What makes a valid instrument?

A valid instrument must be exogeneous and a strong predictor of the explanatory variable of interest. For an instrument to be considered exogenous, it must be uncorrelated with the error term in the regression model. This means it does not have a direct relationship on the dependent variable, but only affects the dependent variable through our endogenous regressor. Our instrument is considered strong when it is a strong predictor of our endogenous regressor (has a high correlation).

Is there a test for instrument validity?

One test for instrument validity is the Hausman test. It is based on the idea that the OLS estimate and IV estimate should give similar results. The null hypothesis of the test is that our regressors and error term are uncorrelated (both OLS and IV give consistent estimators). The alternative hypothesis is that the regressors are correlated with the error term. If the test fails to reject the null hypothesis, you should use OLS since its variance is smaller. If the test rejects the null hypothesis, you should use instrumental variables. However, this test is only informative if the instruments are consistent.

Another test for instrument validity is the Wu test, which again is only informative if you have good instruments. Rejection of the Wu test null hypothesis means that the regressors in the model are endogeneous and the OLS estimates are biased. In that case, instrumental variables approach should be used, if you have a valid instrument.

To test the strength of the instrument, run a “first stage” regression of your instrument on your endogenous regressor.

Xi = alpha_i + beta * Z_i + epsilon_i

The coefficient estimate for beta should be large and statistically significant. However, “large” may be controversial based on the variables of interest, so you should have a valid argument for why your instrumental variable is strong. You should also have a valid argument for why you believe your instrument is exogenous.

Should I use multiple instruments if my instrument is not very good?

No, if you believe your instrument is weak, adding multiple weak instruments to the model will not increase validity. Even though the IV estimate is consistent, if your instrument(s) is weak your estimate will be biased towards OLS, which is not consistent in the case of endogenous regressors. This bias is worse with multiple weak instruments. It is best to only use your strongest instruments.

Can I assume my IV estimate has a normal sampling distribution?

Not necessarily. Weak instruments cause the variance of the IV estimator to be smaller, which means the distribution of the IV estimate is not as well approximated by a normal distribution. The IV estimate’s sampling distribution can deviate from normality in two ways, bias (even if the instrument is valid), and understating the true variability.

What methods are available for modeling IV?

Some methods available for estimating IV coefficients are method of moments (MOM), control functions, and two-stage least-squares (2SLS). MOM is based on minimizing the difference between sample and population “moments”. The 2SLS approach uses two regressions to estimate the IV coefficient, the first regression gets rid of the endogenous part of your regressor, and the second uses those fitted values to obtain the IV estimate. The control function approach is very similar to 2SLS, but works better in the case of nonlinearity.

How will measurement error in my endogeneous regressor affect my IV estimates?

Using an instrumental variable is a way to solve measurement error in your independent variable. As long as the measurement error is uncorrelated with your instrument (and your instrument still meets the exclusion restriction) then your IV estimate will be unbiased.

References

Peter E. Rossi (2014) Invited Paper—Even the Rich Can Make Themselves Poor: A Critical Examination of IV Methods in Marketing Applications. Marketing Science 33(5):655-672

Stehr, Mark. (2019) Drexel University, Lebow College of Business, ECON 940 [PDF]