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  • Weak-instrument–robust tests and confidence intervals

    • Anderson–Rubin

    • Conditional likelihood-ratio

  • Tests and confidence intervals for robust, cluster–robust, and heteroskedasticity- and autocorrelation-consistent VCEs

  • Support for all ivregress estimators: 2sls, liml, and gmm

  • Nonstandard confidence intervals: empty, real line, union of intervals

Do you have weak instruments in your instrumental-variables (IV) regression? Use the new estat weakrobust command to perform reliable inference on endogenous regressors. This command is part of StataNow™.

IV methods allow researchers to estimate causal relationships even when some explanatory variables are endogenous. IV methods exploit other variables—instrumental variables—that are correlated with the endogenous variables but do not themselves suffer from endogeneity.

A well-known problem with IV methods in practice is that when instruments are only weakly correlated with the endogenous regressors, inference can become unreliable even in relatively large samples.

The new estat weakrobust postestimation command after ivregress performs Anderson–Rubin or conditional likelihood-ratio (CLR) tests on the endogenous regressors. estat weakrobust can also construct the associated confidence interval when there is only a single endogenous regressor. These tests and confidence intervals are fully robust to weak instruments.

Let's see it work

We might be interested in modeling the average rental rate (rent) in each U.S. state as a function of average housing values (hsngval) and the proportion of the population living in urban areas (pcturban). Because average housing values are likely to be endogenous, we instrument them using indicator variables for the region in which each state is located (i.region).

. webuse hsng
(1980 Census housing data)

. ivregress 2sls rent pcturban (hsngval = i.region), vce(robust)

Instrumental variables 2SLS regression            Number of obs   =         50
                                                  Wald chi2(2)    =      75.15
                                                  Prob > chi2     =     0.0000
                                                  R-squared       =     0.6692
                                                  Root MSE        =      20.13

rent Coefficient Std. err. z P>|z| [95% conf. interval]
hsngval .0015365 .0005128 3.00 0.003 .0005315 .0025415
pcturban .5149661 .3512175 1.47 0.143 -.1734075 1.20334
_cons 125.7878 11.95326 10.52 0.000 102.3598 149.2157
Endogenous: hsngval Exogenous: pcturban 2.region 3.region 4.region

However, we suspect that our instruments for hsngval are weak. We can perform a test on the coefficient of hsngval that is robust to weak instruments using estat weakrobust.

. estat weakrobust

Weak-instrument–robust test
Model VCE: Robust

 ( 1)  hsngval = 0

Cond. likelihood-ratio (CLR) test =   5.48
                       Prob > CLR = 0.0253

Notes: CLR test reported by default because
       model is overidentified.
       p-value computed by simulation
       (25,000 replications).

estat weakrobust reports a CLR test, which is the default behavior when the model is overidentified (meaning there are more instruments than endogenous regressors). estat weakrobust has accounted for the fact that we have fit the model using a heteroskedasticity-robust variance–covariance estimate (VCE) and reported an appropriate heteroskedasticity-robust version of the CLR test.

We are less confident that the coefficient on hsngval is different from zero (Prob > CLR = 0.0253) compared with the results from ivregress (P>|z| = 0.003).

We can also request a confidence interval robust to weak instruments:

. estat weakrobust, ci

Searching for CI bounds:
Iteration 0:  Grid points = 500
Iteration 1:  Grid points = 1,000

(CI computed using 1,000 grid points on [-.003591, .006664])

Weak-instrument–robust inference
Model VCE: Robust

Coefficient [95% conf. interval]
hsngval .0015365 .0002366 .0028467
Notes: CLR CI reported by default because model is overidentified. Computed using simulation (25,000 replications).

The CLR confidence interval of [.0002, .0028] is noticeably wider than the conventional confidence interval of [.0005, .0025] from the output of ivregress and comes closer to containing 0. Indeed, if we specified a confidence level of 99% by specifying the option level(99), 0 would be included. Accounting for the possibility of weak instruments, we conclude that we cannot rule out a zero coefficient on hsngval.

Tell me more

Read more about inference robust to weak instruments in the Stata Base Reference Manual; see [R] ivregress postestimation.

View all the new features in Stata 18 and, in particular, New in instrumental-variables analysis.

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