E303 Davis
Spring 2005, Problem Set #7
Regression Analysis
Consider the following data
Sales Adv
Yi Xi
3 1
4 2
6 3
5 4
7 5
6 6
5 7
9 8
10 9
9 10
1. Input this data on a spreadsheet. Using the regression option, generate regression predictions. Write your results as an equation, as we did in class. In particular,
(a) Write the regression equation, with estimated coefficients,
(b) Below the regression equation, list in parentheses the standard errors of the coefficient estimates
(c) To the right of the estimated equation write R2 =
^
Yi = 2.733 + 0.667 Xi R2 = .757, adj R2 = .727
(0.827 (0.133) F1,8 = 25 p =.001
MSE = 1.211
^
Equation: ____Y_=___2.733____+ 0.667Xi______R2 = .757
Std Errors (0.827 ) ( 0.133 )
2. Multivariate Regression. Now add to your above regression in a price variable, with values: 8, 7.5, 7.25, 7.25, 6, 6.75, 6, 5, 4.4, 5.2. Estimate the new regression equation. Print regression results. Write out the estimated demand function, as in 1 above.
Equation: ____Y_=__17.68_ + 0.005 – 1.78 ______R2 = 0.868
Std Errors ( 6.19) (0.291 ) ( 0.734)
3. Evaluating regression results: A Descriptive Statistic. With the data you generated in (2) above do the following.
a. Interpret the R2. (In a sentence)
86.8% of the movement in the sales is explained by movements in price and advertising.
______
b. Observe that we can do the same exercises with this information that we did previously. In particular, suppose that advertising expenditures are 10 and that the current price is $6. Calculate the point price elasticity of demand.
^
Yi = 17.68 + 0.005 (10) - 1.78 (6)
= 7.05
Thus
h = -1.78(6)/7.05 = -1.51
.(Note: Parts c, d and e weren’t on your assigned problem, but I would like you to be able to do thems for the examination.)
c. For each coefficient in part a, write the approximate 95% confidence intervals about the estimates. From this data, which parameters are significant explainers of sales?
Estimate Interval Significant?
Intercept 3.05 to 32.31 Yes
Advertising -0.68 to 0.69 No
Price -3.52 to -0.048 Yes
d. What is the MSE of your regression? Using advertising expenditures of 10 and a a price of $6, forecast sales, and calculate an approximate 95% confidence interval for your forecast
MSE = 0.95.
^
A 95% confidence band is Yi + 2(MSE)
^
From b, Yi = 7.05. Thus 7.05 – 2(0.95) to 7.05 + 2(0.95)
Or 5.15 to 8.95
e. Identify two things that would make you more confident of your forecast?
1. You use values for dependent variables like those you have seen previously
2. There is no reason outside the regression to expect the future to look different from the past