Comparing OLS and IV estimates with endogenous dummy regressor and continuous instrument

[Rashesh Shrestha]

I have a question regarding interpretation of coefficients in a model with endogenous dummy variable. The model is Y = a + bD + e, where D is endogenous 0/1 dummy. I have an instrument Z, which is continuous. I run a 2sls model using Stata's "ivregress" command. The concern I have is that the IV estimate of b (call it b_iv) is much larger than the OLS estimate (call it b_ols). Elsewhere, someone mentioned that the correct interpretation of b_iv is a change in Y due to a unit change in instrument Z. Is such an interpretation unique to the case of endogenous dummy regressors?

In the attached example, "formal2" is the endogenous dummy, and "worker" is the continuous instrument. The OLS coefficient is 0.63 and IV coefficient is 1.71. How should I interpret the two coefficients? Please let me know your thoughts!

Thank you.

Rashesh

[Mark Schaffer]

Rashesh:

(1) "... the IV estimate of b (call it b_iv) is much larger than the OLS estimate (call it b_ols)". This is actually how the Durbin-Wu-Hausman test of endogeneity works (it's a vector of contrasts test, and here the length of the vector is 1). If the two estimates of b are similar, then you can interpret this as evidence that both the OLS and IV estimates are consistent; if they're different, you can interpet this as evidence that OLS is inconsistent (since the usual null is that IV is consistent either way). See e.g. the Stata help for estat endogenous for discussion.

(2) "...someone mentioned that the correct interpretation of b_iv is a change in Y due to a unit change in instrument Z". This is wrong, or maybe there's something missing from the claim. (For example, if you multiply your instrument by 10, you'll find the IV estimate of b is unchanged, which of course contradicts the claim.) Maybe you can provide the reference or double-check?

[Rashesh Shrestha]

Dear Mark, Thank you for your response. Regarding the claim about interpretation of IV - I found it in this forum: http://stats.stackexchange.com/quest...iable-question. This is what the author claims: " If you use the continuous instrument, the coefficient of the endogenous variable in the second stage can be interpreted as the causal effect of the endogenous variable due to a one percentage point increase in the instrument," which is not what I initially thought it said. But I am not sure I completely understand the claim as it stands.

I was wondering whether b_iv can still be interpreted as the difference in expected Y given D=1 and Y given D = 0. In a 2SLS framework, we get b_iv by regressing D on Z to get predicted values D_hat. Then regress Y on D_hat. Haven't we changed the nature of regression as D_hat is continuous while D is discrete. Wouldn't that change the interpretation of the coefficient? If we took the second stage literally, b_iv would be the causal impact of one percentage point increase in probability that D takes the value 1...

[Mark Schaffer]

I don't think that's a very helpful way to think about it, to be honest. The intuition is much easier if you ask two questions: (a) What is the interpretation of b? (b) Does my estimation method give me a consistent (or unbiased, similar intuition) estimate of b? The point about asking it this way is that the interpretation of b shouldn't depend on the estimation method. b is b is b. In your setup, if you want to give b a causal interpretation, it's the causal impact of whatever treatment D is. And if the conditions required for the IV estimator to be consistent are satisfied (uncorrelated with the error, correlated with D, etc. etc.) then b_iv is a consistent estimate of this causal impact. No need to mess around with interpretations involving percentage changes in Z and whatnot unless you really want to go there.

[Rashesh Shrestha]

Thank you sharing your thoughts. I agree that the interpretation of b should be the same regardless of whether one uses IV or OLS. I am trying to find an article that uses the same method (continuous IV with dummy endogenous), but with little luck so far...

Thank you again for your help!

[Mark Schaffer]

Standard sources that discuss it are Jeff Wooldridge's Econometric Analysis of Cross Section and Panel Data and (I think) Angrist and Pischke's Mostly Harmless Econometrics. Plus it comes up on Statalist fairly regularly. Here are 3 examples:

http://www.statalist.org/forums/foru...ols-regression
http://www.stata.com/statalist/archi.../msg00188.html
http://www.stata.com/statalist/archi.../msg00042.html

[Rashesh Shrestha]

Thank you for the references, Mark. Wooldridge's book refers to a paper by Currie and Cole (1993, AER) which uses continuous instrument for an endogenous variable. The paper also reports a very large difference between OLS and 2SLS estimates (Table 2)...