# Multiple regression analysis for San Francisco demand

Interpret the coefficient of determination (R2) for the San Francisco demand equation.

What are expected unit sales and sales revenue in a typical market?

Qi = b0 + b1Pi + b2Pxi + b3Adi + b4Ii + uit

To illustrate use of the standard error of the estimate statistic, derive the 95 percent confidence interval for expected unit sales and total sales revenue in a typical market.

See attached file for full problem description.

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#### Solution Preview

Please see attached file.

A. Describe the economic meaning and statistical significance of each individual independent variable included in the San Francisco demand equation.

Variable Economic Meaning Statistical Significance

Price (P)

Competitor Price (Px)

Advertising (Ad)

Income (I)

The regression coefficients give the effect of unit change in the independent variable on the dependent variable (Demand)

Price (P) :

Economic Meaning :

For a unit increase in price the demand decreases by 19875.95363 units

Statistical Significance : Since the significance value (p value ) is less than 0.05 ,the price change have significant impact on demand

Competitor Price (Px)

Economic Meaning :

For a unit increase in Competitor Price (Px) ...

#### Solution Summary

Multiple regression analysis for San Francisco demand equation. The solution contains regression, slope, intercept, correlation, r-square, coefficient of determination, regression coefficients and 95 percent confidence interval for expected unit sales and total sales revenue in a typical market.

Regression (20 Problems) : Multiple Regression Model Building, Averages and Exponential Smoothing, Hypothesis Testing and ANOVA

1. A real estate builder wishes to determine how house size (House) is

influenced by family income (Income), family size (Size), and

education of the head of household (School). House size is measured

in hundreds of square feet, income is measured in thousands of

dollars, and education is measured in years. The builder randomly

selected 50 families and ran the multiple regression. The business

literature involving human capital shows that education influences

an individual's annual income. Combined, these may influence

family size. With this in mind, what should the real estate builder

be particularly concerned with when analyzing the multiple

regression model?

a. Randomness of error terms

b. Collinearity

c. Normality of residuals

d. Missing observations

2. A microeconomist wants to determine how corporate sales are

influenced by capital and wage spending by companies. She

proceeds to randomly select 26 large corporations and record

information in millions of dollars. A statistical analyst discovers

that capital spending by corporations has a significant inverse

relationship with wage spending. What should the microeconomist

who developed this multiple regression model be particularly

concerned with?

a. Randomness of error terms

b. Collinearity

c. Normality of residuals

d. Missing observations

3. The Variance Inflationary Factor (VIF) measures the

a. correlation of the X variables with the Y variable.

b. contribution of each X variable with the Y variable after all

other X variables are included in the model.

c. correlation of the X variables with each other.

d. standard deviation of the slope.

4. In multiple regression, the __________ procedure permits variables

to enter and leave the model at different stages of its development.

a. forward selection

b. residual analysis

c. backward elimination

d. stepwise regression

5. Which of the following is not used to find a "best" model?

a. adjusted r2

b. Mallow's Cp

c. odds ratio

d. all of the above

6. The logarithm transformation can be used

a. to overcome violations of the autocorrelation assumption.

b. to test for possible violations of the autocorrelation

assumption.

c. to change a linear independent variable into a nonlinear

independent variable.

d. to change a nonlinear model into a linear model.

7. The Cp statistic is used

a. to determine if there is a problem of collinearity.

b. if the variances of the error terms are all the same in a

regression model.

c. to choose the best model.

d. to determine if there is an irregular component in a time

series.

8. Which of the following is used to determine observations that have

an influential effect on the fitted model?

a. Cook's distance statistic

b. Durbin-Watson statistic

c. variance inflationary factor

d. the Cp statistic

9. An auditor for a county government would like to develop a model to

predict the county taxes based on the age of single-family houses. A

random sample of 19 single-family houses has been selected, with

the results as shown below (and also in the data file TAXES on your

CD-ROM):

____________________________

Taxes Age of House

925 1

870 2

809 4

720 4

694 5

630 8

626 10

562 10

546 12

523 15

480 20

486 22

462 25

441 25

426 30

368 35

350 40

348 50

322 50

Assuming a quadratic relationship between the age of the house and

the county taxes, which of the following is the best prediction of the

average county taxes for a 20-year old house?

a. $557.30

b. $481.25

c. $480.60

d. $479.15

10. An econometrician is interested in evaluating the relation of

demand for building materials to mortgage rates in Los Angeles and

San Francisco. He believes that the appropriate model is

Y = 10 + 5X1 + 8X2

Where X1 = mortgage rate in %

X2 = 1 if San Francisco, 0 if LA

Y = demand in $100 per capita

Referring to the information above, holding constant the effect of

city, each additional increase of 1% in the mortgage rate would lead

to an estimated increase of ________ in the mean demand.

a. $10

b. $50

c. $60

d. $500

11. Referring to the information in #10 above, the fitted model for

predicting demand in Los Angeles is ________.

a. 10 + 5X1

b. 10 + 13X1

c. 15 + 8X2

d. 18 + 5X2

12. Table 3.1

In Hawaii, condemnation proceedings are underway to enable

private citizens to own the property that their homes are built on.

Until recently, only estates were permitted to own land, and

homeowners leased the land from the estate. In order to comply

with the new law, a large Hawaiian estate wants to use regression

analysis to estimate the fair market value of the land. Each of the

following 3 models were fit to data collected for n = 20 properties, 10

of which are located near a cove.

Model 1: Y = β0 + β1 X1 + β2 X2 + β3 X1X2 + β4 X12 + β5 X12X2 + ε

where Y = Sale price of property in thousands of dollars

X1 = Size of property in thousands of square feet

X2 = 1 if property located near cove, 0 if not using the data

collected for the 20 properties, the following partial output

obtained from Microsoft Excel is shown:

SUMMARY OUTPUT_________________________________________

Regression Statistics

Multiple R 0.985

R Square 0.970

Standard Error 9.5

Observations 20

ANOVA

Df SS MS F Signif F

Regression 5 28324 5664 62.2 0.0001

Residual 14 1279 91

Total 19 29063

Coeff StdError t Stat p-value

Intercept -32.1 35.7 -0.90 0.3834

Size 12.2 5.9 2.05 0.0594

Cove -104.3 53.5 -1.95 0.0715

Size*Cove 17.0 8.5 1.99 0.0661

SizeSq -0.3 0.2 -1.28 0.2204

SizeSq*Cove -0.3 0.3 -1.13 0.2749

Referring to Table 3.1, given a quadratic relationship between sale

price (Y) and property size (X1), what null hypothesis would you test

to determine whether the curves differ from cove and non-cove

properties?

a. H0 : β2 = β3 = β5 = 0

b. H0 : β3 = β5 = 0

c. H0 : β4 = β5 = 0

d. H0 : β2 = 0

13. Referring to Table 3.1, is the overall model statistically adequate

at a 0.05 level of significance for predicting sale price (Y)?

a. No, since some of the t-tests for the individual variables are

not significant.

b. No, since the standard deviation of the model is fairly large.

c. Yes, since none of the β-estimates are equal to 0.

d. Yes, since the p-value for the test is smaller than 0.05.

14. The method of moving averages is used

a. to plot a series.

b. to exponentiate a series.

c. to smooth a series.

d. in regression analysis.

15. When using the exponentially weighted moving average for

purposes of forecasting rather than smoothing,

a. the previous smoothed value becomes the forecast.

b. the current smoothed value becomes the forecast.

c. the next smoothed value becomes the forecast.

d. None of the above.

16. In selecting an appropriate forecasting model, the following

approaches are suggested:

a. Perform a residual analysis.

b. Measure the size of the forecasting error.

c. Use the principle of parsimony.

d. All of the above.

17. To assess the adequacy of a forecasting model, one measure that is

often used is

a. quadratic trend analysis.

b. the MAD.

c. exponential smoothing.

d. moving averages.

18. A model that can be used to make predictions about long-term

future values of a time series is

a. linear trend.

b. quadratic trend.

c. exponential trend.

d. All of the above.

19. You need to decide whether you should invest in a particular stock.

You would like to invest if the price is likely to rise in the long run.

You have data on the daily average price of this stock over the past

12 months. Your best action is to

a. compute moving averages.

b. perform exponential smoothing.

c. estimate a least square trend model.

d. compute the MAD statistic.

20. Which of the following statements about moving averages is not

true?

a. It can be used to smooth a series.

b. It gives equal weight to all values in the computation.

c. It is simpler than the method of exponential smoothing.

d. It gives greater weight to more recent data.

21. The following table contains the number of complaints received in a

department store for the first 6 months of last year.

21. Table 3.2

Month Complaints

January 36

February 45

March 81

April 90

May 108

June 144__

Referring to the Table 3.2 above, if a three-term moving average is

used to smooth this series, what would be the second calculated

term?

a. 36

b. 40.5

c. 54

d. 72