Comparing Slopes

[Olivia Hamilton]

Hi there,

In a trend analysis, I am looking at whether hours worked changed across 15 years. Then I am seeing whether hours worked differs by 4 age groups. I already have coefficients (with p-values) for change in hours from baseline to each successive year, but I would like to see whether there is a significant differences between the slopes of each group (i.e. test whether the slopes for two [or more] independent populations are equal). Any ideas how to do this in Stata?

input float(m_yr m_yrqtr2) double m_hrsov_a float m_age_c
2007 20071   41 3
2007 20071    . 4
2007 20071    . 1
2007 20071    . 3
2007 20071    . 2
2007 20071    . 4
2007 20071    . 3
2007 20071    . 3
2007 20071    . 1
2007 20071    . 3
2007 20071   46 2
2007 20071    . 2
2007 20071    . 1
2007 20071    . 4
2007 20071    . 3
2007 20071    . 2
2007 20071   60 3
2007 20071    . 3
2007 20071    . 4
2007 20071    . 3
2007 20071    . 2
2007 20071    . 2
2007 20071    . 1
2007 20071    . 1
2007 20071    . 2
2007 20071    . 1
2007 20071    . 4
2007 20071   40 2
2007 20071    . 3
2007 20071    . 3
2007 20071    . 3
2007 20071    . 1
2007 20071    . 3
2007 20071    . 3
2007 20071    . 4
2007 20071    . 4
2007 20071 40.5 2
2007 20071    . 2
2007 20071    . 1
2007 20071    . 4
2007 20071    . 2
2007 20071    . 4
2007 20071    . 1
2007 20071    . 2
2007 20071   48 3
2007 20071   55 3
2007 20071    . 2
2007 20071    . 4
2007 20071 25.5 4
2007 20071    . 3
2007 20071    . 2
2007 20071    . 3
2007 20071    . 3
2007 20071    . 2
2007 20071    . 2
2007 20071   56 2
2007 20071    . 2
2007 20071    . 2
2007 20071    . 3
2007 20071   36 4
2007 20071    . 4
2007 20071   50 3
2007 20071    . 2
2007 20071   46 3
2007 20071    . 4
2007 20071    . 4
2007 20071    . 2
2007 20071    . 1
2007 20071    . 4
2007 20071   28 3
2007 20071    . 2
2007 20071    . 4
2007 20071    . 2
2007 20071    . 2
2007 20071    . 2
2007 20071    . 2
2007 20071    . 2
2007 20071    . 1
2007 20071    . 4
2007 20071   40 2
2007 20071    . 2
2007 20071    . 4
2007 20071    . 1
2007 20071    . 3
2007 20071    . 3
2007 20071    . 3
2007 20071    . 3
2007 20071   27 3
2007 20071    . 2
2007 20071    . 1
2007 20071    . 1
2007 20071    . 2
2007 20071   50 2
2007 20071   45 3
2007 20071   40 3
2007 20071   46 4
2007 20071    . 3
2007 20071    . 2
2007 20071    . 2
2007 20071    . 4
end
label values m_yr m_yrla
label values m_hrsov_a m_hrsov_ala
label values m_age_c m_age_cla
label def m_age_cla 1 "Gen. Z", modify
label def m_age_cla 2 "Millennials", modify
label def m_age_cla 3 "Gen. X", modify
label def m_age_cla 4 "Baby Boomers", modify

Thanks in advance for your help!

[Carlo Lazzaro]

Olivia:
what is your regression code?
What is your regessand?
Please clarify. Thanks.

[Olivia Hamilton]

Hi Carlo,

Thanks for responding. It is as follows:

{
    preserve
    
    drop if m_ftemploy_bi == 0
    drop if m_hrsov_a <35

foreach cat of numlist 1/4 {

eststo: areg m_hrsov_a i.m_yr m_wgt m_age /*m_age_sq*/ i.m_sex_bi i.m_mar_c i.m_child_c i.m_cued_bi i.m_ethnic_bi if m_age_c == `cat', absorb (m_region_c) cluster (m_yrqtr2)

esttab using output_age.rtf, replace nogap b(%9.3f) se(%9.3f) parentheses star(* 0.10 ** 0.05 *** 0.01) drop(*cons*)

    }

    restore
    }

StataMP 17

[Carlo Lazzaro]

Olivia:
an idea that springs to my mind is to take a look at -test- and -lincom-.

[Olivia Hamilton]

Hi there, thanks for coming back to me.

I had a little read about those functions, but I wondered how would one use test/lincom when trying to compare differences in the outcome (hours) between different categories in a single variable (age = Gen Z [1]; Millennials [2]; Gen X [3]; Baby Boomers [4])?

Also using lincom, do I have to run the regression and then immediately run lincom? Are the resulting coefficients somehow stored on the backend? #confused

[Andrew Musau]

If I understand your question correctly, you want to compare the predicted outcome across levels of a grouping variable in a model with fixed effects. One approach is to estimate an interacted model. I recommend that you install reghdfe from SSC which allows you to absorb multiple fixed effects variables. You need this as you need to interact the fixed effects variables as well. Here is a simplified version of what you want to do.

sysuse auto, clear
*WHAT YOU ARE DOING NOW
*ssc install reghdfe, clear
 reghdfe mpg weight displacement if foreign==0, absorb(rep78) nocons
 reghdfe mpg weight displacement if foreign==1, absorb(rep78) nocons

*WHAT I SUGGEST
**1. INTERACT EACH CATEGORY OF VARIABLE WITH FIXED EFFECTS
gen rep78xdomestic= rep78#0.foreign
gen rep78xforeign= rep78#1.foreign
**2. ESTIMATE INTERACTD MODEL WITH ROBUST SEs
gen long obsno=_n
reghdfe mpg i.foreign#(c.weight c.displacement), absorb(rep78xdomestic rep78xforeign) nocons cluster(obsno)
**3. GENERATE PREDICTIONS WITH MARGINS AND COMPARE WITH TEST
margins foreign, coefl post
test _b[0bn.foreign]=_b[1.foreign]

Res.:

.  reghdfe mpg weight displacement if foreign==0, absorb(rep78) nocons
(MWFE estimator converged in 1 iterations)

HDFE Linear regression                            Number of obs   =         48
Absorbing 1 HDFE group                            F(   2,     41) =      47.71
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.7942
                                                  Adj R-squared   =     0.7641
                                                  Within R-sq.    =     0.6994
                                                  Root MSE        =     2.3087

------------------------------------------------------------------------------
         mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |  -.0049828   .0011334    -4.40   0.000    -.0072717   -.0026938
displacement |  -.0032801   .0088741    -0.37   0.714    -.0212018    .0146416
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
       rep78 |         5           0           5     |
-----------------------------------------------------+

. 
.  reghdfe mpg weight displacement if foreign==1, absorb(rep78) nocons
(MWFE estimator converged in 1 iterations)

HDFE Linear regression                            Number of obs   =         21
Absorbing 1 HDFE group                            F(   2,     16) =      15.85
                                                  Prob > F        =     0.0002
                                                  R-squared       =     0.6742
                                                  Adj R-squared   =     0.5927
                                                  Within R-sq.    =     0.6646
                                                  Root MSE        =     4.0269

------------------------------------------------------------------------------
         mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |   .0014993   .0077531     0.19   0.849    -.0149366    .0179352
displacement |  -.2597622   .1222911    -2.12   0.050    -.5190078   -.0005166
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
       rep78 |         3           0           3     |
-----------------------------------------------------+

. 
. 
. 
. *WHAT I SUGGEST
. **2. ESTIMATE INTERACTD MODEL WITH ROBUST SEs


. reghdfe mpg i.foreign#(c.weight c.displacement), absorb(rep78xdomestic rep78xforeign) nocons cluster(obsno)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =         69
Absorbing 2 HDFE groups                           F(   4,     57) =      31.07
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.7957
                                                  Adj R-squared   =     0.7563
                                                  Within R-sq.    =     0.6815
Number of clusters (obsno)   =         69         Root MSE        =     2.8958

                                           (Std. Err. adjusted for 69 clusters in obsno)
----------------------------------------------------------------------------------------
                       |               Robust
                   mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-----------------------+----------------------------------------------------------------
      foreign#c.weight |
             Domestic  |  -.0049828   .0014798    -3.37   0.001    -.0079461   -.0020194
              Foreign  |   .0014993   .0073328     0.20   0.839    -.0131844     .016183
                       |
foreign#c.displacement |
             Domestic  |  -.0032801   .0124683    -0.26   0.793    -.0282475    .0216872
              Foreign  |  -.2597622   .1153259    -2.25   0.028    -.4906984   -.0288261
----------------------------------------------------------------------------------------

Absorbed degrees of freedom:
--------------------------------------------------------+
    Absorbed FE | Categories  - Redundant  = Num. Coefs |
----------------+---------------------------------------|
 rep78xdomestic |         6           0           6     |
  rep78xforeign |         4           2           2     |
--------------------------------------------------------+

. 
. **3. GENERATE PREDICTIONS WITH MARGINS AND COMPARE WITH TEST

. 
. margins foreign, coefl post

Predictive margins                              Number of obs     =         69
Model VCE    : Robust

Expression   : Linear prediction, predict()

------------------------------------------------------------------------------
             |     Margin  Legend
-------------+----------------------------------------------------------------
     foreign |
   Domestic  |  -15.75733  _b[0bn.foreign]
    Foreign  |  -46.88705  _b[1.foreign]
------------------------------------------------------------------------------

. 
. test _b[0bn.foreign]=_b[1.foreign]

 ( 1)  0bn.foreign - 1.foreign = 0

           chi2(  1) =   11.30
         Prob > chi2 =    0.0008

.