Department of Business Administration FALL 2007-08 Demand Estimation Demand Estimation by Asst. Prof. Sami Fethi.

Department of Business Administration

FALL 2007-08

Demand EstimationDemand Estimation

by

Asst. Prof. Sami Fethi

2 Managerial Economics © 2007/08, Sami Fethi, EMU, All Right Reserved.

Ch 4 : Demand Estimation

Demand EstimationDemand Estimation

In the previous chapter, we introduced several concepts useful in demand analysis and indicated the key role that product demand plays in most managerial or business decisions.

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.



The Identification ProblemThe Identification Problem

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.


Ch 4 : Demand Estimation The Identification ProblemThe 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.



The Identification ProblemThe Identification Problem

In the previous 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.



Demand Estimation:Demand Estimation:Marketing Research ApproachesMarketing Research Approaches

Consumer SurveysObservational ResearchConsumer ClinicsMarket ExperimentsVirtual ShoppingVirtual Management




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

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.




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.




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.


Ch 4 : Demand Estimation Demand Estimation:Demand Estimation:

Marketing Research ApproachesMarketing Research Approaches 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.



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



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.


Ch 4 : Demand Estimation Scatter Diagram

Scatter diagram shows a positive relationship between the relevant variables and this 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


Ch 4 : Demand Estimation Regression AnalysisRegression 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.



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 FitRegression 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).



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



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.



Regression AnalysisRegression 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.


Ch 4 : Demand Estimation Simple Regression AnalysisSimple 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.



Simple Linear Regression ModelSimple 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

Regression

Line

Dependent (Response) Variable



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

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




Estimation Procedure

1

2

1

( )( )ˆ

( )

n

t tt

n

tt

X X Y Yb

X X

ˆa Y bX


Ch 4 : Demand Estimation Ordinary Least Squares (OLS)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 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


Ch 4 : Demand Estimation Ordinary Least Squares (OLS)Ordinary Least Squares (OLS)

Estimation Example

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



The Equation of Regression LineThe 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.



Crucial AssumptionsCrucial 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.



Tests of Significance:Tests of Significance: Standard ErrorStandard 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:



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


Ch 4 : Demand Estimation Tests of SignificanceTests 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( ) 65.48300.52

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

bt

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




Example Calculation

2

ˆ 2

ˆ( ) 65.48300.52

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

bt

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




Calculation of the t Statistic

ˆ

ˆ 3.536.79

0.52b

bt

s

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

Critical Value (tabulated) at 5% level =2.306



Confidence intervalConfidence intervalWe 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)




Decomposition of Sum of Squares

2 2 2ˆ ˆ( ) ( ) ( )t t tY Y Y Y Y Y

Total Variation = Explained Variation + Unexplained Variation




Decomposition of Sum of Squares



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.




Coefficient of Determination

22

2

ˆ( )

( )t

Y YExplained VariationR

TotalVariation Y Y

2 373.840.85

440.00R



Coefficient of CorrelationCoefficient 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^.




Coefficient of Correlation

2 ˆr R with the signof b

0.85 0.92r

1 1r



Multiple Regression AnalysisMultiple Regression Analysis

Model:

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



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) variable

Independent (Explanatory) variables





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

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




Too complicated by hand! Ouch!


Ch 4 : Demand Estimation Multiple Regression Model: ExampleMultiple 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



Multiple Regression Model: ExampleMultiple 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.




Adjusted Coefficient of Determination

2 2 ( 1)1 (1 )

( )

nR R

n k

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

SST

SSRr ,Y 2

12


Ch 4 : Demand Estimation Interpretation of Coefficient of Multiple Interpretation of Coefficient of Multiple

DeterminationDetermination

212 .9656Y

SSRr

SST

96.56% of the total variation in heating oil can be explained by temperature and amount of insulation

95.99% 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



Testing for Overall SignificanceTesting 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 independentvariable affects Y )

The Null Hypothesis is a Very Strong Statement The Null Hypothesis is Almost Always Rejected




Analysis of Variance and F Statistic

/( 1)

/( )

Explained Variation kF

Unexplained Variation n k

2

2

/( 1)

(1 ) /( )

R kF

R n k



Test for Overall SignificanceTest for Overall SignificanceExcel Output: ExampleExcel 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 - 1

p-value

p-value

k = 3, no of parameters


Ch 4 : Demand Estimation Test for Overall Significance:Test for Overall Significance:

Example SolutionExample Solution

03.89

= 0.05

H0: 1 = 2 = … = k = 0

H1: At least one j 0

= .05

df = 2 and 12

Critical Value:

Test Statistic:

Decision:

Conclusion:

F168.47

Reject at = 0.05.

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



t Test Statistict Test StatisticExcel Output: ExampleExcel 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

bt

S


Ch 4 : Demand Estimation t t Test : Example Solution 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 = -16.1699

Decision:

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



Problems in Regression AnalysisProblems 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



Practical Consequences of MulticollinearityPractical Consequences of Multicollinearity

Large variance or standard errorWider confidence intervalsInsignificant t-ratiosA high R2 value but few significant t-ratiosOLS estimators and their Std. Errors tend to

be unstableWrong signs for regression coefficients


Ch 4 : Demand Estimation MulticollinearityMulticollinearity

How can Multicollinearity be overcome? Increasing number of observationAcquiring additional dataA new sample Using an experience from a previous study Transformation of the variables Dropping a variable from the modelThis is the simplest solution, but the worse

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


Ch 4 : Demand Estimation HeteroskedasticityHeteroskedasticity

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 variables around its mean value remain same at all levels of explanatory variables (equal variance).



Residual Analysis for Homoscedasticity Residual Analysis for Homoscedasticity

Heteroscedasticity Homoscedasticity

SR

X

SR

X

Y

X X

Y



Autocorrelation or serial correlationAutocorrelation 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.



Durbin-Watson StatisticDurbin-Watson Statistic

Test for Autocorrelation

21

2

2

1

( )n

t tt

n

tt

e ed

e

If d=2, autocorrelation is absent.



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 assumption

21

2

2

1

( )n

i ii

n

ii

e eD

e

Should be close to 2.

If not, examine the model for autocorrelation.



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



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 independent)

0H

1H

0 42dL 4-dLdU 4-dU

Reject H0

(positive autocorrelation)

Inconclusive Reject H0

(negative autocorrelation)



Steps in Demand EstimationSteps in Demand Estimation

Model Specification: Identify VariablesCollect DataSpecify Functional FormEstimate FunctionTest the Results



Functional Form SpecificationsFunctional 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



Dummy-Variable ModelsWhen 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



Dummy-Variable ModelsCategorical Explanatory Variable with 2 or More Levels

Yes or No, On or Off, Male or Female,

Use Dummy-Variables (Coded as 0 or 1)

Only Intercepts are Different

Assumes Equal Slopes Across Categories

Regression Model Has Same Form

Can the dependent variable be dummy?



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 House

X1 = Square Footage of House

X2 = 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



Simple and Multiple Regression Compared:Simple and Multiple Regression Compared: ExampleExample



Simple and Multiple Regression Compared: Slope CoefficientsSimple and Multiple Regression Compared: Slope Coefficients



Simple and Multiple Regression Compared: Simple and Multiple Regression Compared: rr22



Example: Adjusted Example: Adjusted rr22 ccan Decreasean Decrease



Regression Analysis in PracticeRegression 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.



Output by Microfit v4.0w 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 *******************************************************************************



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



Dependent Variable LOGN

Explanatory Variables

CON4.9921(5.07)

LOGK 0.40394

(3.10)

LOGW0.0247(2.25)

AD-0.9174(-0.577)

LOGP0.0269(2.70)

LOGWT-0.0539(-2.22)

R2 0.87

0.83

DW 2.16

SER 0.021

X2SC .05165[.820]

X2FF 05687[.812]

X2NORM 1.2819[.527]

X2HET 1.0065[.316]

R2


Ch 4 : Demand Estimation Interpretation

A) 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=2.045

Our 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.



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 are

Ho:R2s=0 (not significant)

H1: R20 (significant)



Diagnostic Tests:

Serial Correlation: Ho:=0 (no autocorrelation)

H1:0 (existence of autocorrelation)

Since CHI-SQ(1)=0.051656< X2=3.841, we accept Ho that

estimate regression does not have first order serial correlation.



Functional Form: Ho:=0 (no misspecification)

H1: 0 (existence 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)=0.051656< X2=3.841, then we accept the null hypothesis that there is no functional misspecification.


Ch 4 : Demand Estimation Normality:

Ho:ut=0 (residuals are normally distributed)

H1:ut0(residuals are not 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.



Heteroscedasticity: Ho:yt

2=2 (homoscedasticity) H1: yt

22(heteroscedasticity) 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.



The EndThe End

Thanks

Department of Business Administration FALL 2007-08 Demand Estimation Demand Estimation by Asst. Prof. Sami Fethi.

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