Problem 1. Consider the following ARMA(1,1) model for some time series data yt = 0.036 + 0.69y_t1 + 0.42u_t1 + u_t

Suppose that you have data for time to t1, i.e. you know that y_t1 = 3.4 , and u_t1 = 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, mean absolute error and percentage of correct sign predictions. Comment on the differing characteristics of the three methods used for evaluating the forecasting accuracy.

c) What procedure might be used to estimate the parameters of an ARMA model? Explain, briefly, how such a procedure operates, and why OLS is not appropriate.

d) Suppose that we find ARCH effects in the data and use a GARCH model to capture the dependence structure in the variance. What procedure might be used to estimate the parameters of a GARCH model? Explain, briefly, how such a procedure operates, and why OLS is not appropriate. Also comment on the possible problems of the procedure you suggest.