Title:

Asymptotic analysis of the 1step recursive Chow test (and variants) in time series model

This thesis concerns the asymptotic behaviour of the sequence of 1step recursive Chow statistics and various tests derived therefrom. The 1step statistics are produced as diagnostic output in standard econometrics software, and are expected to reflect model misspecification. Such misspecification testing is important in validating the assumptions of a model and so ensuring that subsequent inference is correct. Original contributions to the theory of misspecification testing include (i) a result on the pointwise convergence of the 1step statistics; (ii) a result on the extremevalue convergence of the maximum of the statistics; and (iii) a result on the weak convergence of an empirical process formed by the statistics. In Chapter 2, we describe the almost sure pointwise convergence of the 1step statistic for a broad class of time series models and processes, including unit root and explosive processes. We develop an asymptotic equivalence result, and use this to establish the asymptotic distribution of the maximum of a sequence of 1step statistics with normal errors. This allows joint consideration of the sequence of 1step tests via its maximum: the supChow test. In Chapter 3, we use simulation to investigate the power properties of this test and compare it with benchmark tests of structural stability. We find that the supChow test may have advantages when the nature of instability is unknown. In Chapter 4, we consider how the test may be adapted to situations in which the errors cannot be assumed normal. We evaluate several promising approaches, but also note a tradeoff between robustness and power. In Chapter 5 we analyse an empirical process formed from the 1step statistics, and prove a weak convergence result. Under the assumption of normal errors, the limiting distribution reduces to that of a Brownian bridge. The asymptotic approximation appears to works well even in small samples.
