1. You have estimated the following ARMA(1,1) model for some time series data yt =0 . 036+0 .

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

ˆu_t–1 = -1.3

a) Obtain forecasts for the series yfor 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, S_t–1 being 0.0305. Obtain forecasts for the series yfor 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