(b)
Figure xr2.24(b) Histogram of ln(real hammer price)
is not “bell shaped” but it is hardly skewed at all (skewness close to zero). It has been “regularized” by the transformation. This is not necessary for regression, but as you will see in Chapter 3 having data closer to normal makes analysis nice.
Exercise 2.24 (continued)
(c)
Figure xr2.24(c) Observations and log-linear fitted line
The data scatter shows a positive association between and the age of the painting. The fitted OLS regression line passes through the center of the data, as it is designed to do.
(d)
We estimate that each additional year of age increases predicted hammer price by about 2%, other factors held constant.
(e)
In this model, the expected is during non-recession and is in a recession. The estimated regression function during a recession is . We estimate that during a non-recessionary period the average hammer price is $12,867, using , and during a recession we predict the average price to be $4,539, using , more than a 50% reduction.
Exercise 2.25
(a)
Figure xr2.25(a) Histogram of foodaway
The mean of the 1200 observations is 49.27, the 25th, 50th and 75th percentiles are 12.04, 32.56 and 67.60. The histogram figure shows a very skewed distribution, with a mean that is larger than the median. 50% of households spend $32.56 per person or less during a quarter.
(b) Households with a member with an advanced degree spend an average of about $25 more per person than households with a member with a college degree, but not advanced degree. Households with a member with a college degree, but not advanced degree, spend an average of about $9 more per person than households with no members with a college or advanced degree.
NMeanMedian
ADVANCED = 125773.1548.15
COLLEGE = 136948.6036.11
NONE57439.0126.02
Exercise 2.25 (continued)
(c)
Figure xr2.25(c) Histogram of ln(foodaway)
The histogram of ln(FOODAWAY) is much less skewed. There are 178 fewer values of ln(FOODAWAY) because 178 households reported spending $0 on food away from home per person, and ln(0) is undefined. It creates a “missing value” which software cannot use in the regression. If any variable has a missing value in either yi or xi the entire observation is deleted from regression calculations.
(d) The estimated model is
We estimate that each additional $100 household income increases food away expenditures per person of about 0.69%, other factors held constant.
(e)
Figure xr2.25(e) Observations and log-linear fitted line
The plot shows a positive association between ln(FOODAWAY) and INCOMEs.
Exercise 2.25 (continued)
(f)
Figure xr2.25(f) Residuals vs. income
The OLS residuals do appear randomly distributed with no obvious patterns. There are fewer observations at higher incomes, so there is more “white space.”
Exercise 2.26
(a)
We estimate that a household with zero income in the past quarter will spend an average of $13.71 per member on food away from home. This estimate should not be taken too seriously because there are no households with income near zero in the sample. We estimate that each additional $100 household income increases expected food expenditure away from home by 49 cents, holding other factors fixed.