Bootstrap Tests For Unit Root And Seasonal Unit Root

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.

"The application of the sieve bootstrap procedure, which resamples residuals obtained by fitting a finite autoregressvie (AR) approximation to empirical time series, to obtaining prediction intervals for integrated, long-memory, and seasonal time series as well as constructing a test for seasonal unit roots, is considered. The advantage of this resampling method is that it does not require knowledge about the underlying process generating a given time series and has been shown to work well for ARMA processes. We extend the application of the sieve bootstrap to ARIMA and FARIMA processes. The asymptotic properties of the sieve bootstrap prediction intervals for such processes are established, and the finite sample properties are examined by employing Monte Carlo simulations. The Monte Carlo simulation study shows that the proposed method works well for both ARIMA and FARIMA processes. Following the existing sieve bootstrap frame-work for testing unit roots for nonseasonal processes, we propose new bootstrap-based unit root tests for seasonal time series. In this procedure, the bootstrap distributions of the well known Dickey-Hasza-Fuller (DHF) seasonal test statistics are obtained and utilized to determine the critical points for the test. The asymptotic properties of the proposed method are established and a Monte Carlo simulation study is employed to demonstrate that the proposed unit root tests yield higher powers compared to the DHF test. Also, a sieve bootstrap method is implemented to obtaining prediction intervals for time series with seasonal unit roots. The asymptotic properties of the proposed prediction intervals are established and a Monte Carlo simulation study is carried out to examine the finite sample validity. Finally, we derive expressions for the asymptotic distributions of the Dickey-Fuller (DHF) type test statistics, under weakly dependent errors and show that they can be expressed as functional of the standard Brownian motions. Currently, the asymptotic results are available only for non-seasonal time series"--Abstract, leaf v

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.

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.

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.

Three major regression-based seasonal unit root tests: the DHF test introduced by Dickey et al (1984), the HEGY test proposed by Hylleberg et al. (1990) and the Kunst test introduced by Kunst (1997) are compared. The regression model for the DHF test is a reduced form of that for the Kunst test. We modify the Kunst test by using the t-statistic instead of Kunst's proposed joint F-statistic to study the influence of additional variables in the Kunst model. Also, we modify the HEGY test to test the presence of all four quarterly unit roots against the presence of roots 1 and -1. Through the comparison between the DHF test and the modified HEGY test, we find that the DHF test does not have asymptotic power one when the series only have some of the seasonal unit roots but not all of them. We call this case of partial unit roots. The asymptotic distributions derived in the paper provide the explanation of this limitation for the DHF test. Using simulation, we find that the probability that the DHF test will lead researchers to accept the seasonal unit root null hypothesis increases when the series contains more partial unit roots. For the DHF test, the test power depends on the augmented model. We derive limits of the related estimates from two augmented models for the DHF test. Both estimates are inconsistent. The test statistic obtained from the augmented model suggested by Ghysels et al. (1992) has relatively low power. For the HEGY/Kunst test, most limiting distributions for the test statistics depend on the lag augmentation but the test statistics have few problems caused by inconsistent estimates. However, the augmented models for the HEGY/Kunst test have more variables than those for the DHF test. Based on our simulation study results, the inclusion of more variables results in more loss in power when a redundant variable is included, and more sensitivity to the size distortion when the augmented lag length is less than the true lag length.

"Unit root tests are frequently employed by applied time series analysts to determine if the underlying model that generates an empirical process has a component that can be well-described by a random walk. More specifically, when the time series can be modeled using an autoregressive moving average (ARMA) process, such tests aim to determine if the autoregressive (AR) polynomial has one or more unit roots. The effect of economic shocks do not diminish with time when there is one or more unit roots in the AR polynomial, whereas the contribution of shocks decay geometrically when all the roots are outside the unit circle. This is one major reason for economists' interest in unit root tests. Unit roots processes are also useful in modeling seasonal time series, where the autoregressive polynomial has a factor of the form (1-[zeta][superscript s]), and s is the period of the season. Such roots are called seasonal unit roots. Techniques for testing the unit roots have been developed by many researchers since late 1970s. Most such tests assume that the errors (shocks) are independent or weakly dependent. Only a few tests allow conditionally heteroskedastic error structures, such as Generalized Autoregressive Conditionally Heteroskedastic (GARCH) error. And only a single test is available for testing multiple unit roots. In this dissertation, three papers are presented. Paper I deals with developing bootstrap-based tests for multiple unit roots; Paper II extends a bootstrap-based unit root test to higher order autoregressive process with conditionally heteroscedastic error; and Paper III extends a currently available seasonal unit root test to a bootstrap-based one while at the same time relaxing the assumption of weakly dependent shocks to include conditional heteroscedasticity in the error structure"--Abstract, page iv.

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

In this paper, we suggest a new set of regression-based statistics for testing the seasonal unit root null hypothesis. These tests are based on combining conventional Hylleberg et al. (1990)-type seasonal unit root test statistics calculated from both forward and reverse estimation of the auxiliary regression equation. We derive the asymptotic distributions of the new test statistics under the seasonal unit root null hypothesis. We provide finite sample critical values appropriate for the case of quarterly data together with asymptotic critical values, the latter appropriate for any seasonal aspect. Monte Carlo simulation of the finite-sample size and power properties of the new tests reveals that, overall, they perform rather better than extant tests of the seasonal unit root hypothesis.