(d) Assumption SR1 requires that the data pairs in the sample are from the same population. If there are substantial differences between homes on lots and those not on large lots then SR1 will be violated meaning that estimation results on a pooled sample are not reliable. The result in part (c) indicates that there may be large differences between homes on these types of lots. What will be of interest later, in Chapter 3, is whether the difference is statistically significant.
Exercise 2.21
(a)
We estimate that a house that is new, AGE = 0, will have expected price $152,614.40. We estimate that each additional year of age will reduce expected price by $981.20, other things held constant. The expected selling price for a 30-year-old house is .
(b)
Figure xr2.21(b) Observations and linear fitted line
The data show an inverse relationship between house prices and age. The data on newer houses is not as close to the fitted regression line as the data for older homes.
(c)
We estimate that each additional year of age reduces expected price by about 0.75%, holding all else constant.
Exercise 2.21 (continued)
(d)
Figure xr2.21(c) Observations and log-linear fitted line
The fitted log-linear model is not too much different than the fitted linear relationship.
(e) The expected selling price of a house that is 30 years old is . This is about $13,000 less than the prediction based on the linear relationship.
(f) Based on the plots and visual fit of the estimated regression lines it is difficult to choose between the two models. For the estimated linear relationship . For the log-linear model . The sum of squared differences between the data and fitted values is smaller for the estimated linear relationship, by a small margin. This is one way to measure how well a model fits the data. In this case, based on fit alone, we might choose the linear relationship rather than the log-linear relationship.
Exercise 2.22
(a) The regression model is . Under the model assumptions
Thus is the expected total score in regular sized classes, and is the expected total score in small classes. The difference is an estimate of the difference in performance in small and regular sized classes. The model estimates are given in Table xr2-22a, Model (1).
Table xr2-22a
CSMALLNSSE
TOTALSCORECoeff916.441712.17537754300389
Std. err.(3.6746)(5.3692)
READSCORECoeff432.66506.9245775705200
Std. err.(1.4881)(2.1743)
MATHSCORECoeff483.77675.25087751910009
Std. err.(2.4489)(3.5783)
The estimated equation using a sample of small and regular classes (where AIDE = 0) is
Comparing a sample of small and regular classes, we find students in regular classes achieve an average total score of 916.442 while students in small classes achieve an average of . This is a 1.33% increase. This result suggests that small classes have a positive impact on learning, as measured by higher totals of all achievement test scores.
(b) The estimated equations using a sample of small and regular classes are given in Table xr2-22a as Models (2) and (3)
Students in regular classes achieve an average reading score of 432.7 while students in small classes achieve an average of 439.6. This is a 1.60% increase. In math students in regular classes achieve an average score of 483.77 while students in small classes achieve an average of 489.0. This is a 1.08% increase. These results suggests that small class sizes also have a positive impact on learning math and reading.