Find the Autocovariance Function of process $Y_t$

Question

Consider the processes $X_t = \phi X_{t-1} + v_t$ and $Y_t = \phi Y_{t-1} + X_t + e_t$, in which $|\phi| < 1$ and $v_1$ and $e_t$ are non-correlated random errors with zero mean and variances equal to $\sigma^2$. Based on these informations, find the autocovariance function of the process $Y_t$.

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First, I tried to find $E[Y_t]$:

$E[Y_t] = E[\phi Y_{t-1} + X_t + e_t] = \phi E[Y_{t-1}] + E[X_t] = \phi \mu_Y + \mu_X$

And then, I tried to find $Var(Y_t)$:

$Var(Y_t) = Var(\phi Y_{t-1} + X_t + e_t)$

But I am stuck at this point.

Answer 1

The autocovariance between $Y_{t+h}$ and $Y_t$ is defined as: $$\gamma(h)=cov(Y_{t+h},Y_t)=E(Y_{t+h}-E(Y_{t+h}))(Y_t-E(Y_t)).$$ If $Y_t$ is a stationary process, then $E(Y_{t+h})=E(Y_t)=\mu_Y$, therefore $$\gamma(h)=E(Y_{t+h}Y_t)-\mu_Y^2.$$