Why are marginal effects of instrumental variables statistically significant in ivprobit analysis?

[Gina Malek]

Hi,
I am running an ivprobit model and have encountered an issue with the margin command in computing the average marginal effects. I have specified the model as:

ivprobit DV X₁ X₂ ........X_n (IV= Z₁ Z₂ Z₃)

where,

DV is my binary choice dependent (endogenous) variable; X₁ - X_n are my exogenous variables (continuous or binary); IV is my instrumented (endogenous) variable (continuous) and Z₁, Z₂, Z₃ are my additional instruments (continuous).

The model runs in Stata without problems and gives reasonable results and a Wald test of exogeneity of chi2(1) = 10.12 Prob > chi2 = 0.0015 confirming existance of endogeneity.

When I run the margins command (margins, dydx (*) predict (pr) ) to obtain the average marginal effects, the results give reasonable average marginal effects for all exogenous variables (as expected). However, the results also give statistically significant average marginal effects for instrumental variables Z₁, Z₂, Z₃. I find these latter results odd because I expected the instrumental variables to not appear in the marginal effects results.

By looking at the example of marginal effects computation after ivprobit (Example 1 on Page 3 of Stata's " ivprobit postestimation — Postestimation tools for ivprobit" document), I noticed that the instrumental variable (male_educ) shows an average marginal effect of dy/dx = 0 (with an omitted Delta-method Std. Err.). Further, in explaining the marginal effects results, the document mentions that " male_educ has no effect because it appears only as an instrument."

Considering the above, I am confused as to 1) why the instrumental variables in my model (Z₁, Z₂, Z₃) are not omitted from the average marginal effects results table and 2) why they exhibit statistically significant marginal effects.

Wouldn't this mean that instruments Z₁, Z₂, Z₃ are correlated with DV and therefore, not valid? That is also confusing because in testing the validity of instruments, I ran the model with the "twostep" option followed by the "overid" command as follows:

ivprobit DV X₁ X₂ ........X_n (IV= Z₁ Z₂ Z₃), twostep
overid, depvar( DV )

The results gave: Amemiya-Lee-Newey minimum chi-sq statistic 0.241 Chi-sq(1) P-value = 0.8866

As seen, the P value is not significant for the Amemiya-Lee-Newey test which, as I understand, means that instruments are valid (null hypothesis for the "Amemiya-Lee-Newey" test is that instruments are valid, correct?)

Please help me understand what I am doing wrong and why I get statistically significant marginal effects for my instrumental variables.

Thank you in advance.

[Joro Kolev]

This is something very weird, I am able to replicate the problem, and I would also like to know what is going on here, if anybody has an idea...

For the time being I can remark that:

1) No, the fact that -margins- calculates marginal effects with respect to instruments, which is should not, does not mean that your instruments are invalid. By the exclusion restriction on which the estimation of this model is based, instruments do not have a direct effect on the outcome.

2) The problem seems to be innocuous, because the incorrect calculation of marginal effects with respect to instruments does not affect the marginal effects of the included variables in the model.

.  webuse laborsup

.  ivprobit fem_work fem_educ kids (other_inc = male_educ), nolog

Probit model with endogenous regressors         Number of obs     =        500
                                                Wald chi2(3)      =     163.88
Log likelihood = -2368.2062                     Prob > chi2       =     0.0000

----------------------------------------------------------------------------------------------
                             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-----------------------------+----------------------------------------------------------------
                   other_inc |  -.0542756   .0060854    -8.92   0.000    -.0662028   -.0423485
                    fem_educ |    .211111   .0268648     7.86   0.000     .1584569    .2637651
                        kids |  -.1820929   .0478267    -3.81   0.000    -.2758315   -.0883542
                       _cons |   .3672086   .4480724     0.82   0.412    -.5109971    1.245414
-----------------------------+----------------------------------------------------------------
 corr(e.other_inc,e.fem_work)|   .3720375   .1300518                      .0946562    .5958136
              sd(e.other_inc)|   16.66621   .5270318                      15.66461    17.73186
----------------------------------------------------------------------------------------------
Instrumented:  other_inc
Instruments:   fem_educ kids male_educ
----------------------------------------------------------------------------------------------
Wald test of exogeneity (corr = 0): chi2(1) = 6.70        Prob > chi2 = 0.0096

. margins, dydx(*) predict(pr)

Average marginal effects                        Number of obs     =        500
Model VCE    : OIM

Expression   : Probability of positive outcome, predict(pr)
dy/dx w.r.t. : other_inc fem_educ kids male_educ

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   other_inc |  -.0097802   .0014994    -6.52   0.000     -.012719   -.0068414
    fem_educ |   .0623273    .007099     8.78   0.000     .0484135     .076241
        kids |  -.0614265   .0139446    -4.41   0.000    -.0887574   -.0340956
   male_educ |  -.0194406   .0022103    -8.80   0.000    -.0237728   -.0151084
------------------------------------------------------------------------------

. margins, dydx(other fem kids) predict(pr)

Average marginal effects                        Number of obs     =        500
Model VCE    : OIM

Expression   : Probability of positive outcome, predict(pr)
dy/dx w.r.t. : other_inc fem_educ kids

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   other_inc |  -.0097802   .0014994    -6.52   0.000     -.012719   -.0068414
    fem_educ |   .0623273    .007099     8.78   0.000     .0484135     .076241
        kids |  -.0614265   .0139446    -4.41   0.000    -.0887574   -.0340956
------------------------------------------------------------------------------

.

[Gina Malek]

Thank you Joro! Your response was very helpful. At least, I know my instruments are not invalid.
I hope others share their ideas as to why the margins command calculates marginal effects with respect to instruments, while the average marginal effect of the instrumental variable (male_educ) in the example I mentioned above (Example 1 on Page 3 of Stata's " ivprobit postestimation — Postestimation tools for ivprobit" document) is reported as dy/dx = 0 (with an omitted Delta-method Std. Err.).