A Question about the Relationship between the Instrumental Variables and the Instrumented Variable

[Minda Tan]

Hi everyone! I have a quick question about the instrumental variables: I have two instrumental variables in my study. All postestimation tests showed positive results. However, I just found in the 1st stage model IV 2 is not significantly related to my X (the instrumented predictor). Is it a problem?

Thank you!

[Maxence Morlet]

Can you show the 1st stage and outcome equations? Can you tell us which postestimation tests?

If your instrumental variable does not have a significant effect on your endogenous variable in the first stage, that's a bad start... Have you also run exogeneity tests / overidentification tests permitted by the fact that you have two instruments for one endogenous variable?

[Minda Tan]

Can you show the 1st stage and outcome equations? Can you tell us which postestimation tests?

If your instrumental variable does not have a significant effect on your endogenous variable in the first stage, that's a bad start... Have you also run exogeneity tests / overidentification tests permitted by the fact that you have two instruments for one endogenous variable?

Thank you for the reply!! The details are as follows:

The two instrumental variables are "w1pattfrq" (lagged value of w2pattfrq) and "schrpaiols". In the first stage results, the coefficient and error of "schrpaiols" are .0406711 .0291459 respectively.

. ivreg2 lgw2ttscore (patinc=w1pattfrq schrpaiols) stses pexpect household frqsprols, gmm first ffirst r savefirst savefprefix(st4)

-gmm- is no longer a supported option; use -gmm2s- with the appropriate option
gmm = gmm2s robust
gmm robust = gmm2s robust
gmm bw() = gmm2s bw()
gmm robust bw() = gmm2s robust bw()
gmm cluster() = gmm2s cluster()

Stored estimation results
-------------------------
-----------------------------------------------------------------------------
| Dependent Number of
Name | Command variable param. Title
-------------+---------------------------------------------------------------
st4patinc | ivreg2 patinc 7 First-stage regression: patinc
-----------------------------------------------------------------------------

First-stage regressions
-----------------------

First-stage regression of patinc:

OLS estimation
--------------

Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity

Number of obs = 1298
F( 6, 1291) = 51.03
Prob > F = 0.0000
Total (centered) SS = 253.3012327 Centered R2 = 0.1852
Total (uncentered) SS = 345 Uncentered R2 = 0.4018
Residual SS = 206.388123 Root MSE = .3998

------------------------------------------------------------------------------
| Robust
patinc | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
stses | .007181 .0067533 1.06 0.288 -.0060677 .0204296
pexpect | -.0240019 .0103517 -2.32 0.021 -.04431 -.0036939
household | .012675 .0114075 1.11 0.267 -.0097043 .0350543
frqsprols | .0955718 .007904 12.09 0.000 .0800657 .111078
w1pattfrq | -.1208448 .0072453 -16.68 0.000 -.1350586 -.106631
schrpaiols | .0406711 .0291459 1.40 0.163 -.0165073 .0978496
_cons | .298601 .0820164 3.64 0.000 .1377009 .459501
------------------------------------------------------------------------------
Included instruments: stses pexpect household frqsprols w1pattfrq schrpaiols
------------------------------------------------------------------------------
Partial R-squared of excluded instruments: 0.1525
Test of excluded instruments:
F( 2, 1291) = 141.94
Prob > F = 0.0000



Summary results for first-stage regressions
-------------------------------------------

Variable | Shea Partial R2 | Partial R2 | F( 2, 1291) P-value
patinc | 0.1525 | 0.1525 | 141.94 0.0000

NB: first-stage F-stat heteroskedasticity-robust

Underidentification tests
Ho: matrix of reduced form coefficients has rank=K1-1 (underidentified)
Ha: matrix has rank=K1 (identified)
Kleibergen-Paap rk LM statistic Chi-sq(2)=217.12 P-val=0.0000
Kleibergen-Paap rk Wald statistic Chi-sq(2)=285.42 P-val=0.0000

Weak identification test
Ho: equation is weakly identified
Kleibergen-Paap Wald rk F statistic 141.94
See main output for Cragg-Donald weak id test critical values

Weak-instrument-robust inference
Tests of joint significance of endogenous regressors B1 in main equation
Ho: B1=0 and overidentifying restrictions are valid
Anderson-Rubin Wald test F(2,1291)=5.73 P-val=0.0033
Anderson-Rubin Wald test Chi-sq(2)=11.52 P-val=0.0031
Stock-Wright LM S statistic Chi-sq(2)=11.36 P-val=0.0034

NB: Underidentification, weak identification and weak-identification-robust
test statistics heteroskedasticity-robust

Number of observations N = 1298
Number of regressors K = 6
Number of instruments L = 7
Number of excluded instruments L1 = 2

2-Step GMM estimation
---------------------

Estimates efficient for arbitrary heteroskedasticity
Statistics robust to heteroskedasticity

Number of obs = 1298
F( 5, 1292) = 36.44
Prob > F = 0.0000
Total (centered) SS = 78.74777498 Centered R2 = 0.1444
Total (uncentered) SS = 43597.37084 Uncentered R2 = 0.9985
Residual SS = 67.37944948 Root MSE = .2278

------------------------------------------------------------------------------
| Robust
lgw2ttscore | Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
patinc | .1154376 .0353072 3.27 0.001 .0462368 .1846385
stses | .0305582 .0040702 7.51 0.000 .0225807 .0385357
pexpect | .0588527 .0085029 6.92 0.000 .0421874 .0755181
household | .0102323 .0058854 1.74 0.082 -.0013028 .0217675
frqsprols | -.0093272 .0045185 -2.06 0.039 -.0181833 -.0004711
_cons | 5.419693 .058279 93.00 0.000 5.305468 5.533917
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic): 217.117
Chi-sq(2) P-val = 0.0000
------------------------------------------------------------------------------
Weak identification test (Kleibergen-Paap rk Wald F statistic): 141.938
Stock-Yogo weak ID test critical values: 10% maximal IV size 19.93
15% maximal IV size 11.59
20% maximal IV size 8.75
25% maximal IV size 7.25
Source: Stock-Yogo (2005). Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments): 0.732
Chi-sq(1) P-val = 0.3922
------------------------------------------------------------------------------
Instrumented: patinc
Included instruments: stses pexpect household frqsprols
Excluded instruments: w1pattfrq schrpaiols
------------------------------------------------------------------------------