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Essays on Testing Structural Changes and Constant Conditional Dependence Via the Fourier Transform ESSAYS ON TESTING STRUCTURAL CHANGES AND CONSTANT CONDITIONAL DEPENDENCE VIA THE FOURIER TRANSFORM A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Zhonghao Fu May 2017 c 2017 Zhonghao Fu ALL RIGHTS RESERVED ESSAYS ON TESTING STRUCTURAL CHANGES AND CONSTANT CONDITIONAL DEPENDENCE VIA THE FOURIER TRANSFORM Zhonghao Fu, Ph.D. Cornell University 2017 This dissertation consists of three essays on testing structural changes and con- stant conditional dependence via the Fourier transform. The first essay, “A Model-free Consistent Test for Structural Change in Re- gression Possibly with Endogeneity”, proposes a consistent test for structural change in a nonparametric times series regression model based on the Fourier transform. It is well known that structural instability leads to misleading in- ference and imprecise prediction of stationary time series models. I propose a model-free consistent test for structural change in regression by testing the in- stability of the Fourier transform of data. This novel approach avoids smoothed nonparametric estimation of the unknown regression function and so is free of the “curse of dimensionality” problem, especially when the dimension of re- gressors is high. As a result, the proposed test is asymptotically more powerful against a class of local alternatives than Vogt’s (2015) nonparametric test for structural changes, which is the only consistent test for structural changes in a nonparametric regression model in the existing literature. The nonparametric tests of Hidalgo (1995) and Su and Xiao (2008) are asymptotically more pow- erful than the proposed test against certain smooth local alternatives, but they are not consistent tests and are asymptotically less powerful against a class of non-smooth local alternatives. Unlike the existing literature, I allow for endoge- nous and discrete regressors. By using a proper choice of weighting functions for the transform parameters in the Fourier transform, I avoid numerical inte- gration so that the test statistic is easy to compute. The test statistic has a conve- nient asymptotic N(0; 1) distribution under the null hypothesis of no structural change and is consistent against a large class of smooth structural changes as well as abrupt structural breaks with unknown break dates. A Monte Carlo study and an empirical application show that the test performs reasonably well in finite samples. In the second essay titled “Consistent Testing for Structural Change in Time Series Regression Models via the Fourier Transform”, I focus on testing struc- tural changes in a linear time series regression model via the Discrete Fourier Transform (DFT). The intuition is straightforward: if the true model parame- ters are time-varying, then the conventional estimation methods like OLS or 2SLS will fail to estimate the unknown parameters consistently. The estimated residuals will contain the time-varying local feature of model parameters. The Discrete Fourier Transform of estimated residuals will contain this information and reveal it in the frequency domain. One can then infer the existence of struc- tural changes regardless of whether they are smooth or abrupt. Compared to the existing consistent tests for structural change, my test avoids smoothed non- parametric estimation of the unknown time-varying parameters. The rate of the local alternatives that our test can detect is T −1=2. Furthermore, my test is ro- bust to unknown structural change in explanatory variables and instrumental variables, which makes the test widely applicable, especially in macroeconomic models. Simulation studies demonstrate its good finite sample performance. I apply my test to examine the stability of the hybrid New Keynesian Phillips Curve and find evidence of structural changes in 1980 to 2001 which is treated as a stable period by Zhang et al. (2008) and Hall et al. (2012). The third essay, “Testing Constancy of Conditional Joint Dependence”, pro- poses an omnibus test for the constancy of conditional joint dependence on some state variables. The test statistic is constructed by comparing the general- ized conditional covariance function and the generalized unconditional covari- ance function. It detects if the dependence strength between any two random variables varies with a certain factor that we are interested in. I show that by using a special weighting function proposed by Szekely´ et al. (2007), the test statistic can be easily computed without using numerical simulation. Also by nonparametric regression, I show that the test statistic is both computationally and asymptotically invariant to possible high dimensional data. Furthermore, the test statistic is asymptotically pivotal and follows a convenient asymptotic N(0; 1) distribution under the null hypothesis. I also show that the test statis- tic can apply to many other testing frameworks with proper transformation. A simulation study shows that the test works well in finite samples. BIOGRAPHICAL SKETCH Zhonghao Fu was born in Qiqihaer, China in July 1986. He studied Political Science at Renmin University of China and earned his B.A. degree in Law in July 2009. He was admitted to Cornell Institute for Public Affairs and earned his M.P.A. in May 2011. He continued his graduate studies in economics in the Department of Economics at Cornell University and will earn his Ph.D. degree in May 2017. iii This document is dedicated to my wife Yunxuan Zhang, my daughter Linyi Fu and my parents, Hanjun Fu and Lijun Wang. iv ACKNOWLEDGEMENTS I am deeply indebted to my advisor Prof. Yongmiao Hong for his continued support during my Ph.D. studies. His patience, motivation, and excellent guid- ance helped me in all the time of research and writing of this thesis. I would like to express my deepest gratitude to Prof. Nicholas M. Kiefer and Prof. G. Andrew Karolyi, for their helpful discussions and encouragement as well as serving on my committee. I also thank Xin Wang, Prof. Hong’s wife. She treated me like her son and gave me the guidance and encouragement when I need them the most. This thesis could not have been completed without her support. Last but not least, I want to thank my wife, my daughter, and my parents for their love and support. My wife and daughter make my life delightful. My parents have supported me firmly since the beginning. They are my source of strength during my job market period. v TABLE OF CONTENTS Biographical Sketch . iii Dedication . iv Acknowledgements . .v Table of Contents . vi List of Tables . viii List of Figures . ix 1 Introduction 1 1.1 A Model-free Consistent Test for Structural Change in Regression Possibly with Endogeneity . .1 1.2 Consistent Testing for Structural Change in Time Series Regres- sion Models via the Fourier Transform . .5 1.3 Testing Constancy of Conditional Joint Dependence . .9 2 A Model-free Consistent Test for Structural Change in Regression Pos- sibly with Endogeneity 14 2.1 Hypotheses of Interest and Approach . 14 2.2 Nonparametric Testing . 19 2.2.1 Nonparametric Estimation . 19 2.2.2 Test Statistic . 22 2.3 Asymptotic Distribution . 24 2.4 Asymptotic Power . 27 2.5 Weighting Function . 31 2.6 Simulation Studies . 34 2.6.1 The Exogeneity Case . 34 2.6.2 The Endogeneity Case . 38 2.7 Empirical Application . 41 2.8 Conclusion . 43 3 Consistent Testing for Structural Change in Time Series Regression Models via the Fourier Transform 45 3.1 Framework and Approach . 45 3.2 Asymptotic Theory . 53 3.3 Extension to Endogenous Covariates . 63 3.4 Simulation Studies . 71 3.4.1 Exogenous Covariates . 72 3.4.2 Endogenous Covariates . 75 3.5 Empirical Application . 78 3.6 Model Misspecification . 82 3.7 Conclusion . 85 vi 4 Testing Constancy of Conditional Joint Dependence 87 4.1 Basic Framework . 87 4.2 Test Statistic . 90 4.2.1 Estimation for the CCF . 90 4.2.2 Estimation for the UCF . 92 4.2.3 The Test Statistic . 93 4.3 Asymptotic Theory . 95 4.4 Computing the Test Statistic . 99 4.5 Simulation Studies . 104 4.6 Extensions . 105 4.7 Conclusion . 109 A Appendix of Chapter 2 111 A.1 Figures and Tables . 111 A.2 Mathematical Proofs . 118 B Appendix of Chapter 3 161 B.1 Figures and Tables . 161 B.2 Mathematical Proofs . 165 C Appendix of Chapter 4 191 C.1 Figures and Tables . 191 C.2 Mathematical Proofs . 192 Bibliography 222 vii LIST OF TABLES A.1 Empirical size in finite samples . 114 A.2 Empirical power in finite samples . 115 A.3 Empirical rejection rates at the 5% nominal level . 116 A.4 Empirical rejection rates at the 10% nominal level . 116 A.5 Stability test for predictive regressions . 117 B.1 Empirical size with exogenous covariates (T = 100) ........ 161 B.2 Empirical size with exogenous covariates (T = 300) ........ 162 B.3 Empirical power with exogenous covariates (T = 100) ...... 162 B.4 Empirical power with exogenous covariates (T = 300) ...... 163 B.5 Empirical size with endogenous covariates (T = 100) ....... 163 B.6 Empirical size with endogenous covariates (T = 300) ....... 164 B.7 Empirical power with endogenous covariates (T = 100) ...... 164 B.8 Empirical power with endogenous covariates (T = 300) ...... 165 C.1 Empirical size . 191 C.2 Empirical power . 192 viii LIST OF FIGURES A.1 Plot of the real part in Example 2.1.1 . 111 A.2 Plot of the imaginary part in Example 2.1.1 . 111 A.3 Plot of the real part in Example 2.1.2 . 112 A.4 Plot of the imaginary part in Example 2.1.2 . 112 A.5 Plot of the real part in Example 2.1.3 . 113 A.6 Plot of the imaginary part in Example 2.1.3 . 113 ix CHAPTER 1 INTRODUCTION 1.1 A Model-free Consistent Test for Structural Change in Re- gression Possibly with Endogeneity Nonlinearity often exists in economic time series data and various nonlinear time series models have been proposed to capture different forms of nonlinear- ities.
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