aicbic - Akaike and Bayesian information criteria

Syntax

AIC = aicbic(LLF,NumParams) [AIC,BIC] = aicbic(LLF,NumParams,NumObs)

Description

AIC = aicbic(LLF,NumParams) computes the Akaike (AIC) information criteria, using optimized loglikelihood objective function (LLF) values as input. Obtain the LLF values by fitting models of the conditional mean and variance to a univariate return series.

[AIC,BIC] = aicbic(LLF,NumParams,NumObs) computes both the Akaike (AIC) and Bayesian (BIC) information criteria. Perform multiple computations by passing vector values for any input. Since information criteria penalize models with additional parameters, parsimony is the basis of the AIC and BIC model order selection criteria.

Inputs

`LLF`	Vector of optimized loglikelihood objective function (LLF) values associated with parameter estimates of the models to test. Obtain the LLF values from `garchfit`, `garchinfer`, `vgxvarx`, or an Optimization Toolbox function such as `fmincon` or `fminunc`.
`NumParams`	Number of estimated parameters associated with each value in `LLF`. `NumParams` can be a scalar applied to all values in `LLF`, or a vector the same length as `LLF`. All elements of `NumParams` must be positive integers. Use `garchcount` or `vgxcount` to compute `NumParams` values.
`NumObs`	Sample size of the observed return series you associate with each `LLF` value. `NumObs` can be a scalar applied to all values in `LLF`, or a vector the same length as `LLF`. `NumObs` is required to compute `BIC`. All elements of `NumObs` must be positive integers.

Outputs

`AIC`	Vector of AIC statistics associated with each `LLF` objective function value.
`BIC`	Vector of BIC statistics associated with each `LLF` objective function value.

Definitions

The AIC statistic is :

AIC = (–2*LLF) + (2*NumParams).

The BIC statistic is:

BIC = (–2*LLF) + (NumParams * log(NumObs)).

Examples

Calculate and interpret the Akaike Information Criterion for four models. Multiple Time Series Case Study fits four models to FRED data, yielding the following parameters:

LLF1 = -681.4724; LLF2 = -632.3158;
LLF3 = -663.4615; LLF4 = -605.9439;
n1p = 12; n2p = 27; n3p = 18; n4p = 45;

AIC = aicbic([LLF1 LLF2 LLF3 LLF4],[n1p n2p n3p n4p])

AIC =
  1.0e+003 *
    1.3869    1.3186    1.3629    1.3019

The best model has AIC = 1e3 * 1.3019, since this value is smallest.

References

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 1994.