# Forecasting an Accumulated Series Based on Partial Accumulation: A Bayesian Method for Short Series with Seasonal Patterns

By De Alba, Enrique; Mendoza, Manuel | Journal of Business & Economic Statistics, January 2001 | Go to article overview

# Forecasting an Accumulated Series Based on Partial Accumulation: A Bayesian Method for Short Series with Seasonal Patterns

De Alba, Enrique, Mendoza, Manuel, Journal of Business & Economic Statistics

We present a Bayesian solution to forecasting a time series when few observations are available. The quantity to predict is the accumulated value of a positive, continuous variable when partially accumulated data are observed. These conditions appear naturally in predicting sales of style goods and coupon redemption. A simple model describes the relation between partial and total values, assuming stable seasonality. Exact analytic results are obtained for point forecasts and the posterior predictive distribution. Noninformative priors allow automatic implementation. The procedure works well when standard methods cannot be applied due to the reduced number of observations. Examples are provided.

KEY WORDS: Bayesian inference; Prediction; Stable seasonality; Time series.

In certain practical situations the problem of forecasting a future value of a time series must be faced with only a very small amount of data. This situation arises, for example, when a drastic change has occurred with respect to the observational conditions so that most past data become irrelevant. Of course, it also appears when a rather new series is being recorded, as when a new product or service is being launched. In any case, under such circumstances, most of the usual forecasting techniques are no longer applicable. In addition, sometimes the series {[X.sub.i]} to be considered is such that (a) {[X.sub.i]} represents the accumulated value of a positive and continuous variable X over the complete ith period (the ith year, say); (b) for a given, fixed, subinterval of each period it is possible to observe [Y.sub.i], a partial accumulation of X (the accumulated value of X for the first quarter of each year, for example); (c) for each period, we have [Y.sub.i] = [W.sub.i][X.sub.i], where {[W.sub.i]; i = 1, 2 ...} are iid random variables over (0,1). We shall then say that the process has a stable seasonal pattern. A comprehensive review of situations in which this kind of structure arises can be found in Oliver (1987). It is clear that the seasonal pattern may be used in the forecasting process, no matter the amount of past data. However, it seems particularly valuable for the case in which only a few pieces of information are available. We present a Bayesian procedure for forecasting an accumulated value of a positive and continuous-valued time series, based on observation of a partial accumulation. The procedure is directed at obtaining forecasts given a small number of the partially accumulated observations, taking advantage of a seasonal pattern that may be known to exist in the series. The procedure is extremely simple to apply. We provide two examples. One of them is taken from Abraham and Ledolter (1983)--Monthly Average Residential Electricity Usage in Iowa City (Series 3). It is known to be a seasonal series. The second series is Administration Expenses of the Mexican Bank System (Guerrero and Elizondo 1997). As indicated by these authors, this series clearly has a trend.

It is important to emphasize that the procedure is especially useful when both conditions are present--seasonality and a small number of observations. If a large number of observations are available, then standard time series methods can be used to forecast the portion of the period that has not been observed yet. It can then be added to the partial accumulation already available. If few observations are available, then it is very unlikely that standard methods will yield good forecasts. If, in addition to having a small number of observations, it is known that the series is seasonal, then traditional methods such as autoregressive integrated moving average (ARIMA) are at a disadvantage. With monthly data, a typical seasonal autoregressive moving average (ARMA) model will include seasonal lags of order 12, which reduces the effective sample size by 12 (Abraham and Ledolter 1983, p. 283-291; Box and Jenkins 1970). If only 24 observations are available then the effective sample size would be 12. …

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