# Interpreting Poisson regression coefficients

I'm having a little trouble interpreting regression coefficients from a Poisson model.

Wooldridge, for example, says: "The Poisson coefficient implies that $\Delta_{pcnv}=.10$ reduces the expected number of arrests by about 4% [.402(.10) = .0402, and we multiply this by 100 to get the percentage effect]"

But later on, he says "The Poisson coefficient on black implies that, other factors being equal, the expected number of arrests for a black man is estimated to be about $100\cdot[\exp(.661)-1]=93.7\%$ higher than for a white man with the same values for the other explanatory variables."

I'm confused as to when I should just multiply the coefficient by $\Delta_{x_i}$ or use $100\cdot[\exp(x_i)-1]$.

When is it just a change in % and when am I "counting" the number of incidences?

Is it because one is a dummy variable and the other is not?

How would I interpret the following output? Here kids is the number of kids, educ is the number of education years, age is the age, black is a dummy variable and all the rest are year dummies.

. poisson kids educ age agesq black y76 y78 y80 y82 y84

Iteration 0:   log likelihood = -2078.5379
Iteration 1:   log likelihood = -2078.5379

Poisson regression                              Number of obs     =      1,129
LR chi2(9)        =     130.18
Prob > chi2       =     0.0000
Log likelihood = -2078.5379                     Pseudo R2         =     0.0304

------------------------------------------------------------------------------
kids |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ |  -.0424093   .0069363    -6.11   0.000    -.0560043   -.0288143
age |   .1989907   .0545383     3.65   0.000     .0920977    .3058837
agesq |  -.0021678   .0006149    -3.53   0.000     -.003373   -.0009626
black |   .3193816   .0588883     5.42   0.000     .2039627    .4348005
y76 |  -.0831634   .0578159    -1.44   0.150    -.1964805    .0301537
y78 |  -.0641177   .0591386    -1.08   0.278    -.1800273    .0517918
y80 |  -.0691123   .0591021    -1.17   0.242    -.1849502    .0467257
y82 |   -.251696    .057926    -4.35   0.000    -.3652289   -.1381631
y84 |   -.262424   .0599642    -4.38   0.000    -.3799516   -.1448964
_cons |  -2.870521   1.202454    -2.39   0.017    -5.227287   -.5137548


Edit: I checked the link, but I still have the doubts above. When do I just multiply the change by the coefficient and when do I calculate using exponents? When am I calculating a change and when am I "counting" (whether it increases or decreases and by how many "units") the number of kids given a value of a variable?

Edit 2: I apologize. It's Wooldidge's Introductory Econometrics 5th Ed.: A Modern Approach (page 608).

Edit 3: I think my question sort of narrows down to: which of the following would be a correct interpretation? (or are both correct?)

1. Having 10 years of education decreases the number of kids by $e^{(-0.042*10)}=0.657$ kids (units)

2. An increase in one year of education decreases the number of kids by $\left(e^{(-0.042*1)}-1\right )\cdot 100=-4.11\%$ (percent).

• – gung May 3 '18 at 18:47
• "Wooldridge" is an insufficient reference, even if you assume that you're addressing economists (on CV, that's largely wrong). Which book and edition (or paper as the case may be)? Page references would do no harm. – Nick Cox May 3 '18 at 19:18
• I believe it's a ratio, not a count. The coefficients are the difference in the log rate for the effect (+1 year of educ) and the mean log rate, so $e^{-0.042}=0.96$ means each additional year of education results in 0.96 times as many kids. That's my understanding, anyway. – dankernler May 4 '18 at 4:37
• If you were to input the margins command in Stata like: margins, at(educ=12 age=30 agesq=900 black=1 y76=0 y78=0 y80=0 y82=0 y84=1) it gives you a _cons margin value of 2.00619. Is that still a ratio or is that the number of kids for a person with such characteristics? – user197221 May 4 '18 at 17:49

## 1 Answer

Let's say you were hired last year by a firm on a starting salary of 100,000 dollars. After one year of excellent performance on the job, you receive a raise and your new salary is 120,000 dollars. You can say that:

• Compared to last year, your new salary changed by a multiplicative factor of 1.2, since 120,000 = 100,000 x 1.2;
• Compared to last year, your salary increased by 20%, since (1.2 - 1) x 100% = 20%.

Both of the above statements convey the magnitude of your salary increase. However, the first one uses a multiplicative factor to do so (i.e., salary increased by a multiplicative factor of 1.2) while the second one uses a percentage (i.e., salary increased by 20%).

If your performance on the job in the first year was sub-par and your salary decreased from 100,000 dollars to 80,000 dollars, you would say that:

• Compared to last year, your new salary changed by a multiplicative factor of 0.8, since 80,000 = 100,000 x 0.8;
• Compared to last year, your salary decreased by 20%, since (0.8 - 1) x 100% = -20%.

I find this simple example helps a lot when it comes to interpreting estimated coefficients in a Poisson regression model.

The first thing you need to understand for this type of model is that you need to exponentiate the reported model coefficients to get your multiplicative factors. Once you get the multiplicative factors, you can present your results in one or both ways described above.

For example, exponentiate the coefficient of education, -0.0424093, to get the multiplicative factor 0.96. Then you can say that:

All else being equal, the expected number of arrests changes by a multiplicative factor of 0.96 for each additional year of education (i.e., a (1-0.96) x 100% = 4% decrease in expected number of arrests associated with each additional year of education).

If you wanted to report what happens for 10 additional years of education, you would multiply the model coefficient by 10, then exponentiate it to get your multiplicative factor. After that, the interpretation follows the same line as above.