Structural changes and unit roots in non-stationary time series

The problem of testing hypotheses of a unit root and a structural change in one-dimensional time series is considered. A non-parametric two-step method for solution of the problem is proposed. The method is based upon the modified Kolmogorov-Smirnov statistic. At the first step of this method the hypothesis of stationarity of an obtained sample is tested against a unified alternative of a statistical non-stationarity of a time series (a unit root or a structural change). At the second step of the proposed method, in case of rejecting the stationarity hypothesis at the first step, the hypothesis of an unknown structural change is tested against the alternative of a unit root. We prove that probabilities of errors (false classification of hypotheses) of the proposed method converge to zero as the sample size tends to infinity.     

On the interactions of unit roots and exogeneity

The paper considers the impact on estimation and inference of interactions between the existence of unit roots in a data generation process and the presence or absence of weak and strong exogeneity of conditioning variables for the parameters of interest in individual cointegrated linear relationships. The asymptotic distributions of estimators for single equation conditional linear relations are analyzed in conjunction with a Monte Carlo study. The results confirm the important role of weak exogeneity in single equation estimation from integratedcointegrated data; highlight the advantages of using an asymptotic analysis to understand the complicated interactions observed; and reveal the accuracy of the limiting distributions in characterizing finite sample behaviour.     

On the interactions of unit roots and exogeneity

The paper considers the impact on estimation and inference of interactions between the existence of unit roots in a data generation process and the presence or absence of weak and strong exogeneity of conditioning variables for the parameters of interest in individual cointegrated linear relationships. The asymptotic distributions of estimators for single equation conditional linear relations are analyzed in conjunction with a Monte Carlo study. The results confirm the important role of weak exogeneity in single equation estimation from integratedcointegrated data; highlight the advantages of using an asymptotic analysis to understand the complicated interactions observed; and reveal the accuracy of the limiting distributions in characterizing finite sample behaviour.     

Parameter inference for time series with regular and seasonal unit roots

It is common to have both regular and seasonal roots present in many time series data. It may occur that one or both of the roots are just close but not equal to unity. Parameter inference for this situation is considered both when the time series has a finite or an infinite variance. Asymptotic char-acterizations of the test statistics were obtained via functionals of Ornstein-Uhlenbeck processes and Lévy processes. Tabulations for the large sample distributions are obtained. The results will be useful in applications deciding whether both regular and seasonal differencing are needed in fitting a time series model.     

Detecting seasonal unit roots in a structural time series model

In this paper, we propose to detect seasonal unit roots within the context of a structural time series model. Such a model is often found to be useful in practice. Using Monte Carlo simulations, we show that our method works well. We illustrate our approach for several quarterly macroeconomic time series variables.     

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.     

Structural change tests for GEL criteria

This article examines structural change tests based on generalized empirical likelihood methods in the time series context, allowing for dependent data. Standard structural change tests for the Generalized method of moments (GMM) are adapted to the generalized empirical likelihood (GEL) context. We show that when moment conditions are properly smoothed, these test statistics converge to the same asymptotic distribution as in the GMM, in cases with known and unknown breakpoints. New test statistics specific to GEL methods, and that are robust to weak identification, are also introduced. A simulation study examines the small sample properties of the tests and reveals that GEL-based robust tests performed well, both in terms of the presence and location of a structural change and in terms of the nature of identification.     

Testing for unit roots in panel data using a GMM approach

For aggregated time series unit root tests are routinely applied to choose among trend and difference stationary models. Recent work demonstrates that such test can also be applied for testing panel data. However, it is well known that disaggregated data often exhibit a considerable amount of heterogeneity so that standard tests may perform poorly. To account for the heterogeneity in the data we allow for individual specific deterministics, that is, we let the time trends vary across the cross section units. It is shown that standard GMM estimators suggested for the dynamic panel data model may fail to give a valid test procedure. To overcome this difficulty, a modified GMM estimator is suggested. In a Monte Carlo study the finite sample properties of the alternative tests are compared.     

The impact of fat-tailed distributions on some leading unit roots tests

There is substantial evidence that many time series associated with financial and insurance claim data are fat-tailed, with a (much) higher probability of " outliers' compared with the normal distribution. However, standard tests, or variants of them, for the presence of unit roots assume a normal distribution for the innovations driving the series. Application of the former to the latter therefore involves an inconsistency. We assess the impact of this inconsistency and provide information on its impact on inference when innovations are drawn from the Cauchy and sequence of t(v) distributions. A simple prediction that fat tails will uniformly lead to over-sizing of standard tests (because the fatness in the tail translates to the test distribution) turns out to be incorrect: we find that some tests are over-sized but some are under-sized. We also consider size retention and the power of the Dickey-Fuller pivotal and normalized bias test statistics and weighted symmetric versions of these tests. To make the unit root testing procedure feasible, we develop an entropy-based test for some fat-tailed distributions and apply it to share prices from the FTSE100.     

The size of the nonstationary component and its effect on tests for unit roots

We consider a nonstationary time series that is composed of a stationary and nonstationary component. Monte Carlo experiments show that common unit root tests have probabilities of committing a type I error that significantly exceed the level of significance. We find that the probabilities vary according to the relative size of the nonstationary component.     

A sequential approach to testing seasonal unit roots in high frequency data

In this paper we introduce a sequential seasonal unit root testing approach which explicitly addresses its application to high frequency data. The main idea is to see which unit roots at higher frequency data can also be found in temporally aggregated data. We illustrate our procedure to the analysis of monthly data, and we find, upon analysing the aggregated quarterly data, that a smaller amount of test statistics can sometimes be considered. Monte Carlo simulation and empirical illustrations emphasize the practical relevance of our method.     

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.     

Panel data unit roots tests: The role of serial correlation and the time dimension

We investigate the influence of residual serial correlation and of the time dimension on statistical inference for a unit root in dynamic longitudinal data, known as panel data in econometrics. To this end, we introduce two test statistics based on method of moments estimators. The first is based on the generalized method of moments estimators, while the second is based on the instrumental variables estimator. Analytical results for the Instrumental Variables (IV) based test in a simplified setting show that (i) large time dimension panel unit root tests will suffer from serious size distortions in finite samples, even for samples that would normally be considered large in practice, and (ii) negative serial correlation in the error terms of the panel reduces the power of the unit root tests, possibly up to a point where the test becomes biased. However, near the unit root the test is shown to have power against a wide range of alternatives. These findings are confirmed in a more general set-up through a series of Monte Carlo experiments.     

More powerful modifications of unit root tests allowing structural change

It is well known that more powerful variants of Dickey–Fuller unit root tests are available. We apply two of these modifications, on the basis of simple maximum statistics and weighted symmetric estimation, to Perron tests allowing for structural change in trend of the additive outlier type. Local alternative asymptotic distributions of the modified test statistics are derived, and it is shown that their implementation can lead to appreciable finite sample and asymptotic gains in power over the standard tests. Also, these gains are largely comparable with those from GLS-based modifications to Perron tests, though some interesting differences do arise. This is the case for both exogenously and endogenously chosen break dates. For the latter choice, the new tests are applied to the Nelson–Plosser data.     

An LM-type test for idiosyncratic unit roots in the exact factor model with integrated factors

We consider an exact factor model with integrated factors and propose an LM-type test for unit roots in the idiosyncratic component. We show that, for a fixed number of panel individuals (N) and when the number of time points (T) tends to infinity, the limiting distribution of the LM-type statistic is a weighted sum of independent Chi-square variables with one degree of freedom, and when T tends to infinity followed by N tending to infinity, the limiting distribution is standard normal. The results should contribute to the challenging task of deriving likelihood-based unit-root tests in dynamic factor models.     

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.     

Multiple roots of estimating functions

Estimating functions can have multiple roots. In such cases, the statistician must choose among the roots to estimate the parameter. Standard asymptotic theory shows that in a wide variety of cases, there exists a unique consistent root, and that this root will lie asymptotically close to other consistent (possibly inefficient) estimators for the parameter. For this reason, attention has largely focused on the problem of selecting this root and determining its approximate asymptotic distribution. In this paper, however, we concentrate on the exact distribution of the roots as a random set. In particular, we propose the use of higher-order root intensity functions as a tool for examining the properties of the roots and determining their most problematic features. The use of root intensity functions of first and second order is illustrated by application to the score function for the Cauchy location model.     

Distribution of angles between unit vectors and the multiple comparison problem for unit vectors

We suppose that L x i = Fq(μi, KiK) for i=1,...,p, where the independent Fisher distributions on the unit sphere in Rq have modal vectors μi and known concentrations KiK, where K will be large—in fact, to obtain asymptotic distributions, we will let K tend to infinity. The term x'iXj=cos 0ij estimates cos 0ij= μiμy. The asymptotic joint distribution of the terms will be studied when all the vectors μi μ are distinct, and for the special case when all the vectors μi=μ, so that all the terms are zero. These very different results have a variety of applications as well as being interesting in themselves. One of these applications is the 'multiple comparisons problem' for unit vectors. However, it was found necessary to give here a different way of solving this problem.     

Bounded integrated processes and unit root tests

In the framework of integrated processes, the problem of testing the presence of unknown boundaries which constrain the process to move within a closed interval is considered. To analyze this problem, the concept of bounded integrated process is introduced, thus allowing to formally define boundary conditions for I(1) processes. A new class of tests, which are based on the rescaled range of the process, is introduced in order to test the null hypothesis of no boundary conditions. The limit distribution of the test statistics involved can be expressed in terms of the distribution of the range of Brownian functionals, while the power properties are obtained by deriving some asymptotic results for I(1) processes with boundary conditions. Both theoretical and simulation investigations show that range-based tests outperform standard unit root tests significantly when used to detect the presence of boundary conditions. A previous draft of the paper (Cavaliere, 2000) was presented at the 8th World Congress of the Econometric Society, Seattle, 11–16 August 2000. I wish sincerely to thank: Martin Jacobsen for his patience in discussing weak convergence to regulated Brownian motions and his valuable suggestions; the Department of Theoretical Statistics of the University of Copenhagen whose hospitality is gratefully acknowledged; Tommaso Proietti for important suggestions; Silvano Bordignon and partecipants at the CIdE seminar, University of Padua, June 2000; two anonymous referees. Partial financial support from 60% M.U.R.S.T. research grants is acknowledged.