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# chi-square distribution with k degrees of free

Endogeneity Tests
The 2SLS estimator allows the regressor x2i to be endogenous, meaning that x2i
is correlated
with the structural error ei
. If this correlation is zero, then x2i
is exogenous and the structural
equation can be estimated by least-squares. This is a testable restriction. E§ectively, the null
hypothesis is
H0 : E(x2iei) = 0
with the alternative
H1 : E(x2iei) 6= 0:
The maintained hypothesis is E(ziei) = 0. Since x1i
is a component of zi
, this implies E(x1iei) = 0.
Consequently we could alternatively write the null as H0 : E(xiei) = 0 (and some authors do so).
Recall the control function regression (12.61)
yi = x
0
1i 1 + x
0
2i 2 + u
0
2i + “i
=

E

u2iu
0
2i
1
E (u2iei):
Notice that E(x2iei) = 0 if and only if E (u2iei) = 0, so the hypothesis can be restated as H0 : = 0
against H1 : 6= 0. Thus a natural test is based on the Wald statistic W for = 0 in the control
function regression (12.28). Under Theorem 12.9 and Theorem 12.10, under H0; W is asymptotically
chi-square with k2 degrees of freedom. In addition, under the normal regression assumptions the
F statistic has an exact F(k2; n k1 2k2) distribution. We accept the null hypothesis that x2i
is
exogenous if W (or F) is smaller than the critical value, and reject in favor of the hypothesis that
x2i
is endogenous if the statistic is larger than the critical value.
SpeciÖcally, estimate the reduced form by least squares
x2i = b
0
12z1i + b
0
22z2i + ub2i
to obtain the residuals. Then estimate the control function by least squares
yi = x
0
i b + ub
0
2i b + “bi
: (12.65)
Let W, W0 and F = W0=k2 denote the Wald statistic, homoskedastic Wald statistic, and F statistic
for = 0.
Theorem 12.14 Under H0, W
d !
2
k2
. Let c1 solve P

2
k2
 c1

# 

1 . The test ìReject H0 if W > c1 î has asymptotic size .
Theorem 12.15 Suppose ei
jxi
; zi  N

0; 2

. Under H0, F  F(k2; n
k1 2k2). Let c1 solve P (F(k2; n k1 2k2)  c1 ) = 1 . The test
ìReject H0 if F > c1 î has exact size .
Since in general we do not want to impose homoskedasticity, these results suggest that the
most appropriate test is the Wald statistic constructed with the robust heteroskedastic covariance
matrix. This can be computed in Stata using the command estat endogenous after ivregress
when the latter uses a robu