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?


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.

  • $\begingroup$ $var(X_t) = cov(X_t,X_t)$. $\endgroup$
    – Taylor
    Commented Oct 28, 2017 at 22:32

1 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}


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.