Extra Problems, Set #2 Regression Analysis



Extra Problems, Set #2 Regression Analysis Key

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, write

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

i) R2 and adjusted R2

ii) F test statistic, along with its p value

iii) MSE of the estimate.

^

Yi = 2.7333 + .667Xi R2 = .758, Adj. R2 = .727

(0.83) (0.13) F1,8 = 25 (p = .001)

MSE = 1.211

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 part (c) above.

^

Yi = 17.683 + .0049Xi – 1.785Pi. R2 = .868, Adj. R2 = .131 (6.19) (.29) (.73) F2,7 =23.11 (p = .0008)

MSE = 0.954

3. Evaluating regression results. With the data you generated in (2) above do the following.

a. Interpret the R2.

R2 = .868 implies that 86.8% of the variation in data from average sales is explained by the regression.

b. What is the difference between R2 and adjusted R2?

Adjusted R2 is the percentage in the variation of data from the mean of average sales explained by the regression after controlling for the potential loss in explanatory power caused by adding additional independent variables.

c. Goodness of Fit Test. Looking at the F test statistic, can you conclude that the regression estimate explains more of the variation in sales than the simple average of sales? Explain.

F2,7 =23.11 (p = .0008). The F test statistic indicates (roughly) that the regression equation explains 23.11 times more of the variation in the data than it leaves unexplained. The p = .0008 indicates that 99.92% percent of the time a regression that explained no more of the movement in the data than the simple average of sales would explain as much as this estimate.

d. For each coefficient, write the 95% confidence intervals about the estimates. From this data, which parameters are significant explainers of sales?

|Coefficients |Estimate |Standard |Lower 95% |Upper 95% |

| | |Error | | |

|Intercept |17.68362* |6.187328 |3.052921 |32.31431 |

|Y |0.004963 |0.291866 |-0.68519 |0.695116 |

|X |-1.78546* |0.734826 |-3.52305 |-0.04788 |

Price (X) is a significant explainer of sales

(Note, the above is taken from regression output. Your confidence bounds will differ a bit from the above if you add and subtract 2 standard deviations from the parameter estimate. The above is more accurate.)

e. 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.

Y = 17.683 + .0049(10) – 1.785(6) = 7.022. Thus, ( = -1.785(6)/7.022 =-1.525

4. Transform the data you created in question 7 to estimate the equation

ln Y = a + b ln A + c ln P

a. Print regression results, following the style you used in equation 1(c).

ln Yi = 3.31 + 0.228 ln Xi -1.01 ln Pi R2 = .859 Adj. R2 = .818

(1.15) (0.142) (0.525) F2,7 = 21.28 (p=.001)

MSE = 0.16

b. Calculate the price elasticity of demand for this equation.

( = -1.01

b. At a 95% confidence level, can you conclude that the firm is at the point of unitary elasticity?

You cannot conclude that the firm is not at the point of unitary elasticity. The 95% confidence band about price is –2.26 to .22. That range includes –1.

e. What is the advertising elasticity of demand? How confident can you be that advertising increases sales? Justify your answer.

(A = 0.228 The 95% confidence interval extends from –0.11 to .564. This interval includes 0. Therefore we cannot conclude that advertising increases sales.

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