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.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

A Correction Note on Improvement in Variance Estimation Using Auxiliary Information

ABSTRACT

In this paper, the testing problem for homogeneity in the mixture exponential family is considered. The model is irregular in the sense that each interest parameter forms a part of the null hypothesis (sub-null hypothesis) and the null hypothesis is the union of the sub-null hypotheses. The generalized likelihood ratio test does not distinguish between the sub-null hypotheses. The Supplementary Score Test is proposed by combining two orthogonalized score tests obtained corresponding to the two sub-null hypotheses after proper reparameterization. The test is easy to design and performs better than the generalized likelihood ratio test and other alternative tests by numerical comparisons.     

Robust unit root tests with autoregressive errors

ABSTRACT

This article presents a new test for unit roots based on least absolute deviation estimation specially designed to work for time series with autoregressive errors. The methodology used is a bootstrap scheme based on estimating a model and then the innovations. The resampling part is performed under the null hypothesis and, as it is customary in bootstrap procedures, is automatic and does not rely on the calculation of any nuisance parameter. The validity of the procedure is established and the asymptotic distribution of the statistic proposed is proved to converge to the correct distribution. To analyze the performance of the test for finite samples, a Monte Carlo study is conducted showing a very good behavior in many different situations.     

ASYMPTOTIC DISTRIBUTIONS OF SEASONAL UNIT ROOT TESTS: A UNIFYING APPROACH

ABSTRACT

This paper adopts a unified approach to the derivation of the asymptotic distributions of various seasonal unit root tests. The procedures considered are those of Dickey et al. [DHF], Kunst, Hylleberg et al. [HEGY], Osborn et al. [OCSB], Ghysels et al. [GHL] and Franses. This unified approach shows that the asymptotic distributions of all these test statistics are functions of the same vector of Brownian motions. The Kunst test and the overall HEGY F-test are, indeed, equivalent both asymptotically and in finite samples, while the Franses and GHL tests are shown to have equivalent parameterizations. The OCSB and DHF test regressions are viewed as restricted forms of the Kunst-HEGY regressions, and these restrictions may have non-trivial asymptotic implications.     

Tests for the order of integration against higher order integration

Locally best invariant tests for the null hypothesis of I(p) against the alternative hypothesis of I(q), < q, are developed for models with independent normal errors. The tests are semiparametrically extended for models with autocorrelated errors. The method is illustrated by two real data sets in terms of double unit roots. The proposed tests can be used for determining integration orders of nonstationary time series.     

Symmetric Test for Second Differencing in Univariate Time Series

A test for the null hypothesis that a time series has characteristic equations with two unit roots is presented. The test, based on a standard regression computation, is shown to have good power properties when compared to previously existing tests.     

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.     

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.     

A new test for exponentiality against HNBUE alternatives

Abstract

Two families of test statistics for testing the null hypothesis of exponentiality against Harmonic New Better than Used in Expectation (HNBUE) alternatives are proposed. Asymptotic distributions of the test statistics are derived under the null and alternative hypotheses and the consistency of the tests established. Comparison with competing tests are made in terms of Pitman Asymptotic Relative Efficiency (PARE). Simulation studies have been carried out to assess the performance of the tests. Finally, the test has been applied to three real life data sets described in Proschan, Susarla and Van Ryzin and Engelhardt, Bain and Wright.     

A note on a family of criteria for evaluating test statistics

ABSTRACT

In noting that the usual criteria for choosing an optimal test, Uniform Power and Local Power are at opposite ends of a spectrum of dominance criteria, a complete “Power Dominance” family of criteria for classifying and choosing optimal tests on the basis of their power characteristics is identified, wherein successive orders of dominance attach increasing weight to power close to the null hypothesis. Indices of the extent to which a preferred test has superior power characteristics over other members in its class, and an index of the proximity of a test to the envelope function of alternative tests are also provided. The ideas are exemplified using various optimal test statistics for Normal and Laplace population distributions.     

A Five-Decision Testing Procedure to Infer the Value of a Unidimensional Parameter

ABSTRACT

A statistical test can be seen as a procedure to produce a decision based on observed data, where some decisions consist of rejecting a hypothesis (yielding a significant result) and some do not, and where one controls the probability to make a wrong rejection at some prespecified significance level. Whereas traditional hypothesis testing involves only two possible decisions (to reject or not a null hypothesis), Kaiser’s directional two-sided test as well as the more recently introduced testing procedure of Jones and Tukey, each equivalent to running two one-sided tests, involve three possible decisions to infer the value of a unidimensional parameter. The latter procedure assumes that a point null hypothesis is impossible (e.g., that two treatments cannot have exactly the same effect), allowing a gain of statistical power. There are, however, situations where a point hypothesis is indeed plausible, for example, when considering hypotheses derived from Einstein’s theories. In this article, we introduce a five-decision rule testing procedure, equivalent to running a traditional two-sided test in addition to two one-sided tests, which combines the advantages of the testing procedures of Kaiser (no assumption on a point hypothesis being impossible) and Jones and Tukey (higher power), allowing for a nonnegligible (typically 20%) reduction of the sample size needed to reach a given statistical power to get a significant result, compared to the traditional approach.     

Two-Tailed p-Values and Coherent Measures of Evidence

ABSTRACT

In a test of significance, it is common practice to report the p-value as one way of summarizing the incompatibility between a set of data and a proposed model for the data constructed under a set of assumptions together with a null hypothesis. However, the p-value does have some flaws, one being in general its definition for two-sided tests and a related serious logical one of incoherence, in its interpretation as a statistical measure of evidence for its respective null hypothesis. We shall address these two issues in this article.     

Testing for structural changes in linear regressions with time-varying variance

Abstract

This article proposes a nonparametric test for structural changes in linear regression models that allows for serial correlation, autoregressive conditional heteroskedasticity and time-varying variance in error terms. The test requires no trimming of the boundary region near the end points of the sample period, and requires no prior information on the alternative, what it requires is the transformed OLS residuals under the null hypothesis. We show that the test has a limiting standard normal distribution under the null hypothesis, and is powerful against single break, multiple breaks and smooth structural changes. The Monte Carlo experiment is conducted to highlight the merits of the proposed test relative to other popular tests for structural changes.     

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.     

Wild bootstrap seasonal unit root tests for time series with periodic nonstationary volatility

We investigate the behavior of the well-known Hylleberg, Engle, Granger and Yoo (HEGY) regression-based seasonal unit root tests in cases where the driving shocks can display periodic nonstationary volatility and conditional heteroskedasticity. Our set up allows for periodic heteroskedasticity, nonstationary volatility and (seasonal) generalized autoregressive-conditional heteroskedasticity as special cases. We show that the limiting null distributions of the HEGY tests depend, in general, on nuisance parameters which derive from the underlying volatility process. Monte Carlo simulations show that the standard HEGY tests can be substantially oversized in the presence of such effects. As a consequence, we propose wild bootstrap implementations of the HEGY tests. Two possible wild bootstrap resampling schemes are discussed, both of which are shown to deliver asymptotically pivotal inference under our general conditions on the shocks. Simulation evidence is presented which suggests that our proposed bootstrap tests perform well in practice, largely correcting the size problems seen with the standard HEGY tests even under extreme patterns of heteroskedasticity, yet not losing finite sample relative to the standard HEGY tests.     

A simple unit root testing methodology that does not require knowledge regarding the presence of a break

We develop a simple methodology that allows practitioners to test for the presence of a unit root without a priori knowledge regarding the occurrence of a break under the null hypothesis. We use a pre-test that is readily available in the estimated regression used to calculate the unit root statistics, and so our methodology is very easy to implement. The t-statistic corresponding to the impulse dummy variables evaluated at break date estimator is used as a pre-test to ascertain whether a break exists under the null hypothesis. Finite sample simulations show that our methodology yields tests that maintain their size.     

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.     

A modified Mack–Wolfe test for the umbrella alternative problem

ABSTRACT

The Mack–Wolfe test is the most frequently used non parametric procedure for the umbrella alternative problem. In this paper, modifications of the Mack–Wolfe test are proposed for both known peak and unknown peak umbrellas. The exact mean and variance of the proposed tests in the null hypothesis are also derived. We compare these tests with some of the existing tests in terms of the type I error rate and power. In addition, a real data example is presented.     

Convergence of posterior odds

The problem of approximating an interval null or imprecise hypothesis test by a point null or precise hypothesis test under a Bayesian framework is considered. In the literature, some of the methods for solving this problem have used the Bayes factor for testing a point null and justified it as an approximation to the interval null. However, many authors recommend evaluating tests through the posterior odds, a Bayesian measure of evidence against the null hypothesis. It is of interest then to determine whether similar results hold when using the posterior odds as the primary measure of evidence. For the prior distributions under which the approximation holds with respect to the Bayes factor, it is shown that the posterior odds for testing the point null hypothesis does not approximate the posterior odds for testing the interval null hypothesis. In fact, in order to obtain convergence of the posterior odds, a number of restrictive conditions need to be placed on the prior structure. Furthermore, under a non-symmetrical prior setup, neither the Bayes factor nor the posterior odds for testing the imprecise hypothesis converges to the Bayes factor or posterior odds respectively for testing the precise hypothesis. To rectify this dilemma, it is shown that constraints need to be placed on the priors. In both situations, the class of priors constructed to ensure convergence of the posterior odds are not practically useful, thus questioning, from a Bayesian perspective, the appropriateness of point null testing in a problem better represented by an interval null. The theories developed are also applied to an epidemiological data set from White et al. (Can. Veterinary J. 30 (1989) 147–149.) in order to illustrate and study priors for which the point null hypothesis test approximates the interval null hypothesis test. AMS Classification: Primary 62F15; Secondary 62A15