I want to know the first order autocorrelation of a VAR(1) model. That can be calculated as follows $\rho(1)=\frac{\gamma(1)}{\gamma(0)}$ where $\gamma(1)$ denotes the covariance and $\gamma(0)$ denotes the variance.

A VAR(1) model can be described as follows: $$r_t=c+\Gamma r_{t-1}+\varepsilon_t$$ with $\varepsilon_t \sim N(0,\Sigma)$. Furthermore, $r_t$, $r_{t-1}$ and $\varepsilon_t$ all are a $N\times 1$ vector and the parameter matrix $\Gamma$ is a $N\times N$ matrix.

Taking the variance of $r_t$ gives us $\gamma(0) = Var(r_t) = Var(c+\Gamma r_{t-1}+\varepsilon_t)=\Gamma\times Var(r_{t-1})\times \Gamma' + \Sigma$.

So this gives us $\gamma(0) = \Gamma\times \gamma(0)\times \Gamma' + \Sigma$.

For $\gamma(1)$ I get $\gamma(1)=cov(r_t,r_{t-1})=cov(c+\Gamma r_{t-1}+\varepsilon_t,r_{t-1})=\Gamma cov(r_{t-1},r_{t-1})=\Gamma \gamma(0)$.

I just do not know how to implement the last part, i.e. $\rho(1)=\frac{\gamma(1)}{\gamma(0)}$, because of the matrices. I hope someone can help me with that?

I do know that the $\rho(1)$ of an AR(1) model is simply $\phi$ (i.e. the parameter of the lagged variable in an AR(1) model, see Finding the ACF of AR(1) process for instance). For its multivariate case, I also want to know the autocorrelation coefficient, hence this question.


I work with functional time series, in particular with the functional ARMA$(p,q)-$model. If you take a look e.g. at this publication http://arxiv.org/pdf/math/0509256v1.pdf from Mas (2005) or at "Function Spaces" a monographie from D. Bosq (2000) you can see how it works for the functional setting, hence for yours.

Division by $\gamma(0)$ doesn't work in your case, since it's a matrix. But you can consider divison by a matrix as multiplication of the inverse, if it does exist ($\text{det}(\gamma(0)) \neq 0 ?$).

I hope that's helpful.

  • $\begingroup$ Could you please give a little guidance on the paper you refer to in your link? From what page to what page should I read? Is it the paragraph 2.3 " A smoothness condition on the autocorrelation operator"? $\endgroup$ – Eren Jun 19 '16 at 11:16
  • $\begingroup$ Yes. Take a look at equation (2) in chapter 2.1. $\Gamma$ be can interpreted as $\gamma(0)$ in your case and $\Delta$ as $\gamma(1).$ All u gonna do, is to multiply the inverse on both sides, then. $\endgroup$ – Obriareos Jun 19 '16 at 11:37
  • $\begingroup$ I basically wanted to say: you deal with matrices and have to compute inverses. $\endgroup$ – Obriareos Jun 19 '16 at 11:41
  • $\begingroup$ Yeah, I know but I don't really know how to isolate the $\gamma(0)$ part with inverses. That's why I asked this question. $\endgroup$ – Eren Jun 19 '16 at 11:47
  • $\begingroup$ I don't understand, since your able to compute $\gamma^{-1}(0)$ $\endgroup$ – Obriareos Jun 19 '16 at 11:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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