ACF of AR(1) model derivation

Question

AR(1) model: $X_t = \phi X_{t-1} + w_t$

Let $γ(h)$ denote the autocovariance function.

Note that $\gamma(1)=\text{Cov}(X_{t+1},X_t)=\text{Cov}(\phi X_t+w_{t+1}, X_t)=\phi\gamma(0)$

I've read a bunch of different derivations of the autocovariance function for AR(1) model and I still don't understand it. How do we get this part from the above?

$\gamma(1)=\phi\gamma(0)$

From my understanding,

$\gamma(0) = var(X_t) = var(\phi X_{t-1} + w_t) = \phi^2var(X_{t-1})+var(w_t) = \phi^2\gamma(0)+\sigma_w^2$

I'm having trouble getting more than this. It would be nice to have it explained in simple terms.

Answer 1

You left out a vital part of the definition of the AR(1) model: $$X_t=\phi X_{t-1} + w_t$$ where $w_t$ is uncorrelated noise, so $\text{Cov}(w_t, X_{t-1})=0$

\begin{split} \gamma(1)&=\text{Cov}(X_{t+1},X_t)\\ & =\text{Cov}(\phi X_t+w_{t+1}, X_t)\\ & =\text{Cov}(\phi X_t, X_t) + \text{Cov}(w_{t+1}, X_t)\\ &=\phi \text{Cov}(X_t, X_t) + 0\\ &=\phi\gamma(0) \end{split}