Exercise 2.19
(a)
Figure xr2.19(a) Scatter plot of selling price and living area
(b) The estimated linear relationship is
We estimate that an additional 100 square feet of living area will increase the expected home price by $9,893.40 holding all else constant. The estimated intercept −35.9664 would imply that a house with zero square feet has an expected price of $−35,966.40. This estimate is not meaningful in this example. The reason is that there are no data values with a house size near zero.
Figure xr2.19(b) Fitted linear relation
Exercise 2.19 (continued)
(c) The estimated quadratic equation is
The marginal effect is . For a house with 1500 square feet of living area the estimated marginal effect is 2(0.2278)15 = 6.834. We estimate that an additional 100 square feet of living area for a 1500 square foot home will increase the expected home price by $6,834 holding all else constant.
(d)
Figure xr2.19(d) Fitted linear and quadratic relations
The sum of squared residuals for the linear relation is SSE = 1,879,826.9948. For the quadratic model the sum of squared residuals is SSE = 1,795,092.2112. In this instance, the sum of squared residuals is smaller for the quadratic model, one indicator of a better fit.
(e) If the quadratic model is in fact “true,” then the results and interpretations we obtain for the linear relationship are incorrect, and may be misleading.
Exercise 2.20
(a) The estimates are reported in the table below. Of the 1200 homes in the sample, 69 are on large lots. None of the estimated intercepts has a useful interpretation because no houses in the samples have near zero living area. The estimated slope coefficients suggest that for houses on large lots, a 100 square foot increase in house size will increase expected price by $9,763.20, holding all else fixed. For houses not on large lots the estimate is $9,289.70, about $500 less than for houses on large lots. The full sample estimate is $9,893.40, which is between the estimates for homes on large lots and not on large lots.
CLIVAREANSSE
LGELOT = 1Coeff5.01999.763269490972.8
Std. err.(25.6709)(1.0014)
LGELOT = 0Coeff−28.74769.289711311271831.3
Std. err.(3.1374)(0.1884)
AllCoeff−35.96649.893412001879827.0
Std. err.(3.3085)(0.1912)
(b) The estimates are reported in the table below. Of the 1200 homes in the sample, 69 are on large lots. None of the estimated intercepts has a useful interpretation because no houses in the samples have near zero living area. The estimated coefficients of are somewhat different for houses on large lots and those not on large lots.
CLIVAREANSSE
LGELOT = 1Coeff120.70250.172869538400.4
Std. err.(16.6150)(0.0192)
LGELOT = 0Coeff52.25750.236811311128980.3
Std. err.(1.5431)(0.0044)
AllCoeff56.45720.227812001795092.2
Std. err.(1.6955)(0.0043)
To evaluate the differences, it is useful to calculate the slope, . For homes with 2000 square feet of living area the estimated slopes are
Large lots: 6.91128; Not Large lots: 9.471073; All lots: 9.112585
That is, we estimate that for a 2000 square foot home, 100 more square feet of living area, the expected price will increase by $6,911 for homes on large lots, $9,471 for homes not on large lots, and $9,113 based on all lots. The difference between the marginal effect of house size on house price for large lots and not large lots is substantial. The estimate using all the data is close to the estimate on lots that are not large because most of the data comes from such lots.
Exercise 2.20 (continued)
(c)
In this model is the expected price of houses not on large lots, and is the expected price of houses on large lots. Inserting the estimates, we obtain
That is, the expect price of houses on lots that are not large is $117,948.70 and the expected price of houses on large lots is $234,242.80. The expected price on large lots is about twice the expected price of houses on lots that are not large.