Schwarz Information Criteria

Akaike allowed for the existence of other AlC-like criteria that could be derived by making different assumptions regarding the distribution of the data. Schwarz (1978), in a Bayesian context, developed the Bayesian Information Criteria (BIC), which is also called the Schwarz Information Criteria (SIC) or Schwarz s criteria (SC), as... 

Prom (2.62) it can be taken that the model s goodness of fit and the number of parameters used are counterbalanced. Among a set of model specifications, the specification with minimal IC value is recommended. In other words, information criteria aim at minimizing the residuals variance with as few parameters as possible. Often used information criteria for time series models are the Akaike information criterion (AIC)," the Schwarz information criterion (SIC) or the Hannan-Quinn information criterion (HQIC) with the following k functions ... 

Ludden TM, Beal SL, Sheiner LB. Comparison of the Akaike Information Criterion, the Schwarz criterion and the F test as guides to model selection. /PAar-macokinet Biopharm 1994 22 431-45. 

A common form of model selection is to maximize the likelihood that the data arose under the model. For non-Bayesian analysis this is the basis of the likelihood ratio test, where the difference of two -2LL (where LL denotes the log-Ukelihood) for nested models is assumed to be approximately asymptotically chi-squared distributed. A Bayesian approach— see also the Schwarz criterion (36)—is based on computation of the Bayesian information criterion (BIC), which minimizes the KuUback-Leibler KL) information (37). The KL information relates to the ratio of the distribution of the data given the model and parameters to the underlying true distribution of the data. The similarity of the KL information expression (Eq. (5.24)) and Bayes s formula (Eq. (5.1)) is easily seen ... 

Schwarz Bayesian Information Criterion regression parameters (0 Table Rl)... 

When fitting models, the MLE is used to find the optimal fit to the dataset. However, maximizing the log likelihood often results in fitting noise and parameter estimates that are unstable, particularly when the data set is relatively small. This is because MLE trusts too much the observed trends in the, often limited, data (Moons et al., 2004). In order to avoid possible over-fitting, the Bayesian Information Criterion (BIC) was utilized (Schwarz, 1978). BIC is a criterion for model selection, and includes a penalty term for the number of parameters in the model. The BIC is given by the following equation ... 

Structural identification, i.e. selection of the model type and structure, is always an arbitrary research decision. What is helpful is autocorrelation and spectrum analysis (detection of the intervals). Generally, the simplest possible model is chosen. A series of information criteria (algorithms) exist that may help in this process, usually defined as a combination of the model error and the number of model parameters, such as the AIC criterion (Akaike s information criterion), the criterion of the final error of the prediction, Ravelli Vulpiani criterion or Schwarz s BIC criterion (Bayesian information criterion comparison of log likelihood of specific models corrected by the number of estimated parameters and the number of observations). 

---