Hausman specification test and regression based tests (for endogeneity) give different results

[Rosamond Waston]

Hi,

I would like to ask a question regarding my regression. First, my regression is: y=a0+a1x1+a2x2+a4x4+a5x5+a6x6+a7x8+a8d; among these different regressors x2 is suspect as endogeneous, and z3 is IV for x2 (not weak instrument). To test for endogeneity, I run two different test for endogeneity: 1. Hausman Test 2. Regression based test. The code is following:

1. Hausmant test

ivregress 2sls y x1 x4 x5 x6 i.x8 i.d (x2= z3),first 
est store IV
reg y x1 x2 x4 x5 x6 i.d i.x8
est store ols
hausman IV ols,equations(1:1)

The results fail to reject the null implying we have no endogeneity
Test: Ho: difference in coefficients no
> t systematic

chi2(8) = (b-B)'[(V_b-V_B)^
> (-1)](b-B)
= 6.31
Prob>chi2 = 0.6129

2: Regression based test:

reg x2 x1 x4 x5 x6 i.d i.x8 z3  
predict resd, resid
reg y x1 x2 x4 x5 x6 x7 i.d i.x8 resd       //res is significant implying y2 is endogeneous, then we should use IV
drop res

The results clearly show that we have an endogeneity problem (uhat is significant):



. reg y x1 x2 x4 x5 x6 i.x8 i.d uhat

Source SS df MS Number of obs = 322
F(10, 311) = 10.25
Model 56.6758454 10 5.66758454 Prob > F = 0.0000
Residual 171.981689 311 .552995784 R-squared = 0.2479
Adj R-squared = 0.2237
Total 228.657534 321 .712328767 Root MSE = .74364


y Coef. Std. Err. t P>t [95% Conf. Interval]

x1 -.0057789 .0758074 -0.08 0.939 -.1549391 .1433812
x2 .0981171 .0168363 5.83 0.000 .0649896 .1312446
x4 -1.55e-07 2.85e-06 -0.05 0.957 -5.76e-06 5.45e-06
x5 .0000413 .0002984 0.14 0.890 -.0005458 .0006284
x6 .0001767 .0010991 0.16 0.872 -.0019859 .0023393

x8
1 -.0845744 .172997 -0.49 0.625 -.4249669 .255818
2 -.185302 .1184647 -1.56 0.119 -.4183955 .0477916
3 .1617911 .1208343 1.34 0.182 -.075965 .3995472

1.d .0428701 .0952056 0.45 0.653 -.1444583 .2301986
uhat -.0366868 .0183065 -2.00 0.046 -.0727071 -.0006665
_cons 8.890939 1.339391 6.64 0.000 6.255525 11.52635


. test uhat

( 1) uhat = 0

F( 1, 311) = 4.02
Prob > F = 0.0459


Can anyone help me for this problem?
Thank you in advance.

[Joro Kolev]

These two tests are obviously not equivalent: the Hausman test is carried out on 8 degrees of freedom, and the regression based test is carried out on 1 degree of freedom.

I personally have more trust in the regression based test, but I cannot cite any paper showing that it has higher power, which I guess it should because it is on 1 degree of freedom rather than 8.

You can do one more thing, test whether the coefficient on only your variable of interest x2 is statistically different.

Hausman(1) = (Beta_x2_IV - Beta_x2_OLS)/sqrt[se(Beta_x2_IV)^2 + se(Beta_x2_OLS)^2], and compare this statistic to the standard normal distribution.

[Rosamond Waston]

These two tests are obviously not equivalent: the Hausman test is carried out on 8 degrees of freedom, and the regression based test is carried out on 1 degree of freedom.

I personally have more trust in the regression based test, but I cannot cite any paper showing that it has higher power, which I guess it should because it is on 1 degree of freedom rather than 8.

You can do one more thing, test whether the coefficient on only your variable of interest x2 is statistically different.

Hausman(1) = (Beta_x2_IV - Beta_x2_OLS)/sqrt[se(Beta_x2_IV)^2 + se(Beta_x2_OLS)^2], and compare this statistic to the standard normal distribution.

Dear Prof.Joro,

Many thanks for your help. It indeed helps me a lot and I perform the test on "reg y on x2" (only including endogenous variable and ignoring other variables like x1, x3, x4, x5, x6), the results from Hausman and Regression-based tests are indeed the same and indicate there is the endogeneity of x2.

Results from Hausman
Test: Ho: difference in coefficients not systematic

chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B)
= 3.78
Prob>chi2 = 0.0518

However, may I ask how can we explain such special cases? Can we say that the Hausman test has more degree of freedom than regression-based test, thus has lower level of credibility? Is it a reasonable explanation?

Thank you in advance.

Regards,