Appendix C: Bayesian/Schwarz Information Criterion (BIC/SIC)

The Bayesian information criterion (BIC) or Schwarz criterion (SIC) is a measure of the goodness of fit of a statistical model. It is often used as a criterion for model selection among a finite set of models. It is based on the log-likelihood function (LLF) and closely related to Akaike's information criterion.

Similar to AIC, the BIC introduces a penalty term for the number of parameters in the model, but the penalty is larger than one in the AIC.

1. In general, the BIC is defined as:
   
   $$\mathit{BIC}=k\times\ln{n} -2\times\ln(L)$$
   
   Where:
   
   • $k$ is the number of model parameters.
   • $\ln(L)$ is the log-likelihood function for the statistical model.
2. Given any two estimated models, the model with the lower value of BIC is preferred; a lower BIC implies either fewer explanatory variables, better fit, or both.

Remarks

1. It is essential to remember that the BIC can be used to compare estimated models only when the numerical values of the dependent variable are identical for all estimates being compared.
2. BIC has been widely used for model identification in time series and linear regression. It can, however, be applied quite widely to any set of maximum likelihood-based models.