How do I interpret coefficients when the dependent variables is in % and the independent is in log-form?

[Kristina Petrova]

I set up a model with a dependent variable in % (unemployment as a % of total population) and independent variables, which are in logs. For example, I get that the coefficient of log(consumption) is -0.164. How do I interpret the size of this coefficient (percent, percentage points, etc.)? Thanks in advance.

[Justin Niakamal]

Sounds like you have a linear-log model.

The standard interpretation is: the expected change in Y associated with a p% increase in X can be calculated as β * log([100 + p]/100).

di -0.164 * ln(((100+10)/100))
-.01563087

So substituiting B, X, and Y --

A 10% increase in consumption is associated with a -0.0156 decrease in unemployment as a % of total population.

This is how I usually work through these, but others might have a slightly different approach.

[Andrew Musau]

You just state that the coefficient on log consumption is -0.164. Knowing that you have a level-log model, anyone reading this result will understand that if we increase consumption by one percent, we expect that the rate of unemployment will decrease by 0.164/100. If you are unsure of the interpretation, you should always use an example to work it out. Below, I set up a variable y that is in percent (values within [0, 100]) and a positive continuous variable x1.

set obs 20
set seed 05232021
gen y= runiformint(0,100)
gen x1= runiformint(20,10000)
gen lnx1= ln(x1)
*LEVEL-LOG MODEL
regress y lnx1
*PREDICY Y
predict yhat1, xb
*INCREASE X1 BY 1 PERCENT
replace x1= x1+(0.01*x1)
*REPLACE LN(X1)
replace lnx1=ln(x1)
*PREDICY Y AGAIN
predict yhat2, xb
*CALCULATE  DELTA Y & NOTE= COEF(lnx1)/100
gen Dy= yhat2-yhat1
format Dy %9.3f
l y Dy, sep(0)
di %9.3f _b[lnx1]/100

Res.:

. regress y lnx1

      Source |       SS           df       MS      Number of obs   =        20
-------------+----------------------------------   F(1, 18)        =      1.65
       Model |  1294.07318         1  1294.07318   Prob > F        =    0.2153
    Residual |  14118.1268        18  784.340379   R-squared       =    0.0840
-------------+----------------------------------   Adj R-squared   =    0.0331
       Total |     15412.2        19  811.168421   Root MSE        =    28.006

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        lnx1 |   6.692451   5.210244     1.28   0.215    -4.253866    17.63877
       _cons |  -22.18701   42.87929    -0.52   0.611     -112.273    67.89903
------------------------------------------------------------------------------

.
. l y yhat1 yhat2 Dy, sep(0)

     +-----------------------------------+
     |   y      yhat1      yhat2      Dy |
     |-----------------------------------|
  1. |   9   31.59087   31.65746   0.067 |
  2. |   0   33.78894   33.85552   0.067 |
  3. |  48   30.41173   30.47832   0.067 |
  4. |  10   33.71048   33.77707   0.067 |
  5. |  28   11.52255   11.58914   0.067 |
  6. |  49   37.74344   37.81002   0.067 |
  7. |  15   28.97304   29.03963   0.067 |
  8. |  97   39.28468   39.35127   0.067 |
  9. |  61   34.49172    34.5583   0.067 |
 10. |  32   39.10453   39.17113   0.067 |
 11. |  16   37.73306   37.79966   0.067 |
 12. |   8   38.52382   38.59041   0.067 |
 13. |  18   32.40196   32.46855   0.067 |
 14. |   8   8.497664   8.564255   0.067 |
 15. |   6   37.74084   37.80743   0.067 |
 16. |  47   33.54266   33.60925   0.067 |
 17. |  22   30.05893   30.12552   0.067 |
 18. | 100    37.5586    37.6252   0.067 |
 19. |  46   36.02514   36.09173   0.067 |
 20. |  26   33.29537   33.36197   0.067 |
     +-----------------------------------+


.
. di %9.3f _b[lnx1]/100
    0.067

As the dependent variable is in percentage values, a one percent increase in x1 is associated with a 0.067 percentage points increase in y. From the listed values, note that the change is constant across values of y.

[Eric de Souza]

In post #1 above, the expression: "% (unemployment as a % of total population)" doesn't make sense to me. You are taking the level of unemployment, dividing it by the level of the total population, expressing the result as a percentage, and then again taking the percentage of the result. I presume that the % sign outside the parentheses is superfluous.