Question

4. Consider the model for demand for housing: Log Q = β0 + β1 log P + β2 log Y + u Where Q = measure of quantity of housing consumed by each of 3, 120 families per year P = price of unit of housing in family's locality Y = measure of family income The estimated model, (with standard errors in parentheses) is: Log Q = 4.17 - 0.247 log P + 0.96 log Y R2 = .371 (0.11) (0.017) (0.026) a. Interpret the slope coefficients. Test for individual significance at 5% confidence. b. Test whether β2 is statistically different from 1. What is the economic interpretation of such test? Now, suppose that we wish to know whether the demand for housing of blacks differ from that of whites. If we let D represent a dummy variable equal to 1 for black household and 0 otherwise, the expanded model is Log Q = β0 + δ0 D + β1 log P + δ1 D log P + β2 log Y + δ2 D log Y + u The estimated model is (with standard errors in parentheses): Log Q = 4.17 – 0.22 log P + 0.920 log Y + 0.006 D – 0.114 D log P + 0.341 D log Y (0.11) (0.02) (0.031) (0.042) (0.061) (0.120) R2 = 0.380 a. Is there a difference in demand for housing between black and non-black households? Explain the test you use and the decision. b. Which model do you think is better?

Answer 1

a).

Consider the given problem here the estimated model is given by.

=> LogQ = 4.17 - 0.247*LogP + 0.96*LogY, => “b1=(-0.247)”, => if “P” increases by “1%”, => the “Q” decreases by “0.247%”, => it measure the price elasticity of demand. Now, “b2=0.96”, => if “Y” increases by “1%”, => the “Q” decreases by “0.96%”, => it measure the income elasticity of demand.

Now, the “SE” of “b1” is “0.017”, => the “t-statistic” is given by “t=(-0.247/0.017)=(-14.53)”, => the absolute value is more than “2”, => for a large sample “P” is significant at “5%” level of significance.

Similarly, the “SE” of “b2” is “0.026”, => the “t-statistic” is given by “t=(0.96/0.026)=36.92”, => the absolute value is more than “2”, => for a large sample “Y” is also significant at “5%” level of significance.

b).

Now, the estimate value of “b2” is “0.96”, => the H0 is given by, “H0: b2=1”, => the H1: b2 not equal to “1”, => given the “H0” the “t-statistic” is given by.

=> t = (0.96-1)/0.026 = (-1.54), => the absolute value is less than “2”, => for a large sample “b2” is not significantly differ from “1” at “5%” level of significance.

Now, assume that “D” be a dummy variable takes “1” for “black household” and “0” otherwise. So, the new estimated model is given by.

=> LogQ = 4.17 - 0.22*LogP + 0.92*LogY + 0.006*D - 0.114*D*LogP + 0.341*D*LogY.

Now, the coefficient of “D” is “0.006” and its SE is “0.042”, => the “t-statistic” is given by, “t=0.006/0.042=0.14”, => the absolute value is less than “2”, => for a large sample “intercept dummy” is not significant at “5%” level of significance, => both the category should have same intercept term.

Similarly, the coefficient of “D*LogP” is “-0.114” and its SE is “0.061”, => the “t-statistic” is given by, “t=(-0.114/0.061)=(-1.87)”, => the absolute value is less than “2”, => for a large sample “slope dummy D*LogP” is not significant at “5%” level of significance, => both the category should have same intercept term.

Finally, the coefficient of “D*LogY” is “0.341” and its SE is “0.12”, => the “t-statistic” is given by, “t=0.341/0.12=2.84”, => the absolute value is more than “2”, => for a large sample “slope dummy D*LogY” is significant at “5%” level of significance, => both the category should have same intercept term.

So, the demand for housing is differ across “black” and “non-black households”.

b).

Now, the “R^2” value of the former model is “R^2=0.371” and the same for the later model is “R^2=0.38”, => the later model has higher “R^2” value, => the later one is better model compare to the former one.