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.     

Limiting distributions of unconditional maximum likelihood unit root test statistics in seasonal time–series models

Abstract.  The likelihood function of a seasonal model, Y _t  =  ρ Y _{t − d}  +  e _t as implemented in computer algorithms under the assumption of stationary initial conditions is a function of ρ which is zero at the point ρ  = 1. It is a smooth function for ρ in the above seasonal model with a well-defined maximum regardless of the data-generating mechanism. Gonzalez-Farias (PhD Thesis, North Carolina State University, 1992) proposed tests for unit roots based on maximizing the stationary likelihood function in nonseasonal time series. We extend it to seasonal time series. The limiting distribution of seasonal unit root test statistics based on the unconditional maximum likelihood estimators are shown. Models having a single mean, seasonal means, and a single-trend variable across the seasons are considered.     

Likelihood Ratio Tests for Seasonal Unit Roots

This paper proposes regression-based likelihood ratio or F tests for the seasonal unit root hypothesis which fully incorporate the implicit restrictions on the parameters associated with the deterministics. These statistics are similar both exactly and asymptotically with respect to initial values and seasonal drift parameters. The limiting representations of the statistics are presented for a general seasonal aspect. These limiting representations allow those for other scenarios concerning the deterministics to be simply obtained and provide an explanation for the similarity between critical values in apparently quite different cases of interest. We re-examine the seasonal unit root properties of the logarithm of monthly seasonally unadjusted real industrial production in Canada.     

Dickey–Fuller, Lagrange Multiplier and Combined Tests for a Unit Root in Autoregressive Time Series

In this paper we investigate (augmented) Dickey–Fuller (DF) and Lagrange multiplier (LM) type unit root tests for autoregressive time series through comprehensive Monte Carlo simulations. We consider two sorts of null and alternative hypotheses: a unit root without drift versus level stationarity and a unit root with drift versus trend stationarity. The DF-type coef ficient tests are found to show the best overall performance in both cases, at least if the sample size is sufficiently large. How ever, it is also found that the DF and LM tests are roughly complementary with regard to their finite-sample power. We therefore consider combining these two types of unit root tests to obtain ( ad hoc 'but') 'robust' test procedures. Critical values for the proposed tests are provided     

Locally Optimal Tests Against Unit Roots in Seasonal Time Series Processes

This paper builds on the existing literature on tests of the null hypothesis of deterministic seasonality in a univariate time-series process. Under the assumption of independent Gaussian errors, we derive the class of locally weighted mean most powerful invariant tests against unit roots at the zero and/or seasonal frequencies in a seasonally observed process. Representations for the limiting distributions of the proposed test statistics under sequences of local alternatives are derived, and the relationship with tests for corresponding moving average unit roots is explored. We also propose nonparametric modifications of these test statistics designed to have limit distributions which are free of nuisance parameters under weaker conditions on the errors. Our tests are shown to contain existing stationarity tests as special cases and to extend these tests in a number of useful directions.     

Approximate Conditional Unit Root Inference

Based on Cox and Reid (1987) adjustments of likelihood ratio (LR) tests for unit roots in higher-order autoregressive models are proposed. While unit root inference does not fit directly into the framework of Cox and Reid, the ideas are applied in models with multi-dimensional parameters of interest and only asymptotic orthogonality of parameters. The adjustments are very simple to apply in that they are of the degrees of freedom type. Detailed Monte Carlo experiments reveal that, for a wide range of admissible parameter values, adjusted LR statistics approximate the asymptotic percentiles of the unit root distributions at a much faster rate than unadjusted ones. In addition, the proposed adjustments are compared with simulated Bartlett type corrected LR tests. They behave equally well in a reasonable parameter region, while both fail on the boundary of the parameter region where an additional unit root is introduced.     

Range Unit‐Root (RUR) Tests: Robust against Nonlinearities,Error Distributions,Structural Breaks and Outliers

Abstract. Since the seminal paper by Dickey and Fuller in 1979, unit‐root tests have conditioned the standard approaches to analysing time series with strong serial dependence in mean behaviour, the focus being placed on the detection of eventual unit roots in an autoregressive model fitted to the series. In this paper, we propose a completely different method to test for the type of long‐wave patterns observed not only in unit‐root time series but also in series following more complex data‐generating mechanisms. To this end, our testing device analyses the unit‐root persistence exhibited by the data while imposing very few constraints on the generating mechanism. We call our device the range unit‐root (RUR) test since it is constructed from the running ranges of the series from which we derive its limit distribution. These nonparametric statistics endow the test with a number of desirable properties, the invariance to monotonic transformations of the series and the robustness to the presence of important parameter shifts. Moreover, the RUR test outperforms the power of standard unit‐root tests on near‐unit‐root stationary time series; it is invariant with respect to the innovations distribution and asymptotically immune to noise. An extension of the RUR test, called the forward–backward range unit‐root (FB‐RUR) improves the check in the presence of additive outliers. Finally, we illustrate the performances of both range tests and their discrepancies with the Dickey–Fuller unit‐root test on exchange rate series.     

Testing for a Unit Root in Autoregressive Moving-average Models with Missing Data

Testing for a single autoregressive unit root in an autoregressive moving-average (ARMA) model is considered in the case when data contain missing values. The proposed test statistics are based on an ordinary least squares type estimator of the unit root parameter which is a simple approximation of the one-step Newton–Raphson estimator. The limiting distributions of the test statistics are the same as those of the regression statistics in AR(1) models tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Stat. Assoc . 74 (1979), 427–31) for the complete data situation. The tests accommodate models with a fitted intercept and a fitted time trend.     

SOME SIMPLE TESTS OF THE MOVING-AVERAGE UNIT ROOT HYPOTHESIS

Abstract. This paper deals with three test statistics for a moving-average (MA) unit root. The spectral test is based on the estimate of the spectral density at frequency zero. The variance difference statistic compares the sample variance of the integrated series with the estimated variance imposing the MA unit root constraint. Furthermore, Tanaka's score type test statistic is modified to improve the power in higher order models. The asymptotic power of the tests is considered and Monte Carlo experiments are performed to investigate the small sample properties of the tests. Finally, the tests are applied to a number of economic time series to determine the degree of integration.     

A HYBRID BOOTSTRAP APPROACH TO UNIT ROOT TESTS

This article proposes a hybrid bootstrap approach to approximate the augmented Dickey–Fuller test by perturbing both the residual sequence and the minimand of the objective function. Since innovations can be dependent, this allows the inclusion of conditional heteroscedasticity models. The new bootstrap method is also applied to least absolute deviation‐based unit root test statistics, which are efficient in handling heavy‐tailed time‐series data. The asymptotic distributions of resulting bootstrap tests are presented, and Monte Carlo studies demonstrate the usefulness of the proposed tests.     

Testing the Null Hypothesis of Stationarity Against an Autoregressive Unit Root Alternative

We propose a new test for the null hypothesis that a time series is stationary around a deterministic trend. The test is valid under general conditions on stationarity. Asymptotic distributions of the test statistic are derived under both the null and the alternative hypothesis of a unit root. It is shown that the limiting distribution has the classical Kolmogoroff– Smirnoff form. Critical values for the null distribution are calculated. Consistency of the tests is proved. The tests provide a useful complement to the conventional unit root tests.     

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.     

SEARCHING FOR ADDITIVE OUTLIERS IN NONSTATIONARY TIME SERIES*

Abstract. Recently, Vogelsang (1999) proposed a method to detect outliers which explicitly imposes the null hypothesis of a unit root. It works in an iterative fashion to select multiple outlier in a given series. We show, via simulations, that, under the null hypothesis of no outliers, it has the right size in finite samples to detect a single outlier but, when applied in an iterative fashion to select multiple outliers, it exhibits severe size distortions towards finding an excessive number of outliers. We show that his iterative method is incorrect and derive the appropriate limiting distribution of the test at each step of the search. Whether corrected or not, we also show that the outliers need to be very large for the method to have any decent power. We propose an alternative method based on first‐differenced data that has considerably more power. We also show that our method to identify outliers leads to unit root tests with more accurate finite sample size and robustness to departures from a unit root. The issues are illustrated using two US/Finland real‐exchange rate series.     

On the Robustness of Unit Root Tests in the Presence of Double Unit Roots

We examine some of the consequences on commonly used unit root tests when the underlying series is integrated of order two rather than of order one. It turns out that standard augmented Dickey–Fuller type of tests for a single unit root have excessive density in the explosive region of the distribution. The lower (stationary) tail, however, will be virtually unaffected in the presence of double unit roots. On the other hand, the Phillips–Perron class of semi-parametric tests is shown to diverge to plus infinity asymptotically and thus favouring the explosive alternative. Numerical simulations are used to demonstrate the analytical results and some of the implications in finite samples.     

Temporal Aggregation of Seasonally Near‐Integrated Processes

We investigate the implications that temporally aggregating, either by average sampling or systematic (skip) sampling, a seasonal process has on the integration properties of the resulting series at both the zero and seasonal frequencies. Our results extend the existing literature in three ways. First, they demonstrate the implications of temporal aggregation for a general seasonally integrated process with S seasons. Second, rather than only considering the aggregation of seasonal processes with exact unit roots at some or all of the zero and seasonal frequencies, we consider the case where these roots are local‐to‐unity such that the original series is near‐integrated at some or all of the zero and seasonal frequencies. These results show, among other things, that systematic sampling, although not average sampling, can impact on the non‐seasonal unit root properties of the data; for example, even where an exact zero frequency unit root holds in the original data it need not necessarily hold in the systematically sampled data. Moreover, the systematically sampled data could be near‐integrated at the zero frequency even where the original data is not. Third, the implications of aggregation on the deterministic kernel of the series are explored.‐142     

Two Simple Procedures for Testing for a Unit Root When There are Additive Outliers

This paper presents some results on testing for a unit root in the presence of additive outliers. Two procedures are proposed. The first procedure is to ignore the possibility of additive outliers and use modified Phillips–Perron statistics. The second procedure uses a new and very simple outlier detection statistic to identify outliers and then properly adjust standard Dickey–Fuller unit root tests. Simulations show that these procedures are robust to additive outliers in terms of size and power.     

Testing for a unit root under errors with just barely infinite variance

Abstract. This article investigates the problem of testing for a unit root in the case that the error, {u_t}, of the model is a strictly stationary, mixing process with just barely infinite variance. Such errors have the property that for every δ such that 0 ≤ δ < 2, the moments E|u_t|^δ are finite. Under some additional restrictions on the rate of decay of the mixing rates, these errors belong to the domain of the non‐normal attraction of the normal law and obey the invariance principle. This in turn implies that there might be conditions under which the usual Phillips‐type test statistics for unit roots may still converge to the corresponding Dickey–Fuller distributions. In such a case, the unit‐root hypothesis can be tested within an infinite‐variance framework without any modifications to either the tests themselves or the critical values employed. This article derives a necessary and sufficient condition for convergence of the standard test statistics to the Dickey–Fuller distributions. By means of Monte Carlo simulations, the article also shows that this condition is likely to hold in the case that {u_t} is a serially correlated, integrated generalized autoregressive conditionally heteroskedastic (IGARCH) process and the standard unit‐root tests work well.     

TESTING THE ORDER OF DIFFERENCING IN TIME SERIES REGRESSION

Abstract. In this paper we develop a test procedure for detecting overdifferencing or a moving-average unit root in time series regression models with Gaussian autoregressive moving-average errors. In addition to an intercept term the regressors consist of stable or asymptotically stationary variables and non-stationary trending variables generated by an integrated process of order 1. The test of the paper is based on the theory of locally best invariant unbiased tests. Its limiting distribution is derived under the null hypothesis and found to be non-standard but free of unknown nuisance parameters. Asymptotic critical values, which depend on the number of integrated regressors, are obtained by simulation. A limited simulation study is carried out to illustrate the finite sample properties of the test.     

Testing unit roots and long range dependence of foreign exchange

Foreign exchange rate plays an important role in international finance. This article examines unit roots and the long range dependence of 23 foreign exchange rates using Robinson's (1994) test, which is one of the most efficient tests when testing fractional orders of seasonal/cyclical long memory processes. Monte Carlo simulations are carried out to explore the accuracy of the test before implementing the empirical applications.     

Testing Monthly Seasonal Unit Roots With Monthly and Quarterly Information

Abstract. This paper proposes a method for testing seasonal unit roots that combines monthly and quarterly Hylleberg, Engle, Granger and Yoo (HEGY) tests. The new approach is more powerful than the method that does not use quarterly information, i.e. the monthly HEGY test. An empirical illustration of the proposed approach is given for monthly US Industrial Production.