Tomás del Barrio Castro

REGRESSION-BASED SEASONAL UNIT ROOT TESTS

The contribution of this paper is three-fold. Firstly, a characterisation theorem of the sub-hypotheses comprising the seasonal unit root hypothesis is presented which provides a precise formulation of the alternative hypotheses against which regression-based seasonal unit root tests test. Secondly, it proposes regressionbased tests for the seasonal unit root hypothesis which allow a general seasonal aspect for the data and are similar both exactly and asymptotically with respect to initial values and seasonal drift parameters. Thirdly, limiting distribution theory is given for these statistics where, in contrast to previous papers in the literature, in doing so it is not assumed that unit roots hold at all of the zero and seasonal frequencies. This is shown to alter the large sample null distribution theory for regression t-statistics for unit roots at the complex frequencies, but interestingly to not affect the limiting null distributions of the regression t-statistics for unit roots at the zero and Nyquist frequencies and regression Fstatistics for unit roots at the complex frequencies. Our results therefore have important implications for how tests of the seasonal unit root hypothesis should be conducted in practice. Associated simulation evidence on the size and power properties of the statistics presented in this paper is given which is consonant with the predictions from the large sample theory.

The Performance of Lag Selection and Detrending Methods for HEGY Seasonal Unit Root Tests

This paper analyzes two key issues for the empirical implementation of parametric seasonal unit root tests, namely generalized least squares (GLS) versus ordinary least squares (OLS) detrending and the selection of the lag augmentation polynomial. Through an extensive Monte Carlo analysis, the performance of a battery of lag selection techniques is analyzed, including a new extension of modified information criteria for the seasonal unit root context. All procedures are applied for both OLS and GLS detrending for a range of data generating processes, also including an examination of hybrid OLS-GLS detrending in conjunction with (seasonal) modified AIC lag selection. An application to quarterly U.S. industrial production indices illustrates the practical implications of choices made.

On Augmented HEGY Tests for Seasonal Unit Roots

In this paper we extend the large-sample results provided for the augmented Dickey–Fuller test by Said and Dickey ( 1984 , Biometrika 71, 599–607) and Chang and Park ( 2002 , Econometric Reviews 21, 431–447) to the case of the augmented seasonal unit root tests of Hylleberg, Engle, Granger, and Yoo ( 1990 , Journal of Econometrics 44, 215–238), inter alia. Our analysis is performed under the same conditions on the innovations as in Chang and Park ( 2002 ), thereby allowing for general linear processes driven by (possibly conditionally heteroskedastic) martingale difference innovations. We show that the limiting null distributions of the t -statistics for unit roots at the zero and Nyquist frequencies and joint F -type statistics are pivotal, whereas those of the t -statistics at the harmonic seasonal frequencies depend on nuisance parameters that derive from the lag parameters characterizing the linear process. Moreover, the rates on the lag truncation required for these results to hold are shown to coincide with the corresponding rates given in Chang and Park ( 2002 ); in particular, an o ( T 1/2 ) rate is shown to be sufficient.

Testing For Seasonal Unit Roots In Periodic Integrated Autoregressive Processes

This paper examines the implications of applying the Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238) (HEGY) seasonal root tests to a process that is periodically integrated. As an important special case, the random walk process is also considered, where the zero-frequency unit root t -statistic is shown to converge to the Dickey–Fuller distribution and all seasonal unit root statistics diverge. For periodically integrated processes and a sufficiently high order of augmentation, the HEGY t -statistics for unit roots at the zero and semiannual frequencies both converge to the same Dickey–Fuller distribution. Further, the HEGY joint test statistic for a unit root at the annual frequency and all joint test statistics across frequencies converge to the square of this distribution. Results are also derived for a fixed order of augmentation. Finite-sample Monte Carlo results indicate that, in practice, the zero-frequency HEGY statistic (with augmentation) captures the single unit root of the periodic integrated process, but there may be a high probability of incorrectly concluding that the process is seasonally integrated.

Using the HEGY Procedure When Not All Roots Are Present

Empirical studies have shown little evidence to support the presence of all unit roots present in the filter in quarterly seasonal time series. This paper analyses the performance of the Hylleberg, Engle, Granger and Yoo (1990) (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 4 (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 / 2 and two combinations of the previous cases. We show both theoretically and through a Monte-Carlo analysis that the t-ratios and 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 1 t 2 t / 2 .

HEGY Tests in the Presence of Moving Averages

We analyze the asymptotic distributions associated with the seasonal unit root tests of the Hylleberg et al. (1990) procedure for quarterly data when the innovations follow a moving average process. Although both the t- and F-type tests suffer from scale and shift effects compared with the presumed null distributions when a fixed order of autoregressive augmentation is applied, these effects disappear when the order of augmentation is sufficiently large. However, as found by Burridge and Taylor (2001) for the autoregressive case, individual t-ratio tests at the the semi-annual frequency are not pivotal even with high orders of augmentation, although the corresponding joint F-type statistic is pivotal. Monte Carlo simulations verify the importance of the order of augmentation for finite samples generated by seasonally integrated moving average processes.

Cointegration for Periodically Integrated Processes

Integration for seasonal time series can take the form of seasonal periodic or nonperiodic integration. When seasonal time series are periodically integrated, we show that any cointegration is either full periodic cointegration or full nonperiodic cointegration, with no possibility of cointegration applying for only some seasons. In contrast, seasonally integrated series can be seasonally, periodically or nonperiodically cointegrated, with the possibility of cointegration applying for a subset of seasons. Cointegration tests are analyzed for periodically integrated series. A residual-based test is examined, and its asymptotic distribution is derived under the null hypothesis of no cointegration. A Monte Carlo analysis shows good performance in terms of size and power. The role of deterministic terms in the cointegrating test regression is also investigated. Further, we show that the asymptotic distribution of the error-correction test for periodic cointegration derived by Boswijk and Franses (1995, Review of Economics and Statistics 77, 436–454) does not apply for periodically integrated processes.The authors gratefully acknowledge the comments of participants at the conference on Unit Root and Cointegration Testing, University of the Algave, September–October 2005, and they particularly thank two anonymous referees and Helmut LA¼tkepohl (co-editor of this issue of Econometric Theory) for their constructive comments, which have substantially improved the generality of the results in the paper. TomAis del Barrio Castro acknowledges financial support from Ministerio de EducaciA³n y Ciencia SEJ2005-07781/ECON.

The Impact of Persistent Cycles on Zero Frequency Unit Root Tests

In this paper we investigate the impact of non-stationary cycles on the asymptotic and finite sample properties of standard unit root tests. Results are presented for the augmented Dickey-Fuller normalised bias and t-ratio-based tests (Dickey and Fuller, 1979, and Said and Dickey, 1984), the variance ratio unit root test of Breitung (2002) and the M class of unit-root tests introduced by Stock (1999) and Perron and Ng (1996). The limiting distributions of these statistics are derived in the presence of non-stationary cycles. We show that while the ADF statistics remain pivotal (provided the test regression is properly augmented), this is not the case for the other statistics considered and show numerically that the size properties of the tests based on these statistics are too unreliable to be used in practice. We also show that the t-ratios associated with lags of the dependent variable of order greater than two in the ADF regression are asymptotically normally distributed. This is an important result as it implies that extant sequential methods (see Hall, 1994 and Ng and Perron, 1995) used to determine the order of augmentation in the ADF regression remain valid in the presence of non-stationary cycles.

Nonparametric Tests for Periodic Integration

We propose two nonparametric methods to test the null hypothesis of periodic integration, one based on the variance ratio unit root test of Breitung (2002) and the other on the modified Sargan-Bhargava test developed by Stock (1999). The former does not require specification of short-run dynamics, while nevertheless delivering a pivotal limiting distribution; however, the latter requires estimation of the long-run variance. The asymptotic distributions of the new statistics are shown to be invariant to whether the process is periodically integrated or a conventional I(1) process, whereas tests based on the I(1) assumption are not. Further, the new tests do not require nonlinear estimation and can be implemented in a straightforward way. Monte Carlo results show that the variance ratio test has very good finite sample size, but the modified Sargan-Bhargava test has much better power, which can be comparable to that of the parametric likelihood ratio test. Finally, an empirical application illustrates the use of the tests through an analysis of six monthly component Industrial Production Indices for the U.S.

On Augmented Franses Tests for Seasonal Unit Roots

This article extends the results reported in del Barrio Castro, Osborn and Taylor (2012) to the approach followed by Franses (1991a,b) to test for seasonal unit roots, providing the asymptotic representation to the seasonal unit roots tests proposed by Franses for a general number of seasons S.