Interpreting Depend Variable in Percent (i.e. Growth Rate)

[Hannes Boehm]

Dear Stat List,

I know this question has been raised before in various forms, however, I haven't been able to find an encompassing answer that speaks to all parts of the problem.


I have a regression for which the dependent variable is a growth rate and measured in percent. I calculated it as follows: (ln(Y_t) - ln(Y_t-1)) * 100.
You could also derive it like this: ((Y_t - Y_t-1) / Y_t-1) * 100
Both versions correlate at 0.99 and yield nearly the same regression output. My variable is the growth rate of an index. However, you could also imagine the growth of GDP in percent or the sales growth of a firm (it's the general concept I'm interested in).
(Side note: "percent" here really means "growth rate" or "return", not "percentage share" (e.g. expenditure share on GDP) which also gets asked a lot in this context).

I run a OLS panel regression with this growth rate as the dependent variable, here called IndexGrowth. My right hand side variable of interest is Temperature (which is not transformed). Here is the output:

. reghdfe IndexGrowth Temperature, a(Country_ID Region_ID##Month_Year) cl(Country_ID)
(dropped 49 singleton observations)
(MWFE estimator converged in 9 iterations)

HDFE Linear regression                            Number of obs   =      9,963
Absorbing 2 HDFE groups                           F(   1,     53) =       8.94
Statistics robust to heteroskedasticity           Prob > F        =     0.0042
                                                  R-squared       =     0.5220
                                                  Adj R-squared   =     0.4373
                                                  Within R-sq.    =     0.0007
Number of clusters (Country_ID) =         54      Root MSE        =     2.9610

                            (Std. Err. adjusted for 54 clusters in Country_ID)
------------------------------------------------------------------------------
             |               Robust
 IndexGrowth |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
 Temperature |  -.2352437   .0786779    -2.99   0.004    -.3930516   -.0774357
       _cons |   .7727861    .032899    23.49   0.000     .7067991    .8387731
------------------------------------------------------------------------------

Absorbed degrees of freedom:
----------------------------------------------------------------+
            Absorbed FE | Categories  - Redundant  = Num. Coefs |
------------------------+---------------------------------------|
             Country_ID |        54          54           0    *|
   Region_ID#Month_Year |      1446           0        1446     |
----------------------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

.
end of do-file

My question concerns the interpretation of the temperature coefficient. Does it imply that a 1 unit increase in temperature leads to ...
a) a 0.23% decrease of the index level
b) a 0.23% decrease of the index growth rate
c) a 0.23%-point decrease of the index level
d) a 0.23%-point decrease of the index growth rate?


I apologize for this basic question, but l looked at many published articles with growth rates as dependent variables and you find all of these interpretations circulating in the literature. Econometrics textbook cover the "linear-linear", "log-linear", "linear-log" and "log-log" cases to great extends, but percentage variables not as often (or don't say explicitly to which category percentage variables belong).


Therefore, I am grateful for your help!


PS: I am aware that this is not strictly a Stata question, but with so much know-how in econometrics here, I am sure this question can be answered

[Carlo Lazzaro]

Hannes:
I would say something along the lines of d): 1 unit increase in temperature reduces the regressand of about 23 percentage points.

[Hannes Boehm]

Thanks Carlo, I think so too!