Appendix D. 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) |
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Schwarz criterion (SC) |
– 2 |
(l § T) + klog(T) § T |
Hannan-Quinn criterion (HQ) |
– 2 |
(l § T) + 2klog(log(T)) § T |
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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.
For factor analysis models, EViews follows convention (Akaike, 1987), re-centering the criteria by subtracting off the value for the saturated model. The resulting factor analysis form of the information criteria are given by:
Akaike info criterion (AIC) |
(T – k)D § T – (2 § T)df |
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Schwarz criterion (SC) |
(T – k)D § T – (log(T) § T)df |
Hannan-Quinn criterion (HQ) |
(T – k)D § T – (2 ln(log(T)) § T)df |
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where D is the discrepancy function, and df is the number of degrees-of-freedom in the estimated dispersion matrix. Note that EViews scales the Akaike form of the statistic by dividing by T .
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 + t) § T) |
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References—773
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 ).
References
Grasa, Antonio Aznar (1989). Econometric Model Selection: A New Approach, Dordrecht: Kluwer Academic Publishers.
Akaike, H. (1987). “Factor Analysis and AIC,” Psychometrika, 52(3), 317–332.
Lütkepohl, Helmut (1991). Introduction to Multiple Time Series Analysis, New York: Springer-Verlag.
Ng, Serena and Pierre Perron (2001). “Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power,” Econometrica, 69(6), 1519-1554.
774—Appendix D. Information Criteria