Interpretation of regression coefficients
For linear models, with E[y |x] = x'P, the coefficients в are readily interpreted as the effect of a one-unit change in regressors on the conditional mean. For nonlinear models this interpretation needs to be modified. For any model with exponential conditional mean, differentiation yields
^ = Pjexp(x'P), (15.7)
where the scalar Xj denotes the jth regressor. For example, if Sj = 0.25 and exp(x • S) = 3, then a one-unit change in the jth regressor increases the expectation of y by
0. 75 units. This partial response depends upon exp(x'S) which is expected to vary across individuals. It is easy to see that в;- measures the relative change in E [y |x] induced by a unit change in Xj. If Xj is measured on log-scale, в;- is an elasticity.
For purposes of reporting a single response value, a good candidate is an estimate of the average response, n YH=1 dE [ yi|xi]/3xi/- = Sj x n YH= 1exp(x-S). For Poisson regression models with intercept included, this can be shown to simplify
Another consequence of (15.7) is that if, say, в;- is twice as large as ви then the effect of changing the jth regressor by one unit is twice that of changing the kth regressor by one unit.