Elsevier

International Journal of Forecasting

Volume 17, Issue 4, October–December 2001, Pages 607-621
International Journal of Forecasting

On forecasting cointegrated seasonal time series

https://doi.org/10.1016/S0169-2070(01)00085-1Get rights and content

Abstract

We analyze periodic and seasonal cointegration models for bivariate quarterly observed time series in an empirical forecasting study. We include both single equation and multiple equation methods for those two classes of models. A VAR model in first differences, with and without cointegration restrictions, and a VAR model in annual differences are also included in the analysis, where they serve as benchmark models. Our empirical results indicate that the VAR model in first differences without cointegration is best if one-step ahead forecasts are considered. For longer forecast horizons however, the VAR model in annual differences is better. When comparing periodic versus seasonal cointegration models, we find that the seasonal cointegration models tend to yield better forecasts. Finally, there is no clear indication that multiple equations methods improve on single equation methods.

Introduction

In recent years several methods for cointegration analysis of nonadjusted seasonal time series have been developed. If one is willing to assume that the seasonal pattern is approximately constant over time, the vector autoregressive error-correction model [VECM] for first differenced series with constant seasonal dummy parameters can be used. However, tests for various types of changing seasonality often find evidence of a stochastic or changing seasonal pattern over time, see Hylleberg, Engle, Granger and Yoo (1990) [HEGY], Franses (1996), among many others.

In this paper we examine in an empirical forecasting study the relevance of taking care of changing seasonality within multivariate methods for cointegrated seasonal time series. We evaluate three different approaches, see Franses and McAleer (1998), that is nonseasonal cointegration models, seasonal cointegration models and periodic cointegration models. As our empirical study concerns bivariate time series, we also consider single equation methods, and compare these with system methods.

Testing and estimating seasonal cointegration relations can be accomplished in at least two ways. Engle, Granger, Hylleberg and Lee (1993) [EGHL] propose a two-step procedure, which is an extension of the Engle and Granger test for cointegration, whereas Lee (1992) suggests a maximum likelihood method for seasonal cointegration using a seasonal error correction model. Johansen and Schaumburg (1999) introduce a general asymptotic theory for the latter cointegration approach. While cointegration at the zero frequency can be interpreted as evidence of parallel long-run movements among the time series considered, cointegration at the biannual or annual frequencies is viewed as evidence of parallel movements across the corresponding seasonal components of the time series.

An alternative model for changing seasonality in multivariate data extends the periodic integration model, see Franses (1996) and Boswijk, Franses and Haldrup (1997), among others. When the individual time series display periodic features, one may want to consider periodic cointegration. A useful single equation method is proposed in Boswijk and Franses (1995), and it is an extension of the cointegration test approach by Boswijk (1994). A multiple equations method is proposed by Kleibergen and Franses (1999), who consider cointegration in periodic VAR models [PVAR].

The forecasting performance of seasonal cointegration models has been analyzed in Kunst (1993) and in Reimers (1997). Two examples based on real data and a Monte Carlo experiment in Kunst (1993) indicate that the benefits from accounting for seasonal cointegration are quite limited as compared to vector error correction models in first differences with deterministic seasonal dummies included. The main conclusion in Reimers (1997) is that models in first differences produce smaller forecast errors for short horizons, but when longer forecasting periods are considered the seasonal cointegration model appears preferable. Finally, the forecasting performance using different specifications of the single equation periodic cointegration model has been examined in Herwartz (1997), where it is found not to be very successful.

The purpose of the present paper is to compare the two model classes in an empirical forecasting study, which involves seven sets of bivariate quarterly time series, which one may expect to be somehow cointegrated. We aim to shed light on the following issues. Do multiple equations methods for seasonal and periodic cointegration generate better forecasts as compared to their single equation counterparts? And, is one of the two model classes, that is seasonal versus periodic cointegration, preferable in terms of forecasting? We also compare these periodic and seasonal cointegration models with a VECM in first differences with constant seasonal dummy parameters and with an estimated long-run relation included. Additionally, in the analysis we also include a VAR model in first differences with constant seasonal dummy parameters and a VAR model in fourth differences with an intercept included, that is, two models without cointegration. The last model is included as its univariate counterpart has been found to be rather successful in terms of out-of-sample forecasting, even when in sample tests indicate that other models are to be preferred, see Clements and Hendry (1997) and Osborn, Heravi and Birchenhall (1999).

The remainder of this paper is organized as follows. Section 2 gives a brief discussion of the various cointegration approaches. In Section 3, we present the data and Section 4 contains the estimation and forecasting results. The final section presents some concluding remarks.

Section snippets

Cointegration methods

In this section we review four different approaches to cointegration analysis of quarterly time series. Later on, we will contrast these approaches with a VECM with constant seasonal dummy parameters, where the long-run relation is estimated from the data for the bivariate series at hand (denoted as N-ME-1, which means non-seasonal multiple equations of type 1), and with VAR models in first and fourth differences without cointegration (denoted as N-ME-2 and N-ME-3 respectively). Sometimes it

Data

In our forecasting study we consider the logs of quarterly observed time series on consumption and income in the United Kingdom, Sweden, (Western-) Germany, Japan, Italy and the US. The data set for UK covers the time period 1955:1 to 1989:4 (consumption on non-durables and disposable income), whereas the time series for Sweden ranges from the period 1963:1 to 1988:4 (consumption on non-durables and disposable income) and for Germany from 1960:1 to 1988:4 (consumption and disposable income).

Empirical results

In this section we first discuss the in-sample estimation results and then we turn to the out-of-sample forecasting results. To save space we mainly show the test results for cointegration and the forecasting results. Other results can be obtained from the authors upon request. Also, we always consider the 5% significance level.

Concluding remarks

We have analyzed periodic and seasonal cointegration models for bivariate quarterly observed time series in an empirical forecasting study. We included both single equation and multiple equations methods. A VAR model in first differences, with and without cointegration, and a VAR model in annual differences were also included in the analysis, where they served as benchmarks. Our empirical results indicate that the VAR model in annual differences is often preferred, except for one-step ahead

Acknowledgements

We thank the Editor, an Associate Editor and two anonymous referees for helpful comments. We also thank John Wells, Hahn Lee and Pierre Siklos for providing us with the data. Earlier versions of this paper were presented at the ISF2000 conference in Lisbon and aspects of it at various seminars in Portugal and Spain. Additional estimation and diagnostic test results can be obtained from the corresponding author.

Mårten Löf is a Ph.D. student at the Department of Economic Statistics, Stockholm School of Economics. His research focuses on seasonality, cointegration and forecasting.

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    Mårten Löf is a Ph.D. student at the Department of Economic Statistics, Stockholm School of Economics. His research focuses on seasonality, cointegration and forecasting.

    Philip Hans Franses is Professor of Applied Econometrics, affiliated with the Econometric Institute, and Professor of Marketing Research, affiliated with the Department of Marketing and Organization, both at the Erasmus University Rotterdam. His research interests include time series, forecasting, marketing research, empirical finance, environmetrics and political science.

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