Interaction effect between two variables. What is the difference between ## and #

[Jong Ling]

Dear Professors,

I would like to test four interaction effect between variables in random effect model: which are

interaction effect between total and
(i) fd (continuous)
(ii) fceo (dummy, 0 and 1)

AND

interaction effect between id and
(i) dis (continuous)
(ii) fis (continuous)

I noticed that ## and # give different results

. xtreg lnexrem c.fd##c.total i.fceo##c.total c.id##c.dis c.id##c.fis idat rd idrc bs lnta lev laggedroa mv mccg, re
note: total omitted because of collinearity
note: id omitted because of collinearity

Random-effects GLS regression Number of obs = 1395
Group variable: code Number of groups = 279

R-sq: within = 0.1756 Obs per group: min = 5
between = 0.3982 avg = 5.0
overall = 0.3750 max = 5

Random effects u_i ~ Gaussian Wald chi2(19) = 417.87
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000

------------------------------------------------------------------------------
lnexrem | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
fd | .0238029 .6082261 0.04 0.969 -1.168298 1.215904
total | -.3255308 .6907759 -0.47 0.637 -1.679427 1.028365
|
c.fd#c.total | .3000803 1.284027 0.23 0.815 -2.216567 2.816727
|
1.fceo | -.2763074 .2369025 -1.17 0.243 -.7406278 .188013
total | (omitted)
|
fceo#c.total |
1 | .7582793 .4858597 1.56 0.119 -.1939883 1.710547
|
id | -.4038776 .3000665 -1.35 0.178 -.9919971 .1842419
dis | -1.091215 .7278848 -1.50 0.134 -2.517843 .3354127
|
c.id#c.dis | 1.965933 1.654194 1.19 0.235 -1.276228 5.208094
|
id | (omitted)
fis | .2268854 1.233252 0.18 0.854 -2.190245 2.644016
|
c.id#c.fis | -.9240375 2.691288 -0.34 0.731 -6.198865 4.35079
|
idat | -.0009001 .0050679 -0.18 0.859 -.0108329 .0090327
rd | .0819277 .0515627 1.59 0.112 -.0191333 .1829887
idrc | .018694 .1047622 0.18 0.858 -.186636 .224024
bs | .0728553 .0141188 5.16 0.000 .0451829 .1005277
lnta | .4132528 .0309636 13.35 0.000 .3525652 .4739404
lev | -.2044981 .1224848 -1.67 0.095 -.444564 .0355677
laggedroa | .0029332 .0020636 1.42 0.155 -.0011112 .0069777
mv | .0647557 .0215945 3.00 0.003 .0224313 .1070801
mccg | .1190974 .0194464 6.12 0.000 .0809831 .1572116
_cons | 8.732723 .5140285 16.99 0.000 7.725246 9.740201
-------------+----------------------------------------------------------------
sigma_u | .71427335
sigma_e | .32818334
rho | .82569037 (fraction of variance due to u_i)
------------------------------------------------------------------------------




. xtreg lnexrem c.fd#c.total i.fceo#c.total c.id#c.dis c.id#c.fis idat rd idrc bs lnta lev laggedroa mv mccg, re

Random-effects GLS regression Number of obs = 1395
Group variable: code Number of groups = 279

R-sq: within = 0.1732 Obs per group: min = 5
between = 0.3953 avg = 5.0
overall = 0.3722 max = 5

Random effects u_i ~ Gaussian Wald chi2(14) = 413.80
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000

------------------------------------------------------------------------------
lnexrem | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
c.fd#c.total | .5946801 .4266258 1.39 0.163 -.2414911 1.430851
|
fceo#c.total |
0 | .0332837 .3361273 0.10 0.921 -.6255136 .6920811
1 | .2454874 .3161734 0.78 0.437 -.374201 .8651758
|
c.id#c.dis | -.5056158 .5649353 -0.89 0.371 -1.612869 .601637
|
c.id#c.fis | -.6130954 .8468996 -0.72 0.469 -2.272988 1.046797
|
idat | -.0002822 .0049735 -0.06 0.955 -.0100302 .0094657
rd | .0760973 .0512811 1.48 0.138 -.0244119 .1766064
idrc | .0028356 .1006021 0.03 0.978 -.194341 .2000122
bs | .0769993 .013667 5.63 0.000 .0502124 .1037862
lnta | .4148468 .0306303 13.54 0.000 .3548125 .4748812
lev | -.210979 .1220482 -1.73 0.084 -.450189 .028231
laggedroa | .0029897 .0020568 1.45 0.146 -.0010416 .007021
mv | .0652235 .0215337 3.03 0.002 .0230183 .1074287
mccg | .1175888 .019318 6.09 0.000 .0797261 .1554514
_cons | 8.282248 .3928169 21.08 0.000 7.512341 9.052155
-------------+----------------------------------------------------------------
sigma_u | .71332933
sigma_e | .32805932
rho | .82541831 (fraction of variance due to u_i)
------------------------------------------------------------------------------

Could you please advise which one to use? ## or #?

I sincerely look forward to receiving favorable assistance. Thank you.

[Carlo Lazzaro]

Jong Ling:
welcome to the list.
What you're interested is comprhensively covered in -help fvvarlist- and related entry in Stata .pdf manual.
That said, whereas -#- asks Stata to calculate only the interaction between two (or more) predictors, -##- calculates both interaction and the main conditional effects of the predictors included in the intearction itself.
Most of the times (as in your case), researcher should prefer -##- notation.

[Jong Ling]

Dear Professor Carlo,

Many thanks for your prompt response. I sincerely appreciate that.

When trying to create the interaction effect between two variables, I notice that there are several options, such as Interaction (2-way) and 2-way full factorial. May I know what is the difference between these two options and which one is better?

Thank you.

[Carlo Lazzaro]

Jong Ling:
please call Carlo as all on (and many more off) the list do.
The best source for the topic(s) you're interested in is -help fvvarlist- and related entry in Stata .pdf manual.
In brief, the -fvvarlst- operators -i- and -c- tell Stata that the variables included in the interactions are categorical or continuous.

[Jong Ling]

Dear Carlo,

Well noted. Thank you very much.