Exponentially Weighted Moving Average models

Open up any finance textbook and you won't have to look far before you come across exponentially weighted average (EWMA) models. The EWMA is typically used to measure historical volatility, but it can, of course, be used in other contexts. For example, see Muth (1960) for an early discussion of the statistical properties of EWMA models and its use in macroeconomics, in particular, in relation to the adaptive expectations mechanism associated with Milton Friedman.

Based on the amount of online video tutorials and other online content providing explanations of the EWMA, I can only infer that the EWMA is a popular tool for those working in the finance industry. And why not, eh? It's a simple model that can be understood and interpreted by mangers and the like.

Indeed, for some time-series, the EWMA can provide an adequate representation of the data generating mechanism and produce optimal forecasts, however, the EWMA can also produce sub-optimal forecasts when (mis)applied to other series of data. For this reason, using the EWMA simply out of habit may be a bad habit worth kicking.

Before using the EWMA for forecasting, one ought to grant some time to considering whether or not the EWMA is appropriate for the data in question. That is, of course, under the assumption that time is not of the essence and that the practitioners' ultimate goal is to produce optimal forecasts. In other words, let's assume that the practitioner is not under a severe time constraint and faced with the task of updating or developing hundreds of forecasting models for a company.

The claim thus far is that strict use of the EWMA is not the highest standard of forecasting. A justification of this claim now follows.

Consider the following fact:

• The exponentially weighted moving average is algebraically equivalent to an ARIMA(0,1,1) model. 

To put it another way, the EWMA is a particular model within the class of ARIMA models. In fact, there are various types of EWMA models and these happen to be included in the class of ARIMA(0,d,q) models - see Cogger (1974).

Now, recall the fact that properly constructed ARIMA models produce optimal univariate forecasts - meaning that the mean-squared forecast error is smaller than for any other univariate forecast (produced by univariate, linear, fixed coefficient models).

Suppose now that rather than using the EWMA by habit, we instead use the Box-Jenkins method and let the data talk to us. In this situation, there are two possible outcomes; the appropriate model may, or may not, be an ARIMA(0,1,1), or, equivalently, an EWMA.

If it turns out that the appropriate model is an ARIMA(0,1,1) then our forecasts will be optimal. On the other hand, if the appropriate model is not an ARIMA(0,1,1) then our model has not been properly constructed and thus it must be admitted that our forecasts could be improved by specifying a different model from the suite of ARIMA models.