Comparing the difference in coefficients of the same variables in different regression models

[Xiang Ron]

Hello!
I would like to ask about the stata command for comparing the difference in coefficients of the same variables in different regression models.
What I want to do is different from the grouping test performed for categorical variables. Here is the example :
Regression 1: reghdfe y x controls if M==1 ,absorb(id year) vce(cluster id)
Regression 2: reghdfe y x controls if N==1 ,absorb(id year) vce(cluster id)
It should be noted that some of the observations are in both M and N sample; id is the firm fixed effect

The question is: now I need to compare whether the coefficient of X in regression 1 is significantly larger than the coefficient of X in regression 2 (need to find out the two-tailed p-value), how do I write the code in Stata?
Looking forward to your reply, many thanks!

[Andrew Musau]

You can duplicate part of the data and estimate one model with interactions (in this example, the samples don’t overlap, but the idea is similar: https://www.statalist.org/forums/for...erent-outcomes). Alternatively, you can estimate with regress and then use suest followed by test if the number of panels and years is not too large. Note that reghdfe is from https://github.com/sergiocorreia/reghdfe (FAQ Advice #12).

help suest
help test

[George Ford]

Is there some reason to estimate the models separately?

As Andrew suggests, you can estimate with the full sample and have a group interaction on X.

[Xiang Ron]

Thank you for your replies!

In fact my sample is divided into three groups, A, B, and C, where AB is the M=1 group and BC is the N=1 group.

I was discussing my question with other team members today and they thought that the problem could be solved by appending the samples from the M=1 and N=1 groups to a new sample and then using “bdiff” command. Or I can just apply the following calculation to find the z value and then calculate the corresponding t value.

reghdfe y x $z if M==1, absorb(year indid firm) vce(cluster firm)
est store M

reghdfe y x $z if N==1, absorb(year indid firm) vce(cluster firm)
est store N

**计算Z统计量
Z = （β1 - β2）/（SE1^2+SE2^2）^1/2

[Xiang Ron]

You can duplicate part of the data and estimate one model with interactions (in this example, the samples don’t overlap, but the idea is similar: https://www.statalist.org/forums/for...erent-outcomes). Alternatively, you can estimate with regress and then use suest followed by test if the number of panels and years is not too large. Note that reghdfe is from https://github.com/sergiocorreia/reghdfe (FAQ Advice #12).

help suest
help test

Thank you for your reply!

[Xiang Ron]

Is there some reason to estimate the models separately?

As Andrew suggests, you can estimate with the full sample and have a group interaction on X.

Thank you for your reply!

[George Ford]

If M is AB and N is BC, then the two samples are not independent (overlap in B). The error terms between the two regressions/groups would be correlated. Would it make more sense to get coefficients for A, B, and C separately, or just analyze A and C?

I suppose you could estimate for 3 groups, and then lincom a test of weighted average coefficients. ???

This is a tricky one, I think.

[Andrew Musau]

Using suest or the interactions approach is valid for overlapping data. From

help suest

suest combines the estimation results -- parameter estimates and associated (co)variance matrices -- stored under namelist into one parameter vector and simultaneous (co)variance matrix of the sandwich/robust type. This (co)variance matrix is appropriate even if the estimates were obtained on the same or on overlapping data.

[George Ford]

Good to know. Thanks Andrew.

[Xiang Ron]

Using suest or the interactions approach is valid for overlapping data. From

help suest

Dear Andrew,

Thanks for your reply, I ended up using Federico Belotti. the updated suest.ado document for between-group coefficient tests.

However I would like to continue to talk to you about ways to utilize interaction items. The following new setup is what I need to do for my thesis, but when I was doing the regression I realized that the coefficients of my interaction term were omitted.

Now I have three variables: the dependent variable Y(dummy variable), the independent variable X(dummy variable), and the moderating variable Z (dummy variable). Where Z is a dummy variable labeled as a high-low group in the sample X=1 based on whether a continuous variable V is higher than the median of V. It is only possible for Z to equal 1 when X=1, where A high proportion of V is defined as Z=1 (labeled group A), a low proportion of V is defined as Z=0 (labeled group B), and a sample size of X=0 also defines Z as 0 (labeled group C).

My grouping standard is to group according to Z and X, the first group is group A and group C; The second group is group B and group C. I started by comparing the coefficient size of the AC group independent variable X with that of the BC group independent variable X (now tested with suest). However, after I tried your suggestion to introduce the interaction term coefficient for regression, I found that the regression coefficient of X*Z was omitted (My regression command is: reghdfe y c.X##c.Z, absorb(year firm) vce(cluster firm), which made me very confused.

Looking forward to your reply.

Best wishes
Xiang Ron

[Xiang Ron]

The new suest can be downloaded here:
https://gitee.com/arlionn/suest_panel

[Andrew Musau]

As I implied with

You can duplicate part of the data

some data manipulation is required before estimating the model with interactions. Furthermore, you need to interact the fixed effects as well. I will see if I can create a reproducible example later on.

[George Ford]

X/Z are collinear except for group B.

[Andrew Musau]

webuse grunfeld, clear
gen A= inrange(company, 1, 7)
gen B= inrange(company, 4, 10)


*USING REGRESS AND SUEST
regress invest mvalue kstock ib4.company i.year if A
est sto A
regress invest mvalue kstock ib4.company i.year if B
est sto B
suest A B
test _b[A_mean:mvalue]= _b[B_mean:mvalue]

*MODEL WITH INTERACTIONS
expand 2 if A & B, gen(new)
drop A B
gen A= inrange(company, 1, 7) &!new
gen B=!A
gen obsno=_n
reghdfe invest c.A#(c.mvalue c.kstock) c.B#(c.mvalue c.kstock), a(i.company#c.A i.company#c.B i.year#c.A i.year#c.B) cluster(obsno)
test _b[c.A#c.mvalue]= _b[c.B#c.mvalue]

Res.:

. suest A B

Simultaneous results for A, B                              Number of obs = 200

------------------------------------------------------------------------------
             |               Robust
             | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
A_mean       |
      mvalue |   .1220006   .0180759     6.75   0.000     .0865726    .1574286
      kstock |   .3615751   .0547449     6.60   0.000     .2542771    .4688732
             |
     company |
          1  |  -112.8819   63.55125    -1.78   0.076    -237.4401    11.67625
          2  |   105.5867   33.75318     3.13   0.002     39.43164    171.7417
          3  |   -236.953   29.25474    -8.10   0.000    -294.2912   -179.6148
          5  |  -100.1514   24.77684    -4.04   0.000    -148.7131   -51.58966
          6  |   8.768066   8.194366     1.07   0.285    -7.292596    24.82873
          7  |  -42.26754   16.19031    -2.61   0.009    -73.99996   -10.53511
             |
        year |
       1936  |  -24.53739   25.51518    -0.96   0.336    -74.54621    25.47144
       1937  |  -54.25578   33.44403    -1.62   0.105    -119.8049    11.29332
       1938  |  -49.67567   29.06608    -1.71   0.087    -106.6441    7.292786
       1939  |  -92.56549   36.64014    -2.53   0.012    -164.3788   -20.75213
       1940  |  -56.19589   25.70215    -2.19   0.029    -106.5712   -5.820595
       1941  |  -24.36808   24.91802    -0.98   0.328     -73.2065    24.47035
       1942  |  -24.64611   26.43801    -0.93   0.351    -76.46366    27.17145
       1943  |  -50.17774   26.51863    -1.89   0.058    -102.1533    1.797819
       1944  |   -53.1491    30.8346    -1.72   0.085    -113.5838    7.285605
       1945  |  -66.60069   28.90015    -2.30   0.021    -123.2439   -9.957428
       1946  |  -36.14283   29.71565    -1.22   0.224    -94.38443    22.09878
       1947  |  -47.96899   29.03413    -1.65   0.099    -104.8749    8.936865
       1948  |  -47.63102   34.78369    -1.37   0.171    -115.8058    20.54376
       1949  |  -87.04609   33.02942    -2.64   0.008    -151.7826   -22.30962
       1950  |  -91.49044   33.97252    -2.69   0.007    -158.0754   -24.90551
       1951  |  -75.69916   42.68577    -1.77   0.076    -159.3617    7.963414
       1952  |  -76.70162   48.45574    -1.58   0.113    -171.6731    18.26988
       1953  |  -76.57738   50.74942    -1.51   0.131    -176.0444    22.88966
       1954  |  -105.0774   38.05845    -2.76   0.006    -179.6706   -30.48424
             |
       _cons |   14.73764   22.94488     0.64   0.521     -30.2335    59.70878
-------------+----------------------------------------------------------------
A_lnvar      |
       _cons |   8.249905   .1334912    61.80   0.000     7.988267    8.511543
-------------+----------------------------------------------------------------
B_mean       |
      mvalue |   .0681591   .0146953     4.64   0.000     .0393569    .0969614
      kstock |   .0613181   .0250914     2.44   0.015     .0121398    .1104964
             |
     company |
          5  |  -15.26219   15.26227    -1.00   0.317     -45.1757    14.65131
          6  |  -11.04158   7.476444    -1.48   0.140    -25.69514    3.611978
          7  |  -13.36628   12.53193    -1.07   0.286    -37.92841    11.19586
          8  |  -39.52882   5.614395    -7.04   0.000    -50.53283   -28.52481
          9  |  -30.55935   10.49431    -2.91   0.004    -51.12781   -9.990884
         10  |  -33.55412   10.42856    -3.22   0.001    -53.99372   -13.11452
             |
        year |
       1936  |  -.6742191   4.874083    -0.14   0.890    -10.22725    8.878808
       1937  |   1.241771   7.389821     0.17   0.867    -13.24201    15.72555
       1938  |   .0271155   4.413525     0.01   0.995    -8.623235    8.677466
       1939  |  -7.241651   4.826683    -1.50   0.134    -16.70177    2.218473
       1940  |  -3.075184   4.288366    -0.72   0.473    -11.48023    5.329858
       1941  |   7.675197   5.099086     1.51   0.132    -2.318829    17.66922
       1942  |   1.307218   5.306966     0.25   0.805    -9.094245    11.70868
       1943  |   -1.01689   6.408633    -0.16   0.874    -13.57758     11.5438
       1944  |   4.878557   7.658816     0.64   0.524    -10.13245    19.88956
       1945  |    2.86283   5.253673     0.54   0.586     -7.43418    13.15984
       1946  |   3.214002   6.424942     0.50   0.617    -9.378653    15.80666
       1947  |   6.025572   6.667182     0.90   0.366    -7.041864    19.09301
       1948  |   8.060272   5.742374     1.40   0.160    -3.194575    19.31512
       1949  |   2.840691   6.261125     0.45   0.650    -9.430889    15.11227
       1950  |   1.734127   7.173763     0.24   0.809    -12.32619    15.79444
       1951  |   19.82854   10.13134     1.96   0.050    -.0285263     39.6856
       1952  |   19.88965   9.098768     2.19   0.029      2.05639     37.7229
       1953  |   22.96341   9.951782     2.31   0.021     3.458276    42.46854
       1954  |   19.11981   12.32276     1.55   0.121    -5.032355    43.27197
             |
       _cons |   25.95734   10.80931     2.40   0.016     4.771477    47.14321
-------------+----------------------------------------------------------------
B_lnvar      |
       _cons |   5.330049   .1525765    34.93   0.000     5.031004    5.629093
------------------------------------------------------------------------------

. 
. test _b[A_mean:mvalue]= _b[B_mean:mvalue]

 ( 1)  [A_mean]mvalue - [B_mean]mvalue = 0

           chi2(  1) =    5.64
         Prob > chi2 =    0.0175

. 
. 
. 
. *MODEL WITH INTERACTIONS

. 
. expand 2 if A & B, gen(new)
(80 observations created)

. 
. drop A B

. 
. gen A= inrange(company, 1, 7) &!new

. 
. gen B=!A

. 
. gen obsno=_n

. 
. reghdfe invest c.A#(c.mvalue c.kstock) c.B#(c.mvalue c.kstock), a(i.company#c.A i.company#c.B i.year#c.A i.year#c.B) cluster(obsno)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =        280
Absorbing 4 HDFE groups                           F(   4,    222) =      24.16
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.9678
                                                  Adj R-squared   =     0.9594
                                                  Within R-sq.    =     0.6897
Number of clusters (obsno)   =        280         Root MSE        =    45.1112

                                (Std. err. adjusted for 280 clusters in obsno)
------------------------------------------------------------------------------
             |               Robust
      invest | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
c.A#c.mvalue |   .1220006   .0202494     6.02   0.000     .0820949    .1619063
             |
c.A#c.kstock |   .3615751   .0613279     5.90   0.000     .2407158    .4824344
             |
c.B#c.mvalue |   .0681591   .0164624     4.14   0.000     .0357166    .1006016
             |
c.B#c.kstock |   .0613181   .0281086     2.18   0.030     .0059243     .116712
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
 company#c.A |        10           3           7    ?|
 company#c.B |        10           3           7    ?|
    year#c.A |        20           0          20    ?|
    year#c.B |        20           0          20    ?|
-----------------------------------------------------+
? = number of redundant parameters may be higher

. 
. test _b[c.A#c.mvalue]= _b[c.B#c.mvalue]

 ( 1)  c.A#c.mvalue - c.B#c.mvalue = 0

       F(  1,   222) =    4.26
            Prob > F =    0.0403

.

[Xiang Ron]

webuse grunfeld, clear
gen A= inrange(company, 1, 7)
gen B= inrange(company, 4, 10)


*USING REGRESS AND SUEST
regress invest mvalue kstock ib4.company i.year if A
est sto A
regress invest mvalue kstock ib4.company i.year if B
est sto B
suest A B
test _b[A_mean:mvalue]= _b[B_mean:mvalue]

*MODEL WITH INTERACTIONS
expand 2 if A & B, gen(new)
drop A B
gen A= inrange(company, 1, 7) &!new
gen B=!A
gen obsno=_n
reghdfe invest c.A#(c.mvalue c.kstock) c.B#(c.mvalue c.kstock), a(i.company#c.A i.company#c.B i.year#c.A i.year#c.B) cluster(obsno)
test _b[c.A#c.mvalue]= _b[c.B#c.mvalue]

Res.:

. suest A B

Simultaneous results for A, B Number of obs = 200

------------------------------------------------------------------------------
| Robust
| Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
A_mean |
mvalue | .1220006 .0180759 6.75 0.000 .0865726 .1574286
kstock | .3615751 .0547449 6.60 0.000 .2542771 .4688732
|
company |
1 | -112.8819 63.55125 -1.78 0.076 -237.4401 11.67625
2 | 105.5867 33.75318 3.13 0.002 39.43164 171.7417
3 | -236.953 29.25474 -8.10 0.000 -294.2912 -179.6148
5 | -100.1514 24.77684 -4.04 0.000 -148.7131 -51.58966
6 | 8.768066 8.194366 1.07 0.285 -7.292596 24.82873
7 | -42.26754 16.19031 -2.61 0.009 -73.99996 -10.53511
|
year |
1936 | -24.53739 25.51518 -0.96 0.336 -74.54621 25.47144
1937 | -54.25578 33.44403 -1.62 0.105 -119.8049 11.29332
1938 | -49.67567 29.06608 -1.71 0.087 -106.6441 7.292786
1939 | -92.56549 36.64014 -2.53 0.012 -164.3788 -20.75213
1940 | -56.19589 25.70215 -2.19 0.029 -106.5712 -5.820595
1941 | -24.36808 24.91802 -0.98 0.328 -73.2065 24.47035
1942 | -24.64611 26.43801 -0.93 0.351 -76.46366 27.17145
1943 | -50.17774 26.51863 -1.89 0.058 -102.1533 1.797819
1944 | -53.1491 30.8346 -1.72 0.085 -113.5838 7.285605
1945 | -66.60069 28.90015 -2.30 0.021 -123.2439 -9.957428
1946 | -36.14283 29.71565 -1.22 0.224 -94.38443 22.09878
1947 | -47.96899 29.03413 -1.65 0.099 -104.8749 8.936865
1948 | -47.63102 34.78369 -1.37 0.171 -115.8058 20.54376
1949 | -87.04609 33.02942 -2.64 0.008 -151.7826 -22.30962
1950 | -91.49044 33.97252 -2.69 0.007 -158.0754 -24.90551
1951 | -75.69916 42.68577 -1.77 0.076 -159.3617 7.963414
1952 | -76.70162 48.45574 -1.58 0.113 -171.6731 18.26988
1953 | -76.57738 50.74942 -1.51 0.131 -176.0444 22.88966
1954 | -105.0774 38.05845 -2.76 0.006 -179.6706 -30.48424
|
_cons | 14.73764 22.94488 0.64 0.521 -30.2335 59.70878
-------------+----------------------------------------------------------------
A_lnvar |
_cons | 8.249905 .1334912 61.80 0.000 7.988267 8.511543
-------------+----------------------------------------------------------------
B_mean |
mvalue | .0681591 .0146953 4.64 0.000 .0393569 .0969614
kstock | .0613181 .0250914 2.44 0.015 .0121398 .1104964
|
company |
5 | -15.26219 15.26227 -1.00 0.317 -45.1757 14.65131
6 | -11.04158 7.476444 -1.48 0.140 -25.69514 3.611978
7 | -13.36628 12.53193 -1.07 0.286 -37.92841 11.19586
8 | -39.52882 5.614395 -7.04 0.000 -50.53283 -28.52481
9 | -30.55935 10.49431 -2.91 0.004 -51.12781 -9.990884
10 | -33.55412 10.42856 -3.22 0.001 -53.99372 -13.11452
|
year |
1936 | -.6742191 4.874083 -0.14 0.890 -10.22725 8.878808
1937 | 1.241771 7.389821 0.17 0.867 -13.24201 15.72555
1938 | .0271155 4.413525 0.01 0.995 -8.623235 8.677466
1939 | -7.241651 4.826683 -1.50 0.134 -16.70177 2.218473
1940 | -3.075184 4.288366 -0.72 0.473 -11.48023 5.329858
1941 | 7.675197 5.099086 1.51 0.132 -2.318829 17.66922
1942 | 1.307218 5.306966 0.25 0.805 -9.094245 11.70868
1943 | -1.01689 6.408633 -0.16 0.874 -13.57758 11.5438
1944 | 4.878557 7.658816 0.64 0.524 -10.13245 19.88956
1945 | 2.86283 5.253673 0.54 0.586 -7.43418 13.15984
1946 | 3.214002 6.424942 0.50 0.617 -9.378653 15.80666
1947 | 6.025572 6.667182 0.90 0.366 -7.041864 19.09301
1948 | 8.060272 5.742374 1.40 0.160 -3.194575 19.31512
1949 | 2.840691 6.261125 0.45 0.650 -9.430889 15.11227
1950 | 1.734127 7.173763 0.24 0.809 -12.32619 15.79444
1951 | 19.82854 10.13134 1.96 0.050 -.0285263 39.6856
1952 | 19.88965 9.098768 2.19 0.029 2.05639 37.7229
1953 | 22.96341 9.951782 2.31 0.021 3.458276 42.46854
1954 | 19.11981 12.32276 1.55 0.121 -5.032355 43.27197
|
_cons | 25.95734 10.80931 2.40 0.016 4.771477 47.14321
-------------+----------------------------------------------------------------
B_lnvar |
_cons | 5.330049 .1525765 34.93 0.000 5.031004 5.629093
------------------------------------------------------------------------------

.
. test _b[A_mean:mvalue]= _b[B_mean:mvalue]

( 1) [A_mean]mvalue - [B_mean]mvalue = 0

chi2( 1) = 5.64
Prob > chi2 = 0.0175

.
.
.
. *MODEL WITH INTERACTIONS

.
. expand 2 if A & B, gen(new)
(80 observations created)

.
. drop A B

.
. gen A= inrange(company, 1, 7) &!new

.
. gen B=!A

.
. gen obsno=_n

.
. reghdfe invest c.A#(c.mvalue c.kstock) c.B#(c.mvalue c.kstock), a(i.company#c.A i.company#c.B i.year#c.A i.year#c.B) cluster(obsno)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression Number of obs = 280
Absorbing 4 HDFE groups F( 4, 222) = 24.16
Statistics robust to heteroskedasticity Prob > F = 0.0000
R-squared = 0.9678
Adj R-squared = 0.9594
Within R-sq. = 0.6897
Number of clusters (obsno) = 280 Root MSE = 45.1112

(Std. err. adjusted for 280 clusters in obsno)
------------------------------------------------------------------------------
| Robust
invest | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.A#c.mvalue | .1220006 .0202494 6.02 0.000 .0820949 .1619063
|
c.A#c.kstock | .3615751 .0613279 5.90 0.000 .2407158 .4824344
|
c.B#c.mvalue | .0681591 .0164624 4.14 0.000 .0357166 .1006016
|
c.B#c.kstock | .0613181 .0281086 2.18 0.030 .0059243 .116712
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
company#c.A | 10 3 7 ?|
company#c.B | 10 3 7 ?|
year#c.A | 20 0 20 ?|
year#c.B | 20 0 20 ?|
-----------------------------------------------------+
? = number of redundant parameters may be higher

.
. test _b[c.A#c.mvalue]= _b[c.B#c.mvalue]

( 1) c.A#c.mvalue - c.B#c.mvalue = 0

F( 1, 222) = 4.26
Prob > F = 0.0403

.

Thank you so much.