It is the same given the accurate graph.
(f) Since . The sum of squared residuals from the least squares regression including the explanatory variable is . We are able to “fit” the model to the data much better by including the explanatory variable.
Exercise 2.11
(a) We estimate that each additional $100 per month income is associated with an additional 52 cents per person expenditure, on average, on food away from home. If monthly income is zero, we estimate that household will spend an average of $13.77 per person on food away from home.
(b) . We predict that household with $2000 per month income will spend on average $24.17 per person on food away from home.
(c) In this linear relationship, the elasticity is . We estimate that a 1% increase in income will increase expected food expenditure by 0.43% per person.
(d) In this log-linear relationship, the elasticity is .
(e)
It is increasing at an increasing rate. This is shown on Figure xr2.11. Also, the second derivative, the rate of change of the first derivative is . A positive second derivative means that the function is increasing at an increasing rate for all values of x.
Figure xr2.11 Log-linear plot
(f) The number of zeros is 2334 – 2005 = 329. The reason for the reduction in the number of observations is that the logarithm of zero is undefined and creates a missing data value. The software throws out the row of data when it encounters a missing value when doing its calculations.
Exercise 2.12
(a) The model estimates for the two values of x are
We estimate that a household without an advanced degree holder will spend on average $44.96 per month on food away per person. We estimate that a household with an advanced degree holder will spend on average $75.37 per month on food away per person. The coefficient on x is the difference between the average expenditures per month on food away for households with an advanced degree holder and households without an advanced degree holder. The intercept is the average expenditure per month on food away for a household without an advanced degree holder.
(b) In this sample, for households with a member having an advanced degree, their average expenditure on food away from home is $75.37 per person.