The statistic of Vuong likelihood ratio test for compare two
non-nested regression models is defined by
$$T = \frac{1}{\widehat{\omega}^2\,\sqrt{n}}\,\sum_{i=1}^{n}\,
\log\frac{f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})}{
g(y_i \mid \boldsymbol{x}_i,\widehat{\boldsymbol{\gamma}})}$$
where
$$\widehat{\omega}^2 = \frac{1}{n}\,\sum_{i=1}^{n}\,\left(\log \frac{f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})}{g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}})}\right)^2 - \left[\frac{1}{n}\,\sum_{i=1}^{n}\,\left(\log \frac{f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})}{ g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}})}\right)\right]^2$$
is an estimator for the variance of
\(\frac{1}{\sqrt{n}}\,\displaystyle\sum_{i=1}^{n}\,\log\frac{f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})}{g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}})}\),
\(f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})\) and
\(g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}})\)
are the corresponding rival densities evaluated at the maximum likelihood estimates.
When \(n \rightarrow \infty\) we have that \(T \rightarrow N(0, 1)\) in distribution.
Therefore, at \(\alpha\%\) level of significance the null hypothesis of
the equivalence of the competing models is rejected if \(|T| > z_{\alpha/2}\),
where \(z_{\alpha/2}\) is the \(\alpha/2\) quantile of standard normal distribution.
In practical terms, \(f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})\)
is better (worse) than \(g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}})\)
if \(T>z_{\alpha/2}\) (or \(T< -z_{\alpha/2}\)).