Turning point and graphs in culvilinear relationship

[Zeinab Ghamloush]

Hello,

I am trying to examine the effect of X on Y. According to the literature, i tried to test the existence of a culvilinear relationship between the two variables. I ran a GLS regression using X and its square as independent variables with a set of control variables. I found that Y is significantly and negatively affected by X (the coefficient of X is significant and negative = -0.5396), and significantly and positively affected by X square (the the coefficient of X square is significant and positive = 2.7843). According to my result, I found that the turning point is 0.5396/(2*2.7843)= 0.0969.

I have two questions:
1- While trying to plot the curve on Stata, I found that it's totally linear. I tried to plot it using : margins, at (X = (value1 value2 value3...) then marginsplot. I also tried to plot it using the predicted values of Y : twoway qfit Yhat X. However, I always find a linear relation. Is this logic?
2- I also examined the moderating effect of a variable Z, so I included the interaction term between X and Z. Its coefficient is statistically significant. The coefficient of X became -0.4405 and the one of X square became 2.8031. In this case, Can I say that the variable Z moderate the culvilinear relation between X and Y in such a way that the turning point occur at 0.4405/(2*2.8031)= 0.0784?

Your help and support are very appreciated!

Thanks in advance
Zeinab

[Nick Cox]

As there are other predictors present, you have a prediction surface that has more dimensions than a curve. If I understand correctly what you have done, your turning point calculation hinges on the other predictors all being zero.

[Nick Cox]

Picking this up again, a bundle of small comments. Here X is the variable you call by that name.

1. It does not really help to focus on the effect of X --as if its square were held constant -- or conversely on the effect of its square -- as if X were held constant, because the thought experiment makes little sense, as X and its square are yoked together by definition.

2. What is the range of X? A simple graph such as

twoway function -0.5396 * x + 2.7843*x^2

is often indispensable as showing what is going on (always, as in my previous, noting that control variables complicate the issue). In the command above, (lower case) x is generic to twoway function. But you should draw the graph over the observed range of your X

twoway function -0.5396 * x + 2.7843*x^2, range(X)

where in the range() option you should use the variable name actually used it was not X (but again lower case x is understood by Stata outside the options).

a
3. It is quite common that fitting a quadratic "improves" on fitting a straight line, as judged by P-values, but also that one or more of these statements hold:

a. The turning point may be outside the range of the data

b. Over (most of) the observed range, the relationship is monotonic, or nearly so

c. Over (much of) the observed range, the relationship is linear, or nearly so

In short, a quadratic may look good in terms of P-values but it can just be capitalising on curvature yet not validating a turning point.

3. All this may be of little relevance depending on the role of the control variables.

[Zeinab Ghamloush]

Dear Nick,

Thank you for your answer!
My independent variable X is ranged between 0 and 0.253. When I calculate the turning point, I get 0.096 which is inside the range of the data. My full sample consisted of 1407 firm year observations. Among these observations, 1376 have an X below 0.096 and 31 above 0.096. Maybe this means that "over (most of) the observed range, the relationship is monotonic, OR over (much of) the observed range, the relationship is linear". If I try to run two separate regressions, (the first if X<0.096 (1376 observations) and the second if x>0.096 (31 observations)), I get in the first one a negative linear relationship between X and Y, meanwhile in the second one, a positive linear relationship between X and Y.

Now my question is: I need to examine the moderating effect of Z. I already get a significant value of the interaction term. Can I interpret the results of a moderator variable just according to the regression results without being able to plot the relationship and to show how the curve changes according to Z?

Thank you in advance,
Zeinab

[Clyde Schechter]

I'd like to point out that O.P. has never shown us her code for the regression. If she did something like this:

regress Y X X2... // WHERE X2 IS SOME VARIABLE CALCULATED AS X^2
margins, at(X = (0(0.025)0.25)) // OR SOME SIMILAR NUMBERS
marginsplot

then she will, of course, get only a straight-line plot because the -regress- command itself is not suitable for use with -margins-.

She does not show the plot she got: she describes it as linear. If it is actually a perfect straight line, then I would bet that this is the mistake she has made. Because if it were done correctly:

regress Y c.X##c.X... // NO HOMEBREW VARIABLE FOR X^2
margins, at(X = (0(0.025)0.25)) // OR SOME SIMILAR NUMBERS
marginsplot

the graph she gets would have some noticeable curvature and, given that the turning point is clearly inside the data range, there would be a clear turning point. The curvature might be diminished, depending on the aspect ratio of the graph, but no reasonable aspect ratio or axis limitations would make this curvature invisible.

[Nick Cox]

I can't advise well on #4. I will just comment that taking a relationship like this and then jumping to two separate regressions is indeed a big jump. If I were reviewing this, I would want to see a really good substantive story by way of justification.

[Zeinab Ghamloush]

Thank you for your answers, they were really helpful!

My problem was in writing the code (I included X2 calculated as X^2 instead of c.X##c.X). Now I have obtained the noticeable curvature in my graph.
I have another question regarding the moderator effect Z. I should also include it as c.X##c.X##c.Z ?
Actually in my model, I am examining the effect of employee ownership on the debt maturity structure (LTD_W). And the effect of the proportion of employee directors (moderator variable) on this relationship. My code is the following:

xtgls LTD_W c.Employee_Ownership##c.Employee_Ownership##c.Employee_dir LEV_W AM_W MTB_W  Volatility_W F_SIZE_W Liquidity_W ABearnings_W Z_W Ownership SUPERVISORY DUALITY B_IND B_SIZE Managerial_Own i.year i.Sector, panels (hetero) corr(psar1)

I obtained the results bellow:

Cross-sectional time-series FGLS regression

Coefficients:  generalized least squares
Panels:        heteroskedastic
Correlation:   panel-specific AR(1)

Estimated covariances      =       239          Number of obs     =      1,407
Estimated autocorrelations =       239          Number of groups  =        239
Estimated coefficients     =        42          Obs per group:
                                                              min =          3
                                                              avg =   5.887029
                                                              max =          6
                                                Wald chi2(41)     =    3276.25
                                                Prob > chi2       =     0.0000

--------------------------------------------------------------------------------------------------------------------------
                                                   LTD_W | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
---------------------------------------------------------+----------------------------------------------------------------
                                      Employee_Ownership |     -.3182   .1873992    -1.70   0.090    -.6854956    .0490957
                                                         |
               c.Employee_Ownership#c.Employee_Ownership |   2.045527    .925424     2.21   0.027     .2317298    3.859325
                                                         |
                                            Employee_dir |   .0756054   .0440854     1.71   0.086    -.0108004    .1620111
                                                         |
                     c.Employee_Ownership#c.Employee_dir |  -5.046808   1.758768    -2.87   0.004     -8.49393   -1.599686
                                                         |
c.Employee_Ownership#c.Employee_Ownership#c.Employee_dir |   12.41442   9.271574     1.34   0.181    -5.757534    30.58637
                                                         |
                                                LEV_What |   2.590985   .1214419    21.34   0.000     2.352963    2.829007
                                                    AM_W |  -.0000488   .0000552    -0.88   0.377     -.000157    .0000594
                                                   MTB_W |   .0104174   .0051531     2.02   0.043     .0003175    .0205173
                                            Volatility_W |   .2497699   .0411905     6.06   0.000     .1690379    .3305019
                                                F_SIZE_W |   .1228887   .0052074    23.60   0.000     .1126824     .133095
                                             Liquidity_W |    .120484   .0121586     9.91   0.000     .0966536    .1443144
                                            ABearnings_W |  -.0595357   .0091098    -6.54   0.000    -.0773905   -.0416809
                                                     Z_W |  -.0236213   .0015277   -15.46   0.000    -.0266155   -.0206271
                                               Ownership |  -.0071207    .005666    -1.26   0.209    -.0182258    .0039844
                                             SUPERVISORY |  -.0171182   .0037587    -4.55   0.000    -.0244851   -.0097513
                                                 DUALITY |   .0016898   .0029385     0.58   0.565    -.0040696    .0074491
                                                   B_IND |   .0089475   .0084308     1.06   0.289    -.0075765    .0254716
                                                  B_SIZE |    .059591   .0166731     3.57   0.000     .0269123    .0922696
                                          Managerial_Own |  -.0613521   .0116685    -5.26   0.000    -.0842219   -.0384823
                                                         |
                                                    year |
                                                   2015  |   .0103564   .0028996     3.57   0.000     .0046733    .0160395
                                                   2016  |   .0085879   .0035455     2.42   0.015     .0016388    .0155371
                                                   2017  |   .0223616   .0042661     5.24   0.000     .0140002     .030723
                                                   2018  |   .0178842   .0043315     4.13   0.000     .0093945    .0263738
                                                   2019  |   .0001847   .0047865     0.04   0.969    -.0091966     .009566
                                                         |
                                                  Sector |
                           Automobiles & Equipementiers  |  -.0800467   .0106635    -7.51   0.000    -.1009468   -.0591467
              Biens d'équipement & produits industriels  |   .1046952   .0125978     8.31   0.000      .080004    .1293864
                                  Biens de consommation  |   .1853599   .0211858     8.75   0.000     .1438365    .2268834
                                                 Chimie  |    .334314   .0272919    12.25   0.000     .2808228    .3878052
                                  Communication & Média  |  -.1995788   .0166985   -11.95   0.000    -.2323073   -.1668503
                                           Construction  |  -.2363045   .0186244   -12.69   0.000    -.2728077   -.1998012
                                           Distribution  |   .1198596   .0149861     8.00   0.000     .0904874    .1492317
                 Energie conventionnelle & Renouvelable  |   .0404508   .0175385     2.31   0.021      .006076    .0748256
                       Environnement - Biens & Services  |  -.1455987   .0268439    -5.42   0.000    -.1982117   -.0929857
                                     Matières premières  |   .2155741    .019057    11.31   0.000     .1782231    .2529252
                                          Pétrole & Gaz  |    .130032   .0884563     1.47   0.142    -.0433392    .3034033
                                                  Santé  |   .1542059   .0129245    11.93   0.000     .1288744    .1795375
                               Services aux entreprises  |   .0022355   .0119162     0.19   0.851    -.0211198    .0255908
                              Services aux particuliers  |  -.0369837   .0330563    -1.12   0.263    -.1017729    .0278055
                                           Technologies  |   .1381879    .016627     8.31   0.000     .1055996    .1707763
                                             Transports  |  -.1780745    .015963   -11.16   0.000    -.2093613   -.1467876
                                     Télécommunications  |   .0003592   .0391623     0.01   0.993    -.0763976     .077116
                                                         |
                                                   _cons |  -2.281158    .094301   -24.19   0.000    -2.465984   -2.096331
--------------------------------------------------------------------------------------------------------------------------

As you can see, the coefficient of the first interaction term is statistically significant, however the second is not. Can I say that is this case the "employee directors" moderate the relationship between employee ownership and debt maturity (even though the second interaction term is not significant??
I also tried to plot this relation at a different levels of board employee representation using the following code:

margins, at (Employee_Ownership=(0(0.025)0.25) Employee_dir= (0 0 0.076 0.4)) level (90)
marginsplot, xsize(13) ysize(8) scale(1) noci noseparator recast(line) plotopts(lwidth(medthick) lcolor(black) cmissing(n)  connect(direct)) plot1opts( lpattern(dot)) plot2opts(lpattern(dash_dot_dot)) plot3opts(lpattern(dash)) plotregion(fcolor(white)) graphregion(fcolor(white)) xtitle("Employee Ownership", size(medsmall) margin(medium)) ytitle("Debt maturity structure", size(medsmall)  margin(medium)) legend(order(1 "Minimum" 2 "Standard deviation" 3 "Maximum") size(medsmall) rows(4) position(3) title(Level of board employee representation, size(medsmall) color(black))) clegend(region(fcolor(white) lcolor(white)))

I obtained the attached graph.

Is all the above right? Or should I run another regression using only the interaction between employee directors and employee ownership without the interaction between its square?

[Clyde Schechter]

It looks like you are getting it right now.

As for the significance of the coefficient of c.Employee_Ownership#c.Employee_dir but not c.Employee_Ownership#c.Employee_Ownership#c.Employ ee_dir, you should disregard both of those findings. When you introduce a quadratic into a model, it is incorrect to, under any circumstances, interpret either the linear term or the quadratic term separately. Any testing must test them jointly.

One suggestion for your graph. The label "standard deviation" confuses me. What does that refer to? Is that when employee directors is 1 standard deviation above its mean, or 1 standard deviation below its mean? Or something else? And why not include on the graph a curve for when employee directors is at its mean (or median if you prefer)?

[Zeinab Ghamloush]

Thank you Clyde for your answer,
The label "standard deviation" refers to 1 standard deviation above its mean.

"As for the significance of the coefficient of c.Employee_Ownership#c.Employee_dir but not c.Employee_Ownership#c.Employee_Ownership#c.Employ ee_dir, you should disregard both of those findings. When you introduce a quadratic into a model, it is incorrect to, under any circumstances, interpret either the linear term or the quadratic term separately. Any testing must test them jointly." Since I have to disregard both so i can not tell that employee_dir moderates the relation between employee ownership and debt maturity. Now how about introducing only the linear term (c.Employee_Ownership#c.Employee_dir) in my regression? As follows:

xtgls LTD_W c.Employee_Ownership##c.Employee_Ownership c.Employee_Ownership##c.Employee_dir LEV_What AM_W MTB_W  Volatility_W F_SIZE_W Liquidity_W ABearnings_W Z_W Ownership SUPERVISORY DUALITY B_IND B_SIZE Managerial_Own i.year i.Sector, panels (hetero) corr(psar1) force

Here's the results I got:

Cross-sectional time-series FGLS regression

Coefficients:  generalized least squares
Panels:        heteroskedastic
Correlation:   panel-specific AR(1)

Estimated covariances      =       239          Number of obs     =      1,407
Estimated autocorrelations =       239          Number of groups  =        239
Estimated coefficients     =        41          Obs per group:
                                                              min =          3
                                                              avg =   5.887029
                                                              max =          6
                                                Wald chi2(40)     =    3288.55
                                                Prob > chi2       =     0.0000

------------------------------------------------------------------------------------------------------------
                                     LTD_W | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------------------------------------+----------------------------------------------------------------
                        Employee_Ownership |  -.4405077   .1800094    -2.45   0.014    -.7933197   -.0876956
                                           |
 c.Employee_Ownership#c.Employee_Ownership |   2.803077   .8882993     3.16   0.002     1.062042    4.544111
                                           |
                        Employee_Ownership |          0  (omitted)
                              Employee_dir |   .0473646   .0395023     1.20   0.231    -.0300585    .1247876
                                           |
       c.Employee_Ownership#c.Employee_dir |  -2.819888   .8519246    -3.31   0.001    -4.489629   -1.150146
                                           |
                                  LEV_What |   2.586156    .122125    21.18   0.000     2.346796    2.825517
                                      AM_W |  -.0000476   .0000552    -0.86   0.389    -.0001559    .0000607
                                     MTB_W |   .0100306   .0051539     1.95   0.052    -.0000709     .020132
                              Volatility_W |   .2462497   .0411728     5.98   0.000     .1655524    .3269469
                                  F_SIZE_W |   .1223141   .0052054    23.50   0.000     .1121117    .1325166
                               Liquidity_W |   .1184774   .0120563     9.83   0.000     .0948476    .1421073
                              ABearnings_W |  -.0597277   .0090865    -6.57   0.000    -.0775369   -.0419185
                                       Z_W |  -.0235261   .0015262   -15.42   0.000    -.0265174   -.0205349
                                 Ownership |  -.0072356   .0056014    -1.29   0.196    -.0182141    .0037429
                               SUPERVISORY |  -.0165596   .0037546    -4.41   0.000    -.0239184   -.0092007
                                   DUALITY |    .001663   .0029198     0.57   0.569    -.0040597    .0073857
                                     B_IND |     .00914   .0084119     1.09   0.277     -.007347    .0256271
                                    B_SIZE |   .0636612   .0165556     3.85   0.000     .0312128    .0961095
                            Managerial_Own |  -.0621128   .0116235    -5.34   0.000    -.0848945   -.0393311
                                           |
                                      year |
                                     2015  |   .0103537   .0029082     3.56   0.000     .0046537    .0160537
                                     2016  |   .0083379     .00355     2.35   0.019     .0013802    .0152957
                                     2017  |   .0220558   .0042767     5.16   0.000     .0136736    .0304381
                                     2018  |   .0178388   .0043327     4.12   0.000     .0093468    .0263308
                                     2019  |   .0000832   .0047859     0.02   0.986    -.0092969    .0094633
                                           |
                                    Sector |
             Automobiles & Equipementiers  |  -.0797492   .0106822    -7.47   0.000    -.1006859   -.0588125
Biens d'équipement & produits industriels  |   .1048715   .0125479     8.36   0.000      .080278    .1294649
                    Biens de consommation  |   .1861407   .0211326     8.81   0.000     .1447215    .2275599
                                   Chimie  |    .332743   .0278412    11.95   0.000     .2781754    .3873107
                    Communication & Média  |   -.199519   .0166995   -11.95   0.000    -.2322494   -.1667886
                             Construction  |  -.2345983   .0186593   -12.57   0.000    -.2711699   -.1980267
                             Distribution  |   .1184515    .015268     7.76   0.000     .0885268    .1483762
   Energie conventionnelle & Renouvelable  |   .0364963   .0167483     2.18   0.029     .0036702    .0693224
         Environnement - Biens & Services  |   -.147007   .0268181    -5.48   0.000    -.1995696   -.0944444
                       Matières premières  |   .2192051   .0187856    11.67   0.000      .182386    .2560243
                            Pétrole & Gaz  |   .1317676   .0891258     1.48   0.139    -.0429159     .306451
                                    Santé  |    .155168   .0129003    12.03   0.000     .1298839    .1804521
                 Services aux entreprises  |   .0013932   .0118482     0.12   0.906    -.0218288    .0246153
                Services aux particuliers  |  -.0366532   .0330473    -1.11   0.267    -.1014247    .0281184
                             Technologies  |   .1387526    .016597     8.36   0.000     .1062231     .171282
                               Transports  |  -.1775142   .0166866   -10.64   0.000    -.2102193    -.144809
                       Télécommunications  |   .0066068   .0368567     0.18   0.858     -.065631    .0788446
                                           |
                                     _cons |  -2.275307   .0945305   -24.07   0.000    -2.460583    -2.09003
------------------------------------------------------------------------------------------------------------

Is it correct?

[Maarten Buis]

Since I have to disregard both so i can not tell that employee_dir moderates the relation between employee ownership and debt maturity.

That is not what Clyde said. The important part is: "Any testing must test them jointly."