Vuong's closeness test

With strictly non-nested models and iid exogenous variables, model 1 (2) is preferred with significance level α, if the z statistic

${\displaystyle Z={\frac {\operatorname {LR} _{N}(\beta _{ML,1},\beta _{ML,2})}{{\sqrt {N}}\omega _{N}}}}$

with

${\displaystyle \operatorname {LR} _{N}(\beta _{ML,1},\beta _{ML,2})=L_{N}^{1}-L_{N}^{2}-{\frac {K_{1}-K_{2}}{2}}\log N}$

exceeds the positive (falls below the negative) (1 − α)-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, ${\displaystyle \omega _{N}\,}$, is defined by setting ${\displaystyle \omega _{N}^{2}}$ equal to either the mean of the squares of the pointwise log-likelihood ratios ${\displaystyle \ell _{i}\,}$, or to the sample variance of these values, where

${\displaystyle \ell _{i}=\log {\frac {f_{1}(y_{i}\mid x_{i},\beta _{ML,1})}{f_{2}(y_{i}\mid x_{i},\beta _{ML,2})}}.}$

For nested or partially non-nested (overlapping) models the statistic

${\displaystyle 2\operatorname {LR} _{N}(\beta _{ML,1},\beta _{ML,2})\,}$

has to be compared to critical values from a weighted sum of chi squared distributions. This can be approximated by a gamma distribution (in shape-rate form):

${\displaystyle M_{m}(\cdot ,\mathbf {\lambda } )\sim \Gamma (b,p)\,}$

with

${\displaystyle \mathbf {\lambda } =(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}),\,}$

${\displaystyle m=K_{1}+K_{2},\ b={\frac {1}{2}}{\frac {\left(\sum \lambda _{i}\right)^{2}}{\sum \lambda _{i}^{2}}}}$

and

${\displaystyle p={\frac {1}{2}}{\frac {\sum \lambda _{i}}{\sum \lambda _{i}^{2}}}.}$

${\displaystyle \mathbf {\lambda } }$ is a vector of eigenvalues of a matrix of conditional expectations. The computation is quite difficult, so that in the overlapping and nested case multiple authors^[who?] only derive statements from a subjective evaluation of the Z statistic (is it subjectively "big enough" to accept my hypothesis?).