Problem 1 (35 marks). Consider 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 ut-1 = -1.3 a) Obtain forecasts for the series y for times t, t+1, and t+2 using the estimated ARMA model. [8 marks] 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. (12 marks] What procedure might be used to estimate the parameters of an ARMA model? Ex- plain, briefly, how such a procedure operates, and why OLS is not appropriate.[5 marks] 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. (10 marks] Problem 1 (35 marks). Consider 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 ut-1 = -1.3 a) Obtain forecasts for the series y for times t, t+1, and t+2 using the estimated ARMA model. [8 marks] 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. (12 marks] What procedure might be used to estimate the parameters of an ARMA model? Ex- plain, briefly, how such a procedure operates, and why OLS is not appropriate.[5 marks] 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. (10 marks]