Non‐parametric testing for seasonally and periodically integrated processes

This article obtains the asymptotic distributions of the seasonal variance ratio tests proposed by A.M.R. Taylor (2005,Journal of Econometrics 124, 33) when these tests are applied to a periodically integrated process [PI(1)]. In contrast to the situation where the process is seasonally integrated [SI(1)], all test statistics in the PI(1) case are driven by a single stochastic trend and hence follow the distribution obtained by Breitung (2002, Journal of Econometrics 108, 343) for the original (non‐seasonal) variance ratio test. The multivariate non‐parametric cointegration test of Breitung (2002 Journal of Econometrics 108, 343) is also investigated to distinguish between PI and SI processes. A Monte Carlo analysis shows how these results apply in finite samples for both SI and PI processes and an empirical application investigates seasonally unadjusted quarterly US industrial production series.     

Residuals‐based tests for the null of no‐cointegration: an Analytical comparison

Abstract. This article studies the asymptotic distribution of five residuals‐based tests for the null of no‐cointegration under a local alternative when the tests are computed using both ordinary least squares (OLS) and generalized least squares (GLS)‐detrended variables. The local asymptotic power of the tests is shown to be a function of Brownian motion and Ornstein–Uhlenbeck processes, depending on a single nuisance parameter, which is determined by the correlation at frequency zero of the errors of the cointegration regression with the shocks to the right‐hand side variables. The tests are compared in terms of power in large and small samples. It is shown that, while no significant improvement can be achieved by using unit root tests other than the OLS detrended t‐test originally proposed by Engle and Granger (1987), the power of GLS residuals tests can be higher than the power of system tests for some values of the nuisance parameter.     

Seasonal Unit Root Tests Under Structural Breaks*

Abstract. In this paper, several seasonal unit root tests are analysed in the context of structural breaks at known time and a new break corrected test is suggested. We show that the widely used HEGY test, as well as an LM variant thereof, are asymptotically robust to seasonal mean shifts of finite magnitude. In finite samples, however, experiments reveal that such tests suffer from severe size distortions and power reductions when breaks are present. Hence, a new break corrected LM test is proposed to overcome this problem. Importantly, the correction for seasonal mean shifts bears no consequence on the limiting distributions, thereby maintaining the legitimacy of canonical critical values. Moreover, although this test assumes a breakpoint a priori, it is robust in terms of misspecification of the time of the break. This asymptotic property is well reproduced in finite samples. Based on a Monte‐Carlo study, our new test is compared with other procedures suggested in the literature and shown to hold superior finite sample properties.     

Inference about long run canonical correlations

This article proposes methods for testing the null hypothesis that a number of so‐called long run canonical correlations (LRCCs) are zero. Two test statistics are proposed and their limiting distributions are derived under the null hypothesis. The finite sample properties of the tests are illustrated via a number of simulation studies that reveal the asymptotic theory provides a good guidance to behaviour in moderate or large sized samples. It is shown that the statistics provide a natural way for testing the asymptotic independence of two standardized sums. The usefulness of the tests is illustrated via the following examples: inference about cointegrating vector in a particular cointegration model; inference about break points in a cointegration model; moment estimation; parameter estimation in Generalized Method of Moments estimation.     

On LM‐type tests for seasonal unit roots in the presence of a break in trend

This article proposes tests for seasonal unit roots allowing for the presence of a break in the trend slope occurring at an unknown date. In particular, new LM‐type tests are derived based on the framework introduced by Hylleberg et al. (1990). Null asymptotic distributions are derived for the no break case as well as when a break is present in the data‐generating process and a Monte Carlo investigation on the finite sample size and power performance of the new procedures is presented.     

Using the HEGY Procedure When Not All Roots Are Present

Abstract. Empirical studies have shown little evidence to support the presence of all unit roots present in the Δ₄ filter in quarterly seasonal time series. This paper analyses the performance of the Hylleberg, Engle, Granger and Yoo [Journal of Econometrics (1990) Vol. 44, pp. 215–238] (HEGY) procedure when the roots under the null are not all present. We exploit the vector of quarters representation and cointegration relationship between the quarters when factors (1 − L), (1 + L), (1 + L²), (1 − L²) and (1 + L + L² + L³) are a source of nonstationarity in a process in order to obtain the distribution of tests of the HEGY procedure when the underlying processes have a root at the zero, Nyquist frequency, two complex conjugates of frequency π/2 and two combinations of the previous cases. We show both theoretically and through a Monte Carlo analysis that the t‐ratios t and t and the F‐type tests used in the HEGY procedure have the same distribution as under the null of a seasonal random walk when the root(s) is (are) present, although this is not the case for the t‐ratio tests associated with unit roots at frequency π/2.     

IV‐BASED COINTEGRATION TESTING IN DEPENDENT PANELS WITH TIME‐VARYING VARIANCE

The distributions of cointegration tests are affected when the innovation variance varies over time. In panels, one must also pay attention to dependence among units. To obtain a panel cointegration test robust to both heteroskedasticity and dependence, we adapt the nonlinear instruments method proposed for the Dickey–Fuller test by Chang (2002, J Econometrics 110, 261–292) to an error‐correction framework. We show that IV‐based testing of the no error‐correction null in individual equations yields standard normal test statistics when computed with heteroskedasticity‐robust standard errors. The result holds under endogenous regressors, irrespective of the number of integrated covariates and for any variance profile. A non‐cointegration test combining single‐equation tests retains these nice properties. In panels of fixed cross‐sectional dimension, such test statistics from individual units are shown to be asymptotically independent even under dependence, leading to panel tests robust to dependence and heteroskedasticity. The tests perform well in finite panels.     

A New Class of Distribution-Free Tests for Location Parameters

Abstract

A new class of distribution-free tests for the two-sample location problem is proposed. The tests are based on two-sample U-statistics. This class basically generalizes the tests proposed by Deshpande, J.V.; Kochar, S.C. Some competitors of Wilcoxon–Mann–Whitney test for location alternatives. Journal of Indian Statistical Association 1982, 19, 9–18. The proposed class of tests is also extended to the k (≥ 2)-sample problem for testing homogeneity of location parameters against ordered alternatives. This extension is based on linear combinations of two-sample U-statistics. Pitman asymptotic relative efficiencies (AREs) of the members of the proposed class(es) of tests relative to some of the existing tests are computed for a number of underlying distributions. It is shown that the proposed class of tests performs as good as or better than its competitors in literature.     

Testing for co‐integration and nonlinear adjustment in a smooth transition error correction model

This article introduces a testing procedure for cointegration and nonlinear adjustment in a smooth transition vector error correction model. To overcome the unidentified parameters problem under the null of no‐cointegration, the Wald statistic is optimized over the unidentified parameter space. The asymptotic distribution of the test statistic is shown to be non‐standard but nuisance parameter‐free and hence critical values are obtained by simulations, Simulations show that the proposed test outperforms the alternatives in small sample sizes both in terms of size and power. Application to the exchange rate‐monetary fundamentals relationship show that the proposed test works considerably well. This article also finds that nonlinear adjustment dynamics are symmetric for some currencies and therefore the speed of adjustment depends on the size of the deviations and is asymmetric for others, hence, the adjustment dynamics depend not only on the size but also on the sign of the deviations.     

Seasonal Unit Root Tests Based on Forward and Reverse Estimation

In this paper, we suggest a new set of regression-based statistics for testing the seasonal unit root null hypothesis. These tests are based on combining conventional Hylleberg et al . (1990 ) -type seasonal unit root test statistics calculated from both forward and reverse estimation of the auxiliary regression equation. We derive the asymptotic distributions of the new test statistics under the seasonal unit root null hypothesis. We provide finite sample critical values appropriate for the case of quarterly data together with asymptotic critical values, the latter appropriate for any seasonal aspect. Monte Carlo simulation of the finite-sample size and power properties of the new tests reveals that, overall, they perform rather better than extant tests of the seasonal unit root hypothesis.     

A bootstrap test for additive outliers in non‐stationary time series

In this paper we propose a new procedure for detecting additive outliers in a univariate time series based on a bootstrap implementation of the test of Perron and Rodríguez (2003, Journal of Time Series Analysis 24, 193‐220). This procedure is used to test the null hypothesis that a time series is uncontaminated by additive outliers against the alternative that one or more additive outliers are present. We demonstrate that the existing tests of, inter alia, Vogelsang (1999, Journal of Time Series Analysis 20, 237–52) Perron and Rodríguez (2003) and Burridge and Taylor (2006, Journal of Time Series Analysis 27, 685–701) are unable to strike a balance between size and power when the order of integration of a time series is unknown and the time series is driven by innovations drawn from an unknown distribution. We show that the proposed bootstrap testing procedure is able to control size to such an extent that its size properties are comparable with the robust test of Burridge and Taylor (2006) when the distribution of the innovations is not assumed known, whilst maintaining power in the Gaussian environment close to that of the test of Perron and Rodríguez (2003).     

Testing Cointegrating Relationships Using Irregular and Non‐Contemporaneous Series with an Application to Paleoclimate Data

Time series that are observed neither regularly nor contemporaneously pose problems for most multivariate analyses. Common and intuitive solutions to these problems include interpolation and other types of imputation to a higher, regular frequency. However, interpolation is known to cause serious problems with the size and power of statistical tests. Due to the difficulty in dating paleoclimate data such as CO₂ concentrations and surface temperatures, time series of such measurements are observed neither regularly nor contemporaneously. This article presents large‐ and small‐sample analyses of the size and power of cointegration tests of time series with these features and supports the robustness of cointegration of these two series found in the extant literature. Compared to linear or higher‐order polynomial interpolation, step interpolation results in the least size distortion and is therefore recommended.     

Testing the differences between two color measurement probability distributions using Hotelling's T2 test and the permutation test

It is common practice in statistics to test the equality of two population means using, for example, the Student's t test, in the univariate case, or the Hotelling's T² Test in the multivariate case. However, tests on the equality of population means are not well developed for testing the difference between two populations of color measurements. Methods for analyzing populations of spectral reflectance and L*a*b* measurements have been described for applications such as analyzing inter-instrument agreement and repeatability. Methods have also been proposed for the analysis of color differences, but there are little written about techniques for testing whether two samples have the same probability distribution. This article focuses on testing the difference between color measurement probability distributions based on color difference. In addition, a metric is proposed called the threshold for color difference discrimination (TCDD, in units of ΔE), the color difference at which two populations can be considered to have different population distributions. A lower TCDD means smaller color differences between two samples can be resolved. Two parametric tests based on Hotelling's T² test and a nonparametric permutation test were used to determine the TCDD for populations of color measurements with different variances and sample sizes. The TCDD was found to be smaller by tests using the Hotelling's T² statistic, compared with a permutation test performed directly on color difference. It was also found, as expected, that larger sample sizes led to smaller TCDDs, as did smaller population variances.     

Testing for Unit Roots in Monthly Time Series

This paper is concerned with tests for seasonal roots in monthly univariate time series processes. The paper extends the procedures and tables of critical values due to Beaulieu and Miron (Seasonal unit roots in aggregate U.S. data. J. Economet . 55 (1993), 305–28) to obtain tests which are similar ( exactly and a symptotically ) with respect to both the initial values of the process and the possibility of seasonal drifts under the seasonal unit root null hypothesis. We also develop test statistics which test simultaneously for a unit root at each frequency and for a unit root at each of the seasonal frequencies. Representations are derived for the limiting distributions of each of the test statistics proposed in this paper. We illustrate the practical usefulness of the proposed test statistics by a series of empirical applications     

Residual-Based Tests For Fractional Cointegration: A Monte Carlo Study

This paper reports an extensive Monte Carlo study of six residual-based tests of the hypothesis of no cointegration against the alternative of fractional cointegration. Critical values, power and size are simulated and compared. It turns out that the Lobato–Robinson LM test, the Geweke–Porter-Hudak test and the two Phillips–Perron tests exhibit severe size distortions. On the other hand, the augmented Dickey–Fuller test and the modified rescaled range test show only moderate size distortions but are generally less powerful. In addition, the impact and possible treatment of trends in the individual series, and in the cointegration equilibrium, are discussed.     

Fractional cointegration in the presence of linear trends

Abstract. We consider bivariate regressions of nonstationary fractionally integrated variables dominated by linear time trends. The asymptotic behaviour of the ordinary least square (OLS) estimators in this case allows limiting normality to arise at a faster rate of convergence than if the individual series were detrended, increasing in this way the power of the tests for fractional cointegration. We also show that the limiting distribution of the t‐ratio of the slope coefficient depends upon the presence or not of a deterministic trend in the conditional regressor. We introduce the concept of local fractional trend to explain the apparently diverging asymptotic theories that apply when a trend is either present or absent in our set‐up.     

Testing for Cointegration with Temporally Aggregated and Mixed‐Frequency Time Series

We examine the effects of mixed sampling frequencies and temporal aggregation on the size of commonly used tests for cointegration, and we find that these effects may be severe. Matching sampling schemes of all series generally reduces size distortion, and the nominal size is obtained asymptotically only when all series are skip sampled in the same way – for example, end‐of‐period sampling. We propose and analyse mixed‐frequency versions of the cointegration tests in order to control the size when some high‐frequency data are available. Otherwise, when no high‐frequency data are available, we discuss controlling size using bootstrapped critical values. We test stock prices and dividends for cointegration as an empirical demonstration.     

A Sequential and Iterative Testing Procedure to Identify the Nature of a Time Series Generating Process

Abstract

An inferential approach is proposed to identify the nature of the generating process corresponding to a real time series. This new sequential and iterative testing procedure goes beyond the Box and Jenkins methodology for the identification, estimation, and validation of linear data generating processes by investigating the probabilistic structure of non-Gaussian estimated residuals {? _t } for the possible presence of nonlinear serial dependence. The testing procedure aims at indicating the right type of dependence present in a series by means of specific inferential tests on the moments of the generating structure probability distribution. The test statistics adopted are very popular and powerful and encompass a wide range of stochastic nonlinearity alternatives. The U.S. Industrial Production Index series is used to illustrate the iterative testing procedure proposed.     

Testing Stochastic Cycles in Macroeconomic Time Series

A particular version of the tests of Robinson (1994) for testing stochastic cycles in macroeconomic time series is proposed in this article. The tests have a standard limit distribution and are easy to implement in raw time series. A Monte Carlo experiment is conducted, studying the size and the power of the tests against different alternatives, and the results are compared with those based on other tests. An empirical application using historical US annual data is also carried out at the end of the article.     

Testing non‐parametric hypotheses for stationary processes by estimating minimal distances

In this article, new tests for non‐parametric hypotheses in stationary processes are proposed. Our approach is based on an estimate of the L²‐distance between the spectral density matrix and its best approximation under the null hypothesis. We explain the main idea in the problem of testing for a constant spectral density matrix and in the problem of comparing the spectral densities of several correlated stationary time series. The method is based on direct estimation of integrals of the spectral density matrix and does not require the specification of smoothing parameters. We show that the limit distribution of the proposed test statistic is normal and investigate the finite sample properties of the resulting tests by means of a small simulation study.