2. Finance
3. consider the following arma1 process zt zt1...

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

###### Question details

Consider the following ARMA(1) process: , (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, 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 $a\ne \theta$. Calculate the conditional and unconditional variances of ${z}_{t}$ , 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).

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