David F. Hendry 43However, it does matter that selection occurs: the selected model’s estimates do
not have the same properties as if the LDGP equation had been estimated without
any testing. Sampling vagaries entail that some variables which enter the LDGP will
by chance have a samplet^2 <cα(low power). Since they are only retained when
t^2 ≥cα, their estimated magnitudes will be biased away from the origin, and hence
selected coefficients need to be bias corrected, which is relatively straightforward
(see Hendry and Krolzig, 2005). Some variables which are irrelevant will havet^2 ≥
cα(adventitiously significant), where the probability of that event isα
(
N−n∗)whenn∗variables actually matter. Fortunately, bias correction will also drive such
estimates sharply towards the origin. Thus, despite selecting from a large set of
potential variables, nearly unbiased estimates of coefficients and equation standard
errors can be obtained with little loss of efficiency from testing many irrelevant
variables, and some loss for relevant, from the increased value ofcα. The normal
distribution has “thin tails,” so the power loss from tighter significance levels is
usually not substantial, whereas financial variables may have fat tails, so power
loss could be more costly at tighterα.
Impulse saturation is described in Hendryet al.(2008) and Johansen and Nielsen
(2008) as including an indicator for every observation, entered (in the simplest case)
in blocks ofT/2, with the significant outcomes retained. This approach both helps
remove outliers, and is a good example of why testing large numbers of candidate
regressors does not cost much efficiency loss under the null that they are irrelevant.
Settingcα≤ 1 /Tmaintains the average false null retention at one “outlier,” and
that is equivalent to omitting one observation, so is a tiny efficiency loss despite
testing for the relevance ofTvariables. Since all regressors are exact linear functions
ofTimpulses, that effect carries over directly in the independent and identically
distributed (i.i.d.) setting, and in similar ways more generally. Thus,N>Tis not
problematic for automatic model selection, opening the door to large numbers of
new applications.
Since an automatic selection procedure is algorithmic, simulation studies
of its operational properties are straightforward. In the Monte Carlo experi-
ments reported in Hendry and Krolzig (2005), commencing from highly over-
parameterized GUMs (between 8 and 40 irrelevant variables; zero and 8 relevant),
PcGets recovered the LDGP with an accuracy close to what one would expect
if the LDGP specification were known initially, but nevertheless coefficient tests
were conducted. To summarize its simulation-based properties, false rejection fre-
quencies of null hypotheses (measured as retention rates for irrelevant variables)
can be controlled at approximately the desired level; correct rejections of alter-
natives are close to the theoretical upper bound of power (measured as retention
rates for relevant variables); model selection is consistent for a finite model size
as the sample size grows without bound; nearly unbiased parameter estimates can
be obtained for all variables by bias-correction formulae, which also reduce the
mean square errors of adventitiously retained irrelevant variables; and reported
equation standard errors are nearly unbiased estimates of those of the correct spec-
ification (see, e.g., Hendry and Krolzig, 2005). Empirically, automatic Gets selects