The Bayesian information criterion (BIC) or Schwarz criterion (SIC) is a measure of the goodness of fit of a statistical model, and is often used as a criterion for model selection among a finite set of models. It is based on 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.

- 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.

- 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**

- It is important to keep in mind 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.
- 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.

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## References

- D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
- James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896
- Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740
- Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848
- Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568

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