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® E 1 Introduction

I    Running the program

II    Econometric methods

III    Technical details

IV    Appendices

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Chapter 21

Model selectlon criterla

21.1 Inrroduction

In sonie contexts the econometrician chooses between altemative models based on a fornial hy-potliesis test. For example, one might choose a morę generał model o\er a morę restricted one if the restriction in ąuestion can be fonnulated as a testable nuli hypothesis, and the nuli Ls rejected on an appropriate test.

In other contexts one sometimes seeks a criterion for model selection tliat somehow measures the balance between goodness of fu or likelihood, on the one hand, and parsimony on the other. The balandng ls necessary because the addition of extra \ariables to a model cannot reduce tlie degree of fi't or likelihood. and is very likely to increase it somewhat even if the additional variables are not truły relevant to the data-generating process.

The besi known such criterion. for linear models estimated via least squares, is the adjusted R²,

R² = 1 -

SSR/Ot -k) TSS/(it - 1)

where n is the number of obsetrations in the s ample, k denotes the number of parameters esti-mated, and SSR and TSS denote the sum of squared residuals and the toial sum of sąuares for the dependent variable, respectively. Compared to the ordinary coefTicient of determination or unadjusted R²,

r2_,_SSR ^K ~ ¹ TSS

the “adjusted" calculaiion penalizes tlie indusion of additional parameters, other things eąual.

21.2 Information criteria

A morę generał criterion in a similar spirit Ls Akaike's (1974) “Information Criterion’ (AIC). The original formulation of thls measure is

AIC = -2€{0) + 2k

(21.1)

where (■{&) represents the maximum loglikelihood as a function of the vector of parameter esti-maies, 0, and k (as above) denotes the number of “independenily adjusted parameters within the model." In this formulation, with AIC negatively related to the likelihood and posiiively related to the number of piarameters, tlie researcher seeks the minimum AIC.

Tlie AIC can be confusing, in thai several variants of the calculaiion are “in drculation." For exam-ple, Davidson and MacKinnon (2004) present a simplified version,

AIC = ((&) - k

which is just -2 times the original: in this case, obviously, one wams to maximi/.e AIC.

In the case of models estimated by least sąuares, the loglikelihood can be written as

*>«?) =-!i(l + log 2tt - log n) - y log SSR

(21.2)

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