The Akaike information criterion measures the goodness of fit of a statistical model. It describes the trade-off between bias and variance in model construction, or loosely speaking, that of the accuracy and complexity of the model.

The AIC is not a test of the model in the sense of hypothesis testing; instead, it provides a means for comparison among models—a tool for model selection. Given a data set, several candidate models may be ranked according to their AIC, with the model having the minimum AIC being the best. From the AIC values, one may also infer that, e.g., the top two models are roughly in a tie, and the rest are far worse.

- In general, the AIC is defined as: $$\mathit{AIC}=2k-2\times\ln(L)$$ Where:

- $k$ is the number of model parameters.
- $\ln(L)$ is the log-likelihood function for the statistical model.

- For smaller data sets, the AICc applies 2nd order correction: $$ \mathit{AICc}= \mathit{AIC} + \frac{2k(k+1)}{N-k-1} = \frac{2\times N \times k}{N-k-1}-2\times\ln(L) $$Where:

- $N$ is the data sample size.
- $k$ is the number of model parameters.

**Remarks**

- The AIC is not a test on the model in the sense of hypothesis testing; instead, it is a test between models - a tool for model selection.

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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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