International Journal of Forecasting 6 (1990) 559

North-Holland

Comments on:

“ Confidence intervals for non-stationarv J

forecast errors”, by P. Lefraqois *

Chris Chatfield School of Mathematical Sciences, Bath University, Bath, UK

Keyword: Confidence intervals.

The construction of prediction intervals for fore- casts at different lead times is an important practi- cal problem. While theoretical formulae are avail- able for some forecasting models, they are not available for others and a number of approximate formulae are available, some of which date back to work by R.G. Brown in the 1960’s. I am very concerned that one formula, which can often give a very poor approximation, has now appeared twice in the journal in the guise of an exact formula.

The recent paper by Lefrancois (1989) is based entirely on a formula [their equation (l)] which effectively assumes that under certain assumptions (namely that the one-step ahead errors are inde- pendent and normally distributed with zero mean and constant variance) the standard deviation of the k-steps-ahead errors is fi times the standard deviation of the one-step-ahead errors. As far as I am aware, this result was first given by Makrida- kis et al. [1987, equation (1)] who appeared to prove it in an appendix. Unfortunately the “proof” is not rigorous and the result only appears to be true when the underlying process is a random walk and the latest observation is taken to be the forecast for all lead times. The inadequacy of the

* International Journal of Forecasting, 5, no. 4 (1989) pp.

553-557.

fi formula is detailed by Koehler (1990) and also mentioned by Yar and Chatfield (1990, Section 5) but the appearance of Lefrancois (1989) suggests that much more direct comment is needed to prevent the fi formula becoming part of time- series forecasting folklore. While some processes do approximate a random walk, many others do not and then the fi formula ca be seriously in error. Thus the empirical results presented in Lefrancois (1989) and some of those in Makrida- kis et al. (1987) are of dubious value, and the fi formula should be recognised for the inadequate approximation that it is.

References

Koehler, A.B., 1990, “An inappropriate prediction interval”,

International Journal of Forecasting, 6, 551-558 (this issue).

Lefrancois, P., 1989, “Confidence intervals for non-stationary

forecast errors: Some empirical results for the series in the

M-competition”, International Journal of Forecasting, 5,

553-557.

Makridakis, S., M. Hibon, E. Lusk and M. Belhadjali, 1987,

“Confidence intervals: An empirical investigation of the

series in the M-competition”, International Journal of Fore-

casting, 3, 489-508.

Yar, M. and C. Chatfield, 1990, “Prediction intervals for the

Holt-Winters forecasting procedure”, International Journal

of Forecasting, 6, 127-137.

0169-2070/91/$03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)