Interaction effects in ivreg2

Mike Major

Join Date: Jun 2019

Posts: 10
#1

Interaction effects in ivreg2

15 Dec 2023, 19:14

Hi, I am trying to estimate whether the coefficient has changed between two periods using a date dummy (pre- and post-1975). The independent variable is endogenous so I need to use IV. I do not think my implementation is working correctly.

If I do not use interaction effects, when I manually predict the endogenous variables in the first-stage and then apply the second-stage I get the same coefficients as I would from using ivreg2 directly. (Obviously the standard errors are not correct, but for here I am just checking the coeffiicents).

But when I incorporate interactions my results are different. I think it is because the interactions are also being used in the first stage, but i only want to investigate the second stage - and leave the first-stage prediction unchanged.

Here is a reconstructed example:

Code:

use "http://www.stata-press.com/data/r13/wpi1.dta" , clear *Create endogenous variables gen X1 = wpi*2 + runiform()*200 gen X2 = wpi*0.5 + runiform()*500 *Create exogenous variables gen Z1 = X1 + runiform()*200 gen Z2 = X2 + runiform()*300 *Create date dummy gen post1975 = t>tq(1975q1) *Predict first-stage reg X1 Z1 Z2 predict X1_pred reg X2 Z1 Z2 predict X2_pred * Overall regression ivreg2 wpi (X1 X2 = Z1 Z2), first reg wpi X1_pred X2_pred ** Coefficients match, as expected * Interaction regression ivreg2 wpi (X1 X2 c.X1#i.post = Z1 Z2 c.Z1#i.post c.Z2#i.post) post1975 reg wpi X1_pred X2_pred c.X1_pred#i.post1975 post1975 ** Coefficients do not match. Why not?
Tags: None
Andrew Musau

Join Date: Oct 2014

Posts: 10168
#2

15 Dec 2023, 22:07

You have a total of three first stage regressions. The endogenous interaction is a variable in its own right.
Comment
Mike Major

Join Date: Jun 2019

Posts: 10
#3

16 Dec 2023, 01:55

Thanks for clarifying - that means I think I am estimating the wrong model.

My target model is to
a) in a first-stage predict X1 and X2 using Z1 and Z2
b) regress y on the predicted X1 and X2
c) test for a difference in second-stage coefficient on X1, holding the coefficient on X2 constant.

In a non-IV setting, I would just use an interaction, but what we discussed above doesn't seem to apply in a straight-forward way here. Is there a better way to code up this model.

Code:

reg wpi X1_pred X2_pred c.X1_pred#i.post1975 post1975

These are the correct coefficients, but doing it manually means that my inference is wrong. Is there a way to get the desired coeffients using ivreg2 where I can obtain the correct standard errors?
Comment
Andrew Musau

Join Date: Oct 2014

Posts: 10168
#4

16 Dec 2023, 02:11

This is a linear model. So the coefficients on X1 and X2 in the IV regression are marginal effects, i.e. they express how much the outcome is expected to change for a unit change in the predictor, holding the other predictor constant. Or what kind of test do you have in mind?
Comment

Andrew Musau

Join Date: Oct 2014
Posts: 10168

16 Dec 2023, 11:20

#3 was not clear, but rereading #1, I see what you want to do. Here is how you can specify the IV with interactions to test the difference in the estimated coefficient over the two sample periods. Using ## instead of # will give you the differences directly as coefficients without the need to use the test command, but this way is more intuitive.

Code:

use "http://www.stata-press.com/data/r13/wpi1.dta" , clear

*Create endogenous variables 
gen X1 = wpi*2 + runiform()*200
gen X2 = wpi*0.5 + runiform()*500

*Create exogenous variables
gen Z1 = X1 + runiform()*200
gen Z2 = X2 + runiform()*300

*Create date dummy
gen post1975 = t>tq(1975q1)

ivreg2 wpi (X1 X2 = Z1 Z2) if !post1975, robust
ivreg2 wpi (X1 X2 = Z1 Z2) if post1975, robust
gen cons=1
ivreg2 wpi (i.post1975#(c.X1 c.X2) = i.post1975#(c.Z1 c.Z2)) i.post1975#c.cons, nocons robust 
test 0.post1975#c.X1= 1.post1975#c.X1

Res.:

Code:

. 
. ivreg2 wpi (X1 X2 = Z1 Z2) if !post1975, robust

IV (2SLS) estimation
--------------------

Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity

                                                      Number of obs =       61
                                                      F(  2,    58) =     1.07
                                                      Prob > F      =   0.3495
Total (centered) SS     =  2457.769137                Centered R2   =   0.0442
Total (uncentered) SS   =  78222.36125                Uncentered R2 =   0.9700
Residual SS             =  2349.173792                Root MSE      =    6.206

------------------------------------------------------------------------------
             |               Robust
         wpi | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
          X1 |   .0218851   .0179232     1.22   0.222    -.0132437    .0570139
          X2 |   .0073444   .0072797     1.01   0.313    -.0069237    .0216124
       _cons |   29.40602   3.990646     7.37   0.000      21.5845    37.22755
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic):             20.840
                                                   Chi-sq(1) P-val =    0.0000
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               21.927
                         (Kleibergen-Paap rk Wald F statistic):         26.731
Stock-Yogo weak ID test critical values: 10% maximal IV size              7.03
                                         15% maximal IV size              4.58
                                         20% maximal IV size              3.95
                                         25% maximal IV size              3.63
Source: Stock-Yogo (2005).  Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments):         0.000
                                                 (equation exactly identified)
------------------------------------------------------------------------------
Instrumented:         X1 X2
Excluded instruments: Z1 Z2
------------------------------------------------------------------------------

. 
. ivreg2 wpi (X1 X2 = Z1 Z2) if post1975, robust

IV (2SLS) estimation
--------------------

Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity

                                                      Number of obs =       63
                                                      F(  2,    60) =    20.47
                                                      Prob > F      =   0.0000
Total (centered) SS     =  19040.39622                Centered R2   =   0.3260
Total (uncentered) SS   =  522916.7378                Uncentered R2 =   0.9755
Residual SS             =   12832.5333                Root MSE      =    14.27

------------------------------------------------------------------------------
             |               Robust
         wpi | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
          X1 |   .1427594   .0230702     6.19   0.000     .0975427    .1879761
          X2 |   .0206721   .0137092     1.51   0.132    -.0061975    .0475417
       _cons |   43.66184   7.916588     5.52   0.000     28.14562    59.17807
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic):             22.578
                                                   Chi-sq(1) P-val =    0.0000
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               47.073
                         (Kleibergen-Paap rk Wald F statistic):         88.684
Stock-Yogo weak ID test critical values: 10% maximal IV size              7.03
                                         15% maximal IV size              4.58
                                         20% maximal IV size              3.95
                                         25% maximal IV size              3.63
Source: Stock-Yogo (2005).  Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments):         0.000
                                                 (equation exactly identified)
------------------------------------------------------------------------------
Instrumented:         X1 X2
Excluded instruments: Z1 Z2
------------------------------------------------------------------------------

. 
. gen cons=1

. 
. ivreg2 wpi (i.post1975#(c.X1 c.X2) = i.post1975#(c.Z1 c.Z2)) i.post1975#c.cons, nocons robust 

IV (2SLS) estimation
--------------------

Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity

                                                      Number of obs =      124
                                                      F(  6,   118) =   766.94
                                                      Prob > F      =   0.0000
Total (centered) SS     =  112504.7755                Centered R2   =   0.8651
Total (uncentered) SS   =   601139.099                Uncentered R2 =   0.9747
Residual SS             =  15181.70709                Root MSE      =    11.06

---------------------------------------------------------------------------------
                |               Robust
            wpi | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
----------------+----------------------------------------------------------------
  post1975#c.X1 |
             0  |   .0218851   .0179232     1.22   0.222    -.0132437    .0570139
             1  |   .1427594   .0230702     6.19   0.000     .0975427    .1879761
                |
  post1975#c.X2 |
             0  |   .0073444   .0072797     1.01   0.313    -.0069237    .0216124
             1  |   .0206721   .0137092     1.51   0.132    -.0061975    .0475417
                |
post1975#c.cons |
             0  |   29.40602   3.990646     7.37   0.000      21.5845    37.22755
             1  |   43.66184   7.916588     5.52   0.000     28.14562    59.17807
---------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic):              0.000
                                                   Chi-sq(1) P-val =    1.0000
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               22.306
                         (Kleibergen-Paap rk Wald F statistic):          0.000
Stock-Yogo weak ID test critical values:                       <not available>
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments):         0.000
                                                 (equation exactly identified)
------------------------------------------------------------------------------
Instrumented:         0b.post1975#c.X1 1.post1975#c.X1 0b.post1975#c.X2
                      1.post1975#c.X2
Included instruments: 0b.post1975#c.cons 1.post1975#c.cons
Excluded instruments: 0b.post1975#c.Z1 1.post1975#c.Z1 0b.post1975#c.Z2
                      1.post1975#c.Z2
------------------------------------------------------------------------------

. 
. test 0.post1975#c.X1= 1.post1975#c.X1

 ( 1)  0b.post1975#c.X1 - 1.post1975#c.X1 = 0

           chi2(  1) =   17.12
         Prob > chi2 =    0.0000

.

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Interaction effects in ivreg2

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