Z statistic

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[edit] Disambiguation

If you are looking for the z-score or normal score: see standard score.


The Vuong Z statistic is an improvement of the so called information criteria. 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, if the better model is the true model.

With non-nested an iid exogenous variables, model 1 (2) is preferred with signifikance level α, if the statistic

Z=\frac{LR_N(\beta_{ML,1},\beta_{ML,2})} {\sqrt{N}\omega_N} 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 − α)-quantile of the standard normal distribution.

The nominator 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, ..., \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?).

[edit] Further reading

Vuong, Quang H. (1989): Likelihood Ratio Tests for Model Selection and non-nested Hypothesises, in: Econometrica, Vol. 57, Iss. 2, 1989, pages 307-333.

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