2024 – 5 Develop a multiple regression model for auto sales as a function of population and household

 

5. Develop a multiple-regression model for auto sales as a function of population and household income from the following data for 10 metropolitan areas:

 

 

1. Estimate values for b0, b1, and b2 for the following model:

    

   b. Are the signs you find for the coefficients consistent with your expectations? Explain.

    

   c. Are the coefficients for the two explanatory variables significantly different from zero? Explain.

    

   d. What percentage of the variation in AS is explained by this model?

    

   e. What point estimate of AS would you make for a city where INC  $23,175 and POP  128.07? What would the approximate 95 percent confidence interval be?

    

   8. Consider now that you have been asked to prepare a forecast of wholesale furniture sales for the entire United States. You have been given the monthly time-series data in

   the accompanying table:

    

   WFS is wholesale furniture sales in millions of dollars. It is not seasonally adjusted.

   PHS measures new private housing starts in thousands. UR is the unemployment rate

   as a percent. You believe that furniture sales are quite probably related to the general

   state of the economy and decide to test whether the unemployment rate affects furniture sales. You expect that as the unemployment rate rises (and the economy thus shows some sign of difficulty), furniture sales will decline.

    

   a. Summarize the results of your bivariate regression by completing the following

   table:

    

   b. After discussing the results at a staff meeting, someone suggests that you fit a

   multiple-regression model of the following form:

    

   where:

   M1  A dummy variable for January

   M2  A dummy variable for February

   M4  A dummy variable for April

   M9  A dummy variable for September

   M10  A dummy variable for October

    

   Summarize the results in the following table:

    

   • Do the signs of the coefficients make sense?

   • Are the coefficients statistically significant at a 95 percent confidence level

   (one-tailed test)?

   • What percentage of the variation in WFS is explained by the model?

    

   11. The data presented below are for retail sales in the United States quarterly from the

   period 1992Q1 through 2003Q4. Also included is disposable personal income per

   capita (DPIPC) (to use as a measure of the well-being of the economy).

    

    

   a. Develop a regression model of retail sales as a function of the S&P 500. Comment

   on the relevant summary statistics.

    

   b. Estimate a new multiple-regression model using seasonal dummy variables for

   quarters 2, 3, and 4. Additionally, add a time index to account for trend. Comment

   on the relevant statistics of this model. Is this model an improvement on the model

   above? What evidence is there that this second model provides an improvement (no

   improvement)?

    

   c. Square the time index variable and add it to the multiple-regression model above.

   Does the resulting model perform better than either previous model? Explain your

   reasoning.

    

   13. This is a problem in model selection. A “big box” home improvement store has

   collected data on its sales and demographic variables relating to its various stores. The

   cross-sectional data set for these variables is below:

    

    

   where:

    

   Sales  average monthly store sales (in thousands of dollars)

   X1  Households in a 5-mile ring that are do-it-youselfers (in thousands)

   X2  Average monthly advertising expenditures (in dollars)

   X3  Square footage of competitor stores in a 5-mile ring (in thousands)

   X4  Households in a 5-mile ring that are below the poverty level (in hundreds)

   X5  Weighted average daily traffic count at store intersection

    

   a. Begin by estimating a model with independent variables X1, X2, and X3. Comment on the appropriateness of this model and its accuracy.

    

   b. Now add X4 to the model above and again comment on the appropriateness of the model. Has the accuracy improved?

    

   c. Finally, add X5 to the model and again comment on the accuracy of the model. Use the appropriate summary statistics of the three models to suggest which of the five independent variables should be in the model. Advise the “big box” retailer on which characteristics are important when choosing new store locations.