Including an Interaction term leads to insignificant direct effects

[Tom regt]

Dear Members,

I am investigating the effect of Ultra- low cost airline (ULCC) market entry on the airfare of incumbents using quarterly data for the period of 2006-2015. This resulted in an unbalanced three-dimensional panel data set. The dimensions are: airline, market and year-quarter. The independent variable is the airfare of the incumbent (lwaprice). The model used is a log-log model. The independent variables are:
LCCdummy = a dummy variable indicating whether a low-cost carrier (LCC) is present in the market (1 if present)
ULCCdummy = a dummy variable indicating whether an ultra-low-cost carrier (ULCC) is present in the market (1 if present)
LOGLCCmshare = an interaction variable between LCC presence and the LCC market share
LOGULCCmshare = an interaction variable between ULCC presence and the ULCC market share
LOGhhi = the Herfindahl-Hirschman Index as a measure of market concentration
HHI_ULCC = an interaction between the ULCC market presence dummy and the HHI
HHI_LCC = an interaction between the LCC market presence dummy and the HHI
LOGroutetotalpassengers = the total number of passengers enplaned on a route

My question:
I am including the interactions between ULCC market presence and HHI and LCC market presence and HHI to see whether LCC/ULCC market presence occurs more in markets with a high or low market concentration or that there is no significant relationship between the two variables. I am having difficulties with the interpretation of the results. When I perform a regression without the HHI_ULCC and HHI_LCC interactions, then both the HHI and the ULCCdummy are statistically significant. However, when I include the interactions in the regression, then the interactions as well as both the HHI and the ULCCdummy turn out to be insignificant. I do not know how to interpret this result. How is it possible that the HHI and the ULCCdummy become insignificant when an interaction between them is included? Below I have included both regressions.

code:

xtreg lwaprice LCCdummy ULCCdummy LOGLCCmshare LOGULCCmshare LOGhhi LOGroutetotalpassengers, fe vce(robust)

Fixed-effects (within) regression               Number of obs     =     35,711
Group variable: id                              Number of groups  =      1,843

R-sq:                                           Obs per group:
     within  = 0.2802                                         min =          1
     between = 0.1590                                         avg =       19.4
     overall = 0.1669                                         max =         40

                                                F(123,1842)       =      56.55
corr(u_i, Xb)  = -0.1727                        Prob > F          =     0.0000

                                            (Std. Err. adjusted for 1,843 clusters in id)
-----------------------------------------------------------------------------------------
                        |               Robust
               lwaprice |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
------------------------+----------------------------------------------------------------
               LCCdummy |   .0064139   .0171763     0.37   0.709    -.0272732    .0401009
              ULCCdummy |   -.059109   .0235493    -2.51   0.012    -.1052951    -.012923
           LOGLCCmshare |   .0125677   .0063718     1.97   0.049      .000071    .0250643
          LOGULCCmshare |   .0219709   .0084723     2.59   0.010     .0053545    .0385872
                 LOGhhi |   .1669416   .0160189    10.42   0.000     .1355245    .1983588
LOGroutetotalpassengers |  -.0184824   .0067283    -2.75   0.006    -.0316783   -.0052864

xtreg lwaprice LCCdummy ULCCdummy LOGLCCmshare LOGULCCmshare LOGhhi HHI_ULCC HHI_LCC LOGroutetotalpassengers, fe vce(robust)

Fixed-effects (within) regression               Number of obs     =     35,711
Group variable: id                              Number of groups  =      1,843

R-sq:                                           Obs per group:
     within  = 0.2803                                         min =          1
     between = 0.1576                                         avg =       19.4
     overall = 0.1653                                         max =         40

                                                F(125,1842)       =      55.76
corr(u_i, Xb)  = -0.1770                        Prob > F          =     0.0000

                                            (Std. Err. adjusted for 1,843 clusters in id)
-----------------------------------------------------------------------------------------
                        |               Robust
               lwaprice |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
------------------------+----------------------------------------------------------------
               LCCdummy |  -.0332751   .0772272    -0.43   0.667    -.1847371    .1181869
              ULCCdummy |  -.1086207   .0714616    -1.52   0.129     -.248775    .0315335
           LOGLCCmshare |   .0127351   .0064494     1.97   0.048     .0000862     .025384
          LOGULCCmshare |   .0206225   .0083797     2.46   0.014     .0041878    .0370572
                 LOGhhi |   .1615909   .0183192     8.82   0.000     .1256622    .1975195
               HHI_ULCC |   .0146787   .0174125     0.84   0.399    -.0194717     .048829
                HHI_LCC |   .0095671   .0173415     0.55   0.581    -.0244439     .043578
LOGroutetotalpassengers |  -.0186184   .0067036    -2.78   0.006    -.0317659   -.0054708

Is there anyone who can help me out?
Kind regards,
Tom

[Clyde Schechter]

Well, first, there is nothing paradoxical or surprising in this general phenomenon. The "main effects" no longer carry the same meaning when an interaction term is included in the model, so their coefficients have no necessary relationship to the coefficients seen by the variables of the same name in a no-interaction model. So, for example, in the interaction model, ULCCdummy no longer represents the effect of the presence of an ultra-low cost carrier in the market. In fact, in an interaction model there is no such thing as the effect of an ultra-low cost carrier in the market. Rather, in the interaction model, there are an infinite number of effects of an ultra low cost carrier in the market, these effects depending on the value of HHI. What you see as the coefficient of ULCCdummy in your output is nothing more than the effect of the presence of an ultra low cost carrier in the market when the HHI is zero. The fact that it differs from the coefficient of ULCCdummy in a non-interaction model (which is some effect off an ultra low cost carrier in the market averaged, in some sense, over all values of HHI, should not surprise you.

Next, it is not a good idea to use log HHI in the model but create an interaction variable using HHI itself. Such a model is not necessarily mis-specified, but interpreting it would be complicated.

Next, it is not a good idea to create your own interaction terms. You should use Stata's factor-variable notation (-help fvvarlist-) for that. Doing so will enable you to use the -margins- command after you run your regression model. The use of -margins- is especially helpful with interaction models, as interpreting the regression output directly requires calculations that are tedious to do and easy to get wrong. -margins- does all that for you, painlessly and correctly.

And, I will note also that examining statistical significance, a dubious undertaking in any case like this, is even more dubious in an interaction model, where a single test of any of the coefficients involved in the interaction model becomes truly pointless and meaningless. If you feel you must waste your time doing hypothesis tests here, then at least do the right one: -test LOGhhi HHI_ULCC ULCCdummy-, which simultaneously tests all three coefficients. But really, I know little or nothing about the economics of air travel, but is it at all credible to test a null hypothesis that these things have no effect on lwaprice. Doesn't it make more sense to simply estimate how much that effect is and, with a confidence interval, how precise our estimate of that effect is?

So here's what I would do with this (assuming you want an interaction model):

xtreg lwaprice c.logHHI##i.(LCCdummy ULCCdummy) LOGLCCmshare LOGULCCmshare LOGroutetotalpassengers, fe vce(robust)

Then I would pick representative or interesting values of logHHI and get the predicted values of lwaprice and the marginal effects of LCCdummy and ULCCdummy at those value of logHHI. For the sake of illustration, let's assume the interesting values of logHHI are 1 1.25 1.50 1.75 and 2 (I made those up, I have no idea what the HHI actually is):

margins LCCdummy ULCCdummy, at(logHHI = (1 (0.25) 2))
margins, dydx(LCCdummy ULCCdummy) at(logHHI = (1 (0.25) 2))

The -margins- outputs are the real meat of your model and are what you should focus on for interpretation. You might also want to use the -marginsplot- command to graph them for even more insight into what's going on.

[Tom regt]

Thank you very much for your response, this was very useful!

Kind regards,