LATE Theorem with a Continuous Instrument

[Maxence Morlet]

Hi all,

Regarding instrumental variable estimation, I am a bit confused regarding the LATE interpretation in settings with a continuous instrument. With a binary instrument, we obtain a Wald estimator and the first stage is the proportion of compliers, and the second stage estimate is the LATE on compliers.


Card (2001) compiles reasons why IV estimates may be larger in magnitude than OLS estimates in the context of the effect of schooling on wages. One of these reasons is that IV picks up the effect of schooling on the subset of individuals who would benefit the most from it, and links it to the LATE reasoning.


However, do this reasoning and interpretation (second-stage effect and first stage effect) hold with a continuous instrument? I am trying to relate it to the marginal treatment effects literature, with all treatment effects actually representing various weighted averages of marginal treatment effects.

To give an example, the instrument we use is a Bartik-style shift-share instrument, which varies across time and units. It is a fractional variable, and the endogenous variable is also a fractional variable.

I have tried looking into the user written mtefe command, however the treatment variable has to be binary, and I am not sure whether this is the right econometric route to go down...


Our IV estimates are much larger than OLS estimates. One of the reasons is that the first stage coefficient is 0.5, given that the endogenous variable is fractional, a coefficient comprised within the unit interval seems plausible...

Please could someone explain whether Card's (2001) abovementioned interpretation is applicable to this context, with a continuous instrument?

[Joao Santos Silva]

Dear Maxence Morlet,

Would it be possible for you to share the data you are using? Anyway, please do keep in mind that frequently 2SLS estimates do not have a LATE interpretation at all; see here.

Best wishes,

Joao

[Maxence Morlet]

Thank you very much for the paper you suggested. I will give it a read!

Here are my data: (outcome Y, X is the endogenous variable and Z is the instrument). The outcome variable has a limited distribution between -2 and 2. X and Z are both fractional variables.

sum Z X Y

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
Z |     11,377    .1638475    .2345747          0          1
X |     11,377    .0876119    .2654689          0          1
Y |     11,377    -.161402    .4663677         -2          2

input float(Y X Z)
 0         0           0
 -2         0           0
 0         0           0
 0  .8068182    .7068183
-1         1    .4542174
 0         0   .26547596
 0         0   .15186775
-1         0           0
 0         0           0
 1      .125    .1002888
 1 .14285715   .05062258
 1         0    .1002888
 0         0   .07326601
 0         0  .004210526
 0         0  .004210526
-1         1    .7070182
-1         0    .4049807
-1         0   .58612806
-1         0    .4049807
-1         0    .4049807
-1         0    .3845747
-1         1    .7070182
 0         0  .012331825
-1         0   .06135672
 0         0   .00942902
-1         0           0
 0         0           0
 0         0           0
 0         0  .012331825
 0         0  .012331825
 0         0   .06135672
-1         0           0
 0         0           0
 0         0           0
 2         0           0
 0         0  .013035382
 0         0 .0041724616
 0         0  .005972475
 0         0           0
 0         0   .17197452
-1         0   .17197452
 1         0   .06508876
 0         0   .17197452
 0         0           0
 0         0  .004574224
 0         0   .03599965
 0         0  .004574224
 0         0  .017893894
 0         0   .01237955
-1         0  .017893894
-1         0   .05103595
 0         0  .017699115
 0         0           0
 1         0   .08675131
-1         0   .03374839
 0         0   .04884401
 1         0   .03374839
 0         0 .0013805005
 0         0           0
-1         0   .11554386
-1         0   .11554386
 0         0 .0013805005
-1         0 .0013805005
 0         0     .320638
-1  .4210526    .7070182
-1         1    .6237779
 0         0  .011272062
 0         0  .011272062
 0         0           0
 0         0           0
 0         0           0
 0         0           0
 0         0           0
 0         0           0
 0         0           0
 0         0           0
 1         0  .014201437
 1         0  .004222685
 2         0           0
 1         0  .004222685
 0         0  .072082125
 0         0  .072082125
 0         0    .5603871
 0         0    .5603871
 0         1    .8504523
 0        .8    .8504523
 0         0           0
 0         0 .0017256256
 0         0 .0017256256
 1         0           0
 1         0   .03710758
-1         0           0
 0         0           0
 0         0           0
 0         0           0
 0         0           0
 0         0           0
 1         0  .011090573
 0         0   .01930502
 0         0           0

[Maxence Morlet]

I forgot to specify; my question is under the assumption of heterogenous treatment effects.

[Maxence Morlet]

Hi all, really sorry for reviving this thread, just wondering whether anyone has any thoughts?