You have estimated the following ARMA(1,1) model for some time-series data Suppose that you have data for time to t− 1, i.e., you know that yt

−1 = 3.4, and

(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.