Managerial Economics in a Global Economy

Managerial EconomicsManagerial Economics Prof. M. El-SakkaProf. M. El-Sakka CBA. Kuwait University CBA. Kuwait University

Managerial EconomicsManagerial Economics in a Global Economyin a Global Economy

Chapter 4Chapter 4DEMAND ESTIMATIONDEMAND ESTIMATION


Scatter Diagram

Regression AnalysisRegression Analysis

Year X Y

1 10 44

2 9 40

3 11 42

4 12 46

5 11 48

6 12 52

7 13 54

8 13 58

9 14 56

10 15 60


Regression Line: Line of Regression Line: Line of Best FitBest Fit

Regression Line: Regression Line: Minimizes the sum of the Minimizes the sum of the squared vertical squared vertical deviations (edeviations (ett) of each ) of each

point from the regression point from the regression line.line.

Ordinary Least Squares Ordinary Least Squares (OLS) Method(OLS) Method


REGRESSION ANALYSISREGRESSION ANALYSIS

Given the following demand function:Given the following demand function:

Y = A + B1 X + B2 P + B3 I + B4 Pr;Y = A + B1 X + B2 P + B3 I + B4 Pr;

X = selling expenses (advertising) Pr = price of X = selling expenses (advertising) Pr = price of substitutessubstitutes

What we want are estimates of the values of A, B1, B2, What we want are estimates of the values of A, B1, B2, B3, & B4.B3, & B4.

Regression analysis describes the way in which one Regression analysis describes the way in which one variable is related to anothervariable is related to another. It derives an equation that . It derives an equation that can be used to estimate the unknown value of one can be used to estimate the unknown value of one variable on the basis of the known value of the other variable on the basis of the known value of the other variable(s).variable(s).


Simple Regression Model.Simple Regression Model.

the simple regression model takes the following form:the simple regression model takes the following form:

YYii = A + B X = A + B Xii + e + eii;;

Regression analysis assumes that the mean value of Y, Regression analysis assumes that the mean value of Y, given the value of X, is a linear function of X. In other given the value of X, is a linear function of X. In other words, the mean value of the dependent variable is words, the mean value of the dependent variable is assumed to be a linear function of the independent assumed to be a linear function of the independent variable.variable.

Yi is the Yi is the iith observed value of the dependent variable th observed value of the dependent variable and Xi is the and Xi is the iith observed value of the independent th observed value of the independent variable. Essentially evariable. Essentially eii is an error term, that is, a random is an error term, that is, a random amount that is added to A+BXamount that is added to A+BXii (or subtracted if e (or subtracted if eii is is negative).negative).


Because of the presence of the error term, the observed Because of the presence of the error term, the observed values of Yvalues of Yii fall around the population fall around the population regression line regression line

(A+BX(A+BXii), not on it), not on it

Regression analysis assumes that the values of Regression analysis assumes that the values of eeii are are

independent and that their mean value equals zeroindependent and that their mean value equals zero. .


Sample Regression Line (based on a sample) .Sample Regression Line (based on a sample) .

sample regression line (estimated regression line) sample regression line (estimated regression line) describes the average relationship between the dependent describes the average relationship between the dependent variable and independent variable. The general expression variable and independent variable. The general expression of the sample regression line is: of the sample regression line is:

i.e, the value of the dependent variable predicted by the i.e, the value of the dependent variable predicted by the

regression line, regression line, a & b = estimators of A and B.a & b = estimators of A and B.

a = the intercept of the regression linea = the intercept of the regression line b = the slope of the line, measure the change in the b = the slope of the line, measure the change in the

predicted value of Y associated with a one unit increase in predicted value of Y associated with a one unit increase in X.X.

ˆˆ ˆt tY a bX


Method of Least Squares.Method of Least Squares.

Used to determine the values of a and b. Since the Used to determine the values of a and b. Since the deviation of the deviation of the iith observed value of Y from the th observed value of Y from the regression line equals , the sum of the squared regression line equals , the sum of the squared deviations equals:deviations equals:

Where Where nn is the sample size. Using minimization is the sample size. Using minimization technique we can find the values of technique we can find the values of aa and and bb that that minimize this expression, by differentiating these minimize this expression, by differentiating these expression with respect to expression with respect to aa and and bb and by setting and by setting these partial derivatives equal to zero.these partial derivatives equal to zero.

iYiY ˆ

2

1)(

2

1)ˆ(

n

i ibXaiYn

i iYiY


(1)(1)

(2)(2)

solving equations (1) and (2) simultaneously, solving equations (1) and (2) simultaneously, and letting equal the mean value of X in the and letting equal the mean value of X in the sample and equal the mean of Y, we find that;sample and equal the mean of Y, we find that;

and and

n

i ibXaiYa

n

i iYiY

10)(21

2)ˆ(

n

i ibXaiYiXb

n

i iYiY

10)(21

2)ˆ(

b

n Xi X Yi Yi

n

Xi Xii

n

( ) ( )

( )

1

2

1

a Y bX .


given the following data for company X. Given the following results of the table given the following data for company X. Given the following results of the table belowbelow

= 2.533 + 1.504X;= 2.533 + 1.504X;

if Y = the observed value of sales if Y = the observed value of sales

= the computed (estimated) value of sales based on the regression line. = the computed (estimated) value of sales based on the regression line.

from the table Y = 4 when X = 1. from the table Y = 4 when X = 1.

But using the regression line:But using the regression line:

= 2.533 + 1.504(1) = 4.037 (Note there is a difference between the observed = 2.533 + 1.504(1) = 4.037 (Note there is a difference between the observed sales (4) and the estimated sales (4.037).sales (4) and the estimated sales (4.037).

Selling Expenses (X)

Sales (Y)

1 2 4 8 6 5 8 9 7

4 6 8 14 12 10 16 16 12

Y

Y

Y


if X = 0 then Y = 2.533 +1.504(0) = 2.533 (the intercept: if X = 0 then Y = 2.533 +1.504(0) = 2.533 (the intercept: the value of Y that intersects the vertical axis)the value of Y that intersects the vertical axis)

Interpretation: if the firm’s selling expenses = 0, sales Interpretation: if the firm’s selling expenses = 0, sales would be 2.533 million of units, and estimated sales go would be 2.533 million of units, and estimated sales go up 1.504 million units when selling expenses increase up 1.504 million units when selling expenses increase by 1m. by 1m.


Ordinary Least Squares Estimation Using ExcelOrdinary Least Squares Estimation Using Excel

The model:t t tY a bX e

ˆˆ ˆt tY a bX

ˆt t te Y Y

Objective: Determine the slope and intercept that minimize the sum of the squared errors.

2 2 2

1 1 1

ˆˆ ˆ( ) ( )n n n

t t t t tt t t

e Y Y Y a bX


1 10 44 -2 -6 122 9 40 -3 -10 303 11 42 -1 -8 84 12 46 0 -4 05 11 48 -1 -2 26 12 52 0 2 07 13 54 1 4 48 13 58 1 8 89 14 56 2 6 12

10 15 60 3 10 30120 500 106

4910101149

30

Time tX tY tX X tY Y ( )( )t tX X Y Y 2( )tX X

10n

1

12012

10

nt

t

XX

n

1

50050

10

nt

t

YY

n

1

120n

tt

X

1

500n

tt

Y

2

1

( ) 30n

tt

X X

1

( )( ) 106n

t tt

X X Y Y

106ˆ 3.53330

b

ˆ 50 (3.533)(12) 7.60a

Estimation Procedure

1

2

1

( )( )ˆ

( )

n

t tt

n

tt

X X Y Yb

X X

ˆa Y bX


10n 1

12012

10

nt

t

XX

n

1

50050

10

nt

t

YY

n

1

120n

tt

X

1

500n

tt

Y

2

1

( ) 30n

tt

X X

1

( )( ) 106n

t tt

X X Y Y

106ˆ 3.53330

b ˆ 50 (3.533)(12) 7.60a


Tests of SignificanceTests of Significance

Standard Error of the Slope Estimate

2 2

ˆ 2 2

ˆ( )

( ) ( ) ( ) ( )t t

bt t

Y Y es

n k X X n k X X

• A measure of the amount of scatter of individual observations about the regression line.

• It is useful in constructing prediction intervals - that is, intervals within which there is a specified probability that the dependent variable will lie.


if probability is set at 0.95, a very approximate if probability is set at 0.95, a very approximate prediction interval is:prediction interval is:

2se; since se = 0.37022se; since se = 0.3702 if the predicted value of Y is 11, there is a probability if the predicted value of Y is 11, there is a probability

that the firm’s sales will be between:that the firm’s sales will be between: 10.26 (11 – (2 × 0.37))10.26 (11 – (2 × 0.37))

and and 11.74 (11 + (2 × 0.37))11.74 (11 + (2 × 0.37))

Example Calculation2 2

1 1

ˆ( ) 65.4830n n

t t tt t

e Y Y

Y

2

1

( ) 30n

tt

X X

2

ˆ 2

ˆ( ) 65.48300.52

( ) ( ) (10 2)(30)t

bt

Y Ys

n k X X


The t-statistic (significance of individual variables).

Managers need to know whether a particular independent variable influences the dependent variable. The least square estimates of B’s by chance may be positive even if their true values are zero. e.g., B1 = 1.76 i.e., selling expenses have an effect on sales (t=0.0001).

To test whether the true value of B1 is zero we must look at the t-statistic of B1. The t-statistic has a distribution called t-distribution.

All things equal, the bigger the value of t-statistic (in absolute terms), the smaller the probability that the true value of the regression coefficient in question is zero.

In our case, there is only 1 in 10 000 that chance alone

would have resulted in a large t-statistic.


Calculation of the t Statistic

ˆ

ˆ 3.536.79

0.52b

bt

s

Degrees of Freedom = (n-k) = (10 - 2) = 8

Critical Value at 5% level =2.306 ( is significant)^b


Decomposition of Sum of Squares

Total Variation = Explained Variation + Unexplained Variation

Coefficient of Determination

Coefficient of Correlation

2 2 2ˆ ˆ( ) ( ) ( )t t tY Y Y Y Y Y

22

2

ˆ( )

( )t

Y YExplained VariationR

TotalVariation Y Y

2 373.840.85

440.00R

2 ˆr R with the signof b

1 1r 0.85 0.92r


Decomposition of Sum of Squares


Multiple Regression AnalysisMultiple Regression Analysis

2 2 ( 1)1 (1 )

( )

nR R

n k

/( 1)

/( )

Explained Variation kF

Unexplained Variation n k

Model: 1 1 2 2 ' 'k kY a b X b X b X

Adjusted Coefficient of Determination

Analysis of Variance and F Statistic

2

2

/( 1)

(1 ) /( )

R kF

R n k


Problems in Regression AnalysisProblems in Regression Analysis

Multicollinearity.Multicollinearity.

A situation in which two or more independent variables are A situation in which two or more independent variables are very highly correlated. very highly correlated.

Under perfect linear correlation it is impossible to estimate Under perfect linear correlation it is impossible to estimate the regression coefficients.the regression coefficients.

e.g., (perfect linear correlation) e.g., (perfect linear correlation) Y = A + B1 X1i + B2 X2i; where X1i = 3X2i-1Y = A + B1 X1i + B2 X2i; where X1i = 3X2i-1or X1i = 6 + X2ior X1i = 6 + X2i or X1i = 2 + 4X2i;or X1i = 2 + 4X2i;

imperfect linear correlationimperfect linear correlation.. Y = A + B1 X1i + B2 X2i; where; X1 = price, X2 = nominal Y = A + B1 X1i + B2 X2i; where; X1 = price, X2 = nominal

income (p.Q) income (p.Q)


If two independent variables move together in a If two independent variables move together in a rigid fashionrigid fashion, , there is no way to tell how much effect each has separately, there is no way to tell how much effect each has separately, all what we can observe is the effect of both combined. all what we can observe is the effect of both combined.

Consequences of mutlicollinearityConsequences of mutlicollinearity

- High R2 with no significant t-scores- High R2 with no significant t-scores - High simple correlation coefficients (cross correlation - High simple correlation coefficients (cross correlation

matrix)matrix)

How to deal with multicollinearityHow to deal with multicollinearity

- Drop one or more of the multicollinear variables- Drop one or more of the multicollinear variables - Transform the multicollinear variables (e.g. first difference)- Transform the multicollinear variables (e.g. first difference) - Increase sample size- Increase sample size


Serial Correlation (or Autocorrelation)Serial Correlation (or Autocorrelation)

Error terms are not independent, if this year’s error term is Error terms are not independent, if this year’s error term is positive, next year is always positive ( positive serial positive, next year is always positive ( positive serial correlation ), and if this year’s error term is negative, next correlation ), and if this year’s error term is negative, next year’s is always negative. year’s is always negative.

This is a violation of the assumptions underlying regression This is a violation of the assumptions underlying regression analysis. [ should be E(reiej) = 0, if not, the simple analysis. [ should be E(reiej) = 0, if not, the simple correlation between two observations of the error term is not correlation between two observations of the error term is not equal to zero]equal to zero]

Consequences of Serial Correlation Consequences of Serial Correlation

- Increases of the variances of the distributions- Increases of the variances of the distributions - Leads to underestimate the standard errors of the - Leads to underestimate the standard errors of the

coefficients.coefficients.


Detecting Serial correlation.Detecting Serial correlation. - Durbin Watson Test- Durbin Watson Test

Compare the computed DW with the DW tables to show whether d is Compare the computed DW with the DW tables to show whether d is so high so high or so lowor so low, that the hypothesis that there is no serial correlation should be , that the hypothesis that there is no serial correlation should be rejected.rejected.

if d < dL reject the hypothesis of no serial correlation. if d < dL reject the hypothesis of no serial correlation. if d > du accept the hypothesis of no serial correlationif d > du accept the hypothesis of no serial correlation if dL if dL d d du, the test is inconclusive. du, the test is inconclusive.

e.g.; if the hypothesis is that there is a negative serial correlation, we shoulde.g.; if the hypothesis is that there is a negative serial correlation, we should - reject the hypothesis of no serial correlation if d<4-dL- reject the hypothesis of no serial correlation if d<4-dL - accept the hypothesis of no serial correlation if d<4-du- accept the hypothesis of no serial correlation if d<4-du - if 4-du - if 4-du d d 4-dL the test is inconclusive. 4-dL the test is inconclusive.

21

2

2

1

( )n

t tt

n

tt

e ed

e


How to deal with serial correlationHow to deal with serial correlation

- take the difference of the variables- take the difference of the variables - use generalized least squares- use generalized least squares


Steps in Demand EstimationSteps in Demand Estimation

1. 1. Model Specification: Identify VariablesModel Specification: Identify Variables- Identify the independent variables (in reality an empirical issue)- Identify the independent variables (in reality an empirical issue)

2. Specify Functional Form2. Specify Functional Form

- Specify the mathematical form of the equation relating the mean - Specify the mathematical form of the equation relating the mean value of the dependent variable to those of the independent value of the dependent variable to those of the independent variables.variables.

e.g., e.g., Y = f(X,P). Y = f(X,P). This can take the following forms:This can take the following forms:

Y = A + B1 Xi + B2 Pi + ei; B1>0, B2<0Y = A + B1 Xi + B2 Pi + ei; B1>0, B2<0or:or:

log Y = log A + B1 log Xi + B2 log Xi + log ei;log Y = log A + B1 log Xi + B2 log Xi + log ei;

Y AXiB Pi

B ei 1 2 ;


3. Collect your data. Data can be:3. Collect your data. Data can be:

- time series- time series

- cross section- cross section

- cross section/time series (panel)- cross section/time series (panel)

4. Estimate The Function4. Estimate The Function

5. Test the Results5. Test the Results

Managerial Economics in a Global Economy

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