9.3.1 Bayesian Estimation

9.3.1 Bayesian Estimation

Assuming that $u_t\sim$ ind $\mathrm{N}(0,\Sigma )$ , then ${\gamma }_t$ has a density $f(y_t|Y_{t-1},\theta )=$ $\mathrm{N}(y_t;z_{t-1}^{\prime }\beta ,\Sigma )$ , where $\beta =(c,A_1,\dots ,A_p{)}^{\prime },z_{t-1}=(1,y_{t-1}^{\prime },\dots ,y_{t-p}^{\prime }{)}^{\prime }$ and $Y_{t-1}=\{y_{\tau }\},\tau =1,\dots ,t-1$ . The likelihood function for Y can then be written as

$$\mathcal{L}(Y|\beta ,\Sigma )=\frac{1}{(2\pi {)}^{nT/2}}|\Sigma {|}^{-T/2}\exp \left(\frac{-1}{2}\mathrm{tr}\left[{\Sigma }^{-1}(Y-Z\beta {)}^{\prime }(Y-Z\beta )\right]\right),$$

where Y and $Z$ are defined above. Assuming uninformative (diffuse) priors with a uniform prior on $\beta$ and a Jeffrey’s prior on -, we have

$$p(\beta ,\Sigma )\propto |\Sigma {|}^{-(n+1)/2}.$$

Conditional on the data up to time T,

$$\Sigma |Y_T\sim {\mathrm{iW}}_n(S,T-np-1),$$

where $\mathrm{i}{\mathsf{W}}_n$ denotes the inverse Wishart distribution, $S=(Y-Z\hat{\beta }{)}^{\prime }(Y-Z\hat{\beta })$ , and $\hat{\beta }=(Z^{\prime }Z{)}^{-1}{Z}^{\prime }Y$ is the OLS estimate.

Conditional on - and $Y_T,\beta$ follows a normal distribution,5

$$\begin{array}{r}\mathrm{vec}(\beta )|Y_T,\Sigma \sim N(\hat{\beta },\Sigma \otimes (Z^{\prime }Z{)}^{-1}).\end{array}$$

Combining the prior in with the likelihood in, the posterior $\beta |Y_T$ has a t-distribution with mean $\mathrm{vec}(\hat{\mathbf{\beta }})$ , variance proportional to $S\otimes (Z^{\prime }Z{)}^{-1}$ , and degrees of freedom $T-np-1$ . Hence, the marginal posterior of $\dot{\mathbf{\beta }}$ conditioned only on the data has a t-distribution.

What do these results imply for the predictive distribution? At the one-period forecast horizon, $h=1$ , we have

$$y_{T+1}=c+A_1{y}_T+A_2{y}_{T-1}+\cdot \cdot \cdot +A_p{y}_{T-p+1}+u_{T+1},$$

which again follows a t-distribution. For $h=2$ , we have

$$\begin{array}{l}{y}_{T+2}=(A_1^2+A_2)y_T+(A_2{A}_1+A_3)y_{T-1}+\cdot \cdot \cdot +(A_{p-1}{A}_1+A_p)y_{T-p+2}\\ \\ +A_p{A}_1{y}_{T-p+1}+A_{1}c+c+A_1{u}_{T+1}+u_{T+2}.\end{array}$$

This is a nonlinear function of the parameters and so there is no closed-form expression for the density of ${\gamma }_{T+h}$ for $h\ge 2$ . Fortunately the model easily lends itself to generate out-of-sample forecasts. Following (Karlsson, 2013, algorithm 1), this works as follows. To make the j th draw, first a value of ${\Sigma }^j|Y_T$ is generated from the inverse Wishart distribution using. Second, conditional on $Y_T$ ${\Sigma }^j,{\beta }^j$ is drawn from the normal distribution using. Third, $u_{T+1}^j,\dots ,u_{T+h}^j$ are drawn from independent distributions $u_{t+i}^j\sim \mathrm{N}(0,{\Sigma }^j)$ . These draws, along with the parameter draws ${\Sigma }^j,{\beta }^j$ are then used to iteratively generate a single draw of the out-of-sample value of $\dot{\mathbf{y}},$

$$y_{T+h}^j=c^j+\sum _{i=1}^{\mathcal{P}}{A}_i^j{y}_{T+h-i}^j+u_{T+h}^j,$$

where $y_{T+h-i}^j=y_{T+h-i}$ for $i\ge h$ . Repeating these steps J times yields a sample of draws from the posterior predictive distribution $\left\{y_{T+1}^j,\dots ,y_{T+h}^j\right\},j=1,\dots ,J$ These can in turn be used to compute statistics such as the mean forecast,

$${\bar{y}}_{T+h}=J^{-1}\sum _{j=1}^J{y}_{T+h}^j,$$

which minimizes a quadratic loss function. Using this approach we can also construct statistics for the posterior predictive distribution such as the posterior predictive variance or $(1-\alpha )$ % interval forecasts.