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So there is a VAR equation that is to be treated in Bayesian way:

$\mathbb{y} = (X \otimes I_k)\beta + e$ where $\beta$ is parameter.

We give parameters $\beta$ prior multivariate normal distribution with known mean $\beta^{\star}$ and covariance matrix $V_\beta$ and density is then

$f(\beta) = \frac{1}{2\pi}^{k^2p/2}|V_\beta|^{-1/2}\exp(\frac{1}{2}(\beta - \beta^{\star})V_\beta^{-1}(\beta - \beta^{\star}))$.

but I do not get how one gets likelihood function $\ell(\beta|y)$. Can anyone help here?

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First of all I guess that somewhere earlier it was stated that the elements of the vector of residual errors $\mathrm{e}$ iid and obey multivariate normal distribution $\mathrm{e}\sim \mathcal{N}(0, I_T\otimes\Sigma)$. So for the model $\mathrm{y}=(X \otimes I_k){\beta} + \mathrm{e}$: $$\mathrm{y}\sim \mathcal{N}((X \otimes I_k){\beta}, I_T\otimes\Sigma)$$ or its pdf will look like $$\ell(y|\beta)=\left(\frac{1}{2\pi}\right)^{kT/2}| I_T\otimes\Sigma|^{-1/2}\exp\left( -\frac{1}{2}(y-(X\otimes I_k){\beta})' (I_T\otimes\Sigma^{-1})(y-(X\otimes I_k){\beta})\right)$$ Maybe I'm wrong but I think that there is a mistake in notation: it should be $\ell(y|\beta)$ not $\ell(\beta|y)$

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