How one interprets the coefficients in regression models will be a function of how the dependent (y) and independent (x) variables are measured. In general, there are three main types of variables used in econometrics: continuous variables, the natural log of continuous variables, and dummy variables. In the examples below we will consider models with three independent variables:
x1i a continuous variable
ln(x2i) the natural log of a continuous variable
x3i a dummy variable that equals 1 (if yes) and 0 (if no)
Listed below are three models. In each case, the right-hand side variables are the same, but the dependent variables differ. In each of these regressions, the dependent variable will be measured either as a continuous variable, the natural log or a dummy variable. Define the following dependent variables:
y1i a continuous variable
ln(y2i) the natural log of a continuous variable
y3i a dummy variable that equals 1 (if yes) and 0 (if no)
Below each model is text that describes how to interpret particular regression coefficients.
Model 1: y1i = β0 + x1iβ1 + ln(x2i)β2 + x3iβ3 + εi
β1 =∂y1i/∂x1i = a one unit change in x1 generates a β1 unit change in y1i
β2 =∂y1i/∂ln(x2i) = a 100% change in x2 generates a β2 change in y1i
β3 = the movement of x3i from 0 to 1 produces a β3 unit change in y1i
Model 2: ln(y2i) = β0 + x1iβ1 + ln(x2i)β2 + x3iβ3 + εi
β1 =∂ln(y2i)/∂x1i = a one unit change in x1 generates a 100*β1 percent change in y2i
β2 =∂ln(y1i)/∂ln(x2i) = a 100% change in x2 generates a 100*β2 percent change in y2i
β3 = the movement of x3i from 0 to 1 produced a 100*β3 percent change in y2i
Model 3: y3i = β0 + x1iβ1 + ln(x2i)β2 + x3iβ3 + εi
β1 =∂y3i/∂x1i = a one unit change in x1 generates a 100*β1 percentage point change in the probability y3i occurs
β2 =∂y3i/∂ln(x2i) = a 100% change in x2 generates a 100*β2 percentage point change in the probability y3i occurs
β3 = the movement of x3i from 0 to 1 produced a 100*β3 percentage point change in the probability that y3i occurs
