Here are four questions. If you think you are not able to solve them until 3:00 pm Apr.09 (It's today), then just tell me that you can't or else not worth to solve.

1. Let {$1, . . . ,sn} be a sample of size 'n. from the MAH) model, Xi 2 (EH + zt, with {2t} m wmo), and 3 at 0. Assume that the usual sample ACVF and ACF converge to the corresponding model quantities (in prob ability) as \":1 - 00, Le, assume that for all lags h: 'i'Ul-l > THE) and 155(k) > PUB)- Not knowing which model the data came from, suppose we decide to t an ALI-{(1) to it: X1; MM 2 2:, with {2:} N wmo, 02). This is called model misspecication. {a} Obtain the Yule Waiter estimators for ()5 and 02 based on the sample {3:1, . . . , In}. Call them :33 and 6'2, and express them in terms of the sample ACVF and ACF. {b} Show that your YuleWalker estimators in {a} converge [in probability as n ) co] to false parameter values. That is1 show that gin } or and (i2 > , determine the 1values of :1 and ,3, and show that :1 75 l9 and 5 72 r2. 2. Suppose that a sample of n = 200 observations from the causal and invertible ARMA(1,1) model Xt - Xt-1 = 0Zt-1+ Zt, { Zt} ~ IID(0, 0?), results in the MLEs ($, 0) = (0.4, -0.7). (a) Is the estimate d = 0.4 significant? Answer this question by carrying out a level o = 0.05 hypothesis test of the null that d = 0 versus the alternative that o 0. (b) Is the estimate 0 = -0.7 significant? Answer this question by constructing a 95% confidence interval for 0. (c) For a given value of o, is it possible to choose a value for O in the specification of the ARMA(1,1) model so that the resulting MLEs o and O will be approximately uncorrelated (for large n)? Carefully justify your answer.3. Consider the model: Xt = Xt-6+Zt, with {Z{} ~ WN(0, 02) and |0|