Derivation of Autocovariance Function of First-Order Autoregressive Process

Question

In my textbook, the autocovariance of the AR(1) model is derived as such: $$Y_t=\phi Y_{t-1}+e_t$$ After multiplying both sides by $Y_{t-k}(k=1,2,...)$ and take expected values, you get: $$E(Y_{t-k}Y_t)=\phi E(Y_{t-k}Y_{t-1})+E(e_tY_{t-k})$$ which implies that $$\gamma_k=\phi\gamma_{k-1}+E(e_tY_{t-k})$$ However, I don't understand how $E(Y_{t-k}Y_t)$ becomes $\gamma_k$ and how $\phi E(Y_{t-k}Y_{t-1})$ becomes $\phi\gamma_{k-1}$.

Answer 1

Provided there is a centered, weakly stationary solution, with the autocorrelation function $\gamma_k=E(Y_{0}Y_{k}), k\in\mathbb{Z},$ holds

$$ E(Y_{t-k}Y_t) = E(Y_0Y_{t-(t-k)}) = E(Y_0Y_k) = \gamma_k, $$

and also

$$ \phi E(Y_{t-k}Y_{t-1}) = \phi E(Y_0Y_{t-1-(t-k)}) = \phi E(Y_0Y_{k-1}) = \phi\gamma_{k-1}. $$