A difficult question when modeling economic behavior is to decide on what lags should be in the ARIMA model, the ARMA disturbance model, or the dynamic regression model. It is tempting to use hypothesis testing to help make model specification decisions based on the data, but as discussed by Granger, King, and White (1995), there are disadvantages in doing so. They and others recommend the use of a model selection procedure to make these decisions, the most common of which are information criteria (IC). For each of the models under consideration, one calculates the maximized loglikelihood and then penalizes this value to take account of the number of free parameters in the model which we will denote by j. Akaike’s (1973) IC (AIC) uses j as the penalty whereas Schwarz’s (1978) Bayesian IC (BIC) uses j log(n)/2. There are a range of other IC procedures, but these two have become the most popular.
These days, BIC seems to be the favored procedure because it is consistent, which means that as the sample size goes to infinity, the probability that it will choose the correct model from a finite number of models goes to one. An unfortunate consequence of this property is that in small samples, BIC tends to wrongly choose underfitting models and can have very low probabilities of correctly selecting a model which has a large number of free parameters. AIC seems more balanced in this regard in small samples, but can suffer from a tendency to overfit in larger samples.
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