You have estimated the following ARMA(1,1) model for some time series data yt = 0.036 + 0.69yt−1 + 0.42ut−1 + ut Suppose that you have data for time to t−1, i.e. you know that yt−1 = 3.4, and ˆut−1 = −1.3

(a) Obtain forecasts for the series y for times t, t +1, and t +2 using the estimated ARMA model.

(b) If the actual values for the series turned out to be −0.032, 0.961, 0.203 for t, t +1, t +2, calculate the (out-of-sample) mean squared error.

(c) A colleague suggests that a simple exponential smoothing model might be more useful for forecasting the series. The estimated value of the smoothing constant is 0.15, with the most recently available smoothed value, St−1 being 0.0305. Obtain forecasts for the series y for times t, t +1, and t +2 using this model.

(d) Given your answers to parts

(a) to

(c) of the question, determine whether Box–Jenkins or exponential smoothing models give the most accurate forecasts in this application.

AppendixLO1