Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility

A number of recent papers have focused on the problem of testing for a unit root in the case where the driving shocks may be unconditionally heteroskedastic. These papers have, however, taken the lag length in the unit root test regression to be a deterministic function of the sample size, rather than data-determined, the latter being standard empirical practice. We investigate the finite sample impact of unconditional heteroskedasticity on conventional data-dependent lag selection methods in augmented Dickey–Fuller type regressions and propose new lag selection criteria which allow for unconditional heteroskedasticity. Standard lag selection methods are shown to have a tendency to over-fit the lag order under heteroskedasticity, resulting in significant power losses in the (wild bootstrap implementation of the) augmented Dickey–Fuller tests under the alternative. The proposed new lag selection criteria are shown to avoid this problem yet deliver unit root tests with almost identical finite sample properties as the corresponding tests based on conventional lag selection when the shocks are homoskedastic.     

Unit Root Tests under Time-Varying Variances

The paper provides a general framework for investigating the effects of permanent changes in the variance of the errors of an autoregressive process on unit root tests. Such a framework - which is based on a novel asymptotic theory for integrated and near integrated processes with heteroskedastic errors - allows to evaluate how the variance dynamics affect the size and the power function of unit root tests. Contrary to previous studies, it is shown that non-constant variances can both inflate and deflate the rejection frequency of the commonly used unit root tests, both under the null and under the alternative, with early negative and late positive variance changes having the strongest impact on size and power. It is also shown that shifts smoothed across the sample have smaller impacts than shifts occurring as a single abrupt jump, while periodic variances have a negligible effect even when a small number of cycles take place over a given sample. Finally, it is proved that the locally best invariant (LBI) test of a unit root against level stationarity is robust to heteroskedasticity of any form under the null hypothesis.     

Panel Unit Root Tests in the Presence of Cross-Sectional Dependencies: Comparison and Implications for Modelling

Several panel unit root tests that account for cross-section dependence using a common factor structure have been proposed in the literature recently. Pesaran's (2007 Pesaran , M. H. ( 2007 ). A simple panel unit root test in the presence of cross section dependence . Journal of Applied Econometrics 22 : 265 – 312 .[Crossref], [Web of Science ®] , [Google Scholar]) cross-sectionally augmented unit root tests are designed for cases where cross-sectional dependence is due to a single factor. The Moon and Perron (2004 Moon , H. R. , Perron , B. (2004). Testing for a unit root in panels with dynamic factors. Journal of Econometrics 122:81–126.[Crossref], [Web of Science ®] , [Google Scholar]) tests which use defactored data are similar in spirit but can account for multiple common factors. The Bai and Ng (2004a Bai , J. , Ng , S. ( 2004a ). A PANIC attack on unit roots and cointegration . Econometrica 72 : 1127 – 1177 .[Crossref], [Web of Science ®] , [Google Scholar]) tests allow to determine the source of nonstationarity by testing for unit roots in the common factors and the idiosyncratic factors separately. Breitung and Das (2008 Breitung , J. , Das , S. ( 2008 ). Testing for unit roots in panels with a factor structure . Econometric Theory 24 : 88 – 108 .[Crossref], [Web of Science ®] , [Google Scholar]) and Sul (2007 Sul , D. ( 2007 ) . Panel Unit Root Tests Under Cross Section Dependence with Recursive Mean Adjustment . Mimeo : University of Auckland . [Google Scholar]) propose panel unit root tests when cross-section dependence is present possibly due to common factors, but the common factor structure is not fully exploited.

This article makes four contributions: (1) it compares the testing procedures in terms of similarities and differences in the data generation process, tests, null, and alternative hypotheses considered, (2) using Monte Carlo results it compares the small sample properties of the tests in models with up to two common factors, (3) it provides an application which illustrates the use of the tests, and (4) finally, it discusses the use of the tests in modelling in general.     

A Note on Unit Root Tests and GARCH Errors: A Simulation Experiment

This article re-examines the Monte Carlo experiments in Seo (1999 Seo , B. ( 1999 ). Distribution theory for unit root tests with conditional heteroskedasticity . J. Econometrics 91 : 113 – 144 .[Crossref], [Web of Science ®] , [Google Scholar]) for unit root tests with GARCH errors. We report a Monte Carlo study with data generated from various GARCH(1, 1) processes where 0.8 ≤ α + β < 1 and β > α. In this case, the Dickey–Fuller test works better than the Seo test.     

Testing a Unit Root Based on Aggregate Time Series

Many time series encountered in practice are nonstationary, and instead are often generated from a process with a unit root. Because of the process of data collection or the practice of researchers, time series used in analysis and modeling are frequently obtained through temporal aggregation. As a result, the series used in testing for a unit root are often time series aggregates. In this paper, we study the effects of the use of aggregate time series on the Dickey–Fuller test for a unit root. We start by deriving a proper model for the aggregate series. Based on this model, we find the limiting distributions of the test statistics and illustrate how the tests are affected by the use of aggregate time series. The results show that those distributions shift to the right and that this effect increases with the order of aggregation, causing a strong impact both on the empirical significance level and on the power of the test. To correct this problem, we present tables of critical points appropriate for the tests based on aggregate time series and demonstrate their adequacy. Examples illustrate the conclusions of our analysis.     

Unit Root Tests in the Presence of Multi-Variance Break and Level Shifts That Have Power Against the Piecewise Stationary Alternative

Structural breaks in the level as well as in the volatility have often been exhibited in economic time series. In this paper, we propose new unit root tests when a time series has multiple shifts in its level and the corresponding volatility. The proposed tests are Lagrangian multiplier type tests based on the residual's marginal likelihood which is free from the nuisance mean parameters. The limiting null distributions of the proposed tests are the χ²distributions, and are affected not by the size and the location of breaks but only by the number of breaks.

We set the structural breaks under both the null and the alternative hypotheses to relieve a possible vagueness in interpreting test results in empirical work. The null hypothesis implies a unit root process with level shifts and the alternative connotes a stationary process with level shifts. The Monte Carlo simulation shows that our tests are locally more powerful than the OLSE-based tests, and that the powers of our tests, in a fixed time span, remain stable regardless the number of breaks. In our application, we employ the data which are analyzed by Perron (1990), and some results differ from those of Perron's (1990).     

Small-Sample Improved Seasonal Unit Root Tests for Trending and Breaking Series

In this article three unit root tests that allow for a break in both the seasonal mean and linear trend of the data are proposed. The tests, which can be seen as small-sample corrected versions of already known asymptotic tests, are shown to perform very well in simulations, and much better than their asymptotic counterparts.     

门限自回归下的单位根检验可靠性研究

在非对称的门限自回归模型下,由于传统单位根检验式的误设,会导致单位根检验势下降。本文通过一系列的Monte-Carlo模拟表明：非对称性对ADF和PP检验的检验势会产生较大影响,而对其他四种常用的单位根检验势产生的影响较小,也就是说,在非对称的门限自回归下,非对称性对退势单位根检验势产生的影响较小。模拟中也发现：NP单位根检验对TAR模型和持久性都具有稳健性。     

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.     

A Monte Carlo Investigation of Unit Root Tests and Long Memory in Detecting Mean Reversion in I(0) Regime Switching,Structural Break,and Nonlinear Data

The potential observational equivalence between various types of nonlinearity and long memory has been recognized by the econometrics community since at least the contribution of Diebold and Inoue (2001 Diebold, F., Inoue, A. (2001). Long memory and regime switching. Journal of Econometrics 105:131–159.[Crossref], [Web of Science ®] , [Google Scholar]). A large literature has developed in an attempt to ascertain whether or not the long memory finding in many economic series is spurious. Yet to date, no study has analyzed the consequences of using long memory methods to test for unit roots when the “truth” derives from regime switching, structural breaks, or other types of mean reverting nonlinearity. In this article, I conduct a comprehensive Monte Carlo analysis to investigate the consequences of using tests designed to have power against fractional integration when the actual data generating process is unknown. I additionally consider the use of tests designed to have power against breaks and threshold nonlinearity. The findings are compelling and demonstrate that the use of long memory as an approximation to nonlinearity yields tests with relatively high power. In contrast, misspecification has severe consequences for tests designed to have power against threshold nonlinearity, and especially for tests designed to have power against breaks.     

Developments in Maximum Likelihood Unit Root Tests

The exact maximum likelihood estimate provides a test statistic for the unit root test that is more powerful than the usual least-squares approach. In this article, a new derivation is given for the asymptotic distribution of this test statistic that is simpler and more direct than the previous method. The response surface regression method is used to obtain a fast algorithm that computes accurate finite-sample critical values. This algorithm is available in the R package mleur that is available on CRAN. The empirical power of the new test is shown to be much better than the usual test not only in the normal case but also for innovations generated from an infinite variance stable distribution as well as for innovations generated from a GARCH(1,1) process.     

Model-Selection-Based Detection of Unit Root Allowing for Various Trend-Break Types

In the conventional hypothesis-testing approach to the detection of a unit root and a trend break, selections of the outlier type (additive or innovational) and of the break type (jump or kink) are carried out arbitrarily, because there is no generally accepted statistical technique. To overcome this problem, a model-selection approach using the modified Bayesian information criterion (MBIC) is proposed. Whether the observed time series contains a unit root and a trend break is determined as a result of model selection from among alternative models with and without unit root and trend break. The efficacy of the proposed approach is verified using comprehensive simulations.     

M-estimation for Moderate Deviations From a Unit Root

Phillips and Magdalinos (2007 Phillips, P., Magdalinos, T. (2007). Limit theory for moderate deviations from a unit root. J. Econometrics 136:115–130.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a larger neighborhoods of one (called moderate deviations) than the conventional local to unity roots in autoregression models. Least square estimates (LSE) of the serial correlation coefficient were studied and asymptotics were provided. In this article, we investigate the M-estimation of the serial correlation coefficient having moderate deviations from the unit root. For both the near stationary case and explosive case, the Bahadur representations and limits in distribution are given for the M-estimators of the serial correlation coefficient. The limit theory demonstrates that the convergence rates of the M-estimators are the same as that for LSE hence bridging the very different convergence rates of the stationary and unit root cases. The limit theory also facilitates the comparison of the relative asymptotic efficiency among different estimators within the family of M-estimators.     

Testing Parameter Constancy in Unit Root Autoregressive Models Against Multiple Continuous Structural Changes

This article considers tests for logistic smooth transition autoregressive (LSTAR) models accommodating multiple time dependent transitions between regimes when the data generating process is a random walk. The asymptotic null distributions of the tests, in contrast to the standard results in Lin and Teräsvirta (1994 Lin , C. F. J. , Teräsvirta , T. ( 1994 ). Testing the constancy of regression parameters against continuous structural change . Journal of Econometrics 62 : 211 – 228 .[Crossref], [Web of Science ®] , [Google Scholar]), are nonstandard. Monte Carlo experiments reveal that the tests have modest size distortions and satisfactory power against LSTAR models with multiple smooth breaks. The tests are applied to Swedish unemployment rates and the hysteresis hypothesis is over-turned in favour of an LSTAR model with two transitions between extreme regimes.     

STAR模型中的递归退势单位根检验研究

在STAR模型框架下,考虑时间序列具有线性确定性趋势成分,本文建立了一个递归退势单位根检验统计量,推导了其渐近分布;并在考虑初始条件情形下,对递归退势、OLS和GLS退势单位根检验统计量的有限样本性质进行了细致的比较研究。若忽略初始条件的影响,GLS退势和递归退势单位根检验统计量的检验势都显著高于OLS退势。随着初始条件的增大,GLS退势单位根检验统计量的检验势下降得比较厉害,递归退势单位根检验统计量的检验势较为稳定,且在样本量较大情形下更具优势。     

Lagrange Multiplier Tests for Normality Against Seminonparametric Alternatives

In this article, I derive the Lagrange multiplier test of the null hypothesis that a stationary random vector has a (possibly heteroscedastic) normal distribution against the alternative that the distribution is a member of the family with seminonparametric probability density functions considered by Gallant and Tauchen (1989). The test is shown to contain special cases of the moment tests proposed by Newey (1985) and Tauchen (1985). Evidence from a small simulation study is reported, showing that the test has reasonable finite-sample properties in moderately sized samples. The test is applied to the change of price in a treasury-bill data series analyzed by Tauchen and Pitts (1983) and Tauchen (1985).     

Simulating Properties of the Likelihood Ratio Test for a Unit Root in an Explosive Second-Order Autoregression

This paper provides a means of accurately simulating explosive autoregressive processes and uses this method to analyze the distribution of the likelihood ratio test statistic for an explosive second-order autoregressive process of a unit root. While the standard Dickey-Fuller distribution is known to apply in this case, simulations of statistics in the explosive region are beset by the magnitude of the numbers involved, which cause numerical inaccuracies. This has previously constituted a bar on supporting asymptotic results by means of simulation, and analyzing the finite sample properties of tests in the explosive region.     

Simulating Properties of the Likelihood Ratio Test for a Unit Root in an Explosive Second-Order Autoregression

This paper provides a means of accurately simulating explosive autoregressive processes and uses this method to analyze the distribution of the likelihood ratio test statistic for an explosive second-order autoregressive process of a unit root. While the standard Dickey–Fuller distribution is known to apply in this case, simulations of statistics in the explosive region are beset by the magnitude of the numbers involved, which cause numerical inaccuracies. This has previously constituted a bar on supporting asymptotic results by means of simulation, and analyzing the finite sample properties of tests in the explosive region.     

On the Use of GLS Demeaning in Panel Unit Root Testing

One of the most well-known facts about unit root testing in time series is that the Dickey–Fuller (DF) test based on ordinary least squares (OLS) demeaned data suffers from low power, and that the use of generalized least squares (GLS) demeaning can lead to substantial power gains. Of course, this development has not gone unnoticed in the panel unit root literature. However, while the potential of using GLS demeaning is widely recognized, oddly enough, there are still no theoretical results available to facilitate a formal analysis of such demeaning in the panel data context. The present article can be seen as a reaction to this. The purpose is to evaluate the effect of GLS demeaning when used in conjuncture with the pooled OLS t-test for a unit root, resulting in a panel analog of the time series DF–GLS test. A key finding is that the success of GLS depend critically on the order in which the dependent variable is demeaned and first-differenced. If the variable is demeaned prior to taking first-differences, power is maximized by using GLS demeaning, whereas if the differencing is done first, then OLS demeaning is preferred. Furthermore, even if the former demeaning approach is used, such that GLS is preferred, the asymptotic distribution of the resulting test is independent of the tuning parameters that characterize the local alternative under which the demeaning performed. Hence, the demeaning can just as well be performed under the unit root null hypothesis. In this sense, GLS demeaning under the local alternative is redundant.