# Tests of Hypotheses

In the process of proving Theorem 4.3.2, we have in effect shown that asymp­totically

A—A*(G’G)rlG’u, (4.3.28)

where we have put G = (df/d/Пд,. Note that (4.3.28) exactly holds in the linear case because then G = X. The practical consequence of the approximation (4.3.28) is that all the results for the linear regression model (Model 1) are asymptotically valid for the nonlinear regression model if we treat G as the regressor matrix. (In practice we must use G = (3f/dfi’)f, where fi is the NLLS estimator.)

Let us generalize the t and F statistics of the linear model by this principle. If the linear hypothesis Q’fi = c consists of a single equation, we can use the following generalization of (1.5.4):  Q’fi-c

5[Q'(6’6)-lQ]l/2

where ~ means “asymptotically distributed as” and a2 = (T— K)~lST(fi). Gallant (1975a) examined the accuracy of the approximation (4.3.29) by a Monte Carlo experiment using the model

fkP) = P *i. + Рг*ъ + Pa exp (P3xз,). (4.3.30)

For each of the four parameters, the empirical distribution of the left-hand side of (4.3.29) matched the distribution of tT^k reasonably well, although, as we would suspect, the performance was the poorest for A – If Q’P = c consists of q(> 1) equations, we obtain two different approximate F statistics depending on whether we generalize the formula (1.5.9) or the formula (1.5.12). Generalizing (1.5.9), we obtain (4.3.31)

where P is the constrained NLLS estimator obtained by minimizing SAP)
subject to Q’fi= c. Generalizing the formula (1.5.12), we obtain

l=*m-сгтыгуут-с).m T_k) (4332)

Я ST{fi)

These two formulae were shown to be identical in the linear model, but they are different in the nonlinear model. A Monte Carlo study by Gallant (1975b), using the model (4.3.30), indicated that the test based on (4.3.31) has higher power than the test based on (4.3.32).