SVAR: proof of the $(K^2-K)/2$ restrictions and identifiability?

Question

I'm currently using Structural vector autoregressive models by Kevin Kotzé to learn Vector Autoregression. One of the points that it makes is the following:

the number of restrictions that we need to impose is equivalent to the number of terms in the lower (or upper) triangle of the B matrix, which is $(K^2-K)/2$

where our model is the following VAR(1) ($y_t$ is a K-dimensional vector of variables):

$B y_t = \Gamma_0 + \Gamma_1 y_{t-1} + \varepsilon_t$, which leads to the following reduced-form VAR:

$y_t = A_0 + A_1 y_{t-1} + u_t$

In the bivariate case, this seems to make sense as with $K=2$, we can easily solve the system of equations and find all the structural VAR parameters. However, it's not totally obvious to me:

a) Why $(K^2-K)/2$ is the magic number that allows us to deduce the structural parameters - I understand it is equivalent to the upper triangle of B.

b) How we know that given this amount of restrictions, we can always solve for all the structural parameters. I calculated by hand and was able to convince myself for $K=2$, but it's not immediately obvious that higher dimensions will follow similarly.

I'm wondering if there are any proofs or more rigorous illustrations as to why this result can be true for all $K$.

Answer 1

We can use a simple counting argument. In general: the number of additional restrictions = number of unknowns in the structural model - number of estimates in the reduced form.

Consider the structural equation $$ B y_t = \Gamma_0 + \Gamma_1 y_{t-1} + \varepsilon_t. $$ Let us count the number of unknowns.

• The matrix $B$ has $K^2$ elements. However, following the slides you posted here the diagonal elements of $B$ are known to equal 1. So in total $B$ has $K^2 - K$ unknowns.
• The vector $\Gamma_0$ has $K$ unknowns
• The vector $\Gamma_1$ has $K^2$ unknowns
• The errors $\varepsilon_t$ have mean zero and according to the slides, we assume that the covariances between the disturbance terms are also zero. So the only unknown for the variance-covariance matrix here are the variances of which there are $K$.

Adding everything together we have $2 K^2 + K$ unknowns

Now consider the reduced form: $$ y_t = A_0 + A_1 y_{t-1} + u_t. $$ Here $A_0 = B^{-1} \Gamma_0, A_1 = B^{-1} \Gamma_1$ and $u_t = B^{-1} \varepsilon_t$.

Let us count the number of coefficients we can estimate.

• The vector $A_0$ has $K$ coefficients.
• The matrix $A_1$ had $K^2$ coefficients.
• From the error $u_t$ we can estimates its (symmetric) variance-covariance matrix, which has $K + K(K-1)/2$ terms.

In total we have $K + K^2 + K + K(K-1)/2 = K^2 + 2K + K(K-1)/2$ estimates.

So taking the difference between the number of unknowns and the number of estimates, we obtain: $$ 2 K^2 + K - K^2 - 2K - K(K-1)/2 = \frac{K(K-1)}{2}. $$

Note that this is an estimate. In particular, imposing $K(K-1)/2$ additional restrictions will (in general) be necessary to identify all structural coefficients, but it might not be sufficient.