Bootstrap Unit Root Tests

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.

Bootstrap -- Dependent data -- Dickey-Fuller test -- Stationarity -- Unit root tests.

"We consider the use of a sieve bootstrap based on moving average (MA) and autoregressive moving average (ARMA) approximations to test the unit root hypothesis when the true Data Generating Process (DGP) is a general linear process. We provide invariance principles for these bootstrap DGPs and we prove that the resulting ADF tests are asymptotically valid. Our simulations indicate that these tests sometimes outperform those based on the usual autoregressive (AR) sieve bootstrap. We study the reasons for the failure of the AR sieve bootstrap tests and propose some solutions, including a modified version of the fast double bootstrap." --

Unit root process, as a process with stochastic trend and a generalization from random walk, is pervasive in physics, economics, and finance. In the hypothesis test for unit root, bootstrap methods have earned a great deal of attention. This dissertation proposes and investigates various bootstrap unit root tests. Chapter one applies linear process bootstrap to unit root test in order to alleviate the size distortions of unit root tests. While Chapter one focuses on classic unit root tests, which search for stochastic trend, Chapter two tackles seasonal unit root tests, which simultaneously check stochastic trend and stochastic seasonality. In addition, Chapter two takes into consideration seasonal heterogeneity, which pervades in seasonal processes. Specifically, Chapter two offers under seasonal heterogeneity a seasonal AR-sieve bootstrap remedy for a parametric seasonal unit root test and advocates a seasonal block bootstrap solution for a non-parametric test. This dissertation then establishes three bootstrap functional central limit theorems, via which this dissertation shows the consistency of all the aforementioned bootstrap methods.

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.

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.

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.