Suppose time-series data has been generated according to the following process:
where t is independent white noise. Our main interest is consistent estimation of Φ from realizations on y_{t}.
1) Provide conditions for this process to be stationary.
2) From hereonout, assume the process is stationary. Will OLS generally provide you with consistent point estimates of ? Can you give conditions under which it will? Provide the asymptotic distribution of OLS under these assumptions.
3) ARMA processes are generally estimated by ML. Do you have enough information to set up the (marginal or conditional) likelihood function?
4) Use the moment conditions under Question 4 to derive a consistent GMM estimator of (; ).
Does it require knowledge of the distribution of t? Note that we discussed asymptotic properties
of GMM estimators assuming a xed amount of moments.
5) Given your answer to the previous question, derive a lower bound on the variance of your esti-
mator.