1. You have estimated the following ARMA(1,1) model for some time series data *yt*=0*.*036+0*.*69*y _{t}*

_{–}

_{1}+0

*.*42

*u*

_{t}_{–}

_{1}+

*u*Suppose that you have data for time to

_{t}*t*-1, i.e. you know that

*y*

_{t}_{–}

_{1}=3

*.*4, and

ˆ*u _{t}*

_{–}

_{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, *S _{t}*

_{–}

_{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