interpreting interaction effects continuous by categorical variables

[Alex Morgan]

Hi all,

I am having difficulty to interpret interaction effects. I watched some videos and read forums but I am still confused.
Here is my model: Gender inequality= B0+ B1*mosque attendance + B2*gender + B12gendermosqueattendance
Mosque attendance is 0-6 which is the frequency of never attendance to daily attendance, gender is female=0 and male=1. Gender inequality scale is also continuous, so OLS is applied.
Can someone explain and interpret the coefficients of mosque attendance, gender, and interaction. What do those mean?Is there a sole effect of mosque attendance in this model? How do we interpret male and female effects separately? I appreciate if you could be as specific as possible.
Thank you for your time.

[Clyde Schechter]

In your notation:

B1 is the marginal effect of mosque attendance on gender inequality (slope of the gender inequality: mosque attendance relationship) among females (gender = 0).

B2 is the marginal effect of gender on gender inequality (difference in gender inequality between males and females) when mosque attendance = 0

B12 is the difference between the marginal effect of mosque attendance on gender inequality among males and the marginal effect of mosque attenadance on gender inequality among females.

To get the marginal effect of mosque attendance on gender inequality among males, you must add B1 + B12.

The simple way to do this in Stata is to use factor variable notation when you run the regression, and then have the -margins- command produce the separate effects of mosque attendance for males and females.

regress gender_inequality i.gender##c.mosque_attendance
margins gender, dydx(mosque_attendance)

Note that in an interaction model there is no single effect of mosque attendance. There is an effect of mosque attendance for males, and another effect of mosque attendance for females.

For a more general understanding of interactions and the use of the -margins- command I recommend the excellent Richard Williams' https://www3.nd.edu/~rwilliam/stats/Margins01.pdf. It contains worked examples that are similar to yours.

[Alex Morgan]

Thank you very much, sir. This is very helpful.

[Alex Morgan]

Hi again Dr Schechter,

In the model above, I have significant interaction effect (B12= -.12, p<.05). But the effect of B1 is not significant (B1= -.01, not significant). To get the marginal effect of mosque attendance on gender inequality among males, I must add B1+B12. But here, one is significant and the other one is not. It might be other way around, B1 might be significant and B12 might not. So in these cases, how are we going to interpret the results?
Thank you for your time.

[Clyde Schechter]

In general, the American Statistical Association has recommended that the concept of statistical significance and the distinction of p < 0.05 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.

That said, even if you want to continue to work in the statistical significance framework, it is entirely irrelevant and misleading in this context; it was never reasonable to look at interaction interpretation from that perspective. The -margins-output from the commands I suggested in #2 will show you the marginal effects of mosque attendance for each gender. It may well be the case that one of them is "statistically significant" and the other is not. So what? You should report the marginal effects themselves along with their confidence intervals, and, if you wish, their p-value.

[Alex Morgan]

Thank you so much. Happy Christmas.

[Doug Hemken]

Alex,

This suggests that there is not much evidence that mosque attendance has an effect on the outcome measure among females (gender = 0) (using Clyde's emphasis). If you graph this relationship among females it will look like a flat line.

However, you appear to have some evidence that increasing mosque attendance has a negative effect on the outcome measure among males. If you graph this relationship, it will trend down to the right.

You can visualize this with marginsplot after margins.

regress gender_inequality i.gender##c.mosque_attendance
margins gender, at(mosque_attendance=(0/6))
marginsplot

Clyde's suggestion to report marginal effects is, I think, the clearest way to convey the different slopes, but it has the one drawback that it doesn't put the outcomes in relation to each other across genders.

Because this is a simple model, it is worth noting that this same model could be expressed in another, mathematically equivalent, way:

regress gender_inequality i.gender i.gender#c.mosque_attendance

which you can follow with the very same margins and marginsplot commands, and see the equivalence visually. I mention this to make the point that often you should not be too focused on the individual parameters. It is the collection of parameters that describes the model, and they work together. Here, one collection is mathematically just as good as the other, but one is probably going to be easier for you to think about and report. Margins are a very flexible tool for getting answers to questions that involve more than one mathematical parameter.

[Alex Morgan]

Thank you very much for your time.

[Ethan Kilmer]

Hello all,

I am new to Stata and having troubles interpreting the results of an FE regression including an interaction term (categorical by continuous). I selected FE based on the results of Hausman and Mundlak.

For some background, I am assessing the impact of ESG disclosure scores on the cost of capital (WACC is my dependent). I am attempting to identify whether the relationship is more important for firms domiciled in countries with stronger or weaker legal systems. The legal system variable is a dummy, and represents the categorical variable of my interaction term (coded as 0 for weak legal system and 1 for strong). ESG is the continuous variable of my interaction term.

Interestingly, within my baseline regression (excluding the interaction term), ESG has no significant impact on the cost of capital (very high p value). However, with the interaction term included, it seems it does have an impact (see below)... When interpreting the results below, I am confused whether ESG is significant for companies within stronger legal systems (1 of dummy), or if it's significant at all. I would appreciate if you could provide some insight.

Note: I include various control variables which were selected based on my literature review.

regression 1 (without interaction):

xtreg WACC ESG ZSCORE ASG INTCOV CUR LEV PTB ROA TQ BT SIGMA INFL ln_SIZE ln_GDPC COVID i.qdate i.INDUSTRYID i.COUNTRYID, fe vce(cluster ID)
Fixed-effects (within) regression               Number of obs     =      1,019
Group variable: ID                              Number of groups  =         51

R-squared:                                      Obs per group:
     Within  = 0.3404                                         min =         19
     Between = 0.0892                                         avg =       20.0
     Overall = 0.1525                                         max =         20

                                                F(34, 50)         =    1580.21
corr(u_i, Xb) = -0.6522                         Prob > F          =     0.0000

                                    (Std. err. adjusted for 51 clusters in ID)
------------------------------------------------------------------------------
             |               Robust
        WACC | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         ESG |   .0687844   .0563095     1.22   0.228    -.0443165    .1818853
      ZSCORE |    .049747   .0752585     0.66   0.512    -.1014142    .2009082
         ASG |  -.0037423   .0021094    -1.77   0.082    -.0079792    .0004945
      INTCOV |   9.52e-06   1.19e-06     8.02   0.000     7.13e-06    .0000119
         CUR |  -1.103148   1.077268    -1.02   0.311    -3.266904    1.060607
         LEV |  -.1792418   .0924329    -1.94   0.058    -.3648988    .0064152
         PTB |   .1055195   .1391388     0.76   0.452     -.173949     .384988
         ROA |  -.0673705   .0556005    -1.21   0.231    -.1790475    .0443065
          TQ |  -.0484075   .3082389    -0.16   0.876    -.6675236    .5707086
          BT |   .6335186   .4025818     1.57   0.122    -.1750906    1.442128
       SIGMA |   23.94318   16.64045     1.44   0.156    -9.480154    57.36651
        INFL |  -.0159213   .0750541    -0.21   0.833    -.1666719    .1348293
     ln_SIZE |   .0162486   1.345873     0.01   0.990    -2.687016    2.719513
     ln_GDPC |  -5.593552   3.587193    -1.56   0.125    -12.79864    1.611537
       COVID |  -.0000624   .0000542    -1.15   0.256    -.0001713    .0000466
             |
       qdate |
        233  |   .0287675   .6573977     0.04   0.965    -1.291655     1.34919
        234  |   .2091584   .4206399     0.50   0.621    -.6357217    1.054039
        235  |   1.013874   .6674544     1.52   0.135    -.3267474    2.354496
        236  |  -.5172944   .6126361    -0.84   0.402     -1.74781    .7132215
        237  |   .1109379    .651019     0.17   0.865    -1.196672    1.418548
        238  |    .416325   .8141085     0.51   0.611     -1.21886     2.05151
        239  |   .9111516   1.018124     0.89   0.375     -1.13381    2.956113
        240  |  -3.919194   2.148949    -1.82   0.074    -8.235485    .3970975
        241  |  -1.766806   2.538735    -0.70   0.490    -6.866006    3.332394
        242  |   -.380527   2.444412    -0.16   0.877    -5.290273    4.529219
        243  |  -2.178841   2.351864    -0.93   0.359      -6.9027    2.545018
        244  |   2.040107   1.002563     2.03   0.047     .0264004    4.053814
        245  |   5.529303   1.373162     4.03   0.000     2.771225     8.28738
        246  |   3.253068   1.163918     2.79   0.007     .9152706    5.590865
        247  |   3.892572    1.44606     2.69   0.010     .9880753    6.797069
        248  |    1.02296   2.662638     0.38   0.702    -4.325106    6.371026
        249  |   1.445004   2.043709     0.71   0.483    -2.659907    5.549914
        250  |    1.62004    2.68492     0.60   0.549     -3.77278     7.01286
        251  |   2.151947   2.846773     0.76   0.453    -3.565964    7.869858
             |
  INDUSTRYID |
          2  |          0  (omitted)
          3  |          0  (omitted)
          4  |          0  (omitted)
          5  |          0  (omitted)
          6  |          0  (omitted)
          7  |          0  (omitted)
          8  |          0  (omitted)
          9  |          0  (omitted)
             |
   COUNTRYID |
          2  |          0  (omitted)
          3  |          0  (omitted)
          4  |          0  (omitted)
          5  |          0  (omitted)
          6  |          0  (omitted)
          7  |          0  (omitted)
          8  |          0  (omitted)
             |
       _cons |   50.26763   27.51437     1.83   0.074    -4.996608    105.5319
-------------+----------------------------------------------------------------
     sigma_u |  4.8438631
     sigma_e |   3.946532
         rho |  .60102809   (fraction of variance due to u_i)
------------------------------------------------------------------------------

regression 2 (with interaction):

xtreg WACC i.LEGAL ESG i.LEGAL##c.ESG ZSCORE ASG INTCOV CUR LEV PTB ROA TQ BT SIGMA INFL ln_SIZE ln_GDPC COVID i.qdate i.INDUSTRYID i.COUNTRYID, fe vce(cluster ID)

Fixed-effects (within) regression               Number of obs     =      1,019
Group variable: ID                              Number of groups  =         51

R-squared:                                      Obs per group:
     Within  = 0.3513                                         min =         19
     Between = 0.0807                                         avg =       20.0
     Overall = 0.1043                                         max =         20

                                                F(35, 50)         =    1659.43
corr(u_i, Xb) = -0.8423                         Prob > F          =     0.0000

                                    (Std. err. adjusted for 51 clusters in ID)
------------------------------------------------------------------------------
             |               Robust
        WACC | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
     1.LEGAL |          0  (omitted)
         ESG |   .1439323   .0642104     2.24   0.029      .014962    .2729027
         ESG |          0  (omitted)
             |
 LEGAL#c.ESG |
          1  |  -.2264729   .1146294    -1.98   0.054    -.4567127     .003767
             |
      ZSCORE |   .0591991   .0765811     0.77   0.443    -.0946187    .2130168
         ASG |  -.0037025   .0019558    -1.89   0.064    -.0076308    .0002257
      INTCOV |   9.82e-06   1.20e-06     8.21   0.000     7.42e-06    .0000122
         CUR |  -1.129546   1.064909    -1.06   0.294    -3.268478    1.009386
         LEV |  -.1936747   .0967857    -2.00   0.051    -.3880744    .0007251
         PTB |   .1253751   .1378318     0.91   0.367    -.1514683    .4022184
         ROA |  -.0618163   .0535615    -1.15   0.254    -.1693978    .0457651
          TQ |  -.0375138   .3002912    -0.12   0.901    -.6406665    .5656389
          BT |   .6677649   .4083876     1.64   0.108    -.1525057    1.488035
       SIGMA |   23.55815   16.10723     1.46   0.150    -8.794161    55.91047
        INFL |  -.0528759   .0706037    -0.75   0.457    -.1946876    .0889358
     ln_SIZE |    -.19337   1.355216    -0.14   0.887    -2.915402    2.528662
     ln_GDPC |  -2.205258   3.321063    -0.66   0.510    -8.875811    4.465294
       COVID |  -4.20e-06   .0000649    -0.06   0.949    -.0001346    .0001262
             |
       qdate |
        233  |  -.1906105    .641015    -0.30   0.767    -1.478127    1.096906
        234  |  -.1055483   .4136934    -0.26   0.800     -.936476    .7253793
        235  |   .4854662   .7071178     0.69   0.496    -.9348218    1.905754
        236  |  -.6273168    .578256    -1.08   0.283    -1.788778    .5341446
        237  |   -.199369   .5757183    -0.35   0.731    -1.355733    .9569953
        238  |   .0422641   .7130413     0.06   0.953    -1.389921     1.47445
        239  |   .3396056   .9138608     0.37   0.712    -1.495938    2.175149
        240  |  -3.894534   2.071104    -1.88   0.066    -8.054469    .2654006
        241  |  -1.605779   2.412512    -0.67   0.509    -6.451452    3.239893
        242  |  -.5987703   2.432967    -0.25   0.807    -5.485528    4.287987
        243  |  -2.775438    2.44833    -1.13   0.262    -7.693053    2.142177
        244  |   1.877919   1.024538     1.83   0.073    -.1799262    3.935765
        245  |   5.223649   1.365961     3.82   0.000     2.480036    7.967261
        246  |   2.855177   1.157895     2.47   0.017     .5294767    5.180878
        247  |   3.132966   1.431723     2.19   0.033     .2572666    6.008666
        248  |   .3979135   2.626412     0.15   0.880    -4.877391    5.673218
        249  |   1.507441   1.813767     0.83   0.410    -2.135617    5.150498
        250  |   1.657236    2.43503     0.68   0.499    -3.233665    6.548138
        251  |   2.258177   2.553452     0.88   0.381    -2.870583    7.386937
             |
  INDUSTRYID |
          2  |          0  (omitted)
          3  |          0  (omitted)
          4  |          0  (omitted)
          5  |          0  (omitted)
          6  |          0  (omitted)
          7  |          0  (omitted)
          8  |          0  (omitted)
          9  |          0  (omitted)
             |
   COUNTRYID |
          2  |          0  (omitted)
          3  |          0  (omitted)
          4  |          0  (omitted)
          5  |          0  (omitted)
          6  |          0  (omitted)
          7  |          0  (omitted)
          8  |          0  (omitted)
             |
       _cons |   28.69959   25.09056     1.14   0.258    -21.69629    79.09546
-------------+----------------------------------------------------------------
     sigma_u |  7.2590253
     sigma_e |  3.9160179
         rho |  .77457736   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. margins LEGAL,dydx(ESG)

Average marginal effects                                 Number of obs = 1,019
Model VCE: Robust

Expression: Linear prediction, predict()
dy/dx wrt:  ESG

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
ESG          |
       LEGAL |
          0  |          .  (not estimable)
          1  |          .  (not estimable)
------------------------------------------------------------------------------

[Clyde Schechter]

To understand the interaction model you have to do a little calculation:

The coefficient of ESG by itself (.1439...) represents the marginal effect of the ESG score on WACC conditional on LEGAL = 0. (Or it would, were it not for a problem with your model--see below.) The marginal effect of ESG score on WACC conditional on LEGAL = 1 has to be calculated as the coefficient of ESG + coefficient of 1.LEGAL#c.ESG = .1439... - .226... = -.0825406. By contrast in your non-interaction model, the coefficient of ESG represents an overall marginal effect of ESG score on WACC. Since the "coefficient of ESG" represents two entirely different things in the two models, a direct comparison of one with the other is meaningless, and there is no particular reason to expect them to be similar in any way. In particular there is no reason one can't be "significant" and the other "not significant." They are just two different things.

Now, I indicated that these coefficients would work a certain way were it not for a problem. In fact, your model is not capable of estimating these marginal effects at all. You can actually see that because when you ran your -margins LEGAL, dydx(ESG)- command Stata told you that these statistics are not estimable. The cause for that is a colinearity in your model that makes these effects unidentifiable. The problem arises because LEGAL is colinear with the firm fixed effects (id variable). It is colinear with them because each country operates within a single LEGAL environment that does not change over the time period in your study. You cannot have both LEGAL and the firm fixed effects in the model at the same time. Now, since you are interested in whether the marginal effect of ESG differs between LEGAL = 1 and LEGAL = 0 firms, this is not a problem for you: the "pay dirt" in your model is the interaction term 1.LEGAL#c.ESG. The variable LEGAL on its own is of no importance here. So if you revise your model to

xtreg WACC  ESG i.LEGAL#c.ESG ZSCORE ASG INTCOV CUR LEV PTB ROA TQ BT SIGMA INFL ///
    ln_SIZE ln_GDPC COVID i.qdate, fe vce(cluster ID)

you will be good to go, and -margins LEGAL, dydx(ESG) will work for you and give you the needed results correctly.

Note that in addition to removing the direct reference to LEGAL, I also replaced the ## interaction by #. The latter change is required also, because if you leave ##, Stata puts LEGAL back in automatically.

Added: I also removed the country and industry variables because, they, too are colinear with the firm fixed effects--they are time-invariant attributes of the firm. Now, as it happens, they are not colinear with anything else in the model, so leaving them would no real harm. In particular, leaving them in would not render any of the other model parameters unidentifiable. But having them there clutters up the output with a lot of 0 outputs that tell you nothing useful. So best to leave them out.

[Ethan Kilmer]

Thank you for the prompt and comprehensive response, Clyde, I really appreciate your help. I have inputted the new code and Stata outputted the following. I am still struggling to interpret the results however. My understanding is that margins LEGAL, dydx(ESG) suggests that ESG has a significant positive impact on WACC for companies in weak legal environments (0), and the negative impact for strong legal environments (1) is not significant. Is this correct?

Also, I am trying to understand your point of "Since the coefficient of ESG represents two entirely different things in the two models, a direct comparison of one with the other is meaningless". Due to the fact that the marginal effect of ESG on WACC is not significant in the baseline model, and is significant in the model including an interaction, is it fair to say that the relationship between ESG and the cost of capital is only significant for companies in weak legal environments, overall.

xtreg WACC  ESG i.LEGAL#c.ESG ZSCORE ASG INTCOV CUR LEV PTB ROA TQ BT SIGMA INFL ln_SIZE ln_GDPC COVID i.qdate, fe vce(cluster ID)

Fixed-effects (within) regression               Number of obs     =      1,019
Group variable: ID                              Number of groups  =         51

R-squared:                                      Obs per group:
     Within  = 0.3513                                         min =         19
     Between = 0.0807                                         avg =       20.0
     Overall = 0.1043                                         max =         20

                                                F(35, 50)         =    1659.43
corr(u_i, Xb) = -0.8423                         Prob > F          =     0.0000

                                    (Std. err. adjusted for 51 clusters in ID)
------------------------------------------------------------------------------
             |               Robust
        WACC | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         ESG |   .1439323   .0642104     2.24   0.029      .014962    .2729027
             |
 LEGAL#c.ESG |
          1  |  -.2264729   .1146294    -1.98   0.054    -.4567127     .003767
             |
      ZSCORE |   .0591991   .0765811     0.77   0.443    -.0946187    .2130168
         ASG |  -.0037025   .0019558    -1.89   0.064    -.0076308    .0002257
      INTCOV |   9.82e-06   1.20e-06     8.21   0.000     7.42e-06    .0000122
         CUR |  -1.129546   1.064909    -1.06   0.294    -3.268478    1.009386
         LEV |  -.1936747   .0967857    -2.00   0.051    -.3880744    .0007251
         PTB |   .1253751   .1378318     0.91   0.367    -.1514683    .4022184
         ROA |  -.0618163   .0535615    -1.15   0.254    -.1693978    .0457651
          TQ |  -.0375138   .3002912    -0.12   0.901    -.6406665    .5656389
          BT |   .6677649   .4083876     1.64   0.108    -.1525057    1.488035
       SIGMA |   23.55815   16.10723     1.46   0.150    -8.794161    55.91047
        INFL |  -.0528759   .0706037    -0.75   0.457    -.1946876    .0889358
     ln_SIZE |    -.19337   1.355216    -0.14   0.887    -2.915402    2.528662
     ln_GDPC |  -2.205258   3.321063    -0.66   0.510    -8.875811    4.465294
       COVID |  -4.20e-06   .0000649    -0.06   0.949    -.0001346    .0001262
             |
       qdate |
        233  |  -.1906105    .641015    -0.30   0.767    -1.478127    1.096906
        234  |  -.1055483   .4136934    -0.26   0.800     -.936476    .7253793
        235  |   .4854662   .7071178     0.69   0.496    -.9348218    1.905754
        236  |  -.6273168    .578256    -1.08   0.283    -1.788778    .5341446
        237  |   -.199369   .5757183    -0.35   0.731    -1.355733    .9569953
        238  |   .0422641   .7130413     0.06   0.953    -1.389921     1.47445
        239  |   .3396056   .9138608     0.37   0.712    -1.495938    2.175149
        240  |  -3.894534   2.071104    -1.88   0.066    -8.054469    .2654006
        241  |  -1.605779   2.412512    -0.67   0.509    -6.451452    3.239893
        242  |  -.5987703   2.432967    -0.25   0.807    -5.485528    4.287987
        243  |  -2.775438    2.44833    -1.13   0.262    -7.693053    2.142177
        244  |   1.877919   1.024538     1.83   0.073    -.1799262    3.935765
        245  |   5.223649   1.365961     3.82   0.000     2.480036    7.967261
        246  |   2.855177   1.157895     2.47   0.017     .5294767    5.180878
        247  |   3.132966   1.431723     2.19   0.033     .2572666    6.008666
        248  |   .3979135   2.626412     0.15   0.880    -4.877391    5.673218
        249  |   1.507441   1.813767     0.83   0.410    -2.135617    5.150498
        250  |   1.657236    2.43503     0.68   0.499    -3.233665    6.548138
        251  |   2.258177   2.553452     0.88   0.381    -2.870583    7.386937
             |
       _cons |   28.69959   25.09056     1.14   0.258    -21.69629    79.09546
-------------+----------------------------------------------------------------
     sigma_u |  7.2590253
     sigma_e |  3.9160179
         rho |  .77457736   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. margins LEGAL,dydx(ESG)

Average marginal effects                                 Number of obs = 1,019
Model VCE: Robust

Expression: Linear prediction, predict()
dy/dx wrt:  ESG

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
ESG          |
       LEGAL |
          0  |   .1439323   .0642104     2.24   0.025     .0180823    .2697824
          1  |  -.0825405   .0866402    -0.95   0.341    -.2523522    .0872712
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[Clyde Schechter]

My understanding is that margins LEGAL, dydx(ESG) suggests that ESG has a significant positive impact on WACC for companies in weak legal environments (0), and the negative impact for strong legal environments (1) is not significant. Is this correct?

Yes.

Due to the fact that the marginal effect of ESG on WACC is not significant in the baseline model, and is significant in the model including an interaction, is it fair to say that the relationship between ESG and the cost of capital is only significant for companies in weak legal environments, overall.

More or less, yes. I'm hedging on this because I don't know how you defined strong and weak legal environments. If this is a true dichotomy, then your interpretation is correct. But if there is really an underlying continuum of strengths of legal systems, and you have just imposed a cutpoint somewhere and called everything below that weak and everything above it strong, then your conclusion is not justified. After all, it is more likely that the marginal effect of ESG itself varies continuously (or nearly so) with the underlying continuous legal system strength, and had you measured it that way, it is possible that the marginal effect of ESG itself might still be "significant" for some of the legal systems that fell into the "strong" pseudo-category as well.

So it depends on how fine-grained the construct of strength of a legal system is. If it's really as coarse grained as you have modeled it, then your interpretation is correct. But if this coarse-graining has been imposed on a more finely-grained construct, then it may be incorrect.

[Ethan Kilmer]

Following the methodologies of relevant studies on the topic, namely Crifo et al. (2017), Gupta (2018), and Preim and Gabellone (2022), I adopted the Worldwide Governance Indicators (WGI) of The World Bank to measure the quality of each country’s legal system. In line with Preim and Gabellone (2022), for the purpose of the regression, I took the average value of the six indicators (named as LEGALV). This value varies between countries over the years - not time invariant.

To determine whether it is a weak or strong legal environment (among the sampled countries), I took the median value of the overall WGI indicators (LEGALV). And, as you predicted, I defined "weak" as below the median and "strong" as above. This was then coded a dummy variable (named LEGAL) and included in the interaction.

However, if it is more appropriate to do a continuous by continuous interaction term with the LEGALV variable, I may attempt this instead.


Crifo, P., Diaye, M.A., Oueghlissi, R. (2017). The effect of countries’ ESG ratings on their sovereign borrowing costs. The Quarterly Review of Economics and Finance, volume 66, pp.13-20, [online], available at: https://www.sciencedirect.com/scienc...301497#bib0115

Gupta, K. (2018). Environmental Sustainability and Implied Cost of Equity: International Evidence. Journal of Business Ethics, volume 147, pp. 343-365, [online], available at: https://www-jstor-org.elib.tcd.ie/st...5022380?seq=15

Priem, R., Gabellone, A. (2022). The impact of a firm’s ESG score on its cost of capital: can a high ESG score serve as a substitute for a weaker legal environment. Sustainability Accounting, Management and Policy Journal, “Vol. ahead-of-print”, [online],available at: https://www.emerald.com/insight/cont...0254/full/html