Diagnostic Tests for Linear Regression Models
Variable addition tests suggested by Pagan and Hall (1983) consider the additional variables Z of dimension (T x r) and test whether their coefficients are zero using an F-test from the regression
y = X/3 + Zy + u (8.81)
If Ho; y = 0 is true, the model is y = X@ + u and there is no misspecification. The GNR for this restriction would run the following regression:
PX y = Xb + Zc + residuals (8.82)
and test that c is zero. By the FWL Theorem, (8.82) yields the same residual sum of squares as
PX y = PX Zc + residuals (8.83)
Applying the FWL Theorem to (8.81) we get the same residual sum of squares as the regression in (8.83). The F-statistic for y = 0 from (8.81) is therefore identical to the F-statistic for c = 0 from the GNR given in (8.82). Hence, “Tests based on the GNR are equivalent to variable addition tests when the latter are applicable,” see Davidson and MacKinnon (1993, p. 194).
Note also, that the nRU test statistic for H0; y = 0 based on the GNR in (8.82) is exactly the LM statistic based on running the restricted least squares residuals of y on X on the unrestricted set of regressors X and Z in (8.81). If X has a constant, then the uncentered R2 is equal to the centered R2 printed by the regression.
Computational Warning: It is tempting to base tests on the OLS residuals u = P’xy by simply regressing them on the test regressors Z. This is equivalent to running the GNR without the X variables on the right hand side of (8.82) yielding test-statistics that are too small.
Davidson and MacKinnon (1993, p. 195) show that the RESET with yt = Xt@ + yfc+ residual which is based on testing for c = 0 is equivalent to testing for 9 = 0 using the nonlinear model yt = Xte(1 + 9Xt0) + ut. In this case, it is easy to verify from (8.74) that the GNR is
yt — Xt в (1 + 9XtR) = (29(Xt/3)Xt + Xt )b + (Xt 0)2c + residual
 = E k=i вj B(Xij, X) + E S=iY sZis + ui
 The Breusch-Pagan Test
Next, we look at a Lagrange Multiplier test developed by Breusch and Pagan (1980), which tests whether H0; a2^ = 0. The test statistic is given by
 The squared correlation between y and р: R2 = r2y
 The Censored Normal Distribution
Let y* be N(y, a2), then for a constant c, define y = y* if y* > c and y = c if y* < c. Unlike the truncated normal density, the censored density assigns the entire probability of the censored region to the censoring point, i. e., y = c. So that Pr[y = c] = Pr[y* < c] = Ф(^ — y)/a) = Ф(Р*) where c* = (c — y)/a. For the uncensored region the probability of y* remains the same and can be obtained from the normal density.
It is easy to show, see Greene (1993, p. 692) that