Vuong's closeness test

In statistics, the Vuong closeness test is likelihood-ratio-based test for model selection using the Kullback-Leibler information criterion. This statistic makes probabilistic statements about two models. It tests the null hypothesis, that two models (nested, non-nested or overlapping) are as close to the actual model against the alternative that one model is closer. It cannot make any decision whether the "closer" model is the true model.

With non-nested an iid exogenous variables, model 1 (2) is preferred with significance level &alpha;, if the z statistic


 * $$Z=\frac{LR_N(\beta_{ML,1},\beta_{ML,2})} {\sqrt{N}\omega_N}\text{ with }{LR_N(\beta_{ML,1},\beta_{ML,2})}=L^1_N-L^2_N-\frac{K_1-K_2} {2} \log N$$

exceeds the positive (falls below the negative) (1 &minus; &alpha;)-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 denominator $$\sqrt{N}\omega_N$$ equals the sum of squares of $$l_i=f_1(y_1|x_i,\beta_{ML,1})/f_2(y_1|x_i,\beta_{ML,2})\,$$.

For nested or overlapping models the statistic $$2LR_N(\beta_{ML,1}),\beta_{ML,2})\,$$ has to be compared to critical values from a weighted sum of chi squared distributions. This can be approximated by a gamma distribution: $$M_m(.,\bold\lambda)\sim \Gamma(b,p)\,$$ with $$\bold\lambda=(\lambda_1, \lambda_2, \dots, \lambda_m)\,$$, $$m=K_1+K_2\,$$, $$b=\frac 1 2 \frac {\sum\lambda_i} {\sum\lambda_i^2}$$ and $$\frac 1 2 \frac {{(\sum\lambda_i)}^2} {\sum\lambda_i^2}$$.

$$\bold\lambda$$ 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 (is it subjectively "big enough" to accept my hypothesis?).