertia from the standard utility maximization theory is the information-gatheringcost for explaining the phenomenon of the low stock market participation rate among households.In fact, information-gathering costs have recently drawn attention from researchers. Bonaparte andKumar (2013) postulates that politically active individuals are 9% to 25% more likely to participatein the stock market, and greater political activism reduces information-gathering costs and causeshigher stock market participation rates. They find that politically active individuals spend about30 minutes more on news daily and appear more knowledgeable about the economy and the mar-kets. Moreover, Grinblatt, Keloharju, and Linnainmaa (2011) states that high-IQ individuals tendto participate more in the stock market because they have lower costs for processing informationrequired to participate in the stock market. Such information-gathering costs were not observeddirectly. However, the existence of information-gathering costs supports the idea that there existsan unobserved stock market participation cost. Since the stock market participation cost is notobserved, I model the stock market participation cost as a cost to the household utility in Chapter2.3.3 Estimation of the Household Dynamic ModelI apply the estimation procedure described in Chapter 2 to the household data from the PSIDdataset. In the preparation step, I estimate the household income process from the household in-come data from the PSID dataset and stock market return process from the time-series annualizedreturn data of the S&P 500 Index using the ARCH model. In the first step of the estimation pro-cedure, I estimate the household consumption function and conditional choice probability functionof the stock-holding share using the nonparametric kernel regression method. In the second step,I estimate the household utility function using the intertemporal Euler equation as the momentcondition and the estimated household consumption function and conditional choice probabilityfunction of the stock-holding share from the first step. In the third step, I estimate the stockmarket participation and stock-holding share adjustment costs as the disutilities or costs to thehousehold utility using the iterative maximum likelihood estimation method and the estimatedhousehold utility function from the second step. The household panel data from the PSID dataset,25used in the estimation procedure of this chapter, come from the eight survey waves: 2000, 2002,2004, 2006, 2008, 2010, 2012, and 2014. Each wave includes approximately 3,200 households. Thepanel data for household consumption and wealth do not exist before 1998 for the PSID dataset.The survey waves have been collected two years apart from each other since 1998 for the PSIDdataset. The survey wave of 1998 is not used because the previous wave’s stock-holding share ismodeled as one state variable in the household dynamic model described in Chapter 2.3.3.1 Stock Market Return Process and Household Income ProcessHousehold income data from the PSID dataset are used to estimate the household income process.I estimate the persistence parameter, ρy, and the standard deviation of the new exogenous shock,σy, in (2.3) by employing the maximum likelihood estimation method. I scale down the householdincome, consumption, savings, and stock holding from the PSID dataset by the average of thehousehold income data in the 2006 wave of the PSID dataset as the normalization. The persistenceparameter, ρy, is estimated to be approximately 0.88, and the estimated standard deviation of thenew exogenous shock, σy, is estimated to be approximately 0.42.I estimate the stock market return process from the annualized return data of the S&P 500Index since it is the most widely used stock price index in the finance. The data are obtained fromRobert Shiller and Yahoo! Finance. The stationary mean, µr, in (2.1) is estimated as the average ofthe entire dataset. The persistence parameter, ρr, in (2.1) is estimated using the linear regressionmethod. Using the estimations of µr and ρr, I estimate the parameters, α0 and α1, in (2.2) by thelinear regression method. The mean parameter, µr, is estimated to be 0.0851 and the persistenceparameter, ρr, is estimated to be approximately 0.4710. In addition, α0 is estimated to be 0.0368and α1 is estimated to be 0.1253 approximately.3.3.2 Constant Relative Risk Aversion Utility FunctionI present the result of the estimation procedure, defined in Chapter 2, applied to the household datafrom the PSID dataset when the CRRA utility function is assumed as the household preference inTable 3.1. From Table 3.1, we observe that the estimated curvature parameter, σ, of the CRRAutility function is 3.26. The bootstrap standard error for the curvature parameter of the CRRAutility function is 0.11. Moreover, the bias-corrected estimator for the curvature parameter of theCRRA utility function is 3.40. I follow the estimation procedure in Chapter 2. The bootstrap biascorrection successfully eliminates the downward bias of the estimator for the curvature parameterof the CRRA utility function, as described in Chapter 2, Section 2.4.1.The stock market participation and stock-holding share adjustment costs are estimated usingthe estimated household utility function from the second step of the estimation procedure, as inChapter 2, Section 2.3.4. Furthermore, the stock market participation and stock-holding shareadjustment costs are modeled as the utility costs of the household dynamic model in Chapter 2.The stock market participation cost is estimated to be 8.18, and the stock-holding share adjustmentcost is estimated to be 1.85, as shown in Table 3.1. Moreover, the bootstrap standard error for the26stock market participation cost is 0.84, and the bootstrap standard error for the stock-holding shareadjustment cost is 0.03, as shown in Table 3.1. In addition, the bias-corrected estimator for thestock market participation cost is 9.33, and the bias-corrected estimator for the stock-holding shareadjustment cost is 1.87, as shown in Table 3.1. As we can see from the bootstrap standard errorsand the bootstrap 95% confidence intervals for the stock market participation and stock-holdingshare adjustment costs from Table 3.1, the estimation results for stock market participation andstock-holding share adjustment costs are statistically significant.3.3.3 The Quasi-Hyperbolic Discounting PreferenceI present in Table 3.2 the result of the estimation procedure, described in Chapter 2, applied tothe household data from the PSID dataset when the QHD preference is assumed as the householdpreference. The estimated curvature parameter, σ, of the QHD preference is 3.26, and the estimatedshort-run discount factor, δ, of the QHD preference is 0.95. The bootstrap standard error for thecurvature parameter of the QHD preference is 0.11, and the bootstrap standard error for the short-run discount factor of the QHD preference is 0.02. Furthermore, the bias-corrected estimator forthe curvature parameter of the QHD preference is 0.41, and the bias-corrected estimator for theshort-run discount factor of the QHD preference is 0.94. The QHD preference collapses into theCRRA utility function when the short-run discount factor, δ, of the QHD preference is equal to1, as discussed in Chapter 2. We can observe in Table 3.2 that the estimated short-run discountfactor of the QHD preference is not far from 1.The stock market participation and stock-holding share adjustment costs are estimated usingthe estimation result for the QHD preference as the household preference from the second step ofthe estimation procedure, as Chapter 2, Section 2.3.4 illustrates. In addition, the stock marketparticipation and stock-holding share adjustment costs are modeled as the utility costs of thehousehold dynamic model in Chapter 2 when the QHD preference is the household preference. Thestock market participation cost is estimated to be 8.21, and the stock-holding share adjustmentcost is estimated to be 1.85 (see Table 3.2). Furthermore, the bootstrap standard error for thestock market participation cost is 0.86, and the bootstrap standard error for the stock-holdingshare adjustment cost is 0.03, as shown in Table 3.2. Moreover, the bias-corrected estimator forthe stock market participation cost is 9.38, and that for the stock-holding share adjustment cost is1.87, as shown in Table 3.2. As we observe from the bootstrap standard errors and the bootstrap95% confidence intervals for the stock market participation and stock-holding share adjustmentcosts in Table 3.2, the estimation results for the stock market participation and stock-holding shareadjustment costs are statistically significant when the QHD preference is assumed as the householdpreference. By comparing Tables 3.1 and 3.2, I conclude that the estimated curvature parameter,stock participation, and stock-holding share adjustment costs are very similar across the CRRAutility function and the QHD preference.273.3.4 The Model FitIn this section, I discuss the fitness of the household dynamic model defined in Chapter 2 withthe real data. I present the estimation results of the household dynamic model from the PSIDdataset in the previous section. Moreover, I compare the important features of the household datafrom the PSID dataset with the model predictions from the estimated household dynamic model inthis section. Here, I assume the CRRA utility function as the household preference. However, themodel predictions from the household dynamic model with the QHD preference are very similarto those from the household dynamic model with the CRRA utility function, as I discussed in thenext sections.The household consumption data is smoothed out by the kernel regression method to mitigatethe measurement error issue in the consumption data5. Household data from the PSID datasetindicate that stock-holding households experienced almost twice the percentage decline in their con-sumption than what households with no stock holdings experienced between 2006 and 2008. Thisobservation aligns with the predictions from the estimated household dynamic model as shown inTable 3.3. Furthermore, households with no stock holdings and with lower than median wealthexperienced a smaller percentage decline in their consumption compared with stock-holding house-holds with higher than median wealth, based on real data and the model predictions. Stock-holdinghouseholds with a lower than median total income experienced a 3 to 4 times larger percentage dropin their consumption relative to households with no stock holdings and with lower than mediantotal income between 2006 and 2008, based on real data and the model predictions, as shown inTable 3.4. In addition, stock-holding households with higher than median total income experiencedapproximately 1.7 times larger percentage decrease in their consumption between 2006 and 2008as compared with households with no stock holdings and with higher than median total income,as both real data and model predictions demonstrate in Table 3.4. Tables 3.3 and 3.4 both revealthat stock-holding households experienced a much larger percentage decline in their consumptioncompared with households with no stock holdings, based on real data from the PSID dataset andthe model predictions from the estimated household dynamic model.Table 3.5 reveals how the distribution of the stock-holding share among households changedbetween 2006 and 2008. The fraction of stock-holding households whose stock-holding shares werebelow 40% increased between 2006 and 2008 based on both real data and model predictions, asshown in Table 3.5. Meanwhile, the fraction of stock-holding households whose stock-holding shareswere above 40% decreased between 2006 and 2008 based on both real data and model predictions.The household dynamic model predicted a slightly higher fraction of stock-holding householdswhose stock-holding shares were below 40% and a slightly lower fraction of stock-holding householdswhose stock-holding shares were above 40% in both 2006 and 2008 compared with the real data,as shown in Table 3.5. The real household data from the PSID dataset and the predictions fromthe estimated household dynamic model both indicate that the distribution of the stock-holding5Runkle (1991) estimates that 76% variation in the growth rate of food consumption in the PSID dataset is dueto measurement error.28share among stock-holding households shifted downward in 2008 as compared with 2006. I concludethat the estimated household dynamic model captures well the distribution of the stock-holdingshare among stock-holding households except for the stock-holding households whose stock-holdingshares were between 80% and 100%. These stock-holding households usually come from the verytop part of the wealth distribution, as the PSID and the SCF datasets indicate.Households with lower than median wealth, that did not hold any stock in the previous wave ofthe PSID dataset, had a very low participation rate in the stock market in both 2006 and 2008, asshown in Table 3.6. Meanwhile, households with higher than median wealth, that participated inthe stock market in the previous wave of the PSID dataset, had a very high participation rate inthe stock market in both 2006 and 2008. The predictions from the estimated household dynamicmodel and the household data from the PSID dataset both reveal that households that participatedin the stock market in the previous wave of the PSID dataset participated more in the stock marketin the current period in compared with households that did not hold any stock in the previous waveof the PSID dataset, as shown in Table 3.6. Moreover, the household data from the PSID datasetindicate that the highest decrease in the stock market participation rate between 2006 and 2008occurred among households with lower than median wealth that participated in the stock market inthe previous wave of the PSID dataset. The estimated household dynamic model predicted similarchanges in the stock market participation rate among these households. I find that the estimatedhousehold dynamic model captures the stock market participation rate among households well,except for very wealthy households.One possible explanation for why the model does not predict well the stock market participationrate among wealthy households is that there may exist significant heterogeneities in the householdpreference and the stock market participation cost among households. In the household dynamicmodel developed in Chapter 2, I assumed that the household preference and the stock marketparticipation cost are homogeneous among households. If the household preference parameter andthe stock market participation cost for wealthy households are substantially different from thehousehold preference parameter and the stock market participation cost for other households, thehousehold dynamic model, developed in Chapter 2, may not be able to capture well the stock marketparticipation rate among wealthy households. As a potential future research topic, I plan to addressthis issue and construct the extended household dynamic model with the heterogeneous householdpreference and stock market participation cost parameters in the future. The estimation procedurefor this extended household dynamic model may get much more complex than the estimationprocedure employed in Chapters 2 and 3.293.4 The Effects of the Stock Market Collapse and the Zero-LowerBound Monetary Policy on Household Consumption3.4.1 The Effect of the Stock Market Collapse on Household ConsumptionThe percentage change in consumption driven by both the drop in stock market return and theincrease in the expectation of the future volatility of stock market returns between 2006 and 2008is calculated using Equation (3.3), defined asc(·, r2008, σr2008)c(·, r2006, σr2006)− 1 (3.3)where r2006 is the stock market return in 2006; σr2006 is the expected future volatility of the stockmarket return in 2006; r2008 is the stock market return in 2008; σr2008 is the expected future volatilityof the stock market return in 2008; and c(·) is the full model solution consumption function. In thischapter, I use the consumption function from the full model solution, not the nonparametricallyestimated consumption function in the first step of the estimation procedure, to evaluate the effectsof the stock market collapse and the Zero-Lower Bound monetary policy on household consumptiondecision during the GFC. The full model solution consumption function indicates that the dropin the stock market return and the increase in the expected future volatility of the stock marketreturn between 2006 and 2008 decreased the consumption of stock-holding households by 9.28%, onaverage, and the consumption of households with no stock holdings by 0.47%, on average, as shownin Table 3.8. The stock market collapse during the GFC hit the consumption of stock-holdinghouseholds particularly hard.Furthermore, the consumption of stock-holding households with higher stock-holding sharesdeclined more in percentage relative to that of stock-holding households with lower stock-holdingshares, based on the estimated model, as shown in Table 3.9. The reason is that the savings portfolioreturns of stock-holding households, that held relatively higher shares of stocks in their total savings,declined more. This created a larger wealth loss. Therefore, these households decreased theirconsumption more due to the income effect. Figure 3.7 illustrates that the relative change inconsumption, triggered by the drop in the stock market return and the increase in the expectedfuture volatility of the stock market return as a function of a cash-in-hand variable decreasesmore for stock-holding households with higher shares of stock holding when we fix a cash-in-hand variable. In fact, the consumption data from the PSID indicate that the consumption ofstock-holding households with relatively higher stock-holding shares decreased more in percentagecompared with the consumption of stock-holding households with relatively lower stock-holdingshares at the end of the GFC.The consumption of wealthy stock-holding households declined more in percentage as comparedwith the consumption of less wealthy stock-holding households due to the stock market collapseduring the GFC. Table 3.10 shows that the median consumption of stock-holding households withbelow median wealth decreased by 3.22% during the GFC. Meanwhile, the median consumption30of stock-holding households of above median wealth decreased by 8.49% during the GFC. Figure3.7 displays the relative change in consumption as a function of the cash-in-hand variable andthe stock-holding share. According to this figure, the relative change in consumption, triggeredby the drop in the stock market return and the increase in the expected future volatility of thestock market return, decreases more when the cash-in-hand variable (savings plus consumption)increases while fixing the stock-holding share. The empirical evidence from the SCF and the PSIDdatasets indicates that wealthy households tended to participate more in the stock market relativeto less wealthy households. Furthermore, the stock-holding shares tended to be higher for wealthierstock-holding households, on average. Therefore, the stock market collapse during the GFC had ahuge negative impact on the consumption of stock-holding households and an even greater negativeimpact on the consumption of wealthy stock-holding households.3.4.2 The Effect of the Increase in the Expected Future Volatility of StockMarket Returns on Household ConsumptionIn this section, I analyze the effect of the increase in the expected future volatility of stock marketreturns. The significant increase in household expectation of the future volatility of stock marketreturns is estimated by employing the ARCH model. During the GFC, the stock market returndropped sharply and the household expectation of the future volatility of the stock market returnincreased significantly. To disentangle the effects of these two events on household consumption, Idecompose the change in households’ consumption decisions caused by the drop in the stock marketreturn and the increase in the expected future volatility of stock market returns between 2006 and2008 into two parts.c(·, r2008, σr2008)− c(·, r2006, σr2006) =c(·, r2008, σr2008)− c(·, r2008, σr2006) + c(·, r2008, σr2006)− c(·, r2006, σr2006) (3.4)where c(·, r, σr) is a household’s policy function for consumption obtained from the model estima-tion. Equation (3.4) demonstrates that the change in households’ consumption decisions causedby the drop in the stock market return and the increase in the expected future volatility of thestock market return between 2006 and 2008 can be expressed as the sum of two components. Thefirst component, c(·, r2008, σr2008) − c(·, r2008, σr2006), indicates how the change in the stock marketreturn between 2006 and 2008 affected households’ consumption decisions. The second compo-nent, c(·, r2008, σr2006)− c(·, r2006, σr2006), reveals how the change in the expected future volatility ofstock market returns between 2006 and 2008 affected households’ consumption decisions. Thesetwo components are divided by the total change in consumption, c(·, r2008, σr2008)− c(·, r2006, σr2006),driven by the decline in the stock market return and the increase in the expected future volatilityof stock market returns between 2006 and 2008, while keeping the other state variables fixed. This31decomposition is expressed in the following equations:c(·, r2008, σr2008)− c(·, r2008, σr2006)c(·, r2008, σr2008)− c(·, r2006, σr2006)(3.5)c(·, r2008, σr2006)− c(·, r2006, σr2006)c(·, r2008, σr2008)− c(·, r2006, σr2006). (3.6)Equation (3.5) demonstrates the contribution of the change in the stock market return between 2006and 2008 to the total change in consumption between 2006 and 2008, driven by both the decline inthe stock market return and the increase in the expected future volatility of stock market returnsbetween 2006 and 2008. Equation (3.6) illustrates the contribution of the change in the expectedfuture volatility of stock market returns between 2006 and 2008 to the total change in consumptionbetween 2006 and 2008, driven by both the decline in the stock market return and the increase inthe expected future volatility of stock market returns between 2006 and 2008. Adding up Equations(3.5) and (3.6) will equal to 1. This makes the interpretation easier and more comparable acrossdifferent analyses.In Table 3.11, using Equations (3.5) and (3.6), I illustrate how the drop in the stock marketreturn and the increase in the expected future volatility of stock market returns between 2006and 2008 contributed to the change in household consumption. The second (third) column ofTable 3.11 indicates the median contribution of the change in the stock market return (of thechange in the expected future volatility of stock market returns) between 2006 and 2008 to thetotal change in household consumption driven by the stock market collapse for different householdgroups. Table 3.11 reveals that the majority of the decline in household consumption was driven bythe huge drop in the stock market return during the GFC. Nevertheless, the third column of Table3.11 demonstrates that consumption of households with lower than median wealth was affectedsignificantly by the increase in the expected future volatility of stock market returns as comparedwith the consumption of households with higher than median wealth. The median contributionof the increase in the expected future volatility of stock market returns to the total changes inconsumption of households with lower (higher) than median wealth was 7.3% (0.4%), as shown inTable 3.11. This indicates that households with higher wealth are more tolerant of the increase inthe expected future volatility of stock market returns as compared with less wealthy households,as the standard economic theory suggests. Moreover, the consumption of households with lowerthan median total income was affected more by the increase in the expected future volatility ofstock market returns between 2006 and 2008 relative to the consumption of households with higherthan median total income, as shown in Table 3.12. The expected future income of householdswith a higher current total income usually tends to be higher than the expected future income ofhouseholds with a lower current total income. This means that a household with a higher currenttotal income has insurance against bad realizations of the stock market return in the next periodthrough a higher expected total income. Therefore, they do not fear the increase in the expectedvolatility of the stock market return for the next period as much as households with low total32incomes do.The signs are opposite in the second and third columns of Table 3.11. Households’ interestincome from stock holdings declined sharply when the stock market return dropped greatly duringthe GFC, causing huge losses to stock-holding households. A decline in stock market returns affectshousehold consumption through two channels. When the stock market returns drop, income fromthe stock market also decreases. Furthermore, the expected stock market return for the next periodalso decreases according to the ARCH model of the stock market return process. This implies thatthe expected income from the stock market for the next period also decreases. The income effectof the sharp decline in the stock market return during the GFC decreased both household savingsand consumption substantially. In addition, when stock market returns decrease, household savingsbecome less profitable than before, as the household expectation of the stock market return for thenext period drops according to the ARCH model. Therefore, as a substitution effect, householdshave an incentive to consume more instead of saving. Regarding the stock market collapse duringthe GFC, the income effect of the stock market return decline dominated greatly over its substitutioneffect, as empircal evidence reveals huge drops in both household savings and consumption. Insummary, the huge decline in stock market returns during the GFC triggered a significant decreasein household consumption.An increase in household expectation of the future volatility of stock market returns affectshousehold consumption decisions mostly through the substitution effect. The reason is that thechange in the expected future volatility of stock market returns does not have an immediate impacton the current income from stock holdings. When households’ expectation of the future volatilityof stock market returns increases, risk-averse households become fearful and do not find savingsas attractive as before. Consequently, risk-averse households allocate more funds for consumptioninstead of savings as a substitution effect. Fligstein and Rucks-Ahidiana (2016) documents that richfamilies or families that were above the 80th percentile of the wealth distribution actually becamericher after the GFC. Table 3.13 indicates one channel through which rich households recovered wellcompared to less wealthy households after the GFC. The increase in the expected future volatilityof stock market returns did not trigger the substitution effect among very wealthy households. Thismeans that they did not reduce their savings as a response to the increase in the expected futurevolatility of stock market returns during the GFC, contrary to what the less wealthy households did.Once the stock market recovered after the GFC, wealthy households were able to get back to theirformer wealth level faster than less wealthy households did because they did not cut their savingsas much as their less wealthy counterparts as the substitution effect, triggered by the increase inthe expected future volatility of stock market returns.3.4.3 The Effect of the Increase in the Expected Future Volatility of StockMarket Returns on Households’ Stock HoldingIn this section, I evaluate the contribution of the increase in the expected future volatility of stockmarket returns to the change in the distribution of the stock-holding share using the equations33defined below:CCP (·, r2008, σr2008)− CCP (·, r2008, σr2006)CCP (·, r2008, σr2008)− CCP (·, r2006, σr2006)(3.7)CCP (·, r2008, σr2006)− CCP (·, r2006, σr2006)CCP (·, r2008, σr2008)− CCP (·, r2006, σr2006)(3.8)where CCP (·, r, σr) is the conditional choice probability function of the stock-holding share. Equa-tion (3.7) indicates how an increase in the expected future volatility of stock market returns con-tributed to the change in the distribution of the stock-holding share, while Equation (3.8) expressesthe contribution of a decline in the stock market return. Adding up Equations (3.7) and (3.8) willresult in 1 or 100%.In general, the increase in the expected future volatility of stock market returns played a verysmall role in households’ stock market participation decisions during the GFC, as shown in Table3.14. However, the increase in the expected future volatility of stock market returns played amore substantial role in the stock market participation decision of households that held stock inthe previous wave of the PSID relative to those that did not during the GFC. The reason is thathouseholds that held stocks were probably entering the current period with huge income losses, asa result of the large negative shock to the stock market in 2008, as compared with households thatdid not hold stocks in the previous period. Therefore, they had more incentives to withdraw fromthe stock market in the current period, as driven by the decline in the expected stock market returnand the increase in the expectation of future volatility of stock market returns. In addition, theyfeared the increased future volatility of stock market returns more, as they were more reluctant toface a repetition of losses from the stock market in the current and subsequent periods. In summary,the decrease in the households’ stock market participation rate during the GFC was almost entirelydriven by the drop in stock market returns.The median contribution of the increase in expectation of the future volatility of stock marketreturns was approximately 2.3% (2.7%) to the change in the proportion of households whose stock-holding share was below 20% (between 20% and 40%), as triggered by both the decline in thestock market return and the increase in expected future volatility of stock market returns forthe next period, as Table 3.15 illustrates. Meanwhile, the median contribution of the increasein expectation of future volatility of stock market returns was approximately 4.6% (7.0%) to thechange in the proportion of households whose stock-holding share was between 40% and 60%(between 60% and 80%), as triggered by both the decline in stock market returns and the increasein expected volatility of stock market returns for the next period. Therefore, the proportion ofstock-holding households that chose a relatively high stock-holding share was affected relativelymore by the increase in expected future volatility of stock market returns, as illustrated in Table3.15. The likely explanation is that stock-holding households that chose a relatively high level ofstock holding shares were more exposed to the volatility of stock market returns for the next period,as their portfolio return depended more on the next period’s stock market returns, in contrast tostock-holding households that chose a relatively low-level stock holding share.343.4.4 The Effect of the Drop in the Risk-Free Rate on HouseholdConsumption and Stock Holding DecisionsI evaluate the effect of the change in the risk-free rate on household consumption and stock holdingdecisions using the following equations:c(·, rf2008)− c(·, rf2006)c(·, rf2006)(3.9)CCP (·, rf2008)− CCP (·, rf2006)CCP (·, rf2006)(3.10)where c(·, rf ) is the household consumption function; CCP (·, rf ) is the conditional choice proba-bility function of the stock-holding share; rf2006 is the risk-free rate in 2006; and rf2008 is the risk-freerate in 2008. Equation (3.9) demonstrates how the drop in the risk-free rate between 2006 and2008 affected household consumption and Equation (3.10) illustrates how the drop in the risk-freerate between 2006 and 2008 affected the conditional choice probability of the stock-holding share.As shown in Table 3.17, the drop in the risk-free rate, due to the expansionary monetary policy,led to an increase in the stock market participation rate for households with lower than medianwealth by 3.3% to 5.7% depending on the stock-holding status in the previous period. The stockmarket participation rate among households with higher than median wealth actually decreaseddue to the drop in the risk-free rate between 2006 and 2008, as shown in Table 3.17. The estimatedmodel indicates that wealthy households increasingly exited the stock market in response to the dropin the risk-free rate during the GFC, while, households with lower than median wealth increasinglyentered the stock market. Households with lower than median wealth tended to invest in risk-freeassets and not participate in the stock market. The drop in the risk-free rate deteriorated theattractiveness of risk-free assets, as the return from these assets declined during the GFC. As aresult, this might have driven such households to participate increasingly in the stock market. Insummary, the drop in the risk-free rate between 2006 and 2008 increased (decreased) the stockmarket participation rate of households with lower (higher) than median wealth.The drop in the risk-free rate between 2006 and 2008 decreased the consumption of stock-holding households by a much larger percentage, as compared with that of households with nostock holdings during the GFC. The drop in the risk-free rate between 2006 and 2008 triggered a3.7% (13.7%) decrease, on average, in the consumption of households without (with) stock holdings,as shown in Table 3.16. Hence, the regime change of the monetary policy during the GFC triggereda much larger percentage decline in the consumption of stock-holding households versus those withno stock holdings. Stock-holding households, on average, tended to have much more wealth than nostock holding households in average. The real annualized stock market return was −37.3% in 2008.Due to the Zero-Lower Bound monetary policy, the nominal risk-free rate declined to a level veryclose to zero in 2008. Since the inflation rate in 2008 was positive, the real risk-free rate in 2008became negative. For stock-holding households, this negative real risk-free rate, combined with35the huge negative stock market return resulted in tremendous income losses. The model solutionindicates that the marginal change in the consumption function with respect to the risk-free rate∂c(·,xt;rf )∂rfincreases in a cash-in-hand variable, xt. In other words, the MPC,∂c(·,xt;rf )∂xt, increases withthe risk-free rate, rf . Hence, the percentage decrease in consumption of households with relativelyhigher wealth is higher than that of households with relatively lower wealth when the risk-freerate, rf , declines. Household wealth and the cash-in-hand variable (consumption plus savings),xt, are highly positively correlated. The estimated model predicted that the percentage declinein consumption triggered by the permanent drop in the risk-free rate increases in a cash-in-handvariable, xt, as Figure 3.8 illustrates. Therefore, the drop in the risk-free rate triggered a largerpercentage decrease in the consumption of stock-holding households because they tended to havemore wealth than the average household with no stock holdings.According to the estimated model, the drop in the risk-free rate during the GFC triggered aslight increase in the consumption of households at the very bottom of the wealth distribution. Thedrop in the risk-free rate, driven by the monetary policy regime change, triggered, on average, a2.1% increase in the consumption of households whose wealth level was below the 10th percentile,as shown in Table 3.19. These households were most likely struggling to finance their consumption.Their financing cost for consumption decreased due to the drop in the risk-free rate during the GFC,which enabled them to increase their consumption. The Zero-Lower Bound monetary policy had apositive effect on the economy by generating lower borrowing costs for households that needed tofinance their consumption. Nevertheless, the consumption of wealthy households was affected mostsignificantly by the drop in the risk-free rate during the GFC, which, driven by the Zero-LowerBound monetary policy, triggered a large percentage decline in the consumption of households atthe top part of the wealth distribution at that time. From the results discussed above, we canconclude that generally, the drop in the risk-free rate during the GFC triggered larger percentagechanges in household consumption compared with the percentage changes caused by the drop inthe stock market return. The main reason is that the drop in the risk-free rate persisted until theend of 2016, the stock market return bounced back from reaching a low value during the GFC inthe next few years, as Figure 3.9 illustrates.3.4.5 The Effects of the Stock Market Collapse and the Zero-Lower BoundMonetary Policy with the Quasi-Hyperbolic Discounting PreferenceIn this section, I analyze the effects of the stock market collapse and the Zero-Lower Bound mon-etary policy of the GFC on household consumption and stock market participation decisions byassuming the QHD preference as the household preference. Table 3.21 demonstrates that the QHDpreference predicted almost the same percentage declines in household consumption as the aver-age effect, triggered by the stock market collapse during the GFC, as the CRRA utility functionpredicted. The QHD preference predicted that the consumption of stock-holding households withhigher stock-holding shares declined more, percentagewise, relative to the consumption of stock-holding households with lower stock-holding shares, as shown in Table 3.22.36The median increase in expected future volatility of stock market returns between 2006 and 2008accounted for 7.3% (0.45%) of the total change in consumption of households with below (above)median wealth, triggered by the stock market collapse of the GFC, as shown in Table 3.23. Similarto the CRRA utility function, the QHD preference also predicted that the consumption decisions ofwealthy households were not affected by the increase in the expected future volatility of the stockmarket return during the GFC, as shown in Table 3.25. On the contrary, the consumption decisionsof households at the bottom of the wealth distribution were affected significantly, as illustrated inTable 3.25. The extent of these effects is very similar to the predictions of the CRRA utilityfunction.Furthermore, the QHD preference predicted that the drop in the risk-free rate between 2006and 2008 decreased, on average, the consumption of households without (with) stock holdings by3.26% (13.83%). Compared with the prediction of the CRRA utility function, the QHD preferencepredicted a slightly lower percentage decline in household consumption as the average effect. Inaddition, the QHD preference predicted that the drop in the risk-free rate during the GFC increasedthe consumption of households with wealth below the tenth percentile by 5.0%, on average, as shownin Table 3.20. Moreover, the QHD preference predicted a higher percentage increase, on average,in the consumption of households with wealth below the tenth percentile relative to the predictionof the CRRA utility function.In general, the predictions of the QHD preference are very similar to those of the CRRA utilityfunction. However, the main distinction between these predictions regarding the percentage changesin household consumption, driven by the stock market collapse and the monetary policy change, isthat the QHD preference predicted larger percentage increases in the consumption of householdswith wealth below the 10th percentile, triggered by the drop in the risk-free rate between 2006and 2008, as compared with the prediction of the CRRA utility function, as Tables 3.20 and 3.19illustrate. The QHD preference (CRRA utility function) predicted that the drop in the risk-freerate during the GFC increased the consumption of households below the 10th percentile wealthby 5.0% (2.1%), on average. The reason behind this difference is that households with the QHDpreference tended to choose higher consumption for the current period than households with theCRRA utility function because they put more weight on the current period’s utility relative to thefuture utility. I explain this feature of the QHD preference using the following equation:c(dt, dt−1, xt, st; rf )−σ = E{ ∑d′t+1∈DPr{d′t+1|dt, xt+1, st+1; rf}(1 + rf + dt(rt+1 − rf ))[βδ∂c(dt+1, dt, xt+1, st+1; rf )∂xt+1+ β(1− ∂c(dt+1, dt, xt+1, st+1; rf )∂xt+1)]c(dt+1, dt, xt+1, st+1; rf )−σ}(3.11)where c(·) is the household consumption function; dt is the stock-holding share; xt is a cash-in-hand variable; st is a set of the other exogenous state variables; rf is the risk-free rate; Pr{·} is the37conditional choice probability function of the stock-holding share; rt is the stock market return; βis the long-run discount factor; δ is the short-run discount factor; and σ is the curvature parameterof the utility function. Furthermore, I define the effective discount factor by following Harris andLaibson (2002) as[βδ∂c(dt+1, dt, xt+1, st+1; rf )∂xt+1+ β(1− ∂c(dt+1, dt, xt+1, st+1; rf )∂xt+1)]. (3.12)The feature of the QHD preference I discussed above is observed prominently among households atthe bottom of the wealth distribution. The MPC,∂c(dt+1,dt,xt+1,st+1;rf )∂xt+1, is higher for households atthe bottom of the wealth distribution. As we see from Equation 3.12, the effective discount factoris lower for poor households because their MPCs are higher, and the short-run discount factor, δ, islower than 1. This means that households at the bottom of the wealth distribution put less weighton the future marginal utility, as compared with households at the top part of the wealth distri-bution, as shown in Equation 3.11. This extra mechanism of the QHD preference predicted largerpercentage changes in the consumption of households at the bottom of the wealth distribution,driven by the drop in the risk-free rate, as compared with the prediction of the CRRA utility func-tion. In addition, the net worth of households with below the 10th percentile wealth was negative.This means that these households financed their consumption by borrowing. Moreover, in general,they did not participate in the stock market, as the PSID and the SCF datasets demonstrate.Hence, the drop in the risk-free rate lowered the financing costs of consumption. In other words,the decline in the risk-free rate triggered positive income effects among households that borrowedto finance their consumption during the GFC, enabling them to increase their consumption. Ac-cording to Laibson (1998) and Geraats (2005), the QHD preference creates a stronger net incomeeffect when the income effect is positive, as compared with the CRRA utility function. Therefore,the QHD preference predicted a larger percentage increase in the consumption of households belowthe 10th percentile wealth level compared with the prediction of the CRRA utility function, as aresult of the decline in the risk-free rate during the GFC. Meanwhile, the predictions of the QHDpreference are very similar to those of the CRRA utility function for households at the middleand top parts of the wealth distribution regarding the effect of the drop in the risk-free rate onhousehold consumption.3.4.6 The Impact of the Stock Market Collapse and the Zero-Lower BoundMonetary Policy on the Household Consumption DistributionIn this section, I discuss how the stock market collapse and the Zero-Lower Bound monetary policyduring the GFC affected the household consumption distribution. If we look at the relative changesof average consumption in different percentile intervals of the household consumption distributionbetween 2006 and 2008 (second row of Table 3.34), we can observe that the percentage decrease inthe average consumption for the upper percentile intervals was larger than the that for the lowerpercentile intervals. Specifically, the percentage decrease in the average consumption between the3880th and the 100th percentiles was 12.19% based on the PSID dataset. Moreover, the percentagedecrease in the average consumption between the 60th percentile and the 80th percentiles was11.32%. Meanwhile, the percentage decrease in the average consumption between the 20th and the40th percentiles was 7.71%, and the percentage decrease in the average consumption between the 0thand the 20th percentiles was 3.97% based on the PSID dataset, as shown in Table 3.34. However,these relative changes in average consumption do not take into account the fact that householdincome, wealth, and stock-holding share distributions changed between 2006 and 2008. Moreover,I evaluate the effect of the stock market collapse and the Zero-Lower Bound monetary policy onhousehold consumption distribution using the household income, wealth, and stock-holding sharedistributions in 2008 and the estimated consumption function from the household dynamic modeldisplayed in the third row of Table 3.34. I input the stock market returns, the risk-free rates, andthe expected future volatilities, estimated by the ARCH model, of 2006 and 2008 into the estimatedconsumption function from the household dynamic model while fixing the other household statevariables at 2008 values. This way, I can measure the effect of the stock market collapse andthe Zero-Lower Bound monetary policy on the household consumption distribution because in thismethodology, the wealth, income, and stock-holding share distributions do not change between2006 and 2008.Attanasio and Pistaferri (2016) documents that the inequality in food consumption declinedduring the GFC, even though the inequality in food consumption increased over time as a generaltrend based on data from the PSID. Furthermore, Meyer and Sullivan (2017) shows that consump-tion inequality decreased significantly during the GFC. In addition, Meyer and Sullivan (2017)demonstrates that income inequality decreased slightly compared with the decrease in consump-tion inequality during the GFC. The change in income inequality is usually the major explanationfor the change in consumption inequality in the household finance literature. However, the sub-stantial decline in consumption inequality during the GFC cannot be fully explained by the changein income inequality since the income inequality decreased only slightly during the GFC, as thedata from the PSID illustrates. In this chapter, I identify that the significant driving forces behindthis substantial decline in consumption inequality during the GFC were the stock market collapseand the monetary policy switch to the Zero-Lower Bound monetary policy. In fact, Table 3.34indicates that the average consumption between the 80th percentile and the 100th percentile (60thpercentile and the 80th percentile) of the household consumption distribution decreased by 8.82%(7.98%) due to the stock market collapse and the Zero-Lower Bound monetary policy during theGFC. Meanwhile, the average consumption between the 20th percentile and the 40th percentile (0thpercentile and the 20th percentile) of the household consumption distribution decreased by 3.33%(1.07%) due to the stock market collapse and the Zero-Lower Bound monetary policy during theGFC, as shown in Table 3.34. The stock market collapse and the Zero-Lower Bound monetary pol-icy of the GFC triggered larger percentage declines in the consumption of wealthier households asdiscussed in the previous sections. Therefore, consumption inequality among households declinedsubstantially during the GFC as a result of the stock market collapse and the Zero-Lower Bound39monetary policy.3.5 ConclusionThe consumption inequality among households in the United States decreased substantially duringthe GFC. In the existing literature, the change in consumption inequality is usually linked to thechange in income inequality. However, the income inequality among households in the United Statesdecreased slightly as compared with the decrease in consumption inequality among households inthe United States during the GFC. This chapter identifies that the stock market collapse and theZero-Lower Bound monetary policy during the GFC were two of the main driving forces behindthis substantial decrease in consumption inequality among households in the United States duringthe GFC. The huge drop in the stock market return and the decline in the risk-free rate due to theZero-Lower Bound monetary policy during the GFC triggered a large percentage decrease in theconsumption of stock-holding households, that tended to be at the top part of the wealth distri-bution. Meanwhile, the decline in the risk-free rate during the GFC triggered a slight percentageincrease in the consumption of households with negative net worth that were at the bottom part ofthe wealth distribution. Furthermore, this chapter shows that the increase in the expected futurevolatility of stock market returns during the GFC almost did not affect the consumption decisionsof wealthy households. Meanwhile, the consumption decisions of the less wealthy households wereaffected significantly by the increase in the expected future volatility of stock market returns duringthe GFC.3.6 TablesTable 3.1:The Constant Relative Risk Aversion Utility FunctionParameter EstimatorBootstrapstandard errorBootstrap 95%Confidence IntervalBiasCorrectedEstimatorσ 3.26 0.11 (3.22, 3.60) 3.40AC 8.18 0.84 (7.77, 10.58) 9.33PC 1.85 0.03 (1.83, 1.93) 1.87Note: σ denotes the curvature parameter of the Constant Relative Risk Aversionutility function, AC denotes the stock holding share adjustment cost, and PCdenotes the stock market participation cost.40Table 3.2: The Quasi-Hyperbolic Discounting PreferenceParameter EstimatorBootstrapstandard errorBootstrap 95%Confidence IntervalBiasCorrectedEstimatorσ 3.26 0.11 (3.22, 3.60) 3.41δ 0.95 0.02 (0.91, 0.97) 0.94AC 8.21 0.86 (7.78, 10.64) 9.38PC 1.85 0.03 (1.83, 1.93) 1.87Note: σ denotes the curvature parameter of the Quasi-Hyperbolic Discountingpreference, δ denotes the short-run discount factor, AC denotes the stock holdingshare adjustment cost, and PC denotes the stock market participation cost.Table 3.3: The Relative Change in Average Consumptionof Households between 2006 and 2008 WavesWealth level Stock holding status Data Modellower than median no stock holding -5.61% -3.42%lower than median stock holding -12.00% -6.10%higher than median no stock holding -6.93% -6.39%higher than median stock holding -12.54% -13.88%Note: The consumption data are kernel smoothed to mitigate the measure-ment error problem and the consumption data from the 2006 and 2008 wavesof the Panel Study of Income Dynamics dataset are used.Table 3.4: The Relative Change in Average Consumptionof Households between 2006 and 2008 WavesTotal income Stock holding status Data Modellower than median no stock holding -3.76% -4.19%lower than median stock holding -11.82% -17.23%higher than median no stock holding -7.60% -7.74%higher than median stock holding -12.72% -13.28%Note: The consumption data are kernel smoothed to mitigate the measure-ment error problem and the consumption data from the 2006 and 2008 wavesof the Panel Study of Income Dynamics dataset are used.Table 3.5: The Distribution of Stock Holding Share among StockHolding Households(0%, 20%) (20%, 40%) (40%, 60%) (60%, 80%) (80%, 100%)2006 Data 44.57% 18.92% 17.00% 11.14% 8.36%2008 Data 46.06% 22.52% 14.94% 9.31% 7.16%2006 Model 48.51% 24.32% 16.30% 8.56% 2.31%2008 Model 51.43% 25.33% 13.71% 7.43% 2.10%Note: The first row indicates the intervals of the stock holding shares. For example,(0%, 20%) means the interval of the stock holding share where the stock holding shareis between 0% and 20%.41Table 3.6: The Stock Market Participation Rate: 2006 versus 2008Wealth levelStock holding status(previous period)2006 Data 2008 Data 2006 Model 2008 Modellower than median no stock holding 5.86% 4.78% 7.60% 8.51%lower than median stock holding 51.12% 45.22% 43.95% 41.18%higher than median no stock holding 23.54% 22.39% 12.03% 4.58%higher than median stock holding 80.98% 78.78% 53.74% 32.86%Note: The stock market participation rates in 2006 and 2008 for different household groups are displayedin this table.Table 3.7: The Stock Market Participation Rate: 2006 versus 2008Total incomeStock holding status(previous period)2006 Data 2008 Data 2006 Model 2008 Modellower than median no stock holding 8.82% 7.13% 10.59% 7.20%lower than median stock holding 69.55% 66.94% 57.82% 32.79%higher than median no stock holding 16.97% 16.70% 7.35% 7.02%higher than median stock holding 77.21% 71.75% 48.38% 36.24%Note: The stock market participation rates in 2006 and 2008 for different household groups are displayedin this table.Table 3.8: The Average Effect of the StockMarket Collapse on ConsumptionStock holding statusRelative change inconsumptionno stock holding -0.47%stock holding -9.28%all households -3.63%Note: The average effect of the drop in the stock marketreturn and the increase in the expected future volatilityof the stock market return between 2006 and 2008 onhousehold consumption is calculated as using the esti-mated consumption function in this table.42Table 3.9: The Median Effect of the StockMarket Collapse on Stock HoldingHousehold’s ConsumptionStock holding shareRelative change inconsumption(0%, 20%) -6.56%(20%, 40%) -13.79%(40%, 60%) -21.96%(60%, 80%) -27.15%(80%, 100%) -35.10%Note: The median effect of the drop in the stockmarket return and the increase in the expected futurevolatility of the stock market return between 2006 and2008 on stock holding household’s consumption is calcu-lated using the estimated consumption function in thistable.Table 3.10: The Median Effect of theStock Market Collapse onConsumptionWealth levelRelative change inconsumptionbelow median -3.22%above median -8.49%Note: The median effect of the drop in thestock market return and the increase in the ex-pected future volatility of the stock market re-turn between 2006 and 2008 on stock holdinghousehold’s consumption is calculated using theestimated consumption function in this table.Table 3.11: The Median Contributions of the Stock MarketReturn Drop and the Expected Future Volatility Increase tothe Change in Consumption between 2006 and 2008Wealth levelDrop in stockmarket returnIncrease in expected stockmarket return volatilitybelow median 1.0733 -0.0733above median 1.0043 -0.0043Note: The median contributions of the change in the stock market return andthe change in the expected future volatility of the stock market return between2006 and 2008 to the total change in household consumption are calculated usingEquations (3.5) and (3.6).43Table 3.12: The Median Contributions of the Stock MarketReturn Drop and the Expected Future Volatility Increase tothe Change in Consumption between 2006 and 2008Total incomeDrop in stockmarket returnIncrease in expected stockmarket return volatilitybelow median 1.0730 -0.0730above median 1.0062 -0.0062Note: The median contributions of the change in the stock market return andthe change in the expected future volatility of the stock market return between2006 and 2008 to the total change in household consumption are calculated usingEquations (3.5) and (3.6).Table 3.13: The Median Contributions of the Stock MarketReturn Drop and the Expected Future Volatility Increase to theChange in Consumption between 2006 and 2008Wealth levelDrop in stockmarket returnIncrease in expected stockmarket return volatilityabove 80th percentile 1.0009 -0.0009above 70th percentile 1.0002 -0.0002below 40th percentile 1.0736 -0.0736below 30th percentile 1.0737 -0.0737Note: The median contributions of the change in the stock market return and thechange in the expected future volatility of the stock market return between 2006 and2008 to the total change in household consumption are calculated using Equations (3.5)and (3.6).Table 3.14: The Median Contributions of the Stock Market ReturnDrop and the Expected Future Volatility Increase to the Change inStock Market Participation Rate between 2006 and 2008Previous wave’s stockholding statusDrop in stockmarket returnIncrease in expected stockmarket return volatilityno stock holding 0.9997 0.0003stock holding 0.9707 0.0293all households 0.9971 0.0029Note: The median contributions of the change in the stock market return and the changein the expected future volatility of the stock market return between 2006 and 2008 to thetotal change in the stock market participation rate are calculated using Equations (3.7)and (3.8).44Table 3.15: The Median Contributions of the Stock MarketReturn Drop and the Expected Future Volatility Increase tothe Change in Distribution of Stock Holding Share between2006 and 2008Level of stockholding shareDrop in stockmarket returnIncrease in expected stockmarket return volatility(0%, 20%) 0.9773 0.0227(20%, 40%) 0.9728 0.0272(40%, 60%) 0.9539 0.0461(60%, 80%) 0.9301 0.0699Note: The median contributions of the change in the stock market return andthe change in the expected future volatility of the stock market return between2006 and 2008 to the total change in the distribution of the stock holding shareare calculated using Equations (3.7) and (3.8).Table 3.16: The Average Effect of the Dropin the Risk-Free Rate on HouseholdConsumptionStock holding statusRelative change inconsumptionno stock holding -3.68%stock holding -13.74%all households -7.30%Note: The average effect of the drop in the risk-freerate between 2006 and 2008 on household consumptionis calculated using Equation (3.9).Table 3.17: The Median Effect of the Drop in the Risk-Free Rate onthe Stock Market Participation Rate between 2006 and 2008Wealth levelPrevious wave’s stockholding statusRelative change in stockmarket participation ratelower than median no stock holding 3.28%lower than median stock holding 5.77%higher than median no stock holding -3.07%higher than median stock holding -2.19%Note: The median effect of the change in the risk-free rate between 2006 and 2008 onthe stock market participation rate is calculated using Equation (3.10).45Table 3.18: The Median Effect of the Dropin the Risk-Free Rate between 2006 and2008 on Household ConsumptionWealth levelRelative change inconsumptionbelow 20th percentile 0.59%below 15th percentile 0.96%below 10th percentile 1.61%Note: The median effect of the drop in the risk-freerate between 2006 and 2008 on household consumptionis calculated using Equation (3.9).Table 3.19: The Average Effect of theDrop in the Risk-Free Rate between 2006and 2008 on Household ConsumptionWealth levelRelative change inconsumptionbelow 20th percentile 1.20%below 15th percentile 1.59%below 10th percentile 2.11%Note: The average effect of the drop in the risk-freerate between 2006 and 2008 on household consumptionis calculated using Equation (3.9).Table 3.20: The Average Effect of theDrop in the Risk-Free Rate onConsumption of Household with LowWealthWealth levelRelative change inconsumptionbelow 20th percentile 2.66%below 15th percentile 2.53%below 10th percentile 5.01%Note: The average effect of the drop in the risk-freerate between 2006 and 2008 on household consump-tion is calculated using Equation (3.9) by assuming theQuasi-Hyperbolic Discounting Preference as the house-hold preference.46Table 3.21: The Average Effect of the StockMarket Collapse on HouseholdConsumptionStock holding statusRelative change inconsumptionno stock holding -0.45%stock holding -9.24%all households -3.61%Note: The average effect of the drop in the stock marketreturn and the increase in the increase in the expected futurevolatility of the stock market return between 2006 and 2008on household consumption is calculated using Equation (3.9)by assuming the Quasi-Hyperbolic Discounting Preference asthe household preference.Table 3.22: The Median Effect of the StockMarket Collapse on Stock HoldingHousehold’s ConsumptionStock holding shareRelative change inconsumption(0%, 20%) -6.56%(20%, 40%) -13.78%(40%, 60%) -21.94%(60%, 80%) -27.14%(80%, 100%) -35.10%Note: The median effect of the drop in the stockmarket return and the increase in the expected futurevolatility of the stock market return between 2006 and2008 on stock holding household’s consumption is cal-culated using the estimated consumption function byassuming the Quasi-Hyperbolic Discounting Preferenceas the household preference in this table.Table 3.23: The Median Contributions of the Stock MarketReturn Drop and the Expected Future Volatility Increase tothe Change in Consumption between 2006 and 2008Wealth levelDrop in stockmarket returnIncrease in expected stockmarket return volatilitybelow median 1.0731 -0.0731above median 1.0045 -0.0045Note: The median contributions of the change in the stock market return andthe change in the expected future volatility of the stock market return between2006 and 2008 to the total change in household consumption are calculatedusing Equations (3.5) and (3.6) by assuming the Quasi-Hyperbolic DiscountingPreference as the household preference.47Table 3.24: The Median Contributions of the Stock MarketReturn Drop and the Expected Future Volatility Increase tothe Change in Consumption between 2006 and 2008Total incomeDrop in stockmarket returnIncrease in expected stockmarket return volatilitybelow median 1.0728 -0.0728above median 1.0063 -0.0063Note: The median contributions of the change in the stock market return andthe change in the expected future volatility of the stock market return between2006 and 2008 to the total change in household consumption are calculated usingEquations (3.5) and (3.6) by assuming the Quasi-Hyperbolic Discounting Prefer-ence as the household preference.Table 3.25: The Median Contributions of the Stock MarketReturn Drop and the Expected Future Volatility Increase to theChange in Consumption between 2006 and 2008Wealth levelDrop in stockmarket returnIncrease in expected stockmarket return volatilityabove 80th percentile 1.0010 -0.0010above 70th percentile 1.0002 -0.0002below 40th percentile 1.0734 -0.0734below 30th percentile 1.0736 -0.0736Note: The median contributions of the change in the stock market return and thechange in the expected future volatility of the stock market return between 2006 and2008 to the total change in household consumption are calculated using Equations (3.5)and (3.6) by assuming the Quasi-Hyperbolic Discounting Preference as the householdpreference.Table 3.26: The Median Contributions of the Stock MarketReturn Drop and the Expected Future Volatility Increase to theChange in Consumption between 2006 and 2008Total incomeDrop in stockmarket returnIncrease in expected stockmarket return volatilityabove 80th percentile 1.0029 -0.0029above 70th percentile 1.0016 -0.0016below 40th percentile 1.0745 -0.0745below 30th percentile 1.0797 -0.0797Note: The median contributions of the change in the stock market return and thechange in the expected future volatility of the stock market return between 2006 and2008 to the total change in household consumption are calculated using Equations (3.5)and (3.6) by assuming the Quasi-Hyperbolic Discounting Preference as the householdpreference.48Table 3.27: The Median Contributions of the Stock Market ReturnDrop and the Expected Future Volatility Increase to the Change inStock Market Participation Rate between 2006 and 2008Previous wave’s stockholding statusDrop in stockmarket returnIncrease in expected stockmarket return volatilityno stock holding 0.9997 0.0003stock holding 0.9707 0.0293all households 0.9971 0.0029Note: The median contributions of the change in the stock market return and the changein the expected future volatility of the stock market return between 2006 and 2008 tothe total change in the stock market participation rate are calculated using Equations(3.7) and (3.8) by assuming the Quasi-Hyperbolic Discounting Preference as the householdpreference.Table 3.28: The Median Contributions of the Stock MarketReturn Drop and the Expected Future Volatility Increase tothe Change in Distribution of Stock Holding Share between2006 and 2008Level of stockholding shareDrop in stockmarket returnIncrease in expected stockmarket return volatility(0%, 20%) 0.9813 0.0187(20%, 40%) 0.9871 0.0129(40%, 60%) 0.9467 0.0533(60%, 80%) 0.9279 0.0721Note: The median contributions of the change in the stock market return andthe change in the expected future volatility of the stock market return between2006 and 2008 to the total change in the distribution of the stock holding shareare calculated using Equations (3.7) and (3.8) by assuming the Quasi-HyperbolicDiscounting Preference as the household preference.Table 3.29: The Median Effect of the Drop in the Risk-Free Rate onthe Stock Market Participation Rate between 2006 and 2008Wealth levelPrevious wave’s stockholding statusRelative change in stockmarket participation ratelower than median no stock holding 3.28%lower than median stock holding 5.78%higher than median no stock holding -3.07%higher than median stock holding -2.20%Note: The median effect of the change in the risk-free rate between 2006 and 2008 onthe stock market participation rate is calculated using Equation (3.10) by assuming theQuasi-Hyperbolic Discounting Preference as the household preference.49Table 3.30: The Median Effect of the Drop in the Risk-Free Rate onthe Stock Market Participation Rate between 2006 and 2008Total incomePrevious wave’s stockholding statusRelative change in stockmarket participation ratelower than median no stock holding 2.14%lower than median stock holding 1.35%higher than median no stock holding 4.83%higher than median stock holding -1.37%Note: The median effect of the change in the risk-free rate between 2006 and 2008 onthe stock market participation rate is calculated using Equation (3.10) by assuming theQuasi-Hyperbolic Discounting Preference as the household preference.Table 3.31: The Average Effect of the Dropin the Risk-Free Rate on HouseholdConsumptionStock holding statusRelative change inconsumptionno stock holding -3.26%stock holding -13.83%all households -7.06%Note: The average effect of the drop in the risk-freerate between 2006 and 2008 on household consumption iscalculated using Equation (3.9) by assuming the Quasi-Hyperbolic Discounting Preference as the household pref-erence.Table 3.32: The Median Effect of the Dropin the Risk-Free Rate on HouseholdConsumptionStock holding statusRelative change inconsumptionno stock holding -2.43%stock holding -12.28%all households -4.11%Note: The median effect of the drop in the risk-freerate between 2006 and 2008 on household consumption iscalculated using Equation (3.9) by assuming the Quasi-Hyperbolic Discounting Preference as the household pref-erence.50Table 3.33: The Median Effect of the Dropin the Risk-Free Rate on Consumption ofHousehold with Low WealthWealth levelRelative change inconsumptionbelow 20th percentile 0.59%below 15th percentile 0.96%below 10th percentile 1.62%Note: The median effect of the drop in the risk-freerate between 2006 and 2008 on household consump-tion is calculated using Equation (3.9) by assuming theQuasi-Hyperbolic Discounting Preference as the house-hold preference.Table 3.34: The Relative Change of Average Consumption inDifferent Percentile Intervals of the Household ConsumptionDistributionPercentileintervals(0, 20) (20, 40) (40, 60) (60, 80) (80, 100)Consumption Data (PSID) -3.97% -7.71% -10.30% -11.32% -12.19%Stock Market Collapse andZero-Lower Bound Policy-1.07% -3.33% -6.22% -7.98% -8.82%Note: In the second row, the relative changes of average consumptions in differentpercentile intervals of the household consumption distribution between 2006 and 2008are calculated using the observed consumption data from the Panel Study of IncomeDynamics dataset. In the third row, the relative changes of average consumptions indifferent percentile intervals of the household consumption distribution between 2006and 2008 are calculated by using the observed household state variables of 2008 from thePanel Study of Income Dynamics dataset and by inputting the stock market returns,the risk-free rates, and the expected future volatilities, estimated by the AutoregressiveConditional Heteroskedasticity model, of 2006 and 2008 into the estimated consumptionfunction from the household dynamic model.51Table 3.35: The Relative Change of Average Consumption inDifferent Percentile Intervals of the Household ConsumptionDistributionPercentileintervals(0, 20) (20, 40) (40, 60) (60, 80) (80, 100)Consumption Data (PSID) -3.97% -7.71% -10.30% -11.32% -12.19%Estimated Model Prediction -2.07% -7.20% -13.82% -17.95% -13.70%Note: In the second row, the relative changes of average consumptions in different per-centile intervals of the household consumption distribution between 2006 and 2008 arecalculated using the observed consumption data from the Panel Study of Income Dynam-ics dataset. In the third row, the relative changes of average consumptions in differentpercentile intervals of the household consumption distribution between 2006 and 2008 arecalculated using the observed state variables from the Panel Study of Income Dynamicsdataset and the estimated consumption function.3.7 FiguresYear1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018Stock Market Return-0.4-0.3-0.2-0.100.10.20.30.4Figure 3.1: The Annualized Stock Market ReturnNote: The annualized stock market return time-series data of the S&P500 Stock Price Index are adjusted for theinflation. 1% is converted as 0.01. Therefore, x% is converted as 0.01 · x.52Year1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018Stock Market Return-0.15-0.1-0.0500.050.10.150.20.25Figure 3.2: The Expected Stock Market ReturnNote: The expected annual stock market return for the S&P500 Stock Price Index is estimated by the AutoregressiveConditional Heteroskedasticity (ARCH) model. 1% is converted as 0.01. Therefore, x% is converted as 0.01 · x.53Year1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018Stock Market Return Volatility0.190.20.210.220.230.240.25Figure 3.3: The Expected Stock Market Return VolatilityNote: The expected annual stock market return volatility for the S&P500 Stock Price Index is estimated by theAutoregressive Conditional Heteroskedasticity (ARCH) model. 1% is converted as 0.01. Therefore, x% is convertedas 0.01 · x.-80-400408025 50 75 100 125 150 175 200 225 250(a) 2006-100-80-60-40-2002040608025 50 75 100 125 150 175 200 225 250(b) 2008Figure 3.4: The Daily S&P500 Stock Price IndexNote: The daily time-series data of the S&P500 Stock Price Index are filtered using The Hodrick-Prescott (HP)filter to eliminate a longer time-trend and display the high-frequency fluctuations.54.00.01.02.03.04.05.06.0700 01 02 03 04 05 06 07 08 09 10 11 12 13 14(a) The Federal Funds Rate.00.01.02.03.04.05.06.0700 01 02 03 04 05 06 07 08 09 10 11 12 13 14(b) The 3-Month Treasury Bill RateFigure 3.5: The Nominal Federal Funds and 3-Month Treasury Bill RatesNote: The nominal Federal Funds Rate and 3-Month Treasury Bill Rate are displayed in this figure. 1% is convertedas 0.01. Therefore, x% is converted as 0.01 · x.-.04-.03-.02-.01.00.01.02.03.041998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018Annualized ReturnYearFigure 3.6: The 3-Month Treasury BillNote: The 3-Month Treasury Bill Rate is annualized and adjusted for the inflation. 1% is converted as 0.01.Therefore, x% is converted as 0.01 · x.55cash-in-hand variable x t0 5 10 15 20 25 30 35relative change in consumption-0.16-0.14-0.12-0.1-0.08-0.06-0.04-0.020dt=00<dt<=0.20.2<dt<=0.40.4<dt<=0.60.6<dt<=0.80.8<dt<=1Figure 3.7: The Relative Change in ConsumptionNote: The relative change in consumption, triggered by both the drop in the stock market return, rt, and theincrease in the expected future volatility of the stock market return, σrt , between 2006 and 2008, as a function of acash-in-hand variable, xt, is plotted using the estimated consumption function in this figure. Different lines representdifferent levels of stock holding shares. Here, dt denotes the share of stock holding in household savings.56cash-in-hand variable x t0 5 10 15 20 25 30 35relative change in consumption-0.18-0.16-0.14-0.12-0.1-0.08-0.06-0.04-0.020dt=00<dt<=0.20.2<dt<=0.40.4<dt<=0.60.6<dt<=0.80.8<dt<=1Figure 3.8: The Relative Change in ConsumptionNote: The relative change in consumption, triggered by the drop in the risk-free rate, rf , between 2006 and 2008,as a function of a cash-in-hand variable (consumption plus savings), xt, is plotted using the estimated consumptionfunction in this figure. Different lines represent different levels of stock holding shares. Here, dt denotes the share ofstock holding in household savings.Figure 3.9: The Drop in the Stock Market Return versus the Drop in the Risk-Free RateNote: I plot the drop in the stock market return versus the drop in the risk-free rate during the 2007-2008 financialcrisis in this figure. The drop in the risk-free rate during the 2007-2008 financial crisis persisted until the end of 2016due to the Zero-Lower Bound monetary policy. On the other hand, the stock market return quickly bounced backfrom its extremely low level in the 2007-2008 financial crisis in next few years.57Chapter 4Heterogeneous Household Preference4.1 IntroductionIn the representative agent framework, the homogeneous relative risk aversion is assumed acrosshouseholds in general. The relative risk aversion is an important measurement of household’s risktolerance level because it determines the extent of household response to an increase in the futurerisk. When the relative risk aversion of household is zero, such household is called a risk-neutralhousehold. In other words, such households do not react to an increase in the future risk at all.On the other hand, when the relative risk aversion of household is positive, such households seeksto insure itself against the increased future risk in its income. This means that the precautionarymotive for household is higher when its relative risk aversion is larger. In this chapter, I explore thedegree of heterogeneity in the relative risk aversion across households. I am particularly interestedin how the relative risk aversion changes across households with different wealth positions. Thereasons for choosing a household wealth position as the most important household characteristicare that household wealth is one of the most important household characteristics in the householdfinance literature and is the cumulative measure of household earnings over time. In heterogeneousagents models with the incomplete market, households react in different ways to the increasedfuture risk in their expected income depending on their relative risk aversion level. Therefore, theheterogeneity in the relative risk aversion across households can be a very important factor forexplaining the observed differences in households choices such as wealth positions, career choices,investment strategies, etc.In this chapter, I assume the constant relative risk aversion utility function as the householdpreference. Using the standard household intertemporal Euler equation, I estimate the relative riskaversion as the curvature parameter of the constant relative risk aversion utility function since theformula for the relative risk aversion, −cu”(c)u′(c) , equals the curvature parameter, γ, of the constantrelative risk aversion utility function, c1−γ1−γ , when the constant relative risk aversion utility func-tion is assumed as the household preference.6 I estimate the relative risk aversion conditioningon a household wealth position by employing a combination of the extremum and nonparametricNadaraya-Watson kernel regression estimation methods. I present the estimated relative risk aver-sion across households with different wealth positions to demonstrate how the relative risk aversionis correlated with the household wealth position. I find the significant heterogeneity in the relativerisk aversion across households with unlike wealth positions. A hump-shaped relationship pattern6Here, c denotes household consumption and u(·) denotes household utility function.58is found at the bottom part of the wealth distribution between the relative risk aversion and thehousehold wealth position. Furthermore, there exists a significant negative correlation betweenthe relative risk aversion and the household wealth position excluding households at the bottompart of the wealth distribution. These findings are in line with those of several other papers suchas Chiappori and Paiella (2011) and Paravisini, Rappoport, and Ravina (2017) even though theyemploy very different estimation methods.Interestingly, the result of this chapter contradicts implications of existing conventional het-erogeneous agents models. High risk averse individuals own the majority of the total wealth inthe stationary equilibrium in conventional heterogeneous agents models. On the contrary, the re-sult of this chapter indicates that wealthy households are substantially less risk averse than poorhouseholds. I discuss how to extend conventional heterogeneous agents models in a way that theirimplications can be in line with the result found in this empirical paper in the Result and Discus-sion section. Existing conventional heterogeneous agents models do not work well for capturingthe observed wealth distribution, particularly regarding the household wealth distribution in theUnited States. The extension I propose in this chapter may actually improve that aspect of existingconventional heterogeneous agents models to a large extent. This chapter proceeds as follows. Inthe Model and Estimation Methodology section, I formulate the underlining model and explain theestimation procedure in detail. In the Data Description and Summary section, I demonstrate thedatasets used in this chapter and present their summaries. In the Result and Discussion section,I present the estimation result and discuss its implications and comparison with other papers. Inaddition, I discuss the economic interpretation of the result and its relation to existing conventionalmacroeconomic heterogeneous agents models in this section. In the Conclusion section, I presentthe conclusion for this chapter and discuss the potential future research topics based on the resultof this chapter.4.2 Model and Estimation Methodology4.2.1 ModelThe household problem is to choose its optimal consumption and asset holding, given its capitalincome, labor income, and initial value of asset holding, to maximize the present value of itsexpected life time utility asmax{ct}∞t=0∞∑t=0E0βtU(ct) (4.1)subject to the intertemporal budget constraint,ct + at+1 = Rtat + wt (4.2)59where U(·) is the household utility function, β is the discount factor, ct is consumption at theperiod t, at is the asset holding (or wealth) at the period t, Rt denotes the gross return on the assetholding at the period t, and wt is the labor income at the period t. The solution to this model isdefined by the following intertemporal Euler equation:U ′(ct) = Et{βRt+1U ′(ct+1)}. (4.3)In this model, I assume that the household labour supply is inelastic. However, even with theelastic labour supply, (a household chooses its optimal labour supply endogenously), we still obtainthe same intertemporal Euler Equation (4.3) if we have the separability between the consumptionand labour supply in the household utility function as U(ct, lt) = U1(ct) + U2(lt). In this chapter,I assume the constant relative risk aversion utility function as the household preference asU(ct) =c1−γt1− γ , γ ∈ (0,∞) (4.4)where γ is the curvature parameter of the constant relative risk aversion utility function. In con-ventional representative agents models, it is assumed that every agent of the economy has thehomogeneous utility function or curvature parameter, γ, when the constant relative risk aversionutility function is assumed as the household preference, despite the fact that there can exist sig-nificant heterogeneity in the curvature parameter, γ, across households. Households with differentcharacteristics, x, (x is a set of household characteristics such as wealth, gender, age, etc.) mightdiffer by their curvature parameter, γ. If we assume that the curvature parameter, γ, depends onhousehold characteristic, x, then the following version of the constant relative risk aversion utilityfunction represents the preference of household with characteristic, x, asU(ct) =c1−γ(x)t1− γ(x) . (4.5)Then the intertemporal Euler Equation for household i with characteristic, x, is expressed asEt{βRt+1U ′(cit+1)U ′(cit)∣∣∣Xit = x} = 1. (4.6)If we assume the constant relative risk aversion utility function as the household preference, thenthe intertemporal Euler Equation is expressed as follows:Et{βRt+1(cit+1cit)−γ(x)∣∣∣Xit = x} = 1 (4.7)where Xit is the observed characteristic of household i at the period t.60Derivation of the Intertemporal Euler Equation for Consumption Data with TwoYears IntervalI derive the intertemporal Euler Equation for the consumption data with two years interval be-cause the household micro-panel datasets used in the estimation procedure of this chapter conducthousehold surveys every two years even though collected household variables are annual. In otherwords, the observed household consumption data are collected every two years even though theyare annual consumption values. I expand the intertemporal Euler equation (4.7) asEt{βRt+1(cit+1cit)−γ(x) ∣∣∣Xit = x} =Et{βRt+1(cit+1cit)−γ(x)Et+1{βRt+2(cit+2cit+1)−γ(x) ∣∣∣Xit = x}∣∣∣Xit = x} =Et{Et+1{βRt+1(cit+1cit)−γ(x)βRt+2(cit+2cit+1)−γ(x) ∣∣∣Xit = x}∣∣∣Xit = x}.SinceEt+1{βRt+2(cit+2cit+1)−γ(x) ∣∣∣Xit = x} = 1.We obtain the following intertemporal Euler equation:Et{β2Rt+2Rt+1(cit+2cit)−γ(x) ∣∣∣Xit = x} = 1 (4.8)according to the law of iterated expectations.4.2.2 Estimation MethodologyI assume that the discount factor β is the same for all households in this chapter. Since I estimateγ(x) using only the intertemporal Euler Equation (4.8), the discount factor β is not identified. Inother workds, β and γ(x) cannot be estimated jointly from only one moment condition. Therefore,I assume that the discount factor β is the same for all households. To check the robustness of theestimation result, I choose the range of values for the discount factor β covering values estimatedin the existing literature. I estimate γ(x) using the nonparametric estimation method withoutassuming any particular functional form for γ(x). The estimation procedure is summarized asfollows. First, I define yit(γ) for given γ asyit(γ) = β2Rt+2Rt+1(cit+2cit)−γ. (4.9)61Next, the empirical analogy of Et{yit(γ)|Xit = x} is defined asEˆt{yit(γ)|Xit = x} =∑Tt=1∑nti=1 yit(γ)K((Xit − x)/h)∑Tt=1∑nti=1K((Xit − x)/h)(4.10)where K(·) is the kernel function and h is the kernel bandwidth. Eˆt{yit(γ)|Xit = x} consistentlyestimates Et{yit(γ)|Xit = x} as T → ∞, n → ∞, and h → 0 with the standard assumptions. Iderive the following extremum estimator to estimate γ(x) asγˆn(x) = arg minγ∈[0,∞)(Eˆt{yit(γ)|Xit = x} − 1)2. (4.11)I use the Epanechnikov kernel function, K(u) = 34(1 − u2)1(|u| ≤ 1), and find the optimal kernelbandwidth using the cross validation technique.Estimation of the Standard ErrorI define the function G(γ0(x)) asG(γ0(x)) = Et{yit(γ0(x))|Xit = x}− 1 = 0 (4.12)where γ0(x) is the true value of the relative risk aversion for a household with characteristic x.γˆn(x) is the estimator that solves the following equation:Gˆn(γˆn(x)) =∑Tt=1∑nti=1 yit(γˆn(x))K((Xit − x)/h)∑Tt=1∑nti=1K((Xit − x)/h)− 1 = 0. (4.13)The first-order optimality condition for the extremum estimator defined in Equation (4.11) corre-sponds to the Z-estimator in Equation (4.13). To derive the asymptotic normality, first, I employthe mean value expansion for Gˆn(γˆn(x)) around γ0(x) as0 u√nhGˆn(γˆn(x)) =√nhGˆn(γ0(x)) + Gˆ′n(γ˜n(x))√nh(γˆn(x)− γ0(x)), γ˜n(x) ∈ (γˆn(x), γ0(x))⇒√nh(γˆn(x)− γ0(x)) u 1−Gˆ′n(γ˜n(x))√nh(Gˆn(γ0(x))−G(γ0(x))), [∵ G(γ0(x)) = 0]where n is defined as n =∑Tt=1 nt and Gˆ′n(γ) is the first derivative of Gˆn(γ) with respect toγ. Here, γˆn(x) →pγ0(x) means that γ˜n(x) →pγ0(x). Gˆ′n(γ˜n(x)) →p G′(γ0(x)) holds due to thepoint-wise convergence, Gˆ′n(γ0(x)) →p G′(γ0(x)), and the uniform convergence. Moreover, basedon the the asymptotic normality of the Nadaraya-Watson estimator, the following convergence indistribution holds as√nh(Gˆn(γ0(x))−G(γ0(x))) =√nh(1nh∑Tt=1∑nti=1 yit(γ0(x))K((Xit − x)/h)1nh∑Tt=1∑nti=1K((Xit − x)/h)− 1)→dN(0,R(k)σ2(x)f(x))(4.14)62where the roughness parameter for the Epanechnikov kernel function, R(k), is 3/5 and σ2(x) is theconditional variance of the error term, eit(γ0(x)). σ2(x) is consistently estimated asσˆ2n(x) =∑Tt=1∑nti=1 eˆ2it(γˆn(x))K( (Xit−x)h)∑Tt=1∑nti=1K( (Xit−x)h) (4.15)where eˆit(γˆn(x)) is defined as yit(γˆn(x))− 1.Moreover, the probability density function of x, f(x), is estimated asfˆn(x) =1nhT∑t=1nt∑i=1K(Xit − xh). (4.16)The asymptotic normality equation for γˆn(x) is obtained by combining the abovementioned resultsas √nh(γˆn(x)− γ0(x))→dN(0,1(G′(γ0(x)))2R(k)σ2(x)f(x))as n→∞. (4.17)The variance of γˆn(x), σ2γˆn(x), is consistently estimated asσˆ2γˆn(x) =1(Gˆ′n(γˆn(x)))2R(k)σˆ2n(x)fˆn(x)(4.18)following the standard assumptions for the Nadaraya-Watson estimation method. The optimalbandwidth is constructed as C0n−1/5 7. The semiparametric estimator, γˆn(x), has an asymptoticbias of√nh5κ2B(x)/(G′(γ0))2, and B(x) does not depend on n or h. By setting h as C0n−1/5−with a small positive number , this bias converges to zero in the limit, according to the under-smoothing technique in the nonparametric estimation method. Here, I set as 0.01 despite the factthat this creates a small deviation from the optimal bandwidth to eliminate this bias in the limit.4.3 Data Description and Summary4.3.1 Data descriptionIn this chapter, the following two datasets are used to estimate the heterogeneous household prefer-ence parameter when the constant relative risk aversion utility function is assumed as the householdpreference. The first dataset is the Survey on Household Income and Wealth (SHIW) 8 datasetmanaged by the Bank of Italy. The Survey on Household Income and Wealth started in 1977 eventhough it was not a sufficient source for the analysis of this chapter until 1991. Therefore, I usethe survey waves of the Survey on Household Income and Wealth dataset starting from 1991. TheSurvey on Household Income and Wealth is conducted every two years since 1991. Each wave of7Bruce E.Hansen, Lecture Notes on Nonparametrics (2009)8https://www.bancaditalia.it/statistiche/tematiche/indagini-famiglie-imprese/bilanci-famiglie/index.html?com.dotmarketing.htmlpage.language=1 is the link for the detail.63this survey consists of approximately 8000 households and 20000 individuals, distributed over the300 Italian municipilaties. In the next wave of the survey, approximately half of the householdsfrom the previous wave are kept. I construct the dataset by keeping households who are in thetwo adjacent waves of the survey by merging these two adjacent waves. Then, merged datasets arepooled across time. The Survey on Household Income and Wealth contains the aggregate wealthand consumption data for all households. In addition, I conduct the inflation adjustment for thehousehold wealth and consumption data from the Survey on Household Income and Wealth bysetting 1998 as the base year. I use the following waves of the Survey on Household Income andWealth dataset as the household wealth and consumption ratio data: 1991, 1993, 1998, 2000, 2002,2004, 2006, 2008, 2010, and 2012 as a result. Each of these waves includes approximately 4,000households.The second dataset is the Panel Study of Income Dynamics (PSID) 9 dataset managed by theUniversity of Michigan. The survey follows individuals and their descendants over a long periodof time. The Panel Study of Income Dynamics started in 1968. The survey waves of the PanelStudy of Income Dynamics dataset are collected every two years since 1998. Since all householdsfrom the previous survey wave are retained in the next survey wave, I construct the consumptionratio, ct+2/ct, for all households in each wave. Nevertheless, due to the incompleteness of the wealthdata, I use the household wealth data from the following waves: 1983, 1993, 1998, 2000, 2002, 2004,2006, and 2008. Each of these waves includes approximately 5000 households and 18000 individuals.Moreover, the household consumption data are available from the Panel Study of Income Dynamicsdataset since the 1998 wave. Hence, I use the imputed household consumption data for the surveywaves in 1983 and 1993, imputed following Attanasio and Pistaferri (2014). In addition, I conductthe inflation adjustment for the household wealth and consumption data from the Panel Study ofIncome Dynamics dataset by setting 1998 as the base year.4.3.2 Data Summary and Parameter SpecificationTables 4.1 and 4.2 summarize important sample statistics for the Survey on Household Income andWealth and the Panel Study of Income Dynamics datasets. The standard deviations of the wealthand consumption ratio data from the Panel Study of Income Dynamics dataset are larger than thoseof the wealth and consumption ratio data from the Survey on Household Income and Wealth datasetaccording to Tables 4.1 and 4.2. Furthermore, Tables 4.1 and 4.2 demonstrate that the wealth andconsumption ratio data from both datasets are right-skewed to a large extent. The wealth data fromthe Panel Study of Income Dynamics dataset are much more right-skewed than the wealth datafrom the Survey on Household Income and Wealth dataset. Meanwhile, the consumption ratio datafrom the Survey on Household Income and Wealth are more right-skewed than the consumptionratio data from the Panel Study of Income Dynamics. Histograms for the wealth and consumptiondata from the Panel Study of Income Dynamics and the Survey on Household Income and Wealthdatasets are displayed in Figures 4.1, 4.2, 4.3, and 4.4. The annual real interest rate data from9http://psidonline.isr.umich.edu/ is the link for the detail.64the World Bank databank platform for the U.S. and Italy are used as the real interest rate datain this chapter. The inflation rate data (CPI) from the World Bank are used to adjust the wealthand consumption data for the U.S. and Italy to the 1998 USD and Euro values, respectively. Thetime-series annual real interest rate data for the U.S. and Italy are plotted in Figures 4.5 and 4.6,respectively. The time-series annual inflation rate (CPI) data for the U.S. and Italy are plotted inFigures 4.7 and 4.8, respectively.I estimate the heterogeneous curvature parameter, γ(x), of the constant relative risk aversionutility function for each of 51 equally spaced grid points as the value of the discount factor β onthe interval [0.86, 0.96]. The range of values for β chosen in this chapter is equivalent to the rangeof values for β defined on the interval [0.965, 0.99] for the quarterly data because the data used inthis chapter are annual. The heterogeneous relative risk aversion parameter, γ(x), is estimated byconditioning on the household wealth position in this chapter. I define equally spaced 81 grid pointsas the conditioning values for the household wealth position. These grid points are chosen as the 1stgrid point corresponds to mean(X)−std(X), the 41st grid point corresponds to mean(X), and the81st grid point corresponds to mean(X) + std(X). Here, X is the household wealth data, mean(·)is the function that calculates the average of the data, and std(·) is the function that calculates thestandard deviation of the data. All other grid points are equally spaced between these three gridpoints. In this way, the majority of values in the household wealth data are treated as conditioningvalues approximately. The optimal kernel bandwidths are obtained by the cross validation inthis chapter. The obtained optimal kernel bandwidth for the wealth data from the Panel Studyof Income Dynamics dataset is 158,861.43 and the obtained optimal kernel bandwidth for thewealth data from the Survey on Household Income and Wealth dataset is 108,062.48. The optimalbandwidth from the Panel Study of Income Dynamics dataset is approximately 46% higher thanthe optimal bandwidth from the Survey on Household Income and Wealth dataset. From Tables4.1 and 4.2, we observe that the standard deviation of the wealth data from the Panel Study ofIncome Dynamics dataset is approximately 40% higher than the standard deviation of the wealthdata from the Survey on Household Income and Wealth dataset.4.4 The Result and DiscussionI find that the significant heterogeneity in the relative risk aversion exists among households withdifferent wealth positions. The hump-shaped correlation pattern between the relative risk aversionand the household wealth position is estimated at the bottom part of the wealth distribution. First,the relative risk aversion, γ(x), increases with the household wealth postion at the bottom partof the wealth distribution. Then, it decreases sharply with the household wealth position towardthe top end of the wealth distribution as the estimation results for the relative risk aversion, γ(x),illustrate in this section.654.4.1 Estimation Result for the Survey on Household Income and WealthDatasetFigure 4.9 plots the estimation result for the relative risk aversion, γ(x), over the range of thehousehold wealth position when the discount factor β ranges between 0.906 and 0.914. The relativerisk aversion, γ(x), ranges from 0.85 to 0.55 when the discount factor β ranges between 0.906 and0.914. The hump-shaped correlation pattern between the relative risk aversion and the householdwealth position is estimated when the household wealth position ranges between 20, 000 euro and120, 000 euro (adjusted to the 1998 euro value). Starting from 120, 000 euro, there is a steadydecline in the relative risk aversion, γ(x), toward the top end of the wealth axis. Figure 4.10 plotsthe estimation result for the relative risk aversion, γ(x), for all values of the discount factor β onthe interval [0.86, 0.94] over the range of values of the household wealth position. The hump-shapedcorrelation pattern between the relative risk aversion, γ(x), and the household wealth position isestimated for every value of β on the interval [0.86, 0.94] at the bottom part of the wealth axis. Thedecrease in the relative risk aversion, γ(x), toward the top end of the wealth axis, gets less sharpas the value of β increases. The estimated relative risk aversion, γ(x), ranges from 1.265 to 0.159as shown in Figure 4.10. We observe that the relative risk aversion, γ(x), decreases when the valueof β increases from Figure 4.10. This relationship is proven mathematically in Cozzi (2012). Cozzi(2012) demonstrates that the expected consumption growth, ct+1/ct, increases in both the discountfactor β and the relative risk aversion when the constant relative risk aversion utility function isassumed as the household preference.In this chapter, the consumption growth itself is one of the observed variables used in theestimation procedure. Hence, when the value of the discount factor β increases, to offset the increasein the expected consumption growth, driven by the increase in the discount factor β, the relativerisk aversion, γ(x), must decrease. A higher value of the discount factor β implies that householdsaim for a higher consumption in the future, since they value the future utility to a greater extent.This suggests that the expected consumption growth, Et{ ct+1ct−1}, is higher when the value of thediscount factor β is larger. The higher the degree of the relative risk aversion, captured by γ(x)when the constant relative risk aversion utility function is assumed as the household preference, thehigher are the household’s precautionary saving motive. The increase in the relative risk aversioncauses a decrease in the current period’s household consumption and an increase in the householdsavings, due to the precautionary saving motive. Hence, the higher the degree of the relative riskaversion, γ(x), the higher is the expected consumption growth rate, Et{ ct+1ct− 1}, in the currentperiod.Figure 4.11 displays the estimated 95% confidence interval for the relative risk aversion, γ(x),when the value of β is 0.91. Figure 4.12 plots the upper and lower bounds of the estimated 95%confidence interval for the relative risk aversion, γ(x), when the value of β ranges between 0.86 and0.94. The estimated standard error for the relative risk aversion, γ(x), ranges from 0.0289 to 0.0072according to the estimation result. The estimated standard error becomes slightly higher towardthe both ends of the wealth axis as shown in Figure 4.12. Fewer observations have non-zero kernel66weights toward the both ends of the wealth axis. Hence, the estimated variance for the relative riskaversion, γ(x), increases toward the both ends of the wealth axis.4.4.2 Estimation Result for the Panel Study of Income Dynamics DatasetFigure 4.13 plots the estimation result for the relative risk aversion, γ(x), over the range of valuesfor the household wealth position when the discount factor β ranges between 0.906 and 0.914. Theestimated relative risk aversion, γ(x), ranges from 1.15 to 0.22 when the value of β ranges between0.906 and 0.914. The hump-shaped correlation pattern between the relative risk aversion, γ(x),and the household wealth position is estimated when the household wealth position ranges between-150,000 USD and -70,000 USD (at the 1998 value). However, this hump shape is much sharperthan the estimated hump shape from the the Survey on Household Income and Wealth dataset.Beginning from the household wealth position of -70,000 USD, there is a steady decline in γ(x)toward the top end of the wealth axis. Nonetheless, relative to the result from the the Survey onHousehold Income and Wealth dataset, the marginal decline in the relative risk aversion, γ(x), isvery large around the household wealth position of 150,000 USD. Hence, the estimated relative riskaversion, γ(x), from the Panel Study of Income Dynamics dataset for more affluent household ismuch lower than the estimated relative risk aversion, γ(x), from the Survey on Household Incomeand Wealth dataset for household with the same wealth position. This illustrates there might exista significant heterogeneity in the relative risk aversion, γ(x), across different countries.Figure 4.14 plots the estimation result for the relative risk aversion, γ(x), for all values of β onthe interval [0.86, 0.94] over the range of values for the household wealth position. The hump-shapedcorrelation pattern between the relative risk aversion, γ(x), and the household wealth position isestimated for every value of β on the interval [0.86, 0.94] at the bottom part of the wealth axis fromthe Panel Study of Income Dynamics dataset as shown in Figure 4.14. In comparison to the resultfrom the Survey on Household Income and Wealth dataset, this hump shape is very sharp for allvalues of β on the interval [0.86, 0.94]. At around the household wealth position of 150,000 USD,there exists a much larger marginal decline in the relative risk aversion, γ(x), for all values of β onthe interval [0.86, 0.94] compared with the result from the Survey on Household Income and Wealthdataset. Therefore, the relative risk aversion, γ(x), declines much more severely toward the topend of the wealth axis, compared with the result from the Survey on Household Income and Wealthdataset. The estimated relative risk aversion, γ(x), ranges from 1.374 to 0.055 as shown in Figure4.14. Furthermore, another important difference between this result and the result from the Surveyon Household Income and Wealth dataset is that when the value of β increases, the relative riskaversion, γ(x), decreases, but its marginal decrease is much less than the marginal decrease in therelative risk aversion, γ(x), estimated from the Survey on Household Income and Wealth dataset.Figure 4.15 displays the estimated 95% confidence interval for the relative risk aversion, γ(x),when the value of the discount factor β is 0.91. In addition, Figure 4.16 plots the upper and lowerbounds of the estimated 95% confidence interval for the relative risk aversion, γ(x), when the valueof β ranges between 0.86 and 0.94. The estimated standard error for the relative risk aversion,67γ(x), ranges from 0.0759 to 0.0028. The estimated standard error gets higher toward the bothends of the wealth axis because fewer observations have non-zero kernel weights at the both endsof the wealth axis as shown in Figure 4.16. The estimated standard errors from the Panel Study ofIncome Dynamics dataset are much higher, particularly at the bottom part of the wealth axis, incomparison with the estimated standard errors from the Survey on Household Income and Wealthdataset. In the Survey on Household Income and Wealth dataset, the standard errors at the toppart of the wealth axis are higher than the standard errors at the bottom part of the wealth axis.For the most values of the household wealth positon, there exists a significant negative correlationbetween the relative risk aversion, γ(x), and the household wealth position. The hump-shapedcorrelation pattern between the relative risk aversion, γ(x), and the household wealth position isestimated only at the bottom part of the wealth axis as shown in Figure 4.14. Moreover, comparedwith the Survey on Household Income and Wealth dataset, the decline in the relative risk aversion,γ(x), is much sharper for the Panel Study of Income Dynamics dataset when the household wealthposition increases. The largest estimated values of the relative risk aversion, γ(x), are 1.37 and1.26 for the Panel Study of Income Dynamics and the Survey on Household Income and Wealthdatasets respectively.4.4.3 Comparison to the Existing LiteratureThe hump-shaped correlation pattern between the relative risk aversion and the household wealthposition is documented in Chu, Nie, and Zhang (2014). Chu, Nie, and Zhang (2014) finds that therelative risk aversion first increases with the household wealth position and then decreases with thehousehold wealth position over some region of values for the household wealth position. Chu, Nie,and Zhang (2014) estimates this relationship using the experimental survey. The negative correla-tion between the relative risk aversion and the household wealth position is documented in Chiap-pori and Paiella (2011) and Paravisini, Rappoport, and Ravina (2017). Paravisini, Rappoport, andRavina (2017) estimates the relative risk aversion using the data from a person-to-person lendingplatform in the U.S. Paravisini, Rappoport, and Ravina (2017) shows that there exists a negativecorrelation between the relative risk aversion and the wealth position of an investor after controllingfor an investor specific fixed effect. These results are in line with the result of this chapter despitethe fact that these studies employ totally different estimation methodologies.Furthermore, Chiappori and Paiella (2011) indicates that the relative risk aversion is constantor does not vary significantly over time for a household. Moreover, Brunnermeier and Nagel (2008)demonstrates that the relationship between the household wealth position and the asset allocationseems best explained by assuming the constant relative risk aversion for a household. Therefore,these papers support the use of the constant relative risk aversion utility function as the householdpreference since the relative risk aversion for a household does not vary significantly over time. Inthe aforementioned papers, the relationships between the household wealth position and the relativerisk aversion are estimated using the experimental surveys or the linear regression with the cross-sectional or micro-panel data after log-linearizing the intertemporal Euler equation. Meanwhile,68in this chapter, I employ a combination of the nonparametric and extremum estimation methodswhile not log-linearizing the intertemporal Euler equation. The log-linearization is inaccurate sincethe log and expectations operators cannot be interchanged without strong assumptions. On thecontrary, the approach of this chapter avoids this pitfall of the approximation bias since I do notapply any non-linear transformation to the intertemporal Euler equation. Moreover, I use theobserved household consumption and wealth data, not collected in an experimental survey, fromthe Panel Study of Income Dynamics and the Survey on Household Income and Wealth datasets.These points are advantages of the estimation procedure employed in this chapter compared withthe existing literature.I estimate the relative risk aversion by conditioning on a household wealth position for differentvalues of β. When the value of β increases and βRt+1 gets sufficiently closer to one, the estimationresult for the relative risk aversion is no longer smooth in the conditioning variable, householdwealth. It jumps around and becomes negative for some cases. The objective function, that shouldbe close to zero under the estimated value of γ(x) as the Z-Estimator methodology suggests, doesnot approach zero when βRt+1 gets sufficiently close to one. Chamberlain and Wilson (2000) andHuggett (1993) demonstrate that βRt+1 < 1 is a necessary condition for the optimal consumptionsequence and the asset space to be bounded when the household income process is sufficientlystochastic. In this chapter, this result is confirmed in a sense that the objective function to beminimized does not behave well when βRt+1 gets sufficiently close to 1.4.4.4 The Interpretation of the ResultIn this chapter, I find that there exists the negative correlation between the household wealth posi-tion and the relative risk aversion, except the bottom part of the wealth distribution. However, thisresult does not demonstrate the causal relationship between the household wealth position and therelative risk aversion. Low risk averse households may pursue the career of entrepreneurship. Mean-while, high risk averse households may pursue the career of worker. The career of entrepreneurshipmight represent a high risk and high return type of household income process. In conjunctionwith this, the career of worker might represent a low risk and low return type of household incomeprocess. With the heterogeneity in household preferences, households choose different careers de-pending their preferences. Cozzi (2012) concluded that the model with incomplete markets andprecautionary savings, extended to include the heterogeneity in the relative risk aversion or the cur-vature parameter of the constant relative risk aversion utility function and the endogenous choiceof careers with different risk levels in their income processes, could explain better several importantfeatures of the U.S. wealth distribution compared with existing conventional heterogeneous agentsmodels. One major problem with existing conventional heterogeneous agents models is that theycan not generate sufficient inequality in the wealth distribution that is observed in the real data.According to the observed wealth distribution of the U.S., the top 1% holds approximately 30%of the total wealth of the whole country. Meanwhile, existing conventional heteogeneous agentsmodels generate a very low share of the total wealth for the top quintiles of the wealth distribu-69tion. Moreover, another major problem with existing conventional heterogeneous models is thatthe bottom quintiles of the wealth distribution hold too much wealth compared with the observedshare of the total wealth held by the bottom quintiles of the wealth distribution in the real data.Cozzi (2012) shows that those flaws are much reduced by introducing the preference heterogeneityand the self-selection of career choices among households. Cozzi (2012) also demonstrates that theGini index calculated from this extended heterogeneous agents model gets much closer to the Giniindex calculated from the observed wealth distribution for the U.S.The problem with the self-selection of career choices depending on the level of the relative riskaversion is that even though low risk averse households may pursue a high risk and high returntype of career such as an entrepreneurship, they might still end up with a very low level of wealthwith a positive probability since income processes, generated in their chosen careers, are highlyrisky. One possible extension to addressing this problem is as follows. Households have to choosetheir careers in every period. They can stick to careers they have chosen in the previous periodor they can choose different careers. If they stick to careers they have chosen in the previousperiod, income risks of these careers would be relatively less than those of the previous period inthe current period. The justification for this assumption is that households obtain expertises incareers they chose over time as they accumulate more experience or knowledge by sticking to careersover time. Hence, if a household stays in one career for a longer period of time, the income riskarising from that career would decrease over time for such household. In particular, for householdsthat chose an entrepreneurship career, the lessening of the income risk over time would provide anextra motivation to keep pursuing an entrepreneurship career even though they might experiencelarge negative income shocks in the early stages of their enrepreneurship careers. Households whocontinued to pursue entrepreneurship careers for a lengthy period of time will enjoy a high averagereturn with much lesser risk compared with households who just started an entrepreneurship career.Thus, the top quintiles of the stationary wealth distribution might hold much larger shares of thetotal wealth in comparison with existing conventional heterogeneous agents models because thetop quintiles of the wealth distribution would be mostly comprised of households who pursued anentrepreneurship career over long time. This will be in line with the result of this chapter since lowrisk averse households who chose an entrepreneurship career would be at the top quintiles of thewealth distribution. In addition, the wealth distribution generated from this extended model mayreflect the observed wealth distribution much better than those generated from existing conventionalheterogeneous agents models do.The top part of the wealth distribution is comprised of mostly entrepreneurs and stockholdersor investors as the evidence from the Panel Study of Income Dynamics and the Survey of ConsumerFinances datasets suggests. Moreover, the hump-shaped curve of the relative risk aversion at thebottom part of the wealth distribution estimated in this chapter reveals that the bottom part ofthe wealth distribution includes some low risk averse households compared with slightly higherquintiles of the wealth distribution since low risk averse households in high risk and high returncareer can experience hugely negative income shocks with some positive probability when they are70in the early stages of entrepreneurship career. Observed shares of the total wealth at the bottomquintiles of the wealth distribution in the U.S. are much less than shares generated from existingconventional heterogeneous agents models. One way to address this problem is to introduce thewelfare policy targeted for poor households. With such a policy present, households at the bottompart of the wealth distribution will have less incentive to save regarding precautionary savingmotives. Therefore, the welfare policy targeted for poor households may reduce sharply their shareof the total wealth.Furthermore, this chapter points out that there exists a hump-shaped relationship between thehousehold wealth position and the relative risk aversion at the bottom part of the wealth distri-bution. In this chapter, I assume that there is no borrowing constraint so that the intertemporalEuler equation holds with equality for poor households. Nevertheless, if there exist significantborrowing constraints for households at the bottom part of the wealth distribution, the estimatedrelative risk aversion or curvature parameter of the constant relative risk aversion utility functionat the bottom part of the wealth distribution can be different from the true relative risk aversionor curvature parameter of the constant relative risk aversion utility function. For the Survey onHousehold Income and Wealth dataset, only 2% of households in the dataset hold negative netwealth or could be possibly facing a borrowing constraint. Hence, the borrowing constraint mightnot create a big problem regarding the estimation procedure with the Survey on Household Incomeand Wealth dataset. Meanwhile, for the Panel Study of Income Dynamics dataset, 12% of house-holds in the dataset hold negative net wealth. Possibly, the borrowing constraint could presentsome issue regarding the estimation of the relative risk aversion at the bottom part of the wealthdistribution with the Panel Study of Income Dynamics dataset. Tackling this issue is an openchallenge for the future researches.4.5 ConclusionIn this chapter, I estimated the relationship between the relative risk aversion and the householdwealth position. At the bottom part of the wealth distribution, there exists a hump-shaped rela-tionship between the relative risk aversion and the household wealth position. For the Survey onHousehold Income and Wealth dataset, this hump shape is small. Meanwhile, for the Panel Study ofIncome Dynamics dataset, this hump shape is very sharp. For most values of the household wealthposition, there exists a substantially negative relationship between the relative risk aversion andthe household wealth position. Particularly for the Panel Study of Income Dynamics dataset, start-ing from the middle part of the wealth distribution, the estimated relative risk aversion decreasessharply when the household wealth position increases.The abovementioned empirical result contradicts results of existing conventional heterogeneousagents models. In existing conventional heterogeneous agents models, high risk averse individualseventually hold the majority of the total wealth because of their high precautionary saving mo-tives. However, if we add different career options such as a high risk and high return career (such71as entrepreneurship) and a low risk and low return career (paid employees excluding high levelexecutives such as CEO, etc.) and a career choice to existing conventional heterogeneous agentsmodels, high risk averse individuals choose a low risk and low return career and less risk averseindividuals choose a high risk and high return career as a tendency. In addition, if a householdsticks to one career over a long period of time, the risk or uncertainty involved in that career mightshrink over time. Particularly for entrepreneurs, should they retain the entrepreneur career for alonger period of time, they might enjoy a higher average return with a lesser risk. This extension toexisting conventional heterogeneous agents models might generate a significant negative correlationpattern between the household wealth position and the relative risk aversion.Furthermore, several other papers found results similar to the result of this chapter even thoughthey employed totally different estimation methodologies. Some authors employ the log-linearizedintertemporal Euler equation and the linear regression estimation method. However, the expecta-tions operator does not hold anymore after a non-linear transformation on the intertemporal Eulerequation without strong assumptions. In this chapter, I did not apply any non-linear transforma-tion to the intertemporal Euler equation. In addition, some papers use experimental surveys forthe estimation procedure of the relative risk aversion. Meanwhile, this chapter uses the observedconsumption and wealth data for households from the Panel Study of Income Dynamics and theSurvey on Household Income and Wealth datasets.In this chapter, the discount factor, β, is exogenously given and constant across households.The joint estimation of the discount factor, β, and the curvature parameter of the constant relativerisk aversion utility function, γ(x), (the relative risk aversion) is not possible in the framework ofthis chapter. The main reason is that I employ the estimation method similar to the generalizedmethod of moments estimation method. In the generalized method of moments estimation method,the number of the moment conditions has to be more than or equal to the number of parameters tobe estimated. In this chapter, the only one moment condition, the intertemporal Euler equation,is employed for the estimation procedure. Therefore, we cannot identify two parameters with theonly intertemporal Euler equation in this chapter. Nevertheless, I firmly acknowledge that thejoint estimation of the discount factor, β, and the curvature parameter of the constant relative riskaversion utility function, γ(x), is an important future research topic.As another future research topic, I would like to extend existing conventional heterogeneousagents models by adding different career choices and an experience or tenure effect of the careercontinuation to explain the negative relationship between the household wealth position and therelative risk aversion estimated in this chapter. Existing conventional heterogeneous agents modelsdo not work well for capturing the wealth distribution observed in reality. The extension I describedabove could possibly improve that aspect of existing conventional heteogeneous agents models.724.6 TablesTable 4.1: The Survey on Household Income and WealthDataset and the Panel Study of Income Dynamics Dataset:The Summary of the Consumption Ratio DataSample statisticThe Survey on HouseholdIncome and WealthThe Panel Study ofIncome DynamicsMinimum 0.03 0.02Maximum 60.05 70.09Mean 1.03 1.28Median 0.96 1.00Standard deviation 0.58 1.47Skewness 29.56 20.19Number of observations 30230 40184Note: The consumption ratio is calculated as ct+2/ct and ct is the householdconsumption at the period t. The consumption data are adjusted for the inflationby setting 1998 as the base year. .Table 4.2: The Survey on Household Income and WealthDataset and the Panel Study of Income Dynamics Dataset:The Summary of the Wealth DataSample statisticThe Survey on HouseholdIncome and WealthThe Panel Study ofIncome DynamicsMinimum -535000 -815025.3Maximum 22656534.47 26505420.05Mean 214765.3 111229.92Median 130817.49 19639.56Standard deviation 384830.71 534429.29Skewness 13.00 25.11Number of observations 30230 40184Note: The wealth data are adjusted for the inflation by setting 1998 as the baseyear.734.7 Figures0 10 20 30 40 50 60 700500100015002000250030003500Empirical Distribution of The Ratio of Consumptions at t and t+2 periods for SHIWFigure 4.1: The Empirical Distribution of the Consumption Ratio from the Survey on HouseholdIncome and Wealth DatasetNote: The empirical distribution of the consumption ratio between t and t+2 periods from the Survey on HouseholdIncome and Wealth dataset is displayed in this figure and the household consumption data are adjusted for theinflation by setting 1998 as the base year.740 10 20 30 40 50 60 70 80020040060080010001200140016001800Empirical Distribution of The Ratio of Consumptions at t and t+2 periods for PSIDFigure 4.2: The Empirical Distribution of the Consumption Ratio from the Panel Study ofIncome Dynamics DatasetNote: The empirical distribution of the consumption ratio between t and t+2 periods from the Panel Study of IncomeDynamics dataset is displayed in this figure and the household consumption data are adjusted for the inflation bysetting 1998 as the base year.×107-0.5 0 0.5 1 1.5 2 2.50100020003000400050006000700080009000Empirical Distribution of Wealth for SHIWFigure 4.3: The Empirical Distribution of the Household Wealth from the Survey on HouseholdIncome and Wealth DatasetNote: The empirical distribution of household wealth from the Survey on Household Income and Wealth dataset isdisplayed in this figure and the household wealth data are adjusted for the inflation by setting 1998 as the base year.75×107-0.5 0 0.5 1 1.5 2 2.5 3010002000300040005000600070008000900010000Empirical Distribution of Wealth for PSIDFigure 4.4: The Empirical Distribution of the Household Wealth from the Panel Study of IncomeDynamics DatasetNote: The empirical distribution of household wealth from the Panel Study of Income Dynamics dataset is displayedin this figure and the household wealth data are adjusted for the inflation by setting 1998 as the base year.76.01.02.03.04.05.06.07.08.0984 86 88 90 92 94 96 98 00 02 04 06 08 10US annual Real Interest Rate: 1984 - 2010Figure 4.5: The Annual Real Interest Rate Data for the U.S.Note: The annual real interest rate data for the U.S. between 1984 and 2010 are obtained from the World Bankdatabank platform.77.02.04.06.08.10.1292 94 96 98 00 02 04 06 08 10 12 14Italy annual Real Interest Rate: 1992 - 2015Figure 4.6: The Annual Real Interest Rate Data for ItalyNote: The annual real interest rate data for Italy between 1992 and 2015 are obtained from the World Bank databankplatform.78-.01.00.01.02.03.04.05.0684 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14US annual Inflation Rate: 1984 - 2010Figure 4.7: The Annual Inflation Rate Data for the U.S.Note: The annual inflation rate data for the U.S. between 1984 and 2010 are obtained from the World Bank website.79.00.01.02.03.04.05.0692 94 96 98 00 02 04 06 08 10 12 14Italy annual Inflation Rate: 1992 - 2015Figure 4.8: The Annual Inflation Rate Data for ItalyNote: The annual inflation rate data for Italy between 1992 and 2015 are obtained from the World Bank website.80Wealth ×1050 0.5 1 1.5 2 2.5 3 3.5 4 4.5Estimated γ0.50.550.60.650.70.750.80.85Non-Parametric Estimation of CRRA Utility function for Household with different wealth, SHIW β=0.906β=0.908β=0.91β=0.912β=0.914Figure 4.9: The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility FunctionNote: The nonparametric estimation results for the heterogeneous curvature parameter, γ(x), of the ConstantRelative Risk Aversion utility function corresponding to different values of β are displayed in this figure. The Surveyon Household Income and Wealth dataset is used in the estimation procedure.8143.5×105Estimated γ for different β ‘s for different Wealth points, SHIW32.52Wealth1.510.50.860.880.9Different β‘s0.920.941.210.80.60.200.4Estimated γFigure 4.10: The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility FunctionNote: The nonparametric estimation results for the heterogeneous curvature parameter, γ(x), of the ConstantRelative Risk Aversion utility function when β ranges between 0.86 and 0.94 are displayed in this figure. The Surveyon Household Income and Wealth dataset is used in the estimation procedure.82Wealth ×1050 0.5 1 1.5 2 2.5 3 3.5 4 4.5γ0.50.550.60.650.70.750.80.85Point Estimation and 95 percent CI for γ when β = 0.91, SHIW Upper Bound for 95 percent CIPoint EstimationLower Bound for 95 percent CIFigure 4.11: The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility FunctionNote: The nonparametric estimation result for the heterogeneous curvature parameter, γ(x), of the Constant RelativeRisk Aversion utility function when β is 0.91 is displayed in this figure. The point estimator and the 95 percentconfidence interval are shown in this figure. The Survey on Household Income and Wealth dataset is used in theestimation procedure.8343.5×105The Point Estimation and 95% Confidence Interval Results for γ(x)32.52Wealth1.510.50.860.880.9β0.920.9400.20.40.60.811.2Point Estimation and 95% CI Upper and Lower Bounds for γ(x)0.20.30.40.50.60.70.80.911.11.2Figure 4.12: The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility FunctionNote: The nonparametric estimation results for the heterogeneous curvature parameter, γ(x), of the ConstantRelative Risk Aversion utility function when β ranges between 0.86 and 0.94 are displayed in this figure. The pointestimator and the 95 percent confidence interval are shown in this figure. The Survey on Household Income andWealth dataset is used in the estimation procedure.84Wealth ×105-2 -1 0 1 2 3 4γ0.20.30.40.50.60.70.80.911.11.2Non-Parametric Estimation of γ for households with different wealth, PSID β=0.906β=0.908β=0.91β=0.912β=0.914Figure 4.13: The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility FunctionNote: The nonparametric estimation results for the heterogeneous curvature parameter, γ(x), of the ConstantRelative Risk Aversion utility function corresponding to different values of β are displayed in this figure. The PanelStudy of Income Dynamics dataset is used in the estimation procedure.853.5×1053Estimated γ for different β ‘s for different wealth points, PSID2.521.51Wealth0.50-0.5-1-1.50.860.880.9Different β‘s0.920.940.2011.20.60.40.8γFigure 4.14: The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility FunctionNote: The nonparametric estimation results for the heterogeneous curvature parameter, γ(x), of the ConstantRelative Risk Aversion utility function when β ranges between 0.86 and 0.94 are displayed in this figure. The PanelStudy of Income Dynamics dataset is used in the estimation procedure.86Wealth ×105-2 -1 0 1 2 3 4Estimated γ0.20.40.60.811.21.4Point Estimation and 95 percent CI plot for γ when β = 0.91 Upper Bound for 95 percent CIPoint EstimationLower Bound for 95 percent CIFigure 4.15: The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility FunctionNote: The nonparametric estimation result for the heterogeneous curvature parameter, γ(x), of the Constant RelativeRisk Aversion utility function when β is 0.91 is displayed in this figure. The point estimator and the 95 percentconfidence interval are shown in this figure. The Panel Study of Income Dynamics dataset is used in the estimationprocedure.873.5×1053The Point Estimation and 95% Confidence Interval Results for γ(x)2.521.51Wealth0.50-0.5-1-1.50.860.880.9β0.920.940.80.60.200.41.41.21Point Estimation and 95% CI Upper and Lower Bounds for γ(x)0.20.40.60.811.21.4Figure 4.16: The Nonparametric Estimation Result for the Heterogeneous Curvature Parameter,γ(x), of the Constant Relative Risk Aversion Utility FunctionNote: The nonparametric estimation results for the heterogeneous curvature parameter, γ(x), of the ConstantRelative Risk Aversion utility function when β ranges between 0.86 and 0.94 are displayed in this figure. The pointestimator and the 95 percent confidence interval are shown in this figure. The Panel Study of Income Dynamicsdataset is used in the estimation procedure.88Chapter 5ConclusionIn this study, I looked at the identification of household preferences from a micro-panel datasetand the dynamics of household decisions during the GFC. First, I examined the estimation of thehousehold utility function from the micro-panel dataset using the intertemporal Euler equation.Second, I investigated the dynamics of household consumption decisions during the GFC.In Chapter 2, I discussed the identification issue of the household utility function from a micro-panel dataset and proposed a new methodology for estimating the household utility function froma micro-panel dataset using the intertemporal Euler equation. According to existing literature, thelack of variation in the return on household savings arising from the shortness of time dimension ofa household micro-panel dataset created the identification issue of the household utility function.Meanwhile, in Chapter 2, I showed that the extra variation in the return on household savings arisingfrom differences in the portfolio compositions of household savings across households identifies thehousehold utility function successfully from a micro-panel dataset using the intertemporal Eulerequation as the moment condition. Moreover, in Chapter 2, for the first time in the householdfinance literature (to my knowledge), the QHD preference was successfully estimated from a micro-panel dataset using the intertemporal Euler equation. In addition, in Chapter 2, I proposed anew household dynamic model, where the household stock holding behavior is well captured byexplicitly taking into account the hidden stock market participation cost pointed out in recentempirical studies as a potential explanation for the puzzle of low stock market participation rateamong households in the United States.Furthermore, the methodology, developed in Chapter 2, can be utilized to estimate the otherhousehold preferences, for which, the intertemporal Euler equation can be derived explicitly. Asone such example, I implemented the estimation of the constant absolute risk aversion utilityfunction from the micro-panel data using the methodology, discussed in Chapter 2, in the appendixsection of this thesis. As one future research topic, I plan to work on the identification issue of theEpstein-Zin preference from a micro-panel dataset by extending the methodology constructed inChapter 2. Moreover, I am interested in incorporating the household’s home equity holding decisioninto the household’s portfolio composition decision, considered in Chapter 2, as another potentialresearch topic in the future. This extension may create a framework where researchers can analyzethe spillover effects between the housing market crisis and the stock market crisis through theirimpacts on the household’s decision-making process. Finally, I intend to work on the inclusions ofthe future job loss probability and the aggregate fluctuation in the household income process intothe household dynamic model, explored in Chapter 2 of this thesis, as another important extension.89In Chapter 3, the effects of the stock market collapse and the Zero-Lower Bound monetarypolicy during the GFC on household consumption decisions were explored, using the PSID datasetand the household dynamic model developed in Chapter 2. In Chapter 3, I demonstrated thatthe huge drop in the stock market return and the decline in the risk-free rate, driven by the Zero-Lower Bound monetary policy, triggered a significant decrease in consumption inequality amonghouseholds in the United States during the GFC. The huge drop in the stock market return andthe decline in the risk-free rate substantially decreased the consumption of wealthy households.Meanwhile, the drop in the risk-free rate triggered a slight percentage increase in the consumptionof households at the bottom part of the wealth distribution during the GFC. In addition, in Chapter3, I illustrated that the increase in the expected future volatility of the stock market return did notaffect the consumption decisions of wealthy households, while the consumption decisions of poorhouseholds were affected significantly by the increase in the expected future volatility of the stockmarket return during the GFC.In Chapter 4, the heterogeneity in household preference was examined by employing a com-bination of the extremum and nonparametric estimation methods. In Chapter 4, the significantheterogeneity in household preference across households was found by assuming the CRRA utilityfunction as the household preference. At the bottom end of the wealth distribution, a hump-shapedrelationship pattern was found between the curvature parameter of the CRRA utility function andthe household wealth level. 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Iskhakov (2012): “A generalized endogenous grid method fordiscrete-continuous choice,” Discussion paper.Sousa, R. (2007): “Wealth Shocks and Risk Aversion,” NIPE Working Papers 28/2007, NIPE -Universidade do Minho.van Rooij, M., A. Lusardi, and R. Alessie (2011): “Financial literacy and stock marketparticipation,” Journal of Financial Economics, 101(2), 449–472.Vissing-Jørgensen, A., and O. P. Attanasio (2003): “Stock-Market Participation, Intertem-poral Substitution, and Risk-Aversion,” American Economic Review, 93(2), 383–391.94Appendix AAppendix to Chapter 2A.1 The Definitions of the Household Stock Holding Share andthe Household Portfolio ReturnThe household stock holding share, dit, is calculated as a household’s stock holding value dividedby a household’s total savings excluding a value of home equity. Then, the household stock holdingshare, dit, is discretized as grid points on the interval, [0, 1]. The number of grid points is denotedby K. In this study, K is set as 6. The computational burden of the model estimation dependson K. The grid points are defined as D = {dk}Kk=1 where dk is equal to k−1(K−1) . If dit is largerthan k−1(K−1) and less than or equal tok(K−1) , dit is set as dk. Lastly, if dit is equal to 0, dit is setas d1 or zero. This discretization of household stock holding share is an approximation of the truehousehold stock holding share in a sense that when the number of grid points, K, increases, thedistance between the true household stock holding share and the nearest grid point of stock holdingshare will shrink.The household portfolio return of household savings is defined asrˆit+1 = rf + dit(rt+1 − rf ) = (1− dit)rf + ditrt+1rt+1 = (1− ρr)µr + ρrrt + rt+1rˆit+1(dit−1, rˆit, dit, rt+1) = rf + dit{(1− ρr)µr + ρr[(rˆit − rf )/dit−1 + rf ] + rt+1 − rf}(A.1)where rˆit denotes the household portfolio return of household savings, rt denotes the stock marketreturn, and rf denotes the risk-free rate. The stock market return, rt, varies only over timeregarding a household micro-panel dataset. On the other hand, the household portfolio return,rˆit, described in Equation (A.1), varies over time and across households. This cross sectionalvariation in a household portfolio return, rˆit, enables the inter-temporal Euler equation estimationmethod to estimate a household utility function successfully from a micro-panel dataset becausethe household portfolio return , rˆit, has a much more variation than the variation in the stockmarket return variable, rt, thanks to the heterogeneity in a household portfolio composition acrosshouseholds.95A.2 The Derivations of the Inter-Temporal Euler EquationsHere, I demonstrates the derivations of the inter-temporal Euler equations for the Constant RelativeRisk Aversion utility function and the Quasi-Hyperbolic Discounting preference.A.2.1 The Constant Relative Risk Aversion Utility FunctionI define the Lagrangian function asL1 ={U(cit)−AC · (dit − dit−1)2 − PC · 1{dit 6= 0}+ βEV¯ (dit, xit+1, sit+1) (A.2)+ λ((xit − cit)(1 + rf + dit(rt+1 − rf )) + yit+1 − xit+1)}.According to Rust (1987), the household ex ante value function is expressed asV¯ (dit−1, xit, sit) = ln( ∑d′t∈Dexp(V (d′t, dit−1, xit, sit)))+ γ (A.3)where γ is the Euler-Mascheroni constant. Therefore, ∂V¯ (dit,xit+1,sit+1)∂xit+1 is expressed as∑dit∈Dexp(V (dt, dit−1, xit, sit))∑d′t∈D exp(V (d′t, dit−1, xit, sit))∂V (dit, xit+1, sit+1)∂xit+1(A.4)by using the standard differentiation formulas. The Envelope theorem states that ∂V (dit,xit+1,sit+1)∂xit+1is equal to U ′(cit+1). As noted by Rust (1987), the conditional choice probability function of thestock holding share, Pr{dit|dit−1, xit, sit}, is expressed asexp(V (dit, dit−1, xit, sit))∑d′t∈D exp(V (d′t, dit−1, xit, sit)).I take the derivative of the Lagrangian function w.r.t. cit as∂L1∂cit= U ′(cit)− λ(1 + rf + dit(rt+1 − rf )). (A.5)Then, I take the derivative of the Lagrangian function w.r.t. xit+1 as∂L1∂xit+1= βE∂V¯ (dit, xit+1, sit+1)∂xit+1− λ. (A.6)Furthermore, U ′(cit) is expressed as c−σit when the Constant Relative Risk Aversion utility functionis assumed as the household preference. By combining the equations derived above, I obtain the96following inter-temporal Euler equation:c(dit, dit−1, xit, sit)−σ = βEsit+1{ ∑d′it+1∈DPr{d′it+1|dit, xit+1, sit+1}(1 + rf + dit(rt+1 − rf ))c(d′it+1, dit, xit+1, sit+1)−σ}(A.7)where c(·) is the household consumption function.A.2.2 The Quasi-Hyperbolic Discounting PreferenceThe household optimization problem is described asW (dit, dit−1, xit, sit) = maxc(dit,dit−1,xit,sit){U(c(dit, dit−1, xit, sit))−AC · (dit − dit−1)2 − PC · 1{dit 6= 0}(A.8)+ δβEV¯ (dit, xit+1, sit+1)},V (dit, dit−1, xit, sit) = U(c(dit, dit−1, xit, sit))−AC · (dit − dit−1)2 − PC · 1{dit 6= 0} (A.9)+ βEV¯ (dit, xit+1, sit+1).The value function, W (dit, dit−1, xit, sit), can be expressed asW (dit, dit−1, xit, sit) = δV (dit, dit−1, xit, sit) + (1− δ)U(c(dit, dit−1, xit, sit)). (A.10)I define the Lagrangian function asL2 ={U(c(dit, dit−1, xit, sit))−AC · (dit − dit−1)2 − PC · 1{dit 6= 0}+ δβEV¯ (dit, xit+1, sit+1)(A.11)+ λ((xit − cit)(1 + rf + dit(rt+1 − rf )) + yit+1 − xit+1)}.I follow all steps from Section A.2.1. The main difference is that, in this case, the Envelope theoremstates that ∂W (dit,dit−1,xit,sit)∂xit is equal to U′(c(dit, dit−1, xit, sit)). Therefore, the following equationholds asU ′(c(dit, dit−1, xit, sit)) = δ∂V (dit, dit−1, xit, sit)∂xit+ (1− δ)U ′(c(dit, dit−1, xit, sit))∂c(dit, dit−1, xit, sit)∂xit.(A.12)U(c) is defined as c1−σ1−σ and the first derivative of U(c), U′(c), is obtained as c−σ. Now we take thederivatives of the Lagrangian function, L2, w.r.t. cit and xit+1. Then, by using Equation (A.12)97and rearranging, we obtain the following inter-temporal Euler equation asc(dit, dit−1, xit, sit)−σ = E{ ∑d′it+1∈DP{d′it+1|dit, xit+1, sit+1}(1 + rf + dit(rt+1 − rf ))[βδ∂c(d′it+1, dit, xit+1, sit+1)∂xit+1+ β(1− ∂c(d′it+1, dit, xit+1, sit+1)∂xit+1)]c(d′it+1, dit, xit+1, sit+1)−σ}.(A.13)A.3 The Detail of the Estimation ProcedureA.3.1 The Estimation of the Utility FunctionThe inter-temporal Euler equation is approximated as follows:Qit(σ) = βE∑dit+1∈DPr{dit+1|dit, xit+1, sit+1} × ...× (1 + rf + dit(rit+1 − rf ))c(dit+1, dit, xit+1, sit+1)−σc(dit, dit−1, xit, sit)−σ− 1, (A.14)Qˆit(σ) = β1NsNs∑j=1∑dit+1∈DPˆ r{dit+1|dit, xjit+1, sjit+1} × ...× (1 + rf + dit(rjit+1 − rf )) cˆ(dit+1, dit, xjit+1, sjit+1)−σcˆ(dit, dit−1, xit, sit)−σ− 1. (A.15)In Equation (A.15), sjit+1 = (rjit+1, yjit+1, rjit+1) is drawn randomly conditioning on (dit, xit, sit).Subsequently, by averaging over {sjit+1}Nsj=1, I approximate the expectation operator in Equation(A.14). When Ns increases, it decreases an approximation error of the expectation operator. How-ever, it significantly increases the computational burden since I estimate c(dit+1, dit, xjit+1, sjit+1)and Pr(dit+1|dit, xjit+1, sjit+1) nonparametrically for each i and t, and for each choice of dit+1 ∈ D.This part is similar to the Simulated Method of Moments (SMM) estimation methodology. At thetrue parameter σ0, Qit(σ0) is equal to 0 since Qit(σ) is a representation of the inter-temporal Eulerequation. In this study, I construct the feasible approximation of Qit(σ) as Qˆit(σ) as in Equation(A.15). I find that Ns greater than 5 performs better compared to Ns less than 5.E{β1NsNs∑j=1∑dit+1∈DPr{dit+1|dit, xjit+1, sjit+1} × ...× (1 + rf + dit(rjit+1 − rf ))c(dit+1, dit, xjit+1, sjit+1)−σ0c(dit, dit−1, xit, sit)−σ0− 1}= 0.Here, cˆ(dit, dit−1, xit, sit) is a nonparametric estimator of c(dit, dit−1, xit, sit) defined as98cˆ(dit, dit−1, xit, sit) =∑Nl=1∑Tq=1 clq1{dlq = dit, dlq−1 = dit−1}K(xlq−xithx)K(slq−siths)∑Nl=1∑Tq=1 1{dlq = dit, dlq−1 = dit−1}K(xlq−xithx)K(slq−siths). (A.16)Moreover, cˆ(dit+1, dit, xjit+1, sjit+1) is a nonparametric estimator of c(dit+1, dit, xjit+1, sjit+1) de-fined ascˆ(dit+1, dit, xjit+1, sjit+1) =∑Nl=1∑Tq=1 clq1{dlq = dit+1, dlq−1 = dit}K(xlq−xjit+1hx)K(slq−sjit+1hs)∑Nl=1∑Tq=1 1{dlq = dit+1, dlq−1 = dit}K(xlq−xjit+1hx)K(slq−sjit+1hs). (A.17)In addition, Pˆ r(dit+1|dit, xjit+1, sjit+1) is a nonparametric estimator of Pr(dit+1|dit, xjit+1, sjit+1)defined asPˆ r(dit+1|dit, xjit+1, sjit+1) =∑Nl=1∑Tq=1 1{dlq = dit+1, dlq−1 = dit}K(xlq−xjit+1hx)K(slq−sjit+1hs)∑Nj=1∑Tq=1 1{dlq−1 = dit}K(xlq−xjit+1hx)K(slq−sjit+1hs). (A.18)Here, K( slq−siths)and K( slq−sjit+1hs)are defined as follows:K(slq − siths)= K(rq − rthr)K(ylq − yithy)K(rq − rthr),K(slq − sjit+1hs)= K(rq − rjit+1hr)K(ylq − yjit+1hy)K(rq − rjit+1hr).The Nadarya-Watson Kernel Regression estimation method is employed to estimate c(dit+1, dit, xjit+1, sjit+1)andPr(dit+1|dit, xjit+1, sjit+1) in this study, even though the use of other nonparametric estimationmethods will not pose any issue. In fact, I conducted estimations using the series estimationmethod (the nonparametric polynomial estimation method) and the result was very similar. I alsobelieve that using other types of prediction methods such as machine learning methods might bepossible.QT (σ) =1NN∑i=11TT∑t=1Qˆit(σ). (A.19)The two-step estimator σˆ satisfies the following condition.QT (σˆ) =1NN∑i=11TT∑t=1Qˆit(σˆ) = 0.The difference of this estimator from the usual GMM estimator is that in the first step, consumptionfunctions, c(dit, dit−1, xit, sit) and c(dit+1, dit, xjit+1, sjit+1), and the conditional choice probabilities,Pr{dit+1|dit, xjit+1, sjit+1}, are estimated nonparametrically. In the second step, QT (σˆ) defined inequation (A.19), the empirical analogue of the inter-temporal Euler equation, is constructed usingthese nonparametric estimators. Finally, the two-step estimator σˆ solves the equation QT (σˆ) = 0.In more detailed way, it can be understood as a three-step estimation method because nonparamet-ric estimations of the consumption function and the conditional choice probability function of stock99holding share are conducted before the inter-temporal Euler equation estimation of a householdutility function.A.3.2 The Estimation of the Full ModelThe conditional choice probability of household stock holding share, dit, is expressed asPr(dit|dit−1, xit, sit; θ) = exp(V (dit, dit−1, xit, sit; θ)∑dit∈D exp(V (dit, dit−1, xit, sit; θ)).The log-likelihood functions will be defined in the following way asB = {dit, dit−1, xit, sit}N, Ti=1,t=1Lˆ1(θ;B) =N∑i=1T∑t=1log[ exp(V (dit, dit−1, xit, sit; θ))∑d′t∈D exp(V (d′t, dit−1, xit, sit; θ))](A.20)Lˆ2(θ, θg;B) =N∑i=1T∑t=1log[exp(c(dit,dit−1,xit,sit)1−σ1−σ −AC · (dit − dit−1)2 − PC · 1{dit 6= 0}+ βEV¯ (dit, xit+1, sit+1; θg))∑d′t∈D exp(c(d′t,dit−1,xit,sit)1−σ1−σ −AC · (d′t − dit−1)2 − PC · 1{d′it 6= 0}+ βEV¯ (d′t, x′it+1, sit+1; θg))](A.21)θ is (AC,PC) and θg is (ACg, PCg). The log-likelihood function, Lˆ1(θ;B), is straightforwardto understand. Using an analytic form expression of the conditional choice probability function,Pr(dit|dit−1, xit, sit; θ), we can form the log-likelihood function as in Equation (A.20). For eachcandidate value of θ, the household value function V (d′t, dit−1, xit, sit; θ) is solved. Using the so-lution for V (d′t, dit−1, xit, sit; θ), the objective function, Lˆ1(θ;B), is evaluated for each candidatevalue of θ. The Maximum Likelihood estimator, θL1 , is obtained by maximizing the objective func-tion, Lˆ1(θ;B), with respect to θ. This approach mimics the conventional Maximum LikelihoodEstimation method (MLE) more even though it might be inefficient compared to the approachusing Lˆ2(θ, θg;B) as the objective function. The log-likelihood function, Lˆ2(θ, θg;B), is more oftenused as the objective function in the conditional choice probability estimation literature. Accord-ing to the conditional choice probability estimation literature, the Maximum Likelihood estimationmethod using Lˆ2(θ, θg;B) as an objective function is computationally faster and more accuraterelative to the Maximum Likelihood estimation method using Lˆ1(θ;B) as an objective function.Thus, I used Lˆ2(θ, θg;B) as an objective function in this thesis. I explain the estimation procedurefor obtaining the Maximum Likelihood estimator that uses Lˆ2(θ, θg;B) as an objective function indetail as follows:1. σˆ estimated in the Euler equation estimation procedure illustrated through Equations (A.14)to (A.19) is used in the place of σ in objective function Lˆ2(θ, θg;B).2. Given an initial guess of θg, household value functions are solved.3. Using the solved household value function, V (d′t, dit−1, xit, sit; θ), the objective function,100Lˆ2(θ, θg;B), is constructed and maximized with respect to θ and the maximizer of the ob-jective function, Lˆ2(θ, θg;B), is denoted by θ˜.4. Next, θ˜ will be a new guess for θg.5. Repeat the steps from 2 to 4, untill ||θ˜ − θg|| < etol is satisfied. Here, etol is the tolerancelevel for the convergence and || · || is the Euclidean norm.The household value function is solved by employing the Endogenous Grid Method. Initial guessesfor the household value function and the consumption function are constructed in a similar way asthe terminal period’s household value function and consumption function are constructed in the life-cycle model. These initial guesses for the household value function and the consumption functionare not posing any issues to the convergence of solution of the household dynamic model even thoughthe convergence is not guaranteed for arbitrary choices of initial guesses for the household valuefunction and the consumption function. The Endogenous Grid Method is often viewed as fasterthan the other computational methods for solving a household dynamic model in ComputationalEconomics literature.A.3.3 The Bootstrapping for the Standard Error and the Bias CorrectionThe bootstrapping method is employed to obtain the standard error, the confidence interval, andthe bias-corrected estimator in this study. I follow the econometrics textbook written by BruceE.Hansen to construct the bootstrap standard error, the bootstrap confidence interval, and thebootstrap bias-corrected estimator. Detail on the bootstrapping method is discussed extensivelyin the econometrics textbook written by Bruce E.Hansen. One disadvantage of boostrapping isthat it is computationally very costly because the estimation procedure has to be repeated for eachbootstrap sample.The bootstrap standard error is computed asVˆ ?n =1BB∑b=1(ψˆ?b − ψˆ?)2,s?(ψˆ) =√Vˆ ?n .The bootstrap confidence interval is constructed as[ψˆ − qˆ?(.975), ψˆ − qˆ?(.025)].Here, qˆ?(α) is the αth sample quantile of the simulated statistics {T ?1 , ..., T ?B} which are obtainedfrom all boostrap samples.101The bootstrap bias-corrected estimator is calculated as follows:τˆ? =1BB∑b=1(ψˆ?b − ψˆ)= ψˆ? − ψˆ,ψ˜? = ψˆ − τˆ?= 2ψˆ − ψˆ?.Here, τˆ? is the estimator of the bias and ψ˜? is the bootstrap bias-corrected estimator. ψ represents{σ,AC, PC} in the equations defined above.102Appendix BAppendix to Chapter 3B.1 Data DescriptionIn Chapter 3, I used the Panel Study of Income Dynamics (PSID) dataset for the estimation ofthe household dynamic model. The reason is that the PSID dataset is the micro-panel dataset,collected every two years between 1998 and 2016, and includes the household consumption, income,and wealth data. Therefore, it is well suited for the estimation procedure of the household dynamicmodel developed in Chapter 2. Meanwhile, the household consumption data from the PSID datasetare known to accommodate substantial measurement errors. In addition, the PSID dataset includesrelatively fewer households from the top part of the wealth distribution in comparison to the Surveyof Consumer Finances (SCF) dataset. Hence, I used the SCF dataset for evaluating the effects of thestock market collapse and the Zero-Lower Bound monetary policy on household decisions in Chapter3. The SCF dataset has the panel structure only for the 2006 and 2008 waves. Moreover, the SCFdataset does not include the household consumption data. I evaluated the effects of the stockmarket collapse and the Zero-Lower Bound monetary policy on household consumption decisionusing the consumption function from the full model solution. Thus, the lack of the householdconsumption data was not an issue for the purpose of evaluating the effects of the stock marketcollapse and the Zero-Lower Bound monetary policy on household consumption decision.B.1.1 Household Consumption DataThe household consumption data from the PSID dataset are widely known to contain substantialmeasurement errors. In the first step of the estimation procedure for estimating the householddynamic model, the household consumption function is nonparametrically estimated using thehousehold consumption data from the PSID dataset in Chapter 3. By assuming that the measure-ment error is independent of the household state variables, I avoid the measurement error issuein the household consumption data using the nonparametric estimation method. Furthermore,the household consumption data from the PSID dataset are well approximated by the log-normaldistribution as Figure B.1 demonstrates. The household consumption data from the PSID datasetinclude the food consumption, electricity bill, home insurance bill, heating bill, water bill, miscel-laneous utilities bill, car insurance bill, car repair expenses, vehicle fuel expenses, vehicle parkingexpenses, bus fare expenses, taxi fare expenses, miscellaneous transportation fare expenses, edu-cation related expenses, miscellaneous education related expenses, childcare expenses, and medicalexpenses.1030.00001.00002.00003.00004Density0 100000 200000 300000Household consumption level (1998 USD value)Figure B.1: Histogram for The Household Consumption Data from the PSID Dataset(CPI-adjusted to the 1998 USD)B.1.2 Household Wealth DataThe household wealth data from the PSID dataset include the value of the farm or business, thechecking and savings account balances, the value of the stock holding, the total value of vehicles,the value of the other assets, and the balance of the individual retirement account (IRA). Moreover,the household wealth data from the PSID dataset contain the following debts: the credit card andstore card debts, the outstanding student loans, the outstanding medical bills, the outstanding legalbills, the loans from relatives, and the other debts. Furthermore, the value of the home equity isincluded in the household wealth data from the PSID dataset. In the household dynamic model ofChapter 3, I exclude the home equity value from the household wealth data. In addition, I classifythe stock holding in the household wealth as the risky asset holding and the remaining value in thehousehold wealth as the risk-free asset holding. I use the household wealth data from the PSIDdataset for estimating the household dynamic model in Chapter 3.The household wealth data from the PSID dataset do not include the tax-deferred retirementsavings account (401(K)). On the other hand, the household wealth data from the Survey of Con-sumer Finances (SCF) dataset include the tax-deferred retirement savings account (401(K)). Fur-thermore, the 2006 and 2008 waves of the SCF dataset contain more households than the PSIDdataset. In addition, the SCF dataset is known for representing the real wealth distribution of theU.S. better than the PSID dataset. Therefore, I used the SCF dataset for evaluating the effectsof the stock market collapse and the Zero-Lower Bound monetary policy on household decisions inChapter 3. I present the histogram of the household wealth data from the 2008 wave of the SCFdataset in Figure B.2.104Household Wealth Level (in 1000 dollars, CPI-adjusted to the 1998 USD value)-2000 0 2000 4000 6000 8000 10000 12000 14000 16000Frequency0100020003000400050006000Figure B.2: Histogram for The Household Wealth Data from the SCF Dataset (2008 Wave)B.1.3 The Distribution of the Stock-Holding ShareI demonstrate the distribution of the stock-holding share among stock-holding households from thePSID dataset in Table B.1. Approximately, 30% of households participate in the stock market inany time point in the U.S. in average if we look at the PSID and SCF datasets.Table B.1: The Distribution of the Stock Holding Share among Households from thePanel Study of Income Dynamics DatasetNo stock holding (0%, 20%) (20%, 40%) (40%, 60%) (60%, 80%) (80%, 100%)No stock holding 88.35% 32.86% 27.32% 24.81% 27.90% 28.64%(0%, 20%) 05.79% 43.49% 25.67% 15.95% 12.26% 10.33%(20%, 40%) 02.24% 11.93% 23.45% 20.34% 10.46% 11.11%(40%, 60%) 01.58% 06.79% 12.37% 21.19% 18.34% 12.05%(60%, 80%) 01.09% 02.92% 07.62% 11.63% 19.57% 14.71%(80%, 100%) 00.95% 02.01% 03.58% 06.09% 11.47% 23.16%Note: The first column of this table expresses the current period’s household stock holding share choice and the firstrow of this table demonstrates the previous period’s household stock holding share choice.From the diagonal of Table B.1, we can observe that there exists a strong persistence in house-hold stock holding share choices in the current and previous periods. Furthermore, we can seethat 88.35% of households, who did not participate in the stock market in the previous period,choose not to participate in the stock market in the current period. Moreover, 24.81% to 32.86% ofhouseholds, who participated in the stock market in the previous period, choose not to participatein the stock market in the current period.105B.1.4 The Stock Market Return and the Risk-Free Rate DataIn Chapter 3, the real annualized return of the S&P500 stock market price index is used as thesource for the stock market return data. The S&P500 stock market price index is widely used ineconomic researches as the measure of the stock price movement. Moreover, the S&P500 measuresthe performances of the stock prices for the 500 large companies listed in the stock exchange marketin the U.S. In Chapter 3, dividends are included in the calculation of the real annualized return ofthe S&P500 stock market price index, and the inflation rate is taken into account to obtain thereal return. Furthermore, the real annualized 3-month Treasury Bill rate is used as the risk-freerate in Chapter 3. The 3-month Treasury Bill rate is broadly used as the measure of the risk-freerate in economic researches. In 2008, the 3-month Treasury Bill rate is pushed to the zero lowerbound as a result of the Zero-Lower Bound monetary policy. And it stayed at the zero lower bounduntil 2016.B.2 The Estimation Result for the Constant Absolute RiskAversion Utility FunctionIn this section, I demonstrate that the Constant Absolute Risk Aversion utility function can beestimated from a micro-panel dataset using the methodology developed in this thesis. When theConstant Absolute Risk Aversion utility function is assumed as the household preference, the house-hold optimization problem can be described asV (dit, dit−1, xit, sit) = maxcit{1− e−σcitσ−AC · (dit − dit−1)2 − PC · 1{dit 6= 0}+ βEV¯ (dit, xit+1, sit+1)}(B.1)subject to the inter-temporal budget constraint,xit+1 = (xit − cit)(1 + rf + dit(rt+1 − rf )) + yit+1where the Constant Absolute Risk Aversion utility function is defined as 1−e−σcσ . As noted by Rust(1987), we obtain the following expressions asP{dit|dit−1, xit, sit} = exp(V (dit, dit−1, xit, sit))∑d′it∈D exp(V (d′it, dit−1, xit, sit)),V¯ (dit−1, xit, sit) = ln( ∑d′it∈Dexp(V (d′it, dit−1, xit, sit)))+ γ.106By exactly following all steps from Section A.2.1, the inter-temporal Euler equation for the ConstantAbsolute Risk Aversion utility function is derived as follows:e−σc(dit,dit−1,xit,sit) = βE{ ∑d′it+1∈DP{d′it+1|dit, xit+1, sit+1}(1 + rf + dit(rt+1 − rf ))e−σc(dit+1,dit,xit+1,sit+1)}(B.2)where the first derivative of the Constant Absolute Risk Aversion utility function is e−σc. I presentthe estimation result of the household dynamic model with the Constant Absolute Risk Aversionutility function from the Panel Study of Income Dynamics dataset in the following table.Table B.2: The Estimation Result: The Constant Absolute RiskAversion Utility FunctionParameter EstimatorBootstrapstandard errorBootstrap 95%Confidence IntervalBias CorrectedEstimatorσ 5.48 0.20 (5.24, 6.08) 5.65AC 3.48 0.07 (3.31, 3.63) 3.48PC 1.71 0.01 (1.68, 1.73) 1.71Note: σ denotes the parameter of the Constant Absolute Risk Aversion utility function,AC denotes the stock holding share adjustment cost, and PC denotes the stock marketparticipation cost.I estimate the Constant Absolute Risk Aversion utility function using the exact same procedurethat is used for estimating the Constant Absolute Risk Aversion utility function and the Quasi-Hyperbolic Discounting preference in this thesis.107
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Title | Essays on household preference and dynamic decisions |
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Enkhbaatar, Tsenguun |
Publisher | University of British Columbia |
Date Issued | 2020 |
Description | This study examines the identification of household preference from a micro-panel dataset and the effects of the stock market collapse and the Zero-Lower Bound monetary policy on household consumption decisions during the 2007-2008 global financial crisis (GFC). In the first chapter, I propose a procedure for estimating the household utility function from a micro-panel dataset using the intertemporal Euler equation. Here, I demonstrate that the household utility function can be estimated accurately from a micro-panel dataset using the intertemporal Euler equation by taking into account the differences in the portfolio compositions of household savings across households. In addition, I construct a new household dynamic model in which the household stock-holding behavior is modeled by explicitly taking into account the hidden stock market participation cost pointed out in recent empirical studies as a potential explanation for the low stock market participation rate among households in the United States. In the second chapter, I explore the effects of the stock market collapse and the Zero-Lower Bound monetary policy on household consumption decisions during the GFC. The huge drop in the stock market return and the decline in the risk-free rate due to the Zero-Lower Bound monetary policy triggered a large percentage decrease in the consumption of wealthy households in this period in the United States. Meanwhile, the consumption of households at the bottom part of the wealth distribution, which generally do not participate in the stock market, increased slightly in percentage due to the decline in the risk-free rate during the GFC in the United States. As a result, the stock market collapse and the Zero-Lower Bound monetary policy generated a substantial decrease in consumption inequality among households. In the third chapter, I estimate the heterogeneity in household preference by employing a combination of the extremum and nonparametric estimation methods and find significant heterogeneity in the preferences across households in the United States and Italy. This estimated heterogeneity in household preference can be used to explain the observed wealth distribution and the differences in career and investment choices across households in future research. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2020-12-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0395343 |
URI | http://hdl.handle.net/2429/76835 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
GraduationDate | 2021-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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