If an equation has more than one independent variable we have to careful when we

Question

If an equation has more than one independent variable we have to careful when we interpret the regression coefficients of that equation Think, for example, about how might build an per 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 each public schools, but the faster enrollment is growing less there would be to spend on each pupil. Thus, a reasonable equation for pupil spending would include at least two variables: income and enrollment growth: S_i = beta_0 + beta_1 Y_1 + beta_2 G_1 + element1 where: Si = educational dollars spent per public school student in the ith state Y_1 = per capita income in the ith state (in dollars) G_i = in percent growth of public school enrollment in the ith state State the economic meaning of the coefficients of Y and G. If we were to estimate Equation 1.24, what signs would you expect the coefficients of Y and G to have? Why? Silva and Sonstelie estimated a cross-sectional model of per student spending by state that is very similar to Equation 1.24:^12 _i = - 183 + 0.1422Y_i - 5926G_i N = 49 Do these estimated coefficients correspond to your expectations? Explain Equation 1.25 in common sense terms. 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?

Explanation / Answer

The given equation is S= b0 +b1y1 +b2g1 +e0

the coefficients are usually interpreted in terms of effects on dependent variable (s) for change in independent variables y1 and g1 . so this means that for every unit change in y1 , s would change by an effect of b1

similarly for every change in g1 , S would change by an effect of b2

as y is per capita income and g is percent change in educational enrollments, so if these values increase we would expect the dependent variable to increase as well, as we would expect to see a positive relationship between the independent and dependent variables.S is educational spent , so we shoudl expect to see an increase in the spent if the y and g increase

c) while Y corresponds to our estimates in terms of the positive relationship , G gives us a negative rleationship which means that as the percentage growth in enrollments decreases the spent on education would increase , this is not consistent woth what we would expect to see

S= -183 +0.1422Y1 - 5926G1

d) There is no data in the quesiton to arrive at the actual coefficient values , but the simple answer is Yes. If we are using absolute values we would get the coefficients in absolute terms , to convert this into percentage points , you can simply convert this into percentage equation by deviding the equation by 100

S= -183 +0.1422Y1 - 5926G1

would become

S= -1.83 +0.001422Y1 - 59.26 G1

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