12.2.2 Bayesian Approaches

12.2.2 Bayesian Approaches

Given a conditional likelihood $p_Y(y_{t+1}|z_t,\beta )$ and a set of priors on the parameters, Bayesian methods can be used to construct posterior distributions for $\beta$ and hence construct a predictive distribution $p_Y(y_{t+1}|z_t)$ . A uniform prior on $\beta$ results in the MLE as the posterior mean; for a normal prior the posterior mode is used as an asymptotic approximation to the mean of $\beta .$ Following Albert and Chib (1993), a more typical approach is to use Gibbs sampling to directly estimate a distribution for some latent variable $y_{t+1}^{\ast }$ underlying the observed outcome $y_{t+1}$ . Using a normal prior for $\beta$ and replacing the unknown $y_{t+1}^{\ast }$ with draws from the sampler, we can characterize properties of the posterior density such as its mode in ways similar to those for the normal linear regression model discussed in chapter 5. Expressions for the posterior mean of $\beta$ would replace the unknown outcomes with such draws. Geweke and Whiteman (2006) give a brief discussion of this point.

A number of papers on event forecasting have adopted Bayesian methods. Chauvet and Potter (2005) use Bayesian methods to predict recessions conditional on information from the term structure of interest rates. Let ${\gamma }_t=\{0,1\}$ be an indicator variable that tracks recessions $(y_t=1)$ and expansions $(y_t=0)$ , while ${\mathbf{y}}_t^{\ast }$ is the underlying, latent state of the economy. We then have

$$\begin{array}{r}{y}_t=0\mathrm{if}{y}_t^{\ast }<0,\\ y_t=1\mathrm{if}{y}_t^{\ast }\ge 0.\end{array}\text{\hspace{1em}(12.13)}$$

In turn, ${\mathbf{y}}_t^{\ast }$ follows the process

$${\mathbf{y}}_{t+1}^{\ast }={\beta }_0+{\beta }_1{\mathbf{S}}_t+{\epsilon }_{t+1},{\epsilon }_{t+1}\sim \mathrm{N}(0,{\sigma }_{\epsilon }^2),\text{\hspace{1em}(12.14)}$$

where $S_t$ is a variable such as the spread between 10-year and 3-month Treasury rates. It follows from (12.13) and (12.14) that the probability that the economy is in a recession next period is $\mathrm{Pr}(y_{t+1}^{\ast }\ge 0|S_t,\beta )=\Phi ({\beta }_0+{\beta }_1{S}_t)$ , where $\Phi$ is the standard Gaussian cumulative distribution function.

Chauvet and Potter also entertain a dynamic probit specification that changes (12.13) to

$$y_{t+1}^{\ast }={\beta }_0+{\beta }_1{S}_t+{\beta }_2{y}_t^{\ast }+{\epsilon }_{t+1},\text{\hspace{1em}(12.15)}$$

for $|{\beta }_2|<1$ . Chauvet and Potter (2005) use the Gibbs sampler to evaluate the likelihood function and estimate the parameters of this model. An empirical analysis of the 2001 US recession suggests that information in the yield curve helped to predict an economic slowdown and that probit models that do not account for persistence in different phases of the business cycle succeeded in predicting the recession with a high probability.

Dueker (2005) proposes an approach to generating dynamic forecasts of qualitative variables embedded in a $\mathrm{VAR}$ , labeled Qual VAR. The approach is based on a dynamic probit specification that includes lagged values of the dependent variable. Let $y^{\ast }$ be some continuous latent variable that determines the sign of the binary dependent variable, $y\in \{0,1\}$ , and assume that ${\mathbf{y}}_t^{\ast }$ is generated by a dynamic probit model very similar to that in (12.13),

$${\mathbf{y}}_{t+1}^{\ast }=\rho {\mathbf{y}}_t^{\ast }+{\beta }^{\prime }{\mathbf{z}}_t+{\mathbf{\epsilon }}_{t+1},{\mathbf{\epsilon }}_t\sim \mathrm{N}(0,1),\text{\hspace{1em}(12.16)}$$

where $z_{t-1}$ is a set of predictors. Dueker (2005) embeds this specification within a Qual VAR for $Y_t=(X_t^{\prime }{y}_t^{\ast }{)}^{\prime }$ :

$$\sum _{i=0}^P{A}_i{Y}_{t-i}=\mu +{\epsilon }_t,{\epsilon }_t\sim \mathrm{N}(0,\Sigma ).$$

Dueker uses MCMC estimation to draw sequences of values for $A=(A_1,\dots ,A_{},\dots ,A_{})$ $A_{\mathbf{p}})$ , , and $y^{\ast }$ , using the following blocks:

$$\begin{array}{r}p(A^{i+1}|\{{\mathbf{y}}_t^{\ast (i)}{\}}_{i=1}^T,\{{\mathbf{z}}_t{\}}_{t=1}^T,{\Sigma }^{(i)})\sim \mathrm{Normal},\end{array}$$

$$p({\Sigma }^{(i+1)}|\{y_t^{\ast (i)}{\}}_{i=1}^T,\{z_t{\}}_{t=1}^T,A^{(i+1)})\sim \mathrm{InvertedWishart},\text{\hspace{1em}(12.17)}$$

$$\begin{array}{r}\mathcal{P}(y_t^{\ast (i+1)}|\{y_j^{\ast (i+1)}{\}}_{j<t},\{y_k^{\ast (i)}{\}}_{k>t},\{z_t{\}}_{t=1}^T,A^{(i+1)},{\Sigma }^{(i+1)})\sim \mathrm{TruncatedNormal}.\end{array}$$

In an empirical exercise to forecast US recessions where y is the NBER’s binary recession indicator, Dueker finds that a Qual VAR model did well in predicting the 2001 US recession out-of-sample. One advantage of this approach is that it generates recession probabilities not only for the next period but for multiple periods ahead in time.

12.2.3 Nonparametric Approaches

Nonparametric approaches for estimating distributional forecasts of binary variables parallel those discussed in chapter 11 adapted to the binary forecasting problem, the key difference being that the probability forecasts must be constrained to lie on the unit interval.

For example, the local linear or local polynomial regressions from chapter 11 could be applied after truncating forecasts outside the unit interval to equal 0 or 1. As an alternative, Gozalo and Linton (2000) suggest replacing the local linear polynomial with the probit or logit objective function. Their approach retains the property of the logit and probit models that it falls between 0 and 1, but allows the model to vary with the predictors. Estimation of the model parameters

involves computing

$$\hat{\beta }=\arg \underset{\beta }{\min }\sum _{t=0}^{T-1}{\left(y_{t+1}-\varphi ({\beta }^{\prime }(z_t-z_T))\right)}^{2}K(z_t-z_T),\text{\hspace{1em}(12.18)}$$

where $\varphi$ is a function restricted to the 0–1 interval. The forecasting model depends on $z_T$ which, as discussed in chapter 11, may be useful especially when the true functional form is unknown.