Exercise 1: Arithmetic of OLS The point of this exercise is to demonstrate some of the arithmetic properties of residuals, fitted values, and the R2 statistic discussed in Section 2.3.  Run the...

Exercise 1: Arithmetic of OLS
The point of this exercise is to demonstrate some of the arithmetic properties of
residuals, fitted values, and the R2 statistic discussed in Section 2.3.
 Run the regression of weight on height. Interpret the slope and intercept.
 Create the residuals and confirm they have mean zero (see Eq 2.14)
 Create the fitted values (Yhats), and confirm that the average of Yhat is the
same as the average of Y
 Confirm that the residuals are uncorrelated with X and with fitted values
 Confirm that R2 is the ratio of the variance of the fitted values to var(Y)
 Confirm that R2 is the square of the correlation between fitted values and Y
 Calculate SSE, SSR, and SST and confirm that these are same as Stata reports in
the upper left as “Model”, “Residual” and “Total”. Recall that SSE = (n1)*Var(Yhat)
Exercise 2: Conditional Means and the Standard Error of Regression (SER)
The point of this exercise is to demonstrate that regression is about calculating
conditional expectations; you will also calculate the standard error of the regression
(SER) and the standard error of
1
ˆ

using the formula from Section 2.5.
 Run a regression of height on female
 Use summarize to find the average height for men and confirm that it equals the
intercept
 Use summarize to find the average height for women and confirm that the
regression coefficient on female is the difference between the male and female
average heights
 Generate the residuals and apply Eq. 2.61 to find the standard error of the
regression (SER). Confirm this comes out the same as what Stata reports as the
Root Mean Square Error (Root MSE).
 Use this result to find the standard error of the parameter estimate for height
and confirm that this is what Stata reports. Recall that SSTx = (n-1)*Var(x).
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Exercise 3: Units of Measurement
This exercise comes from Section 2.4, and looks at the effect of changing units of
measurement.
 Download and open the dataset CEOSAL1.DTA.
 Study Example 2.3, and recreate the regression results. State carefully what
the slope coefficient means, specifying the units of measurement.
 Now follow the steps described for equation 2.40. Generate a variable that
measures salary in dollars instead of 1000s of dollars. (So multiply it by
1000). Confirm that if you use this new variable as the y-variable, the
estimated slope and intercept both increase by a factor of 1000. State
carefully what the slope coefficient means, specifying the units of
measurement. Does this transformation change the interpretation of the
regression results? Does R2 change?
 Now generate a variable that measures roe in decimals instead of percentages.
(So divide it by 100). Confirm that if you use this new variable as the xvariable,
the intercept does not change, but the slope increases by a factor of
100. State carefully what the slope coefficient means, specifying the units of
measurement. Does this transformation change the interpretation of the
regression results? Does R2 change?
Exercise 4: The Log-Level Functional Form
This exercise also comes from Section 2.4, and looks at the effect of changing the
functional form of a regression, using logs for the y-variable only. A linear
relationship between education and log(wage) is a nonlinear relationship between
education and wage (see Figure 2.6).
 Download the dataset WAGE1.DTA, and recreate the regression results from
Example 2.4.
 Generate a variable that measures the hourly wage in logarithms, and recreate
the results from Example 2.10.
These two regressions have fundamentally different interpretations: in one case we
impose an assumption that each year of schooling raises wages by a constant dollar
amount; in the second case we assume each year of schooling raises wages by a constant
percentage amount. This makes more sense: a year of grad school should be worth more
than a year of elementary school.