Appendix E. Information Criteria

As part of the output for most regression procedures, EViews reports various information criteria. The information criteria are often used as a guide in model selection (see for example, Grasa 1989).

The Kullback-Leibler quantity of information contained in a model is the distance from the “true” model and is measured by the log likelihood function. The notion of an information criterion is to provide a measure of information that strikes a balance between this measure of goodness of fit and parsimonious specification of the model. The various information criteria differ in how to strike this balance.

Definitions

The basic information criteria are given by:

Akaike info criterion (AIC)	− 2	( l ⁄ T ) + 2( k ⁄ T )

Schwarz criterion (SC)	− 2	( l ⁄ T ) + klog ( T) ⁄ T
Hannan-Quinn criterion (HQ)	− 2	( l ⁄ T ) + 2klog ( log ( T) ) ⁄ T

Let l be the value of the log of the likelihood function with the k parameters estimated using T observations. The various information criteria are all based on –2 times the average log likelihood function, adjusted by a penalty function.

In addition to the information criteria described above, there are specialized information criteria that are used in by EViews when computing unit root tests:

	Modified AIC (MAIC)	− 2			( l ⁄ T ) + 2			( ( k + τ) ⁄ T )

	Modified SIC (MSIC)	− 2			( l ⁄ T ) + ( k + τ )log ( T) ⁄ T
	Modified Hannan-Quinn (MHQ)	− 2			( l ⁄ T ) + 2			( k + τ )log ( log ( T) ) ⁄ T

where the modification factor τ is computed as:
	τ =		2		˜ 2	⁄ σ	2
	τ =	α		Σ yt − 1		⁄ σ				(E.1)
	˜			t				˜
	˜							˜	as defined in
for yt ≡ yt when computing the ADF test equation (17.43), and for yt									as defined in

(“Autoregressive Spectral Density Estimator” beginning on page 528) when estimating the frequency zero spectrum (see Ng and Perron, 2002, for a discussion of the modified information criteria).

Note also that:

972—Appendix E. Information Criteria

•The Hannan-Quinn criterion is reported only for binary, ordered, censored, and count models.

•The definitions used by EViews may differ slightly from those used by some authors. For example, Grasa (1989, equation 3.21) does not divide the AIC by n . Other authors omit inessential constants of the Gaussian log likelihood (generally, the terms involving 2π ).

While very early versions of EViews reported information criteria that omitted inessential constant terms, the current version of EViews always uses the value of the full likelihood function. All of your equation objects estimated in earlier versions of EViews will automatically be updated to reflect this change. You should, however, keep this fact in mind when comparing results from frozen table objects or printed output from previous versions.

•For systems of equations, where applicable, the information criteria are computed using the full system log likelihood. The log likelihood value is computed assuming a multivariate normal (Gaussian) distribution as:

	TM		T	ˆ	(E.2)
	TM		T	ˆ
l = − ---------			( 1 + log 2π) − --log	Ω
2			2
where
	ˆ		ˆ ˆ
	ˆ		ˆ ˆ
	Ω	= det( Σ ′ ⁄ T)			(E.3)
	Ω	= det( Σ ′ ⁄ T)			(E.3)

M is the number of equations. Note that these expressions are only strictly valid when you there are equal numbers of observations for each equation. When your system is unbalanced, EViews replaces these expressions with the appropriate summations.

Using Information Criteria as a Guide to Model Selection

As a user of these information criteria as a model selection guide, you select the model with the smallest information criterion.

The information criterion has been widely used in time series analysis to determine the appropriate length of the distributed lag. Lütkepohl (1991, Chapter 4) presents a number of results regarding consistent lag order selection in VAR models.

You should note, however, that the criteria depend on the unit of measurement of the dependent variable y . For example, you cannot use information criteria to select between a model with dependent variable y and one with log( y ).

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