Vuong's closeness test


In statistics, the Vuong closeness test is a likelihood-ratio-based test for model selection using the Kullback–Leibler information criterion. This statistic makes probabilistic statements about two models. They can be nested, non-nested or overlapping. The statistic tests the null hypothesis that the two models are equally close to the true data generating process, against the alternative that one model is closer. It cannot make any decision whether the "closer" model is the true model.
With non-nested models and iid exogenous variables, model 1 is preferred with significance level α, if the z statistic
with
exceeds the positive -quantile of the standard normal distribution. Here K1 and K2 are the numbers of parameters in models 1 and 2 respectively.
The numerator is the difference between the maximum likelihoods of the two models, corrected for the number of coefficients analogous to the BIC, the term in the denominator of the expression for Z,, is defined by setting equal to either the mean of the squares of the pointwise log-likelihood ratios, or to the sample variance of these values, where
For nested or overlapping models the statistic
has to be compared to critical values from a weighted sum of chi squared distributions. This can be approximated by a gamma distribution:
with
and
is a vector of eigenvalues of a matrix of conditional expectations. The computation is quite difficult, so that in the overlapping and nested case many authors only derive statements from a subjective evaluation of the Z statistic.
Vuong's test for non-nested models has been used to compare a zero-inflated model to its non-zero-inflated counterpart. Wilson argues that such use of Vuong's test is invalid as a non-zero-inflated model is not strictly non-nested in its zero-inflated counterpart