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Multiple Regression Analysis

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A prospective home buyer engaged the services of a consultant to develop a multiple regression model to predict mortgage interest rates by using the yield on long-term treasury bonds and the type of mortgage. There are two types of mortgages that the buyer was considering: variable rate (coded 1) and fixed rate (coded 0). All were 30 year mortgages. Data were collected for 20 mortgages and are presented in the table below.

Mortgage Rate % Treasury Bond Yield % Mortgage Type Mortgage Rate % Treasury Bond Yield % Mortgage Type
6.9 5.85 1 6.95 5.95 1
7.1 6.05 1 8.25 6.9 1
7.5 6.05 1 6.3 6.25 1
7.35 5.35 0 6.65 5.4 0
6.3 3.95 0 7.05 5.3 0
8.2 5.9 0 7.05 7.6 1
7.6 6.2 1 7.15 7.05 1
7.25 6.4 1 6.35 5.05 1
9.15 6.45 0 6.6 5.65 0
4.95 4.3 1 8.9 7.25 0

SUMMARY OUTPUT

Regression Statistics
Multiple R 0.800224
R Square 0.640359
Adjusted R Square 0.598048
Standard Error 0.605622
Observations 20

ANOVA
df SS MS F Significance F
Regression 2 11.10214 5.551071 15.13467 0.000168
Residual 17 6.235233 0.366778
Total 19 17.33738

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 2.988741 0.917049 3.259086 0.00462 1.053934 4.923548
Treasury Bond Yield % 0.80199 0.157649 5.087194 9.13E-05 0.46938 1.134601
Mortgage Type -0.96512 0.28665 -3.36691 0.003661 -1.5699 -0.36035
(Economics for Management and Economics, Watson, Billingsley, Croft and Huntsberger, Fifth Edition, 1993, Page 688)

(a) Copy and paste the data from this document to an Excel file. Select Mortgage Rate as the dependent variable and Treasury Bond yield and mortgage type as the independent variables. Conduct multiple regression using Excel. Paste the output report below. Note: Follow the instructions given in module 5 to conduct simple regression. At the step where you specify the input data range, instead of selecting the data for one independent variable, select data for all the independent variables.
(b) Write the equation from the regression output report. If you are using symbols in the equation for the variables, do define the symbols before using the symbols in the equation.
(c) Provide clear and complete interpretation of the coefficients b1 and b2 in the equation. There is no need to interpret b0. Note: Use actual variable names and numbers in answering your question. b1 and b2 are slopes is not sufficient answer.
(d) What is the value of R2 for this model? Do you think that the model does a good job of explaining the variation in wages? Why or why not?
(e) Set up the hypotheses to test whether the model is significant. Is the regression model significant at 0.05 as the level of significance? What does this mean?
(f) Set up the hypotheses to test for each of the regression coefficients individually and perform the test at the 0.05 level of significance.
(g) What average mortgage rate do you predict for mortgages of variable rate type (mortgage type = 1) if the treasury bond yield is 6%. Provide a clear and complete interpretation of the prediction.

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Solution Summary

The solution provides step by step method for the calculation of multiple regression model. Formula for the calculation and Interpretations of the results are also included.

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Multiple Regression Analysis, Time Series Analysis

See attached data file.

Background:

One day, after reporting the performance of the company to the shareholders, the CEO of A. Fictitious & Co. decided that he would like to quantify the impact of the company's expenditures has on how much sales it generates. In other words, he would like to know if the company increases the amount spent on marketing by one dollar, how large of an increase (or decrease) in sales would be expected? The major categories of expenditures and how much was spent in each are known for the company, as is the total sales generated per quarter for the last five years. He has the data file with the relevant data sent to you, and asks you to do the multiple-regression analysis to find out the answer to his questions. Oh, and he also asks you to do a time-series analysis on the total sales per quarter and forecast the amount of sales expected in the future.

Part I. Multiple Regression:

1. Look over the expenditure categories that the CEO gave you. Check to see if there interaction between the category Capital Equipment and Materials, and the category Salary and Benefits. Namely, do a multiple regression model with quarterly sales as the y-variable and the four expenditure categories given in the Data set as the x-variables. Do a second multiple regression model with four expenditure categories plus an interaction term between the category Capital Equipment and Materials, and the category Salary and Benefits. After comparing the two models, which is the better model? Can you conclude whether the two categories are independent of each other?

2. Based on your analysis in Question 1, write down the best-fit multiple regression equation for this problem with quarterly sales as the y-variable and the expenditure categories as the x-variables; do not forget the interaction term if there is one. Define each variable in the equation.

3. Answer the CEO's question. Namely, for each the four categories of expenditure (marketing, R&D, equipment and supplies, and salaries and benefits), if the CEO increases spending in one category by one million dollars (holding the others fixed), how much increase in sales should he expect? If you found an interaction term, explain its effects as well.

4. Being a very cautious person, you decide to also give the CEO the confidence intervals for the rates of increase your calculated in Question 3. Calculate the 95% confidence intervals for the slopes you calculated in Question 2, including the interaction term if you found one.

5. Being even more cautious - you are reporting the results to the CEO, after all - you decide to do a residual error analysis by applying the F-Test on the entire regression model. Do so, and interpret the results.

Part II. Time-series Analysis

The CEO noticed that he has five years of quarterly sales data in hand, and they form a time series. He decided to also ask you to perform time-series analysis on it, and use it to forecast what future sales are expected to be at the end of 1Q 2009.

6. Plot the quarterly sales as a function of time in your Excel data spreadsheet. From the shape of these graphs, and any analysis that you think is needed, determine what type of trend model is best suitable for this data. Write down the equation for the trend model, and define and explain each of the variables as it applies to this problem.

7. Do a regression analysis on the data for the trend model you decided on in Question 6, and determine the parameters for the model.

8. Answer the CEO's question. Tell him how much sales are expected to be at the end of 1Q 2009. Be careful, and also include the 95% confidence interval for this number.

Part III. Conclusions

9. Write a report to the CEO of your findings. Which expenditure has the largest impact on sales, and which one has the least impact on sales? How fast do you expect sales to increase in time?

10. When tasked to do this analysis, the CEO made a number of assumptions about the data, and what can be extrapolated from it. List down the assumptions he made, and criticize each assumption. Criticize also the conclusions that you drew from your analysis for your report to the CEO. As a starting point, remember that the data covered a period of five years.

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