International Journal of Forecasting 15 (1999) 409–419 www.elsevier.com / locate / ijforecast Forecasting using a periodic transfer function: with an application to the UK price of ferrous scrap * Kevin Albertson , Jonathan Aylen Department of Economics, University of Salford, Salford M54 WT, UK Abstract The familiar concept of cointegration enables us to determine whether or not there is a long-run relationship between two integrated time series. However, this may not capture short-run effects such as seasonality. Two series which display different seasonal effects can still be cointegrated. Seasonality may arise independently of the long-run relationship between two time series or, indeed, the long-run relationship may itself be seasonal. The market for recycled ferrous scrap displays these features: the US and UK scrap prices are cointegrated, yet the local markets exhibit different forms of seasonality. The paper addresses the problem of using both cointegrating and seasonal relationships in forecasting time series through the use of periodic transfer function models. We consider the problems of testing for cointegration between series with differing seasonal patterns and develop a periodic transfer function model for the US and UK scrap markets. Forecast comparisons with other time series models suggest that forecasting efficiency may be improved by allowing for periodicity but that such improvement is by no means guaranteed. The correct specification of the periodic component of the model is critical for forecast accuracy.  1999 Elsevier Science B.V. All rights reserved. Keywords: Cointegration; Ferrous scrap; Forecasting competition; Periodic transfer function; Seasonality; Recycling 1. Introduction influences, but still be related in the long run. So seasonality must also be modelled if we are to The well-known concept of cointegration allows forecast such series accurately and efficiently. us to assess whether there is a long-run relationship In this paper, we address the problem of using between two integrated time series. However, such a both long-run cointegrating and short-run seasonal relationship may not capture short-run or seasonal relationships when forecasting industrial times series. effects in the series. Two series may display different We take account of the two effects by using periodic seasonal effects, yet still be cointegrated. Seasonality transfer function models. Testing for cointegration in time series can arise independently of long-run and exogeneity when series display differing season- cointegrating relationships between them or from the al patterns raises particular difficulties. We address long-run relationships themselves. For instance, two these issues by using Franses’ (1996b) periodic commodity prices can be subject to different climatic cointegration tests and by adapting standard ex- ogeneity tests to allow for the nature of our data. Having established that our suggested model is *Corresponding author. Tel.: 144-161-295-5000; fax: 144- 161-295-5992. consistent with both theory and data, we estimate its 0169-2070 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S0169-2070(99)00020-5