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?

Answers 1

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.

October 10, 2018 20:54 PM

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