1. HW 21 Key

2. 23:41 Home Prices. In order to help clients determine the price at which their house is likely to sell, a realtor gathered a sample of 150 purchase transactions in her area during a recent three- month period. For the response in the model, use the price of the home ($thousand). As explanatory variables, use the number of sq ft (thousands) and the number of bathrooms.

3. 23:41 a a. Examine scatterplots of the response versus the two explanatory variables as well as the scatterplot between the explanatory variables. Do you notice any unusual features in the data? Do the relevant plots appear straight enough for multiple regression?

4. 23:41 a a. There seem to be some leveraged outliers. The scatterplots seem straight enough to run regression.

5. 23:41 b b. Fit the indicated multiple regression and show a summary of the estimated features of the model.

6. 23:41 c c. Does the estimated model appear to meet the conditions for the use of the MRM? Linear – yes Lurking Variables – probably some Normal – yes Equal Variance – no, funnel Independence - yes

7. 23:41 d d. Does this estimated model explain statistically significant variation in the prices of homes? Our model explains 53% of variation. F stat of 84 means the slopes are not 0.

8. 23:41 e e. Compare the marginal slope for the number of bathrooms to the partial slope. Explain why these are so different, and show a confidence interval for each. Marginal Partial The partial controls for the sq ft variable.

9. 23:41 f f. A homeowner asked the realtor if she should spend $40,000 to convert a walk-in closet into a small bathroom in order to increase the sale price of her home. What does your analysis indicate? Don’t do it. Our estimate is that the additional bathroom will add $14,000 (interval of -$8,000 to $38,000), so a cost of $40,000 is to great.

10. 23:43 R&D Expenses. This data table contains accounting and financial data that describe 324 companies operating in the information sector. The variables include the expenses on research and development (R&D), total assets of the company, and the cost of goods sold (COGS). All columns are reported in millions of dollars; the variables are recorded in millions, so 1,000=1 billion. Use natural logs of all variables rather than the originals.

11. 23:43 a a. Examine scatterplots of the log of spending on R&D versus the log of total assets and the log of COGS. Then consider the scatterplot of the log of total assets versus the log of COGS. Do you notice any unusual features in the data? Do the relevant plots appear straight enough for multiple regression?

12. 23:43 a a. Nothing seems out of the ordinary. They all look rather linear.

13. 23:43 b b. Fit the indicated multiple regression and show a summary of the estimated features of the model.

14. 23:43 c c. Does the estimated model appear to meet the conditions for the use of the MRM? Linear – yes Lurking Variables – probably Normal – no Equal Variance – no Independence - yes

15. 23:43 d d. Does the fit of this model explain statistically significantly more variation in the log of spending on R&D than a model that uses the log of assets alone? (with logs, these powers are partial elasticities) The model explains 63% of variation. The f stat is over 270, therefore the slopes are not 0.

16. 23:43 d d. Yes, t=-2.08 for log COGS, p<.05

17. 23:43 e e. Interpret the slope for the log of the COGS in the equation estimated by the fitted model in part b. Include the confidence interval in your calculation. A one percent positive change in COGS will cause a -.14% change (.14% decrease) in R&D expenses.

18. 23:43 f f. The marginal elasticity of R&D spending with respect to COGS is about.6. Why is the partial elasticity in the multiple regression for COGS so different? Is it really that different? The explanatory power of COGS is encompassed by the other explanatory variable of assets. Yes, these are very different. The CI for the marginal elasticity is.53-.67, significantly positive and not overlapping with the CI for the partial elasticity. The marginal elasticity is positive because of the size effect: firms with substantial assets spend more on R&D and have higher COGS. MRM removes this indirect effect and shows that higher spending for materials is associated with less funds left for R&D.