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.
With strictly non-nested models and iid exogenous variables, model 1 (2) is preferred with significance level α, if the z statistic with exceeds the positive (falls below the negative) (1 − α)-quantile of the standard normal distribution.
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,
, or to the sample variance of these values, where For nested or partially non-nested (overlapping) models the statistic has to be compared to critical values from a weighted sum of chi squared distributions.
The computation is quite difficult, so that in the overlapping and nested case many authors[who?]
Vuong (1989) gives two examples of strictly non-nested models: Vuong (1989) also gives an intuitive example of partially non-nested (aka overlapping) models: