A Locally Optimal Seasonal Unit-Root Test

This article proposes a locally best invariant test of the null hypothesis of seasonal stationarity against the alternative of seasonal unit roots at all or individual seasonal frequencies. An asymptotic distribution theory is derived and the finite-sample properties of the test are examined in a Monte Carlo simulation. My test is also compared with the Canova and Hansen test. The proposed test is superior to the Canova and Hansen test in terms of both size and power.     

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.     

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.     

Heteroscedasticity Robust Panel Unit Root Tests

This article proposes new unit root tests for panels where the errors may be not only serial and/or cross-correlated, but also unconditionally heteroscedastic. Despite their generality, the test statistics are shown to be very simple to implement, requiring only minimal corrections and still the limiting distributions under the null hypothesis are completely free from nuisance parameters. Monte Carlo evidence is also provided to suggest that the new tests perform well in small samples, also when compared to some of the existing tests. Supplementary materials for this article are available online.     

TESTS FOR SEASONAL DIFFERENCING WITH AN UNKNOWN BREAK-POINT

Some Lagrange multiplier tests for seasonal differencing are proposed; their main objective is to avoid over-differencing due to structural change. The null hypothesis is either the presence of both regular and seasonal unit roots or the presence of a seasonal unit root. Alternative hypotheses allow for stationarity around a possible structural change where the break-point is unknown. The location of the structural change is estimated using the proposed procedures, the asymptotic distribution of the test statistics under the null hypothesis is derived and some useful percentiles are tabulated. An illustrative example based on the Canadian Consumer Price Index is presented.     

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.     

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.     

Seasonality Tests

This article modifies and extends the test against nonstationary stochastic seasonality proposed by Canova and Hansen. A simplified form of the test statistic in which the nonparametric correction for serial correlation is based on estimates of the spectrum at the seasonal frequencies is considered and shown to have the same asymptotic distribution as the original formulation. Under the null hypothesis, the distribution of the seasonality test statistics is not affected by the inclusion of trends, even when modified to allow for structural breaks, or by the inclusion of regressors with nonseasonal unit roots. A parametric version of the test is proposed, and its performance is compared with that of the nonparametric test using Monte Carlo experiments. A test that allows for breaks in the seasonal pattern is then derived. It is shown that its asymptotic distribution is independent of the break point, and its use is illustrated with a series on U.K. marriages. A general test against any form of permanent seasonality, deterministic or stochastic, is suggested and compared with a Wald test for the significance of fixed seasonal dummies. It is noted that tests constructed in a similar way can be used to detect trading-day effects. An appealing feature of the proposed test statistics is that under the null hypothesis, they all have asymptotic distributions belonging to the Cramér–von Mises family.     

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.     

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.     

Unit Roots,Level Shifts,and Trend Breaks in Per Capita Output: A Robust Evaluation

Determining whether per capita output can be characterized by a stochastic trend is complicated by the fact that infrequent breaks in trend can bias standard unit root tests towards nonrejection of the unit root hypothesis. The bulk of the existing literature has focused on the application of unit root tests allowing for structural breaks in the trend function under the trend stationary alternative but not under the unit root null. These tests, however, provide little information regarding the existence and number of trend breaks. Moreover, these tests suffer from serious power and size distortions due to the asymmetric treatment of breaks under the null and alternative hypotheses. This article estimates the number of breaks in trend employing procedures that are robust to the unit root/stationarity properties of the data. Our analysis of the per capita gross domestic product (GDP) for Organization for Economic Cooperation and Development (OECD) countries thereby permits a robust classification of countries according to the “growth shift,” “level shift,” and “linear trend” hypotheses. In contrast to the extant literature, unit root tests conditional on the presence or absence of breaks do not provide evidence against the unit root hypothesis.     

Testing for the Equality of Two Autoregressive Functions Using Quasi-Residuals

This article discusses the problem of testing the equality of two nonparametric autoregressive functions against one-sided alternatives. The heteroscedastic errors and stationary densities of the two independent strong mixing strictly stationary time series can be possibly different. The article adapts the idea of using sum of quasi-residuals to construct the test and derives its asymptotic null distributions. The article also shows that the test is consistent for general alternatives and obtains its limiting distributions under a sequence of local alternatives. Then a Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. We also compare the test to an existing lag matched test theoretically and by Monte Carlo experiments.     

F-tests for seasonal differencing with a break-point

The classification between stochastic trend stationarity and deterministic broken trend stationarity is important because incorrect inferences can follow if a stationary series with a broken trend is incorrectly classified as integrated. In this paper, we consider joint tests for regular and seasonal unit roots null hypothesis against broken trend stationarity alternatives where the location of the break is known or unknown. Based on the F-test proposed by Hasza and Fuller (1982, Ann. Statist. 10, 1209–1216), we develop testing procedures for distinguishing these two types of process. The asymptotic distributions of test statistics are derived as functions of Wiener processes. A response surface regression analysis directed to relating the finite sample distributions and the breaking position is studied. Simulation experiments suggest that the power of the test is reasonable. The testing procedure is illustrated by the Canadian consumer price index series.     

Panel Seasonal Unit Root Test: Further Simulation Results and An Application to Unemployment Data

Summary: In this paper the seasonal unit root test of Hylleberg et al. (1990) is generalized to cover a heterogenous panel. The procedure follows the work of Im, Pesaran and Shin (2002) and is independently proposed by Otero et al. (2004). Test statistics are given and critical values are obtained by simulation. Moreover, the properties of the tests are analyzed for different deterministic and dynamic specifications. Evidence is presented that for a small time series dimension the power is low even for increasing cross section dimension. Therefore, it seems necessary to have a higher time series dimension than cross section dimension. The test is applied to unemployment data in industrialized countries. In some cases seasonal unit roots are detected. However, the null hypotheses of panel seasonal unit roots are rejected. The null hypothesis of a unit root at the zero frequency is not rejected, thereby supporting the presence of hysteresis effects. * The research of this paper was supported by the Deutsche Forschungsgemeinschaft. The paper was presented at the workshop “Unit roots and cointegration in panel data” in Frankfurt, October 2004 and in the poster-session at the EC2 meeting in Marseille, December 2004. We are grateful to the participants of the workshops and an anonymous referee for their helpful comments.     

Nonparametric likelihood inference for general autoregressive models

This paper shows how nonparametric likelihood inference for autoregressive models can be based on the family of “empirical” Cressie–Read statistics. The results of the paper apply to possibly nonstationary autoregressive models with innovations that form a martingale difference sequence, and can accommodate multiple and complex unit roots, as well as deterministic components. As an application, the paper considers nonparametric likelihood-based tests for seasonal unit roots and for double unit roots. Monte Carlo evidence seems to suggest that the proposed tests have competitive finite sample properties.     

TESTING FOR PERIODIC STATIONARITY

This paper proposes a test for the null hypothesis of periodic stationarity against the alternative hypothesis of periodic integration. We derive the limiting distribution of the test statistic and its characteristic function, which are the same as those of the test developed in Kwiatkowski, Phillips, Schmidt and Shin.[15] We find that some parameters, which we must assume under the alternative, have an important effect on the limiting power, so we should choose such parameters carefully. A Monte Carlo simulation reveals that the test has reasonable power but may be affected by the lag truncation parameter that is used for the correction of nuisance parameters.     

Unit roots and double smooth transitions

Techniques for testing the null hypothesis of difference stationarity against stationarity around some deterministic function have received much attention. In particular, unit root tests where the alternative is stationarity around a smooth transition in a linear trend have recently been proposed to permit the possibility of non-instantaneous structural change. In this paper we develop tests extending such an approach in order to admit more than one structural change. The analysis is motivated by time series that appear to undergo two smooth transitions in the linear trend, and the application of the new tests to two such series (average global temperature and US consumer prices) highlights the benefits of this double transition extension.     

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).     

Testing for unit roots in time series with nearly deterministic seasonal variation

This paper addresses the problem of testing for the presence of unit autoregressive roots in seasonal time series with negatively correlated moving average components. For such cases, many of the commonly used tests are known to have exact sizes much higher than their nominal significance level. We propose modifications of available test procedures that are based on suitably prewhitened data and feasible generalized least squares estimators. Monte Carlo experiments show that such modifications are successful in reducing size distortions in samples of moderate size.     

Unit root econometrics and economic nonlinearities

This paper concentrates on some shortcomings of contemporary unit root econometric methodology (testing for cointegration, common roots and stationarity) where the dynamics of an economy are described by a nonlinear process. It is shown that, in such circumstances, traditionally applied unit root econometrics may not lead to interpretable or statistically significant results. Two cases of such nonlinearities are discussed: (i) a stochastically nonlinear data generating process and (ii) a time-varying parameters cointegrating relation, typical of an economic reform process. It is shown that case (i) consists of a wide family of economic processes and in most such cases the results of standard unit root tests are not directly interpretable. Case (ii) does not result in a (conventionally understood) error-correction representation of a cointegrated process. Some Monte Carlo experiments evaluate the validity of cointegration tests in situations where there is a change in the cointegration parameter and from cointegration regime to noncointegration and vice versa. A simple method of estimation through simulation is proposed and its finite-sample properties examined.