Question: consider the following arma1 process zt zt1...
Consider the following ARMA(1) process: , (1)
where εt is a zero-mean white noise process with variance , and assume |α|, |θ| < 1 and , which together make sure is covariance stationary.
(a) Calculate the conditional and unconditional means of , that is, and
(b) Set α = 0. Derive the autocovariance and autocorrelation function of this process for all lags as functions of the parameters θ and σ.
(c) Assume now . Calculate the conditional and unconditional variances of , that is, and.
Hint: for the unconditional variance, you might want to start by deriving the unconditional covariance between the variable and the innovation term, i.e., .
(d) Derive the autocovariance and autocorrelation for lags of 1 and 2 as functions of the parameters of the model. Hint: use the hint of part (c).