If an equation has more than one independent variable, we have to be careful when we interpret the regression coefficients of that equation. Think, for example, about how you might build an equation to explain the amount of money that different states spend per pupil on public education. The more income a state has, the more they probably spend on public schools, but the faster enrollment is growing, the less there would be to spend on each pupil. Thus, a reasonable equation for per pupil spending would include at least two variables: income and enrollment growth:

S_i = β₀ + β₁Y_i + β₂G_i + ε_i

where: S_i = educational dollars spent per public school student in the ith state

Y_i = per capita income in the ith state (in dollars)

G_i = the percent growth of public school enrollment in the ith state

a. State the economic meaning of the coefficients of Y and G.

b. If we were to estimate Equation 1.24, what signs would you expect the coefficients of Y and G to have? Why?

c. Silva and Sonstelie estimated a cross-sectional model of per student spending by state that is very similar to Equation 1.24:12

Ŝ_i = – 183 + 0.1422Yi - 5926G_i

N = 49

Do these estimated coefficients correspond to your expectations? Explain Equation 1.25 in common sense terms.

d. The authors measured G as a decimal, so if a state had a 10 percent growth in enrollment, then G equaled .10. What would Equation 1.25 have looked like if the authors had measured G in percentage points, so that if a state had 10 percent growth, then G would have equaled 10?