Elsevier

International Journal of Forecasting

Volume 16, Issue 1, January–March 2000, Pages 111-116
International Journal of Forecasting

Notes
Forecasting the levels of vector autoregressive log-transformed time series

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Abstract

In this paper we give explicit expressions for the forecasts of levels of a vector time series when such forecasts are generated from (possibly cointegrated) vector autoregressions for the corresponding log-transformed time series. We also show that simply taking exponentials of forecasts for logged data leads to substantially biased forecasts. We illustrate this using a bivariate cointegrated vector series containing US GNP and investments.

Introduction

In the empirical time series analysis of economic variables it is common practice to transform the data using natural logarithms prior to the construction of econometric models that are often used for forecasting. Some of the motivations for this strategy are that this log-transformation reduces the impact of outliers, that first differenced log-transformed data correspond to growth rates, and that it reduces the often observed increasing variance of trending time series. Once a model has been constructed, and the parameters have been estimated, one can make forecasts for the log-transformed data. In some cases, however, one is interested in the forecasts of the levels of the time series (i.e. the untransformed data) instead of (functions of the) log-transformed data. In that case, as is well known from the results in Granger and Newbold (1976) for univariate time series, simply taking exponentials of the forecasts of the logged data yields biased forecasts. For the class of the univariate autoregressive [AR] model, Granger and Newbold (1976) derive expressions for unbiased forecasts of the levels. In the present paper we extend their results to the practically very relevant class of vector autoregressive [VAR] time series models. VAR models are often used in empirical economics to generate out-of-sample forecasts since their parameters are easy to estimate, and especially since such models provide a simple framework for the analysis of cointegration, see for example Johansen (1995). In Section 2 we give explicit expressions for the out-of-sample forecasts of the levels of m time series when these series (in log-transformed format) are modeled by a VAR model of order p. In Section 3, we give an empirical example concerning a bivariate US series containing GNP and investments, where we take into account that the log-transformed series are cointegrated. We conclude our paper in Section 4 with some remarks.

Section snippets

Forecasting levels

In this section we present explicit expressions for the forecasts of the levels of a time series, when the log-transformed vector time series follows a vector autoregressive model. To motivate our paper, consider the univariate time series Xt, for which one analyses Yt with the latter being the series in logs, that is, Yt=log Xt, where log denotes the natural logarithmic transformation. Suppose that the log-transformed series can be modelled as Yt=Mt+ηt where Mt denotes the conditional

An application

In this section we apply the expressions obtained in the previous section to a two-dimensional time series (X1(t), X2(t))′. X1 is the real GNP of the US and X2 is the real gross domestic investment series. The data are given in Pindyck and Rubinfeld (1991, chapter 12). Quarterly observations are available from the first quarter of 1947 until the first quarter of 1988. We will use observations until the fourth quarter of 1980 to estimate our VAR model for the log-transformed data, and we leave

Concluding remarks

In this paper we presented explicit expressions for forecasts for the levels of a vector time series when a VAR model was used for the log-transformed data. We showed that exponentials of the forecasts for logged data are biased, as could also be observed from our empirical forecasts from a bivariate cointegrated vector autoregressive time series model containing US GNP and investment. Our results can be practically relevant in case one aims to forecast the levels of a vector time series.

Acknowledgements

We thank the editor and two anonymous referees for their helpful comments. Miguel A. Ariño has received partial financial support from CIIF, Centro Internacional de Investigación Financiera (International Center for Financial Research).

Biographies: Miguel A. ARIÑO is Professor of Statistics and Decision Sciences at IESE, Universidad de Navarra. He has a Ph.D. in Mathematics from Universidad de Barcelona. He has publications in several mathematical and economic journals. His current research interests are in the areas of time series analysis and econometrics.

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Cited by (0)

Biographies: Miguel A. ARIÑO is Professor of Statistics and Decision Sciences at IESE, Universidad de Navarra. He has a Ph.D. in Mathematics from Universidad de Barcelona. He has publications in several mathematical and economic journals. His current research interests are in the areas of time series analysis and econometrics.

Philip Hans FRANSES is Professor of Applied Econometrics at the Econometric Institute of the Erasmus University Rotterdam. One of his research interests is modelling and forecasting time series. On this topic he has published in various journals and books.

The Notes section of the International Journal of Forecasting contains commentary on the theory and practice of forecasting in the form of communications to the journal such as research notes, teaching tips, practitioners’ and consultants’ views, and other contributions, especially those that attempt to bridge the gap between new developments in the methodology of forecasting and their practical application. Contributions to this section of the journal can be submitted to any of the four editors.

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