Interpreting interaction term in random effects model

[Noor Hend]

Hello!

I am investigating whether implementing GSCMP (green practices) and EMS (environmental management systems) simultaneously positively influences the relationship between GSCMP and firm performance. Therefore, I added an interaction term in my model to test if there is a moderating effect. However, I'm not sure how to interpret the outcome; Can I interpret the GSCMP and EMS variables separately, or should I only look at the interaction coefficient? In my example, the coefficient is negative, would that imply a negative moderating effect? However, the effect is not significant does this mean that there is no support for a moderating effect at all? The variable EMS can be either 1 (=implemented EMS) or 0(=no EMS)


Thank you in advance!

Model (1) without interaction term

 xtreg TobinsQ_w laggedGSCMP Firmrisk_w Firmsize_w i.Industry i.year, re

Random-effects GLS regression                   Number of obs      =      3704
Group variable: ID                              Number of groups   =       463

R-sq:  within  = 0.1526                         Obs per group: min =         8
       between = 0.1937                                        avg =       8.0
       overall = 0.1788                                        max =         8

                                                Wald chi2(15)      =    682.88
corr(u_i, X)   = 0 (assumed)                    Prob > chi2        =    0.0000

------------------------------------------------------------------------------
   TobinsQ_w |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
 laggedGSCMP |  -.0372598   .0850066    -0.44   0.661    -.2038697      .12935
  Firmrisk_w |  -.4236717   .1316738    -3.22   0.001    -.6817476   -.1655957
  Firmsize_w |  -.2410935   .0255676    -9.43   0.000    -.2912051   -.1909818
             |
    Industry |
          2  |   -.558424   .2066768    -2.70   0.007    -.9635031   -.1533448
          3  |    .632852   .1774375     3.57   0.000      .285081    .9806231
          4  |  -.3235721   .2409018    -1.34   0.179     -.795731    .1485868
          5  |  -.5079201   .1782467    -2.85   0.004    -.8572772   -.1585631
          6  |   -.283431   .1584239    -1.79   0.074    -.5939362    .0270742
             |
        year |
       2008  |  -.6494589   .0436928   -14.86   0.000    -.7350951   -.5638226
       2009  |  -.4367351   .0436323   -10.01   0.000    -.5222528   -.3512174
       2010  |  -.2601445   .0436846    -5.96   0.000    -.3457647   -.1745243
       2011  |  -.3809431   .0442207    -8.61   0.000     -.467614   -.2942722
       2012  |  -.2867888   .0444518    -6.45   0.000    -.3739128   -.1996647
       2013  |   .1048752   .0444915     2.36   0.018     .0176735    .1920769
       2014  |   .1389063    .044855     3.10   0.002     .0509922    .2268205
             |
       _cons |    4.23682   .2465329    17.19   0.000     3.753625    4.720016
-------------+----------------------------------------------------------------
     sigma_u |  .86607694
     sigma_e |  .65909614
         rho |  .63325553   (fraction of variance due to u_i)
------------------------------------------------------------------------------

Model 2 with interaction term

  xtreg TobinsQ_w c.laggedGSCMP##i.laggedEMS Firmrisk_w Firmsize_w i.Industry i.year, re

Random-effects GLS regression                   Number of obs      =      3704
Group variable: ID                              Number of groups   =       463

R-sq:  within  = 0.1527                         Obs per group: min =         8
       between = 0.1935                                        avg =       8.0
       overall = 0.1787                                        max =         8

                                                Wald chi2(17)      =    682.62
corr(u_i, X)   = 0 (assumed)                    Prob > chi2        =    0.0000

-----------------------------------------------------------------------------------------
              TobinsQ_w |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------------+----------------------------------------------------------------
            laggedGSCMP |  -.0258513   .1159604    -0.22   0.824    -.2531295    .2014269
            1.laggedEMS |  -.0137214   .0542992    -0.25   0.801    -.1201458    .0927031
                        |
laggedEMS#c.laggedGSCMP |
                     1  |  -.0117569   .1467287    -0.08   0.936    -.2993398    .2758261
                        |
             Firmrisk_w |   -.424106   .1317266    -3.22   0.001    -.6822853   -.1659267
             Firmsize_w |  -.2401096   .0258406    -9.29   0.000    -.2907562   -.1894631
                        |
               Industry |
                     2  |  -.5561707   .2070759    -2.69   0.007    -.9620319   -.1503094
                     3  |   .6347316   .1777639     3.57   0.000     .2863208    .9831425
                     4  |  -.3226728   .2412485    -1.34   0.181    -.7955111    .1501655
                     5  |  -.5064302   .1785666    -2.84   0.005    -.8564144    -.156446
                     6  |  -.2812923    .158771    -1.77   0.076    -.5924778    .0298932
                        |
                   year |
                  2008  |  -.6490783   .0437131   -14.85   0.000    -.7347544   -.5634021
                  2009  |  -.4359109   .0437189    -9.97   0.000    -.5215983   -.3502235
                  2010  |  -.2593825   .0437514    -5.93   0.000    -.3451336   -.1736313
                  2011  |  -.3797915   .0443713    -8.56   0.000    -.4667576   -.2928254
                  2012  |  -.2856803   .0445877    -6.41   0.000    -.3730705   -.1982901
                  2013  |   .1056095    .044617     2.37   0.018     .0181619    .1930571
                  2014  |   .1419346   .0457825     3.10   0.002     .0522025    .2316667
                        |
                  _cons |   4.229072   .2482444    17.04   0.000     3.742522    4.715622
------------------------+----------------------------------------------------------------
                sigma_u |  .86731741
                sigma_e |  .65914867
                    rho |  .63388308   (fraction of variance due to u_i)
-----------------------------------------------------------------------------------------

[Clyde Schechter]

Here is how I would interpret the interaction model. The expected value of theTobinsQ_w:laggedGSCMP slope is 0.012 (to three decimal places) units lower when laggedEMS = 1 than when laggedEMS = 0. However, the confidence interval suggests that this difference is plausibly anywhere between -0.299 and +0.276. Consequently, the data do not contain enough information to provide a precise estimate of this difference, and we cannot even be certain in which direction it runs, or, perhaps, whether it might be 0.

To get a feel for how this plays out in your data, I would follow the regression model with:

margins laggedEMS, at(laggedGSCMP = (list of interesting values of GSCMP))
marginsplot

Replace list of interesting values of GSCMP with a list of values that span the range of GSCMP. The -marginsplot- command will then show you two curves of the TobinsQ_w:laggedGSWCMP relationship, one for laggedEMS = 1 and the other for laggedEMS = 0. It will then be visually clear whether the two curves are sufficiently separated to matter for practical purposes, or whether they are more or less the same curve with minimal separation. You may also find that the curves are close together for some values of GSCMP but separate meaningfully for other values of GSCMP. The graph will enable you to truly understand what's going on in ways that coefficients and formulas never will.

Finally, I call your attention to the recent position paper by the American Statistical Association calling for the abandonment of the concept of statistical significance. https://www.tandfonline.com/doi/full...5.2019.1583913 Interpret your results in terms of estimated effects and ranges of uncertainty around them. Do not think of them as "significant" or "not significant," and, certainly, do not confuse "not significant" with "no effect."

[Noor Hend]

Thank you Clyde, very useful insights. Attached the margins plot which shows that they are more or less the same curve with minimal separation. I was wondering if it is also possible to plot the marginal effects of the interaction term on Tobin's Q to graphical show the effect of implementing GSCM and EMS together on financial performance?

Thank you in advance!