Numerical distribution functions for seasonal unit root tests

Highlights

• A generalisation of HEGY seasonal unit root tests was presented in Smith, Taylor and del Barrio Castro (2009).

• We use a response surface regressions approach to calculate $P$-values for the HEGY statistics.

• They can be used for any seasonal periodicity, sample size and autoregressive order.

• A Gretl function package is provided for applying the tests and calculating their $P$-values.

Abstract

It is often necessary to test for the presence of seasonal unit roots when working with time series data observed at intervals of less than a year. One of the most widely used methods for doing this is based on regressing the seasonal difference of the series over the transformations of the series by applying specific filters for each seasonal frequency. This provides test statistics with non-standard distributions. A generalisation of this method for any periodicity is presented and a response surface regressions approach is used to calculate the $P$-values of the statistics whatever the periodicity and sample size of the data. The algorithms are prepared with the Gretl open source econometrics package and two empirical examples are presented.

Introduction

Unit roots may cause severe problems in a regression model if they are not properly dealt with: this may imply inconsistent coefficient estimators and non-standard distributions for significance tests and for forecast intervals. There have been many papers on testing for unit roots since the book by Fuller (1976), which introduced the test currently known as the Augmented Dickey–Fuller test, ADF (see also  Dickey and Fuller, 1981). Apart from the ADF test, other noteworthy tests are Phillips and Perron (1988), the KPSS test for stationarity by Kwiatkowski et al. (1992) and the ADF–GLS test by Elliott et al. (1996), all of which have become widely used by empirical economists. However, when working with time series data observed at intervals shorter than a year, the presence of unit roots should be tested for, not only in the long run but also in seasonal cycles. Over the last thirty years various methods have been proposed for testing for seasonal unit roots. For example, Hasza and Fuller (1982) and Dickey et al. (1984) propose joint tests for all seasonal unit roots, and in later papers Osborn et al. (1988) and in particular Hylleberg et al. (1990) (hereinafter HEGY) propose tests that would deal with each seasonal and zero frequency root to be considered separately. There are also interesting tests of seasonal stability by Canova and Hansen (1995), which also consider each frequency individually. The HEGY tests are not very difficult to implement and have therefore become widely used among empirical economists.

One of the problems with most of the unit root tests mentioned above is that their statistics have non-standard distributions, so in practice the values in the tables published need to be interpolated to compare them with the values calculated or simulate the empirical distributions for exactly the same model and the same sample size that is being used. MacKinnon (1994) uses simulation methods and response surface regressions to estimate the asymptotic distributions of a large number of unit roots and cointegration tests at zero frequency (long run). MacKinnon (1996) then extends these results, providing a way of approximating small sample distributions too.

Harvey and van Dijk (2006) apply MacKinnon’s method, using response surface regressions to provide a simple way of obtaining critical values and $P$ values for any sample size and any order of lags of the endogenous variable in the regression for the HEGY tests mentioned above. As in the original HEGY article, all this is for quarterly data.

In the 21st century it seems quite anachronistic to have to use statistical tables. Empirical economists normally use computers for calculations, so ideally they should have a computer algorithm to calculate $P$-values instead of having to look at statistical tables. The main objective of this paper is to obtain a generalisation of the method by Harvey and van Dijk for calculating the $P$ values of the HEGY statistics whatever the seasonal periodicity, $S$, and sample size $T$ of the data. Seasonal periodicity $S$ is defined as the number of values of the series observed in each year (it is sometimes convenient to change the reference period from one year, for example, to one day if data are hourly, $S=24$, or one week if data are daily, $S=7$, but for ease of exposition I will continue to refer to the reference period as the ‘year’).

For this approach to be practical it needs to be possible to implement it in a computer algorithm. As a complement to this paper an algorithm prepared in Gretl is provided (see http://gretl.sourceforge.net). Gretl is a cross-platform software package for econometric analysis. It is open source, free software: anyone may redistribute it and/or modify it under the terms of the GNU General Public License (GPL). This makes Gretl a very good econometric package in terms of the replicability of its calculations and results (Baiocchi, 2007).