6.11.1 Schwarz Information Criterion

6.11.1 Schwarz Information Criterion

The Schwarz Bayesian information criterion (BIC) ranks models by their posterior probabilities. To make this idea operational, we require model priors, $\pi ({\mathbf{M}}_{\mathbf{k}})$ , that sum to 1 across all models: ${\sum }_{M_k\in {\mathcal{M}}_K}\pi (M_k)=1$ . We also need priors over the parameters of each model, $\pi ({\beta }_k)$ . With these in place, the posterior probability of each model, $M_k$ , is given by

$$\pi (M_k,{\beta }_k|Z)=\frac{p({\beta }_k|Z,M_k)\pi ({\beta }_k)}{p_Z(Z)}\pi (M_k),\text{\hspace{1em}(6.45)}$$

where $p({\beta }_k|Z,M_k)$ is the likelihood for model $k,$ and $p_Z(Z)$ is the marginal density for the data, Z. For notational convenience we use $Z$ to denote the entire data set used to construct the forecasting model, $e.g.,\{y_t,z_t{\}}_{t=1}^T$ . To obtain the model posterior given the data, ${\beta }_k$ must be integrated out. To do this we employ an approximation that allows the normal prior to be used as an approximate conjugate for any distribution.

Using a two-term expansion for the likelihood around the maximum likelihood estimate ${\hat{\beta }}_k$ , we have

$$\begin{array}{r}\ln p({\beta }_k|Z)\approx \ln p({\hat{\beta }}_k|Z)+\frac{1}{2}({\hat{\beta }}_k-{\beta }_k{)}^{\prime }(TI({\hat{\beta }}_k))({\hat{\beta }}_k-{\beta }_k),\end{array}\text{\hspace{1em}(6.46)}$$

where $TI(\widehat{{\beta }_k})$ is the average Fisher information evaluated at the sample estimate scaled by the sample size T, which is an $O_P(1)$ term. Exponentiating (6.46) and integrating over ${\beta }_k$ we get

$$\begin{array}{l}\int p({\beta }_k|Z)\pi ({\beta }_k)d{\beta }_k\\ \approx \int p({\hat{\beta }}_k|Z)\exp \left\{-\frac{1}{2}({\hat{\beta }}_k-{\beta }_k{)}^{\prime }(TI({\hat{\beta }}_k))({\hat{\beta }}_k-{\beta }_k)\right\}\pi ({\beta }_k)d{\beta }_k\\ =p({\hat{\beta }}_k|Z)\int \exp \left\{-\frac{1}{2}({\hat{\beta }}_k-{\beta }_k{)}^{\prime }(TI({\hat{\beta }}_k))({\hat{\beta }}_k-{\beta }_k)\right\}\pi ({\beta }_k)d{\beta }_k.\end{array}\text{\hspace{1em}(6.47)}$$

This approximate result allows the normal distribution to become a conjugate. Setting the prior over the coefficients to $\pi ({\beta }_k)=1$ and evaluating the integral in (6.47) yields

$$\int P({\beta }_k|Z)\pi ({\beta }_k)d{\beta }_k\approx \hat{p}({\hat{\beta }}_k|Z)(2\pi {)}^{n_k/2}|TI({\hat{\beta }}_k){|}^{-1/2}.$$

Using this result in the expression for the posterior probability in (6.45), we have

$$\begin{array}{r}\int \pi (M_k,{\beta }_k)d{\beta }_k\approx \pi (M_k)\frac{\hat{P}({\hat{\beta }}_k|Z)(2\pi {)}^{n_k/2}|TI({\hat{\beta }}_k){|}^{-1/2}}{\hat{P}z(Z)}\\ =\pi (M_k)\frac{\hat{P}({\hat{\beta }}_k|Z)(2\pi /T{)}^{n_k/2}|I({\hat{\beta }}_k){|}^{-1/2}}{\hat{P}z(Z)}.\end{array}\text{\hspace{1em}(6.48)}$$

Taking logarithms in (6.48),

$$\begin{array}{l}\ln \int \pi ({\mathcal{M}}_k,{\beta }_k)d{\beta }_k\\ \approx \ln \pi ({\mathcal{M}}_k)-\ln ({\gamma }_Z(Z))+\ln (\gamma ({\hat{\beta }}_k|Z))+\frac{n_k}{2}\ln (2\pi /T)-\frac{1}{2}\ln |\mathrm{I}({\hat{\beta }}_k)|.\end{array}\text{\hspace{1em}(6.49)}$$

The logarithm of the marginal distribution of Z is the same for each model and so can be ignored in model rankings. Maximizing (6.49) is therefore the same as minimizing

$$-2\ln \pi (M_k)-2\ln (p({\hat{\beta }}_k|Z))+n_k\ln (T/2\pi )+\ln |I[({\hat{\beta }}_k)|.\text{\hspace{1em}(6.50)}$$

As the sample size, T , gets large, the first and last terms in (6.50) are bounded and so remain “small” and can be ignored, while the second and third terms get larger.

This suggests choosing the model that minimizes the following criterion:

$$-2\ln (p({\hat{\beta }}_k|Z))+n_k\ln (T),\text{\hspace{1em}(6.51)}$$

or, as is more common in practice, minimizing (6.51) divided by the sample size,

$$\begin{array}{r}\mathrm{BIC}(k)=-2T^{-1}\ln (p({\hat{\beta }}_k|Z))+n_k\ln (T)/T.\end{array}\text{\hspace{1em}(6.52)}$$

Choosing the model that minimizes the expression in (6.51) or (6.52) is thus equivalent, in large samples, to selecting the model with the highest posterior probability.