*Question:* consider the following arma1 process zt zt1...

###### Question details

Consider the following ARMA(1) process: ${\mathit{z}}_{\mathbf{t}\mathbf{}}\mathbf{=}\mathbf{}\mathit{\gamma}\mathbf{}\mathbf{+}\mathbf{}\mathit{\alpha}{\mathit{z}}_{\mathbf{t}\mathbf{-}\mathbf{1}}\mathbf{}\mathbf{+}\mathbf{}{\mathit{\epsilon}}_{\mathbf{t}}\mathbf{}\mathbf{+}\mathbf{}\mathit{\theta}{\mathit{\epsilon}}_{\mathbf{t}\mathbf{-}\mathbf{1}}$, (1)

where εt is a zero-mean white noise process with variance ${\sigma}^{2}$ , and assume |α|, |θ| < 1 and $\alpha +\theta \ne 0$, which together make sure ${z}_{t}$ is covariance stationary.

(a) Calculate the conditional and unconditional means of ${z}_{t}$ , that is, ${E}_{t-1}[{z}_{t}]$ and $E\left[{z}_{t}\right].$

(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 $a\ne \theta $. Calculate the conditional and unconditional variances of ${z}_{t}$ , that is, $Va{r}_{t-1}[{z}_{t}]$ and$Var[{z}_{t}]$.

Hint: for the unconditional variance, you might want to start by deriving the unconditional covariance between the variable and the innovation term, i.e., $Cov[{z}_{t},{\epsilon}_{t}]$.

(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).