Advanced Econometrics Takeshi Amemiya
Tests of Hypotheses, Prediction, and Computation
8.3.1 Tests of Hypotheses
Suppose we want to test a hypothesis of the form h(a) = 0 in model (8.1.1), where h is a ^-vector of nonlinear functions. Because we have not specified the distribution ofY„ we could not use the three tests defined in (4.5.3), (4.5.4), and (4.5.5) even if we assumed the normality of u. But we can use two test statistics: (1) the generalized Wald test statistic analogous to (4.5.21), and (2) the difference between the constrained and the unconstrained sums of squared residuals (denoted SSRD). Let a and St be the solutions of the unconstrained and the constrained minimization of (8.1.2), respectively. Then the generalized Wald statistic is given by
Wald = j~ h(a)'[H(G'Pw£)-1H/]-1h(a), (8.3.1)
where ft = [dh/da'k and 6 = [df/Sa']*, and the SSRD statistic is given by
SSRD = -~[SA*)~S7{a)]- (8.3.2)
Both test statistics are asymptotically distributed as chi-square with q degrees of freedom.
Gallant and Jorgenson (1979) derived the asymptotic distribution (a noncentral chi-square) of the two test statistics under the assumption that a deviates from the hypothesized constraints in the order of T~m.
As an application of the SSRD test, Gallant and Jorgenson tested the hypothesis of homogeneity of degree zero of an equation for durables in the two-equation translog expenditure model of Joigenson and Lau (1978).
The Wald and SSRD tests can be straightforwardly extended to the system of equations (8.2.1) by using NL3S in lieu of NL2S. As an application of the SSRD test using NL3S, Gallant and Jorgenson tested the hypothesis of symmetry of the matrix of parameters in the three-equation translog expenditure model of Jorgenson and Lau (1975).
If we assume (8.1.22) in addition to (8.1.1) and assume normality, the model is specified (although it is a limited information model); therefore the three tests of Section 4.5.1 can be used with NLLI. The same is true of NLFI in model (8.1.22) under normality. The asymptotic results of Gallant and Holly (1980) given in Section 4.5.1 are also applicable if we replace <72(G'G)-1 in the right-hand side of (4.5.26) by the asymptotic covariance matrix of the NLFI estimator.
