Test Exercise 3
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Goals and skills being used:
• Apply the Akaike Information Criterion (AIC) for model selection.
• Examine large-sample behavior of AIC.
• Link AIC to F-test discussed in multiple regression lectures.
This test exercise is of a theoretic nature. The exercise is based on Exercise 5.2c of ‘Econometric Methods with
Applications in Business and Economics’. The question ofinterest is how the decision whether or not to include a
group of variables differs based on AIC from that based on the F-test. We will stepwise show that for large samples
selection based on AIC corresponds to an F-test with a critical value of approximately 2.
(a) Consider the usual linear model, wherey = X β + ε. We now compare two regressions, which differ in how
many variables are included in the matrixX . In the full (unrestricted) model p1 regressors are included. In the
restricted model only a subset ofp0 < p1 regressors are included.
Show that the smallest model is preferred according to the AIC if
< e n ( p1 − p0 ) .
(b) Argue that for very large values ofn the inequality of (a) is equal to the condition
s02 − s12
< ( p1 − p0 ).
Use that ex ≈ 1 + x for small values of x .
(c) Show that for very large values ofn the condition in (b) is approximately equal to
eR eR − eU eU
< ( p1 − p0 ),
where eR is the vector of residuals for the restricted model withp0 parameters and eU the vector of residuals
for the full unrestricted model withp1 parameters.
(d) Finally, show that the inequality from (c) is approximately equivalent to an F-test with critical value 2, for large
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