(d)
Figure xr2.17(d) Observations and quadratic fitted line
(e) The estimated elasticity is
For a 2000 square foot house, we estimate that a 1% increase in house size will increase expected price by 0.882%, holding all else fixed.
(f) The residual plots are
Figures xr2.17(f) Residuals from linear and quadratic relations
Exercise 2.17(f) (continued)
In both models, the residual patterns do not appear random. The variation in the residuals increases as SQFT increases, suggesting that the homoskedasticity assumption may be violated.
(g) The sum of square residuals linear relationship is 5,262,846.9. The sum of square residuals for the quadratic relationship is 4,222,356.3. In this case the quadratic model has the lower SSE. The lower SSE means that the data values are closer to the fitted line for the quadratic model than for the linear model.
Exercise 2.18
(a) The histograms for PRICE and are below. The distribution of PRICE is skewed with a long tail to the right. The distribution of is more symmetrical
Figures xr2.18(a) Histograms for PRICE and ln(PRICE)
(b) The estimated log-linear model is
The estimated slope can be interpreted as telling us that a 100 square foot increase in house size increases predicted price by approximately 3.6%, holding all else fixed. The estimated intercept tells us little as is. But suggests that the predicted price of a zero square foot house is $80,953. This estimate has little meaning because in the sample there are no houses with zero square feet of living area.
For a 2000 square foot house the predicted price is
The estimated slope is
Exercise 2.18 (continued)
The predicted price of a house with 2000 square feet of living area is $166,460.10. We estimate that 100 square foot size increase for a house with 2000 square feet of living area will increase price by $6,000, holding all else fixed. This is the slope of the tangent line in the figure below.
Figure xr2.18(b) Observations and log-linear fitted line
(c) The residual plot is shown below. The residual plot is a little hard to interpret because there are few very large homes in the sample. The variation in the residuals appears to diminish as house size increases, but that interpretation should not be carried too far.
Figure xr2.18(c) Residuals from log-linear relation
(d) The summary statistics show that there are 189 houses close to LSU and 311 houses not close to LSU in the sample. The mean house price is $10,000 larger for homes close to LSU, and the homes close to LSU are slightly smaller, by about 100 square feet. The range of the data is smaller for the homes close to LSU, and the standard deviation for those homes is half the standard deviation of homes not close to LSU.
Exercise 2.18 (continued)
CLOSE = 1CLOSE = 0
STATSPRICESQFTPRICESQFT
N189189311311
mean256.629826.59011246.351827.70267
sd108.58788.735512200.350511.05563
min110105010
max90059.73137091.67
(e) The estimates for the two sub-samples are
CSQFTNSSE
CLOSE = 1Coeff4.76370.026918914.2563
Std. err.(0.0645)(0.0023)
CLOSE = 0Coeff4.20190.040231136.6591
Std. err.(0.0528)(0.0018)
For homes close to LSU we estimate that an additional 100 square feet of living space will increase predicted price by about 2.69% and for homes not close to LSU about 4.02%.
(f) Assumption SR1 implies that the data are drawn from the same population. So the question is, are homes close to LSU and homes not close to LSU in the same population? Based on our limited sample, and using just a simple, one variable, regression model it is difficult to be very specific. The estimated regression coefficients for the sub-samples are different, the question we will be able to address later is “Are they significantly different.” Just looking at the magnitudes is not a statistical test.