Department of Business Administration FALL 20 10 -11 Demand Estimation by Assoc. Prof. Sami Fethi.

Department of Business Administration

FALL 2010-11 Demand Estimation

byAssoc. Prof. Sami Fethi

2

© 2004, Managerial Economics, Dominick Salvatore © 2010/11, Sami Fethi, EMU, All Right Reserved.

Ch 4 : Demand Estimation

Demand Estimation

To use these important demand relationship in decision analysis, we need empirically to estimate the structural form and parameters of the demand function-Demand Estimation.

Qdx= (P, I, Pc, Ps, T)

(-, + , - , +, +) The demand for a commodity arises from the consumers’

willingness and ability to purchase the commodity. Consumer demand theory postulates that the quantity demanded of a commodity is a function of or depends on the price of the commodity, the consumers’ income, the price of related commodities, and the tastes of the consumer.

3


Ch 4 : Demand Estimation Demand Estimation

In general, we will seek the answer for the In general, we will seek the answer for the following qustions:following qustions:

How much will the revenue of the firm change after increasing the price of the commodity?

How much will the quantity demanded of the commodity increase if consumers’ income increase

What if the firms double its ads expenditure? What if the competitors lower their prices? Firms should know the answers the

abovementioned questions if they want to achieve the objective of maximizing thier value.

4



The Identification Problem

The demand curve for a commodity is generally estimated from market data on the quantity purchased of the commodity at various price over time (i.e. Time-series data) or various consuming units at one point in time (i.e. Cross-sectional data).

Simply joinning priced-quantity observations on a graph does not generate the demand curve for a commodity. The reason is that each priced-quantity observation is given by the intersection of a different and unobserved demand and supply curve of commodity.

In other words, The difficulty of deriving the demand curve for a commodity from observed priced-quantity points that results from the intersection of different and unobserved demand and supply curves for the commodity is referred to as the identification problem.

5


Ch 4 : Demand Estimation The Identification Problem In the following demand

curve, Observed price-quantity data points E1, E2, E3, and E4, result respectively from the intersection of unobserved demand and supply curves D1

and S1, D2 and S2, D3 and S3, and D4 and S4. Therefore, the dashed line connecting observed points E1, E2, E3, and E4 is not the demanded curve for the commodity. The derived a demand curve for the commodity, say, D2, we allow the supply to shift or to be different and correct, through regression analysis, for the forces that cause demand curve D2 to shift or to be different as can be seen at points E2, E'2. This is done by regression analysis.

6



Demand Estimation: Marketing Research Approaches

Consumer Surveys Observational Research Consumer Clinics Market Experiments

These approaches are usually covered extensively in marketing courses, however the most important of these are consumer surveys and market experiments.

7




o Consumer surveys: These surveys require the questioning of a firm’s customers in an attempt to estimate the relationship between the demand for its products and a variety of variables perceived to be for the marketing and profit planning functions.

These surveys can be conducted by simply stopping and questioning people at shopping centre or by administering sophisticated questionnaires to a carefully constructed representative sample of consumers by trained interviewers.

8




Major advantages: they may provide the only information available; they can be made as simple as possible; the researcher can ask exactly the questions they want

Major disadvantages: consumers may be unable or unwilling to provide reliable answers; careful and extensive surveys can be very expensive.

9




Market experiments: attempts by the firm to estimate the demand for the commodity by changing price and other determinants of the demand for the commodity in the actual market place.

10




Major advantages: consumers are in a real market situation; they do not know that they being observed; they can be conducted on a large scale to ensure the validity of results.

Major disadvantages: in order to keep cost down, the experiment may be too limited so the outcome can be questionable; competitors could try to sabotage the experiment by changing prices and other determinants of demand under their control; competitors can monitor the experiment to gain very useful information about the firm would prefer not to disclose.

11



Purpose of Regression Analysis

Regression Analysis is Used Primarily to Model Causality and Provide Prediction Predict the values of a dependent

(response) variable based on values of at least one independent (explanatory) variable

Explain the effect of the independent variables on the dependent variable

The relationship between X and Y can be shown on a scatter diagram

12



Scatter Diagram

It is two dimensional graph of plotted points in which the vertical axis represents values of the dependent variable and the horizontal axis represents values of the independent or explanatory variable.

The patterns of the intersecting points of variables can graphically show relationship patterns.

Mostly, scatter diagram is used to prove or disprove cause-and-effect relationship. In the following example, it shows the relationship between advertising expenditure and its sales revenues.

13 Managerial Economics © 2008/09, Sami Fethi, EMU, All Right Reserved.


Scatter Diagram

Scatter Diagram-Example

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

14


Ch 4 : Demand Estimation Scatter Diagram

Scatter diagram shows a positive relationship between the relevant variables. The relationship is approximately linear.

This gives us a rough estimates of the linear relationship between the variables in the form of an equation such as

Y= a+ b X

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Regression Analysis

In the equation, a is the vertical intercept of the estimated linear relationship and gives the value of Y when X=0, while b is the slope of the line and gives an estimate of the increase in Y resulting from each unit increase in X.

The difficulty with the scatter diagram is that different researchers would probably obtain different results, even if they use same data points. Solution for this is to use regression analysis.

16



Regression Analysis

Regression analysis: is a statistical technique for obtaining the line that best fits the data points so that all researchers can reach the same results.

Regression Line: Line of Best Fit Regression Line: Minimizes the sum of the

squared vertical deviations (et) of each point from the regression line.

This is the method called Ordinary Least Squares (OLS).

17


Ch 4 : Demand Estimation Regression Analysis

In the table, Y1 refers actual or observed sales revenue of $44 mn associated with the advertising expenditure of $10 mn in the first year for which data collected.

In the following graph, Y^1 is the

corresponding sales revenue of the firm estimated from the regression line for the advertising expenditure of $10 mn in the first year.

The symbol e1 is the corresponding vertical deviation or error of the actual sales revenue estimated from the regression line in the first year. This can be expressed as e1= Y1- Y^

1.

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

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Ch 4 : Demand Estimation Regression Analysis

In the graph, Y^1

is the corresponding sales revenue of the firm estimated from the regression line for the advertising expenditure of $10 mn in the first year.

The symbol e1 is the corresponding vertical deviation or error of the actual sales revenue estimated from the regression line in the first year. This can be expressed as e1= Y1- Y^

1.

19



Regression Analysis

Since there are 10 observation points, we have obviously 10 vertical deviations or error (i.e., e1 to e10). The regression line obtained is the line that best fits the data points in the sense that the sum of the squared (vertical) deviations from the line is minimum. This means that each of the 10 e values is first squared and then summed.

20



Simple Regression Analysis

Now we are in a position to calculate the value of a ( the vertical intercept) and the value of b (the slope coefficient) of the regression line.

Conduct tests of significance of parameter estimates.

Construct confidence interval for the true parameter.

Test for the overall explanatory power of the regression.

21



Simple Linear Regression Model

Regression line is a straight line that describes the dependence of the average average value value of one variable on the other

ii iY X

Y Intercept SlopeCoefficient

Random Error

Independent (Explanatory) Variable

RegressionLine

Dependent (Response) Variable

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Ordinary Least Squares (OLS)

Model: t t tY a bX e

ˆˆ ˆt tY a bX

ˆt t te Y Y

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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

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Estimation Procedure

1

2

1

( )( )ˆ

( )

n

t tt

n

tt

X X Y Yb

X X

ˆa Y bX

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Ch 4 : Demand Estimation Ordinary Least Squares (OLS)

Estimation Example

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 1210 15 60 3 10 30

120 500 106

491010114930

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

10n

1

120 1210

nt

t

XXn

1

500 5010

nt

t

YYn

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

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Ch 4 : Demand Estimation Ordinary Least Squares (OLS)

Estimation Example

10n 1

120 1210

nt

t

XXn

1

500 5010

nt

t

YYn

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

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The Equation of Regression Line

The equation of the regression line can be constructed as follows:

Yt^=7.60 +3.53 Xt

When X=0 (zero advertising expenditures), the expected sales revenue of the firm is $7.60 mn. In the first year, when X=10mn, Y1

^= $42.90 mn.

Strictly speaking, the regression line should be used only to estimate the sales revenues resulting from advertising expenditure that are within the range.

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Crucial Assumptions

Error term is normally distributed. Error term has zero expected value

or mean. Error term has constant variance in

each time period and for all values of X.

Error term’s value in one time period is unrelated to its value in any other period.

29



Tests of Significance: Standard Error

To test the hypothesis that b is statistically significant (i.e., advertising positively affects sales), we need first of all to calculate standard error (deviation) of b^.

The standard error can be calculated in the following expression:

30



Tests of Significance

Standard Error of the Slope Estimate

2 2

ˆ 2 2

ˆ( )( ) ( ) ( ) ( )

t tb

t t

Y Y es

n k X X n k X X

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Ch 4 : Demand Estimation Tests of Significance

Example Calculation

2 2

1 1

ˆ( ) 65.4830n n

t t tt t

e Y Y

2

1

( ) 30n

tt

X X

2

ˆ 2

ˆ( ) 65.4830 0.52( ) ( ) (10 2)(30)

tb

t

Y Ys

n k X X

1 10 44 42.90

2 9 40 39.37

3 11 42 46.43

4 12 46 49.96

5 11 48 46.43

6 12 52 49.96

7 13 54 53.49

8 13 58 53.49

9 14 56 57.02

10 15 60 60.55

1.10 1.2100 4

0.63 0.3969 9

-4.43 19.6249 1

-3.96 15.6816 0

1.57 2.4649 1

2.04 4.1616 0

0.51 0.2601 1

4.51 20.3401 1

-1.02 1.0404 4

-0.55 0.3025 9

65.4830 30

Time tX tYtY ˆ

t t te Y Y 2 2ˆ( )t t te Y Y 2( )tX X

Yt^=7.60 +3.53 Xt =7.60+3.53(10)= 42.90

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Example Calculation

2

ˆ 2

ˆ( ) 65.4830 0.52( ) ( ) (10 2)(30)

tb

t

Y Ys

n k X X

2

1

( ) 30n

tt

X X

2 2

1 1

ˆ( ) 65.4830n n

t t tt t

e Y Y

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Calculation of the t Statistic

ˆ

ˆ 3.53 6.790.52

b

bts

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

Critical Value (tabulated) at 5% level =2.306

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Confidence interval

We can also construct confidence interval for the true parameter from the estimated coefficient.

Accepting the alternative hypothesis that there is a relationship between X and Y.

Using tabular value of t=2.306 for 5% and 8 df in our example, the true value of b will lies between 2.33 and 4.73

t=b^+/- 2.306 (sb^)=3.53+/- 2.036 (0.52)

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Decomposition of Sum of Squares

2 2 2ˆ ˆ( ) ( ) ( )t t tY Y Y Y Y Y

Total Variation = Explained Variation + Unexplained Variation

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Decomposition of Sum of Squares

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Coefficient of Determination

Coefficient of Determination: is defined as the proportion of the total variation or dispersion in the dependent variable that explained by the variation in the explanatory variables in the regression.

In our example, COD measures how much of the variation in the firm’s sales is explained by the variation in its advertising expenditures.

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Coefficient of Determination

22

2

ˆ( )( )t

Y YExplained VariationRTotalVariation Y Y

2 373.84 0.85440.00

R

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Coefficient of Correlation

Coefficient of Correlation (r): The square root of the coefficient of determination.

This is simply a measure of the degree of association or co-variation that exists between variables X and Y.

In our example, this mean that variables X and Y vary together 92% of the time.

The sign of coefficient r is always the same as the sign of coefficient of b^.

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Coefficient of Correlation

2 ˆr R with the signof b

0.85 0.92r

1 1r

41



Multiple Regression Analysis

Model:

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

42



Relationship between 1 dependent & 2 or more independent variables is a linear function

1 2i i i k ki iY X X X

Y-intercept Slopes Random error

Dependent (Response) variableIndependent (Explanatory) variables


43




X2

Y

X1Y|X = 0 + 1X 1i + 2X 2i

0

Y i = 0 + 1X 1i + 2X 2i + i

ResponsePlane

(X 1i,X 2i)

(O bserved Y )

i

44




Too complicated by hand!

Ouch!

45



Multiple Regression Model: Example

Develop a model for estimating heating oil used for a single family home in the month of January, based on average temperature and amount of insulation in inches.

Oil (Gal) Temp Insulation275.30 40 3363.80 27 3164.30 40 1040.80 73 694.30 64 6

230.90 34 6366.70 9 6300.60 8 10237.80 23 10121.40 63 331.40 65 10

203.50 41 6441.10 21 3323.00 38 352.50 58 10

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Ch 4 : Demand Estimation Multiple Regression Model: Example

0 1 1 2 2i i i k kiY b b X b X b X Coefficients

Intercept 562.1510092X Variable 1 -5.436580588X Variable 2 -20.01232067

Excel Output

1 2ˆ 562.151 5.437 20.012i i iY X X

For each degree increase in temperature, the estimated average amount of heating oil used is decreased by 5.437 gallons, holding insulation constant.

For each increase in one inch of insulation, the estimated average use of heating oil is decreased by 20.012 gallons, holding temperature constant.

47




Adjusted Coefficient of Determination

2 2 ( 1)1 (1 )( )nR Rn k

Regression StatisticsMultiple R 0.982654757R Square 0.965610371Adjusted R Square 0.959878766Standard Error 26.01378323Observations 15

SSTSSRr ,Y 2

12

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Interpretation of Coefficient of Multiple Determination

212 .9656Y

SSRrSST

96.56% of the total variation in heating oil can be explained by temperature and amount of insulation95.99% of the total fluctuation in heating oil can be explained by temperature and amount of insulation after adjusting for the number of explanatory variables and sample size

2adj .9599r

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Testing for Overall Significance

Shows if Y Depends Linearly on All of the X Variables Together as a Group

Use F Test Statistic Hypotheses:

H0: …k = 0 (No linear relationship) H1: At least one i ( At least one independent

variable affects Y ) The Null Hypothesis is a Very Strong Statement The Null Hypothesis is Almost Always Rejected

50




Analysis of Variance and F Statistic

/( 1)/( )

Explained Variation kFUnexplained Variation n k

2

2

/( 1)(1 ) /( )

R kFR n k

51


Ch 4 : Demand Estimation Test for Overall Significance

Excel Output: Example

ANOVAdf SS MS F Significance F

Regression 2 228014.6 114007.3 168.4712 1.65411E-09Residual 12 8120.603 676.7169Total 14 236135.2

k -1= 2, the number of explanatory variables and dependent variable

n - 1p-value

k = 3, no of parameters

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Ch 4 : Demand Estimation Test for Overall Significance:

Example Solution

03.89

= 0.05

H0: 1 = 2 = … = k = 0H1: At least one j 0 = .05df = 2 and 12Critical Value:

Test Statistic:

Decision:

Conclusion:

F 168.47

Reject at = 0.05.

There is evidence that at least one independent variable affects Y.

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t Test StatisticExcel Output: Example

Coefficients Standard Error t Stat P-valueIntercept 562.1510092 21.09310433 26.65094 4.77868E-12Temp -5.436580588 0.336216167 -16.1699 1.64178E-09Insulation -20.01232067 2.342505227 -8.543127 1.90731E-06

t Test Statistic for X2 (Insulation)

t Test Statistic for X1 (Temperature)

i

i

b

btS

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Ch 4 : Demand Estimation t Test : Example Solution

Does temperature have a significant effect on monthly consumption of heating oil? Test at = 0.05.

H0: 1 = 0

H1: 1 0

df = 12

Critical Values:

Test Statistic:t Test Statistic = -16.1699Decision:Reject H0 at = 0.05.

Conclusion:There is evidence of a significant effect of temperature on oil consumption holding constant the effect of insulation.

Reject HReject H 00

.025 .025

-2.1788 2.17880

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Problems in Regression Analysis

Multicollinearity: Two or more explanatory variables are highly correlated.

Heteroskedasticity: Variance of error term is not independent of the Y variable.

Autocorrelation: Consecutive error terms are correlated.

Functional form: Misspecified by the omission of a variable

Normality: Residuals are normally distributed or not

56



Practical Consequences of Multicollinearity

Large variance or standard error Wider confidence intervals Insignificant t-ratios A high R2 value but few significant t-

ratios OLS estimators and their Std. Errors

tend to be unstable Wrong signs for regression coefficients

57



Multicollinearity

How can Multicollinearity be overcome? Increasing number of observation Acquiring additional data A new sample Using an experience from a previous study Transformation of the variables Dropping a variable from the model This is the simplest solution, but the worse

one referring an economic model (i.e., model specification error)

58



Heteroskedasticity

Heteroskedasticity: Variance of error term is not independent of the Y variable or unequal/non-constant variance. This means that when both response and explanatory variables increase, the variance of response variables does not remain same at all levels of explanatory variables (cross-sectional data).

Homoscedasticity: when both response and explanatory variables increase, the variance of response variable around its mean value remains same at all levels of explanatory variables (equal variance).

59



Residual Analysis for Homoscedasticity Residual Analysis for Homoscedasticity

Heteroscedasticity Homoscedasticity

SR

X

SR

X

Y

X X

Y

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Autocorrelation or serial correlation

Autocorrelation: Correlation between members of observation ordered in time as in time series data (i.e., residuals are correlated where consecutive errors have the same sign).

Detecting Autocorrelation: This can be detected by many ways. The most common used is DW statistics.

61



Durbin-Watson Statistic

Test for Autocorrelation

21

2

2

1

( )n

t tt

n

tt

e ed

e

If d=2, autocorrelation is absent.

62



Residual Analysis for IndependenceResidual Analysis for Independence

The Durbin-Watson Statistic– Used when data is collected over time to detect

autocorrelation (residuals in one time period are related to residuals in another period)

– Measures violation of independence assumption2

12

2

1

( )n

i ii

n

ii

e eD

e

Should be close to 2. If not, examine the model for autocorrelation.

63



Residual Analysis for IndependenceResidual Analysis for Independence

Not Independent

Independent

e e

TimeTime

Residual is Plotted Against Time to Detect Any Autocorrelation

No Particular PatternCyclical Pattern

Graphical Approach

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Accept H0

(no autocorrelation)

Using the Durbin-Watson StatisticUsing the Durbin-Watson Statistic

: No autocorrelation (error terms are independent) : There is autocorrelation (error terms are not)

0H

1H

0 42dL 4-dLdU 4-dU

Reject H0

(positive autocorrelation)

Inconclusive Reject H0

(negative autocorrelation)

65



Steps in Demand Estimation

Model Specification: Identify Variables Collect Data Specify Functional Form Estimate Function Test the Results

66



Functional Form Specifications

Linear Function:

Power Function:

0 1 2 3 4X X YQ a a P a I a N a P e

1 2( )( )b bX X YQ a P P

Estimation Format:

1 2ln ln ln lnX X YQ a b P b P

67



Dummy-Variable Models

When the explanatory variables are qualitative in nature, these are known as dummy variables. These can also defined as indicators variables, binary variables, categorical variables, and dichotomous variables such as variable D in the following equation:

eDcIcPccQ xx ......3210

68



Dummy-Variable Models

Categorical Explanatory Variable with 2 or More LevelsYes or No, On or Off, Male or Female, Use Dummy-Variables (Coded as 0 or 1)Only Intercepts are DifferentAssumes Equal Slopes Across CategoriesRegression Model Has Same FormCan the dependent variable be dummy?

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0 1 1 2 0 1 1ˆ (0)i i iY b b X b b b X

0 1 1 2 0 2 1 1ˆ (1) ( )i i iY b b X b b b b X

Dummy-Variable ModelsDummy-Variable Models

Given:Y = Assessed Value of HouseX1 = Square Footage of HouseX2 = Desirability of Neighbourhood =Desirable (X2 = 1)

Undesirable (X2 = 0)

0 if undesirable 1 if desirable

0 1 1 2 2i i iY b b X b X

Same slopes

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Simple and Multiple Regression Compared: Example

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Regression Analysis in Practice

Suppose we have an Employment (Labor Demand) Function as follows:

N=Constant+K+W+AD+P+WT N: employees in employment K: capital accumulation W: value of real wages AD: aggregate deficit P: effect of world manufacturing exports on

employment WT: the deviation of world trade from trend.

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Output by Microfit v4.0w

Ordinary Least Squares Estimation ******************************************************************************* Dependent variable is LOGN 39 observations used for estimation from 1956 to 1994 ******************************************************************************* Regressor Coefficient Standard Error T-Ratio[Prob] CON 4.9921 .98407 5.0729[.000] LOGK .040394 .012998 3.1078[.004] LOGW .024737 .010982 2.2526[.032] AD -.9174E-7 .1587E-6 .57798[.567] LOGP .026977 .0099796 2.7032[.011] LOGWT -.053944 .024279 2.2219[.034] ******************************************************************************* R-Squared .82476 F-statistic F( 6, 33) 20.8432[.000] R-Bar-Squared .78519 S.E. of Regression .012467 Residual Sum of Squares .0048181 Mean of Dependent Variable 10.0098 S.D. of Dependent Variable .026899 Maximum of Log-likelihood 120.1407 DW-statistic 1.8538 *******************************************************************************

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Ch 4 : Demand Estimation Diagnostic Tests

******************************************************************************* * Test Statistics * LM Version * F Version * ******************************************************************************* * * * * * A:Serial Correlation *CHI-SQ( 1)= .051656[.820]*F(1,30)=.039788[.843]** * * * * B:Functional Form *CHI-SQ( 1)= .056872[.812]*F(1,30)=.043812[.836]* * * * * * C:Normality *CHI-SQ( 2)= 1.2819[.527]* Not applicable * * * * * * D:Heteroscedasticity *CHI-SQ( 1)= 1.0065[.316]*F( 1,37)=.98022[.329]* ******************************************************************************* A:Lagrange multiplier test of residual serial correlation B:Ramsey's RESET test using the square of the fitted values C:Based on a test of skewness and kurtosis of residuals D:Based on the regression of squared residuals on squared fitted values

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Dependent Variable: LOGNExplanatory Variables

CON 4.9921(5.07)

LOGK 0.40394

(3.10)

LOGW0.0247(2.25)

AD -0.9174(-0.577)

LOGP 0.0269(2.70)

LOGWT -0.0539(-2.22)

R2 0.87

0.83

DW 2.16

SER 0.021X2

SC .05165[.820]X2

FF 05687[.812]X2

NORM 1.2819[.527]X2

HET 1.0065[.316]

R2 bar

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Ch 4 : Demand Estimation Interpretation

t-test (individual significance)t-test (individual significance)Let’s first see the significance of each variable;n=39k=6 and hence d.f.=39-6=33=0.05 (our confidence level is 95%). With =0.05 and d.f.=33, ttab=2.045Our Hypothesis are: Ho:s=0 (not significant)

H1: s0 (significant) This is t- distribution and using this distribution, you can decide whether individual t-values (calculated or estimated) of the existing variables are significant or not according to the tabulated t-values as appears in the fig above.

79



F-test (overall significance)F-test (overall significance) Our result is F(6,33)=20.8432k-1=5 and n-k=33= 0.05 (our confidence level is 95%). With = 0.05 and F(6,33), the Ftab=2.34 Our hypothesis areHo:R2s=0 (not significant)H1: R20 (significant)

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Diagnostic Tests:

Serial Correlation: Ho:=0(existence of autocorrelation )H1:0 (no autocorrelation) Since CHI-SQ(1)=0.051656< X2=3.841, we reject Ho that estimate regression does not have first order serial correlation or autocorrelation.

81



Functional Form: Ho:=0 (existence misspecification)H1: 0 (no of misspecification) The estimated LM version of CHI-SQ is 0.0568721 and with = 0.05 the tabular value is X2=3.841. Because CHI-SQ (1)=0.0568721< X2=3.841, then we reject the null hypothesis that there is functional misspecification.

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Normality:

Ho:ut=0 (residuals are not normally distributed)

H1:ut0(residuals are normally distributed)

Our estimated result of LM version

for normality is CHI-SQ(2)=1.28191, and the tabular value with 2 restrictions with = 0.05 is X2=5.991.

Since CHI-SQ(2)=1.28191< X2=5.991, the test result shows that the null hypothesis of normality of the residuals is accepted.

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Heteroscedasticity: Ho:yt

2=2 (heteroscedasticity) H1: yt

22(homoscedasticity) LM version of our result for

the heteroscedasticity is CHI-SQ(1)=1.00651 and table critical value with 1 restriction with = 0.05 is X2=3.841. Since CHI-SQ(1)=1.00651< X2=3.841, we accept the null hypothesis that error term is constant for all the independent variables.

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The EndThe End

Thanks

Department of Business Administration FALL 20 10 -11 Demand Estimation by Assoc. Prof. Sami Fethi.

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Transcript of Department of Business Administration FALL 20 10 -11 Demand Estimation by Assoc. Prof. Sami Fethi.