Comparing coefficients of interaction term across subgroups (reghdfe)

[Jongeun Park]

Hi, is there any way to compare the coefficients of interaction term across different subgroups using reghdfe?

Suppose I'm interested in the effects of union in the city on wage (ln_wage), which makes me interact "union" with "city" variable. In addition, I'd like to do subgroup analysis by marriage (msp) and test whether the coefficients of interaction term (union#c_city) in two groups, which are -0.0196 and -0.0298 in the example code below, are significantly different to each other. How can I compare these two interaction coefficients?

webuse nlswork, clear

reghdfe ln_wage union##c_city if msp==1, absorb(idcode year)

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
     1.union |   .1045442    .011149     9.38   0.000     .0826896    .1263987
    1.c_city |   .0331573   .0132385     2.50   0.012     .0072068    .0591078
             |
union#c_city |
        1 1  |  -.0196805   .0190602    -1.03   0.302    -.0570431     .017682
             |
       _cons |    1.72928   .0048373   357.49   0.000     1.719798    1.738763
------------------------------------------------------------------------------


reghdfe ln_wage union##c_city if msp==0, absorb(idcode year)
------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
     1.union |   .1249048    .015938     7.84   0.000     .0936593    .1561504
    1.c_city |   .0331801   .0160409     2.07   0.039     .0017327    .0646274
             |
union#c_city |
        1 1  |  -.0298477   .0213081    -1.40   0.161     -.071621    .0119256
             |
       _cons |    1.72628   .0083392   207.01   0.000     1.709931    1.742628
------------------------------------------------------------------------------

[Andrew Musau]

reghdfe is from https://github.com/sergiocorreia/reghdfe (FAQ Advice #12).

webuse nlswork, clear
reghdfe ln_wage union##c_city if msp==1, absorb(idcode year)
reghdfe ln_wage union##c_city if !msp, absorb(idcode year)

*JOINT ESTIMATION
reghdfe ln_wage i.msp#(c.union##c.c_city), absorb(i.msp#idcode i.msp#year) cluster(idcode) nocons
*WALD TEST
test _b[0b.msp#c.union#c.c_city] = _b[1.msp#c.union#c.c_city]

Res.:

. reghdfe ln_wage union##c_city if msp==1, absorb(idcode year)
(dropped 686 singleton observations)
(MWFE estimator converged in 8 iterations)

HDFE Linear regression                            Number of obs   =     11,225
Absorbing 2 HDFE groups                           F(   3,   8758) =      38.06
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.7484
                                                  Adj R-squared   =     0.6776
                                                  Within R-sq.    =     0.0129
                                                  Root MSE        =     0.2597

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
     1.union |   .1045442    .011149     9.38   0.000     .0826896    .1263987
    1.c_city |   .0331573   .0132385     2.50   0.012     .0072068    .0591078
             |
union#c_city |
        1 1  |  -.0196805   .0190602    -1.03   0.302    -.0570431     .017682
             |
       _cons |    1.72928   .0048373   357.49   0.000     1.719798    1.738763
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
      idcode |      2453           0        2453     |
        year |        12           1          11     |
-----------------------------------------------------+

. 
. reghdfe ln_wage union##c_city if !msp, absorb(idcode year)
(dropped 764 singleton observations)
(MWFE estimator converged in 9 iterations)

HDFE Linear regression                            Number of obs   =      6,549
Absorbing 2 HDFE groups                           F(   3,   4932) =      31.94
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.8017
                                                  Adj R-squared   =     0.7367
                                                  Within R-sq.    =     0.0191
                                                  Root MSE        =     0.2444

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
     1.union |   .1249048    .015938     7.84   0.000     .0936593    .1561504
    1.c_city |   .0331801   .0160409     2.07   0.039     .0017327    .0646274
             |
union#c_city |
        1 1  |  -.0298477   .0213081    -1.40   0.161     -.071621    .0119256
             |
       _cons |    1.72628   .0083392   207.01   0.000     1.709931    1.742628
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
      idcode |      1603           0        1603     |
        year |        12           1          11     |
-----------------------------------------------------+

. 
. 
. 
. *JOINT ESTIMATION

. 
. reghdfe ln_wage i.msp#(c.union##c.c_city), absorb(i.msp#idcode i.msp#year) cluster(idcode) nocons
(dropped 1450 singleton observations)
(MWFE estimator converged in 10 iterations)

HDFE Linear regression                            Number of obs   =     17,774
Absorbing 2 HDFE groups                           F(   6,   3392) =      18.68
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.7691
                                                  Adj R-squared   =     0.7002
                                                  Within R-sq.    =     0.0149
Number of clusters (idcode)  =      3,393         Root MSE        =     0.2543

                                     (Std. err. adjusted for 3,393 clusters in idcode)
--------------------------------------------------------------------------------------
                     |               Robust
             ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
---------------------+----------------------------------------------------------------
         msp#c.union |
                  0  |   .1249048   .0203871     6.13   0.000     .0849326    .1648771
                  1  |   .1045442   .0149483     6.99   0.000     .0752356    .1338528
                     |
        msp#c.c_city |
                  0  |   .0331801   .0231686     1.43   0.152    -.0122458     .078606
                  1  |   .0331573   .0186464     1.78   0.075    -.0034021    .0697167
                     |
msp#c.union#c.c_city |
                  0  |  -.0298477   .0257005    -1.16   0.246    -.0802378    .0205424
                  1  |  -.0196805   .0228557    -0.86   0.389    -.0644928    .0251317
--------------------------------------------------------------------------------------

Absorbed degrees of freedom:
------------------------------------------------------+
  Absorbed FE | Categories  - Redundant  = Num. Coefs |
--------------+---------------------------------------|
   msp#idcode |      4056        4056           0    *|
     msp#year |        24           1          23     |
------------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

. 
. *WALD TEST

. 
. test _b[0b.msp#c.union#c.c_city] = _b[1.msp#c.union#c.c_city]

 ( 1)  0b.msp#c.union#c.c_city - 1.msp#c.union#c.c_city = 0

       F(  1,  3392) =    0.09
            Prob > F =    0.7648

[Jongeun Park]

Awesome! Thank you so much, Andrew Musau. It totally works.