Logit and probit regressions  (Page 5/9)

(Assumption: . nested in full) Prob>chi2 = 0.0061

The interpretation of these results is that the omitted variables are statistically significant at the 0.6 percent level. The phrase “(Assumption: . nested in full)” tells you the name of the regression is the unrestricted model (full) and offers you a hyperlink to call this regression up to the screen.

The lagrange multiplier test

The intuition behind the Lagrange multiplier (LM) test (or score test) is that the gradient of the log of the likelihood function is equal to zero at the maximum of the likelihood function. The gradient is a vector of first-derivatives. In this case it is a vector of the first-derivatives with respect to each parameter estimate $\left(\text{i}\text{.e}\text{., }{\widehat{\beta }}_i\right).$ To obtain the ML estimate, we have to set these first-derivatives equal to zero. If the null hypothesis in (2.9) is correct, then maximizing the log of the likelihood function for the restricted model is equivalent to maximizing the log of the likelihood function with the constraint specified by the null hypothesis. The LM test measures how close the Lagrangian multipliers of this constrained maximization problem are to zero—the closer they are to zero, the more likely that the null hypothesis can be rejected.

Economists generally do not make use of the LM test because the test is complicated to compute and the LR test is a reasonable alternative. Thus, as a practical matter the Wald test and the LR test are reasonable alternative test statistics to use to test most linear restrictions on the parameters. Moreover, since the calculations are relatively easy, it may make sense to calculate both test statistics to be sure they produce consistent conclusions. However, when the sample size is small, the LM test probably is preferred.

Goodness-of-fit measures

The standard measure of goodness-of-fit in the linear OLS regression model is $R^2.$ No such measure exists for non-linear models like the logit model. Several potential alternatives have been developed in the literature and are known collectively as pseudo - R ² . Many of these measures are discussed in McFadden (1974), Amemiya (1981), and Maddala (1983). In case any reader really cares about the pseudo-- $R^2,$ a practical approach is to report the value that the computer program reports.