Compare coefficients of lagged independent variables across two regression models

[Carla Jansen]

Hello everyone,

As part of my master's thesis, I am measuring the effect of the total number of AI professors on AI patent grants within US counties. I am using a random effects panel regression with years from 2000 to 2020, standard errors clustered at the county level.

As part of the thesis, I hypothesise that the effect of professors on patents becomes stronger after several years.
Therefore, I want to test the effect of the independent variable (professors) with different time lags, i.e. 0, 3, 5, 7 and 9 years, and see when it is the strongest.
Since the independent variable has the same metric in each model (=number of professors), is it possible to simply compare the effects across models? I have already tried the regression with the original and standardised independent variables, which gave the same results that the effect is strongest after 7 years.

I am sorry if this question might sound silly to you, but I have not been able to find literature that has helped me with this exact question regarding time lags.

Thank you a lot in advance!
Kind regards
Carla

[Carlo Lazzaro]

Carla:
welcome to this forum.
I reply to your question via tyhe following toy-example:

. use "https://www.stata-press.com/data/r18/nlswork.dta"
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)

. xtreg ln_wage i.year L2.grade L3.grade, re vce(cluster idcode)
note: L3.grade omitted because of collinearity.

Random-effects GLS regression                   Number of obs     =      5,798
Group variable: idcode                          Number of groups  =      2,583

R-squared:                                      Obs per group:
     Within  = 0.1110                                         min =          1
     Between = 0.2502                                         avg =        2.2
     Overall = 0.1922                                         max =          6

                                                Wald chi2(6)      =    1102.92
corr(u_i, X) = 0 (assumed)                      Prob > chi2       =     0.0000

                             (Std. err. adjusted for 2,583 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
        year |
         72  |  -.0006183   .0112874    -0.05   0.956    -.0227412    .0215046
         73  |   .0090073   .0124871     0.72   0.471     -.015467    .0334817
         75  |  -.0192709   .0133118    -1.45   0.148    -.0453615    .0068197
         80  |   .0791781   .0146177     5.42   0.000      .050528    .1078283
         85  |   .1821825   .0156395    11.65   0.000     .1515295    .2128354
             |
       grade |
         L2. |   .0848134    .003201    26.50   0.000     .0785396    .0910872
         L3. |          0  (omitted)
             |
       _cons |   .6010253    .040907    14.69   0.000      .520849    .6812015
-------------+----------------------------------------------------------------
     sigma_u |  .30883404
     sigma_e |  .23925736
         rho |  .62493035   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. test L2.grade=L3.grade

 ( 1)  L2.grade - oL3.grade = 0

           chi2(  1) =  702.05
         Prob > chi2 =    0.0000

.

In addition, I would consider interactng time with -professors- (provided that I'm right in guessing that -professors- is the predictor you're most interested in).
See -fvvarlist- notation for interactions and categorical variables.