intersection variable

[amera amery]

reghdfe cases_normed i.days#bus_policy_open_risk cases_normed_delay_14days cases_normed_delay_14days_growth residential workplaces transit_stations parks grocery_and_pharmacy retail_and_recreation  if days<=368 & days>=331  , absorb( fips  )

I am trying to build the previous regression where bus_policy_open_risk has two values (0,1) the results as follows:

 reghdfe cases_normed i.days#bus_policy_open_risk  cases_normed_delay_14days cases_normed_delay_14days_growth residential workplaces tr
> ansit_stations parks grocery_and_pharmacy retail_and_recreation  if days<=368 & days>=331   , absorb( fips  )
(dropped 78 singleton observations)
(MWFE estimator converged in 1 iterations)

HDFE Linear regression                            Number of obs   =      2,988
Absorbing 1 HDFE group                            F(  83,   2402) =       3.61
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.6040
                                                  Adj R-squared   =     0.5075
                                                  Within R-sq.    =     0.1108
                                                  Root MSE        =     0.4728

--------------------------------------------------------------------------------------------------
                    cases_normed | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
---------------------------------+----------------------------------------------------------------
       days#bus_policy_open_risk |
                          331 1  |   .2096405   .1333903     1.57   0.116    -.0519315    .4712124
                          332 0  |   .0290436   .1178141     0.25   0.805    -.2019843    .2600714
                          332 1  |   .1167609    .136788     0.85   0.393    -.1514738    .3849956
                          333 0  |   .0202103   .1238841     0.16   0.870    -.2227206    .2631411
                          333 1  |   .0876057   .1318282     0.66   0.506    -.1709031    .3461145
                          334 0  |  -.0836872   .1223314    -0.68   0.494    -.3235732    .1561988
                          334 1  |   .3293634   .1349964     2.44   0.015      .064642    .5940849
                          335 0  |  -.0519824   .1225741    -0.42   0.672    -.2923444    .1883795
                          335 1  |   .0569438   .1304069     0.44   0.662     -.198778    .3126656
                          336 0  |  -.0497269   .1156959    -0.43   0.667    -.2766011    .1771473
                          336 1  |   .3096333   .1357965     2.28   0.023     .0433429    .5759236
                          337 0  |  -.0742253   .1197005    -0.62   0.535    -.3089522    .1605016
                          337 1  |   .0582338   .1327688     0.44   0.661    -.2021195    .3185871
                          338 0  |  -.0234937   .1142326    -0.21   0.837    -.2474985     .200511
                          338 1  |   .1724191   .1303567     1.32   0.186    -.0832041    .4280424
                          339 0  |  -.0593665   .1191785    -0.50   0.618    -.2930699    .1743368
                          339 1  |   .1652005   .1308348     1.26   0.207    -.0913603    .4217612
                          340 0  |  -.1055072   .1156711    -0.91   0.362    -.3323326    .1213183
                          340 1  |   .1114776    .133643     0.83   0.404    -.1505899    .3735451
                          341 0  |  -.1084851   .1178873    -0.92   0.358    -.3396563    .1226862
                          341 1  |   .0508871   .1362215     0.37   0.709    -.2162367    .3180108
                          342 0  |  -.0646933   .1205707    -0.54   0.592    -.3011266      .17174
                          342 1  |   .1925672   .1317682     1.46   0.144     -.065824    .4509584
                          343 0  |  -.0475884   .1174408    -0.41   0.685    -.2778842    .1827073
                          343 1  |   .1629395   .1286369     1.27   0.205    -.0893112    .4151902
                          344 0  |  -.0611853   .1277101    -0.48   0.632    -.3116188    .1892482
                          344 1  |   .1395029    .137892     1.01   0.312    -.1308967    .4099024
                          345 0  |  -.0902414   .1253265    -0.72   0.472    -.3360007    .1555178
                          345 1  |   .2101073   .1293424     1.62   0.104     -.043527    .4637415
                          346 0  |   .0101026   .1336115     0.08   0.940    -.2519032    .2721085
                          346 1  |   .1752553   .1273663     1.38   0.169    -.0745039    .4250144
                          347 0  |  -.0532696   .1222337    -0.44   0.663    -.2929639    .1864247
                          347 1  |   .1535804   .1298833     1.18   0.237    -.1011146    .4082755
                          348 0  |  -.0573576   .1159042    -0.49   0.621    -.2846401    .1699249
                          348 1  |   .2491161   .1292221     1.93   0.054    -.0042823    .5025146
                          349 0  |  -.0163862   .1191614    -0.14   0.891    -.2500559    .2172835
                          349 1  |   .1651863   .1297673     1.27   0.203    -.0892812    .4196537
                          350 0  |  -.1191376   .1331604    -0.89   0.371    -.3802588    .1419836
                          350 1  |   .1687508    .130931     1.29   0.198    -.0879987    .4255003
                          351 0  |  -.0480022   .1182763    -0.41   0.685    -.2799363    .1839319
                          351 1  |   .1326414   .1288148     1.03   0.303    -.1199583     .385241
                          352 0  |   .0030494   .1195856     0.03   0.980    -.2314522    .2375509
                          352 1  |   .2942891   .1336999     2.20   0.028     .0321101    .5564682
                          353 0  |   .0189665   .1278907     0.15   0.882    -.2318211    .2697541
                          353 1  |   .1465909   .1319872     1.11   0.267    -.1122296    .4054114
                          354 0  |   -.110522   .1299275    -0.85   0.395    -.3653036    .1442596
                          354 1  |   .1523915   .1312036     1.16   0.246    -.1048925    .4096754
                          355 0  |  -.0756447   .1262508    -0.60   0.549    -.3232166    .1719271
                          355 1  |    .060433   .1279538     0.47   0.637    -.1904783    .3113443
                          356 0  |  -.0957646   .1245305    -0.77   0.442    -.3399629    .1484336
                          356 1  |   .0408801    .133175     0.31   0.759    -.2202697    .3020299
                          357 0  |    .138325   .1254909     1.10   0.270    -.1077567    .3844067
                          357 1  |  -.0411404   .1399316    -0.29   0.769    -.3155396    .2332589
                          358 0  |  -.1219758    .139442    -0.87   0.382    -.3954148    .1514633
                          358 1  |    .109354   .1297075     0.84   0.399    -.1449962    .3637043
                          359 0  |   .0039393   .1240928     0.03   0.975    -.2394007    .2472793
                          359 1  |   .2560763    .135518     1.89   0.059     -.009668    .5218206
                          360 0  |  -.0592402   .1347542    -0.44   0.660    -.3234868    .2050064
                          360 1  |   .5000361   .1370191     3.65   0.000     .2313483     .768724
                          361 0  |  -.1226208   .1245707    -0.98   0.325     -.366898    .1216565
                          361 1  |     .15382   .1383023     1.11   0.266    -.1173843    .4250243
                          362 0  |  -.2125093   .1265772    -1.68   0.093    -.4607211    .0357025
                          362 1  |   .0132899   .1403392     0.09   0.925    -.2619086    .2884884
                          363 0  |  -.2454007   .1270654    -1.93   0.054      -.49457    .0037685
                          363 1  |  -.1819625   .1400782    -1.30   0.194    -.4566491    .0927242
                          364 0  |   .1367787    .130235     1.05   0.294    -.1186059    .3921632
                          364 1  |  -.0606629   .1295566    -0.47   0.640    -.3147172    .1933914
                          365 0  |  -.1406556   .1307284    -1.08   0.282    -.3970078    .1156966
                          365 1  |   .0483386   .1360251     0.36   0.722       -.2184    .3150773
                          366 0  |   .0631108   .1278295     0.49   0.622    -.1875567    .3137783
                          366 1  |  -.0712824    .132783    -0.54   0.591    -.3316635    .1890987
                          367 0  |   .0174997   .1255504     0.14   0.889    -.2286986    .2636979
                          367 1  |  -.0645931    .140157    -0.46   0.645    -.3394342     .210248
                          368 0  |  -.1548754   .1278592    -1.21   0.226    -.4056013    .0958504
                          368 1  |     .01671   .1327696     0.13   0.900    -.2436448    .2770649
                                 |
       cases_normed_delay_14days |   .2925883    .022953    12.75   0.000     .2475785    .3375981
cases_normed_delay_14days_growth |   .0009109   .0002954     3.08   0.002     .0003316    .0014902
                     residential |  -.0551442   .6125502    -0.09   0.928    -1.256326    1.146037
                      workplaces |  -.1953761   .2616483    -0.75   0.455    -.7084558    .3177037
                transit_stations |   .1590853   .0618105     2.57   0.010     .0378778    .2802927
                           parks |  -.0212484   .0286065    -0.74   0.458    -.0773444    .0348475
            grocery_and_pharmacy |  -.0191809   .1138943    -0.17   0.866    -.2425221    .2041602
           retail_and_recreation |  -.2590651   .1554198    -1.67   0.096    -.5638359    .0457056
                           _cons |    .182201   .8551501     0.21   0.831    -1.494707    1.859109
--------------------------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
        fips |       503           0         503     |
-----------------------------------------------------+

if I try to build same regression but without intersection term and by add condition to the end of regression one when bus_policy_open_risk==0 and other when bus_policy_open_risk==1 as follows:

reghdfe cases_normed i.days    cases_normed_delay_14days cases_normed_delay_14days_growth residential workplaces transit_stations parks grocery_and_pharmacy retail_and_recreation  if days<=368 & days>=331   &bus_policy_open_risk==0 , absorb( fips  )


reghdfe cases_normed i.days    cases_normed_delay_14days cases_normed_delay_14days_growth residential workplaces transit_stations parks grocery_and_pharmacy retail_and_recreation  if days<=368 & days>=331   &bus_policy_open_risk==1 , absorb( fips  )

after comparing results with intersection variable when it is =1 or 0 it is different. I am wondering which regression is more accurate use intersection variable or adding condition to the regression?

[Leonardo Guizzetti]

First a clarification of terminology. What you have called an intersection term, is actually an interaction term. (It is nothing more than than the multiplied value between two or more variables.)

If I understand you correctly, you are comparing one model which includes a 2-way interaction between -bus_policy_open_risk- and -days-. You expect that running the reduced regression model (that is, without this interaction) on the separate subsets of -bus_policy_open_risk- will provide you with equivalent results. Sometimes people do this as a way of performing a stratified analysis, however these two approaches will only be equivalent when the model contains only the terms involved in the interaction and nothing else. In the case of having additional covariates, as in your example, there is no reason to expect them to the the same or even similar. A truly stratified (or subgroup) analysis using a single, "big" model would need to have an interaction with -bus_policy_open_risk- and every covariate in the model.

[amera amery]

Thank you. Yes it is typo. should be interaction. I mean running model with interaction term between bus_policy_open_risk- and -days should be the same as running two regression: first when bus_policy_open_risk=0 and other when bus_policy_open_risk=1 is it true?

[Leonardo Guizzetti]

Amera, thank you, I did understand your question correctly, and my answer to you in #2 applies. Short answer, no, they will not generally be the same.

[amera amery]

So, which one is more accurate to study the effect of bus_policy_open_risk and days on the dependent variable?

[Leonardo Guizzetti]

It's impossible for me to stay since I don't know your research area. Generally speaking, conducting separate regressions will give you the most flexibility in what you are modeling, but the interaction model may provide substantively similar results for your particular research question. Flexibility in modeling ability alone does not automatically mean more accurate, and indeed could result in worse results because at some point, the model becomes overfit and all you are doing is modeling noise.

Speaking of which, looking at your model it's not clear to me why you are modeling -days- as a categorical variable. Have you considered modeling -days- as a linear variable (-c.days-) or perhaps a as a polynomial? That could give you a better idea of the time trends in each subgroup and greatly simplify your model.

[amera amery]

Thank you. days in my model are categorical. i am studying event based model for -7 to 30 days of ordering mask event. and bus_policy_open_risk is 0 or 1 means that the business_closed policy is implemented or not during days (-7 to 30)