# VAR coefficients

Say I have a VAR(p) model without any noise (a multidimensional AR model without noise). How would I go about calculating the coefficients that are MSE optimal? Is there an extension to the Yule Walker equations for multidimensional data?

Edit: Maybe an equation like the following would help?

$$\hat Y_{k+1}^{}=AY_{k}$$

where any general $$\ Y_{i}=[Y_{1} \ Y_{2}\ Y_{3}\ Y_{4}\ ...... \ Y_{p}]^{T}$$ and I want to estimate the optimal A in MSE sense. Such a problem should fit under a Weiner filtering approach in my opinion. Would using the matrix orthogonality principle be sufficient to solve this?

• Contrary to what you are implying, an AR model does contain a noise component. E.g. in an AR(1): $y_t=c+\varphi_1 y_{t-1}+\varepsilon_t$, it is $\varepsilon_t$. Given that, can you specify more precisely what your VAR(p) model without any noise is? May 13, 2022 at 7:55
• @RichardHardy I updated my question and made it clearer. May 14, 2022 at 3:34

$$Y_{k+1}=AY_{k}$$ where any general $$\ Y_{i}=[Y_{1} \ Y_{2}\ Y_{3}\ Y_{4}\ ...... \ Y_{p}]^{T}$$ and I want to estimate the optimal A in MSE sense.
So you have an exact, deterministic, linear relationship. The only thing that is missing is the coefficient values in $$A$$. Then equation-by-equation OLS will return you exact estimates, and they will be MSE-optimal.
I am trying to figure out a problem where this is perhaps not even $$Y_{k+1}$$ but rather maybe just an estimate of it.
If you change your problem by making $$Y_{k+1}$$ only observable with some error while $$Y_1,\dots,Y_k$$ are observable without error, you get a multivariate multiple regression. There is no MSE-optimal estimator for that. If you were to restrict your attention to linear estimators and the error variance were given, it would be a form of ridge regression (a univariate case is discussed in Dave Giles' blog "A Regression "Estimator" that Minimizes MSE". But that is perhaps too restrictive.