International Journal of Forecasting 5 (1989) 231-240 231 North-Holland The effect of additive from ARIMA models outliers on the forecasts Johannes LEDOLTER University of Iowa, Iowa City, IA 52242, USA Abstract: Assume that a time series of length n = T + k includes an additive outlier at time T and suppose this fact is ignored in the estimation of the coefficients and the calculation of the forecasts. In this paper we derive the resulting increase in the mean square of the /-step-ahead forecast error. We show that this increase is due to (i) a carry-over effect of the outlier on the forecast, and (ii) a bias in the estimates of the autoregressive and moving average coefficients. Looking at several special cases we find that this increase is rather small provided that the outlier occurs not too close to the forecast origin. In such cases the point forecasts are largely unaffected. Our conclusion concerning the width of the prediction intervals is different, however. Since outliers in a time series inflate the estimate of the innovation variance, we find that the estimated prediction intervals are quite sensitive to additive outliers. Keywords: ARIMA model, Forecasting, Mean square forecast error, Outliers, Prediction intervals, Time series. I. Introduction In standard time series forecasting little attention is usually given to individual, and perhaps outlying observations. Although methods for the detection and the adjustment of outlying observations are now available [see Tsay (1988), for example], it is still very common that forecasts from ARIMA or exponential smoothing models are based on observations that have not been adjusted for outliers. In fact, many comparative forecasting studies do not make allowances for the possibility of spurious observations. What are the consequences of ignoring occasional outliers? How do unrecognized outliers affect the forecasts? How do they affect the width of the prediction intervals? These are some of the questions that are addressed in this paper. These questions could be investigated empirically. For example, one could use the 1001 series of the M-competition [see Makridakis et al. (1982)] and compare the forecasts from time series or smoothing methods before and after adjustments for outliers in the historical data base have been made. Another approach is to make certain simplifying assumptions and look at these questions theoretically. It is this second approach that is adopted in this paper. We assume that a time series (X,) is generated by an autoregressive integrated moving average, ARIMA (p, d, q), process O(B)(1 -B)aXt=O(B)a,, (1) where B is the backshift operator such that BmX~ = Xt_m, q~(B) = 1 - ~IB - .,. -~ppB p is the autoregres- sive operator, 0(B)= 1- O~B-...-OqB q is the moving average operator, and {a,} is a sequence of independent normal random variables with mean zero and variance o 2. For stationarity and invertibility it is assumed that the roots of the autoregressive and moving average operators are outside the unit circle. 0169-2070/89/$3.50 ~5 1989, Elsevier Science Publishers B.V. (North-Holland)