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634 Chapter 14 Introduction to Time Series Regression and Forecasting

homoskedastic, then F has a x12 asymptotic distribution; if not, it has some other asymp­
totic distribution. Thus Pr3BIC(2) - BIC(1) 6 04 = Pr5T 3BIC(2) - BIC(1)4 6 06
= Pr5-T ln31 + F > (T - 2)4 + (lnT ) 6 06 = Pr5T ln31 + F > (T - 2) 4 7 lnT6.
As T increases, T ln[1 + F>(T - 2)] - F ¡p 0 [a consequence of the logarithmic
approximation ln(1 + a) ≅ a, which becomes exact as a ¡ 0]. Thus Pr3BIC(2) -
BIC(1) 6 04 ¡ Pr(F 7 lnT ) ¡ 0, so Pr( pn = 2) ¡ 0.

AIC

In the special case of an AR(1) when zero, one, or two lags are considered, (i) applies to
the AIC where the term lnT is replaced by 2, so Pr(pn = 0) ¡ 0. All the steps in the
proof of (ii) for the BIC also apply to the AIC, with the modification that lnT is replaced
by 2; thus Pr3AIC(2) - AIC(1) 6 04 ¡ Pr(F 7 2) 7 0. If ut is homoskedastic, then
Pr(F 7 2) ¡ Pr(x12 7 2) = 0.16, so Pr(pn = 2) ¡ 0.16. In general, when pn is chosen
using the AIC, Pr( pn 6 p) ¡ 0, but Pr( pn 7 p) tends to a positive number, so Pr( pn = p)
does not tend to 1.