9.3.3 Alternative Priors

9.3.3 Alternative Priors

The Litterman priors are quite restrictive and so are often relaxed. Extensions allow for a nondiagonal prior variance–covariance matrix -, and also allow the priors on $\beta$ and - to be correlated. Bayesian posterior computation can be handled by methods such as the Gibbs sampler. The conjugate combination of a normal prior distribution on $\beta$ and an independent Wishart distribution on - does not give a closed-form solution for the means of the posterior but yields a posterior distribution that is easy to simulate from. Karlsson (2013) and Kadiyala and Karlsson (1997) examine and extend conventional priors by introducing a more general prior which they call the Extended Natural Conjugate (ENC) prior which relaxes the requirements on the prior on $\beta .$ In evaluations, there seem to be modest gains from this relaxation of the restrictions on prior distributions. While most of these methods can be thought of as shrinkage methods, they do not differentiate between parameters by removing some while keeping others. To remedy this, George, Sun, and Ni (2008) use a mixture prior with mixture weights estimated from the data. The mixture is between a very tight prior and a less tight prior, so if the estimation method gives a large weight to the tight prior the coefficient is effectively set to 0.

A number of improvements to the specification of the priors on β appear to have been useful in empirical work. In the context of macroeconomic forecasting, rather than employing the random walk or ad hoc priors, Ingram and Whiteman (1994) and Del Negro and Schorfheide (2004) suggest setting priors in a manner that implies shrinking towards the coefficients that arise from a linearized general equilibrium model of the macroeconomy. The idea is that a real business cycle (RBC) model can be used to generate implied variances and covariances of the data. These can be combined to obtain implied VAR parameter values, which form the priors for the OLS coefficients. The methods then shrink the OLS estimates towards the implied values. Ingram and Whiteman (1994) develop a model that implies a VAR with one lag, while priors on lags of higher order are set to 0. This is not strictly a fully Bayesian approach since the priors are applied to the reduced-form regression rather than to the likelihood for the data. Ingram and Whiteman call it a “limited information Bayesian” approach and find that their method results in forecasts comparable to those based on a Litterman prior but better than those from an unrestricted VAR.

Villani (2009) suggests an alternative approach to eliciting priors. Motivated by economic models, Villani places informative priors directly on the steady state of the model rather than on the coefficients of the VAR. Villani uses the unconditional means of the variables in the VAR to model the steady state.

Wright (2013) instead suggests using long-run forecasts from survey data to pin down the steady state. Using the reparameterized VAR,

$$(y_t-\mu )=A_1(y_{t-1}-\mu )+\cdot \cdot \cdot +A_p(y_{t-p}-\mu )+u_t,$$

where $E\left[u_t{u}_t^{\prime }=\Sigma \right]$ , Wright suggests using the priors

$$\mathrm{vec}(A_{1t}^{\prime },\dots ,A_{pt}^{\prime })\sim N({\theta }_{At},{\Omega }_A),$$

$${\mu }_t\sim \mathrm{N}({\theta }_{\mu t},{\Omega }_{\mu t}),$$

$$\begin{array}{r}{\Sigma }_t\sim |{\Sigma }_t{|}^{-(n+1)/2}.\end{array}$$

Notice that the parameters are allowed to vary with t, the point in time where the model is estimated and the forecast is generated. Specifically, Wright (2011) proposes to set the prior mean ${\theta }_{\mu t}$ at the mean of the most recent long-run forecasts obtained from the Blue Chip survey. For variables such as CPI inflation, growth in industrial production, and T-bill yields, Wright reports notable gains in out-of-sample forecast performance from using this approach relative to both VARs estimated by least squares or Bayesian VARs based on the Minnesota prior.