How to interpret coefficients when with ln transformed dependent variable?

[Patrick James Lim]

Here's the result of my regression together with the code used.

. xtreg lnGDP Dummy CronyNW Crony RuleofLaw GovFreedom TradeOpen Democracy WI lnPop UnemploymentILO

Random-effects GLS regression                   Number of obs      =       506
Group variable: A                               Number of groups   =        68

R-sq:  within  = 0.7761                         Obs per group: min =         1
       between = 0.7621                                        avg =       7.4
       overall = 0.8242                                        max =         9

                                                Wald chi2(10)      =   1671.44
corr(u_i, X)   = 0 (assumed)                    Prob > chi2        =    0.0000

---------------------------------------------------------------------------------
       lnGDP |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
         Dummy |  -.0510804   .0128683    -3.97   0.000    -.0763018    -.025859
        CronyNW |  -.0005677    .000128    -4.43   0.000    -.0008186   -.0003167
          Crony |   .0051423   .0005505     9.34   0.000     .0040633    .0062213
      RuleofLaw |   .0029894   .0002964    10.08   0.000     .0024084    .0035704
     GovFreedom |   -.001993    .001483    -1.34   0.179    -.0048997    .0009137
      TradeOpen |   .0106172   .0010782     9.85   0.000      .008504    .0127305
      Democracy |  -.0020406   .0013353    -1.53   0.126    -.0046578    .0005766
             WI |   .0015382   .0007525     2.04   0.041     .0000634     .003013
          lnPop |   .8786396   .0401595    21.88   0.000     .7999285    .9573506
UnemploymentILO |   -.016465   .0019734    -8.34   0.000    -.0203328   -.0125973
          _cons |   11.52465   .6950353    16.58   0.000      10.1624    12.88689
----------------+----------------------------------------------------------------
        sigma_u |  .55074607
        sigma_e |  .06532987
            rho |  .98612438   (fraction of variance due to u_i)
---------------------------------------------------------------------------------

My dependent variable is lnGDP which is generated by - gen lnGDP = ln(GDP) -
What I believe to be true right now is that an increase in 1 unit of variable increase lnGDP by (Coef.)%
so the Dummy variable has a -5.10804% effect on lnGDP

However, searching online indicates that this is wrong.
If so, I'm wondering how do I correctly interpret the coefficients in the results especially with the Dummy variable.

Also, is it possible to interpret the Coefficients in terms of percentage?
For example, I would like to know how much % change in lnGDP would an increase by 1 unit of the CronyNW variable cause.

[Rich Goldstein]

many years ago, I wrote -logdummy- and published a short piece in the stb for just this purpose; use -findit- to find and download; if you want to read the original article, it was in STB 5 which you can download for free from Stata's website; note that I have not used it in years as I generally prefer other forms of regression (e.g., poisson) and do not know whether it will work with whatever version of Stata you are currently using

[Clyde Schechter]

It's just some algebra.

[code]
lnGDP(x=1) = lnGDP(x=0) - 0.051
GDP(x = 1) = exp[lnGDP(x=0) - 0.051] = GDP(x=0)*exp(-0.051)
GDP(x=1) = GDP(x=0) * 0.950 (to 3 decimal places)
[code]
So that is a 5% smaller GDP.

[Jeff Wooldridge]

To follow up on Clyde, where are you getting that your original interpretation is faulty? Sure, it's an approximation, but for small to moderate changes it's not a bad one. From a practical perspective, the difference between -5.1% and 5.0% is small.

Besides, the calculation exp(-.0510804) - 1 = -.04979773 is the estimated change going from zero to 1. If you start at 1 and go to zero, the proportionate change is

exp(.0510804) - 1 = .0524075. You can see that, in absolute value, the approximation .0510804 falls between .04979773 and .0524075.

The effect of CronyNW is even much smaller, and any exact calculation will get you very close to the approximation. But are you sure you only want to increase CronyNW by one unit? The drop in GDP is only about .057%.

[John Mullahy]

Patrick: Beyond the helpful observations offered by Rich, Clyde, and Jeff, a couple references might be helpful for further reading. A classic reference is Halvorsen & Palmquist, 1980, "The interpretation of dummy variables in semi-logarithmic equations." American Economic Review 70: 474-475. A more recent one is Giles, 2011, "Interpreting Dummy Variables in Semi-logarithmic Regression Models: Exact Distributional Results." Econometric Working paper EWP1101. University of Victoria, Canada.

[Patrick James Lim]

Thank you guys for clarifying it for me and also for the additional references. Truly appreciated.

Jeff: One reason that led me to believe it was wrong is that when I do xtreg on these variables, I get the results above.
However, when I use
reg lnGDP Dummy CronyNW Crony RuleofLaw GovFreedom TradeOpen Democracy WI lnPop UnemploymentILO if Year == 2015 instead, I get these results:

. reg lnGDP Dummy CronyNW Crony RuleofLaw GovFreedom TradeOpen WI lnPop UnemploymentILO if Year == 2015

      Source |       SS       df       MS              Number of obs =      66
-------------+------------------------------           F(  9,    56) =   36.40
       Model |  136.735354     9  15.1928171           Prob > F      =  0.0000
    Residual |  23.3750216    56    .4174111           R-squared     =  0.8540
-------------+------------------------------           Adj R-squared =  0.8305
       Total |  160.110375    65  2.46323654           Root MSE      =  .64607

---------------------------------------------------------------------------------
          lnGDP |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
         Dummy |  -.4376796   .1929351    -2.27   0.027    -.8241751   -.0511841
        CronyNW |   .0002328   .0021365     0.11   0.914    -.0040471    .0045127
          Crony |   .0047803   .0083029     0.58   0.567    -.0118523     .021413
      RuleofLaw |    .032902   .0064169     5.13   0.000     .0200474    .0457566
     GovFreedom |  -.0145972   .0105761    -1.38   0.173    -.0357837    .0065892
      TradeOpen |   .0017649   .0101066     0.17   0.862    -.0184811    .0220109
             WI |  -.0094197   .0055164    -1.71   0.093    -.0204703     .001631
          lnPop |   .7457445   .0772646     9.65   0.000     .5909649     .900524
UnemploymentILO |   .0025634   .0176787     0.15   0.885    -.0328513    .0379781
          _cons |   12.98681   1.729195     7.51   0.000     9.522812     16.4508

Additionally, I tried regressing it on other years separately (My panel data is from 2006-2016 btw) and none of them made the Coefficient of Dummy variable to drop below 0.10.
So I thought my interpretations were incorrect.

But now you've all cleared it up, I'm wondering what may have caused the Dummy variable coefficient to drop significantly when using xtreg?
Should I post another thread for this? Thank you all

[Carlo Lazzaro]

Patrick:
as a sidelight, there are also coefficients flipping their sign and loosing statistical significance (for what it worths) when you move from -xtreg, re- to -regress-...if Year==2015. Have you already checked heteroskedasticity and (the so called) omitted variable bias in your last code?

[Clyde Schechter]

-regress- and -xtreg- estimate different things. There is no reason to expect that their results will be the same, or even remotely similar. -xtreg, re- incorporates panel level effects, thereby adjusting, at least to some extent, for omitted variable bias. -regress- does not. Notice in your -xtreg, re- output that rho is quite large. That means that these individual level effects are not ignorable.

[Patrick James Lim]

I think I may have a problem with heteroskedasticity

. estat ovtest

Ramsey RESET test using powers of the fitted values of lnGDPppp
       Ho:  model has no omitted variables
                 F(3, 147) =      1.10
                  Prob > F =      0.3510

. estat hettest

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity 
         Ho: Constant variance
         Variables: fitted values of lnGDPppp

         chi2(1)      =    13.18
         Prob > chi2  =   0.0003

. estat imtest, white

White's test for Ho: homoskedasticity
         against Ha: unrestricted heteroskedasticity

         chi2(53)     =     82.30
         Prob > chi2  =    0.0061

Cameron & Trivedi's decomposition of IM-test

---------------------------------------------------
              Source |       chi2     df      p
---------------------+-----------------------------
  Heteroskedasticity |      82.30     53    0.0061
            Skewness |       9.72      9    0.3739
            Kurtosis |       0.02      1    0.8754
---------------------+-----------------------------
               Total |      92.04     63    0.0099
---------------------------------------------------

Any thoughts on what should I do?

[Carlo Lazzaro]

Patrick:
you can go -robust- with -regress-.
However, keep in mind Clyde's wise advice about the difference betweem -regress- and -xtreg-.

[Patrick James Lim]

Will do. Thank you all for the advice!