Whitehouse, Emily J.
(2017)
Robust methods in univariate time series models.
PhD thesis, University of Nottingham.
Abstract
The size and power properties of a hypothesis test typically depend on a series of factors which are unobservable in practice. A branch of the econometric literature therefore considers robust testing methodologies that achieve good size-control and competitive power across a range of differing circumstances. In this thesis we discuss robust tests in three areas of time series econometrics: detection of explosive processes, unit root testing against nonlinear alternatives, and forecast evaluation in small samples.
Recent research has proposed a method of detecting explosive processes that is based on forward recursions of OLS, right-tailed, Dickey-Fuller [DF] unit root tests. In Chapter 2 an alternative approach using GLS DF tests is considered. We derive limiting distributions for both mean-invariant and trend-invariant versions of OLS and GLS variants of the Phillips, Wu and Yu (2011) [PWY] test statistic under a temporary, locally explosive alternative. These limits are dependent on both the value of the initial condition and the start and end points of the temporary explosive regime. Local asymptotic power simulations show that a GLS version of the PWY statistic offers superior power when a large proportion of the data is explosive, but that the OLS approach is preferred for explosive periods of short duration. These power differences are magnified by the presence of an asymptotically non-negligible initial condition. We propose a union of rejections procedure that capitalises on the respective power advantages of both OLS and GLS-based approaches. This procedure achieves power close to the effective envelope provided by the two individual PWY tests across all settings of the initial condition and length of the explosive period considered in this chapter. We show that these results are also robust to the point in the sample at which the temporary explosive regime occurs. An application of the union procedure to NASDAQ daily prices confirms the empirical value of this testing strategy.
Chapter 3 examines the local power of unit root tests against globally stationary exponential smooth transition autoregressive [ESTAR] alternatives under two sources of uncertainty: the degree of nonlinearity in the ESTAR model, and the presence of a linear deterministic trend. First, we show that the Kapetanios, Shin and Snell (2003) [KSS] test for nonlinear stationarity has local asymptotic power gains over standard Dickey-Fuller [DF] tests for certain degrees of nonlinearity in the ESTAR model, but that for other degrees of nonlinearity, the linear DF test has superior power. Second, we derive limiting distributions of demeaned, and demeaned and detrended KSS and DF tests under a local ESTAR alternative when a local trend is present in the DGP. We show that the power of the demeaned tests outperforms that of the detrended tests when no trend is present in the DGP, but deteriorates as the magnitude of the trend increases. We propose a union of rejections testing procedure that combines all four individual tests and show that this captures most of the power available from the individual tests across different degrees of nonlinearity and trend magnitudes. We also show that incorporating a trend detection procedure into this union testing strategy can result in higher power when a large trend is present in the DGP. An empirical application of our proposed union of rejections procedures to energy consumption data in 180 countries shows the value of these procedures in practical cases.
In Chapter 4 we show that when computing standard Diebold-Mariano-type tests for equal forecast accuracy and forecast encompassing, the long-run variance can frequently be negative when dealing with multi-step-ahead predictions in small, but empirically relevant, sample sizes. We subsequently consider a number of alternative approaches to dealing with this problem, including direct inference in the problem cases and use of long-run variance estimators that guarantee positivity. The finite sample size and power of the different approaches are evaluated using an extensive Monte Carlo simulation exercise. Overall, for multi-step-ahead forecasts, we find that the recently proposed Coroneo and Iacone (2015) test, which is based on a weighted periodogram long-run variance estimator, offers the best finite sample size and power performance.
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