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  • Two interaction terms

    Hello! I have a regression model that requires two interaction terms. I am however a bit unsure how to correctly interpret the coefficients? My variables are as follow

    1) INV_W --> change in investment spending
    2) CR_POM (dummy variable) --> measuring if firms are close to a credit rating change or not
    3) Tech_Industry ( dummy variable) --> first interaction term to distinguishing between firms in technology industry and the ones not
    4) INV_G (dummy variable) --> second interaction term to distinguish between investment grade firms and the ones below
    5) Size, Profitability are control variables


    My regression and output are as follows:

    Code:
     regress INV_W CR_POM Tech_Industry INV_G Int_CR_POM_Tech Int_CR_POM_INV_G Int_Tech_INV_G Size Profitability, vce(hc3)
    
    Linear regression                                      Number of obs =    5715
                                                           F(  8,  5706) =    6.80
                                                           Prob > F      =  0.0000
                                                           R-squared     =  0.0089
                                                           Root MSE      =  1.4518
    
    ----------------------------------------------------------------------------------
                     |             Robust HC3
               INV_W |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
    -----------------+----------------------------------------------------------------
              CR_POM |   .0820362   .0840089     0.98   0.329    -.0826532    .2467256
       Tech_Industry |  -.0754513   .0922812    -0.82   0.414    -.2563574    .1054549
               INV_G |   .0118881   .0831091     0.14   0.886    -.1510372    .1748135
     Int_CR_POM_Tech |   .0329208   .0867673     0.38   0.704    -.1371761    .2030178
    Int_CR_POM_INV_G |  -.0782386   .0891753    -0.88   0.380     -.253056    .0965788
      Int_Tech_INV_G |   .1185679   .0921215     1.29   0.198    -.0620253    .2991611
                Size |  -.1483267   .0293412    -5.06   0.000    -.2058467   -.0908068
       Profitability |   1.027661   .2402928     4.28   0.000     .5565956    1.498726
               _cons |   1.619802   .2798224     5.79   0.000     1.071244     2.16836
    ----------------------------------------------------------------------------------
    I am currently unsure if for instance coefficient for CR_POM would indicate the firms that are close to a credit rating change, but excluding firms in tech industry as well as the ones in investment grade? Additionally, I if I would like to arrive at a coefficient to see if the effect on net investments is stronger for firms that are close to a credit rating change within investment grade in contrast to rest of the firms. Also if this effect is more prominent in Tech industries. Overall, I am wondering how adding up the coefficients can be interpreted, for instance adding up CR_POM + Int_CR_POM_Tech or alterantivly CR_POM + Int_CR_POM + Int_Tech_INV_G. Thank you in advance!!




  • #2
    Hello Elizabeth,

    As can be seen in the output, the interaction terms are insignificant. Thus, they could be eliminated from the model. Note, (a) inclusion of the interactions requires strong theory, and (b) the interaction term must be (i) significant, and (ii) the percent of explained variation attributable to the interaction term (effect size) must also be significant. When the interactions are present, the interpretation of the main effects is conditional on the mean value of the interaction term. Having removed the interactions, you would be able to interpret the main effects in a straightforward way.

    Hope this helps.

    On a side note: the R squared appears to be very small -- did you check the model specification (i.e., tested the assumptions of OLS)?
    Last edited by Anton Ivanov; 08 Jun 2015, 17:18.

    Comment


    • #3
      I see your point, but in general in my case it is also important to show that the interaction has no significance. What I am actually looking to fins out is how to exactly interpret the results, even if not significant. And more specifically the meaning of adding up coefficients (using -lincom-) in order to arrive at full extent of the possible relationships! Thanks!

      Comment


      • #4
        Well, in this case I would suggest checking a book by Aiken, L. S., West, S. G., & Reno, R. R. (1991). Multiple regression: Testing and interpreting interactions. Sage.

        As for "adding up" coefficients, I am not familiar with that, so hopefully other members would help. Good luck!

        Comment


        • #5
          Thanks! It is quite urgent though, so not sure if i will be able to get the book in time so if anyone else can point me into right direction that would be amazing!
          Last edited by Elizabete Lurie; 08 Jun 2015, 17:34.

          Comment


          • #6
            Hi Elizabeth,

            When trying to understand interaction terms, I also find it very useful to take derivatives. For example, let's say you're interested in the effect of being an "investment-grade" firm. The derivative of would simply be:

            0.011881 - 0.0782386*CR_POM + 0.1185679*Tech

            (I don't think I missed any... Investment grade appears three times, correct?) So, to find the marginal effect of investment grade, you need to insert some value for CR_POM and some value for tech (1 if the firm is also a tech firm and 0 if not).

            Similarly, the effect of being a tech firm would be:

            -0.0754513 + 0.0329208*CR_POM + 0.1185679*INV_G

            Also, I am going to add a slight addendum to Anton's statement, and say that if you have a theoretical reason to think the interaction terms should be in there, then they should stay, whether they are significant or not. I hope this helps!

            Josh

            Comment


            • #7
              Thanks! I think it does help, at least it feels I a bit more clearer. Just to check if I get it correctly, If I was looking for effect on firm net investments for all firms close to a credit rating change regardless of industry or if investment grade, I would add up CR_POM + Int_CR_POM_Tech + Int_CR_POM_INV_G , correct?? And alternatively if I was looking for for the difference in effect on investment spending firms have if close to an upgrade when investment grade in comparison to remaining firms. Namely, If I wanted to say firms that are close to a credit rating change and also an investment grade class are likely to reduce their investment spending by xx% less than remaining firms would I add up Int_CR_POM_INV_G + Int_Tech_INV_G ? Thank you so much!

              Comment


              • #8
                Elizabete,

                Be careful about adding up coefficients with interaction terms. You can't quite find the effect on firm net investments for all firms simultaneously. The effect of being close to a credit rating change is found in three separate variables in your model: CR_POM, INT_CR_POM_Tech, and INT_CR_POM_INV_G. In this case, CR_POM represents the effect of being close to a change for non-tech, non-investment-grade firms (because if POM_Tech==1 then INT_CR_POM_Tech does not equal zero, and if INV_G==1, then INT_CR_POM_INV_G does not equal zero). Similarly, CR_POM + INT_CR_POM_Tech is the effect of being close to a credit rating change for non-investment grade tech firms, CR_POM + INT_CR_POM_INV_G is the effect of being close to a change for non-tech investment-grade firms, and CR_POM + INT_CR_POM_Tech + INT_CR_POM_INV_G is the effect of being close to a credit rating change for investment-grade tech firms.

                In order for interaction terms to not enter into an interpretation, they need to be equal to zero. Clearly, if you are interested in a variable that is not interacted, then the derivative of all interaction terms are, by definition, zero. However, when you are interested in a variable taht is interacted with other variables, make sure to pay careful attention to when the interaction term is equal to zero and when it is not.

                HTH,

                Josh

                Comment


                • #9
                  Thanks a lot! Josh! Just wondering then, when I follow your line of though would not adding up CR_POM + INT_CR_POM_TECH + INT_CR_POM_INV_G lead to all firms that are close to a credit rating change or am I mistaken? Thanks!

                  Comment


                  • #10
                    Not quite. Again, adding them all up is the effect for a specific subset of firms: investment-grade tech firms.

                    Comment


                    • #11
                      But does not CR_POM represent firms that are close to a credit rating change, but neither in tech industry nor investment grade? Thanks

                      Comment


                      • #12
                        Correct. So CR_POM is the effect of being near a change for non-tech, non-investment grade firms. CR_POM + INT_CR_POM_TECH is the effect for non-investment-grade tech firms only, etc.

                        Comment


                        • #13
                          Just wondering then, when I follow your line of though would not adding up CR_POM + INT_CR_POM_TECH + INT_CR_POM_INV_G lead to all firms that are close to a credit rating change or am I mistaken?
                          Elizabete, it's not really different from the model with a single interaction. There is no such thing as the effect of CR_POM in a model that contains interactions. The inclusion of the interaction stipulates that no such thing exists. You will only reap frustration if you attempt to find some combination of coefficients that estimates this non-existent effect. There are several (four in this case) different effects of CR_POM conditional on the values of tech and investment-grade. Each can be estimated by an appropriate combination of the CR_POM coefficient and the interaction coefficients, as Joshua has already shown you.

                          You can, if you want, calculate an "average" effect of CR_POM, but that average effect depends on whether you want to take the values of tech and investment-grade as they came in the data, or set them to mean values, or assume some other joint distribution for them--so that average effect is really rather ill-defined.

                          Comment


                          • #14
                            Thank you so much! This leads me to believe that my initial understanding of interaction term was flawed. To make sure I understand it correctly. If we would only consider one interaction term, such as Tech_Industry. Then in that scenario CR_POM would indicate the effect for firms close to a credit rating change, excluding tech industries, whilst Int_CR_POM_Tech, would measure the difference between tech firms and other when close to a credit rating change. And lastly, the CR_POM + Int_CR_POM_Tech would indicate the effect for only tech firms? Thank you in advance!

                            Comment


                            • #15
                              Usually examples help a lot, but let us state this more general in terms of \(y\), \(x\) and \(z\). In the model

                              \[
                              y = \beta_0 + \beta_1 * x + \beta_2 * z + \beta_3 * (x*z)
                              \]

                              \(\beta_1\) is the effect of \(x\) given \(z=0\). Obviously, \(\beta_2\) is interpreted in the same way, meaning that it represents the effect of \(z\) conditional on \(x=0\). The coefficient of the interaction term, \(\beta_3\), is then the difference in these effects. In other words, the effect of \(x\) differs by \(\beta_3 * z\) and is, thus, dependent on \(z\). This is also true for the effect of \(z\), which in turn differs by \(\beta_3 * x\) for any given level of \(x\).

                              This is what we usually do, speaking about effects. Adding up the coefficients, as you seem to have in mind, is, in my view, better interpreted in terms of predicted values.

                              For example, the expected level of \(y\) for \(x\) and \(z\) both \(=0\), is \(\beta_0\). The expected level of \(y\) for \(x = 1\) and \(z=0\) is \(\beta_0 + \beta_1\). Staying with \(x=1\) but now increasing \(z\) by one unit, the expected value for \(y\) is given by \(\beta_0 + \beta_1 + \beta_2 + \beta_3\). And so on ...

                              So I guess in your example, it is not clear what you think

                              CR_POM + Int_CR_POM_Tech
                              should give. If you want the expected value for

                              all firms close to a credit rating change in all industries
                              (given Size and Profitability) then you need

                              Code:
                              _cons + CR_POM + Tech_Industry + Int_CR_POM_Tech
                              because

                              all firms [...] in all industries
                              means Tech_Industry (= 1) and non-Tech_Industry (i.e. _cons), and

                              close to a credit rating change
                              means CR_POM (= 1) and since you expect the difference between close to credit change firms and not close to credit change firms to differ by industry (or, equivalent, that the differences between industry differ by closeness to credit change) you also need to include the interaction effect.

                              I hope this helps.

                              Best
                              Daniel
                              Last edited by daniel klein; 09 Jun 2015, 05:53. Reason: fixing TeX

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