Testing in binary response index models
Denote a binary response index model as: formula_1, formula_2 where formula_3. Description. This type of model is applied in many economic contexts, especially in modelling the choice-making behavior. For instance, formula_4 here denotes whether consumer formula_5 chooses to purchase a certain kind of chocolate, and formula_6 includes many variables characterizing the features of consumer formula_5 . Through function formula_8 , the probability of choosing to purchase is determined. Now, suppose its maximum likelihood estimator (MLE) formula_9 has an asymptotic distribution as formula_10 and there is a feasible consistent estimator for the asymptotic variance formula_11 denoted  as formula_12 . Usually, there are two different types of hypothesis needed to be tested in binary response index model. The first type is testing the multiple exclusion restrictions, namely, testing formula_13. If the unrestricted MLE can be easily computed, it is convenient to use the Wald test whose test statistic is constructed as: formula_14 Where "D" is a diagonal matrix with the last "Q" diagonal entries as 0 and others as 1. If the restricted MLE can be easily computed, it is more convenient to use the Score test (LM test). Denote the maximum likelihood estimator under the restricted model  as formula_15 and define formula_16 and formula_17, where formula_18. Then run the OLS regression formula_19 on formula_20, where formula_21 and formula_22. The LM statistic is equal to the explained sum of squares from this regression and it is asymptotically distributed as formula_23. If the MLE can be computed easily under both of the restricted and unrestricted models, Likelihood-ratio test is also a choice: let formula_24 denote the value of the log-likelihood function under the unrestricted model and let formula_25 denote the value under the restricted model, then formula_26 has an asymptotic formula_23 distribution. The second type is testing  a nonlinear hypothesis about formula_28, which can be represented as formula_29 where formula_30 is a Q×1 vector of possibly nonlinear functions satisfying the differentiability and rank requirements. In most of the cases, it is not easy or even feasible to compute the MLE under the restricted model when formula_30 include some complicated nonlinear functions. Hence, Wald test is usually used to deal with this problem. The test statistic is constructed as: formula_32 where formula_33 is the Q×N Jacobian of formula_30 evaluated at formula_9. For the tests with very general and complicated alternatives, the formula of the test statistics might not  have the exactly same representation as above. But we can still derive the formulas as well as its asymptotic distribution by Delta method and implement Wald test, Score test or Likelihood-ratio test. Which test should be used is determined by the relative computation difficulty of the MLE under restricted and unrestricted models.