Unit Root Testing in Heteroscedastic Panels Using the Cauchy Estimator

The Cauchy estimator of an autoregressive root uses the sign of the first lag as instrumental variable. The resulting IV t-type statistic follows a standard normal limiting distribution under a unit root case even under unconditional heteroscedasticity, if the series to be tested has no deterministic trends. The standard normality of the Cauchy test is exploited to obtain a standard normal panel unit root test under cross-sectional dependence and time-varying volatility with an orthogonalization procedure. The article’s analysis of the joint N, T asymptotics of the test suggests that (1) N should be smaller than T and (2) its local power is competitive with other popular tests. To render the test applicable when N is comparable with, or larger than, T, shrinkage estimators of the involved covariance matrix are used. The finite-sample performance of the discussed procedures is found to be satisfactory.     

Normal tests for unit roots based on instrumental variable estimators

For estimating unit roots of autoregressive processes, we introduce a new instrumental variable (IV) method which discounts large values of regressors corresponding to the unit roots. Based on the IV estimator, we propose new unit root tests whose limiting null distributions are standard normal. Observation at time t is adjusted for mean recursively by the sample mean of observations up to the time t. The powers of the proposed tests are better than those of the Dickey–Fuller tests and are comparable to those of the tests based on the weighted symmetric estimator, which are known to have the best power against stationary alternatives.     

Nonlinear IV panel unit root testing under structural breaks in the error variance

The paper examines the behavior of a generalized version of the nonlinear IV unit root test proposed by Chang (2002) when the series’ errors exhibit nonstationary volatility. The leading case of such nonstationary volatility concerns structural breaks in the error variance. We show that the generalized test is not robust to variance changes in general, and illustrate the extent of the resulting size distortions in finite samples. More importantly, we show that pivotality is recovered when using Eicker-White heteroskedasticity-consistent standard errors. This contrasts with the case of Dickey-Fuller unit root tests, for which Eicker-White standard errors do not produce robustness and thus require computationally costly corrections such as the (wild) bootstrap or estimation of the so-called variance profile. The pivotal versions of the generalized IV tests – with or without the correct standard errors – do however have no power in $1/T$ -neighbourhoods of the null. We also study the validity of panel versions of the tests considered here.     

Asymptotic properties of the bootstrap unit root test statistic under possibly infinite variance

ABSTRACT

In this article, the unit root test for the AR(1) model is discussed, under the condition that the innovations of the model are in the domain of attraction of the normal law with possibly infinite variances. By using residual bootstrap with sample size m < n (n being the size of the original sample), we bootstrap the least-squares estimator of the autoregressive parameter. Under some mild assumptions, we prove that the null distribution of the unit root test statistic based on the least-square estimator of the autoregressive parameter can be approximated by using residual bootstrap.     

Nonparametric two-sample tests for dispersion differences based on placements

Two different two-sample tests for dispersion differences based on placement statistics are proposed. The means and variances of the test statistics are derived, and asymptotic normality is established for both. Variants of the proposed tests based on reversing the X and Y labels in the test statistic calculations are shown to have different small-sample properties; for both pairs of tests, one member of the pair will be resolving, the other nonresolving. The proposed tests are similar in spirit to the dispersion tests of both Mood and Hollander; comparative simulation results for these four tests are given. For small sample sizes, the powers of the proposed tests are approximately equal to the powers of the tests of both Mood and Hollander for samples from the normal, Cauchy and exponential distributions. The one-sample limiting distributions are also provided, yielding useful approximations to the exact tests when one sample is much larger than the other. A bootstrap test may alternatively be performed. The proposed test statistics may be used with lightly censored data by substituting Kaplan-Meier estimates for the empirical distribution functions.     

Power of One-Sample Location Tests Under Distributions with Equal Lévy Distance

In this article, we study the power of one-sample location tests under classical distributions and two supermodels which include the normal distribution as a special case. The distributions of the supermodels are chosen in such a way that they have equal distance to the normal as the logistic, uniform, double exponential, and the Cauchy, respectively. As a measure of distance we use the Lévy metric. The tests considered are two parametric tests, the t-test and a trimmed t-test, and two nonparametric tests, the sign test and the Wilcoxon signed-rank tests. It turns out that the power of the tests, first of all, does not depend on the Lévy distance but on the special chosen supermodel.     

Robust inference for predictability in smooth transition predictive regressions

This article provides a novel test for predictability within a nonlinear smooth transition predictive regression (STPR) model where inference is complicated due not only to the presence of persistent, local to unit root, predictors, and endogeneity but also the presence of unidentified parameters under the null of no predictability. In order to circumvent the unidentified parameters problem, t? statistic for the predictor in the STPR model is optimized over the Cartesian product of the spaces for the transition and threshold parameters; and to address the di?culties due to persistent and endogenous predictors, the instrumental variable (IVX) method originally developed in the linear cointegration testing framework is adopted within the STPR model. Limit distribution of this statistic (i.e., sup?t^IVX test) is shown to be nuisance parameter-free and robust to the local to unit root and endogenous regressors. Simulations show that sup?t^IVX has good size and power properties. An application to stock return predictability reveals presence of asymmetric regime-dependence and variability in the strength and size of predictability across asset-related (e.g., dividend/price ratio) vs. other (e.g., default yield spread) predictors.     

Bootstrap procedures for variance breaks test in time series with a changing trend

In this article, we consider the problem of testing for variance breaks in time series in the presence of a changing trend. In performing the test, we employ the cumulative sum of squares (CUSSQ) test introduced by Inclán and Tiao (1994, J.?Amer.?Statist.?Assoc., 89, 913 ? 923). It is shown that CUSSQ test is not robust in the case of broken trend and its asymptotic distribution does not convergence to the sup of a standard Brownian bridge. As a remedy, a bootstrap approximation method is designed to alleviate the size distortions of test statistic while preserving its high power. Via a bootstrap functional central limit theorem, the consistency of these bootstrap procedures is established under general assumptions. Simulation results are provided for illustration and an empirical example of application to a set of high frequency real data is given.     

Bootstrapping p Values and Power in the First-Order Autoregression: A Monte Carlo Investigation

The small-sample behavior of the bootstrap is investigated as a method for estimating p values and power in the stationary first-order autoregressive model. Monte Carlo methods are used to examine the bootstrap and Student-t approximations to the true distribution of the test statistic frequently used for testing hypotheses on the underlying slope parameter. In contrast to Student's t, the results suggest that the bootstrap can accurately estimate p values and power in this model in sample sizes as small as 5–10.     

Testing stationarity and trend stationarity against the unit root hypothesis

In this paper we propose a family of relativel simple nonparametrics tests for a unit root in a univariate time series. Almost all the tests proposed in the literature test the unit root hypothesis against the alternative that the time series involved is stationarity or trend stationary. In this paper we take the (trend) stationarity hypothesis as the null and the unit root hypothesis as the alternative. The order differnce with most of the tests proposed in the literature is that in all four cases the asymptotic null distribution is of a well-known type, namely standard Cauchy. In the first instance we propose four Cauchy tests of the stationarity hypothesis against the unit root hypothesis. Under H1 these four test statistics involved, divided by the sample size n, converge weakly to a non-central Cauchy distribution, to one, and to the product of two normal variates, respectively. Hence, the absolute values of these test statistics converge in probability to infinity 9at order n). The tests involved are therefore consistent against the unit root hypothesis. Moreover, the small sample performance of these test are compared by Monte Carlo simulations. Furthermore, we propose two additional Cauchy tests of the trend stationarity hypothesis against the alternative of a unit root with drift.     

Nonstationarity and Level Shifts With an Application to Purchasing Power Parity

This study considers testing for a unit root in a time series characterized by a structural change in its mean. The analysis is in the spirit of Perron (1990a), who showed that the existence of such a shift in a stationary time series biases the usual tests for a unit root toward nonrejection. The approach is, however, different given that we suppose the date of the change to be unknown. The statistic of interest is then the minimal t statistic over all possible breakpoints in regressions similar to those proposed by Perron (1990a). Other related statistics are also discussed. We derive and tabulate the asymptotic distributions of interest. Most of the emphasis, however, is given to the tabulation of finite-sample critical values using simulation experiments. Particular attention is given to the effect, on the finite-sample critical values, of various procedures to select the appropriate order of the estimated autoregressions. We apply the tests to analyze the issue of purchasing power parity between the United States and the United Kingdom and also between the United States and Finland, whose real exchange rates are characterized by apparent shifts in level when using particular price indexes.     

A simulation study of balanced and antithetic bootstrap resampling methods

An extensive simulation study is conducted to compare the performance between balanced and antithetic resampling for the bootstrap in estimation of bias, variance, and percentiles when the statistic of interest is the median, the square root of the absolute value of the mean, or the median absolute deviations from the median. Simulation results reveal that balanced resampling provide better efficiencies in most cases; however, antithetic resampling is superior in estimating bias of the median. We also investigate the possibility of combining an existing efficient bootstrap computation of Efron (1990) with balanced or antithetic resampling for percentile estimation. Results indicate that the combination method does indeed offer gains in performance though the gains are much more dramatic for the bootstrap t statistic than for any of the three statistics of interest as described above.     

Wilcoxon's signed‐rank statistic: what null hypothesis and why it matters

In statistical literature, the term ‘signed‐rank test’ (or ‘Wilcoxon signed‐rank test’) has been used to refer to two distinct tests: a test for symmetry of distribution and a test for the median of a symmetric distribution, sharing a common test statistic. To avoid potential ambiguity, we propose to refer to those two tests by different names, as ‘test for symmetry based on signed‐rank statistic’ and ‘test for median based on signed‐rank statistic’, respectively. The utility of such terminological differentiation should become evident through our discussion of how those tests connect and contrast with sign test and one‐sample t‐test. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

An LM-type test for idiosyncratic unit roots in the exact factor model with integrated factors

We consider an exact factor model with integrated factors and propose an LM-type test for unit roots in the idiosyncratic component. We show that, for a fixed number of panel individuals (N) and when the number of time points (T) tends to infinity, the limiting distribution of the LM-type statistic is a weighted sum of independent Chi-square variables with one degree of freedom, and when T tends to infinity followed by N tending to infinity, the limiting distribution is standard normal. The results should contribute to the challenging task of deriving likelihood-based unit-root tests in dynamic factor models.     

ON STUDENTIZING AND BLOCKING METHODS FOR IMPLEMENTING THE BOOTSTRAP WITH DEPENDENT DATA

Gōtze & Kūnsch (1990) announced that a certain version of the bootstrap percentile-t method, and the blocking method, can be used to improve on the normal approximation to the distribution of a Studentized statistic computed from dependent data. This paper shows that this result depends fundamentally on the method of Studentization. Indeed, if the percentile-t method is implemented naively, for dependent data, then it does not improve by an order of magnitude on the much simpler normal approximation despite all the computational effort that is required to implement it. On the other hand, if the variance estimator used for the percentile-t bootstrap is adjusted appropriately, then percentile-t can improve substantially on the normal approximation.     

Goodness-of-fit testing of a weak memoryless property of life distributions

A life distribution is said to have a weak memoryless property if its conditional probability of survival beyond a fixed time point is equal to its (unconditional) survival probability at that point. Goodness‐of‐fit testing of this notion is proposed in the current investigation, both when the fixed time point is known and when it is unknown but estimable from the data. The limiting behaviour of the proposed test statistic is obtained and the null variance is explicitly given. The empirical power of the test is evaluated for a commonly known alternative using Monte Carlo methods, showing that the test performs well. The case when the fixed time point t₀ equals a quantile of the distribution F gives a distribution‐free test procedure. The procedure works even if t₀ is unknown but is estimable.     

Testing for a unit root in a nonlinear quantile autoregression framework

The nonlinear unit root test of Kapetanios, Shin, and Snell (2003 Kapetanios, G., Shin, Y., Snell, A. (2003). Testing for a unit root in the nonlinear STAR framework. Journal of Econometrics 112:359–379.[Crossref], [Web of Science ®] , [Google Scholar]) (KSS) has attracted much recent attention. However, the KSS test relies on the ordinary least squares (OLS) estimator, which is not robust to a heavy-tailed distribution and, in practice, the test suffers from a large power loss. This study develops three kinds of quantile nonlinear unit root tests: the quantile t-ratio test; the quantile Kolmogorov–Smirnov test; and the quantile Cramer–von Mises test. A Monte Carlo simulation shows that these tests have significantly better power when an innovation follows a non-normal distribution. In addition, the quantile t-ratio test can reveal the heterogeneity of the asymmetric dynamics in a time series. In our empirical studies, we investigate the unit root properties of U.S. macroeconomic time series and the real effective exchange rates for 61 countries. The results show that our proposed tests reject the unit roots more often, indicating that the series are likely to be asymmetric nonlinear reverting processes.     

The sensitivity of robust unit root tests

The power properties of the rank-based Dickey–Fuller (DF) unit root test of Granger and Hallman [C. Granger and J. Hallman, Nonlinear transformations of integrated time series, J. Time Ser. Anal. 12 (1991), pp. 207–218] and the range unit root tests of Aparicio et al. [F. Aparicio, A. Escribano, and A. Siplos, Range unit root (RUR) tests: Robust against non-linearities, error distributions, structural breaks and outliers, J. Time Ser. Anal. 27 (2006), pp. 545–576] are considered when applied to near-integrated time series processes with differing initial conditions. The results obtained show the empirical powers of the tests to be generally robust to smaller deviations of the initial condition of the time series from its underlying deterministic component, particularly for more highly stationary processes. However, dramatic decreases in power are observed when either the mean or variance of the deviation of the initial condition is increased. The robustness of the rank- and range-based unit root tests and their higher power results relative to the seminal DF test have both been noted previously in the econometrics literature. These results are questioned by the findings of the present paper.     

Nonparametric tests for conditional independence using conditional distributions

The concept of causality is naturally defined in terms of conditional distribution, however almost all the empirical works focus on causality in mean. This paper aims to propose a nonparametric statistic to test the conditional independence and Granger non-causality between two variables conditionally on another one. The test statistic is based on the comparison of conditional distribution functions using an L₂ metric. We use Nadaraya–Watson method to estimate the conditional distribution functions. We establish the asymptotic size and power properties of the test statistic and we motivate the validity of the local bootstrap. We ran a simulation experiment to investigate the finite sample properties of the test and we illustrate its practical relevance by examining the Granger non-causality between S&P 500 Index returns and VIX volatility index. Contrary to the conventional t-test which is based on a linear mean-regression, we find that VIX index predicts excess returns both at short and long horizons.     

The Disappointing Properties of GLS-Based Unit Root Tests in the Presence of Structural Breaks

Abstract

It is well known that prior application of GLS detrending, as advocated by Elliot et al. [Elliot, G., Rothenberg, T., Stock, J. (1996). Efficient tests for an autoregressive unit root. Econometrica 64:813–836], can produce a significant increase in power to reject the unit root null over that obtained from a conventional OLS-based Dickey and Fuller [Dickey, D., Fuller, W. (1979). Distribution of the estimators for autoregressive time series with a unit root. J. Am. Statist. Assoc. 74:427–431] testing equation. However, this paper employs Monte Carlo simulation to demonstrate that this increase in power is not necessarily obtained when breaks occur in either level or trend. It is found that neither OLS nor GLS-based tests are robust to level or trend breaks, their size and power properties both deteriorating as the break size increases.