Estimating Demand

Outline

•Where do demand functions come from?

•Sources of information for demand estimation

•Cross-sectional versus time series data

•Estimating a demand specification using the ordinary least squares (OLS) method.

•Goodness of fit statistics.

The goal of forecasting

To transform available data into equations that provide the best possible forecasts of economic variables—e.g., sales revenues and costs of production—that are crucial for management.

Demand for air travel Houston to Orlando

Q = 25 + 3Y + PO – 2P

Recall that our demand function was estimated as follows:

[4.1]

Where Q is the number of seats sold; Y is a regional income index; P0 is the fare charged by a rival airline, and P is the airline’s own fare.

Now we will explain how

we estimated this demand

equation

Questions managers should ask about a forecasting equations

1. What is the “best” equation that can be obtained (estimated) from the available data?

2. What does the equation not explain?

3. What can be said about the likelihood and magnitude of forecast errors?

4. What are the profit consequences of forecast errors?

How do get the data to estimate demand forecasting equations?

•Customer surveys and interviews.

•Controlled market studies.

•Uncontrolled market data.

Campbell’s soup estimates demand functions from data

obtained from a survey of more than 100,000 consumers

Survey pitfalls Sample bias Response bias Response accuracy Cost

Time -series data: historical data--i.e., the data sample consists of a series of daily, monthly, quarterly, or annual data for variables such as prices, income , employment , output , car sales, stock market indices, exchange rates, and so on.

Cross-sectional data: All observations in the sample are taken from the same point in time and represent different individual entities (such as households, houses, etc.)

Types of data

Year Month Day Won per Dollar1997 3 10 8771997 3 11 880.51997 3 12 879.51997 3 13 880.51997 3 14 881.51997 3 17 8821997 3 18 8851997 3 19 8871997 3 20 886.51997 3 21 8871997 3 24 8901997 3 25 891

Time series data: Daily observations, Korean Won per dollar

Student ID Sex Age Height Weight

777672431 M 21 6’1” 178 lbs.

231098765 M 28 5’11” 205 lbs.

111000111 F 19 5’8” 121 lbs.

898069845 F 22 5’4” 98 lbs.

000341234 M 20 6’2” 183 lbs

Example of cross sectional data

Estimating demand equations using regression analysis

Regression analysis is a statistical technique that allows us to quantify the relationship between a

dependent variable and one or more independent or “explanatory” variables.

Y

X0

X and Y are notperfectly correlated.However, there is on average a positiverelationshipbetween Y and X

X1 X2

Regression theory

1

Y1

E(Y|X1)

Y

X0 X1

E(Y |Xi) = 0 + 1Xi

1 = Y1 - E(Y|X1)

We assume that expected conditional values

of Y associated with alternative values of X

fall on a line.

Our model is specified as follows:

Q = f (P) where Q is ticket sales and P is the fare

Specifying a single variable model

Q is the dependent variable—that is, we

think that variations in Q can be explained by

variations in P, the “explanatory” variable.

ii PQ 10

0 and 1 are called parameters or population parameters.

We estimate these parameters using the data we have available

iii PQ 10

Estimating the single variable model

[1]

[2]

Since the datapoints are unlikely to fall

exactly on a line, (1)must be modified

to include a disturbanceterm (εi)

Estimated Simple Linear Regression Estimated Simple Linear Regression EquationEquation

The The estimated simple linear regression estimated simple linear regression equationequation

0 1y b b x 0 1y b b x

• is the estimated value of is the estimated value of yy for a given for a given xx value. value.yy• bb11 is the slope of the line. is the slope of the line.• bb00 is the is the yy intercept of the line. intercept of the line.

• The graph is called the estimated regression line.The graph is called the estimated regression line.

Estimation Process

Regression ModelRegression Modelyy = = 00 + + 11xx + +

Regression EquationRegression EquationEE((yy) = ) = 00 + + 11xx

Unknown ParametersUnknown Parameters00, , 11

Sample Data:Sample Data:x yx y

xx11 y y11

. .. . . .. . xxnn yynn

bb00 and and bb11

provide estimates ofprovide estimates of00 and and 11

EstimatedEstimatedRegression EquationRegression Equation

Sample StatisticsSample Statistics

bb00, , bb11

0 1y b b x 0 1y b b x

Least Squares Method

Least Squares Criterion

min (y yi i )2min (y yi i )2

where:where:

yyii = = observedobserved value of the dependent variable value of the dependent variable

for the for the iith observationth observation^yyii = = estimatedestimated value of the dependent variable value of the dependent variable

for the for the iith observationth observation

Slope for the Estimated Regression Equation

1 2

( )( )

( )i i

i

x x y yb

x x

1 2

( )( )

( )i i

i

x x y yb

x x

Least Squares Method

yy-Intercept for the Estimated Regression -Intercept for the Estimated Regression EquationEquation

Least Squares MethodLeast Squares Method

0 1b y b x 0 1b y b x

where:where:xxii = value of independent variable for = value of independent variable for iithth observationobservation

nn = total number of observations = total number of observations

__yy = mean value for dependent variable = mean value for dependent variable

__xx = mean value for independent variable = mean value for independent variable

yyii = value of dependent variable for = value of dependent variable for iithth observationobservation

Line of best fit

The line of best fit is the one that minimizes the

squared sum of the vertical distances of the sample points from the

line

1. Specification

2. Estimation

3. Evaluation

4. Forecasting

The 4 steps of demand estimation using regression

Year and Average Number AverageQuarter Coach Seats Fare

97-1 64.8 25097-2 33.6 26597-3 37.8 26597-4 83.3 24098-1 111.7 23098-2 137.5 22598-3 109.6 22598-4 96.8 22099-1 59.5 23099-2 83.2 23599-3 90.5 24599-4 105.5 24000-1 75.7 25000-2 91.6 24000-3 112.7 24000-4 102.2 235

Mean 87.3 239.7Std. Dev. 27.9 13.1

Table 4-2

Ticket Prices and Ticket Sales along an Air Route

Simple linear regression begins by plotting Q-P values on a scatter diagram to determine if there exists an approximate linear relationship:

Scatter plot diagram

Passengers

16014012010080604020

Fare

290

280

270

260

250

240

230

220

210

Scatter plot diagram with possible line of best fit

Average One-way Fare

7

6

5

4

3

2

$ 2 0

2 0

2 0

2 0

2 0

2 0

Demand curve: Q = 330- P

500 100 150

Number of Seats Sold per Flight

Note that we use X to denote the explanatoryvariable and Y is the dependent variable.

So in our example Sales (Q) is the “Y” variable and Fares (P) is the “X” variable.

Q = Y

P = X

Computing the OLS estimators

We estimated the equation using the statistical software package SPSS. It generated the following output:

Coefficientsa

478.690 88.036 5.437 .000

-1.633 .367 -.766 -4.453 .001

(Constant)

FARE

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: PASSa.

Reading the SPSS Output

From this table we see that our estimate of 0 is 478.7 and our estimate

of 1 is –1.63.

Thus our forecasting equation is given by:

ii PQ 63.17.478ˆ

Step 3: Evaluation

Now we will evaluate the forecasting equation using standard goodness of fit statistics, including:

1. The standard errors of the estimates.

2. The t-statistics of the estimates of the coefficients.

3. The standard error of the regression (s)

4. The coefficient of determination (R2)

•We assume that the regression coefficients are normally distributed variables.

•The standard error (or standard deviation) of the estimates is a measure of the dispersion of the estimates around their mean value.

•As a general principle, the smaller the standard error, the better the estimates (in terms of yielding accurate forecasts of the dependent variable).

Standard errors of the estimates

The following rule-of-thumb is useful: The standard error of the regression coefficient should be less than half of the size of the corresponding regression coefficient.

2ˆˆ 11 ss

2

22ˆ1

i

i

xkn

es

Note that:

XXx ii

1sLet denote the standard error of our estimate of 1

Thus we have:

Where:

and

iii QQe ˆ

and

k is the number of estimated coefficients

Computing the standard error of 1

Coefficientsa

478.690 88.036 5.437 .000

-1.633 .367 -.766 -4.453 .001

(Constant)

FARE

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: PASSa.

By reference to the SPSS output, we see that the standard error of our estimate

of 1 is 0.367, whereas the (absolute value)our estimate of 1 is 1.63 Hence our estimate is about 4 ½

times the size of its standard error.

The SPSS output tells us that the t statistic for the the fare coefficient (P)

is –4.453 The t test is a wayof comparing the errorsuggested by the null

hypothesis to the standard error of the estimate.

To test for the significance of our estimate of 1, we set the following null hypothesis, H0, and the alternative hypothesis, H1

H0: 1 0

H1: 1 < 0

The t distribution is used to test for statistical significance of the estimate:

45.4049.0

063.1ˆ

1ˆ

11

s

t

The t test

The coefficient of determination, R2, is defined as the proportion of the total variation in the dependent variable (Y) "explained" by the regression of Y on the independent variable (X). The total variation in Y or the total sum of squares (TSS) is defined as:

n

i

i

n

i

i yYYTSS1

22

1

The explained variation in the dependent variable(Y) is called the regression sum of squares (RSS) and is given by:

n

i

i

n

i

i yYYRSS1

22

1

ˆˆ

Note: YYy ii

Coefficient of determination (R2)

What remains is the unexplained variation in the dependent variable or the error sum of squares (ESS)

n

i

i

n

i

i eYYESS1

22

1

ˆ

We can say the following:

•TSS = RSS + ESS, or

•Total variation = Explained variation + Unexplained variationR2 is defined as:

n

i

i

n

i

i

n

i

i

n

i

i

y

e

y

y

RSS

ESS

TSS

RSSR

1

2

1

2

1

2

1

2

2 1ˆ

1

We see from the SPSS model summary table that R2 for this model is .586

ANOVAb

6863.624 1 6863.624 19.826 .001a

4846.816 14 346.201

11710.440 15

Regression

Residual

Total

Model1

Sum ofSquares df

MeanSquare F Sig.

Predictors: (Constant), FAREa.

Dependent Variable: PASSb.

Model Summary

.766a .586 .557 18.6065Model1

R R SquareAdjusted R

Square

Std. Errorof the

Estimate

Predictors: (Constant), FAREa.

Note that: 0 R2 1

If R2 = 0, all the sample points lie on a horizontal line or in a circle

If R2 = 1, the sample points all lie on the regression line

In our case, R2 0.586, meaning that 58.6 percent of the variation in the dependent variable (consumption) is explained by the regression.

Notes on R2

This is not a particularly good fit based on R2 since 41.4 percent of the variation in the dependent variable is unexplained.

The standard error of the regression (s) is given by:

kn

e

s

n

i

i

1

2

Standard error of the regression

The model summary tells us that s = 18.6

Regression is based on the assumption that the error term is normally distributed, so that 68.7% of the actual values of the dependent variable (seats sold) should be within one standard error ($18.6 in our example) of their fitted value.

Also, 95.45% of the observed values of seats sold should be within 2 standard errors of their fitted values (37.2).

Model Summary

.766a .586 .557 18.6065Model1

R R SquareAdjusted R

Square

Std. Errorof the

Estimate

Predictors: (Constant), FAREa.

Step 4: Forecasting

ii PQ 63.17.478ˆ

Recall the equation obtained from the regression results is :

Our first step is to perform an “in-

sample” forecast.

At the most basic level, forecasting consists of inserting forecasted values

of the explanatory variable P (fare) into the forecasting equation to obtain forecasted values of the

dependent variable Q (passenger seats sold).

Year and Predicted Actual Quarter Sales (Q*) Sales (Q) Q* - Q (Q* - Q)sq

97-1 64.8 70.44 5.64 31.8197-2 33.6 45.94 12.34 152.2897-3 37.8 45.94 8.14 66.2697-4 83.3 86.77 3.47 12.0498-1 111.7 103.1 -8.6 73.9698-2 137.5 111.26 -26.24 688.5498-3 109.6 111.26 1.66 2.7698-4 96.8 119.43 22.63 512.1299-1 59.5 103.1 43.6 1900.9699-2 83.2 94.94 11.74 137.8399-3 90.5 78.61 -11.89 141.3799-4 105.5 86.77 -18.73 350.8100-1 75.7 70.44 -5.26 27.6700-2 91.6 86.77 -4.83 23.3300-3 112.7 86.77 -25.93 672.3600-4 102.2 94.94 -7.26 52.71

Sum of Squared Errors 4846.80

In-Sample Forecast of Airline Sales

In-Sample Forecast of Airline Sales

Year/Quarter

00.300.199.399.198.398.197.397.1

Pass

engers

160

140

120

100

80

60

40

20

Actual

Fitted

Our ability to generate accurate forecasts of the dependent variable depends on two factors:

•Do we have good forecasts of the explanatory variable?

•Does our model exhibit structural stability, i.e., will the causal relationship between Q and P expressed in our forecasting equation hold up over time? After all, the estimated coefficients are average values for a specific time interval (1987-2001). While the past may be a serviceable guide to the future in the case of purely physical phenomena, the same principle does not necessarily hold in the realm of social phenomena (to which economy belongs).

Can we make a good forecast?

Single Variable Regression Using Excel

We will estimate an equation and use it to

predict home prices in two cities. Our data set is on

the next slide

City Income Home Price

Akron, OH 74.1 114.9

Atlanta, GA 82.4 126.9

Birmingham, AL 71.2 130.9

Bismark, ND 62.8 92.8

Cleveland, OH 79.2 135.8

Columbia, SC 66.8 116.7

Denver, CO 82.6 161.9

Detroit, MI 85.3 145

Fort Lauderdale, FL 75.8 145.3

Hartford, CT 89.1 162.1

Lancaster, PA 75.2 125.9

Madison, WI 78.8 145.2

Naples, FL 100 173.6

Nashville, TN 77.3 125.9

Philadelphia, PA 87 151.5

Savannah, GA 67.8 108.1

Toledo, OH 71.2 101.1

Washington, DC 97.4 191.9

•Income (Y) is average family income in 2003

•Home Price (HP) is the average price of a new or existing home in 2003.

Model Specification

YbbHP 10

Scatter Diagram: Income and Home Prices

80

100

120

140

160

180

200

50 60 70 80 90 100 110

Income

Ho

me

Pri

ces

Regression Statistics

Multiple R 0.906983447

R Square 0.822618973

Adjusted R Square 0.811532659

Standard Error 11.22878416

Observations 18

  CoefficientsStandard

Error t Stat

Intercept -48.11037724 21.58459326 -2.228922114

Income 2.332504769 0.270780116 8.614017895

ANOVA

  df SS

Regression 19355.71550

2

Residual 162017.36949

8

Total 17 11373.085

Excel Output

YHP 33.211.48

City Income Predicted HP

Meridian, MS 59,600 $ 138,819.89

Palo Alto, CA 121,000 $ 281,881.89

Equation and prediction