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.     

Joint Hypothesis Tests for a Unit Root When There is a Break in the Innovation Variance

Abstract. We develop extensions of the Dickey–Fuller F‐statistics for the joint null hypothesis of a unit root that allows for a break in the innovation variance. Our statistics are based on the modified generalized least squares (GLS) strategy outlined in Kim, Leybourne and Newbold [Journal of Econometrics (2002) Vol. 109, pp. 365–387] that requires estimation of the break‐date and corresponding pre‐break and post‐break variances. We derive the asymptotic distribution of the new F‐statistics, tabulate their finite sample and asymptotic critical values, and present finite sample simulation evidence regarding their size and power.     

The power of unit root tests against nonlinear local alternatives

This article extends the analysis of local power of unit root tests in a nonlinear direction by considering local nonlinear alternatives and tests built specifically against stationary nonlinear models. In particular, we focus on the popular test proposed by Kapetanios et al. (2003, Journal of Econometrics 112, 359–379) in comparison to the linear Dickey–Fuller test. To this end, we consider different adjustment schemes for deterministic terms. We provide asymptotic results which imply that the error variance has a severe impact on the behaviour of the tests in the nonlinear case; the reason for such behaviour is the interplay of non‐stationarity and nonlinearity. In particular, we show that nonlinearity of the data generating process can be asymptotically negligible when the error variance is moderate or large (compared to the ‘amount of nonlinearity’), rendering the linear test more powerful than the nonlinear one. Should however the error variance be small, the nonlinear test has better power against local alternatives. We illustrate this in an asymptotic framework of what we call persistent nonlinearity. The theoretical findings of this article explain previous results in the literature obtained by simulation. Furthermore, our own simulation results suggest that the user‐specified adjustment scheme for deterministic components (e.g. OLS, GLS, or recursive adjustment) has a much higher impact on the power of unit root tests than accounting for nonlinearity, at least under local (linear or nonlinear) alternatives.     

Testing for a Unit Root in Noncausal Autoregressive Models

This work develops maximum likelihood‐based unit root tests in the noncausal autoregressive (NCAR) model with a non‐Gaussian error term formulated by Lanne and Saikkonen (2011, Journal of Time Series Econometrics 3, Issue 3, Article 2). Finite‐sample properties of the tests are examined via Monte Carlo simulations. The results show that the size properties of the tests are satisfactory and that clear power gains against stationary NCAR alternatives can be achieved in comparison with available alternative tests. In an empirical application to a Finnish interest rate series, evidence in favour of an NCAR model with leptokurtic errors is found.     

Testing for EGARCH Against Stochastic Volatility Models

Abstract. It is shown that the EGARCH model is the degenerate case of Danielsson's [Journal of Econometrics (1994) Vol. 61, pp. 375–400] stochastic volatility model where the disturbance of the transition equation of conditional volatility has zero variance. The Lagrange multiplier test statistic is obtained for the EGARCH model against the stochastic volatility model by expressing the degenerate density under the null hypothesis by the Dirac delta function. The finite sample performance of the test is studied in a small Monte Carlo experiment.     

A Modified Nonparametric Prewhitened Covariance Estimator

Abstract. This paper proposes a fully modified version of the spectral matrix estimator (and the long‐run variance estimator as a special case) proposed originally by Xiao and Linton [Journal of Time Series Analysis (2002) Vol. 23, pp. 215–250], and derives its asymptotic results. A striking feature of the modified spectral matrix estimator is to achieve the convergence rate of O(T ^?8/9) in the mean squared error (MSE), which is usually achieved under the fourth‐order spectral window. However, this estimator does not sacrifice the positive definiteness of the resulting estimate for the rate improvement; it is Hermitian and positive definite in finite samples by construction. The faster convergence rate is established by a multiplicative bias correction of the crude spectral estimator under the second‐order spectral window. The approximations to some sensible definitions of the MSE of the estimator and the bandwidths that minimize the asymptotic MSEs are also derived. Monte Carlo results indicate that for a wide variety of processes the modified spectral matrix estimator reduces the bias without inflating the variance and thus improves the MSE, compared with the crude, bias‐uncorrected estimator.     

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     

A FIXED‐ b TEST FOR A BREAK IN LEVEL AT AN UNKNOWN TIME UNDER FRACTIONAL INTEGRATION

In this paper, we propose a test for a break in the level of a fractionally integrated process when the timing of the putative break is not known. This testing problem has received considerable attention in the literature in the case where the time series is weakly autocorrelated. Less attention has been given to the case where the underlying time series is allowed to be fractionally integrated. Here, valid testing can only be performed if the limiting null distribution of the level break test statistic is well defined for all values of the fractional integration exponent considered. However, conventional sup‐Wald type tests diverge when the data are strongly autocorrelated. We show that a sup‐Wald statistic, which is standardized using a non‐parametric kernel‐based long‐run variance estimator, does possess a well‐defined limit distribution, depending only on the fractional integration parameter, provided the recently developed fixed‐b asymptotic framework is applied. We give the appropriate asymptotic critical values for this sup‐Wald statistic and show that it has good finite sample size and power properties.     

Local Whittle estimation of multi‐variate fractionally integrated processes

This article derives a semi‐parametric estimator of multi‐variate fractionally integrated processes covering both stationary and non‐stationary values of d. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the multi‐variate local Whittle estimator of Shimotsu (2007) to cover non‐stationary values of d. Consistency and asymptotic normality is shown for d ∈ (?1/2,∞). A simulation study illustrates the performance of the proposed estimator for relevant sample sizes. Empirical justification of the proposed estimator is shown through an empirical analysis of log spot exchange rates. We find that the log spot exchange rates of Germany, United Kingdom, Japan, Canada, France, Italy and Switzerland against the US Dollar for the period January 1974 until December 2001 are well decribed as I(1) processes.     

A computationally convenient unit root test with covariates,conditional heteroskedasticity and efficient detrending

When testing for a unit root in a time series, in spite of the well‐known power problem of univariate tests, it is quite common to use only the information regarding the autoregressive behaviour contained in that series. In a series of influential papers, Elliott et al. (Efficient tests for an autoregressive unit root, Econometrica 64, 813–836, 1996), Hansen (Rethinking the univariate approach to unit root testing: using covariates to increase power, Econometric Theory 11, 1148–1171, 1995a) and Seo (Distribution theory for unit root tests with conditional heteroskedasticity, Journal of Econometrics 91, 113–144, 1999) showed that this practice can be rather costly and that the inclusion of the extraneous information contained in the near‐integratedness of many economic variables, their heteroskedasticity and their correlation with other covariates can lead to substantial power gains. In this article, we show how these information sets can be combined into a single unit root test.     

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.     

Assessing Persistence In Discrete Nonstationary Time‐Series Models

Abstract. The aim of this paper is to examine the application of measures of persistence in a range of time‐series models nested in the framework of Cramer (1961) . This framework is a generalization of the Wold (1938) decomposition for stationary time‐series which, in addition to accommodating the standard I(0) and I(1) models, caters for a broad range of alternative processes. Two measures of persistence are considered in some detail, namely the long‐run impulse‐response and variance‐ratio functions. Particular emphasis is given to the behaviour of these measures in a range of non‐stationary models specified in discrete time. We document the conflict that arises between different measures, applied to the same model, as well as conflict arising from the use of a given measure in different models. Precisely which persistence measures are time dependent and which are not, is highlighted. The nature of the general representation used also helps to clarify which shock the impulse‐response function refers to in the case of models where more than one random disturbance impinges on the time series.     

RESIDUAL AUTOCOVARIANCES AND UNIT ROOT TESTS BASED ON INSTRUMENTAL VARIABLE ESTIMATORS FROM TIME SERIES REGRESSION MODELS

Abstract. Hall (Testing for a unit root in the presence of moving average errors. Biometrika 76 (1989), 49–56; Joint hypothesis tests for a random walk based on instrumental variable estimators. J. Time Ser. Anal. 13 (1992), 29–45), Pantula and Hall (Testing for unit roots in autoregressive moving average models:an instrumental variable approach. J. Econometrics 48 (1991), 325–53) and Lee and Schmidt (Unit root tests based on instrumental variable estimation. Int. Econ. Rev. 39 (1994), 449–62) proposed instrumental variable (IV) based tests for a unit root in an ARMA(p+ 1, q) time series. To perform the tests it is essentially necessary to know (p, q) but in many cases this information is unknown. In practice a natural solution to this problem is to estimate (p, q) from the data using a strategy based on the residual autocovariances from the IV regression. In this paper we examine the properties of these residual autocovariances under various assumptions about the true nature of the time series. This analysis allows us to propose a model selection procedure which has desirable asymptotic and finite sample properties whether the time series is stationary or possesses a unit root. A sideproduct of our analysis is that we extend Box and Pierce's (Distribution of residual autocorrelations in autoregressive integrated moving average time series models. J. Am. Statist. Assoc. 65 (1970), 1509–26) analysis of the least squares residual autocorrelations to the residual autocovariances from IV regressions.     

CUSUM of Squares‐Based Tests for a Change in Persistence

Abstract. Using standardized cumulative sums of squared sub‐sample residuals, we propose a new ratio‐based test of the null hypothesis that a time series exhibits no change in its persistence structure [specifically that it displays constant I(1) behaviour] against the alternative of a change in persistence from trend stationarity to difference stationarity, or vice versa. Neither the direction nor location of any possible change under the alternative hypothesis need be assumed known. A key feature of our proposed test which distinguishes it from extant tests for persistence change [certain of which test the null hypothesis of constant I(0) behaviour while others, like our proposed test, test the null hypothesis of constant I(1) behaviour] is that it displays no tendency to spuriously over‐reject when applied to series which, although not constant I(1) series, do not display a change in persistence [specifically are constant I(0) processes]. Moreover, where our ratio test correctly rejects the null of no persistence change, the tail in which the rejection occurs can also be used to identify the direction of change since, even in relatively small samples, the test almost never rejects in the right [left] tail when there is a change from I(0) to I(1) [I(1) to I(0)]. Again this useful property is not shared by existing tests. As a by‐product of our analysis, we also propose breakpoint estimators which are consistent where the timing of the change in persistence is unknown.     

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.     

Recursive adjustment,unit root tests and structural breaks

In this paper, we introduce unit root tests for time series with a potential structural break computed from test regressions in which the deterministic components have been recursively adjusted. We present finite sample critical values as well as Monte Carlo results on the size and power performance of the new procedures, and compare these with other available tests in the literature, such as OLS and quasi‐differenced based tests (see, for instance, Perron, (1997) Perron and Rodriguez, (2003) and Carrion‐i‐Silvestre et al. (2009) ). The small sample behaviour of the tests is evaluated in a known and an unknown break date context allowing for negligible and non‐negligible initial conditions. In the unknown break date case, two break date estimation procedures are considered, one based on the minimum unit root t‐statistic and the other based on the minimum sum of squared residuals obtained from a regression on a set of deterministic variables. The size and power performance of the recursive adjustment based procedure in the unknown break date case is encouraging. A further result of this paper relates to the aditional finite sample evidence on the performance of quasi‐differenced unit root tests, complementing the results in Perron and Rodriguez (2003) .     

A Nonparametric Test for Weak Dependence Against Strong Cycles and its Bootstrap Analogue

Abstract. We examine a test for the hypothesis of weak dependence against strong cyclical components. We show that the limiting distribution of the test is a Gumbel distribution, denoted G(·). However, since G(·) may be a poor approximation to the finite sample distribution, being the rate of the convergence logarithmic [see Hall Journal of Applied Probability (1979) , Vol. 16, pp. 433–439], inferences based on G(·) may not be very reliable for moderate sample sizes. On the other hand, in a related context, Hall [Probability Theory and Related Fields (1991) , Vol. 89, pp. 447–455] showed that the level of accuracy of the bootstrap is significantly better. For that reason, we describe an approach to bootstrapping the test based on Efron's [Annals of Statistics (1979) , Vol. 7, pp. 1–26] resampling scheme of the data. We show that the bootstrap principle is consistent under very mild regularity conditions.     

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.     

Time‐Transformed Unit Root Tests for Models with Non‐Stationary Volatility

Abstract. Conventional unit root tests are known to be unreliable in the presence of permanent volatility shifts. In this paper, we propose a new approach to unit root testing which is valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. The new tests are based on a time transformation of the series of interest which automatically corrects their form for the presence of non‐stationary volatility without the need to specify any parametric model for the volatility process. Despite their generality, the new tests perform well even in small samples. We also propose a class of tests for the null hypothesis of stationary volatility in (near‐) integrated time‐series processes.     

Testing for nonlinear deterministic components when the order of integration is unknown

We consider testing for the presence of nonlinearities in the deterministic component of a time series, approximating the potential nonlinear behaviour using a Fourier function expansion. In contrast to procedures that are currently available, we develop tests that are robust to the order of integration, in the sense that they are asymptotically correctly sized regardless of whether the stochastic component of the series is stationary or contains a unit root. The tests we propose take the form of Wald statistics based on cumulated series, together with a correction factor to line up the asymptotic critical values across the I(0) and I(1) environments. The local asymptotic power and finite sample properties of the tests are evaluated using various different correction factors. We envisage that the testing procedure we recommend should be very useful to applied researchers wishing to draw robust inference regarding the presence of nonlinear deterministic components in a series.