LECTURE 4_140929

DEMAND ESTIMATION

BUSINESS SCHOOLUNIVERSITI PUTRA MALAYSIA

Lecture 4

Prepared byDatuk Dr Muzahet Masruri

2014

DEMAND ESTIMATION

• REGRESSION ANALYSIS

• CORRELATION ANALYSIS

• The previous lecture we discussed the theory of demand, including the concepts of price elasticity, income elasticity, and cross-elasticity of demand.

• This lecture looks at estimating demand for the firm’s product.

• Demand estimation is essential for a manager as decisions to enter new market, decisions concerning production, inventory plans, pricing and investment strategies are all depend on demand estimation.

INTRODUCTION

Two approaches in estimating demand:

• MARKETING RESEARCH TECHNIQUES

• STATISTICAL ESTIMATION OF THE DEMAND FUNCTION

ESTIMATION OF DEMAND

Customer surveys and interviews.Involve questioning a sample of consumers to determine factors, such as their purchase intent, willingness to pay, sensitivity to price change & awareness of advertising campaigns

Consumer focus group• A research technique employing close observation among target

consumers• The experimental group of consumers are give a small amount of

money to buy certain item. The consumers are closely observed discussing the choices they made & why

Market experiment in test store

Examine the way consumers behave in controlled purchase environment, involving different stores with different kind of attractions and advertisements

MARKET RESEARCH TECHNIQUES

Kiosks Sale Advertising Price Income

('000 litre) (RM'000) (RM/litre) (RM'000 m)1 160 150 15.00 19.002 220 160 13.50 17.503 140 50 16.50 14.004 190 190 14.50 21.005 130 90 17.00 15.506 160 60 16.00 14.507 200 140 13.00 21.508 150 110 18.00 18.009 210 200 12.00 18.50

10 190 100 15.50 20.00

PETRON : PETROL SALE BY REGION, 2013Example of demand model

• As a new entrée in the Malaysian market, Petron is attempting to develop a demand model for its petrol sale in 10 petrol kiosks.

• Petron feels the most important variables affecting the petrol sales are advertising (A), selling price (P) and disposable income (Y)

• Data on advertising and sales are obtained from the market research, while the disposable income from Department of Statistics.

STATISTICAL ESTIMATION OF THE DEMAND FUNCTION

• Econometrics is normally used technique in the measurement and estimating the economic relationship and variables in the demand function.

• The techniques used are:

• Regression analysis:• simple linear regression• multiple linear regression

• Correlation analysis

REGRESSION LINE

• Before using regression analysis, let us look at the regression line

• Suppose there is some underlying data on the relation between a dependent variable, Y, and some explanatory variable, X. When the values of X and Y are plotted, they appear as points A, B, C, D, E, and F in Figure 1. Clearly, the points do not lie on a straight line, or even a smooth curve. 0

Y

X

A B

C

E

FD

Figure 1: Scatter Plot Diagram

• The job of the econometrician is to find a smooth curve or line that does a “good” job of approximating the points.

• For example, suppose the econometrician believes that, on average, there is a linear relation between Y and X,

• But, for any line drawn through the points, there will be some discrepancy between the actual points and the line.

0

Y

X

A B

C

E

F

êê

A

êD

êE

Regression line

D

Figure 2: Regression Line

• For example, points A and D actually lie above the line, while points C and E lie below it.

• The deviations between the actual points and the line are given by the distance of the dashed lines, namely êA, êC, êD, and êE.

• This is called as stochastic disturbance or random error, denoted by ê.

• Therefore, the regression line (the true relationship between Y and X) is,

Y = a + bX + e0

Y

X

A B

C

E

F

êêC

êD

êE

Regression line

D

Figure 2: Regression Line

Where ê is the deviation (standard deviation) of the observed Y

SIMPLE LINEAR REGRESSION

A statistical technique for finding the relationship between two variables - dependent variable and independent variable.

The formula is given: Y = a + bX + e

Y : dependent variable, amount to be determined;a : constant value; y-intercept;b : slope (regression coefficients), or parameters to be

estimated (it measures the impact of independent variable);

X : independent variable used to explain the variation in the dependent variable;

e : random error

The next step, we need to estimate the regression coefficients:

b = slope (regression coefficients), or parameters to be estimated (it measures the impact of independent variable);

a = constant value; y-intercept;

How to determine the value of a and b?

• There are several method for determining the value of a and b.• The best known and widely used is the method of least squares ( by

squaring the errors, positive and negative errors cumulate without cancelling each other.

• Using calculus, the value of a and b that minimize the sum of square deviations, is estimated as follows:

where and are the means of X and Y ( i.e. = ∑x/n and ∑y/n )

b = n∑𝑥𝑖𝑦𝑖− ∑𝑥𝑖∑𝑦𝑖n∑ 𝑥𝑖2 − (∑𝑥𝑖 )2

a = 𝑦ො�� - b𝑥ො��

QUESTION

1. As a new entrée in the Malaysian market, PETRON is attempting to develop a demand model for its petrol sale in 10 petrol kiosks in Kuala Lumpur. PETRON predict that only promotional expenditures will affect the petrol sale, as shown in the Table in the next slide.

a) Using Simple Linear Regression, calculate the estimated petrol sale from the kiosks

observed (show your calculation in a worksheet)

b) Based on the result of the estimation, what advise you may offer to the top management on the PETRON future sale in Malaysia.

= ⁄n = 1,25010 = 125 = ∑ ⁄n = 1,750 ⁄10 = 175

• Suppose PETRON predict that only promotional expenditures will affect the petrol sale.

• The estimated slope of the regression line is calculated using the equation as follows:

b = n∑𝑥𝑖𝑦𝑖− ∑𝑥𝑖∑𝑦𝑖n∑ 𝑥𝑖2 − (∑𝑥𝑖 )2

a = 𝑦ො�� - b𝑥ො��

b = = 0.433962

The intercept is estimated as follows:a = 175 – 0.433962 (125) = 120.75475

Therefore, the Simple Linear Regression model for the PETRON sale is estimated as follows:

Y = 120.755 + 0.434 X

PETRON: PETROL SALE BY REGION, 2013Region Promotion Sale

(RM'000) ('000/litre)(1) (2) (3) (4) (5) (6)

1 150 160 24000 22500 256002 160 220 35200 25600 484003 50 140 7000 2500 196004 190 190 36100 36100 361005 90 130 11700 8100 169006 60 160 9600 3600 256007 140 200 28000 19600 400008 110 150 16500 12100 225009 200 210 42000 40000 44100

10 100 190 19000 10000 361001250 1750 229100 180100 314900

ଶݔ ݔଶݕ ݕݕݔ

ݕ∑ ݕݔ∑ σݔଶ σݔଶ∑ݔ

ADVISE TO TOP MANAGEMENT

• Since the coefficient of promotional expenditure (X) is positive, it shows that the promotional activities has a positive effect on the petrol sale in PETRON kiosks.

Y = 120.755 + 0.434 X

• The coefficient of X (0.434) indicates that for one unit increase in X (RM1,000) in promotional expenditure, expected sales (Y) will increase by 0.434 (1,000) = 434 litres in the region.

• So, PETRON should continue the promotional activities to increase sale.

MULTIPLE LINEAR REGRESSION

• A linear relationship containing two or more independent variables.

• The dependent variable Y is hypothesized to be a function of independent variable X1, X2 …… Xm and to be in the form,

Y = α + β1X1 + β2X2 + …… + βmXm + ɛ

In the PETRON example, Y is hypothesized to be a function of three variables – promotional expenditure (A), price (P) and household income (M),

Q = α + β1A + β2P + …… + β3M + ɛ

Result from the calculation, we obtained the regression equation as follows:

Y = 310.245 + 0.008A - 12.202P + 2.677M

The coefficient of P variable indicates that a RM1.00 price increase will reduce expected sale of petrol by:

-12.202 X 1,000 = 12,202 litres in 10 petrol kiosks.

CORRELATION ANALYSIS

Statistical evaluation of the regression results

• Regression results are based on a sample.

• How confident are we that these results are truly reflective of population?

• Cautious must be exercise in using regression model for prediction when the value of independent variable lies outside the range of observation from which the model was estimated

• Measure of the accuracy of estimation is needed to test the statistical significance of the estimated regression coefficients.

• A measure of accuracy of estimation with the regression equation can be obtained by calculating the standard deviation of the errors (also known as standard error of estimation)

• The standard deviation of ei is based on the sum square error, ∑ normalized by the number of observation minus 2.

STANDARD DEVIATION

Se = ට∑𝑒𝑖2𝑛−2 = ට∑ሺ𝑦𝑖 –𝑎 − 𝑏𝑥𝑖ሻ 2𝑛−2

Or, simplified

Se = ටσ 𝑦𝑖2− 𝑎∑𝑦𝑖 −𝑏∑𝑥𝑖𝑦𝑖 𝑛−2

Se =

= 22,799 or standard error of 22,799

• If the observations are tightly clustered about the regression line, the value of Se will be small and prediction error will tend to be small.

• Conversely, if the deviation of Se is large, both Se and prediction error will be large.

• In the case of PETRON, substituting the data from the Table, yields:

COEFFICIENT OF DETERMINATION OR R SQUARRED (R2)

R2 measure the proportion of the variation in dependent variable that is explained in the regression line (independent variables) and ranges in value from 0 to 1

• R2 = 0 variation in the dependent variable cannot be explained at all by ⇒the variation in the independent variable.

• R2 = 1 all of the variation in the dependent variable can be explained by ⇒the independent variables

• For statistical analysis, the closer R2 is to one, the better the regression equation; i.e., the greater the explanatory power of the regression equation

• Low value of R2 indicates the absence of some important variables from the model.

Substituting the figures from PETRON data, we have:

= = 0.519

The figure shows that the regression model with promotional expenditure as the sole independent variable, explain about 52 percent of the variations in petrol sale

R2 =

The estimation of R2 is written as follows:

t-Test

t-test is conducted by computing t-value or t-statistic for each of the estimated coefficient, to test the impact of each variable separately.

t =

• The rule of twowe can say that estimated coefficient is statistically significant (has an impact on the dependent variable) if t-value is greater than or equal to 2.

LECTURE 4_140929

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