The Binary Response Model Regressio

Davidson and MacKinnon (1984) suggest a modified version of the Gauss-Newton regression (GNR) considered in Chapter 8 which is useful in the context of a binary response model described in (13.5).2 In fact, we have shown that this model can be written as a nonlinear regression

Уі = F (x’i в)+ Ui (13.24)

with ui having zero mean and var(ui) = Fi(1 — Fi). The GNR ignoring heteroskedasticity yields (yi — Fi) = fix’ib + residual

where b is the regression estimates when we regress (y — Fi) on fix’i.

Подпись: (Vi — Fi) y/Fi(1 — Fi) Подпись: fi л/Fi (1 — Fi) Подпись: xib + residual Подпись: (13.25)

Correcting for heteroskedasticity by dividing each observation by its standard deviation we get the Binary Response Model Regression (BRMR):

image546 Подпись: (13.26)

For the logit model with fi = Лі(1 — Лі), this simplifies further to

Подпись: (13.27)Подпись: xib + residualFor the probit model, the BRMR is given by Vi — фг Фі

л/Фі(1 — Фі ) ^фі(1 — фі)

Like the GNR considered in Chapter 8, the BRMR given in (13.25) can be used for obtaining parameter and covariance matrix estimates as well as test of hypotheses. In fact, Davidson and MacKinnon point out that the transpose of the dependent variable in (13.25) times the matrix of regressors in (13.25) yields a vector whose typical element is exactly that of S(/3) given in (13.17). Also, the transpose of the matrix of regressors in (13.25) multiplied by itself yields a matrix whose typical element is exactly that of I(в) given in (13.21).

Подпись: Vi — Fi Fi (1 — Fi) Подпись: fi Fi (1 — Fi) Подпись: x/bi + Подпись: г j= x'i2b2 + residual Fi(1 — Fi) Подпись: (13.28)

Let us consider how the BRMR is used to test hypotheses. Suppose that в = (ві, в2) where ві is of dimension к — r and в2 is of dimension r. We want to test Ho; в2 = 0. Let в = (ві, 0) be the restricted MLE of в subject to Ho. In order to test Ho, we run the BRMR:

Подпись: Vi — V л/V/ — V) Подпись: fi(a) л/V/ — V) image557 Подпись: (13.29)

where xi = (xii, xi2) has been partitioned into vectors conformable with the corresponding partition of в. Also, F = F(xiв) and fi = f (xiв). The suggested test statistic for Ho is the explained sum of squares of the regression (13.28). This is asymptotically distributed as x2 under Ho.3 A special case of this BRMR is that of testing the null hypothesis that all the slope coefficients are zero. In this case, xii = 1 and ві is the constant a. Problem 2 shows that the restricted MLE in this case is FF(a) = V or a = F-i(V), where V is the proportion of the sample with Vi = 1. Therefore, the BRMR in (13.25) reduces to

Note that V(1 — V) is constant for all observations. The test for b2 = 0 is not affected by dividing the dependent variable or the regressors by this constant, nor is it affected by subtracting a constant from the dependent variable. Hence, the test for b2 = 0 can be carried out by regressing Vi on a constant and xi2 and testing that the slope coefficients of xi2 are zero using the usual least squares F-statistic. This is a simpler alternative to the likelihood ratio test proposed in the previous section and described in the empirical example in section 13.9. For other uses of the BRMR, see Davidson and MacKinnon (1993).

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