Does the Box-Cox transformation help in forecasting macroeconomic time series?

Tommaso, Proietti and Helmut, Luetkepohl (2011): Does the Box-Cox transformation help in forecasting macroeconomic time series?

Abstract

The paper investigates whether transforming a time series leads to an improvement in forecasting accuracy. The class of transformations that is considered is the Box-Cox power transformation, which applies to series measured on a ratio scale. We propose a nonparametric approach for estimating the optimal transformation parameter based on the frequency domain estimation of the prediction error variance, and also conduct an extensive recursive forecast experiment on a large set of seasonal monthly macroeconomic time series related to industrial production and retail turnover. In about one fifth of the series considered the Box-Cox transformation produces forecasts significantly better than the untransformed data at one-step-ahead horizon; in most of the cases the logarithmic transformation is the relevant one. As the forecast horizon increases, the evidence in favour of a transformation becomes less strong. Typically, the na¨ıve predictor that just reverses the transformation leads to a lower mean square error than the optimal predictor at short forecast leads. We also discuss whether the preliminary in-sample frequency domain assessment conducted provides a reliable guidance which series should be transformed for improving significantly the predictive performance.

• Item Type: MPRA Paper
• Language: English
• Keywords: Forecasts comparisons; Multi-step forecasting; Rolling forecasts; Nonparametric estimation of prediction error variance.
• Subjects: C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Atkinson, A. C. (1973). Testing transformations to normality, Journal of the Royal Statistical Society B, 35, 473–479.

Bardsen, G. and Luetkepohl, H. (2011). Forecasting levels of log variables in vector autoregressions, International Journal of Forecasting, forthcoming.

Bell, W. R. and S. C. Hillmer (1983). Modeling time series with calendar variation. Journal of the American Statistical Association, 78, 526–534.

Box G. E. P. and Cox D.R. (1964). An analysis of transformations (with discussion), Journal of the Royal Statistical Society B, 26, 211–246.

Box, G.E.P. and Cox D.R. (1982). An analysis of transformation revisited, rebutted, Journal of the American Statistical Association, 77, 209–210.

Box, G. E. P. and Jenkins, G. M. (1970). Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco. Box G. E. P. and Jenkins, G.M. (1973). Some comments on a paper by Chatfield and Prothero and on a review by Kendall, Journal of the Royal Statistical Society A, 136, 337-352.

Cameron, M.A. (1978). The prediction variance and related statistics for stationary time series, Biometrika, 65, 283–296

Chatfield C., and Prothero, D. L. (1973). Box-Jenkins seasonal forecasting: problems in a casestudy, Journal of the Royal Statistical Society A, 136, 295–315.

Cleveland, W.S. and Devlin, S.J. (1982). Calendar effects in monthly time series: modeling and adjustment, Journal of the American Statistical Association, 77, 520–528.

Davis, H.T. and Jones, R.H. (1968). Estimation of the innovation variance of a stationary time series, Journal of the American Statistical Association, 63, 141–149.

Diebold, F.X. and Kilian, L. (2001). Measuring predictability: theory and macroeconomic applications, Journal of Applied Econometrics, 16, 657–669.

Diebold, F.X. and Mariano, R.S. (1995). Comparing predictive accuracy, Journal of Business & Economic Statistics, 13, 253–263.

Engle, R.F. (1974). Band spectrum regression, International Economic Review, 15, 1–11.

Fernandes, M. and Grammig J. (2006). A family of autoregressive conditional duration models, Journal of Econometrics, 130, 1–23.

Goncalves, S. and Meddahi, N. (2011). Box-Cox transforms for realized volatility, Journal of Econometrics, 160, 129–144.

Granger, C.W.J. and Newbold, P. (1976). Forecasting transformed time series, Journal of the Royal Statistical Society B, 38, 189-203.

Guerrero, V. M. (1993). Time-series analysis supported by power transformations, Journal of Forecasting, 12, 37–48.

Hannan, E.J., and Nicholls, D.F. (1977). The estimation of prediction error variance, Journal of the American Statistical Association, 72, 834-840.

Harvey, D., Leybourne, S. and Newbold, P. (1997). Testing the equality of prediction mean squared errors, International Journal of Forecasting, 13, 281–291.

Higgins, M.L. and A. Bera (1992). A class of nonlinear ARCH models. International Economic Review, 33, 137-158.

Hopwood, W.S., McKeown, J.C. and Newbold P. (1981). Power transformations in time-series models of quarterly earnings per share, Accounting Review, 56, 927–933.

Luetkepohl, H. and Xu, F. (2011). Forecasting annual inflation with seasonal monthly data: using levels versus logs of the underlying price index, Journal of Time Series Econometrics, 3(1), article 7.

Luetkepohl, H. and Xu, F. (2011a). The role of the log transformation in forecasting economic variables, Empirical Economics, forthcoming.

Nelson, C.R. (1976). The interpretation of R2 in autoregressive-moving average time series models, American Statistician, 30, 175–180.

Nelson, H.L. and Granger, C.W.J. (1979). Experience with using the Box-Cox transformation when forecasting economic time series, Journal of Econometrics, 1979, 57–69.

Neyman J.N. and Scott E.L. (1960). Correction for bias introduced by a transformation of variables, Annals of Mathematical Statistics, 31, 3, pp. 643–655.

Pankratz, A. and Dudley, U. (1987). Forecasts of power-transformed series, Journal of Forecasting, 6, 239–248.

Pascual, L., Romo, J. and Ruiz, E. (2005). Bootstrap prediction intervals for power-transformed time series, International Journal of Forecasting, 21, 219–235.

Proietti, T. and Riani, M. (2009). Transformations and seasonal adjustment, Journal of Time Series Analysis, 30, 47–69.

Tukey, J. W. (1957) On the comparative anatomy of transformations, Annals of Mathematical Statistics, 28, 602-632.

Tunnicliffe Wilson, G. (1973). Discussion of the paper by Dr Chatfield and Dr Prothero, Journal of the Royal Statistical Society A, 136, 315–319.