ESSAYS ON TESTING STRUCTURAL CHANGES AND CONSTANT CONDITIONAL DEPENDENCE VIA THE FOURIER TRANSFORM

Abstract

This dissertation consists of three essays on testing structural changes and constant conditional dependence via the Fourier transform. The first essay, "A Model-free Consistent Test for Structural Change in Regression 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 inference and imprecise prediction of stationary time series models. I propose a model-free consistent test for structural change in regression by testing the instability 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 regressors 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 powerful 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 endogenous and discrete regressors. By using a proper choice of weighting functions for the transform parameters in the Fourier transform, I avoid numerical integration so that the test statistic is easy to compute. The test statistic has a convenient 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 structural changes in a linear time series regression model via the Discrete Fourier Transform (DFT). The intuition is straightforward: if the true model parameters 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 structural changes regardless of whether they are smooth or abrupt. Compared to the existing consistent tests for structural change, my test avoids smoothed nonparametric estimation of the unknown time-varying parameters. The rate of the local alternatives that our test can detect is Root-T. Furthermore, my test is robust 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", proposes an omnibus test for the constancy of conditional joint dependence on some state variables. The test statistic is constructed by comparing the generalized conditional covariance function and the generalized unconditional covariance 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 statistic can apply to many other testing frameworks with proper transformation. A simulation study shows that the test works well in finite samples.