Bootstrap Tests For Unit Root Ar1 Models

Testing for a unit root is now an essential part of time series analysis. This volume provides a critical overview and assessment of tests for a unit root in time series, developing the concepts necessary to understand the key theoretical and practical models in unit root testing.

Many economic theories depend on the presence or absence of a unit root for their validity, making familiarity with unit roots extremely important to econometric and statistical theory. This book introduces the literature on unit roots in a comprehensive manner to empirical and theoretical researchers in economics and other areas.

In this paper, we consider a sieve bootstrap for the test of a unit root in models driven by general linear processes. The given model is first approximated by a finite autoregressive integrated process of order increasing with the sample size, and then the method of bootstrap is applied for the approximated autoregression to obtain the critical values for the usual unit root tests. The resulting tests, which may simply be viewed as the bootstrapped versions of Augmented Dickey-Fuller (ADF) unit root tests by Said and Dickey (1984), are shown to be consistent under very general conditions. The asymptotic validity of the bootstrap ADF unit root tests is thus established. Our conditions are significantly weaker than those used by Said and Dickey. Simulations show that bootstrap provides substantial improvements on finite sample sizes of the tests.

This article proposes a hybrid bootstrap approach to approximate the augmented Dickey-Fuller test by perturbing both the residual sequence and the minimand of the objective function. Since innovations can be dependent, this allows the inclusion of conditional heteroscedasticity models. The new bootstrap method is also applied to least absolute deviation-based unit root test statistics, which are efficient in handling heavy-tailed time-series data. The asymptotic distributions of resulting bootstrap tests are presented, and Monte Carlo studies demonstrate the usefulness of the proposed tests.

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This study leads to the following conclusions: (i) augmented DF tests are always preferred to standard DF tests; (ii) the sieve bootstrap performs better than the block bootstrap; (iii) difference-based tests appear to have slightly better size properties, but residual-based tests appear more powerful. We show that two sieve bootstrap tests based on residuals remain asymptotically valid. In contrast to the literature which focuses on a comparison of the bootstrap tests with an asymptotic test, we compare the bootstrap tests among themselves using response surfaces for their size and power in a simulation study. In this article, we study and compare the properties of several bootstrap unit-root tests recently proposed in the literature. The tests are Dickey Fuller (DF) or Augmented DF, based either on residuals from an auto-regression and the use of the block bootstrap or on first-differenced data and the use of the stationary bootstrap or sieve bootstrap. We extend the analysis by interchanging the data transformations (differences vs. residuals), the types of bootstrap and the presence or absence of a correction for autocorrelation in the tests. We show that two sieve bootstrap tests based on residuals remain asymptotically valid. In contrast to the literature which focuses on a comparison of the bootstrap tests with an asymptotic test, we compare the bootstrap tests among themselves using response surfaces for their size and power in a simulation study. This study leads to the following conclusions: (i) augmented DF tests are always preferred to standard DF tests; (ii) the sieve bootstrap performs better than the block bootstrap; (iii) difference-based tests appear to have slightly better size properties, but residual-based tests appear more powerful.