IV回归系数比OLS大很多咋回事, 怎么办呢?

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问题：IV估计的系数比OLS大了很多很多，有什么问题吗？

I am estimating an instrumental variable model by using -ivreg2 in Stata. The IV estimates are way more than OLS estimate (4.006567).Could you advise what could be possible reasons for such a blown up coeff (49.97424 )? or my interpretation of coeff is incorrect.

areg agr_share roaddum ims_hab_pop, robust absorb(dcode)

Woordbridge回答：

For the IV estimates to be at all convincing, first run the reduced form,

areg roaddum rank agr_share  ims_hab_pop, absorb(dcode) robust

and look at the sign of the coefficient on rank and its t statistic. Hopefully you have a predicted sign for the effect of rank on roaddum. If the sign is what you expect, is rank strongly statistically significant? The statistics for weak identification provide this information, too, but you should estimate the reduced form, anyway. It is easier to see what is happening.

伍德里奇怀疑有弱工具变量问题，又或者这个工具变量压根就不是外生的。

My guess is the instrument is weak. Or, the IV could actually be endogenous (which cannot be tested, unfortunately).

下面这个问题也是与此相关的，IV回归的系数相对于OLS很大的原因？

I am doing an OLS analysis, and get a reasonable economic magnitude. I also do a 2sls analysis, still get significant results, but the economic magnitude is way above reasonable. Is there a justification for arguing that the 2sls confirms that the variables are indeed significant, but that we should not look at the economic magnitude of the 2sls, and instead, look at those of the OLS? Thanks

下面这个是非常细致的回答，尤其需要关注的是“局部平均处理效应”的异质性问题。

This is a good question, even though it got hounded on here. A couple of points.

First, the exclusion restriction for iV is untestable. Which is worth noting only because your IV estimate, which is significantly larger than your OLS estimate, is consistent so long as the instrument is exogenous, and that is an untestable assumption.

Second, it's entirely possible that the IV estimate to be larger than the OLS estimate because IV is estimating the local average treatment effect. Say that you are instrumenting for education and estimating the returns to schooling. OLS is estimating the average treatment effect (or trying to) over the entire population. What is IV estimating? It is estimating the local average treatment effect. So let's say that your instrument shifts the behavior of a subgroup of individuals for whom the returns to education are larger than average. Then IV estimates will be larger than OLS estimates because of heterogeneity in the population you're studying.

With that in mind, I will now somewhat contradict what I just said. This is basically where we are in the field, so you will have to probably defend those estimates against what I'm about to say.

You cannot prove the the instrument is excludable, sadly. But, the only known test of the exclusion restriction (which I already said doesn't exist) is a rule of thumb like what you're showing. Consider a simple diagram where you have Z as an instrument, X1 as the endogenous variable (schooling) and Y the outcome (salary). The exclusion restriction requires Z->X1 and X1->Y, but Z does not cause Y directly or through some other mediated variable, like X2.

If, though, Z->X2 and X2->Y, as well as Z->X1->Y, then you could have two biased estimates in OLS and 2SLS. There's a backdoor path from X1 to Y through X1<-Z->X2->Y. And when you estimate IV, you are estimating a first stage of X1 on Z, and a reduced form of Y on Z. And the IV estimator is the ratio of those two. But the reduced form model of YZ is biased as it may be that Z is causing Y through X2, which you've ruled out by assumption.

The IV estimate is going to be larger than OLS because the numerator>denominator in the IV model, which is Cov(Y,Z)/Cov(X1,Z). It's either that the numerator is blowing up, because there is a connection between Z and X1, or the first stage is weak. You probably checked whether the first stage is weak, so it's possible your untestable assumption is simply being violated.

This is hard because recall, it's entirely possible that LATE>ATE if there are heterogenous subpopulations. And in fact, the work of the last decade or so has focused a lot on that very possibility. Yet, when you have a blown up 2SLS coefficient compared to the OLS coefficient, I think people have a feeling that probably your instrument isn't exogenous in the first place.

继续追问：

Then what is the answer? OLS gives ATE. Under additional assumptions, IV gives LATE. How can the two be so wildly different? What does it mean? That the endogeneity is too hot to handle.

进一步用例子讲LATE：

Say you have two groups of workers, both uneducated. If group 1 goes to college, their wages increase by 10%. If group 2 goes to college their wages increase by 20%.

Now assume that the first group goes to college, and you estimate the ATT which would yield 10%.

Now assume that your instrument shifts the behavior of the second group only. Say that the second group was a discriminated minority, and that was the reason they didn't goto college. Now you shift them into college through some mandatory program, or hefty subsidies. As the first group was already going to college, your IV doesn't identify the ATT as your instrument is always only identifying the parameter associated with the behavior of the group affected by the instrument. So if you used IV, and it was valid, you'd estimate a 20% return to schooling.

But notice, that's only with regards to the affected sub-group. The ATT is 10% as noted. The LATE was twice as high, and that was only because the subgroup had such large returns.

I think that if you found something like this, you really will have your work cut out for you to defend it. There's a reason that Heckman says we typically care about the ATT moreso than the ATU (treatment effect on untreated). If you can't find it for the treatment group who self-select into treatment, then it's odd that you'd find an even larger effect for the untreated group. 

Economic theory would suggest the individuals with the highest returns to schooling are going to schooling, unless they face some different set of unobserved constraints or incentives -- say, liquidity constraints are different or discrimination is different. For certain applications, I think readers are probably primed to expect this -- like labor markets with discrimination -- but even then, there's a point of credibility one has to overcome since as I noted earlier the same large coefficients on 2SLS could be generated by an invalid instrument.

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