Tests for Stationarity in Series with Endogenously Determined Structural Change*

We consider tests of the null hypothesis of stationarity against a unit root alternative, when the series is subject to structural change at an unknown point in time. Three extant tests are reviewed which allow for an endogenously determined instantaneous structural break, and a related fourth procedure is introduced. We further propose tests which permit the structural change to be gradual rather than instantaneous, allowing the null hypothesis to be stationarity about a smooth transition in linear trend. The size and power properties of the tests are investigated, and the tests are applied to four economic time series.     

London calling: Nonlinear mean reversion across national stock markets

This paper revisits empirical evidence of mean reversion of relative stock prices in international stock markets. We implement a strand of univariate and panel unit root tests for linear and nonlinear models of 18 national stock indices from 1969 to 2016. Our major findings are as follows. First, we find strong evidence of nonlinear mean reversion of the relative stock price with the UK index as the reference, calling attention to the stock index in the UK, but not with the US index. Our results imply an important role of the local common factor in the European stock markets. Second, panel tests yield no evidence of linear and nonlinear stationarity when the cross-section dependence is considered, which provides conflicting results from those of the univariate tests. Last, we show how to understand these results via dynamic factor analysis. When the stationary common factor dominates nonstationary idiosyncratic components in small samples, panel tests that filter out the stationary common factor may yield evidence against the stationarity null hypothesis in finite samples. We corroborate this conjecture via extensive Monte Carlo simulations.     

Rank tests for short memory stationarity

We propose a rank-test of the null hypothesis of short memory stationarity possibly after linear detrending.     

Evidence on purchasing power parity from univariate models: the case of smooth transition trend‐stationarity

Recent research has found that trend‐break unit root tests derived from univariate linear models do not support the hypothesis of long‐run purchasing power parity (PPP) for US dollar real exchange rates. In this paper univariate smooth transition models are utilized to develop unit root tests that allow under the alternative hypothesis for stationarity around a gradually changing deterministic trend function. These tests reveal statistically significant evidence against the null hypothesis of a unit root for the real exchange rates of a number of countries against the US dollar. However, restrictions consistent with long‐run PPP are rejected for some of the countries for which a rejection of the unit root hypothesis is obtained. Copyright © 2005 John Wiley & Sons, Ltd.     

Testing for Unit Roots with Breaks: Evidence on the Great Crash and the Unit Root Hypothesis Reconsidered

Nelson and Plosser (1982), in a classic paper, failed to find strong evidence against the null hypothesis of a generating process with a unit autoregressive root for thirteen US macroeconomic time series. Perron (1989) claimed that such evidence was available for a majority of these series if the alternative hypothesis was of trend stationarity with a break in 1929. Zivot and Andrews (1992) treated the break date as endogenous, then finding strong evidence agcainst the null for a minority of these series. Our own analysis extends theirs by permitting a break under the null as well as the alternative hypothesis, and allowing for the sequential nature of the testing. Our empirical findings complete the circle. We find no strong evidence against the unit root hypothesis for any of the thirteen Nelson–Plosser series.     

A Comparative Study of Unit Root Tests with Panel Data and a New Simple Test

The panel data unit root test suggested by Levin and Lin (LL) has been widely used in several applications, notably in papers on tests of the purchasing power parity hypothesis. This test is based on a very restrictive hypothesis which is rarely ever of interest in practice. The Im–Pesaran–Shin (IPS) test relaxes the restrictive assumption of the LL test. This paper argues that although the IPS test has been offered as a generalization of the LL test, it is best viewed as a test for summarizing the evidence from a number of independent tests of the sample hypothesis. This problem has a long statistical history going back to R. A. Fisher. This paper suggests the Fisher test as a panel data unit root test, compares it with the LL and IPS tests, and the Bonferroni bounds test which is valid for correlated tests. Overall, the evidence points to the Fisher test with bootstrap-based critical values as the preferred choice. We also suggest the use of the Fisher test for testing stationarity as the null and also in testing for cointegration in panel data.     

Variance ratio tests of the seasonal unit root hypothesis

In this paper we develop tests of the seasonal (quarterly) unit root null hypothesis which reject in favour of stationarity for small values of certain variance ratio statistics, similar to those used by Canova and Hansen (J. Bus. Econom. Statist. 13 (1995) 237) in a different testing context. We demonstrate that our proposed statistics have pivotal limiting distributions under both the null and near seasonally integrated alternatives even when we allow for the possibility of both weak dependence and periodically heteroscedastic behaviour in the driving shocks. This is in contrast to the popular regression-based lag-augmented seasonal unit root tests of Hylleberg et al. (J. Econometrics 44 (1990) 215). A simulation study into the finite sample size and power properties of our proposed tests suggests that they display far superior size properties and, overall, broadly comparable power properties to the corresponding tests of Hylleberg et al. (J. Econometrics 44 (1990) 215), implemented with data-based lag augmentation. The results for the variance ratio tests at the seasonal harmonic frequency are particularly encouraging.     

Panel Stationarity Test with Structural Breaks*

In this paper, we extend the heterogeneous panel data stationarity test of Hadri [Econometrics Journal, Vol. 3 (2000) pp. 148–161] to the cases where breaks are taken into account. Four models with different patterns of breaks under the null hypothesis are specified. Two of the models have been already proposed by Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175]. The moments of the statistics corresponding to the four models are derived in closed form via characteristic functions. We also provide the exact moments of a modified statistic that do not asymptotically depend on the location of the break point under the null hypothesis. The cases where the break point is unknown are also considered. For the model with breaks in the level and no time trend and for the model with breaks in the level and in the time trend, Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175] showed that the number of breaks and their positions may be allowed to differ across individuals for cases with known and unknown breaks. Their results can easily be extended to the proposed modified statistic. The asymptotic distributions of all the statistics proposed are derived under the null hypothesis and are shown to be normally distributed. We show by simulations that our suggested tests have in general good performance in finite samples except the modified test. In an empirical application to the consumer prices of 22 OECD countries during the period from 1953 to 2003, we found evidence of stationarity once a structural break and cross‐sectional dependence are accommodated.     

Fluctuation Tests for a Change in Persistence*

In this paper, we develop a set of new persistence change tests which are similar in spirit to those of Kim [Journal of Econometrics (2000) Vol. 95, pp. 97–116], Kim et al. [Journal of Econometrics (2002) Vol. 109, pp. 389–392] and Busetti and Taylor [Journal of Econometrics (2004) Vol. 123, pp. 33–66]. While the exisiting tests are based on ratios of sub‐sample Kwiatkowski et al. [Journal of Econometrics (1992) Vol. 54, pp. 158–179]‐type statistics, our proposed tests are based on the corresponding functions of sub‐sample implementations of the well‐known maximal recursive‐estimates and re‐scaled range fluctuation statistics. Our statistics are used to test the null hypothesis that a time series displays constant trend stationarity [I(0)] behaviour against the alternative of a change in persistence either from trend stationarity to difference stationarity [I(1)], or vice versa. Representations for the limiting null distributions of the new statistics are derived and both finite‐sample and asymptotic critical values are provided. The consistency of the tests against persistence change processes is also demonstrated. Numerical evidence suggests that our proposed tests provide a useful complement to the extant persistence change tests. An application of the tests to US inflation rate data is provided.     

Are You All Normal? It Depends!

The assumption of normality has underlain much of the development of statistics, including spatial statistics, and many tests have been proposed. In this work, we focus on the multivariate setting and first review the recent advances in multivariate normality tests for i.i.d. data, with emphasis on the skewness and kurtosis approaches. We show through simulation studies that some of these tests cannot be used directly for testing normality of spatial data. We further review briefly the few existing univariate tests under dependence (time or space), and then propose a new multivariate normality test for spatial data by accounting for the spatial dependence. The new test utilises the union-intersection principle to decompose the null hypothesis into intersections of univariate normality hypotheses for projection data, and it rejects the multivariate normality if any individual hypothesis is rejected. The individual hypotheses for univariate normality are conducted using a Jarque–Bera type test statistic that accounts for the spatial dependence in the data. We also show in simulation studies that the new test has a good control of the type I error and a high empirical power, especially for large sample sizes. We further illustrate our test on bivariate wind data over the Arabian Peninsula.     

Measure for Measure: Exact F Tests and the Mixed Models Controversy

We consider exact F tests for the hypothesis of null random factor effect in the presence of interaction under the two factor mixed models involved in the mixed models controversy. We show that under the constrained parameter ( CP ) model, even in unbalanced data situations, MSB/MSE (in the usual ANOVA notation) follows an exact F distribution when the null hypothesis holds. We also obtain an exact F test for what is generally (and erroneously) assumed to be an equivalent hypothesis under the unconstrained parameter ( UP ) model. For unbalanced data, such a corresponding test statistic does not coincide with MSB/MSAB (the test statistic advocated for balanced data cases). We compute the power of the exact test under different imbalance patterns and show that although the loss of power increases with the degree of imbalance, it still remains reasonable from a practical point of view.     

The Yen Real Exchange Rate may be Stationary after all: Evidence from Non‐linear Unit‐root Tests*

The empirical literature that tests for purchasing power parity (PPP) by focusing on the stationarity of real exchange rates has so far provided, at best, mixed results. The behaviour of the yen real exchange rate has most stubbornly challenged the PPP hypothesis and deepened this puzzle. This paper contributes to this discussion by providing new evidence on the stationarity of bilateral yen real exchange rates. We employ a non‐linear version of the augmented Dickey–Fuller test, based on an exponentially smooth‐transition autoregressive model (ESTAR) that enhances the power of the tests against mean‐reverting non‐linear alternative hypotheses. Our results suggest that the bilateral yen real exchange rates against the other G7 and Asian currencies were mean reverting during the post‐Bretton Woods era. Thus, the real yen behaviour may not be so different after all but simply perceived to be so because of the use of a restrictive alternative hypothesis in previous tests.     

Multiple Testing for No Cointegration under Nonstationary Volatility

With cointegration tests often being oversized under time‐varying error variance, it is possible, if not likely, to confuse error variance non‐stationarity with cointegration. This paper takes an instrumental variable (IV) approach to establish individual‐unit test statistics for no cointegration that are robust to variance non‐stationarity. The sign of a fitted departure from long‐run equilibrium is used as an instrument when estimating an error‐correction model. The resulting IV‐based test is shown to follow a chi‐square limiting null distribution irrespective of the variance pattern of the data‐generating process. In spite of this, the test proposed here has, unlike previous work relying on instrumental variables, competitive local power against sequences of local alternatives in 1/T‐neighbourhoods of the null. The standard limiting null distribution motivates, using the single‐unit tests in a multiple testing approach for cointegration in multi‐country data sets by combining P‐values from individual units. Simulations suggest good performance of the single‐unit and multiple testing procedures under various plausible designs of cross‐sectional correlation and cross‐unit cointegration in the data. An application to the equilibrium relationship between short‐ and long‐term interest rates illustrates the dramatic differences between results of robust and non‐robust tests.     

Tests for multiple change points under ordered alternatives

We consider the problem of testing the null hypothesis of no change against the alternative of multiple change points in a series of independent observations when the changes are in the same direction. We extend the tests of Terpstra (1952), Jonckheere (1954) and Puri (1965) to the change point setup. We obtain the asymptotic null distribution of the proposed tests. We also give approximations for their limiting critical values and tables of their finite sample Monte Carlo critical values. The results of Monte Carlo power studies conducted to compare the proposed tests with some competitors are reported. This research was supported by research grant SS045 of Kuwait University. Acknowledgments. We wish to thank the two referees for their comments and suggestions which have significantly improved the presentation. We are particularly thankful to one of the referees for suggesting the test statistics T_n1 ^* (k) and T_n2 ^* (k).     

Trend and Initial Condition in Stationarity Tests: The Asymptotic Analysis

In this article, we investigate the behaviour of stationarity tests proposed by Müller [Journal of Econometrics (2005) Vol. 128, pp. 195–213] and Harris et al. [Econometric Theory (2007) Vol. 23, pp. 355–363] with uncertainty over the trend and/or initial condition. As different tests are efficient for different magnitudes of local trend and initial condition, following Harvey et al. [Journal of Econometrics (2012) Vol. 169, pp. 188–195], we propose decision rule based on the rejection of null hypothesis for multiple tests. Additionally, we propose a modification of this decision rule, relying on additional information about the magnitudes of the local trend and/or the initial condition that is obtained through pre‐testing. The resulting modification has satisfactory size properties under both uncertainty types.     

Testing for Fractional Integration Versus Short Memory with Structural Breaks*

Although it is commonly accepted that most macroeconomic variables are non‐stationary, it is often difficult to identify the source of the non‐stationarity. Integrated processes and short‐memory models with trending components, possibly affected by structural breaks, imply similar features in the data and, accordingly, are hard to distinguish. The goal of this article is to extend the classical testing framework of I(1) versus I(0) + trends and/or breaks by considering a more general class of models under the null hypothesis: fractionally integrated (FI) processes. The asymptotic properties of the proposed tests are derived and it is shown that they are very well‐behaved in finite samples. An illustration using US inflation data is also provided.     

Testing conditional independence via empirical likelihood

We construct two classes of smoothed empirical likelihood ratio tests for the conditional independence hypothesis by writing the null hypothesis as an infinite collection of conditional moment restrictions indexed by a nuisance parameter. One class is based on the CDF; another is based on smoother functions. We show that the test statistics are asymptotically normal under the null hypothesis and a sequence of Pitman local alternatives. We also show that the tests possess an asymptotic optimality property in terms of average power. Simulations suggest that the tests are well behaved in finite samples. Applications to some economic and financial time series indicate that our tests reveal some interesting nonlinear causal relations which the traditional linear Granger causality test fails to detect.     

Testing stationarity of functional time series

Economic and financial data often take the form of a collection of curves observed consecutively over time. Examples include, intraday price curves, yield and term structure curves, and intraday volatility curves. Such curves can be viewed as a time series of functions. A fundamental issue that must be addressed, before an attempt is made to statistically model such data, is whether these curves, perhaps suitably transformed, form a stationary functional time series. This paper formalizes the assumption of stationarity in the context of functional time series and proposes several procedures to test the null hypothesis of stationarity. The tests are nontrivial extensions of the broadly used tests in the KPSS family. The properties of the tests under several alternatives, including change-point and I(1)$I\left(1\right)$, are studied, and new insights, present only in the functional setting are uncovered. The theory is illustrated by a small simulation study and an application to intraday price curves.     

Analysis of Divergence in Loglinear Models When Expected Frequencies are Subject to Linear Constraints

Consider the loglinear model for categorical data under the assumption of multinomial sampling. We are interested in testing between various hypotheses on the parameter space when we have some hypotheses relating to the parameters of the models that can be written in terms of constraints on the frequencies. The usual likelihood ratio test, with maximum likelihood estimator for the unspecified parameters, is generalized to tests based on -divergence statistics, using minimum -divergence estimator. These tests yield the classical likelihood ratio test as a special case. Asymptotic distributions for the new -divergence test statistics are derived under the null hypothesis.     

Testing of unit root and other nonstationary hypotheses in macroeconomic time series

Recently proposed tests for unit root and other nonstationarity of Robinson (1994a) are applied to an extended version of the data set used by Nelson and Plosser (1982). Unusually, the tests are efficient (against appropriate parametric alternatives), the null can be any member of the I(d) class, and the null limit distribution is chi-squared. The conclusions vary substantially across the 14 series, and across different models for the disturbances (which, also unusually, include the Bloomfield spectral model). Overall, the consumer price index and money stock seem the most nonstationary, while industrial production and unemployment rate seem the closest to stationarity.