In the context of ARMA models, a model minimizing an AIC or related statistic will have a certain number of parameters defining the order of the model. Each of these parameters may or may not be statistically significant. Tables where figures in parentheses under each parameter estimate are the standard errors of the estimates are often seen in the literature. With these standard errors, obtained from either the observed or expected (Fisher) information, statistical testing for significance can be done (using either p-values or confidence intervals). For example, one readily gets p-values associated with z-ratios obtained by dividing estimated coefficients by their estimated standard errors.
While the paper does not seem to detail the approach to model selection taken, I think it is as simple as not selecting the model minimizing the AIC statistic if at least one of the coefficients is not significantly different from zero. For example, using
print(lh300<-arima(lh, order = c(3,0,0)))
print(lh100<-arima(lh, order = c(1,0,0)))
we first check the significance of the lh300 fit and find that p-values of the ar2 and ar3
coefficients are too large to consider the coefficients statistically different from zero. The second minimizer of the AIC is the lh100 fit with all of its coefficients significant.
This procedure can be questioned on at least two viewpoints.
First, following Chatfield JRSS(1995)[vol.158,p.441]
"least-squares theory is known to not apply when the same data are used to formulate
and fit a model so that estimation follows model selection".
Consequently, the p-values cannot be considered accurate enough to guide the decision to disregard a model. Furthermore, the subjective 5 per cent mentioned in the question is clearly inconsistent with the objective spirit of the Minimum AIC Estimate procedure. Indeed, on the first page of his most-cited 1974 paper, Akaike wrote that
"By the introduction of MAICE the problem of statistical identification is explicitly
formulated as a problem of estimation and the need of the subjective judgement required
in the hypothesis testing procedure for the decision on the levels of significance is
So to answer my own question, introducing subjectivity into what was meant to remove
it is wrong.
Second, as in linear regression modelling, that a coefficient is insignificant does not automatically mean it should be eliminated from the model. Furthermore, the idea to consider models whose values of the AIC statistic are within a certain distance of the minimum AIC value as somewhat inferior to the model minimizing the AIC is wrong. The number 2 is often mentioned for that distance in the literature, although Chatfield once
mentioned 4 and people who worked closely with Akaike mentioned 1. I remember Gavin Simpson expressed similar ideas on this (or related) Q&A website. Formal references can be made to Brockwell & Davis; Ruppert, Wand and Carrol as well as other well-known book-length sources of statistical knowledge. In our example, the data support lh300 and lh100 equally well, but lh100 has an advantage of being simpler.
So to answer my own question from another perspective, to consider models not minimizing the AIC statistic only if the model minimizing it happened to contain insignificant parameters is wrong.