# Conjugate priors for dynamic model $x_{t+1}=Ax_{t}+\eta_t$

by An old man in the sea.   Last Updated October 10, 2018 21:19 PM

What conjugate priors do we have for the model(multivariate) $$x_{t+1}=Ax_{t}+\eta_t$$, where $$\eta_t\overset{iid}\sim N(0,\Sigma)$$?

I was thinking of using $$\tilde{x}=Diag[x_1,...,x_{n-1}]$$, $$\tilde{y}=Diag[x_2,...,x_{n}]$$ and simply use the usual conjugate priors for the Bayesian multivariate linear regression.

Will there be any problem, if I do this?

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### Bottom Line

You're right. You can rewrite the model as a multivariate linear regression and simply use the conjugate prior for that model.

### Some Detail

The model you wrote down is a vector autoregression. You wrote it with a single lag and no intercept term, but the general model with $$p$$ lags is

$$\underbrace{\mathbf{y}_t}_{n\times 1}=\underbrace{\mathbf{B}_0}_{n\times 1}+\underbrace{\mathbf{B}_1}_{n\times n}\mathbf{y}_{t-1}+...+\mathbf{B}_p\mathbf{y}_{t-p}+\underbrace{\mathbf{u}_t}_{n\times 1},\quad\mathbf{u}_t\sim N(\mathbf{0},\,\underbrace{\boldsymbol{\Sigma}}_{n\times n}).$$ If for each $$t$$ you define the vector $$\underbrace{\mathbf{x}_t}_{(np+1)\times 1} = [\mathbf{y}'_{t-1} \quad \mathbf{y}'_{t-2} \quad...\quad \mathbf{y}'_{t-p}\quad 1]'$$ and the matrix $$\underbrace{\mathbf{B}}_{(np+1)\times n}=\begin{bmatrix}\mathbf{B}'_1\\\mathbf{B}'_2\\\vdots\\\mathbf{B}'_p\\\mathbf{B}'_0\end{bmatrix}$$ then you can rewrite the VAR model as a multivariate regression: $$\mathbf{y}'_t=\mathbf{x}'_t\mathbf{B}+\mathbf{u}'_t,\quad\mathbf{u}_t\sim N(\mathbf{0},\,\boldsymbol{\Sigma}).$$ The conjugate prior for $$\mathbf{B}$$ and $$\mathbf{\Sigma}$$ is the Matrix Normal Inverse Wishart Distribution. That is $$\mathbf{\Sigma}\sim IW(d_0,\, \mathbf{\Psi}_0) \\ \mathbf{B}\,|\,\mathbf{\Sigma}\sim MN(\bar{\mathbf{B}}_0,\, \mathbf{\Omega}^{-1}_0,\,\mathbf{\Sigma}).$$

$$d_0$$, $$\mathbf{\Psi}_0$$, $$\bar{\mathbf{B}}_0$$, and $$\mathbf{\Omega}^{-1}_0$$ are prior hyperparameters that you set as the researcher.

bamts
October 10, 2018 20:54 PM