ikiki22i110i X...XXY ε+β++β+β+β=

εβ +=

=

XY

ni ,...,1

jji

i

X

Y β=∂∂

k22221

k11211

2

1

X...XX1

X...XX1

XY

Y

Y

=

=MMMM

62

Estimation of regression parameters:-- Least squares-- Maximum likelihood

1nxn

2

1

1x)1k(k

1

0

)1k(nxnk2n1n1nxn X...XX1Y

ε

εε

=ε

β

ββ

=β

+

+

MM

MMMM

Endogenous variables --Dependent variables, values of which are determined within the system

Exogenous variables --Variables determined outside the system butinfluence the system by affecting the values of the endogenous variables

63

of the endogenous variables

Structural parameters --Estimated using econometric techniques andrelevant data

Lagged endogenous variables

Lagged exogenous variables

Predetermined variables

(1) Degrees-of Freedom Problem

Available data imply do not include enough observations to allow adequate or reasonable estimates of the model

64

(2) Collinearity Problem

Tendency of the data to bunch or move together, e.g., in time-series data, the variables tend to exhibit the same trends, cyclical and secular, over time; interdependence among regressors in the model

(3) Serial Correlation Problem

Predominantly a problem with time-series data; underlying changes occur very slowly over time to the extent that the stochastic disturbance term represents conditions relevant to the model but not explicitly accounted for (such as omitted variables); serial correlation represents the

65

variables); serial correlation represents the dependence of stochastic disturbance terms in different periods.

(4) Structural Change Problem

Parameters may not be constant over time

(5) Errors-in-Measurement Problem

Data subject to various inaccuracies or biases; for example, GDP figures are revised periodically. Such changes necessitate data manipulations to make them comparable and

66

manipulations to make them comparable and

consistent over time.

• Structural specification versus forecasting

• Functional form

• Single-equation versus multi-equation specifications

• Time and effort to construct models

• Choice of econometric or time-series models

67

“It will be remembered that the 70 translators of the Septuagint were shut up in 70 separate rooms with the Hebrew text and brought out with them, when they emerged, 70 identical translations. Would the same miracle be vouchsafed if 70 multiple correlators were shut up with the same statistical material?”

-- John Maynard Keynes

� Model specification entails the expression of theor etical constructs in mathematical terms

� This phase of applied econometrics constitutes the model building activity

� In essence, model specification is the translation of theoretical constructs into mathematical/statistical forms.

68

constructs into mathematical/statistical forms.� Fundamental principles in model building

� The principle of parsimony (other things the same, simple models generally are preferable to complex models, especially in forecasting)

� The shrinkage principle (imposing restrictions eith er on estimated parameters or on forecasts often improves model performance)

� The KISS principle (Zellner, 2001), “Keep it Sophis tically Simple.”

� Often in applied econometrics, analysts have interest in estimating demand relationships, particularly for commodities

� Analysts may wish to estimate the demand for

69

� Analysts may wish to estimate the demand for cosmetic products, automobiles, various food products, or various beverages.

� The Demand Curve shows the theoretical relation between price and quantity demanded, holding all other factors constant.

� Axes: price is on y -axis, quantity on the x -axis

70

� Axes: price is on y -axis, quantity on the x -axis

� Example: Demand curve for Lipton tea,

Q=2500 – 500P

� Key question: How are these numbers obtained?

Average price per package

Demand Curve

Q = 2,500 – 500P

P = 5 – .002Q

5

4

3

71

2

1

500 1000 1500 2000 2500

packages of Lipton tea

� Movement along a given demand curve reflects a change in price and quantity of the commodity in question.

� Answer the following question: What happens to the quantity demanded if the price changes (exogenously) holding all other factors constant?

72

holding all other factors constant?

� Law of demand:

� The demand curve is negatively sloped.

� An increase (a decrease) in own price leads to a decrease (an increase) in quantity demanded.

� A very important empirical finding in economics.

� Prices of other products

� Income

� Advertising (positive information)

� Food Recalls or Food Scares (negative

73

� Food Recalls or Food Scares (negative information)

� Health and nutrition factors

� Lifestyle factors

� Tastes and preferences

1. Q = a-bP2. Q = a0 – a1P + a2I + a3A + a4PS

own-price effect income effect advertising price of substitute product

74

� The coefficients a 0, a1, a2, a3, and a 4 are labeled the demand parameters.

� We expect certain signs and magnitudes of the demand parameters according to economic theory.

� Different versions of the econometric model for applied analysis are possible.

own-price effect(-)

income effect(+)

advertising effect

(+)

price of substitute product(+)

� The starting point then rests on the development of a linear regression model.

� Philosophical differences exist among statisticians and applied econometric analysts.

� Attention centered on signs and magnitudes of the coefficients inherent in the model.

75

coefficients inherent in the model.

� Model specifications in applied econometrics are non-unique.

� Model specifications provide the blueprint for action.

� The effect of unionization on earnings hypothesis:� Unionization increases real wages

lnw = 0.28lnQ + 0.77Z + 0.84lnP + 0.228U- 0.16UZ(0.056) (0.28) (0.32) (0.094) (1.10)

-1.94UlnP + CONSTANT(0.99)

� R2= 0.891

76

� R2= 0.891� W = relative wages (average hourly compensation in

unionized industries relative to compensation in nonunionized industries)

� Q = relative value of output� Z = unemployment rate� P = measure of expected inflation� U = measure of the extent of union membership� UZ, UlnP = interaction variables

� Specification based on change rather than levels, e .g., estimation of a demand equation for new automobiles in the United States

85.0R

PS827.0)m/p(234.0S507.0I106.0115.0A

2)261.0()088.0()086.0()011.0(

=

−∆−∆−∆+=∆

77

� A = annual retail sales of new passenger automobile s� change in annual retail sales� change in real disposable income� change in the stock of passenger automobiles� change in average retail price deflated by the

average duration of automobile loans� PS= dummy variable to take account of years of sever e

production shortages

=∆A

=∆I

=∆S

=∆ )m/p(

Special type of algebraic model, namely stochastic (as opposed to deterministic)

� linear or nonlinear in parameters

� inherently linear versus nonlinear models

78

� inherently linear versus nonlinear models

� additive stochastic disturbance term that plays� the role of a chance medium

In general, each equation of an econometric model, other than definitions, equilibrium conditions, and identities, is assumed to contain an additive disturbance term.

An inherently linear model is a model which is linear in parameters.

βββ

ε

++=

=

kk

k

XXXgXXXgyF

XXXFY

),...,,(),...,,()(

),,...,,(

212221110

21

79

Polynomial model, exponential model, reciprocal model, semi-logarithmic model, interaction model

εσσσσ

εσ

εβ

σσσ

+++++=

=

+++

kk

k

kkk

XXXY

XXXYge

XXXg

k

ln...lnlnlnln

)exp(....,.

),...,,(...

22110

210

21

21

Show that by appropriate transformations of variables (e.g. logarithms, reciprocals) the following nonlinear models relating y to the explanatory variables x1 and x2 and the stochastic disturbance term u can be converted to models that are linear in parameters

u22

11 exAxY ββ=1.

(Constant elasticity)

80

122110

22110

)uxx(Y

)uxxexp(Y

−+β+β+β=

+β+β+β=

(Constant elasticity)

2.(exponential)

3.(reciprocal)

Stochastic, a random variable

Statistical distribution often normal

Captures:

81

Captures:

1. Omission of the influence of other variables

2. Measurement error

1. Model linear in parameters

2.

3.(a) Homoscedasticity(b) No serial or autocorrelation

0)(E =ε

IE T 2)( σεε =

82

(b) No serial or autocorrelation

4. X is a nx(k+1) matrix of fixed numbers

5. Rank of X = r(X) = k+1 < n

6.

[ ] [ ] 0EXXE TT =ε=ε⇒

),0(~ 2IN σε

1. Computational cost

2. Least squares

3. R2

4. Unbiasedness

5. Minimum variance

6. Minimum mean squared error

83

6. Minimum mean squared error

7. Asymptotic properties

8. Maximum likelihood

Choice of parameter estimates that maximize the probability of observing the data; requires assumptions of a statistical distribution

Often the normal distribution is the statistical distribution of choice

84

Residuals

Minimization of the sum of squared residuals

OLS estimator1T2OLS

T1T

)XX(s)ˆ(VAR

YX)XX(ˆ−

−

=β=β

Not sensitive to outliers

More generally these estimators are not sensitive to departures from assumptions

Minimize the sum of the absolute values of the

85

Minimize the sum of the absolute values of the residuals:

MAD (Mean Absolute Deviation)

LAR (Least Absolute Residuals)

LAE (Least Absolute Error)

1. Sum of squared residuals (SSE)

2. Regression sum of squares (SSR)

3. Corrected total sum of squares (SST)

( )2

2ˆ

∑

∑=

−=

i

ii

eSSE

YYSSE

( )2ˆ∑ −= YYSSR i

86

3. Corrected total sum of squares (SST)

4. SSE + SSR = SST

5. R2 = SSR/SST

6. F-statistics, t-statistics

7. Tests of hypotheses, construction of confidence intervals

( )2

∑ −= YYSST i

)1)(1/1(1

/1/

2

22

22

2

R

RR

RknnR

SSTSSESSTSSRR

≥

−−−−−=

−==

can be negative

If k>1, then

87

)/)1))((1/(()1/(

)/(

0...:

221,

210

kknRRknSSE

kSSRF

H

R

knk

k

−−−=−−

=

====

−−

βββ

can be negative

• N refers to the number of observations

• K refers to the number of parameters to be estimated, not counting the intercept

� is the “penalty factor”

MSEpT

T

T

e

pT

Ts

T

eMSE

T

tt

T

tt

−=

−=

=

∑

∑

=

=

1

2

2

1

2

pT

T

−

Useful way of discerning

alternative model specifications; for each specification,

calculate the associated model

selection statistics.

88

is the “penalty factor”

� Akaike Information Criterion (AIC)

� Schwarz or Bayesian Information Criterion(SIC) or (BIC)

� p is the number of parameters to be estimated� T is the number of observations

pT−

MSEeAIC T

p

=2

MSETBICSIC T

p

=)(

selection statistics. Then the smallest statistic yields the most appropriate

model specification.

Model 1 Model 2 Model 3

AIC 19.35 15.83 17.15

89

Which model to choose?

SIC 19.37 15.86 17.17

1. Ordinary regression coefficients

2. Beta or standardized regression coefficients

90

3. Elasticities

4. Partial correlation coefficients

kki*

2x

2i2*2

1x

1i1*1

Y

i

)XX(...

S

)XX(

S

)XX(

S

)YY(

−β++

−β+

−β=−

91

Interpretation: A beta coefficient of .25 means a 1 standard deviation change in the exogenous variable will lead to a .25 standard deviation chan ge in the dependent variable

Y

xjj

*j

xk

kki*k

S

Sˆˆ

S

)XX(...

β=β

−β++

Standardized regression coefficients

92

coefficientsInterpretation?

Elasticity (for a linear functional form):

� Interpretation: An elasticity of -.4 means a 1 percent change in the exogenous variable will lead to a -.4 percent change in the dependent variable.

Y

X

X

Y

Y

XE j

j

jjj

∂∂== β

93

� The elasticity corresponds to a measure of sensitivity of the percentage change in Y due to a one percent change in the explanatory variable Xj. This estimate of the elasticity of Y with respect to Xj is a function of , Xjt, and Yt. This estimate varies with the respective observations of Xjt, and Yt. Often, to come up with a single estimate of this elasticity, we employ the sample means.

jβ

VariableEstimated OLS

CoefficientMean Elasticity

PSHRIMP -0.03223 611.37 -2.8749

PFIN -0.02731 441.68 -1.7599

94

PSHELL1 0.01590 661.74 1.5351

ADSHRIMP -0.00101 46.33 -0.0068

ADFIN -0.01398 118.29 -0.2413

ADSHELL1 0.00848 11.49 0.0142

QSHRIMP ------ 6.85 ------

�Demand Estimation :�Dependent variable: Q, tile sales (in 1000 of cases)

�Right-hand side variables: tile price per case (p), income per capita I (in 1000 of $), and advertising expenditure A(in 1000 $) .

Q = ββββ0 + ββββ1P + ββββ2I + ββββ3A + uRegression:

95

� Interpretation :�β1 measures the effects of the tile price on the tile

consumption, holding all other factors fixed

�β2 represents the effects of income, holding all other factors fixed

�β3 represents the effects of advertising, holding all other factors fixed

(1) What is the impact of a price change on tile sales?

(2) What is the impact of a change in income on til e sales?

(3) What is the impact of a change in advertising

96

(3) What is the impact of a change in advertising expenditures on tile sales?

� Calculation of own-price elasticity?

� Calculation of income elasticity?

� Calculation of advertising elasticity?

� Suppose we wish to see whether or not the dependent variable and one independent variable are related after netting out the effect of any other independent variables in the model.

97

� Partial correlation coefficient between Yand Xj -- the effect of Xj on Y not accounted for by the other variables in the model.

Calculation:

(1) Eliminating the linear effect of the remaining X’s on Y

(2) Eliminate the linear effect of the remaining X’s on

(3) Let **

kk

1j1j1j1j22110j

kk

1j1j1j1j22110

XXX,YYY

Xˆ...

XˆXˆ...XˆXˆˆX

Xˆ...

XˆXˆ...XˆXˆˆY

−=−=

α++α+α++α+α+α=

α++α+α++α+α+α=

++−−

++−−

98

(3) LetCorr -partial correlation coefficient, -1,1

The partial correlation coefficient of .53 on varia ble implies that 28% (.53 2) of the variance in the dependent variable not accounted for by the other independent variables is accounted for by

jj*

j* XXX,YYY −=−=

)X,Y( *j

*

jX

jX

99

Circled entries are the squared partial correlation coefficients. The partial correlation coefficients are respectively:

PSHRIMP → -.7010 ADSHRIMP → -.0151

PFIN → -.2068 ADFIN → -.2932

PSHELL1 → .2272 ADSHELL1 → .0430

100

101

102

Variable Name

Estimated Coefficient

Partial Correlation

Standardized Coefficient

Elasticity at Means

PPAG -206.92 -0.796 -0.6017 -2.8388

PRAG 31.629 0.235 0.1309 0.4080

103

PRAG 31.629 0.235 0.1309 0.4080

TOTEXP 0.10728 0.737 0.6007 0.7090

CONSTANT 366.10

Goodness-of-fit Test (F-test)

Single Coefficients (t- or F-tests)

Linear Combinations of Coefficients (F-tests)

104

Nonlinear Combinations of Coefficients (χ2-tests)

Joint Tests of Several Regression Coefficients (F-tests)

Equality of Coefficients of Different Regressions (Chow Test) (F-test)

0 oneleast at : 0: 10 ≠= jHH ββ

ANOVA (Analysis of Variance) TableSource DF SS MS

(degrees of freedom) (sum of squares) (mean square)

105

1/

/1, −−

=−− knSSE

kSSRF knk

regressionresidualtotal 1

1/ˆ1

/ˆ 2

−

−−−−−

−

n

knSSEYXYYkn

kSSRYnYXk

TTT

TT

β

β

SSR → regression sum of squares

SSE → error sum of squares

N → # of observations

K → # of parameters to be estimated, not including the intercept

:

:

01

00

ββ

ββ

≠

=

H

H

j

j

106

A percent confidence interval for

)1( 100

)(ˆ

/)ˆ(

ˆ2/),1(

ˆ01

α

β

ββ

βα

β

−

±

−=

−−

−−

j

j

St

St

knj

jkn

jβ

107

� Estimation entails two classes, point and interval

� Point estimators yield a single numerical estimate

� But, the estimates we obtain usually do not have precisely the same value from sample to sample.

108

precisely the same value from sample to sample.

� Characterize estimates of structural parameters in terms of intervals, an explicit recognition that parameter estimates indeed are random variables.

� Construction of confidence intervals

� Another way to use classical statistical testing is to construct a confidence interval using the same critical value as was used for a two-sided test

� A (1 - αααα) % confidence interval is defined as

( ) percentile -1 theis where, ˆˆ

•± csec

αββ

109

� Depending on the sample size and the assumptions about the distribution of the error term, the critical value also can be constructed using the standard normal instead of the t distribution.

( )ondistributi ain

percentile 2

-1 theis where, ˆˆ

1−−

•±

kn

jj

t

csec ββ

110

111

� 95% confidence intervals for the OLS parameter esti mates

� t90, .025 = 1.987

� So, for PSHRIMP, parameter estimate = -.03223 and

standard error = .00346

� 95% CI = -.03223 ± .00346(1.987)

:

) :

01

000

kj

iii

kj

H

aH

γββ

γβγββ

≠+

==+ ∑(more generally,

112

)ˆ,ˆ(2)ˆ()ˆ(

)ˆ,ˆ(2)ˆ()ˆˆ(

)ˆˆ(

)ˆˆ(

201

kjkj

jijijiiia

kj

kjkn

COVVARVARS

COVaaVARaSS

t

kj

ii

ββββ

βββββ

γββ

ββ

β

++=

+=+

−+=

+

∑−− ∑ ∑ ∑

Reject H 0 : Test that the estimated coefficients of Pfin

and Pshell1 are jointly equal to zero.

Reject H 0 : Test that the estimated coefficients of Adfin and Adshell1 shell are jointly

equal to zero.

113

Reject H 0: Test that the estimated coefficients of

Pshrimp and Pshell1 are the same.

Reject H 0 : Test that the estimated coefficients of Pfin

and Pshell1 are the same.

Unrestricted Model (UR)

Restricted Model

ε+β++β+β+β= kk22110 X...XXY

0...:

...

10

22110

===

+++++=

+−

−−

H

XXXY

kqk

qkqk

ββεββββ

114

( )1/)1(

/

1/

/)(

2

22

1,

1,

−−−−=

−=−≥⇒

=−−

−=

≥

−−

−−

knR

qRRF

SSRSSRSSESSE

SSRSSR

SSTSST

knSSE

qSSESSEF

SSESSE

UR

RURknq

RURURR

RUR

RUR

UR

URRknq

URR

Tests involving the equality of coefficients of dif ferent regressions

Calculate

)1kN(df)N,...,1i(

X...XXY)1( ikiki22i110i

−−=ε+β++β+β+β=

)1kM(df)M,...,1J(

X...XXY)2( jkjkj22j110j

−−=

ε+σ++σ+σ+σ=

)22( −−++= kMNdfSSESSESSE

115

Calculate

Pooling Under

)22( 21 −−++= kMNdfSSESSESSEUR

kk2211000 ,...,,,:H σ=βσ=βσ=βσ=β

0H→

)22/(

1/)(

),...,11(

...

22,1

1121211101

−−++−=

+=+++++=

−−++ kMNSSE

kSSESSEF

SSE

MN

XXXY

UR

URRkMNk

R

kk εαααα

116

� Split the sample in two pieces at various break points.

� Reason for doing so due to the fact that the exact break point often is unknown.

� Basic test for structural change

� Make sure there exists a sufficient number of

117

� Make sure there exists a sufficient number of observations before and after a “potential” break point in order to conduct the Chow test.

� The break point specified on the Chow = (n) option is such that the first subsample consists of n – 1 observations and the second subsample starts from the n th observation and runs to the last observation of the entire sample.

� Sequential Chow Test:Split the sample in two pieces at every point

118

� Recursive Estimation:

Run a series of regressions by adding one observation per regression

N + M = 52K = 7

Sample 1 Sample 2

14 38 Reject H 0

119

K + 1

N + M – 2K – 2

14 38 Reject H 0

19 33 Reject H 0

24 28 Reject H 0

29 23 Reject H 0

N + K = 36K = 6

Sample 1 Sample 2

7 29Fail to

Dependent variable lnpcg

120

K + 1

N + M – 2K – 2

7 29Fail to Reject H 0

12 24 Reject H 0

21 15 Reject H 0

121

122

� RESET – Regression Specification Error Test:� Run additional regressions of the dependent variable on

the independent variables and on powers of included in the same regression

� The Reset Test is an F-test: Y

Adequacy of Model Specification

123

� The Reset Test is an F-test:� Test whether the coefficients on the variables are

(jointly) zero

Y

Y

Ramsey Reset Test indicates a problem

124

indicates a problem with model

specification

125

� Linear Functional Form

� Linear in Logs Functional Form

tktkt22t110t X...XXY ε+α++α+α+α=

Y

X iiαElasticity:

tktktltt vXbXbXbbY +++++= ln...lnlnln 2210

126

� Semi-Log Functional Form

� Semi-Log Functional Form

ibElasticity:

tktkt22t110t uXc...XcXccYln +++++=

ii XcElasticity:

tktkt22t110t wXlnd...XlndXlnddY +++++=Elasticity:

Y

di

� Box-Cox Transformation

� Linear Form

� Double Logarithmic Form (linear in logs functional

tkktk2t221t1100t )(X...)(X)(X)(y ε+λβ++λβ+λβ+β=λ

λ−λ /)1Z(Z transformed to

1... k210 =λ==λ=λ=λ

127

� Double Logarithmic Form (linear in logs functional form)

� Semi-Log Form

� Semi-Log Form

0... k210 =λ==λ=λ=λ

1...;0 k210 =λ==λ=λ=λ

0...;1 k210 =λ==λ=λ=λ

Linear Functional Form

128

Linear in Logs

Functional form

Double Logarithmic

129

Double Logarithmic

Functional Form

-0.3224 = own-price elasticity

0.1589 = cross-price elasticity

0.0651 = income elasticity

Semi-Logarithmic

Functional form

130

Semi-Logarithmic

Functional form

131

132

133