Interaction versus margins dydx pwcompare

[Deepika Deshpande]

I am researching the effect of Geographic Segment Diversification (GSD) on firm performance (EBIT ROA) for US industrial companies for the period 2000-2019. This period includes the financial crisis (2008, 2009.) Hence, I included an interaction term to see how the crisis impacts the quadratic relationship between GSD and EBIT_ROA and find that the interaction impact is significant (below.)

. xtreg Ln_EBIT_ROA Ln_Revenue Ln_LTD_to_Sales Ln_Intangible_Assets  CoAge wGDPpc wCPI wDCF wExpgr
>  wGDPgr wCons Ln_PS_RD c.l1.Ln_GSD##c.l1.Ln_GSD##ib2.crisis if  CoAge>=0 & NATION=="UNITED STATE
> S" & NATIONCODE==840 & FSTS>=10 & FSTS <=100 & GENERALINDUSTRYCLASSIFICATION ==1 & Year_<2020 & 
> Year_<YearInactive & Discr_GS_Rev!=1, fe cluster(n_WSID)

Fixed-effects (within) regression               Number of obs     =      1,080
Group variable: n_WSID                          Number of groups  =        215

R-sq:                                           Obs per group:
     within  = 0.1280                                         min =          1
     between = 0.0043                                         avg =        5.0
     overall = 0.0123                                         max =         19

                                                F(17,214)         =          .
corr(u_i, Xb)  = -0.7239                        Prob > F          =          .

                                             (Std. Err. adjusted for 215 clusters in n_WSID)
--------------------------------------------------------------------------------------------
                           |               Robust
               Ln_EBIT_ROA |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
---------------------------+----------------------------------------------------------------
                Ln_Revenue |   .5231021   .1499161     3.49   0.001     .2276008    .8186035
           Ln_LTD_to_Sales |   -.125481   .0419333    -2.99   0.003    -.2081362   -.0428258
      Ln_Intangible_Assets |  -.1103335   .0615829    -1.79   0.075    -.2317202    .0110532
                     CoAge |  -.0029363   .0166989    -0.18   0.861    -.0358517    .0299791
                    wGDPpc |   .0000298   .0000218     1.37   0.172    -.0000131    .0000727
                      wCPI |   .0060675   .0254869     0.24   0.812      -.04417     .056305
                      wDCF |   1.36e-13   1.27e-13     1.07   0.285    -1.14e-13    3.85e-13
                    wExpgr |   .0126791   .0125128     1.01   0.312    -.0119851    .0373433
                    wGDPgr |   .0115004   .0300052     0.38   0.702    -.0476431     .070644
                     wCons |  -2.27e-14   4.36e-14    -0.52   0.603    -1.09e-13    6.32e-14
                  Ln_PS_RD |    -.04777   .0474308    -1.01   0.315    -.1412614    .0457213
                           |
                    Ln_GSD |
                       L1. |    -.49074   .2583853    -1.90   0.059    -1.000046    .0185662
                           |
       cL.Ln_GSD#cL.Ln_GSD |    .177507   .1103887     1.61   0.109    -.0400813    .3950953
                           |
                    crisis |
                        1  |   .0000289   .1170896     0.00   1.000    -.2307677    .2308255
                        3  |   -.244763   .1386875    -1.76   0.079    -.5181314    .0286055
                           |
          crisis#cL.Ln_GSD |
                        1  |  -.0697625   .1898751    -0.37   0.714    -.4440274    .3045024
                        3  |  -.1822128   .2083067    -0.87   0.383    -.5928084    .2283829
                           |
crisis#cL.Ln_GSD#cL.Ln_GSD |
                        1  |  -.2631727   .1049846    -2.51   0.013    -.4701091   -.0562364
                        3  |  -.2041293   .0970556    -2.10   0.037    -.3954366   -.0128219
                           |
                     _cons |   -13.0135   2.823174    -4.61   0.000    -18.57829   -7.448709
---------------------------+----------------------------------------------------------------
                   sigma_u |  1.1013267
                   sigma_e |  .59130596
                       rho |  .77623771   (fraction of variance due to u_i)
--------------------------------------------------------------------------------------------

However, when I run margins dydx pwcompare, there seems to be no statistical difference in the marginal effects (below). I am not clear how to interpret these two results. Any help would be appreciated. Thanks!

. margins crisis, dydx(cL.Ln_GSD) pwcompare

Pairwise comparisons of average marginal effects

Model VCE    : Robust                           Number of obs     =      1,080

Expression   : Linear prediction, predict()
dy/dx w.r.t. : L.Ln_GSD

--------------------------------------------------------------
             |   Contrast Delta-method         Unadjusted
             |      dy/dx   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
L.Ln_GSD     |
      crisis |
     2 vs 1  |   -.140872   .1625424     -.4594492    .1777053
     3 vs 1  |  -.1597067   .2006283     -.5529309    .2335176
     3 vs 2  |  -.0188347   .1449543     -.3029399    .2652705
--------------------------------------------------------------

[Clyde Schechter]

The statistically significant interaction terms you are seeing in the regression output involve only the quadratic terms. The implication of that, if you believe in null hypothesis significance testing, is that the relationship is quadratic and not just linear. But the interactions between crisis and the linear terms are not at all statistically significant. So you cannot conclude one way or another from that whether you have a statistically significant interaction between crisis and the (overall quadratic) effect of Ln_GSD. When modeling variables quadratically, you must never interpret the linear term(s) alone or the quadratic term(s) alone. They must always be interpreted jointly. The -margins- output does that.

[Deepika Deshpande]

Thank you Clyde. If I understand correctly, would it mean that the interaction effect varies at different points i.e. I should be using "margins dydx" at different points rather than one consolidated "margins dydx" command? i.e.

should I use: something like

margins crisis, dydx(cL.Ln_GSD) at(l1.Ln_GSD=(-4.839976 (1).2830217))

instead of using

margins crisis, dydx(cL.Ln_GSD)

[Clyde Schechter]

would it mean that the interaction effect varies at different points i.e.

Well, that is true, but it is not what I was getting at. The point I was making in #2 is that interpreting the significance of the quadratic terms (of the interaction, or even outside an interaction) without the linear terms is always a meaningless exercise.

I should be using "margins dydx" at different points rather than one consolidated "margins dydx" command? i.e.

Perhaps. It depends on your research question. In a quadratic model, the marginal effect will always be a linear function of the independent variable. One consequence of that is that there will always be some value of the independent variable where the marginal effect is zero (the value that corresponds to the peak or trough of the upside-down or upright U). And there will always be a range of values around that where the marginal effect is small. It also implies that you can also find values of the independent variable that give very large marginal effects. The question is whether those values of the independent variable are realistically seen in real life, are meaningful, and are of interest.

When you use -margins, dydx(indep_var)- without specifying an -at()- option, you get what is known as the average marginal effect. It is an average of the marginal effects at all the values of indep_var weighted according to their frequency in the data. Depending on what you are trying to get it in the real world, it may be the right statistic to look at. Or not. In other situations looking at the marginal effects at other specified values may be more relevant to the purposes at hand. While you have given the general context of your research in #1, you do not provide enough specifics for me to advise you on that.

[Deepika Deshpande]

Thank you Clyde. Your response was very helpful.