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.     

Rethinking the Univariate Approach to Panel Unit Root Testing: Using Covariates to Resolve the Incidental Trend Problem

In an influential article, Hansen showed that covariate augmentation can lead to substantial power gains when compared to univariate tests. In this article, we ask if this result extends also to the panel data context? The answer turns out to be yes, which is maybe not that surprising. What is surprising, however, is the extent of the power gain, which is shown to more than outweigh the well-known power loss in the presence of incidental trends. That is, the covariates have an order effect on the neighborhood around unity for which local asymptotic power is negligible.     

同期相关面板数据结构突变单位根检验的统计性质 ——中国CPI指数平稳性的经验证据*

 内容提要：本文首先研究了同期相关面板数据外生同期截距突变同质面板单位根检验的统计性质。研究发现对于大面板数据该检验具有良好的实际检验水平,面板数据的大小、同期相关程度、结构突变位置和结构突变幅度等因素对该检验的检验功效具有显著影响,而且ρSUR检验比τSUR检验有更理想的检验效果。其次,利用该检验对中国省级CPI指数的平稳性进行了经验分析,发现中国省级CPI指数是趋势结构突变的平稳过程。     

Maximum Likelihood Unit Root Testing in the Presence of GARCH: A New Test with Increased Power

The literature on testing the unit root hypothesis in the presence of GARCH errors is extended. A new test based upon the combination of local-to-unity detrending and joint maximum likelihood estimation of the autoregressive parameter and GARCH process is presented. The finite sample distribution of the test is derived under alternative decisions regarding the deterministic terms employed. Using Monte Carlo simulation, the newly proposed ML t-test is shown to exhibit increased power of relative to rival tests. Finally, the empirical relevance of the simulation results is illustrated via an application to real GDP for the UK.     

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.     

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.     

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.     

非对称单位根检验和非参数方差比

基于三种退势方法较详细研究了方差比检验在非对称单位根检验中的适用性,并通过MC模拟揭示了其检验势性质。结果表明：在不含趋势项的TAR下,两机制TAR数据落在第一机制的比率是影响方差比检验势的重要因素,且比率越高检验势也越高;三机制TAR中落在中间机制的数据比率会影响检验势,随着比率增加检验势呈下降趋势,但程度不大。在含趋势的TAR下,由于趋势项在数据生成过程中具有支配作用,各种检验势会随着趋势设定的不同而不同。数据在不同机制之间的转换概率越高,则ROLS和RDM退势较OLS退势具有明显优势。     

Modified Stationarity Tests With Data-Dependent Model-Selection Rules

We describe some simple methods for improving the performance of stationarity tests (i.e., tests that have a stationary null and a unit-root alternative). Specifically, we increase the rate of convergence of the test under the unit-root alternative from O _p(T) to O _p (T ²), then suggest an optimal method of selecting the order of the autoregressive component in the fitted autoregressive integrated moving average model on which the test is based. Simulation evidence suggests that these modifications work well. We apply the modified procedure to U.S. monthly macroeconomic data and uncover new evidence of a unit root in unemployment.     

Statistical Adequacy and the Testing of Trend Versus Difference Stationarity

Abstract

The debate on whether macroeconomic series are trend or difference stationary, initiated by Nelson and Plosser [Nelson, C. R.; Plosser, C. I. (1982). Trends and random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics10:139–162] remains unresolved. The main objective of the paper is to contribute toward a resolution of this issue by bringing into the discussion the problem of statistical adequacy. The paper revisits the empirical results of Nelson and Plosser [Nelson, C. R.; Plosser, C. I. (1982). Trends and random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics10:139–162] and Perron [Perron, P. (1989). The great crash, the oil price shock, and the unit root hypothesis. Econometrica57:1361–1401] and shows that several of their estimated models are misspecified. Respecification with a view to ensuring statistical adequacy gives rise to heteroskedastic AR(k) models for some of the price series. Based on estimated models which are statistically adequate, the main conclusion of the paper is that the majority of the data series are trend stationary.     

Statistical Adequacy and the Testing of Trend Versus Difference Stationarity

The debate on whether macroeconomic series are trend or difference stationary, initiated by Nelson and Plosser [Nelson, C. R.; Plosser, C. I. (1982). Trends and random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics10:139-162] remains unresolved. The main objective of the paper is to contribute toward a resolution of this issue by bringing into the discussion the problem of statistical adequacy. The paper revisits the empirical results of Nelson and Plosser [Nelson, C. R.; Plosser, C. I. (1982). Trends and random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics10:139-162] and Perron [Perron, P. (1989). The great crash, the oil price shock, and the unit root hypothesis. Econometrica57:1361-1401] and shows that several of their estimated models are misspecified. Respecification with a view to ensuring statistical adequacy gives rise to heteroskedastic AR(k) models for some of the price series. Based on estimated models which are statistically adequate, the main conclusion of the paper is that the majority of the data series are trend stationary.     

A Likelihood Ratio Test for Idiosyncratic Unit Roots in the Exact Factor Model with Integrated Factors

We consider an exact factor model with unobservable common stochastic trends imposed by non-stationary factors, and study, by simulation, the power of the likelihood ratio test for unit roots in the idiosyncratic components. The power of the test is compared with the analogous Lagrange multiplier test and the Fisher-type test proposed by Bai and Ng. The results suggest that the benefit of the likelihood ratio test is in panels with a small cross-section.     

Likelihood Ratio Test Against Stochastic Order in Three Way Contingency Tables

Trend tests in dose-response have been central problems in medicine. The likelihood ratio test is often used to test hypotheses involving a stochastic order. Stratified contingency tables are common in practice. The distribution theory of likelihood ratio test has not been full developed for stratified tables and more than two stochastically ordered distributions. Under c strata of m × r tables, for testing the conditional independence against simple stochastic order alternative, this article introduces a model-free test method and gives the asymptotic distribution of the test statistic, which is a chi-bar-squared distribution. A real data set concerning an ordered stratified table will be used to show the validity of this test method.     

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.     

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.     

Testing for a Unit Root in ARIMA Processes

A unit root has important long-run implications for many time series in economics and finance. This paper develops a unit-root test of an ARIMA(p-1, 1, q) with drift null process against a trend-stationary ARMA(p, q) alternative process, where the order of the time series is assumed known through previous statistical testing or relevant theory. This test uses a point-optimal test statistic, but it estimates the null and alternative variance-covariance matrices that are used in the test statistic. Consequently, this test approximates a point-optimal test. Simulations show that its small-sample size is close to the nominal test level for a variety of unit-root processes, that it has a robust power curve against a variety of stationary alternatives, that its combined small-sample size and power properties are highly competitive with previous unit-root tests, and that it is robust to conditional heteroskedasticity. An application to post-Second World War real per capita gross domestic product is provided.     

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.     

The Performance of Panel Unit Root and Stationarity Tests: Results from a Large Scale Simulation Study

This paper presents results on the size and power of first generation panel unit root and stationarity tests obtained from a large scale simulation study. The tests developed in the following papers are included: Levin et al. (2002), Harris and Tzavalis (1999), Breitung (2000), Im et al. (1997, 2003), Maddala and Wu (1999), Hadri (2000), and Hadri and Larsson (2005). Our simulation set-up is designed to address inter alia the following issues. First, we assess the performance as a function of the time and the cross-section dimensions. Second, we analyze the impact of serial correlation introduced by positive MA roots, known to have detrimental impact on time series unit root tests, on the performance. Third, we investigate the power of the panel unit root tests (and the size of the stationarity tests) for a variety of first order autoregressive coefficients. Fourth, we consider both of the two usual specifications of deterministic variables in the unit root literature.