Testing Covariance Stationarity

In this paper, we show that the widely used stationarity tests such as the Kwiatkowski Phillips, Schmidt, and Shin (KPSS) test have power close to size in the presence of time-varying unconditional variance. We propose a new test as a complement of the existing tests. Monte Carlo experiments show that the proposed test possesses the following characteristics: (i) in the presence of unit root or a structural change in the mean, the proposed test is as powerful as the KPSS and other tests; (ii) in the presence of a changing variance, the traditional tests perform badly whereas the proposed test has high power comparing to the existing tests; (iii) the proposed test has the same size as traditional stationarity tests under the null hypothesis of stationarity. An application to daily observations of return on U.S. Dollar/Euro exchange rate reveals the existence of instability in the unconditional variance when the entire sample is considered, but stability is found in subsamples.     

A robust test for autocorrelation in the presence of a structural break in variance

It has been known that when there is a break in the variance (unconditional heteroskedasticity) of the error term in linear regression models, a routine application of the Lagrange multiplier (LM) test for autocorrelation can cause potentially significant size distortions. We propose a new test for autocorrelation that is robust in the presence of a break in variance. The proposed test is a modified LM test based on a generalized least squares regression. Monte Carlo simulations show that the new test performs well in finite samples and it is especially comparable to other existing heteroskedasticity-robust tests in terms of size, and much better in terms of power.     

Variance Shifts,Structural Breaks,and Stationarity Tests

This article considers the problem of testing the null hypothesis of stochastic stationarity in time series characterized by variance shifts at some (known or unknown) point in the sample. It is shown that existing stationarity tests can be severely biased in the presence of such shifts, either oversized or undersized, with associated spurious power gains or losses, depending on the values of the breakpoint parameter and on the ratio of the prebreak to postbreak variance. Under the assumption of a serially independent Gaussian error term with known break date and known variance ratio, a locally best invariant (LBI) test of the null hypothesis of stationarity in the presence of variance shifts is then derived. Both the test statistic and its asymptotic null distribution depend on the breakpoint parameter and also, in general, on the variance ratio. Modifications of the LBI test statistic are proposed for which the limiting distribution is independent of such nuisance parameters and belongs to the family of Cramér–von Mises distributions. One such modification is particularly appealing in that it is simultaneously exact invariant to variance shifts and to structural breaks in the slope and/or level of the series. Monte Carlo simulations demonstrate that the power loss from using our modified statistics in place of the LBI statistic is not large, even in the neighborhood of the null hypothesis, and particularly for series with shifts in the slope and/or level. The tests are extended to cover the cases of weakly dependent error processes and unknown breakpoints. The implementation of the tests are illustrated using output, inflation, and exchange rate data series.     

Testing for stationarity in heterogeneous panel data in the presence of cross-section dependence

The panel variant of the KPSS tests developed by Hadri [Hadri, K., 2000, Testing for stationarity in heterogeneous panels. Econometrics Journal, 3, 148–161] for the null of stationarity suffers from size distortions in the presence of cross-section dependence. However, applying the bootstrap methodology, we find that these tests are approximately correctly sized.     

The effects of additive outliers on the seasonal KPSS test: a Monte Carlo analysis

This article builds on the test proposed by Lyhagen [The seasonal KPSS statistic, Econom. Bull. 3 (2006), pp. 1–9] for seasonal time series and having the null hypothesis of level stationarity against the alternative of unit root behaviour at some or all of the zero and seasonal frequencies. This new test is qualified as seasonal-frequency Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test and it is not originally supported by a regression framework.

The purpose of this paper is twofold. Firstly, we propose a model-based regression method and provide a clear illustration of Lyhagen's test and we establish its asymptotic theory in the time domain. Secondly, we use the Monte Carlo method to study the finite-sample performance of the seasonal KPSS test in the presence of additive outliers. Our simulation analysis shows that this test is robust to the magnitude and the number of outliers and the statistical results obtained cast an overall good performance of the test finite-sample properties.     

Bootstrap LR tests of stationarity,common trends and cointegration

This paper considers the likelihood ratio (LR) tests of stationarity, common trends and cointegration for multivariate time series. As the distribution of these tests is not known, a bootstrap version is proposed via a state- space representation. The bootstrap samples are obtained from the Kalman filter innovations under the null hypothesis. Monte Carlo simulations for the Gaussian univariate random walk plus noise model show that the bootstrap LR test achieves higher power for medium-sized deviations from the null hypothesis than a locally optimal and one-sided Lagrange Multiplier (LM) test that has a known asymptotic distribution. The power gains of the bootstrap LR test are significantly larger for testing the hypothesis of common trends and cointegration in multivariate time series, as the alternative asymptotic procedure – obtained as an extension of the LM test of stationarity – does not possess properties of optimality. Finally, it is shown that the (pseudo-)LR tests maintain good size and power properties also for the non-Gaussian series. An empirical illustration is provided.     

Effect of neglected deterministic seasonality on unit root tests

Whenever deterministic seasonality is ignored, the distribution of the Dickey-Fuller test is shifted to the left, with lower dispersion at the same time. When accounting for serial correlation, the distortions become less predictable. A Monte Carlo study confirms that the (augmented) Dickey-Fuller test without seasonal dummies is oversized and has little power at the same time, due to the need of lag augmentation. The effect of neglecting seasonal deterministics on the KPSS test for stationarity depends on the way the long-run variance is estimated. This is a shorter version of a working paper containing additional experimental evidence and the proofs of the propositions. The working paper is available online under http://www.wiwi.uni-frankfurt.de/~deme/ends_urt.pdf.     

时间序列平稳性分类识别研究

平稳性检验是时间序列回归分析的一个关键问题,已有的检验方法在处理海量时间序列数据时显得乏力,检验准确率有待提高。采用分类技术建立平稳性检验的新方法,可以有效地处理海量时间序列数据。首先计算时间序列自相关函数,构建一个充分非必要的判定准则;然后建立序列收敛的量化分析方法,研究收敛参数的最优取值,并提取平稳性特征向量;最后采用k-means聚类建立平稳性分类识别方法。采用一组模拟数据和股票数据进行分析,将ADF检验、PP检验、KPSS检验进行对比,实证结果表明新方法的准确率较高。     

Modified stationarity tests with improved power in small samples

In a recent paper Kwiatkowski et al. (1992) propose the so-called KPSS statistic for testing the null hypothesis of stationarity against the alternative of a unit root process. The statistic employs a spectral estimator which can be shown to diverge with increasing sample size, given the alternative is true. Here, we suggest a modified spectral estimator which is shown to stabilize for moving average models. It is shown that this test statistic uniformly outperforms the KPSS statistic in an MA(1) model. Furthermore, a two-step nonparametric correction procedure is suggested, giving a test statistic with similar asymptotic properties as the original KPSS statistic. However, in small samples this correction performs better especially in detecting large random walk components. This paper was written while the author was a post-doctoral fellow at the University of Amsterdam. The author likes to thank Peter Boswijk, Inge van den Doel, Noud van Giersbergen and Jan F.Kiviet for their help during that time. Moreover, I would like to thank an anonymous referee for a number of helpful comments.     

中、日、韩长期购买力平价的非线性检验——基于1994-2009年的经验分析

购买力平价理论是一种基本的汇率决定理论,主要是检验实际汇率是否稳定。以往的研究主要是在线性模型的框架下,应用ADF和PP检验考察购买力平价是否成立。尝试在非线性的研究框架下,基于ESTAR模型应用KPSS检验方法对中、日、韩三国的长期购买力平价进行检验。实证分析结果表明,基于非线性的KPSS检验方法发现日本的实际汇率为非平稳时间序列,而中国和韩国的实际汇率为平稳序列。这说明长期中日本的购买力平价不成立,而中国和韩国的购买力平价成立。     

The effect of spillover on the Granger causality test

In this paper, we investigate the properties of the Granger causality test in stationary and stable vector autoregressive models under the presence of spillover effects, that is, causality in variance. The Wald test and the WW test (the Wald test with White's proposed heteroskedasticity-consistent covariance matrix estimator imposed) are analyzed. The investigation is undertaken by using Monte Carlo simulation in which two different sample sizes and six different kinds of data-generating processes are used. The results show that the Wald test over-rejects the null hypothesis both with and without the spillover effect, and that the over-rejection in the latter case is more severe in larger samples. The size properties of the WW test are satisfactory when there is spillover between the variables. Only when there is feedback in the variance is the size of the WW test slightly affected. The Wald test is shown to have higher power than the WW test when the errors follow a GARCH(1,1) process without a spillover effect. When there is a spillover, the power of both tests deteriorates, which implies that the spillover has a negative effect on the causality tests.     

傅里叶型平滑结构突变下Dickey-Fuller检验的功效——兼论Shibor的平稳性

通过推导Dickey-Fuller检验功效函数,研究表明:即使中小型的傅里叶型结构突变,都会严重影响Dickey-Fuller检验的功效,从而使得含傅里叶型平滑结构突变的平稳过程被误判为单位根过程。使用3、6、9个月期和一年期Shibor日度数据发现:传统的ADF、PP、DF-GLS和KPSS几乎都指出Shibor是单位根过程;考虑平滑结构突变的单位根检验则在1%的显著性水平下拒绝了单位根的原假设,这表明Shibor是含结构突变的平稳过程。因此,预测Shibor和理解其动态行为必须考虑其结构突变特征。     

Unit root tests and dramatic shifts with infinite variance processes

A model which explains data that is subject to sudden structural changes of unspecified nature is presented. The structural shifts are generated by a random walk component whose innovations belong to the normal domain of attraction of a symmetric stable law. To test the model against the stationarity case, several non-parametric, and regression-based statistics are studied. The non-parametric tests are a generalization of the variance ratio test to innovations with heavy-tailed distributions. The tests are consistent and shown to have good finite sample size and power properties and are applied to a set of economic variables.     

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.     

Robust Stationarity Tests in Seasonal Time Series Processes

This article builds on the existing literature on (stationarity) tests of the null hypothesis of deterministic seasonality in a univariate time series process against the alternative of unit root behavior at some or all of the zero and seasonal frequencies. This article considers the case where, in testing for unit roots at some proper subset of the zero and seasonal frequencies, there are unattended unit roots among the remaining frequencies. Monte Carlo results are presented that demonstrate that in this case, the stationarity tests tend to distort below nominal size under the null and display an associated (often very large) loss of power under the alternative. A modification to the existing tests, based on data prefiltering, that eliminates the problem asymptotically is suggested. Monte Carlo evidence suggests that this procedure works well in practice, even at relatively small sample sizes. Applications of the robustified statistics to various seasonally unadjusted time series measures of U.K. consumers' expenditure are considered; these yield considerably more evidence of seasonal unit roots than do the existing stationarity tests.     

An alternative method for evaluating stationarity in transition models

Transition models are an important framework that can be used to model longitudinal categorical data. A relevant issue in applying these models is the condition of stationarity, or homogeneity of transition probabilities over time. We propose two tests to assess stationarity in transition models: Wald and likelihood-ratio tests, which do not make use of transition probabilities, using only the estimated parameters of the models in contrast to the classical test available in the literature. In this paper, we present two motivating studies, with ordinal longitudinal data, to which proportional odds transition models are fitted and the two proposed tests are applied as well as the classical test. Additionally, their performances are assessed through simulation studies. The results show that the proposed tests have good performance, being better for control of type-I error and they present equivalent power functions asymptotically. Also, the correlations between the Wald, likelihood-ratio and the classical test statistics are positive and large, an indicator of general concordance. Additionally, both of the proposed tests are more flexible and can be applied in studies with qualitative and quantitative covariates.     

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.     

A Modified Kolmogorov–Smirnov Test for Normality

In this article we propose an improvement of the Kolmogorov-Smirnov test for normality. In the current implementation of the Kolmogorov-Smirnov test, given data are compared with a normal distribution that uses the sample mean and the sample variance. We propose to select the mean and variance of the normal distribution that provide the closest fit to the data. This is like shifting and stretching the reference normal distribution so that it fits the data in the best possible way. A study of the power of the proposed test indicates that the test is able to discriminate between the normal distribution and distributions such as uniform, bimodal, beta, exponential, and log-normal that are different in shape but has a relatively lower power against the student's, t-distribution that is similar in shape to the normal distribution. We also compare the performance (both in power and sensitivity to outlying observations) of the proposed test with existing normality tests such as Anderson–Darling and Shapiro–Francia.     

A comparison of some conditional and unconditional exact tests for 2x2 contingency tables

In 1935, R.A. Fisher published his well-known “exact” test for 2x2 contingency tables. This test is based on the conditional distribution of a cell entry when the rows and columns marginal totals are held fixed. Tocher (1950) and Lehmann (1959) showed that Fisher s test, when supplemented by randomization, is uniformly most powerful among all the unbiased tests UMPU). However, since all the practical tests for 2x2 tables are nonrandomized - and therefore biased the UMPU test is not necessarily more powerful than other tests of the same or lower size. Inthis work, the two-sided Fisher exact test and the UMPU test are compared with six nonrandomized unconditional exact tests with respect to their power. In both the two-binomial and double dichotomy models, the UMPU test is often less powerful than some of the unconditional tests of the same (or even lower) size. Thus, the assertion that the Tocher-Lehmann modification of Fisher's conditional test is the optimal test for 2x2 tables is unjustified.     

LSTAR-GARCH模型的单位根检验

LSTAR模型的单位根检验往往易忽视其条件方差的时变性,实际上,对许多经济变量尤其是金融变量建立LSTAR模型后,经常发现其条件方差存在GARCH效应。针对LSTAR-GARCH模型的平稳性检验,本文构建了检验统计量tNG,之后在极大似然估计的基础上,推导出tNG的渐近分布,通过蒙特卡洛模拟方法得到该统计量的渐近临界值,并在此基础上研究了tNG检验的检验功效。在与刘雪燕和张晓峒(2009)提出的tNL检验、Ling等(2003)提出的tLG检验以及DF单位根检验进行比较后,发现tNG检验具备明显优势。