AIC and BIC are two information criteria that measures the quality of a model as a tradeof between its goodness of fit and its simplicity.
The AIC or Akaike information criterion is a quality measure of a model. It is based on the likelihood function and it deals with the trade-off between the goodness of fit of the model and the simplicity of the model.
Suppose that we have a statistical model of some data. Let \(k\) be the number of estimated parameters in the model. Let \(\hat{L}\) be the maximum value of the likelihood function for the model. Then the AIC value of the model is the following:
\[AIC = 2k - 2 \log(\hat{L})\]A lower AIC is better.
The BIC or Bayesian information criterion is a quality measure of a model. It is based on the likelihood function and it deals with the trade-off between the goodness of fit of the model and the simplicity of the model.
Suppose that we have a statistical model of some data. Let \(k\) be the number of estimated parameters in the model and \(n\) the number of data points. Let \(\hat{L}\) be the maximum value of the likelihood function for the model. Then the BIC value of the model is the following:
\[BIC = k \log(n) - 2 \log(\hat{L})\]A lower BIC is better.
AIC and BIC are generally used in time series as there are the best quality metric for models. Cross-validation is more complex in time series (due to time dependance) and time series models are more computationally intensive thant other types of models.
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