Vuong test

The Vuong test is a statistical model selection test based on Bayesian information criterion. It is named after the mathematician Quang H. Vuong who proposed the test in 1989.

Theoretical procedure

The Vuong test tests the null hypothesis that two models - regardless of whether they are hierarchical, non-hierarchical or overlapping - are equally close to the true distribution against the counter hypothesis that one model is closer to it. However, it does not make any statement that the better model is really the true model. Assuming non-hierarchical and identically and independently distributed explanatory variables, model 1 (or model 2) is preferred at the level of significance if the test variable ${\displaystyle \alpha }$

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

With

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

exceeds (or falls below) the (negative) quantile of the standard normal distribution. The numerator size is the difference between the maximum log-likelihoods of the two model estimates, which is corrected by the number of coefficients, analogously to Bayesian information criterion ; the denominator size corresponds to the sum of the squares of ${\displaystyle (1-\alpha )}$ ${\displaystyle {\sqrt {N}}\omega _{N}}$

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

For hierarchical and overlapping models, the test statistic is

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

compared with corresponding critical quantities from a weighted sum of chi-square distributions . This can be approximated using a gamma distribution :

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

With

${\displaystyle \mathbf {\lambda } =(\lambda _{1},\lambda _{2},\dotsc ,\lambda _{m})}$, ,   And   . ${\displaystyle m=K_{1}+K_{2}}$ ${\displaystyle b={\frac {1}{2}}{\frac {\sum \lambda _{i}}{\sum \lambda _{i}^{2}}}}$ ${\displaystyle p={\frac {1}{2}}{\frac {{(\sum \lambda _{i})}^{2}}{\sum \lambda _{i}^{2}}}}$

Here is the vector of the eigenvalues ​​of a matrix of conditional expectation values . However, its derivation is quite difficult, so that statements in the overlapping case are usually only made on the basis of subjectively sufficiently large values. ${\displaystyle \mathbf {\lambda } }$

literature

• Quang H. Vuong: Likelihood Ratio Tests for Model Selection and non-nested Hypotheses, in: Econometrica, Vol. 57, Iss. 2, 1989, pp. 307-333, JSTOR 1912557 .