a. If forecasts are based on simple exponential smoothing, with x`t denoting the smoothed value of the series at time t, show that the error made in forecasting xt, standing at time (t – 1), can be written as follows:
et = xt – x`t-1
b. Hence, show that we can write x`t = xt – (1 – a)et, from which we see that the most recent observation and the most recent forecast error are used to compute the next forecast.