How to interpret non-standardised coefficients?

[Hellie Flint]

Hi all,

I am looking at the effect of gender inequality (measured by GII which is a 0 to 1 scale, where higher values indicate greater inequality) on economic growth (measured by GDP per capita growth, as a %). As the two variables aren't measured in the same form I was wondering how I interpret the effect of GII on growth when looking at my coefficients, and also how to interpret the interaction term between GII and income (measured by natural log of GDP per capita) given I have interacted two variables measured differently?

xtreg Growth lagGII lagIncomeln GII_Income i.Year, fe robust

Fixed-effects (within) regression               Number of obs     =      2,276
Group variable: CountryID                       Number of groups  =        114

R-sq:                                           Obs per group:
     within  = 0.1394                                         min =         18
     between = 0.0410                                         avg =       20.0
     overall = 0.0255                                         max =         20

                                                F(22,113)         =      14.49
corr(u_i, Xb)  = -0.9338                        Prob > F          =     0.0000

                            (Std. Err. adjusted for 114 clusters in CountryID)
------------------------------------------------------------------------------
             |               Robust
      Growth |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      lagGII |  -50.08883   12.19639    -4.11   0.000    -74.25208   -25.92559
 lagIncomeln |  -7.186864   1.354488    -5.31   0.000     -9.87035   -4.503379
  GII_Income |   5.313213   1.206568     4.40   0.000     2.922784    7.703641
             |
        Year |
       1997  |  -.3565464   .3871003    -0.92   0.359    -1.123462    .4103691
       1998  |  -.9939671   .4270054    -2.33   0.022    -1.839942   -.1479923
       1999  |   -.847875   .4163987    -2.04   0.044    -1.672836    -.022914
       2000  |  -.0046979   .4400977    -0.01   0.992    -.8766109    .8672151
       2001  |  -.9246834   .4146138    -2.23   0.028    -1.746108   -.1032587
       2002  |     -.5211   .4825815    -1.08   0.283    -1.477181    .4349809
       2003  |   .2614466   .4714257     0.55   0.580    -.6725327    1.195426
       2004  |   1.514311     .45475     3.33   0.001     .6133698    2.415253
       2005  |   1.131297   .4277166     2.64   0.009      .283913     1.97868
       2006  |   2.066314   .4628423     4.46   0.000      1.14934    2.983288
       2007  |   2.169376   .4984575     4.35   0.000     1.181842    3.156911
       2008  |   .4962147   .5488308     0.90   0.368     -.591118    1.583548
       2009  |  -2.813979   .4936407    -5.70   0.000     -3.79197   -1.835987
       2010  |   1.726426   .4774053     3.62   0.000     .7805997    2.672252
       2011  |   1.158964   .6006279     1.93   0.056    -.0309881    2.348916
       2012  |   .9286711   .5992796     1.55   0.124    -.2586098    2.115952
       2013  |   .7256153   .6654279     1.09   0.278    -.5927175    2.043948
       2014  |   1.278339   .5844081     2.19   0.031     .1205209    2.436156
       2015  |   .8467109   .6811747     1.24   0.216     -.502819    2.196241
             |
       _cons |   69.04605    12.6388     5.46   0.000     44.00631    94.08578
-------------+----------------------------------------------------------------
     sigma_u |  5.5939725
     sigma_e |  3.3023023
         rho |  .74156902   (fraction of variance due to u_i)
------------------------------------------------------------------------------

Is there a way I can work this out or is it easier to standardise my coefficients, if so how do I do this?

Many thanks,

Hellie

[Clyde Schechter]

I fear I am missing something and not interpreting your question properly, because interacted variables (except for quadratic terms, which are interactions of a variable with itself) are usually measured differently. In fact, in my entire life I can't recall a single instance where two interacted variables were measured in the same units. In any case, it makes no difference what units they are measured in, the interpretation of interactions is the same. So I have a feeling I may be misunderstanding your question. Nevertheless, I'll tell you how I interpret your interaction.

I'm making one assumption here: the variable you call GII_Income is, in fact, the interaction (pproduct) of lagGII and lagIncomeln.

So, the coefficient of lagGII, -50.0883 is the slope of the Growth:lagGII relationship when lagIncomeln is 0. Since the latter variable is log-transformed, one could also say that it is the slope of the Growth:lagII relationship when the lagged value of Income itself is 1. (Depending on the units in which Income is measured, this may or may not ever happen in reality.)

The coefficient of lagIncomeln, -7.186864, is the slope of the Growth:lagIncomeln relationship when lagGII is 0.

The interaction coefficient of GII_Income, 5.313213 says that the expected value of the Growth:lagGII slope grows at a rate of 5.313213 percentage points per lagGII unit in association with every unit increase in lagIncomeln. Equivalently, you could say that the expected value of the Growth:lagIncomeln slope grows at a rat4e of 5.313213 percentagepoints per unit increase in lagGII.

That's a lot of verbiage, and it's not easy to follow and remain. To really see what's going on in this kind of model, it is best to actually see it with your eyes. So first, revise your model using factor variable notation, and then use -margins- and -marginsplot-.

xtreg Growth c.lagGII##c.lagIncomeln i.Year, fe vce(cluster CountryID)
margins, at(lagGII = (0(.2)1) lagIncomeln = (list_of_interesting_values_of_this_variable)
marginsplot, xdimension(lagGII) name(by_lagGII, replace)
marginsplot, xdimension(lagIncomeln) name(by_lagIncomeln, replace)

In the above code replace list_of_interesting_values_of_this_variable) by a numlist that spans the range of typical and interesting values of lagIncomeln, with a few points in between, just as 0(.2)1 spans the complete 0 to 1 range of GII with both endpoints and 4 values between.

Almost forgot your final question. No, please don't standardize the variables. That will just make things an order of magnitude harder to understand!

[Hellie Flint]

Thank you Clyde this was such a helpful and insightful answer!