Seasonal Unit Root Tests Research Articles

ASYMPTOTIC DISTRIBUTIONS OF SEASONAL UNIT ROOT TESTS: A UNIFYING APPROACH

ABSTRACT This paper adopts a unified approach to the derivation of the asymptotic distributions of various seasonal unit root tests. The procedures considered are those of Dickey et al. [DHF], Kunst, Hylleberg et al. [HEGY], Osborn et al. [OCSB], Ghysels et al. [GHL] and Franses. This unified approach shows that the asymptotic distributions of all these test statistics are functions of the same vector of Brownian motions. The Kunst test and the overall HEGY F-test are, indeed, equivalent both asymptotically and in finite samples, while the Franses and GHL tests are shown to have equivalent parameterizations. The OCSB and DHF test regressions are viewed as restricted forms of the Kunst-HEGY regressions, and these restrictions may have non-trivial asymptotic implications.

A Note on Seasonal Unit Root Tests

The seasonal unit root tests make it possible to determine the nature of the deterministic and stochastic seasonal fluctuations. In Section 2, we define the main seasonal time series models and the seasonal integration notion. Section 3 describes the HEGY test procedure.

Recursive and rolling regression-based tests of the seasonal unit root hypothesis

This paper is concerned with rolling and recursive regression-based implementations of tests for seasonal unit roots in a univariate time series process. These tests are based on changing subsamples of the data and thus allow one to test the conventional fixed seasonal unit root hypothesis against the alternative that the process under investigation admits a stable autoregressive root over part, if not all, of the sample at either the zero or seasonal frequencies. Asymptotic critical values are provided together with representations for the limiting distributions of these test statistics. A finite sample size and power study of the proposed test statistics is also reported together with a discussion on the problem of lag truncation selection in the context of rolling and recursive test regressions. An application of the proposed test statistics to seasonally unadjusted U.K. consumers’ expenditure on tobacco is considered.

APPROXIMATION TO THE LIMITING DISTRIBUTION OF t- AND F-STATISTICS IN TESTING FOR SEASONAL UNIT ROOTS

Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238), Beaulieu and Miron (1993, Journal of Econometrics 55, 305–328), Ghysels, Lee, and Noh (1994, Journal of Econometrics 62, 415–442), Smith and Taylor (1998, Journal of Econometrics 85, 269–288; 1999, Journal of Time Series Analysis 20, 453–476; 1999, Discussion paper 99-15 in economics, University of Birmingham), and Taylor (1998, Journal of Time Series Analysis 19, 349–368) have developed a method of testing for seasonal unit roots of zero and nonzero frequencies. They propose to use t- and F-statistics as criteria that are obtained from an auxiliary regression and find their limiting distributions as the number of observations becomes large. Their limiting distributions are expressed by means of Brownian motions. In this paper the moment generating functions associated with the limiting distributions are derived, and it is shown, as in Nabeya (2000, Econometric Theory 16, 200–230), that the limiting distribution of t is well approximated by a distribution given in Gram–Charlier series. The limiting distribution of F is also well approximated by another type of distribution.

On the Properties of Regression-Based Tests for Seasonal Unit Roots in the Presence of Higher-Order Serial Correlation

We analyze the behavior of widely used regression-based tests for seasonal unit roots when the shocks are serially correlated. We show, in the quarterly case, that the common assumption that serial correlation may be accommodated by augmenting the test regression with appropriate lagged seasonal differences is only partially correct. The limiting null distributions of t statistics for unit roots at the zero and Nyquist frequencies are corrected by the lag augmentation, but those of t statistics at the harmonic seasonal frequency are not. Fortunately, the joint F-type tests at the harmonic frequency, which are in widespread use, do remain pivotal and should therefore supplant the individual t statistics in applied work. That the latter are indeed badly behaved in finite samples, while the F-type tests are correctly sized, is demonstrated by a Monte Carlo experiment.

On regression-based tests for seasonal unit roots in the presence of periodic heteroscedasticity

We analyse regression-based tests for seasonal unit roots when the shocks are periodically heteroscedastic. We show, using the case of quarterly data to illustrate, that the limiting marginal null distributions of tests for unit roots at the zero and Nyquist frequencies are unaffected by the periodic heteroscedasticity. However, tests at the harmonic seasonal frequencies are shown to be either unaffected or to display a shift in their limiting distribution, depending on the specific nature of the periodic heteroscedasticity. In extreme cases certain of these limiting distributions are degenerate while others are simple functions of the well-known Dickey–Fuller distributions. Monte Carlo evidence shows that the asymptotic theory provides a very good prediction for the finite sample behaviour of the unit root test statistics. We show that tests with approximately correct size may be obtained by simulating their sampling distributions using periodic variance parameters estimated from the data in hand. Though laborious, this procedure seems to be the best available, since more conservative approaches sacrifice significant power.

NEAR SEASONAL INTEGRATION

This paper presents asymptotic results for the seasonal unit root test proposed by Hylleberg, Engle, Granger and Yoo (1990, Journal of Econometrics 44, 215–238) in a near integration context. The findings are important in that they provide the asymptotic power functions of the Hylleberg et al. statistics when the characteristic roots of a seasonal process are local to unity. These conclusions extend the available asymptotic results for this test and serve as a framework for the potential development of more powerful test procedures.

Measurement Errors and Outliers in Seasonal Unit Root Testing

Frequently, seasonal and non-seasonal data (especially macro time series) are observed with noise. For instance, the time series can have irregular abrupt changes and interruptions following as a result of additive or temporary change outliers caused by external circumstances which are irrelevant for the series of interest. Equally, the time series can have measurement errors. In this paper we analyse the above types of data irregularities on the behaviour of seasonal unit roots. It occurs that in most cases outliers and measurement errors can seriously affect inference towards the rejection of seasonal unit roots. It is shown how the distortion of the tests will depend upon the frequency, magnitude, and persistence of the outliers as well as on the signal to noise ratio associated with measurement errors. Some solutions to the implied inference problems are suggested.

Semiparametric tests for seasonal unit roots based on a semiparametric feasible GLSE

Semiparametric extensions of the seasonal unit root tests for the model of Dickey et al. (1984, J. Amer. Statist. Assoc. 79, 355) are proposed. Development of semiparametric extensions based on the ordinary least-squares estimator (OLSE) is impossible for the regression of Dickey et al. (1984) since the limiting null distribution of the OLSE-based test statistic is entangled with nuisance parameters under the usual normalization. This is in contrast with the successful development of the OLSE-based semiparametric unit root tests of Phillips (Econometrica 55 (1987) 277). To overcome the difficulty, we propose tests based on a feasible generalized least-squares estimator (GLSE), instead of the OLSE, and the spectral decomposition of the generalized sum of products of the regressor variables. The key advantage of the proposed method is that one can construct the feasible GLSE and hence tests of the seasonal unit root without specifying a parametric model for the error process.

Gaussian tests for seasonal unit roots based on Cauchy estimation and recursive mean adjustments

We propose tests for seasonal unit roots whose limiting null distributions are always standard normal regardless of the period of seasonality and types of mean adjustments. The seasonal models of Dickey, Hasza and Fuller (1984. Journal of American Statistical Association 79, 355–367) (DHF) and Hylleberg, Engle, Granger and Yoo (1990. Journal of Econometrics 44, 215–238) (HEGY) are considered. For estimating parameters related to the seasonal unit roots, regressor signs are used as instrumental variables while recursive sample means are used for adjusting the seasonal means. In addition to normality of the limiting null distributions, in seasonal mean models, the recursive mean adjustment provides the new tests with locally higher powers than those of the existing tests of DHF and HEGY based on the ordinary least-squares estimators. If data have a strong linear time trend, the recursive mean adjustment is a source of both power gains of some tests for local alternatives and power losses of all tests for other alternatives. Limiting normality allow evaluation of p-values and testing joint significance of subsets of seasonal unit roots.

Explorations in monetary cliometrics: the Reichsbank: 1876-1920

The seasonal unit root tests make it possible to determine the nature of the deterministic and stochastic sea sonal fluctuations. In this paper, we apply this method to the original monthly series of the Reichsbank monetary stock (constructed in weekly data with 2160 observations) and emphasize deterministic seasonal fluctuations with notably a strong seasonality at the beginning and at the end of the year. This statistical result is closely related to the turning points detected by the historical analysis.

Performance of seasonal unit root tests for monthly data

This paper uses Monte Carlo simulations to analyze the performance of several seasonal unit root tests for monthly time series. The tests are those of Dickey, Hasza and Fuller (DHF), Hylleberg, Engle, Granger and Yoo (HEGY), and Osborn, Chui, Smith and Birchenhall (OCSB). The unit root test of Dickey and Fuller (DF) is also considered. The results indicate that users have to be particularly cautious when applying the monthly version of the HEGY test. In general, the DHF and OCSB tests are preferable in terms of size and power, but these procedures may impose invalid restrictions. An empirical illustration is undertaken for UK two-digit industrial production indicators.

Seasonal Unit Root Tests with Structural Breaks in Deterministic Seasonality

The seasonal root tests of Hylleberg et al (1990) are extended using the sequential approach of Zivot and Andrews (1992). This paper presents Monte Carlo evidence to support a sequential approach to estimation and critical values are estimated. It is demonstrated that non‐stationary data with structurally unstable deterministic seasonality can lead to low power in standard tests for seasonal roots. The sequential tests are applied to US agricultural price data and macroeconomic data and compared with the standard tests. Seasonal roots are rejected in all series.

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.

Modelling the Seasonality of Hotel Nights in Denmark by County and Nationality

Danish time series data on hotel nights exhibit a strong seasonal pattern, with peaks during the summer holiday season and slumps during the winter. However, over the past 30 years holiday preferences have changed from a single summer holiday into several vacation periods spread over the year. This should be reflected in the seasonal pattern of the demand for holiday accommodation such as hotel nights. Consequently, the nature of the seasonal process should have become more stochastic and less deterministic. The paper examines this hypothesis by the use of univariate econometric modelling of seasonality on hotel nights by county as well as by nationality using monthly time series data for Denmark. Special attention is devoted to testing for seasonal unit roots, ie a varying and changing pattern. A research strategy for analysing seasonality including graphical tools is presented. It is found that unit roots are present in many time series on hotel nights at the county level as well as by nationality.

Seasonal unit roots and forecasts of two-digit European industrial production

Monthly industrial production in important sectors of the German, French and UK economies are shown to exhibit very strong seasonality, such that typically 80% or more of the variation in monthly growth can be attributed to seasonal effects. Seasonal unit root test results imply that most of these series should be modelled using conventional first differences with the inclusion of monthly dummy variables, rather than as specifications involving other levels of differencing. However, when the post-sample forecast accuracy is compared for various models, annual difference specifications often produce the most accurate forecasts at horizons of up to a year. First difference models appear to be most accurate at short forecast horizons for series where seasonality is particularly marked. The study also examines the impact of updating coefficient estimates and model respecification within the forecast period.

Tests for Stochastic Seasonality Applied to Daily Financial Time Series

We develop tests for seasonal unit roots for daily data by extending the methodology of Hylleberg et al. (‘Seasonal Integration and Cointegration’, Journal of Econometrics, Vol. 44 (1990), No. 1–2, pp. 215–238) and apply our tests to UK and US daily stock market indices. We also investigate a suggestion by Franses and Romijn (‘Periodic Integration in Quarterly Macroeconomic Variables’, International Journal of Forecasting, Vol. 9 (1993), No. 4, pp. 467–476) and Franses (‘A Multivariate Approach to Modelling Univariate Seasonal Time Series’, Journal of Econometrics, Vol. 63 (1994), No. 1, pp. 133–151) and create a price series for each day of the week and test for cointegration amongst these series. Our Monte Carlo experiments indicate that the Hylleberg et al. procedure is robust to autoregressive conditional heteroscedasticity type errors, while the Franses and Romijn procedure is less so. Finally, we employ Harvey’s (Time Series Models, Hemel Hempstead, Harvester Wheatsheaf, 1993) basic structural model to test for the presence of stationary stochastic seasonality. Our results suggest that we can reject the existence of seasonal unit roots at the daily frequency in both of these markets; however, we do find evidence of stationary stochastic seasonality.

A model selection strategy for time series with increasing seasonal variation

We propose a model selection strategy for time series with increasing seasonal variation. This strategy amounts to a selection of the most appropriate differencing filter to obtain a stationary time series without using a Box-Cox transformation. Hence, it is based on a sequence of tests for nonseasonal and seasonal unit roots. Through Monte Carlo replications, we provide new tables of critical values for the various test statistics. We apply our methods, which can be automated, to six example series and find that the results compare favorably to those of an expert.

Additional critical values and asymptotic representations for seasonal unit root tests

This paper is concerned with tests for seasonal unit roots in a univariate time series process. The paper extends the procedures and tables of critical values due to Hylleberg et al. (1990) and Ghysels et al. (1994) to obtain tests which are similar (exactly and asymptotically) with respect to both the initial values of the process and the possibility of differential seasonal drift under the null hypothesis of a seasonal unit root. Representations are derived for the limiting distributions of the test statistics in this and other cases of interest. These representations provide an explanation for the similarity between critical values for the statistics in different scenarios. The seasonal unit root properties of real seasonally unadjusted UK non-durable consumption expenditure are re-examined.

Seasonality and unit roots: the demand for fruits

Abstract We apply seasonal unit root tests to apple and pear price and quantity data. We then develop a method for testing shifts in amplitude and/or phase of the seasonal cycles. The results have implications to econometric specifications of models which use short-run data (quarterly, monthly).