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Chapter2l. Modelselectioncriteria

subsdtuilng (21.2) lnto (21.1) we gei

aic = n(l + log2Tr - logn) + n logssR + 2 k whlch can alsobe wrltten as

AIC = nlog (-yr ,1 + 2 k + n( l + Iog2ir)

172

(21.3)

Some authors slmpllfy the formula for the case of models esdmated Ma least sąuares. For lnstance. William Greene wrltes

^=44)4    ⁽²ⁱ-⁴»

Thls varlant can be derlved from (21.3) by dlvidlng ihrough by n and subtracdng the oonsiant l + log2tt. Tliat ls. wrldng AlCc for the version glven by Greene, we have

Al Cc =yj-AIC-(l + log 2 TT)

Flnally, Ramanathan gives a further vanant:

_A,„ /SSR\ ^AICR = (—)

,2 kin

whlch ls the exponential of the one glven by Greene.

Greil began by uslng the Ramanathan varlant, but sińce version 1.3.1 the program has used the odginał Akalke fonnula (21.1), and morę speclflcally (21.3) for models esumated Ma least sąuares.

Ahhough the Akalke cdterlon ls deslgned to favor parsimony, arguably lt does not go far enough ln that directlon. For lnstance, lf we ha\e two nested models with k - l and k parameiers respec-dvely, and lf the nuli h>poihesis that parameter k eąuals o ls true, m large samples the AIC will nonetheless tend to select the less parsimonlous model about IG percent of the dme isee DaMdson and MacKinnon, 2004, chapter 13).

An altemathe to the Aic whlch avoids thls problem ls the Schwarz (1978) “Bayesian lnfomiaUon crlierlon" (BIC). The BIC can be wrltten (ln llne wlth Akalke’s formuladon of the AIC) as

BlC =    +k\ogn

The muldpllcatlon of k by logn ln the BIC means that the penalty for addlng extra parameters grows wlth the sample size. Thls ensures that, asymptotlcally, one will not select a larger model over a conecily speclfled parsimonlous model.

A further aliernadve to Aic, whlch agam tends to select morę parsimonlous models ihan Aic, ls the Hannan-Qulnn alterlon or HQC (Hannan and qulnn, 1979). wrltten conslsienily wlth the formulatlons above, thls ls

HQC = -2((d) + 2k log log n

The Hannan-Qulnn calculatlon ls based on the lawof the lterated logadthm (notę that the last term ls the log of the log of the sample slze). The authors argue that thelr procedurę provides a “su-ongly consisieni esdmadon procedurę for tlie order of an autoregression", and that ‘compared to other strongly consisieni procedures thls procedurę will underesilmate the order to a lesser degree."

Gretl reports tlie aic, BIC and HQC (calculated as explalned above) for most sorts of models. The key point ln lnterpretlng these values ls to know whether they are calculated sudi that smaller values are better, or such that larger values are beuer. m gretl, smaller values are better: one wams to mlnlmlze the chosen crlterlon.

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