inverted U-shape combined with moderation

[Jana Schue]

Hello everyone,

for the analysis of my data I identified a inverted U-shape as suitable for my model. In this analysis q_tot_w is my depenent variable and rdallinances is my indepeneent variable. All others are controles.

Code:

xtreg q_tot_w rdalliances rdalliances_2 rdi_w adi_w ln_emp1 i.fyear, fe vce(robust)

Results:

Fixed-effects (within) regression               Number of obs     =    115,742
Group variable: gvkey                           Number of groups  =     12,206

R-squared:                                      Obs per group:
     Within  = 0.0110                                         min =          1
     Between = 0.0031                                         avg =        9.5
     Overall = 0.0060                                         max =         29

                                                F(33,12205)       =      22.01
corr(u_i, Xb) = -0.0039                         Prob > F          =     0.0000

                              (Std. err. adjusted for 12,206 clusters in gvkey)
-------------------------------------------------------------------------------
              |               Robust
      q_tot_w | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
  rdalliances |   .0747057   .0315854     2.37   0.018     .0127934     .136618
rdalliances_2 |  -.0019935   .0006235    -3.20   0.001    -.0032157   -.0007714
        rdi_w |    .016515   .0201425     0.82   0.412    -.0229675    .0559974
        adi_w |   .3026396   1.067127     0.28   0.777    -1.789098    2.394378
      ln_emp1 |   -.100483   .0419701    -2.39   0.017    -.1827509    -.018215
              |
        fyear |
        1990  |  -.0357344   .1047133    -0.34   0.733     -.240989    .1695203
        1991  |   .2004181   .1113381     1.80   0.072    -.0178221    .4186584
        1992  |    .302907   .1146206     2.64   0.008     .0782324    .5275816
        1993  |   .4831456   .1191374     4.06   0.000     .2496174    .7166738
        1994  |   .1656343   .1182963     1.40   0.161    -.0662451    .3975137
        1995  |   .4850309   .1265344     3.83   0.000     .2370033    .7330584
        1996  |   .4427113   .1277652     3.47   0.001     .1922714    .6931513
        1997  |    .599052   .1404002     4.27   0.000     .3238454    .8742585
        1998  |   .4566188   .1450601     3.15   0.002     .1722781    .7409595
        1999  |   .8430929   .1569679     5.37   0.000     .5354109    1.150775
        2000  |   .4531411   .1462793     3.10   0.002     .1664105    .7398716
        2001  |   .2124873   .1421871     1.49   0.135     -.066222    .4911965
        2002  |   -.225915   .1417812    -1.59   0.111    -.5038287    .0519986
        2003  |   .2964072   .1437541     2.06   0.039     .0146263    .5781881
        2004  |   .4820569   .1420315     3.39   0.001     .2036527    .7604611
        2005  |   .4623861    .142104     3.25   0.001     .1838398    .7409323
        2006  |   .4261785   .1401344     3.04   0.002     .1514928    .7008642
        2007  |     .27914   .1430544     1.95   0.051    -.0012693    .5595494
        2008  |  -.5448981   .1412114    -3.86   0.000    -.8216948   -.2681014
        2009  |  -.2937102   .1423348    -2.06   0.039    -.5727089   -.0147116
        2010  |  -.1315857   .1405104    -0.94   0.349    -.4070083     .143837
        2011  |  -.2652228   .1419427    -1.87   0.062     -.543453    .0130073
        2012  |  -.2629306   .1425723    -1.84   0.065    -.5423949    .0165338
        2013  |   .1241235   .1466388     0.85   0.397    -.1633118    .4115588
        2014  |   .0497599   .1461983     0.34   0.734     -.236812    .3363318
        2015  |  -.1963734   .1449494    -1.35   0.176    -.4804972    .0877504
        2016  |  -.0650551   .1526762    -0.43   0.670    -.3643246    .2342144
        2017  |   .0982877   .1625534     0.60   0.545    -.2203428    .4169182
              |
        _cons |   1.910016   .3591531     5.32   0.000     1.206019    2.614013
--------------+----------------------------------------------------------------
      sigma_u |  5.7788823
      sigma_e |  3.0242235
          rho |  .78501117   (fraction of variance due to u_i)
-------------------------------------------------------------------------------

Now I want to implement a moderation into this model, but I am not shure how to do this. After some research I found different approaches for linear regressions but it's hard for me to transfer this knowledge to a inverted U-shaped model. "rdalliances" is my independent variable and "munificence" is my moderator.

This is my acutal idea about the code, but I am really not shure about this and my results are not like I expected them. Probably I made a misetake.

xtreg q_tot_w cl.rdalliances##cl.rdalliances##cl.munificence_w rdi_w adi_w ln_emp1 , fe vce(robust)

Can anyone confirm the code or help me with the correct one?

Best regards,
Jana

[Clyde Schechter]

Well, the code you propose in #1 and it is very close to what you say you want. But why are you using a cl.prefix instead of just c.? You didn't say anything in your description of the problem to suggest that you need to use lagged variables, and you did not do so in the original, unmoderated, model. So why would you introduce lags here? I also notice that in the original model you include i.fyear, but you do not include it in the proposed moderated model. Again, absent some specific reason for making that change (e.g. if munificence is colinear with the fyear indicators), if fyear belonged in the original model it should be kept in the moderated model.

I think what you want is actually simpler:

xtreg q_tot_w c.rdalliances##c.rdalliances##c.munificence_w rdi_w adi_w ln_emp1 i.fyear , fe vce(robust)

[Jana Schue]

Clyde:
thanks for your quick and helpful answer!
You are right the lagged version should not be included.

I am pretty new to an analysis with moderators. Maybe you could answer me a second question?

Code:

xtreg q_tot_w c.rdalliances##c.rdalliances##c.munificence_w rdi_w adi_w ln_emp1 , fe vce(robust)

Results:

 xtreg q_tot_w c.rdalliances##c.rdalliances##c.munificence_w rdi_w adi_w ln_emp1 , fe vce(robu
> st)

Fixed-effects (within) regression               Number of obs     =    115,742
Group variable: gvkey                           Number of groups  =     12,206

R-squared:                                      Obs per group:
     Within  = 0.0025                                         min =          1
     Between = 0.0121                                         avg =        9.5
     Overall = 0.0093                                         max =         29

                                                F(8,12205)        =       6.98
corr(u_i, Xb) = 0.0310                          Prob > F          =     0.0000

                                              (Std. err. adjusted for 12,206 clusters in gvkey)
-----------------------------------------------------------------------------------------------
                              |               Robust
                      q_tot_w | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
------------------------------+----------------------------------------------------------------
                  rdalliances |  -.0118459   .0352631    -0.34   0.737    -.0809673    .0572754
                              |
  c.rdalliances#c.rdalliances |   -.000316   .0008218    -0.38   0.701    -.0019268    .0012948
                              |
                munificence_w |   .1201944   .0255507     4.70   0.000      .070111    .1702779
                              |
c.rdalliances#c.munificence_w |     .15641   .0520391     3.01   0.003     .0544051    .2584149
                              |
  c.rdalliances#c.rdalliances#|
              c.munificence_w |  -.0029884   .0018257    -1.64   0.102    -.0065671    .0005903
                              |
                        rdi_w |   .0200534    .020213     0.99   0.321    -.0195673    .0596742
                        adi_w |   .1926644   1.073564     0.18   0.858     -1.91169    2.297019
                      ln_emp1 |  -.1323882   .0432347    -3.06   0.002    -.2171352   -.0476413
                        _cons |   2.233968   .2977856     7.50   0.000     1.650261    2.817675
------------------------------+----------------------------------------------------------------
                      sigma_u |  5.7599915
                      sigma_e |  3.0368196
                          rho |   .7824924   (fraction of variance due to u_i)
-----------------------------------------------------------------------------------------------

To make a statement about the moderation in the inverted U-shaped model I only fokus on the 5th row of the results (here coefficient -.0029884 and p-value 0.102 )? Is this correct? Or do I also need some other rows to interpret if there is a moderation effect?
So in this case I could not confirm that there is a moderation effect since the p-value is bigger than 0.05 (I test on 5% significance niveau).


Best regards,
Jana

[Maarten Buis]

If you have an inverted U-shape (or I would just call it a quadratic) then you no longer have one effect of rdallinances, but many. Whether those many effects change rdallinances when munificence_w changes can no longer have a simple yes or no. You could test whether both the 4th and 5th row are jointly significant. Better yet, just look at a graph and rely on the much more important interoccular trauma test^*): http://maartenbuis.nl/wp/inter_quadr/inter_quadr.html

^*) inter = between, occular = eyes, interoccular trauma = it hits you between the eyes: If you look at a graph and the pattern is so obvious that it hits you between the eyes, then it is there and worth talking about and it has passed the interoccular trauma test. If it is not that obvious then it failed the interoccular trauma test. Just using a good dose of common sense is often more useful when answering statistical questions than fancy tests.