Empirical Likelihood with Applications in Time Series

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Abstract

This thesis investigates the statistical properties of Kernel Smoothed Empirical Likelihood (KSEL, e.g. Smith, 1997 and 2004) estimator and various associated inference procedures in weakly dependent data. New tests for structural stability are proposed and analysed. Asymptotic analyses and Monte Carlo experiments are applied to assess these new tests, theoretically and empirically. Chapter 1 reviews and discusses some estimation and inferential properties of Empirical Likelihood (EL, Owen, 1988) for identically and independently distributed data and compares it with Generalised EL (GEL), GMM and other estimators. KSEL is extensively treated, by specialising kernel-smoothed GEL in the working paper of Smith (2004), some of whose results and proofs are extended and refined in Chapter 2. Asymptotic properties of some tests in Smith (2004) are also analysed under local alternatives. These special treatments on KSEL lay the foundation for analyses in Chapters 3 and 4, which would not otherwise follow straightforwardly. In Chapters 3 and 4, subsample KSEL estimators are proposed to assist the development of KSEL structural stability tests to diagnose for a given breakpoint and for an unknown breakpoint, respectively, based on relevant work using GMM (e.g. Hall and Sen, 1999; Andrews and Fair, 1988; Andrews and Ploberger, 1994). It is also original in these two chapters that moment functions are allowed to be kernel-smoothed after or before the sample split, and it is rigorously proved that these two smoothing orders are asymptotically equivalent. The overall null hypothesis of structural stability is decomposed according to the identifying and overidentifying restrictions, as Hall and Sen (1999) advocate in GMM, leading to a more practical and precise structural stability diagnosis procedure. In this framework, these KSEL structural stability tests are also proved via asymptotic analysis to be capable of identifying different sources of instability, arising from parameter value change or violation of overidentifying restrictions. The analyses show that these KSEL tests follow the same limit distributions as their counterparts using GMM. To examine the finite-sample performance of KSEL structural stability tests in comparison to GMM's, Monte Carlo simulations are conducted in Chapter 5 using a simple linear model considered by Hall and Sen (1999). This chapter details some relevant computational algorithms and permits different smoothing order, kernel type and prewhitening options. In general, simulation evidence seems to suggest that compared to GMM's tests, these newly proposed KSEL tests often perform comparably. However, in some cases, the sizes of these can be slightly larger, and the false null hypotheses are rejected with much higher frequencies. Thus, these KSEL based tests are valid theoretical and practical alternatives to GMM's.

Keyword(s)

(Generalised) Empirical Likelihood, Time Series, Structural Stability Testing, Generalised Method of Moments, Kernel Smoothing, Monte Carlo

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