Manual 2SLS as simply as humanly possible

[Chuck Hotelier]

Hi all,

I have what I think is a very simple question but I have not seen a generalizable answer to this on StataList.

if you want to run a 2sls, you can do so very easily with Procedure 1:

ivreg y1 x1 (x2 = z1 z2)

But if I want to do so in two steps as below in Procedure 2, the standard errors will obviously be wrong.

reg x2 x1 z1 z2
predict x2hat, xb
reg y1 x1 x2hat

Is there a quick (or less quick) way to force the standard errors in Procedure 2 to the correct values taken from Procedure 1? Similarly, what if the regression I wanted to complete manually was

ivreg y1 x1 (x2 = z1 z2), r

Would that have a simple fix as well? Hope I'm not violating any StataList Guidelines here!
Thanks so much in advance!

Best,
Chuck

[Andrew Musau]

You can bootstrap the procedure, and the resulting bootstrap standard errors should be similar to the estimated robust standard errors.

[Chuck Hotelier]

Hi Andrew, this is informative. I've heard of bootstrapping, but what is the procedure in stata to do this for this simple example? Or at least good resources on this? Thank you for your time.

Best,
Chuck

[Chuck Hotelier]

Hi Andrew, this is informative. I've heard of bootstrapping, but what is the procedure in stata to do this for this simple example? Or at least good resources on this? I suppose my question is really this --

Is it sufficient for me to just change the last stage of my regression to

ivreg y1 x1 x2hat, vce(bootstrap, reps(1000))

Thank you for your time.

Best,
Chuck

[Jeff Wooldridge]

Chuck, that won't do it, for two reasons. The first is easy to fix: you want reg in place of ivreg. The second is fundamental: the command will not account for the first-step estimation, which means it will roughly give you the same standard errors as if you do

reg x2 x1 z1 z2
predict x2hat
reg y x1 x2hat, robust

Unfortunately, adding "robust" does not account for the sampling error in the first-stage regression.

What I've always done is write a short program that does both estimation steps, and then call the program in a bootstrap routine. You have to do both steps with each resampling of the data.

[Jeff Wooldridge]

I should say that I'm not a fan of doing this "by hand," and I don't see why a researcher would do this unless you have some sort of split-sample situation.

[Andrew Musau]

You certainly can use the boostrap prefix with ivregress. But #1 suggests that you want to run the IV regression in 2 stages. Here is how, replicating the results of ivregress. Also, as stated in #2, the bootstrap standard errors would be similar to those obtained using the -robust- option.

*BOOTSTRAPPING TWO-STAGE PROCEDURE
cap prog drop pboot
program pboot
args depvar endog exog1 iv1
regress `endog' `iv1' `exog1'
tempvar res
predict `res', res
regress `depvar' `endog' `exog1' `res'
end


webuse hsng2, clear
bootstrap _b[hsngval] _b[pcturban], reps(50) seed(02202024): pboot rent hsngval pcturban faminc

*IVREGRESS WITH BOOTSTRAP SEs
bootstrap, reps(50) seed(02202024): ivregress 2sls rent pcturban (hsngval = faminc)

*ROBUST SEs
ivregress 2sls rent pcturban (hsngval = faminc), r

Res.:

. *BOOTSTRAPPING TWO-STAGE PROCEDURE

.
. cap prog drop pboot

.
. program pboot
  1.
. args depvar endog exog1 iv1
  2.
. regress `endog' `iv1' `exog1'
  3.
. tempvar res
  4.
. predict `res', res
  5.
. regress `depvar' `endog' `exog1' `res'
  6.
. end

.
.
.
.
.
. webuse hsng2, clear
(1980 Census housing data)

.
. bootstrap _b[hsngval] _b[pcturban], reps(50) seed(02202024): pboot rent hsngval pcturban faminc
(running pboot on estimation sample)

Bootstrap replications (50): .........10.........20.........30.........40.........50 done

Bootstrap results                                           Number of obs = 50
                                                            Replications  = 50

      Command: pboot rent hsngval pcturban faminc
        _bs_1: _b[hsngval]
        _bs_2: _b[pcturban]

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
             | coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       _bs_1 |   .0031938   .0007289     4.38   0.000     .0017653    .0046224
       _bs_2 |  -.5064118   .5925213    -0.85   0.393    -1.667732    .6549086
------------------------------------------------------------------------------

.
.
.
. *IVREGRESS WITH BOOTSTRAP SEs

.
. bootstrap, reps(50) seed(02202024): ivregress 2sls rent pcturban (hsngval = faminc)
(running ivregress on estimation sample)

Bootstrap replications (50): .........10.........20.........30.........40.........50 done

Instrumental variables 2SLS regression            Number of obs   =         50
                                                  Wald chi2(2)    =      47.89
                                                  Prob > chi2     =     0.0000
                                                  R-squared       =     0.2887
                                                  Root MSE        =     29.517

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
        rent | coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
     hsngval |   .0031938   .0007289     4.38   0.000     .0017653    .0046224
    pcturban |  -.5064118   .5925213    -0.85   0.393    -1.667732    .6549086
       _cons |   113.8143    18.8149     6.05   0.000      76.9378    150.6909
------------------------------------------------------------------------------
Endogenous: hsngval
Exogenous:  pcturban faminc

.
.
.
. *ROBUST SEs

.
. ivregress 2sls rent pcturban (hsngval = faminc), r

Instrumental variables 2SLS regression            Number of obs   =         50
                                                  Wald chi2(2)    =      32.55
                                                  Prob > chi2     =     0.0000
                                                  R-squared       =     0.2887
                                                  Root MSE        =     29.517

------------------------------------------------------------------------------
             |               Robust
        rent | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
     hsngval |   .0031938    .000738     4.33   0.000     .0017474    .0046402
    pcturban |  -.5064118   .5428297    -0.93   0.351    -1.570339    .5575149
       _cons |   113.8143   21.62169     5.26   0.000     71.43659    156.1921
------------------------------------------------------------------------------
Endogenous: hsngval
Exogenous:  pcturban faminc

.

Depending on how many instruments you have, change the bootstrap program accordingly.

[Michael Cho]

I should say that I'm not a fan of doing this "by hand," and I don't see why a researcher would do this unless you have some sort of split-sample situation.

Hi Jeff,

Thanks for sharing your insights I have two follow-up questions, and I really appreciate if you have any suggestions or comments.

1. Calculation of (adjusted) R2 in 2SLS.
It is recommended to use a software package with 2SLS command rather than explicitly carry out the two-step procedure (Wooldridge 2002). In doing so, 2SLS software will compute 2SLS residuals, instead of residuals from the second-stage OLS regression. The main difference is that whether we use predicted X or actual X when we calculate the residuals. As such, it is possible that RSS > TSS, and then R2 is negative. (In case I didn't discuss the problem clearly, please see https://www.stata.com/support/faqs/s...least-squares/). My question is that under what conditions we'd better report R2 (or adjusted R2) for 2SLS results and under what conditions a negative R2 (or adjusted R2) is acceptable? Can we just not report R2 if it is negative?

2. Doing 2SLS by hand.
If we do 2SLS by hand, the R2 (or adjusted R2) are positive. Is it reasonable to report R2 (or adjusted R2) from the two-stage OLS analyses? If yes, I guess it is better to use bootstrap to calculate SE and R2 (or adjusted R2).


Thank you for your guidance and look forward to your reply.


Best,
Michael