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  • Margins Plots with Two Dummy Variables (and a quadratic term)

    Hello,

    I am running a regression to understand how the financial crisis has impacted the effect of Geographic Diversification (GSD lagged) on Firm Performance (EBIT_ROA.) I am using US data over the period 2000-2019. I have defined 2 dummy variables Precrisis (1 for years 2000-2006, 0 otherwise) and PostCrisis (1 for years 2010-2019, 0 otherwise.) The crisis years 2007-09 are considered the reference period.

    The code I am using is:

    Code:
    xtreg Ln_EBIT_ROA Ln_Revenue Ln_TA Ln_Curr_Ratio CoAge wGDPpc c.l1.Ln_GSD##c.l1.Ln_GSD##PreCrisis c.l1.Ln_GSD##c.l1.Ln_GSD##PostCrisis, fe cluster(n_CUSIP)
    I would like to reconfirm if my operationalization of the crisis dummy variables is correct for the purpose of my study.

    Also, I would like to run marginsplots for the PreCrisis, PostCrisis and crisis periods. I presume the dummy variables would take the following values for these periods:
    For the Precrisis period, Precrisis =1
    For the Crisis period, Precrisis =0 and PostCrisis =0
    For the Postcrisis period, Postcrisis =1

    Any suggestions on how I can run this marginsplot would be helpful. Thank you.

  • #2
    No, it's not quite right. You can't have separate Precrisis and Postcrisis indicator ("dummy") variables like this: -margins- will not realize that they are mutually exclusive. You need a single three level variable to cover all three time periods.

    Code:
    label define era 1 "Pre Crisis" 2 "Crisis" 3 "Post Crisis"
    gen era:era = 1 if year < 2006
    replace era = 2 if inrange(year, 2007, 2009)
    replace era = 3 if year > 2009 & !missing(year)
    xtreg Ln_EBIT_ROA Ln_Revenue Ln_TA Ln_Curr_Ratio CoAge wGDPpc c.l1.Ln_GSD##c.l1.Ln_GSD##ib2.era , fe cluster(n_CUSIP)
    margins era, at(l1.Ln_GSD = (2(1)6)) // SUBSTITUTE APPRPORIATE RANGE OF VALUES FOR Ln_GSD HERE
    marginsplot
    Note: the ib2. prefix will assure that the Crisis period is used as the reference for the era variable.

    Comment


    • #3
      Thank you so much, Clyde. This is extremely helpful. (I was struggling with this command since I knew that the original command I had put in would not know how to recognize the PreCrisis and Post Crisis were exclusive periods. Your response has been very helpful.)

      If I may ask another question: Is there any nlcom command whereby I can establish the maximum points of the 3 inverted-U curves (precrisis, crisis, postcrisis)?

      (In case not, I can calculate it manually. However, just wondering if such a command existed for these 3 era values.)

      Comment


      • #4
        Code:
        forvalues i = 1/3 {
            nlcom -0.5*(_b[c.l1.Ln_GSD] + _b[c.l1.Ln_GSD#`i'.era]) / (_b[c.l1.Ln_GSD#c.l1.Ln_GSD] + _b[c.l1.LnGSD#c.l1.Ln_GSD#`i'.era])
        }

        Comment


        • #5
          Thank you very much. This works.

          Comment


          • #6
            Further to the suggestion in #2, I created a variable called "era" and marked the pre-crisis years (2000-06) as era=1, crisis years (2007-09) as era=2, and post-crisis years (2010-19) as era=3. I used era=2 as the reference period and obtained the following results. I would like to confirm if my interpretation (below) is correct:
            1. During the Crisis, the impact of Geographic Diversification on Performance is significant and negative (P-value = 0.056)
            2. Pre-Crisis and Post-Crisis periods have a non-significant interaction effect on the reference year impact of Geographic Diversification on Performance
            Also, if I need to find out the main effect of cL.Ln_GSD#cL.Ln_GSD (without including the impact of the crisis i.e. era=2), how can I obtain it? Should I be re-running the regression without era?

            I do hope I have worded my questions clearly. Thank you.

            Code:
            . xtreg Ln_EBIT_ROA Ln_Revenue Ln_LTD_to_Sales Ln_Intangible_Assets  CoAge wGDPpc wCPI wDCF wExpg
            > r wGDPgr wCons Ln_PS_RD c.l1.Ln_GSD##c.l1.Ln_GSD##ib2.era if  CoAge>=0 & NATION=="UNITED STATES
            > " & NATIONCODE==840 & FSTS>=10 & GENERALINDUSTRYCLASSIFICATION ==1 & Year_<2020 & Year_<YearIna
            > ctive & Discr_GS_Rev!=1, fe cluster(n_CUSIP)
            
            Fixed-effects (within) regression               Number of obs     =      1,094
            Group variable: n_CUSIP                         Number of groups  =        217
            
            R-sq:                                           Obs per group:
                 within  = 0.1154                                         min =          1
                 between = 0.0136                                         avg =        5.0
                 overall = 0.0122                                         max =         19
            
                                                            F(17,216)         =          .
            corr(u_i, Xb)  = -0.7433                        Prob > F          =          .
            
                                                     (Std. Err. adjusted for 217 clusters in n_CUSIP)
            -----------------------------------------------------------------------------------------
                                    |               Robust
                        Ln_EBIT_ROA |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
            ------------------------+----------------------------------------------------------------
                         Ln_Revenue |   .5670965   .1408313     4.03   0.000     .2895169     .844676
                    Ln_LTD_to_Sales |  -.0996159   .0370059    -2.69   0.008    -.1725548   -.0266769
               Ln_Intangible_Assets |  -.1249349   .0596555    -2.09   0.037    -.2425164   -.0073534
                              CoAge |  -.0047677   .0162806    -0.29   0.770    -.0368569    .0273214
                             wGDPpc |   .0000303   .0000223     1.36   0.176    -.0000137    .0000743
                               wCPI |  -.0065127   .0236511    -0.28   0.783    -.0531291    .0401037
                               wDCF |   1.45e-13   1.20e-13     1.21   0.228    -9.18e-14    3.83e-13
                             wExpgr |   .0257288   .0142087     1.81   0.072    -.0022765    .0537342
                             wGDPgr |  -.0196886    .033677    -0.58   0.559    -.0860662    .0466891
                              wCons |  -2.75e-14   4.16e-14    -0.66   0.509    -1.10e-13    5.46e-14
                           Ln_PS_RD |  -.0304176   .0457847    -0.66   0.507    -.1206596    .0598244
                                    |
                             Ln_GSD |
                                L1. |   -.783652    .259358    -3.02   0.003    -1.294849   -.2724555
                                    |
                cL.Ln_GSD#cL.Ln_GSD |  -.1412219   .0734569    -1.92   0.056     -.286006    .0035622
                                    |
                                era |
                                 1  |  -.0815845   .1083007    -0.75   0.452     -.295046    .1318769
                                 3  |  -.3074317   .1204217    -2.55   0.011    -.5447837   -.0700797
                                    |
                      era#cL.Ln_GSD |
                                 1  |   .1511042   .2147916     0.70   0.483    -.2722517    .5744601
                                 3  |   .0818235   .2057448     0.40   0.691     -.323701     .487348
                                    |
            era#cL.Ln_GSD#cL.Ln_GSD |
                                 1  |   .0625535   .0817691     0.77   0.445    -.0986139     .223721
                                 3  |   .0570302   .0689316     0.83   0.409    -.0788345    .1928949
                                    |
                              _cons |  -13.25324   2.766799    -4.79   0.000    -18.70662   -7.799859
            ------------------------+----------------------------------------------------------------
                            sigma_u |  1.0536876
                            sigma_e |  .59365848
                                rho |   .7590529   (fraction of variance due to u_i)
            -----------------------------------------------------------------------------------------

            Comment


            • #7
              Your interpretations are not correct.

              Interpretation these results is complicated, and is best done using the -margins- command. It can only be derived from the regression output through calculations that are laborious and error prone; best not to do it by hand. Almost nothing in the regression output is directly interpretable by itself.

              You do not explain your variables in the post. I'll assume that by "geographic diversification" you are referring to the variable Ln_GSD and that by performance you are referring to Ln_EBIT_ROA as that seems the most sensible way to relate your narrative to the output in the absence of an explanation. Is that correct? If so, running
              Code:
              margins era, dydx(cL.Ln_GSD)
              will give you the average marginal effect of Ln_GSD on Ln_EBIT_ROA in each of the three eras. It is averaged over the joint distribution of all of the model's explanatory variables.

              Also, if I need to find out the main effect of cL.Ln_GSD#cL.Ln_GSD (without including the impact of the crisis i.e. era=2), how can I obtain it? Should I be re-running the regression without era?
              Within this model, there is no such thing as "the main effect of cL.Ln_GSD#cL.Ln_GSD." In fact, in no model is there any such effect. In a model without the era variable, then there is an effect of cL.Ln_GSD, but a quadratic term neverhas a marginal effect of its own (unless there is no linear or other order term in the model--which is very unusual), and that marginal effect is not given by the coefficient of L.Ln_GSD. Again, you need the -margins- command to avoid tedious and error prone calculations.

              But first, there is the question of whether a model without era is a valid model. By comparing the marginal effects in each era, computed with the -margins- command shown above, you can decide if that is true. If the three marginal effects are meaningfully different, then it does not make sense to run a model without the era variable, nor is it reasonable to even talk about "the" marginal effect of L.Ln_GSD. But if they are equal for practical purposes, then you could simplify the model by eliminating the era variable, and in the simplified model you could just run -margins, dydx(cL.Ln_GSD)- to get the average marginal effect of L.Ln_GSD.

              Another thing you can do with the existing model is calculate the average marginal effect of L.Ln_GSD taking into account the differences across eras:
              Code:
              margins, dydx(cL.Ln_GSD)
              This will give you the average marginal effect, a weighted average over all three eras.

              Comment


              • #8
                Thanks, Clyde (#7). Your response is very helpful.

                Apologies for not clarifying the variables - the assumption you made in the statement below is correct (i.e. Ln_GSD is the Log of the Geographic Segment Diversification and Ln_EBIT_ROA is the Log of the EBIT based ROA):

                You do not explain your variables in the post. I'll assume that by "geographic diversification" you are referring to the variable Ln_GSD and that by performance you are referring to Ln_EBIT_ROA as that seems the most sensible way to relate your narrative to the output in the absence of an explanation. Is that correct?
                I ran the command you suggested and I get the following. Clearly, Lagged Ln_GSD has a significant marginal effect on performance for all three values of "era" i.e. pre-crisis, crisis, and post-crisis (since the P-values are 0.007, 0.003, and 0.001.)

                Code:
                . margins era, dydx(cL.Ln_GSD)
                
                Average marginal effects                        Number of obs     =      1,094
                Model VCE    : Robust
                
                Expression   : Linear prediction, predict()
                dy/dx w.r.t. : L.Ln_GSD
                
                ------------------------------------------------------------------------------
                             |            Delta-method
                             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
                -------------+----------------------------------------------------------------
                L.Ln_GSD     |
                         era |
                          1  |  -.5672611   .2109349    -2.69   0.007     -.980686   -.1538362
                          2  |  -.6664523   .2248132    -2.96   0.003    -1.107078   -.2258265
                          3  |   -.631958   .1836315    -3.44   0.001    -.9918691   -.2720469
                ------------------------------------------------------------------------------
                I would like to check if these marginal effects are significantly different. I understand there is a command called "test" in stata that I can use. However, I am unable to get the correct syntax. (Apologies for my limited knowledge.) In case you could help with that, I would be grateful. Thanks.

                Comment


                • #9
                  I have an additional question pertaining to my post above (#8). As per my model, the relationship between Geographic Diversification (Ln_GSD) and Performance (EBIT_ROA) is a curvilinear relationship i.e. an inverted U i.e. it is positive at lower values of Ln_GSD and negative at high values of Ln_GSD. I infer that from the fact that the coefficient of the quadratic term Ln_GSD##Ln_GSD is negative.

                  Does that mean that I should be getting different values of dy/dx at different values of Ln_GSD? If so, I should probably write the code so as to get dy/dx at different values of Ln_GSD.

                  Comment


                  • #10
                    Re #8. Yes, you can do that with -lincom-, but it is simpler to just re-run the -margins- command adding the -pwcompare- option. It will give you all 3 pairwise comparisons among the eras in a single command.

                    Re #9. Yes, you should identify a range of values of Ln_GSD that span the realistic range of that variable in real life and then get marginal effects at those values. For purposes of demonstration, let's suppose that realistic values range from 1 to 10. Then you would do this as:

                    Code:
                    margins era, dydx(cL.Ln_GSD) at(cL.Ln_GSD = (1(1)10))
                    and you might want to get a graph of the results with
                    Code:
                    marginsplot, xdimension(cL.Ln_GSD)
                    which will give you three curves (one for each era) plotting the marginal effect vs the starting value of Ln_GSD.

                    Comment


                    • #11
                      Thank you very much, Clyde (#10.) Very helpful. I will try these now.

                      I have one last question. This pertains to the various p-values I obtain under the 3 scenarios I have mentioned below. I understand that scenarios 1 and 2 contain two different hypotheses and hence the p-values will obviously be different. In my view, if the p-values I get in 2 are significant (which they are at 5% confidence), I should interpret that that "GSD has a significant impact on performance for all 3 era values". I hope that understanding is correct? Secondly, under point 3 below, I get different p-values for different values of Ln_GSD - some are significant and some are not. So, I am a bit confused as to how I should interpret that?

                      Here are the 3 scenarios:


                      Scenario 1: Coefficients of Ln_GSD and Ln_GSD squared in the original regression:
                      Code:
                       
                       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.era if  CoAge>=0 & NATION=="UNITED STATES" & NATIONCODE==840 & FSTS>=10 & GENERALINDUSTRYCLASSIFICATION ==1 & Year_<2020 & Year_<YearInactive & Discr_GS_Rev!=1, fe cluster(n_CUSIP)
                      Scenario 2: the 3 p-values of the marginal effects for the 3 era values


                      Code:
                      . margins era, dydx(cL.Ln_GSD)
                      
                      Average marginal effects                        Number of obs     =      1,094
                      Model VCE    : Robust
                      
                      Expression   : Linear prediction, predict()
                      dy/dx w.r.t. : L.Ln_GSD
                      
                      ------------------------------------------------------------------------------
                                   |            Delta-method
                                   |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
                      -------------+----------------------------------------------------------------
                      L.Ln_GSD     |
                               era |
                                1  |  -.5672611   .2109349    -2.69   0.007     -.980686   -.1538362
                                2  |  -.6664523   .2248132    -2.96   0.003    -1.107078   -.2258265
                                3  |   -.631958   .1836315    -3.44   0.001    -.9918691   -.2720469
                      ------------------------------------------------------------------------------


                      Scenario 3: the various p-values I obtain when I run

                      Code:
                       
                       . margins era, dydx(l1.Ln_GSD)  at(l1.Ln_GSD=(-4.839976 (0.3).2830217))  Average marginal effects                        Number of obs     =      1,094 Model VCE    : Robust  Expression   : Linear prediction, predict() dy/dx w.r.t. : L.Ln_GSD  1._at        : L.Ln_GSD        =   -4.839976  2._at        : L.Ln_GSD        =   -4.539976  3._at        : L.Ln_GSD        =   -4.239976  4._at        : L.Ln_GSD        =   -3.939976  5._at        : L.Ln_GSD        =   -3.639976  6._at        : L.Ln_GSD        =   -3.339976  7._at        : L.Ln_GSD        =   -3.039976  8._at        : L.Ln_GSD        =   -2.739976  9._at        : L.Ln_GSD        =   -2.439976  10._at       : L.Ln_GSD        =   -2.139976  11._at       : L.Ln_GSD        =   -1.839976  12._at       : L.Ln_GSD        =   -1.539976  13._at       : L.Ln_GSD        =   -1.239976  14._at       : L.Ln_GSD        =    -.939976  15._at       : L.Ln_GSD        =    -.639976  16._at       : L.Ln_GSD        =    -.339976  17._at       : L.Ln_GSD        =    -.039976  18._at       : L.Ln_GSD        =     .260024  ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+---------------------------------------------------------------- L.Ln_GSD     |      _at#era |         1 1  |   .1289582   .3057765     0.42   0.673    -.4703526    .7282691         1 2  |   .5833692   .5783048     1.01   0.313    -.5500874    1.716826         1 3  |   .1131435   .5338097     0.21   0.832    -.9331043    1.159391         2 1  |   .0817573    .277921     0.29   0.769    -.4629579    .6264724         2 2  |   .4986362   .5371029     0.93   0.353    -.5540661    1.551338         2 3  |   .0626285    .497079     0.13   0.900    -.9116285    1.036886         3 1  |   .0345562   .2507241     0.14   0.890     -.456854    .5259665         3 2  |    .413903   .4963946     0.83   0.404    -.5590125    1.386818         3 3  |   .0121134   .4607087     0.03   0.979     -.890859    .9150858         4 1  |  -.0126448   .2244255    -0.06   0.955    -.4525107    .4272211         4 2  |   .3291698   .4563121     0.72   0.471    -.5651853    1.223525         4 3  |  -.0384016   .4247912    -0.09   0.928    -.8709771    .7941739         5 1  |  -.0598458   .1993809    -0.30   0.764    -.4506252    .3309335         5 2  |   .2444367   .4170358     0.59   0.558    -.5729385    1.061812         5 3  |  -.0889166    .389452    -0.23   0.819    -.8522286    .6743953         6 1  |  -.1070468    .176126    -0.61   0.543    -.4522474    .2381537         6 2  |   .1597036   .3788168     0.42   0.673    -.5827636    .9021708         6 3  |  -.1394317   .3548639    -0.39   0.694    -.8349521    .5560887         7 1  |  -.1542479    .155466    -0.99   0.321    -.4589557    .1504599         7 2  |   .0749704   .3420094     0.22   0.826    -.5953557    .7452965         7 3  |  -.1899468   .3212694    -0.59   0.554    -.8196232    .4397296         8 1  |  -.2014489   .1385666    -1.45   0.146    -.4730345    .0701367         8 2  |  -.0097628   .3071218    -0.03   0.975    -.6117104    .5921849         8 3  |  -.2404618   .2890153    -0.83   0.405    -.8069215    .3259978         9 1  |  -.2486499   .1269387    -1.96   0.050    -.4974452    .0001454         9 2  |  -.0944959   .2748858    -0.34   0.731    -.6332622    .4442704         9 3  |  -.2909768   .2586038    -1.13   0.261    -.7978309    .2158772        10 1  |  -.2958509   .1220978    -2.42   0.015    -.5351582   -.0565437        10 2  |   -.179229   .2463447    -0.73   0.467    -.6620557    .3035976        10 3  |  -.3414919   .2307642    -1.48   0.139    -.7937815    .1107977        11 1  |   -.343052   .1248359    -2.75   0.006    -.5877258   -.0983781        11 2  |  -.2639622    .222922    -1.18   0.236    -.7008813     .172957        11 3  |   -.392007   .2065395    -1.90   0.058    -.7968169     .012803        12 1  |   -.390253   .1346916    -2.90   0.004    -.6542437   -.1262623        12 2  |  -.3486953   .2063682    -1.69   0.091    -.7531695    .0557789        12 3  |   -.442522    .187337    -2.36   0.018    -.8096958   -.0753482        13 1  |   -.437454   .1502709    -2.91   0.004    -.7319796   -.1429284        13 2  |  -.4334284   .1984099    -2.18   0.029    -.8223047   -.0445522        13 3  |   -.493037   .1748198    -2.82   0.005    -.8356775   -.1503966        14 1  |   -.484655   .1700076    -2.85   0.004    -.8178639   -.1514462        14 2  |  -.5181616   .2000755    -2.59   0.010    -.9103024   -.1260209        14 3  |  -.5435521   .1704668    -3.19   0.001    -.8776609   -.2094433        15 1  |  -.5318561   .1926279    -2.76   0.006    -.9093999   -.1543122        15 2  |  -.6028947   .2111373    -2.86   0.004    -1.016716   -.1890732        15 3  |  -.5940671   .1748888    -3.40   0.001    -.9368429   -.2512914        16 1  |  -.5790571    .217233    -2.67   0.008    -1.004826   -.1532883        16 2  |  -.6876279   .2302451    -2.99   0.003      -1.1389   -.2363558        16 3  |  -.6445822   .1874659    -3.44   0.001    -1.012009   -.2771558        17 1  |  -.6262581   .2432211    -2.57   0.010    -1.102963   -.1495535        17 2  |   -.772361   .2556007    -3.02   0.003    -1.273329    -.271393        17 3  |  -.6950972   .2067148    -3.36   0.001    -1.100251   -.2899437        18 1  |  -.6734591   .2701935    -2.49   0.013    -1.203029   -.1438896        18 2  |  -.8570942   .2855445    -3.00   0.003    -1.416751   -.2974373        18 3  |  -.7456123   .2309734    -3.23   0.001    -1.198312   -.2929127 ------------------------------------------------------------------------------


                      Comment


                      • #12
                        Well, I am among those who support the American Statistical Association's recommendation that the concept of statistical significance be abandoned. See https://www.tandfonline.com/doi/full...5.2019.1583913 for the "executive summary" and https://www.tandfonline.com/toc/utas20/73/sup1 for all 43 supporting articles. Or https://www.nature.com/articles/d41586-019-00857-9 for the tl;dr. So I would not ask myself whether the effects in each era are "significant" or not. For those who believe in statistical significance, the answer would be yes, but since I don't know how to make any sense out of what that means, I don't go there.

                        The question I would ask myself is, how big an effect would matter in the real world. And where does that threshold of importance lie with respect to the confidence intervals around those effects. Do the confidence intervals lie entirely on one side of that threshold of importance--if so we can describe these effects as big enough to matter, or too small to matter. If the threshold of importance lies in the confidence interval, then we can say that the data do not tell us whether these effects are large enough to matter or not, although if the threshold is near one end of the confidence interval, we might say that the data lean is in a certain direction.

                        As for the results undelying your third question, this is territory where even those who still like the notion of statistical significance would recognize that it simply doesn't apply here. These calculations show you how the relationship between the marginal effects and L.ln_GSD play out in each era. When working in interaction models like this, these effects are going to vary considerably. Some will be very large, some will be very small. In fact, in principle, it is always mathematically possible to find a value of L.Ln_GSD to give you any pre-specified marginal effect result here--though the corresponding values of L.Ln_GSD may not be realistic. So these numbers just have to be understood in their own right--again, graphing them is probably the most helpful way to see what is going on. But applying statistical significance criteria to them is entirely meaningless, even for those who believe statistical significance can be a useful concept.

                        Comment


                        • #13
                          Hi Clyde, Just to elaborate, I saw a detailed discussion on my scenario 3 here (https://www.statalist.org/forums/for...s-significance).

                          I am not sure I fully understand and hence would like to share the code and marginsplot I obtained for scenario 3 (- the marginsplot is included as an attachment.)

                          The question I am wondering about is that the p-values for certain L.Ln_GSD _at#era are significant and some are not. So, does that mean that the marginal impact of Ln_GSD is significant only in some situations? (I hope my question is worded clearly.)


                          Code:
                          . margins era, dydx(l1.Ln_GSD)  at(l1.Ln_GSD=(-4.839976 (1).2830217))
                          
                          Average marginal effects                        Number of obs     =      1,094
                          Model VCE    : Robust
                          
                          Expression   : Linear prediction, predict()
                          dy/dx w.r.t. : L.Ln_GSD
                          
                          1._at        : L.Ln_GSD        =   -4.839976
                          
                          2._at        : L.Ln_GSD        =   -3.839976
                          
                          3._at        : L.Ln_GSD        =   -2.839976
                          
                          4._at        : L.Ln_GSD        =   -1.839976
                          
                          5._at        : L.Ln_GSD        =    -.839976
                          
                          6._at        : L.Ln_GSD        =     .160024
                          
                          ------------------------------------------------------------------------------
                                       |            Delta-method
                                       |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
                          -------------+----------------------------------------------------------------
                          L.Ln_GSD     |
                               _at#era |
                                  1 1  |   .1289582   .3057765     0.42   0.673    -.4703526    .7282691
                                  1 2  |   .5833692   .5783048     1.01   0.313    -.5500874    1.716826
                                  1 3  |   .1131435   .5338097     0.21   0.832    -.9331043    1.159391
                                  2 1  |  -.0283785    .215916    -0.13   0.895     -.451566    .3948091
                                  2 2  |   .3009255   .4431197     0.68   0.497    -.5675732    1.169424
                                  2 3  |  -.0552399   .4129399    -0.13   0.894    -.8645872    .7541073
                                  3 1  |  -.1857152    .143694    -1.29   0.196    -.4673502    .0959198
                                  3 2  |   .0184817    .318498     0.06   0.954     -.605763    .6427264
                                  3 3  |  -.2236234   .2995907    -0.75   0.455    -.8108104    .3635635
                                  4 1  |   -.343052   .1248359    -2.75   0.006    -.5877258   -.0983781
                                  4 2  |  -.2639622    .222922    -1.18   0.236    -.7008813     .172957
                                  4 3  |   -.392007   .2065395    -1.90   0.058    -.7968169     .012803
                                  5 1  |  -.5003887   .1772789    -2.82   0.005    -.8478491   -.1529284
                                  5 2  |   -.546406   .2027678    -2.69   0.007    -.9438236   -.1489884
                                  5 3  |  -.5603904   .1709716    -3.28   0.001    -.8954887   -.2252922
                                  6 1  |  -.6577255   .2611119    -2.52   0.012    -1.169495   -.1459555
                                  6 2  |  -.8288498   .2751412    -3.01   0.003    -1.368117    -.289583
                                  6 3  |  -.7287739   .2224248    -3.28   0.001    -1.164719   -.2928293
                          ------------------------------------------------------------------------------
                          
                          . marginsplot
                          
                            Variables that uniquely identify margins: L.Ln_GSD era
                          
                          . graph save "Graph" "/Users/deepikadeshpande/Downloads/marginsplot.gph"
                          (file /Users/deepikadeshpande/Downloads/marginsplot.gph saved)
                          Attached Files

                          Comment


                          • #14
                            As I have said, I do not see how one can attach any meaning to the concept of statistical significance in this context. What you see is that the marginal effect declines as L.Ln_GSD increases. But the lines, in theory, extend all the way to infinity at both the left and the right. So you can always find values of L.Ln_GSD that will give you a large positive marginal effect, other values that will give you an effect close to zero, other values that will give you an effect that is negative. That is exactly what it means to use a quadratic model. In your graph you have, presumably, chosen a range of values of L.Ln_GSD to look at that is meaningful in the real world--these are values that occur in real life under normal, and perhaps slightly extraordinary, circumstances. So what you see is that over the range of meaningful values of L.Ln_GSD, the marginal effects range from about +0.5 down to about -0.5 in era 2. In eras 1 and 3, they start out pretty close to zero, and soon become negative, ending up at about -0.4 on the right side of the graph. But in all three eras, the marginal effect decreases as L.Ln_GSD increases, more steeply in era 2 than in the other eras. The fact that some are "significant" and some are not doesn't mean anything at all--that will always be the case in any quadratic model.

                            It is fair to say that the confidence intervals are pretty wide here, so the data are not giving you a really precise handle on these marginal effects. Only at the right hand end of the graph do they fall clearly into negative territory for all three eras. So if the negative vs positive distinction is important in real world terms here, you can say that the data give clearly negative estimates at higher values of L.Ln_GSD, and are more ambiguous for lower values, say, below about -1.84
                            Last edited by Clyde Schechter; 31 Jul 2021, 17:27.

                            Comment


                            • #15
                              Thanks Clyde. Noted. I will ignore the p-values pertaining to the various scenarios and Ln_GSD values.

                              Sorry for asking another related question. If I need to answer the question "Does Ln_GSD have a significant marginal effect on performance" for the 3 era values, should I instead refer to the p-values in the below table? (- I think these are computed at the average levels.)

                              Code:
                              . margins era, dydx(l1.Ln_GSD) 
                              
                              Average marginal effects                        Number of obs     =      1,094
                              Model VCE    : Robust
                              
                              Expression   : Linear prediction, predict()
                              dy/dx w.r.t. : L.Ln_GSD
                              
                              ------------------------------------------------------------------------------
                                           |            Delta-method
                                           |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
                              -------------+----------------------------------------------------------------
                              L.Ln_GSD     |
                                       era |
                                        1  |  -.5672611   .2109349    -2.69   0.007     -.980686   -.1538362
                                        2  |  -.6664523   .2248132    -2.96   0.003    -1.107078   -.2258265
                                        3  |   -.631958   .1836315    -3.44   0.001    -.9918691   -.2720469
                              ------------------------------------------------------------------------------

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