7.5.1 Exponentially Weighted Moving Average

7.5.1 Exponentially Weighted Moving Average

The exponentially weighted moving average (EWMA) model computes a moving average of yt using exponentially declining weights over the entire history of the data. The forecast, or “trend,” is

$$f_{t+1|t}=\frac{(1-\delta )}{(1-{\delta }^t)}\sum _{s=0}^{t-1}{\delta }^s{y}_{t-s}.\text{\hspace{1em}(7.45)}$$

We can motivate this as a weighted average, $\begin{array}{r}{f}_{t+1|t}=c_t{\sum }_{s=0}^{t-1}{\delta }^s{y}_{t-s}\end{array}$ , where the scaling factor $c_t$ is chosen to ensure that the weights sum to 1 (Abraham and Ledolter, 1983). The sequence of forecasts $\{f_{t+1|t}\}$ is $\{y_1,(1+\delta {)}^{-1}(y_2+\delta y_1)$ , $(1+\delta +{\delta }^2{)}^{-1}(y_3+\delta y_2+{\delta }^2{y}_1,\dots ),\dots \}$

The EWMA model in (7.45) relates to the simple smoothing method (7.44) in the sense that they are equivalent in the limit as $t\to \infty {.}^{11}$ To see this, note that

$$\begin{array}{r}{f}_{t+1\mid t}-\delta f_{t\mid t-1}=\frac{(1-\delta )}{(1-{\delta }^t)}\sum _{s=0}^{t-1}{\delta }^s{y}_{t-s}-\frac{(1-\delta )}{(1-{\delta }^{t-1})}\delta \sum _{s=0}^{t-2}{\delta }^s{y}_{t-1-s}\\ =\frac{(1-\delta )}{(1-{\delta }^t)}{y}_t+\left(\frac{(1-\delta )}{(1-{\delta }^t)}-\frac{(1-\delta )}{(1-{\delta }^{t-1})}\right)\sum _{s=1}^{t-1}{\delta }^s{y}_{t-s}.\end{array}\text{\hspace{1em}(7.46)}$$

Figure 7.6: Forecasts of the inflation rate and stock returns generated by exponentially weighted moving average (EWMA) models with different smoothing factors (δ).

Since $0<\delta <1$ , lim ${}_{\cdot j\to \infty }{\delta }^j=0$ , and so we have, approximately,

$$f_{t+1|t}-\delta f_{t|t-1}=(1-\delta )y_t,$$

which is identical to (7.44).

Figure 7.6 shows forecasts of the quarterly rate of inflation and quarterly stock returns generated by three EWMA models with $\delta =0.5,\mathrm{0.8.}$ , and 0.999. The bigger is $\delta ,$ , the smoother the forecast as the weights are spread out more evenly on past observations. In fact, comparing these forecasts with the random walk forecast $(f_{t+1|t}=y_t)$ and the prevailing mean $(f_{t+1|t}=t^{-1}{\sum }_{\tau =1}^t{y}_{\tau })$ shown in the right window of the figure, it is clear that the EWMA forecasts get increasingly similar to a random walk forecast, the smaller is δ, while the EWMA forecasts increasingly resemble the prevailing mean forecast, the closer is δ to unity.12 Given the importance of δ to the forecast, one might use methods such as cross validation to select this parameter optimally, i.e., to generate the smallest average loss, given the loss function. We consider this in an empirical application in chapter 8.

7.5.2 Discounted Least Squares

Brown (1962) considered the discounted least squares loss function,

$$L(y_1,\dots ,y_t,\mu )=\sum _{s=0}^{t-1}{\delta }^s(y_{t-s}-\mu {)}^2,$$

as an estimator for the mean of a series. The first-order condition for $\mu ,$ computed at time t, is

$$\sum _{s=0}^{t-1}{\delta }^s(y_{t-s}-\hat{\mu })=0,$$

which can be rearranged to give

$$\begin{array}{l}\hat{\mu }={\left(\sum _{s=0}^{t-1}{\delta }^s\right)}^{-1}\sum _{s=0}^{t-1}{\delta }^s{y}_{t-s}\\ =\frac{(1-\delta )}{(1-{\delta }^t)}\sum _{s=0}^{t-1}{\delta }^s{y}_{t-s}.\end{array}$$

This is identical to the EWMA model in (7.45).

Hence the discounted least squares path for the means (which can be employed as the forecasts) is the same as that for the EWMA and hence approximately the same as the simple updating formula (7.44).