Exponential smoothing

Given $Y_1, ..., Y_T$ , want to predict $Y_{T+1}$ , denoted by $\hat{Y}_{T+1}$ , and can measure the quality with the MSE $ = E(Y_{T+1} - \hat{Y}_{T+1})^2$ .

Suppose underlying process has constant mean, $\mu$ , which we don’t know but can estimate from the sample. If instead of being constant the mean is slowly varying function of $t$ , then it makes sense to use a weighted mean with recent values having greater weights.

\[ \hat{Y}_{T+1} = c(Y_t + \omega Y_{T-1} + \omega^2 Y_{T-2} + ...) \]

For some $|\omega| \lt 1$ (known as the discounting coefficient), with weights summing to 1, ie. $\hat{Y}_{T+1} = (1- \omega)(Y_t + \omega Y_{T-1} + \omega^2 Y_{T-2} + ...)$

Updating

Can define exponential smoothing recursively: $\hat{Y}_{T+1} = \hat{Y}_T + \alpha(Y_T - \hat{Y}_T)$ . Often use $\alpha = 1 - \omega$ to make formula even simpler. Since $|\omega| \lt 1$ , values in remote past do not affect predictions to any great extent, and procedure is relatively insenstive to value of $\omega$ .

In order to get started need a value for $Y_1$ . eg. $Y_1$ , $\bar{Y}$ , predict backwards from later values.