multiple regression analysis

Class Outline

• Multiple Regression Analysis• Application of Regression– Substitute goods VS. Complimentary goods

• Group Exercise: Best Foods VS. Kraft

Multiple Regression Analysis

Example – Sales Data ContinuedMarket ID Sales Price Competitor Price

1 228 2.2 2.22 216 2.7 2.93 223 2.4 2.44 207 2.9 2.65 216 2.8 2.46 247 2.2 2.57 233 2.0 2.28 249 2.3 2.79 239 2.1 2.4

10 209 2.7 2.411 214 2.8 2.412 236 2.6 3.013 218 2.6 2.114 191 2.9 2.215 223 2.6 3.0

Example – Sales Data

SALESCOMPETITOR

PRICE

ADVERTISIGNPROMOTION

COUPONDISPLAY

••••••

PRICE

• SALES = f ( Price, Competitor Price, Other factors )

• Assumptions of Regression Model 1. Linear Relationship Between SALES and PRICE2. Linear Relationship Between SALES and

COMPETITOR PRICE3. Other factors follow N( )2,

),0(~

,CPricePriceSALES2

21

Ni

iiii

Competitor Price

• Using data, we make inferences on , , and .• Our best guess on using the sample data: a• Our best guess on using the sample data: b1

• Our best guess on using the sample data: b2

• Determine a, b1, and b2 by minimizing the sum of squared errors

1

iiii CPricePriceSALES 21

2

1

2

Use of Regression Model

1. Prediction / Forecastingeg.) Price = 3; CPrice = 2Exp. Sales=284.86–46.60*3+22.40*2+ Expected Value of ε

=284.86–46.60*3+22.40*22. Relationship between variables

One Unit Increase in Price 46.60 Units Decrease in Expected Sales

One Unit Increase in CPrice 22.40 Units Increase in Expected Sales

Sales=284.86–46.60*Price+22.40*CPrice+ε

=0

Exercise

• Use “Regression Exercise 3.xlsx => Multiple Regression 1”

• Use Excel “Solver” and “Data Analysis”

In-Class Exercise• Use “Regression Exercise 3.xlsx” Multiple Regression

2• Q1: Estimate a, b1,and b2• Q2: Compute the average of errors• Q3: Compute the expected sales when Price=3; CPrice=2 • Q4: Compute the expected sales when Price=2; CPrice=3• Q5: Compute the R-Square• Q6: Perform the same regression analysis using “Excel

Data Analysis”

Regression StatisticsMultiple R 0.85R Square 0.73Adjusted R Square 0.68Standard Error 7.83

Observations 15.00

ANOVA df SS MS F Significance F

Regression 2.00 1984.27992.1

3 16.19 0.00Residual 12.00 735.33 61.28

Total 14.00 2719.60

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 419.95 37.40 11.23 0.00 338.46 501.45Price -42.80 8.30 -5.15 0.00 -60.89 -24.70

Cprice 4.39 9.74 0.45 0.66 -16.82 25.60

Application of Regression ModelSubstitute Good VS. Complimentary Good

• Substitute goods: replace each other in use Margarine and butter Tea and coffeeSales_Tea = a + b1 * Price_Tea + b2 * Price_Coffee + ε

• Complimentary goods: complement each other in useHotdog and hotdog bunHardware and softwareSales_Hard = a + b1 * Price_Hard + b2 * Price_Soft + ε

+ or - ?

+ or - ?

Application of Regression ModelSubstitute Good VS. Complimentary Good

• Coke vs. Pepsi• Coke vs. Sierra Mist (?)

• Why important? – Identify _________________

Samuel Adams – Brewer & Patriot

• Relationship between Beer and Tea: Substitute goods• Sales_Beer = a + b1 * Price_Beer + b2 * Price_Tea + ε• b2: ( + ) or ( - ) ?• Tea supply ↓ Tea price ↑ Sales_Beer ?• For Sam, Good or Bad ?

Group ExerciseAnalysis of Mayonnaise Market

Best Foods VS. KraftStrategic Pricing

Group Exercise: Best Foods VS. Kraft• Use “PHXMayoData.xlsx”• 173 weeks (2002-2005)• A grocery store in Phoenix area• Sales and Prices of Best Foods (BF) Mayo and Kraft (KR)

Mayo

Week Sales_BF Sales_KR Price_BF Price_KR1 455 135 1.61 1.022 530 63 1.34 1.293 527 41 1.38 1.634 418 71 1.44 1.535 380 34 1.62 1.71: : : : :

Group Exercise: Best Foods VS. Kraft

• Q1: Compute average sales and average prices for both brands. What can we infer about this market from these numbers?

Use “=average( )” Best Foods Kraft

Average Sales 350 73Average Price 1.63 1.48

Group Exercise: Best Foods VS. Kraft• Q2: Perform regression analysis– Model1: Sales_BF = a + b1* Price_BF + b2* Price_KR + Error– Model2: Sales_KR = a + b1* Price_BF + b2* Price_KR + Error Use “Data Analysis – Regression”

Model 1

Model 2

• Q3: Interpret the results – Model1 (Best Foods)

Sales_BF = a + b1* Price_BF + b2* Price_KR + ε

• Q3: Interpret the results – Model2 (Kraft)

Sales_KR = a + b1* Price_BF + b2* Price_KR + ε

Group Exercise: Best Foods VS. Kraft

• Q4: Compute the expected sales of both brands when Price_BF = average of Price_BF’sPrice_KR = average of Price_KR’s

Sales_BF = 900 - 393 * Price_BF + 61* Price_KR + ε

Sales_KR = 155 + 55 * Price_BF – 116* Price_KR + ε

Group Exercise: Best Foods VS. Kraft

Best Foods KraftAverage Sales 350 73Average Price 1.63 1.48

Exp. Sales_BF = 900 - 393 * 1.63 + 61* 1.48 = 350

Exp. Sales_KR = 155 + 55 * 1.63 – 116* 1.48 = 73

Group Exercise: Best Foods VS. Kraft• Q5: Now assume that Best Foods decrease its price

by $0.1. What will happen to the sales of both brands?

Best Foods KraftAverage Sales 350 73Average Price 1.63 1.48

Exp. Sales_BF = 900 - 393 * 1.53 + 61* 1.48 = 389 (+11%)

Exp. Sales_KR = 155 + 55 * 1.53 – 116* 1.48 = 68 (-8%)

1.53

Group Exercise: Best Foods VS. Kraft• Q6: Now assume that Kraft decrease its price by $0.1.

What will happen to the sales of both brands?

Best Foods KraftAverage Sales 350 73Average Price 1.63 1.48

Exp. Sales_BF = 900 - 393 * 1.63 + 61* 1.38 = 344 (-2%)

Exp. Sales_KR = 155 + 55 * 1.63 – 116* 1.38 = 85 (+16%)

1.38

Group Exercise: Best Foods VS. Kraft

Best Foods Kraft Total

Average Sales 350 73 423

Best Foods Price ↓ $0.1389 68 457

(+11%) (-8%) (+8%)

Kraft Price ↓ $0.1344 85 429

(-2%) (+16%) (+1%)

Group Exercise: Best Foods VS. Kraft• Q7: Now assume that the cost of BF is $1. What is the

BF’s expected profit?Exp.Profit = Exp.Sales * ( Price – Cost )

Coefficients Standard Error t StatIntercept 900.80 58.06 15.52Price_BF -392.88 32.88 -11.95Price_KR 61.25 23.29 2.63

Best Foods KraftAverage Price 1.63 1.48

Exp.Sales 350 = Exp.Profit 221=

1

2

3

4 51 2 3+ +X XX ( - 1)

4

4

5

• Q8: What is the optimal price that maximizes the BF’s profit? Hint: Use “Solver”

Best Foods KraftAverage Price 1.76 1.48

Exp.Sales 299Exp.Profit 228

Optimal Solution