Advanced Econometrics
ADVANCED
ECONOMETRICS
SAJID ALI KHAN
1
Advanced Econometrics
ADVANCED
ECONOMETRICS
SAJID ALI KHAN
M.Phil. Statistics AIOU, Islamabad
M.Sc. Statistics AJKU, Muzaffarabad
PRINCIPAL
GREEN HILLS POSTGRADUATE COLLEGE
RAWALAKOT AZAD KASHMIR
E.Mail: sajid.ali680@gmail.com
Mobile: 0334-5439066
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Advanced Econometrics
CONTENTS
Chapter: 1. Econometrics
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
1
Introduction
Mathematical and statistical relationship
Goals of econometrics
Types of econometrics
Methodology of econometrics
The role of the computer
Exercise
Chapter: 2. Simple Linear Regression
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
2.10.
2.11.
2.12.
2.13.
2.14.
The nature of the regression analysis
Data
Method of ordinary least squares
Properties of least square regression line
Assumptions of ordinary least square
Properties of least squares estimators small/ large sample
Variance of disturbance term 𝑼𝒊
Distribution of dependent variable Y
Maximum likelihood method
Goodness of fit test
Mean prediction
Individual prediction
Sampling distributions and confidence interval
Exercise
Chapter: 3. Multiple Linear Regression and Correlation
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
44
Introduction
Properties of GLR
Polynomial
Exercise
Chapter: 5. Dummy Variables
5.1.
5.2.
5.3.
5.4.
36
Multiple linear regression
Coefficient of multiple determination
Adjusted 𝑹𝟐
Cobb-Douglas production function
Partial correlation
Testing multiple regression (F-test)
Relation between 𝑹𝟐 𝒂𝒏𝒅 𝑭
Exercise
Chapter: 4. General Linear Regression
4.1.
4.2.
4.3.
4.4.
6
53
Nature of dummy variables
Dummy variable trap
Uses of dummy variables
Exercise
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Advanced Econometrics
Chapter: 1
ECONOMETRICS
1.1: INTRODUCTION
Econometrics is the field of economics that concerns itself
with the application of mathematical statistics and the tools of
statistical inference to the empirical measurement of relationships
postulated by economic theory.
the quantitative measurement and analysis of actual economic and
business phenomena. Econometrics is a fascinating set of
techniques that allows the measurements and analysis of economic
trends.
Econometrics, the result of a certain outlook on the role of
economics, consists of the application of mathematical statistics to
economic data to lend empirical support to the models constructed
by mathematical economics and to obtain numerical results.
Econometrics may be defined as the quantitative analysis of actual
economic phenomena based on the concurrent development of
theory and observation, related by appropriate methods of
inference.
Econometrics may be defined as the social science in which
the tools of economic theory, mathematics and statistical inference
are applied to the analysis of economic phenomena. Econometrics
is concerned with the empirical determination of economic laws.
Frisch (1933) and his society responded to an
unprecedented accumulation of statistical information. They saw a
need to establish a body of principles that could organize what
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Advanced Econometrics
 ANALYSIS: Econometrics aims primarily at the verification
of economic theories. In this case we say that the purpose of
the research is analysis that is obtaining empirical evidence to
test the explanatory power of economic theories.
1.4: TYPES OF ECONOMETRICS
Econometrics may be divided into two broad categories:
 THEORETICAL ECONOMETRICS
Theoretical econometrics is concerned with the development of
appropriate methods for measuring economic relationship specified
by econometric models. Since the economic data or observations
of real life and not derived from controlled experiments, so
econometrics methods have been developed for such non
experimental data.
 APPLIED ECONOMETRICS
In applied econometrics we use the tools of theoretical
econometrics to study some special field of economics and business,
such as the production function, investment function, demand and
supply function, etc.
Applied econometric methods will be used for estimation of
important quantities, analysis of economic outcomes, markets or
individual behavior, testing theories, and for forecasting. The last of
these is an art and science in itself, and the subject of a vast library of
sources.
1.5: METHODOLOGY OF ECONOMETRICS
Traditional econometric methodology has the following main
points:
1. Statement of theory or hypothesis.
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Advanced Econometrics
2.
3.
4.
5.
6.
7.
8.
Specification of the mathematical model of the theory.
Specification of the statistical or econometric model.
Obtaining the data.
Estimation of the parameters of the econometric model.
Hypothesis testing.
Forecasting or prediction.
Using the model for control or policy purpose.
1. Statement of Theory or Hypothesis
Keynes stated, the fundamental psychological law is men
(women) are disposed as a rule and on average, to increase their
consumption as their income but not as much as the increase in
their income.
2. Specification of the Mathematical Model
Although Keynes postulated a positive relationship
between consumption and income, a mathematical economist
might suggest the following form of consumption function:
X
0< <1
Where:
3. Specification of the Econometric Model of Consumption
The inexact relationship between economic variables, the
econometrician would modify the deterministic consumption
function as follows:
+ X+u
known as the disturbance, error term or random
(stochastic) variable.
4. Obtaining Data
To estimate the econometric model that is to obtain the
, we need data. e.g
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Advanced Econometrics
Year
2004
2005
2006
Y
55
58
60
X
67
70
72
5. Estimation of the Econometric Model
Regression analysis technique to obtain the estimates of the
model. Thus
6. Hypothesis Testing
Assuming that the fitted model is a reasonably good
approximation of reality, we have to develop suitable criteria to
find out whether the estimates obtained in accord with the
expectations of the theory that is being tested.
7. Forecasting or Prediction
If the chosen model does not refute the hypothesis or theory
under consideration, we may use it to predict the future value of
the dependent, or forecast variable Y on the basis of known or
expected future value of the explanatory or predictor variable X.
8. Use of the Model for Control or Policy Purposes
An estimated model may be used for control, or policy
purposes. By appropriate fiscal and monetary policy mix, the
government can manipulate the control variable X to produce the
desired level of the target variable Y.
1.6: THE ROLE OF THE COMPUTER
Regression software packages, such as
EVIEWS, SAS, SPSS, STATA, SHAZAM etc.
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MINITAB,
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1.7: Exercise
1. What is econometrics? How many types
econometrics.
2. Discuss the methodology of econometrics.
3. Differentiate between statistics and mathematics.
4. What are the goals of econometrics?
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of
Advanced Econometrics
Chapter: 2
SIMPLE LINEAR REGRESSION
2.1: THE NATURE OF REGRESSION ANALYSIS
2.1.1: HISTORICAL ORIGIN OF THE TERM REGRESSION
The term regression was introduced by Francis Galton.
Galton found that there was a tendency for tall parents to have tall
children and for short parents to have short children, the average
height of children born of parents of a given height tended to move
e average height in the population as a
whole.
2.1.2: THE MODERN INTERPRETATION OF REGRESSION
Regression analysis is concerned with the study of
dependence of one variable on one or more other variable variables
with a view to estimating the mean value of the former in terms of
the known or fixed values of the latter.
TERMINOLOGY AND NOTATION
Dependent variable
Independent variable
Explained
Explanatory
Predictand
Predictor
Regressand
Regressor
Response
Stimulus
Endogenous
Exogenous
Controlled
Control
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Advanced Econometrics
2.2: DATA
Collection of information or facts and figures is called data.
2.2.1: TYPES OF DATA
There are three types of data.
 Time Series Data: A time series is a set of observations on the
values that a variable takes at different times. Such data may be
collected at regular time intervals, such as daily, weekly, monthly,
quarterly and yearly.
 Cross-Section Data: Cross-Section data are data on one or more
variables collected at the same point in time, such as the census of
population conducted by the Census Bureau every 10 years.
 Pooled Data: In pooled, or combined, data are elements of both
time series and cross-section data.
 Panel, Longitudinal, or Micro panel Data: This is a
special type of pooled data in which the same crosssectional unit is surveyed over time.
2.3: METHOD OF ORDINARY LEAST SQUARES
The method of ordinary least squares is the sum of squares of
observed
The estimated model is
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Advanced Econometrics
Then the residual sum of squares is
a bX
eq. (A)
( 1)
a bX)
a bX)
+b
+b
eq. (1)
Minimizing
a bX
( X)
a bX)
eq. (2)
+
̅
̅
b̅
̅
̅
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Advanced Econometrics
=(
)
=
+b
+b
b
b{
b=
2.4: PROPERTIES OF LEAST SQUARE REGRESSION LINE





2.5:
It passes through mean points ( ̅ ,
The mean value of residual = 0.
The residual are uncorrelated with predicted
The residual are uncorrelated with predicted
.
.
THE ASSUMPTIONS UNDERLYING THE
METHOD OF LEAST SQUARES: THE
CLASSICAL LINEAR REGRESSION
MODEL
1. Linear Regression Model
The regression model is linear in the parameter. That is
= +
+
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Advanced Econometrics
2. X Value are Fix in Repeated Sampling
Values taken by the regression X are considered fixed in
repeated samples. More technically, X is assumed to be
nonstochastic.
3. Zero Mean Value of Disturbance Term 𝒊
Given the value of X, the mean or expected value of random
disturbance term is zero. Technically the conditional mean value
of is zero. That is
E[ ⁄ ]=0
4. Homoscedasticity or Equal Variances of 𝑼𝒊
Given the value of X, the variance
is the same for all
observation. That is the conditional variance of are identical.
[ ⁄ ]= E[
⁄ ] = E[
⁄ ]=
5. No Autocorrelation between the Disturbance Term 𝑼𝒊
Given any two X values
and
between any two and
[
⁄
]=E[{
⁄
[
⁄
]= E[ ⁄ ][ ⁄ ]
[
⁄
]= 0
][{
6. Zero Covariance between 𝑼𝒊 and 𝒊
][
(
) = E[
[
]
(
)=E
(
)=E
(
)=0
( ⁄ )}]
]
E
=0
E
7. The Number of Observations” n” Must be Greater than the
Number of Parameter to be Estimated
greater than the number of explanatory variables.
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Advanced Econometrics
8. Variability in X Values
The X values in a given sample must not all be the same.
Technically variance of X must be a finite positive number.
9. The Regression Model is Correctly Specified
Alternatively, there is no specification bias error in the
model used in empirical analysis.
10. There is No Perfect Multicollinearity
There is no perfect linear relationship among the
explanatory variables.
2.6: PROPERTIES OF LEAST SQUARES ESTIMATORS
2.6.1: SMALL SAMPLE PROPERTIES OF THE LEAST SQUARES ESTIMATORS
I. Unbiasedness: An estimator is said to be unbiased if the
expected value is equal to the true population parameter.
II. Least Variance: An estimate is best when it has the smallest
variance as compared with any other estimate obtained from other
econometric methods.
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Advanced Econometrics
II.
parameter b, if the asymptotic mean of ̂ is equal to be b.
That is
[̂ ]
Consistency: An estimator ̂ is a consistent estimator of the
true population parameter b, if it satisfies two conditions:
(a) ̂ Must be asymptotically unbiased.
That is
[̂ ]
(b) The variance of ̂ must approach zero as n tends to
infinity. That is
[̂ ]
III.
2.6*
Asymptotic Efficiency: An estimator ̂ is an
asymptotically efficient estimator of the true population
parameter b, if
(a) ̂ is consistent.
̂ has a smaller asymptotic variance as compared with any
other consistent estimator.
GAUSS MARKOV THEOREM
STATEMENT:
Least squares theory was put forth by Gauss in 1809 and
minimum variance approach to the estimators of
was proposed
by Markov in 1900. Since determining of minimum variance linear
unbiased estimator involves both the concepts, the theorem is
known as Gauss-Markov theorem. It can be stated as follows:
Let
be n independent variables with mean
and variance. The minimum variance linear unbiased estimators of
the regression coefficients are (j=1,2,..,k).
Under the terms and conditions imposed above, the
minimum variance linear unbiased estimators of the regression
coefficients are identically the same as the least square estimators.
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Advanced Econometrics
The combination of the above two statements is known as
Gauss-Markov theorem. i.e. the least square estimators of
and
are best, linear, unbiased estimators (BLUE).
PROOF:
We use the model,
Y=
FOR
 LINEARITY:
̅
=
̅
=
=
=
=
Where
=
are nonstochastic weight,
=
This is linear function of sample observations
 UNBIASEDNESS:
=
=
=
Properties of is
1.
2.
eq. (1)
=
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Advanced Econometrics
3.
Put these results in eq. (1).
=
= +
E
=E
E
=
+
Which shows that
is an unbiased estimator of
 Variance of :
By definition
) = E[
.
]
]
) = E[
]
) = E[
from eq. (2)
]
) = E[
(
)=
, (
)=
)=
(
)
̅
And
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)
)
Advanced Econometrics
FOR
 LINEARITY:
̅
̅
*
̅ +
+
Which is linear function of sample observations .
̅
Where
 UNBIASEDNESS:
+
+
Taking expectation on both sides
E( )= +
E( )=
,
.
 Variance of :
By definition
) = E[
) = E[
) = E[
) = E[
]
]
]
from eq. (2)
]
(
)=
, (
)
)=
)=
*
)=
*
21
+
̅
̅ +
)
Advanced Econometrics
)=
*
)=
*
̅
̅
̅
+
+
,
2.6** MINIMUM VARIANCE PROPERTY OF LEAST SQUARE
ESTIMATORS
Suppose
is any other linear unbiased estimator of
Taking expectation on both sides
E
=
E
=
 Variance of
= E[
= E[
= E[
]
]
]
rom eq. 2.
]
= E[
=[
]
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Advanced Econometrics
=
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Advanced Econometrics
Variance of
= E[
= E[
= E[
]
]
]
... From eq. 2.
]
= E[
=[
]
=
, (
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Advanced Econometrics
+
=
For sample
̅
By subtraction
̅
̂=
̂
Making substitution in . Using eq. 1 & eq, 2.
̂
Applying sum and squares on both sides.
[
]
Taking expectation on both sides.
E
]+E
=E[
. ..eq.
]
Now, E[
E[
]
*
[
+
]
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Advanced Econometrics
E[
]
E[
]
E[
]
E[
]
E[
]
(
)
=
E
E
* (
E
*
E
[
E
[
E
[
E
E
27
)
+[
+[
]
]
]
]
]
Advanced Econometrics
Put eq.
.
(
)=
(
) =
(
) =
(
) =
(
)
E
=
This shows that
2.8: DISTRIBUTION OF DEPENDENT VARIABLE Y
Let

+
Mean of
:
[ ]
[
[ ]
[ ]
+
 Variance of :
+ ]
+
)
[
]
[
]
 The shape of the distribution
and by assumption of OLS. We assume that distribution of
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Advanced Econometrics
normal and we also know that any linear function of normal
variable is also normal.
Since
2.9: MAXIMUM LIKELIHOOD ESTIMATORS
𝟐
OF ,
(
)=∏
(
)
√
(
)
and equating zero.
Differentiate eq.(A) w.r.t
=
⁄
2
0=
0=
0=
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Advanced Econometrics
=
2
0=
0=
0=
=0
[
=
0=
0=
(
[
(
(
)
)
]
]
)
0=
0=
0=
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Advanced Econometrics
Which is biased estimator of
.
Taking expectations on both sides.
(
)
(
)
(
)
(
(
)
)
Hence M.L.E of
is bias estimator. But M.L.E of
2.10: TEST OF GOODNESS OF FIT
𝟐
The ratio of explained variation to the total variation is called
the coefficient of determination. The varies between 0 and 1.
Total Variation = Unexplained Variation + Explained Variation
̅
(
̂) + ( ̂
In deviation form:
̂
̂
Where
̅
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̅)
Advanced Econometrics
̂)
(
2.11: MEAN PREDICTION
Where
E(
(̂)
(̂ )
(̂ )
(̂ )
̅
*
̅
+
(
̅
(̂ )
32
̅
)
Advanced Econometrics
(
̂ )
[
(
̂ )
[
(
̂ )
[
(
̂ )
(
̂ )
(
̂ )
]
]
]
By definition variance of prediction error is:
(
̂ )
[(
(
̂ )
[
(
̂ )
(
̂ )
(
̂ )
̂ )
̂ )]
(
]
[
]
̅
*
[
]
+
[
(
̅
̂ )
(
̂ )
[̅
(
̂ )
̅
(
̂ )
̅
33
̅
)
̅
(
*
]
̅]
+
Advanced Econometrics
2.13: SAMPLING DISTRIBUTIONS AND CONFIDENCE INTERVAL
Use z-test if
is known or n is large, otherwise
we use t-test.
.
√
Z=
/
and
with (n
√
√
(
̅
(
) )
And
√
(
)
Confidence Interval for
√
(
√
:
(
,
̅
)
)
Confidence Interval for
√
,
Confidence Interval for Mean Prediction:
√
(
*
̅
+
Confidence Interval for Individual Prediction:
̂
Confidence Interval for
√
*
:
34
̅
+
:
Advanced Econometrics
Example: Given data
X 30
Y 50
60
80
90
120
120
130
150
180
i) Estimate the model Y=
ii) Estimate Y when X = 60.
iii) Test the significance of
.
iv) 95% confidence interval of
v) Estimate
vi) Estimate mean and individual prediction when
vii)
and r.
Solution:
X
30
60
90
120
150
450
i)
Y
50
80
120
130
180
560
Y=
XY
1500
4800
10800
15600
27000
59700
𝟐
𝟐
900
3600
8100
14400
22500
49500
2500
6400
14400
16900
32400
72600
𝒊
̅
̅
̅
̅
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.
Advanced Econometrics
e) Critical region:
| |
f) Conclusion:
Since our calculated value less than table value so
we accept , and may conclude that null hypothesis is
better than alternative hypothesis.
Testing for
a)
b) Choose level of significance at
c) Test statistic
with n-2 d.f.
√
̂
d) Computation:
√
e)
iv)
Critical region:
| |
f) Conclusion:
Since our calculated value greater than table
value so we reject , and may conclude that
alternative hypothesis is better.
95% confidence interval for
⁄
√
(
19.3
19.3
37
̅
)
Advanced Econometrics
90% confidence interval for
⁄
√
:
̂
0.7947
v)
Covariance:
̅
vi)
Mean prediction:
When
(̂ )
̅
*
+
(̂ )
*
+
(̂ )
[
]
(̂ )
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Advanced Econometrics
Individual prediction:
When
(̂ )
̅
*
+
(̂ )
*
+
(̂ )
[
]
(̂ )
vii)
𝟐
and r :
Total Variation = Unexplained Variation + Explained Variation
̅
(
̂) + ( ̂
̅)
In deviation form:
̂
(
Unexplained Variation
̂)
̅
⁄
⁄
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Advanced Econometrics
√
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Advanced Econometrics
2.14: Exercise
1.
2.
3.
4.
Discuss the nature of regression analysis.
What are the different types of data for economic analysis?
State and prove Gauss-Markov theorem.
Prove that
̅
5. Prove that
6.
7.
8.
9.
E( ̂ ) =
Find the ML estimates of least square regression line.
Given the data:
X
2
3
1
5
9
Y
4
7
3
9
17
i.
Estimate the model Y=
by OLS.
ii.
Find the variance of
.
iii.
.
The following marks have been obtained by a class of students
in economics:
X 45 55 56 58 60 65 68 70 75 80 85
Y 56 50 48 60 62 64 65 70 74 82 90
1. Find the equation of the lines of regression.
2. Test the significance of
.
3. 98% confidence interval of
.
A sample of 20 observations corresponding to the model
gave the following data:
(a) Estimate
and calculate estimates of variance of
your estimates.
(b) Find 95% confidence interval for . Explain the mean
value of Y corresponding to a value of X fixed at X = 10.
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Advanced Econometrics
Chapter: 3
MULTIPLE LINEAR REGRESSION AND
CORRELATION
3.1: Multiple Linear Regression
It investigates the dependence of one variable (dependent
variable) on more than one independent variables, e.g. production
of wheat depends upon fertilizer, land condition, temperature,
water etc.
Y=
Normal equations are:
̅
̅
0
[
̅ ̅
1
{
√
42
} ]
Advanced Econometrics
[
]
And
0
[
1
{
} ]
√
[
]
or
√
3.2: Coefficient of Multiple Determinations
Co-efficient of multiple determinations is the proportion of
variability due to independent variable
and dependent
variable Y of total variation.
̂
̅
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Advanced Econometrics
3.3: Adjusted 𝑹𝟐
The important property of
that it is non-decreasing.
That is including the explanatory variable. Value of
increasing
and do not decrease to adjust this we are adjusted ̅ .
̅
̅
3.4: COBB-DOUGLAS PRODUCTION FUNCTION
The Cobb-Douglas Production function, in its stochastic
form, may be expressed as
Where Y = output,
,
capital input
U = stochastic disturbance term, e = base of natural logarithm
The relationship between output and two inputs is nonlinear.
Using log-transformation we obtain linear regression model in the
parameters.
Where
and
.
3.5: Partial Correlation
If there are three variables Y,
. Then the
correlation between Y and
is called partial correlation. The
simple partial correlation co-efficient is the measure of strength of
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Advanced Econometrics
linear relationship between Y and
after removing the linear
influence of from Y and is denoted by
.
=
√(
)√(
)
3.6: TESTING THE OVERALL SIGNIFICANCE OF A
MULTIPLE REGRESSION (The F-test)
 Hypothesis
 Choose level of significance at
 Test statistic to be used:
with
 Computations:
̅
Total SS =
(
Residual SS =
̂)
Explained SS = Total SS
S. O. V
d. f
Regression k
Residual
n
Total
n
SS
Explained
Residual
Total
45
MS
F
⁄
⁄
Advanced Econometrics
Example: Given the following data:
Y
i.
ii.
iii.
5
1
2
7
3
4
8
9
3
10
8
10
Estimate
them.
Find
and ̅ .
Test the goodness of fit.
and interpret
Solution:
i.
Estimate
𝟐
𝑼𝒊
𝟐
𝟐
𝟐
𝟐
𝟐
Y
5
7
8
10
1
3
9
8
2
4
3
10
5
21
72
80
10
28
24
100
2
12
27
80
1
9
81
64
4
16
9
100
25
49
64
100
30
21
19
178
162
121
155
129
238
𝟐
𝟐
̂
Normal equations are:
Solving these equations, we get
47
𝟐
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̂
ii.
Find 𝑹𝟐 and 𝑹𝟐
(
̅
⁄
⁄
61
̅
̅
iii.
Testing
a)
b)
c) Test statistic
48
̂)
Advanced Econometrics
⁄
with d.f.
⁄
d) Computation
⁄
⁄
e) Critical region
f) Since our calculated value less than table value so
we accept null hypothesis.
49
Advanced Econometrics
3.8: Exercise
1.
2.
3.
4.
5.
Differentiate between simple and multiple regression.
Write note on
and ̅ .
Discuss the Cobb-Douglas production function.
How the overall significance of regression is tested?
Consider the following data:
Y 40 30 20 10 60 50 70 80 90
50 40 30 80 70 20 60 50 40
20 10 30 40 80 30 50 10 60
iv.
Estimate
and interpret
them.
v.Find
and ̅ .
vi.
Test the goodness of fit.
vii.
Find variance of
6. Use the following data:
Y
5.5
190
49
6.5
170
58
8.0
210
55
7.5
170
58
7.0
190
55
5.0
180
49
6.0
200
46
6.5
210
46
a. Estimate
by OLS.
b. Test overall significance of regression model.
c. Find adjusted coefficient of multiple correlation.
d. Find
.
50
Advanced Econometrics
Chapter: 4
GENERAL LINEAR REGRESSION (GLR)
4.1: INTRODUCTION
The general linear regression is an extension of simple
linear regression and it involves more than one independent
variables.
Let we
relationship exist between a variable
and K
explanatory
variables
, then regression model is:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
It may be written as a matrix notation
[ ]
[
[ ]
[
]
][
51
]
[
]
[
]
Advanced Econometrics
Assumptions of GLR:
1.
[ ]
[ ]
Taking expectation on both sides
[ ]
[ ]
[
2. Variance
( )
]
[
]
[
( )
[
]
]
( )
[
]
52
Advanced Econometrics
̂
̂
Since ̂
̂
̂
̂ ̂
is scalar, therefore it is equal its transpose i.e.
̂
̂
̂
̂ ̂
̂̂
Minimize with respect to ̂ and equating zero.
̂
̂
̂
̂
̂
54
̂
̂
Advanced Econometrics
̂
̂
3. Minimum Variance: By definition
( ̂)
[̂
( ̂ )][ ̂
( ̂)
[̂
( ̂ )]
][ ̂
]
Using eq. (1) we get
̂
̂
( ̂)
[
][
]
( ̂)
[
][
]
( ̂)
( ̂)
[
[
( ̂)
[
]
]
]
( ̂)
Example: Given
Y
X
i)
ii)
4
5
6
7 8
2
3
4
5 7
Calculate SLR estimate using GLR technique.
Also find their variance and covariance.
55
Advanced Econometrics
Solution:
Y
4
5
6
7
8
30
X
2
3
4
5
7
21
𝟐
XY
8
15
24
35
56
138
𝟐
2
9
16
25
49
103
16
25
36
49
64
190
i)
̂
̂
̂
∑
[
]
∑
[
]
*
|
|
|
*
+
∑
+
|
|
|
*
+
Now
̂
ii)
̂
*
̂
[
+*
+
̂
]
Variance-covariance
̂
56
*
+
[
̂
]
Advanced Econometrics
̂
[
]*
+
[
]
̂
( ̂)
̂
( ̂)
( ̂)
*
*
+
+
0
̂
̂
̂
̂
̂
̂
1
4.3: POLYNOMIAL
Any algebraic expression in which the degree
-negative i.e. positive or zero is known as
polynomial. E.g.
Y=
 PLYNOMIAL REGRESSION
It is a simple multiple linear regression, where
explanatory variables are all powers of a single variable. E.g
second degree polynomial variable in which
It is called polynomial regression model in one regression. If
Then thi
variables. The Kth order polynomial in one variable is:
57
Advanced Econometrics
Polynomial regression model is used where the relationship
between the response variable and explanatory variable is
curve linear.
58
Advanced Econometrics
4.4: Exercise
1. Discuss general linear regression.
2. State the assumptions under which OLS estimates are best,
linear and unbiased in general linear regression.
3. Prove that:
a) ̂
b)
( ̂)
4. Define polynomial regression.
5. Given the data:
X
15
20
30 50 100
Y
20
40
60 80 120
Find:
i)
ii)
( ̂)
iii)
90% confidence interval of ̂ .
̂
iv)
Test the hypothesis when
.
v)
Estimate Y when X=200.
vi)
And ̅ .
6. Consider the GLR model with the following data:
Y
3
7
5
9
7
11
8
10
5
3
9
3
Find:
i)
ii)
( ̂)
iii)
90% confidence interval of ̂
̂
iv)
Test the hypothesis when
.
̅
v)
And .
7. Given the following information in deviation form:
*
*
+,
+
59
Advanced Econometrics
̅
, ̅
, ̅
,
a) Find the estimates of ̂
variances and covariance.
b) How would you estimate ̂
̂
c) Test the hypothesis that
d)
And ̅ .
8. Given the following data:
2
3
8
10
12
16
19
20
22
25
1
5
6
8
10
13
17
21
23
24
̂ . Also find their
̂
3
4
7
6
11
14
18
20
25
27
Find:
a) Estimate the model in deviation form
b)
( ̂)
c) 95% confidence interval of ̂ and ̂ .
̂
d) Test the hypothesis when
.
̅
e)
And .
60
.
.
Advanced Econometrics
Chapter: 5
DUMMY VARIABLES
Econometric models are very flexible as they allow for the
use of both qualitative and quantitative explanatory variables. For
the quantitative response variable each independent variable can
either a quantitative variable or a qualitative variable, whose levels
represent qualities and can only be categorized. Examples of
qualitative variables may be male and female, black and white etc.
But for a qualitative variable, a numerical scale does not exist. We
must assign a set of levels to qualitative variable to account for the
effect that the variable may have on the response, then we use
dummy variables.
A dummy variable is a variable which we construct to
describe the development or variation of the variable under
5.1: NATURE OF DUMMY VARIABLES
In regression analysis dependent variable is
affected not only by quantitative variables but also by qualitative
variables. For example income, output, height, temperature etc, can
be quantified on some well define scales. Similarly religion,
nationality, strikes, earthquakes, sex etc, are qualitative in nature.
These all variables affect on dependent variable. In
order to study these variables, we quantified the qualitative
varia
are called dummy variables. Dummy variables are also called
Indicator, Binary, Categorical variables.
EXAMPLE:
Where
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Advanced Econometrics
Suppose
Sex
3000
Female
0
4000
Male
1
5000
Female
0
6000
Male
1
Using OLS method. There is only one dummy variable in the model.
̂
Mean salary of Female College Professors:
( ⁄
)
Mean salary of Male College Professors:
( ⁄
)
5.2: DUMMY VARIABLE TRAP
If an indicator variable has k categories, that is k-1 dummy
variables, otherwise the situation of perfect multicollinearity arises
and the researcher will fall into the dummy variable trap.
We consider a model
Where
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Advanced Econometrics
This model is an example of dummy variable trap. There is a
rule of introducing a dummy variable. If a qualitative variable have
introduce only (m ) variable (dummy). If this
rule is not followed we say that there is trap of dummy variable.
EXAMPLES:
 Sex has two categories F and M that is m = 2. If we introduce
m
dummy variable, we follow the rule of
introducing dummy variables. If we introduce 2 dummy
variables then we say there is dummy variable trap.
 Suppose there are three categories of color as white, black and
red. Then m = 3. If we not introduce m
dummy variables, then there will be dummy variable trap.
5.3: USES OF DUMMY VARIABLES
a) Dummy variables used as alternate for qualitative factors.
b) The dummy variables can be used to deseasonalize the time
series.
c) Dummy variables are used in spline function.
d) Interaction effects can be measured by using dummy
variables.
e) Dummy variables are used for determining the change of
regression coefficient.
f) Dummy variables are used as categorical regressors.
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Advanced Econometrics
5.4: Exercise
1. What are the dummy variables? Discuss briefly the
features of the dummy variable regression model.
2. Discuss the uses of dummy variables.
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Advanced Econometrics
Chapter: 6
AUTO-REGRESSIVE AND DISTRIBUTED-LAG
MODEL
6.1: DISTRIBUTED-LAG MODEL
In regression analysis involving time-series data, If the
regression model includes not only the current but the lagged (past)
lag-model. That is,
Represent a distributed lag-model.
6.2: AUTO-REGRESSIVE MODEL
If the model includes one or more lagged values of
the dependent variable among its explanatory variables, it is called
an auto-regressive model. That is,
Represent an auto-regressive model. Auto-regressive models
are also known as dynamic models. Auto-regressive and
distributed-lag models are used extensively in econometric
analysis.
6.3: LAG
In economics the dependence of a variable Y
(dependent variable) on other variables (explanatory variable) is
rarely instantaneous (happen immediately). Very often Y responds
to X with a laps of time, such a laps of time is called a lag.
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Advanced Econometrics
6.4: REASONS SOURCES OF LAGS
There are three main reasons of lags.
1. Psychological Reasons: Due to the force of habit people do
not change their consumption habits immediately following a
price decrease or an income increase. For example those who
become instant millionaires by winning lotteries may not
change their life styles. Given reasonable time, they may learn
to live with their newly acquired fortune.
2. Technological Reason: Technological reason is the major
source of lags. In the field of economics if the drop in price is
expected to be temporary firms may not substitute labor,
especially if they expected that after the temporary drop, the
price of capital may increase beyond the previous levels. For
example, since the introduction of electronic pocket calculators
dramatically decrease as a result consumers for the calculators
may hesitate to buy until they have time to look into the
features and prices of all the competing brands. Moreover they
may hesitate to buy in the expectation of further decrease in
price.
3. Institutional Reason: These reasons also contribute to lags.
For example, those who have placed funds in long term saving
accounts for fixed durations such as 1 year, 3 year or 7 year are
may be such that higher yields are available elsewhere.
Similarly, employers often given their employees a choice
among several health insurance plans, but one a choice is made
on employee may not switch to another plan for at least one
year.
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Advanced Econometrics
6.5: TYPES OF DISTRIBUTED LAG MODEL
There are two types of distributed lag model:
1. Infinite Distributed Lag Model
In case of infinite distributed lag model we do not
specify the length of the lag. It means that how for back
into the past we want to go: e.g
2. Finite Distributed Lag Model
In case of finite distributed lag model we specify the
length of lag: e.g
6.6: ESTIMATION OF DISRIBUTED LAG MODEL
We use the following methods for estimation of
distributed lag model.
1) Ad Hoc Estimation Method.
2) Koyck Estimation Method.
3) Almon Approach Method.
1) Ad Hoc Estimation Method
This is the approach taken by Alt and Tinbergen.
They suggest estimating,
One may proceed sequentially under this method, first we
regress
then regress
and
and so on. This
sequential procedure stops when the regressive coefficients of
the lagged variables start becoming statistically insignificant and
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Advanced Econometrics
or the coefficients of at least one of the variables. Changes sign
from positive to negative or vice versa.
2) Koyck Approach
This method is used in case of finite distributed lag model.
Under this method we assume that
are all of the same sign.
Koyck assume they decline geometrically as follows:
.
.
.
.
.
.
decay of the distributed lag where 1
is known as the speed of
adjustment. As the distributed lag model is:
From eq. (A) we substitute
we get
...eq. (C)
Lagging one period, we get
Subtracting eq. (D) from eq. (C), we get
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Advanced Econometrics
It is also regressive model, so we can apply OLS method to
model (E) and get
,
,
using them we can fined
In a sense of multicollinearity is resolved by replacing
By a single variable
. But note the
following features of Koyck transformation.
 Koyck model is transformend into auto regressive model from
distributed lag model.
 It gives biased and inconsistent estimator.


3) Almon Approach to Distributed Lag Models
If
coefficients do not decline geometrically, They
increase at first and then decrease it is assumed that
follow a
cyclical pattern. In this situation we apply Almon approach.
To illustrate Almon technique, we use the finite distributed
lag model.
This may be written as:
69
Advanced Econometrics
6.7: Exercise
1. Differentiate between auto-regressive and distributed-lag models.
2. What is Lag? Discuss the sources of lags.
3. Discuss the different methods of distributed-lags model.
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Advanced Econometrics
Chapter: 7
MULTICOLLINEARITY
7.1: Collinearity
In a multiple regression model with two independent
variables, if there is linear relationship between independent
variables, we say that there is collinearity.
7.2: Multicollinearity
If there are more than two independent variables and they
are linearly related, this linear relationship is called
multicollinearity.
Multicollinearity
arises
from
the
presence
of
interdependence among the regressors in a multivariable equation
system. The departure of orthognality in the set of regressors in a
measure of multicollinearity. It means the existence of a perfect or
exact linear relationship among some or all explanatory variables.
When the explanatory variables are perfectly correlated, the
method of least squares breaks down.
7.3: Sources of Multicollinearity
 The data collection method employed for example,
sampling over a limited range of the values taken by the
regressors in the population.
 Constraints on the model or in the population being
sampled. In the regression of electricity consumption (Y)
on income ( ) and house size ( ) there is a physical
constraints in the population in that families with higher
income generally larger homes than families with lower
income.
 Model specification: For example adding polynomial
terms to a regression model, especially when the range of
the variable is small.
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Advanced Econometrics
 An Over determined Model: This happens when the
model has more explanatory variables than the number of
observations. This could happen in medical research,
where there may be a small number of patients about
whom information is collected on a large number of
variables.
 An additional reason for multicollinearity, especially in
time series data may be that the regressors included in the
model share a common trend, that is they all increase or
decrease over time. Thus in the regression of consumption
expenditure on income, wealth and population, the
regressors income, wealth and population may all be
growing over time at more or less the same rate leading to
collinearity among these variables.
7.4: TYPES OF MULTICOLLINEARITY
There are two types of multicollinearity.
 Perfect Multicollinearity
Relates to the situation where explanatory variables are
perfectly linearly related with each other. Simply when
correlation between two explanatory variables is exactly one i.e.
). This situation is called perfect multicollinearity.
 Imperfect Multicollinearity
If the correlation coefficient between two explanatory
variables is not equal to one but close to one approximately 0.9,
it is called high multicollinearity. If
approximately 0.5,
it is called moderate and if
it is called low
multicollinearity. Both are troublesome because it cannot be
easily detected.
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Advanced Econometrics
7.5: ESTIMATION IN THE PRESENCE OF PERFECT
MULTICOLLINEARITY
The three variable regression model using deviation form as
̂
̂
(
̂
(
)
)(
(
̂
(
And
)
)
)(
)
Assume that
-zero constant. Then
(
̂
(
)
)(
)
(
̂
(
)
)(
(
[
̂
(
)
(
(
)
)
)
[(
̂
(
)
)]
) ]
[ ]
[ ]
̂
.
Similarly,
(
̂
(
)
)(
(
̂
(
̂
)
)
)(
)
(
[
[(
73
(
(
)
)
)
(
(
)
) ]
)]
Advanced Econometrics
[ ]
̂
[ ]
̂
(̂ )
(
(̂ )
)(
(
) (
)(
(̂ )
(̂ )
)
) (
(
)(
[(
)
)
)
(
(
)
) ]
(̂ )
(̂ )
(̂ )
.
Similarly,
(̂ )
(̂ )
(
)(
(
)(
(̂ )
(̂ )
(̂ )
74
) (
)
) (
(
)(
[(
)
)
(
)
(
) ]
)
Advanced Econometrics
(̂ )
(̂ )
.
̂
̂
Put
̂
̂
̂
̂
̂
Where ̂
̂
̂
Regression in y on x is:
̂
Therefore, although we can estimate ̂ uniquely, but there is
no way to estimate ̂ ̂ uniquely. Hence in the case of perfect
̂
multicollinearity the variance and standard error of ̂
individually are infinite.
7.6: CONSEQUENCES OF MULTICOLLINEARITY
1) The estimate of the coefficient of statistical unbiased,
even multicollinearity is strong. The sample property
of unbiased of the estimate does not require that the
estimate seriously imprecise.
2) If the intercorrelation between the explanatory is
perfect. Then the estimates of the coefficient are
indeterminate.
75
Advanced Econometrics
Proof:
The three variable regression model using
deviation form as
̂
̂
(
̂
(
̂
)
)(
(
(
And
)
)
)(
)
Assume that
̂
-zero constant. Then
(
)
(
)(
)
(
̂
(
̂
)
)
(
)
[
)
[(
̂
(
)(
(
(
(
)
)
)]
) ]
[ ]
[ ]
̂
.
Similarly,
̂
(
(
)
)(
)
(
̂
(
̂
[
[(
̂
[ ]
[ ]
76
)
(
)(
)
(
)
)
(
)
(
(
)
) ]
)]
Advanced Econometrics
̂
3) If the intercorrelation of the explanatory is perfectly
one. Then the standard error of these estimate become
infinitely large.
Proof:
If
, the standard error the estimate become
infinitely large in the two variable model:
0
1
*
⁄
*
⁄
[
+
+
]
⁄√
*
+
Putting
*
+
* +
Infinitely large.
Similarly:
77
Advanced Econometrics
*
+
*
⁄
*
⁄
[
+
+
]
⁄√
*
+
Putting
*
+
* +
Infinitely large
4) In case of strong multicollinearity regression
coefficients are determinate but their standard errors
are large.
Proof:
*
+
Put
*
78
+
Advanced Econometrics
*
+
[
]
In case of
If
*
+
*
+
*
+
5) In case of multicollinearity the confidence interval
becomes wider.
6) In the presence of multicollinearity the t-test will be
misleading.
7) In the presence of multicollinearity prediction is not
accurate.
7.7: DETECTION OF MULTICOLLINEARITY
1. The Farrar and Glauber Test of Multicollinearity
A statistical test for multicollinearity has been developed by
Farrar and Glauber. It is really a set of three tests.
a) The first test is a 𝟐 test for the detection of the
existence and the severity of multicollinearity in a function
including several explanatory variables.
Procedure:
i.
ii.
iii.
.
Choose level of significance at
Test statistic to be used
*
+
79
Advanced Econometrics
iv.
Computations: where
is the value of the
standardized correlation determinant. K is
number of explanatory variables.
v.
Critical Region:
vi.
Conclusion: Reject
if our calculated value is
greater than table value. Otherwise accept.
b) The second test is an F-test for locating which
variables are multicollinear.
Procedure:
i.
ii. Choose level of significance at
iii. Test statistic to be used
with
d.f
iv. Computations:
Compute the multiple correlation coefficients
among the explanatory variables.
v. Critical Region: F
vi. Conclusion:
Reject
if our calculated value is greater than
table value. Otherwise accept.
c) The third test is a t-test for finding out the pattern
of multicillinearity that is for determining which variables are
responsible for the appearance of the multicollinear variable.
Procedure:
i.
ii.
iii.
Choose level of significance at
Test statistic to be used
√
√
with
80
Advanced Econometrics
iv.
Computations:
Computed the partial
coefficients.
v.
Critical Region: | |
correlation
vi.
Conclusion:
Reject
if our calculated value
is greater than table value. Otherwise
accept.
2. High Pair Wise Correlation among Regressors
Multicollinearity exists if the pair wise or zero order
coefficients between the two regressors are very high.
3.
Eigen Value and Condition Number
A condition number K is defined as
If K is between 100 and 1000, There is moderate to
strong multicollinearity and if exceeds 1000 there is severe
multicollinearity.
The condition index defined as
√
If
is the condition effect lie between 10 and 30 then
there is moderate to strong multicollinearity and if it exceed
30 there is severe multicollinearity.
4. Tolerance and Variance Inflation Factor
As
the coefficient of determination in the regression
of regressors
on the remaining regressor in the model
increases towards that is as the collinearity with the other
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Advanced Econometrics
regressor increases VIF all the increases and the limit it can
be infinite.
VIF
(
)
Tolerance can also
multicollinearity. That is
be
(
Tolerance
used
to
detect
the
)
5. High 𝑹𝟐 but Few Significant t-Ratios
If
is high the F-test in most cases will reject the
hypothesis that the partial correlation coefficients are
simultaneously equal to zero, but the individual t-test will
show that non are very few of the partial slope of coefficients
are statistically different from zero. This is the symptom of
multicollinearity.
6. Some Other Multivariate Methods
Like Principal Component Analysis (PCA), Factor
Analysis (FA) and Ridge Regression can also be used for
detection of multicollinearity.
7.8. REMEDIAL MEASURES OF MULTICOLLINEARITY
i. A Prior Information
Suppose we consider the model
Where Y = Consumption,
Income and wealth variable tends to be highly collinear.
Suppose
that is the rate of change of
consumption with respect to wealth one tended the
corresponding rate with respect to income. We can then run
the regression
82
Advanced Econometrics
Where
Once we obtain
we can estimate
postulated relationship between
and .
from the
ii. Combining Cross-sectional and Time Series Data
A variant of the extraneous are a priori information
technique is the combination of cross-sectional and time
series data known as pooling the data. The combination of
cross-sectional and time series data may be a situation of
reduction of multicollinearity.
iii. Dropping a Variable or Variables
When faced with severe multicollinearity one of the
simplest things to do is to drop one of the collinear variables.
In dropping a variable
from the model we may be
committing a specification bias or specification error.
iv. Transformation of Variables
One way of minimizing this dependence is to proceed as
follows:
If the above
we have
is arbitrary, therefore
is known as first difference form.
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Advanced Econometrics
The first difference regression model often reduces the
severity of multicollinearity.
v. Additional or New Data
Since multicollinearity is a sample feature, it is possible
that in another sample involving the same variables.
Multicollinearity may not be as serious as in the first sample.
Sometimes simply increasing the size of slope may reduce the
multicollinearity problem.
vi. Other Methods
Multivariate statistical technique such as factor analysis
and principal components or other techniques such as ridge
regression are often implied to solve the problem of
multicollinearity.
84
Advanced Econometrics
7.9: Exercise
1)
2)
3)
4)
Explain the problem of multicollinearity and its types.
Explain the methods for detection of multicollinearity.
Describe the consequences of multicollinearity.
How would you proceed for estimation of parameters in
the presence of perfect multicollinearity?
5) Define any four methods for removal of multicollinearity.
6) Apply Farrar and Glauber test to the following data:
6
6
6.5
7.6
9
40.1 40.3 47.5
58
64.7
5.5
4.7
5.2
8.7
17.1
108
94
108
99
93
7) Find severity, location and pattern of multicollinearity to
the following data:
85
Advanced Econometrics
Chapter: 8
HETEROSCEDASTICITY
8.1. NATURE OF HETEROSCEDASTICITY
One of the important assumptions of the classical linear
regression model is that the variance of each disturbance term is
equal to . This is the assumption of homoscedasticity.
[ ]
Symbolically,
If this assumption of the homoscedasticity is fail that is:
[ ]
[
]
may all
be different.
DIFFERENCE BETWEEN HOMOSCEDASTICITY AND
HETEROSCEDASTICITY
Homoscedasticity is the situation in which the probability
distributions of the disturbance term remain same overall
is the same for all values of the explanatory variables.
Heteroscedasticity is the situation in which the probability
distributions of the disturbance term does not remain the same over
each
is not the same for all the values of the explanatory
variables.
8.1.1. Reasons of Heteroscedasticity
i. Error Learning Model
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Advanced Econometrics
As people learn their error of behavior become smaller
over time. In this case
is expected to decrease, e.g. as the
number of hours of typing practice increases. The average
number of typing errors as well as their variances decreases.
ii. Data Collection Technique
Another reason of heteroscedasticity is the collection of
data techniques. Improvement of data collection techniques
is likely to decrease.
iii. Variance in Cross-Section and Time Series Data
In cross-sectional data the variance is greater than as
compared to the time series data variance. Because in crosssectional data, one usually deals with numbers of population
at a given point in time.
iv. Due to Specification Error
The heteroscedasticity problem is also arises from
specification errors, due to that error the variance tends to
variate.
8.2. OLS ESTIMATION OF HETEROSCEDASTICITY
Let us we use two variable model
[
=
=
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= +
E
=E
E
=
+
Which shows that is still unbiased estimator of
the presence of heteroscedasticity.
Variance of
, even in
:
By definition
) = E[
]
]
) = E[
]
) = E[
]
) = E[
(
)=
By assumption of heteroscedasticity
, (
)
)
)=
)=
In the presence of heteroscedasticity, we observed that
OLS estimator is still linear, unbiased and consistent but not
BLUE, that is
is not efficient, because has not minimum
variance in the class of unbiased estimator in the presence of
heteroscedasticity.
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Advanced Econometrics
8.3: CONSEQUENCES OF HETEROSCEDASTICITY
1) The OLS estimators in the presence of heteroscedasticity
are still linear, unbiased and consistent.
2) In the presence of heteroscedasticity the OLS estimators
are not BLUE, that is they have not minimum variance in the
class of unbiased estimators.
3) In the presence of heteroscedasticity the confidence
interval of OLS estimators are wider.
4)
misleading.
8.4: DETECTION OF HETEROSCEDASTICITY
1. The Park Test
Professor Park suggested that
is same function of the
explanatory variable . The functional form is
Where is the stochastic disturbance term.
Taking In on both sides. We get
Since
is generally not known. Park suggests using
as a proxy and running the following regression
If
turns out to be statistically significant it means
heteroscedasticity is present in the data, otherwise does not
present it.
Two stages of Park test:
Stage 1: we run the OLS and obtain .
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Advanced Econometrics
Stage 2: again we run OLS with
as a dependent variable.
2. Glejser Test
Glejser test is similar in spirit to Park test. The
difference is that Glejser suggests as many as six functional
forms while Park suggested only one functional form.
Furthermore Glejser used absolute values of
. Glejser used
the following functional forms to detect heteroscedasticity.
I.
| |
II.
| |
√
III.
| |
( )
IV.
| |
(
V.
| |
√
VI.
| |
√
√
)
Stages of Glejser test:
Stage 1: Fit a model Y on X and compute .
Stage 2: Take the absolute value of
and then
regress
with X using any one of functional form.
3. Spearman Rank Correlation Test
Rank correlation co-efficient can be used to detect
heteroscedasticity. That is
Step 1: State hypothesis
,
Step 2: Fit the regression of Y on X and compute .
Step 3: Taking the absolute values of . Rank both | | and
X according to ascending or descending order then compute
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Advanced Econometrics
Where
| |
Step 4: For n
√
with
d.f.
√
| |
Step 5: C.R
⁄
Step 6: Conclusion: As usual.
4. Goldfeld Quandt Test
This test is applicable to large samples. The observations
must be at least twice as many as the parameters to be
estimated.
Step 1. State null and alternative hypothesis.
Step 2. Choose level of significance at
Step 3. Test statistic to be used
(
)
(
)
With
(
)
(
)
Step 4. Computation: Where C is central observations
omitted and K is number of parameters estimated.
i.
We arrange the observations in ascending
or descending order of magnitude.
ii.
of central observations which we omitted
one fourth of the observations for n>30.
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iii.
The remaining (n-c) observations are
divided into two sub samples of equal
size , one including the small values of
iv.
We fit a separate regression lines to each
sub samples, we obtain the sum of
squared residuals from each of them. That
is
.
Compute the value of F.
v.
Step 5. C.R:
Step 6. Conclusion:
Since our calculated value is greater than
table value. So we reject null hypothesis and may
conclude that there is heteroscedasticity.
8.5: REMEDIAL MEASURES OF HETEROSCEDASTICITY
There are two approaches of remediation:
(a) When
is known.
(b) When
is not known.
𝟐
(a) When 𝒊 is known
The most straight forward correcting method of
heteroscedasticty, when
is known by means of weighted
least squares for the estimator, thus obtained for BLUE. i.e
Dividing by
on both sides.
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(b) When
𝟐
𝒊
is unknown
We consider two variable regression model.
That is
Now we consider several assumptions about
the pattern of heteroscedasticity.
I.
The error variance proportional to
.
Proof: Dividing original model by
Where
. That is
.
is the disturbance term.
Taking squaring and expectation on both sides.
( )
(
Hence the variance of
II.
)
is homoscedastic.
The error variance proportional to
.
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. That is
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Proof: The original model can be transform as:
√
Where
√
√
√
is the disturbance term.
√
Taking squaring and expectation on both sides.
(
√
(
Hence the variance of
III.
)
)
is homoscedastic.
The error variance proportional to the squares of the
[
] .
Proof: The original model can be transform as:
Where
is the disturbance term.
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Advanced Econometrics
Taking squaring and expectation on both sides.
(
(
[
)
)
]
[
[
Hence the variance of
IV.
]
]
is homoscedastic.
A log transformation such as:
Reduces heteroscedasticity, when compared with the
regression:
.
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Advanced Econometrics
8.9: Exercise
a) Define
Heteroscedasticity?
What
are
the
consequences of the violation of the assumption of
Homoscedasticity?
b) Review suggested approaches to estimation of a
regression
model
in
the
presence
of
Heteroscedasticity.
c) Discuss the three methods for detection of
Heteroscedasticity.
d) What are the solutions of Heteroscedasticity?
e) Apply Goldfeld and Quandt test on the following
data to test whether there is heteroscedasticity or not.
X
Y
20
18
25
17
23
16
18
10
26
8
27
15
29
16
31
20
22
18
27
17
32
19
35
18
40
26
f) Given
Year
2000
2001
2002
2003
2004
2005
2006
2007
2008
Y
3.5
4.5
5.0
6.0
7.0
9.0
8.0
12.0
14.0
15
20
30
42
50
54
65
8.5
90
16
13
10
7
7
5
4
3.5
2
-0.16
0.43
0.12
0.22
-0.50
1.25
-1.31
-0.43
1.07
g) Consider the model:
Using the data below apply Park-Glejser test?
Year
2002
2003
2004
2005
Y
37
48
45
36
96
X
4.5
6.5
3.5
3.0
41
25
39
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Advanced Econometrics
Chapter: 9
AUTOCORRELATION
9.1: INTRODUCTION
Autocorrelation refer to a case in which the error term in
one time period is correlated with the error term in any other time
ordered in time as in case of time series data or space as in case of
crossOne of the assumptions of linear regression model is
that there is zero correlation between error terms. That is
(
)
If the above assumption is not satisfied than there is
autocorrelation, that is if the value of in any particular period is
correlated with its own preceding value or values. Therefore it is
known as the autocorrelation or serial correlation. That
is (
)
. Autocorrelation is a special case of correlation.
Autocorrelation is referring to the relationship not between two
different variables but between the successive values of the same
variable.
Autocorrelation:
Lag correlation of a given series with itself is called
autocorrelation, thus correlation between two time series such as
is called autocorrelation.
Serial Correlation:
Lag correlation between two different series is called
serial correlation, thus correlation between two different series
such as
is called serial correlation.
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9.2. REASONS OF AUTOCORRELATION
There are several reasons which become the cause of
autocorrelation.
1) Omitting Explanatory Variables:
Most of the economic variables are generally tend
to be auto correlated. If an auto correlated variable has been
excluded from the set of explanatory variables, its influence
be auto correlated.
2) Miss Specification of the Mathematical Model:
If we have adopted a mathematical form which
differs from the true form of the relationship, the
show serial correlation.
3) Specification Bias:
Autocorrelation also arises due to specification bias,
arises from true variables excluded from model and wrong
use of functional form.
4) Lags:
Regression models using lagged values in time
series data occur relatively often in economics, business
and some fields of engineering. If we neglect the lagged
term from the autoregressive model, the resulting error
term will reflect a systematic pattern and therefore
autocorrelation will be present.
5) Data Manipulation:
For empirical analysis, the raw data are often
manipulated. Manipulation introduces smoothness into the
raw data by dampening the fluctuations. This manipulation
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Advanced Econometrics
leads to a systematic pattern and therefore, autocorrelation
will be there.
9.3. OLS ESTIMATION IN THE PRESENCE OF
AUTOCORRELATION
 Mean:
Taking expectations on both sides
[
]
[
]
[ ]
 Variance: By definition:
[
[ ]
[
]
]
[
]
, r=0, 1, 2, 3...
[
]
The expression in brackets is a sum of a geometric
progression of infinite term.
Where is first term of geometric progression and
common ratio, when | |
, the formula reduce to
By using this formula, we get
*
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Advanced Econometrics
Where
 Covariance:
[
[
][
]
]
Given that
[
]
[
]
[
]
[ ]
[
]
[
]
[ [
]
[
[
*
]
]
[
*
]
]
(
)+
+
Similarly:
In general
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Advanced Econometrics
9.4. CONSEQUENCES OF AUTOCORRELATION
Following are the consequences of OLS method in
the presences of autocorrelation.
1. The least square estimator is unbiased even when the
residuals are correlated.
2. With autocorrelation values of the disturbance term
the OLS variance of the parameter are likely to be
larger than those of other econometric models, so they
do not have the minimum variance that is BLUE.
3. If the values of are auto correlated the prediction
based on ordinary least square estimates will be
inefficient in the sense that they will have larger
variances as compared to others.
4.
likely to give misleading conclusion.
5.
correlated.
9.5. DETECTION OF AUTOCORRELATION
1. Durbin Watson d-Statistic
This test was developed by Durbin and Watson to
examine whether autocorrelation exist in a given situation or
(
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Advanced Econometrics
Where
then
*
+
Which is simply the ratio of the sum of squared
differences in successive residuals to RSS (residual sum of
square) is called Durbin Watson d-Statistic. It is noted that
in the numerator of the d-statistic, the number of
observations in (N ) because one observation is lost in
taking successive differences.
Assumption of Durbin Watson d-Statistic
1. The regression model includes the intercept term.
2.
-stochastic or
fixed in repeated sampling.
3.
nerated by the first order
auto regressive scheme i.e.
4. The regression model does not include lag values of the
dependent variable Y.
5. There is no missing observation in the data.
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9.6. REMEDIAL MEASURES OF AUTOCORRELATION
There are two types of remedial measures, when
is
known and when is unknown.
I.
When is known
The problem of autocorrelation can be easily
solved, if the coefficient of first order
autocorrelation is known.
II.
When is not known
There are different ways of estimating
i.
ii.
The First-Difference Method
DurbinWatson d-Statistic
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Advanced Econometrics
9.7: Exercise
1) What is autocorrelation? Discuss its consequences.
2) Differentiate between autocorrelation and serial
correlation. What are its various sources?
3) How can one detect each autocorrelation?
4) In the presence of autocorrelation how can one
obtain efficient estimates?
5) Describe briefly Durbin Watson d-statistic.
6) Apply Durbin Watson d-statistic to the following
data:
Y
X
2
1
1.37
2
2
0.46
2
3
0.45
1
4
-2.36
3
5
1.27
5
6
-0.81
6
7
-0.09
6
8
-1.00
10
9
2.08
10
10
1.17
10
11
0.27
12
12
1.36
15
13
3.44
10
14
-2.46
11
15
2.37
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Chapter: 10
SIMULTANEOUS EQUATION MODELS
10.1: INTRODUCTION
There are two types of Simultaneous Equation Models
1. Simultaneous Equation Models
2. Recursion Equation Models
1. Simultaneous Equation Models
When the independent variable in one equation is also an
independent variable in some other equation we call it
simultaneous equations system or model. The variable entering a
simultaneous equation models are two types:
i .Endogenous variable
ii. Exogenous variable
i. Endogenous variable
The variable whose values are determined within the model
is called Endogenous variable
ii. Exogenous variable
The variable whose values are determined outside the
model is called exogenous variable. These variables are treated as
nonstochastic.
2. Recursion Equation Models
In this model one dependent variable may be a function of
other dependent variable but other dependent variable might not be
the function variable.
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10.2: SYSTEM OF SIMULTANEOUS EQUATION
between Y and X by using single equation.
We must use a multi-equation model which we include
separate equations in which m Y and X, would appear as an
endogenous variable although that might appear as explanatory
variable in other equation of the model.
10.3: Simultaneous Equation Bias
It refers to the overestimation or underestimations of the
structural parameters obtain from the applications the OLS to the
structural equations. This bias result because these endogenous
variables of the system which are also explanatory variables or
correlated with the error term.
Structural Equations and Parameters
Structural equations describe the structure of an economy
or behaviors are some economic agents such as consumer or
producer. There is only on structural equation for each of the
endogenous variable of the system.
The coefficients of the structural equations are called
structural parameters and express the direct effect of each
explanatory variable on the dependent variable.
Reduced Form Equations
These are equations obtained by solving the system of
structural equations so as to express each endogenous variable as a
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Advanced Econometrics
function of only the exogenous variables of the function. Since the
endogenous variable of the system are uncorrelated with error
term, so OLS gives consistent reduced form parameters estimate.
These measure the total direct and indirect effect of a change in the
exogenous variables on the endogenous variables and may be used
to obtain consistence structural parameter.
Example:
Considering Keynesian model of consumption and income
function:
Here
and
are endogenous variables and
exogenous variable both are structural equations
Putting eq (i) in eq (ii).
Putting eq (*) in eq (i).
[
]
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Here
and
are two structural parameters,
are four reduced form coefficients.
10.4: Methods of Estimation in Simultaneous Equation
Models
The most common methods are:
1)
2)
3)
4)
5)
6)
Direct Least Square (DLS)
Indirect Least Square (ILS)
Two stage least square (2SLS)
Three stage least square(3SLS)
Instrumental variable method(IV)
Least variance ratio method(LVR)
1. Direct Least Square Method (DLS)
In this method, we estimate the structural parameter by
applying OLS directly to the structural equation. This method does
not require complete knowledge of the structural system. In this
system, we express all the endogenous variables as a function of all
predetermined variables of the system and we apply ordinary least
square non restriction. Because it does not take into account any
information on the structural parameters.
2. Indirect Least Square Method (ILS)
There is definite relationship between the reduced form
coefficients and the structural parameters it is thus possible first to
obtain estimates of the structural parameters by any econometric
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Advanced Econometrics
technique and then substitute. These estimates into the system of
parameters relationship to obtain
Advantages of ILS
1)
2) Structural changes occur continuously over time.
3) Extraneous information is same structural parameters may
become available from other studies.
Disadvantages of ILS
1) It does not give the standard error of the estimate of the
structural parameters.
2) It cannot be used to calculate unique and consistent structural
parameter estimates from the reduced form coefficients from
the over identify equations of a simultaneous equation models.
Assumption of ILS method
1) Structural equation must be exact identified.
2) ILS method should satisfied first six stochastic assumptions of
OLS method i.e.

is random.


(
)



If ILS method satisfied this assumptions and estimates of ILS
are BLUE estimators.
3) Micro variables should be correctly aggregative.
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Advanced Econometrics
Question: Show that ILS estimator
and
are consistent estimators.
Proof:
Consider Keynesian model
Reduced forms are
. (1)
[
And,
]
Then
̅
̅
And,
̅
̅
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Advanced Econometrics
Subtracting eq (3) from eq (1).
̅
̅
̅
̅
Subtracting eq (4) from eq (2).
̅
̅
̅
̅
̅
̅
We know that
̅
[
̅ ]
̅
Putting the value of
̅
*,
-
̅
*
̅ +
̅ +
̅
̅
Similarly
[
*{
̅
̅ ]
̅
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Advanced Econometrics
̅
̅
̂
̂
[
̂
]
[
]
̂
Applying limit n
,
,
i.e. constant
̂
̂
̂
Similarly
̂
̅
̅
*
̂
̂
+
̅
[
+
̅
[
̅
]
]
̅
̂
Applying limit n
̅*
,
,
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i.e. constant,
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Advanced Econometrics
̅
̂
̂
̂
Hence proved ̂
and ̂
are consistent estimators of
and .
3. The Method of Two Stage Least Square (2SLS)
This method was discovered by Theil and Basmann. It is a
method of estimating consistent structural parameter for the exact
or over identified equations of a simultaneous equation system. For
exactly identified equation Two Stages Least Squares gives the
same result as of ILS. Two Stages Least Squares estimation
involves the application of OLS in two stages.
Stage 1:
In the first stage each endogenous variable is regressed on
all the predetermined variable of the system. At this stage we get
the new reduced form equation.
Stage11:
In the second stage predicted values rather than the actual
values of endogenous are used to estimate the structural equation of
the model. That is, we obtain the estimates ̂ . From stage first and
replacing ̂ in the original equation by the estimated ̂ and then
apply OLS to the equation thus transformed.
The predicted values of the endogenous variable are uncorrected
with the error term which will give us two stages least square
parameters estimates.
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Advanced Econometrics
Advantages of 2SLS with respect to ILS
1) 2SLS can be used to get consistent structural parameter
estimates for the over identified as well as exactly identified
equation in a system of simultaneous equation.
2) 2SLS gives the standard error of the estimate structural
parameter directly while ILS does not provide it.
3) 2SLS is very useful. It is the simplest and one of the best and
most common of all the simultaneous equation estimates.
Properties of 2SLS estimator
1) The 2SLS gives the biased estimator for small sample.
2) For large sample 2SLS estimates are unbiased that is biased
will be zero as n
3) A 2SLS estimate gives the asymptotically efficient estimator.
4) 2SLS estimates are consistent.
Question: Find out the 2SLS estimate and show that in case of exactly
identified 2SLS is same as ILS.
Proof:
We use the simple Keynesian model
Reduced forms are:
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Advanced Econometrics
[
]
Estimated equation of (3)
̂
̂
̂
̂
(̂
̂
̂)
̅ ⁄
̅ ⁄
̅
̂
̂
̅
Residual
̂
̂
̂
̂
Putting equation (4) in equation (1)
̂
̂
̂
̂
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Advanced Econometrics
̂
̂
̂
̂
Since ̂ involves only endogenous variable which is
independently distributed with and .Then application of OLS
will give us consistent estimate.
̂
̂
̂̂
̂
̂ +
̅
̂
̂
[
̅
̂
[
̅ ̂
*
̂
[
̂
̂
̂
̂
̂
̂
̅
̂
̂
̂̂
(̂
̂)
̂̂
̂
̂
̂̂
̂
̂̂
̂
̂̂
̂
̅ ]
̂
̅ ]
̅
[
̂
̅ ]
̂
̅
̂
̅
̂
̂
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̂
̅
̂
̅ ]
Advanced Econometrics
̂
̂
̂
̂
̂
̂
̂
It means that 2SLS and ILS are same in case of exactly identified.
̅
̂
̅
̂
̂
̂
̂
̂
̅
= ̅+ ̅
̅
̂
̅
̂
̅ + ̅)
̅
̅
̂
̂
̅
̂
̅
̅
̂
̂
̅
̅
Hence proved.
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Advanced Econometrics
4. Three Stage Least Square Method (3SLS)
3SLS is a system method. It is applied to all the equations
at the same time and gives estimates of all the parameters
simultaneous. This method is logical extension of two stages least
square method. Under this method we apply OLS method in three
successive stages. It uses more information than single equation
technique.
The first two stages of 3SLS are same as 2SLS. We deal
with the reduced form of all the equation of the system. 3SLS is
the application of GLS (Generalized Least Squares). It means that
we apply OLS method to a set of transformed equations in which
the transformation is obtained from reduced form residuals of the
previous stage.
5. Method of Instrumental Variable (IV)
The instrumental variable method is a single equation
method being applied to one equation of system at a time. It has
been developed as a solution of the simultaneous equation bias and
is appropriate for over identified model.
The instrumental variable method attains the reduction of
appropriate exogenous variable (as instrument). The estimates
obtains from this method is consistent for large sample and biased
for small sample.
Procedure of IV Method
Step I:
An instrumental variable is an exogenous variable located
somewhere in a system of simultaneous equation which satisfies
the following condition:
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1) It must strongly correlated
2) It must truly exogenous
3) If more than one instrumental variable is to be used in the same
structural equation they must be least correlated.
Step II:
Multiplying the structural equation through by the each of
instrument variable form the equation we obtain the estimator of
the structural parameter
Properties of IV
1) For small sample estimator of structural parameter are baised.
2) For large sample the estimates of structural parameter are
consistent.
3) The estimates are not asymptotically efficient.
Assumption of IV method
1) Exogenous variable used as instrumental variable.
2)
of OLS.
3) The exogenous variable must not be multicollinear.
4) The structural function must be identified.
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Advanced Econometrics
10.5: Exercise
1) What is meant by simultaneous equations model? Discuss.
2) Show that OLS estimates are biased in simultaneous
equations problems.
3) Differentiate between endogenous and exogenous
variables.
4) Write short notes on following:
i.
Indirect Least Squares Method
ii.
Instrumental Variable Method
iii. Two Stage Least Squares Method
iv.
Three Stage Least Squares Method
5) Show that ILS estimates are consistent estimators.
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Chapter: 11
IDENTIFICATION
11.1 INTRODUCTION
By identification, we mean whether numerical estimates of
the parameters of the structural equation can be obtained from the
estimated reduced form equations.
If this can be done, we say that the particular equation is
identified. If it is not possible then we say that the equation under
consideration is unidentified or under identified.
In econometric theory there are two possible equations of
identification.
1) Equation under identified
2) Equation identified
1) Equation Under Identified
If the numerical estimates of the parameters of structural
equation cannot be obtain from the estimated reduced form coefficient then we say that the equation under consideration is
unidentified or under identified.
An equation is under identified if its statistical shape is not
unique if it is impossible to estimate all the parameters of an
equation with any econometric technique then equation is
under identified.
A system is called under identified when one or more
equations are under identified.
Example: Consider the following demand and supply model
with equilibrium condition.
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Solution:
(
Put eq (*) in
[
]
Four structural parameters are
from
structural equations of 1 and 2.We have two reduced form
0
1 from the reduced form equations a & b.
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These reduced form equations contain all four structural
parameters. So there is no way in which the four structural
unknown parameters can be estimated from only two reduced form
coefficients. So the system of equation is unidentified or under
identified.
2) Equation Identified
If numerical estimates of the parameters of a structural
equation can be obtained from the estimated reduced form
coefficients then we say that equation is identified
If an equation has a unique statistical solution we may say
that equation is identified.
Identification is a problem of model formulation and
identified equation may be exactly (just) identified or over
identified.
a. Exact (Just) Identification
An identified equation is said to be exactly identified if
unique numerical values of the structural parameters can be
obtained.
Example: Consider the following demand and supply model with
equilibrium condition.
Solution:
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(
Put eq (*) in
*
We
+
have
six structural parameters that are
and six reduced form coefficients that are
here we obtain unique solution of structural
parameters. So the system of equation is exactly identified.
b. Over Identification
An equation is said to be over identified if more than one
numerical value can be obtained for some of the parameters of the
structural equations.
Example: Consider the following demand and supply model with
equilibrium condition.
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Solution:
(
Put eq (*) in
[
We
]
have
seven structural parameters that are
but there are eight reduced form
coefficients that are
The number of
equation are greater than the number of unknown parameters as a
result we may get more than one numerical value for some of the
parameters of the structural equations. So the system of equation is
over identified.
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11.2 RULES FOR IDENTIFICATION
Identification may be established either by examination of the
specification of the structural model or by the examination of the
reduced form of the model.
1) Examination of Structural Model
It is simpler and more useful method for identification.
2) Examination of Reduced form Determinant
This approach for finding the identification is
comparatively confusing and difficult to compute because we
first find the reduced form of the structural models and study
the determinants.
11.3 CONDITIONS OF IDENTIFICATION
There are two conditions which must be fulfilling for an
equation to be identified.
1) The Order Condition of the Identification
This condition is based on a counting rule of the variables
included and excluded from the particular equation. It is a
necessary but not sufficient condition for the identification of an
equation.
Definition:
variables (endogenous and exogenous) excluded from it must be
equal to or greater than the number of endogenous variables in the
model less one


If
If
The equation is just or exact identified.
It is over identified.
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Advanced Econometrics
Where M = number of endogenous variables in the model or
system.
m= number of endogenous variables in a given equation.
K = number of pre-determined or exogenous variables in the
model or system.
k = number of predetermined or exogenous variables in a given
equation.
Example: Consider the following demand and supply function.
Apply order condition.
Solution:
Q and P are endogenous variables. I is exogenous
variable. Apply order condition.
K=1 , M=2
For eq (1).
k=1 , m=2
So demand function is unidentified.
For eq (2).
k=0 , m=2
So supply function is just identified.
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Advanced Econometrics
Example: Consider the following demand and supply function.
Apply order condition.
Solution:
Q and P are endogenous variables. I, R,
variables. Apply order condition.
are exogenous
K=3 , M=2
For eq (1).
k=2 , m=2
So demand function is exact identified.
For eq (2).
k=1 , m=2
So supply function is over identified.
2) The Rank Condition for Identification
The order condition is necessary but not sufficient
condition for identification. Sometime the order condition is
satisfied but it happens that an equation is not identified.
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Advanced Econometrics
Therefore we required another condition for identification
is the rank condition which is sufficient condition for
identification.
Rank Condition
The rank condition states that in a system of G equations,
particular equation is identified if and only if (iff) it is possible to
construct at least one none zero determinants of order (G-1) from
the coefficient of variables excluded from that particular equation
but contained in the other equation of the model.
Procedure of Rank Condition
a) Write down the equations in tabular form.
b) Strike out (exclude) the coefficient of the row in which
the equation under consideration appears.
c) Also strike out the columns corresponding to those
coefficients in step (b) which are none zero.
d) The entries left in the table will give only the coefficient
of variables included in the system but not in the equation
under consideration.
Example: Given the following equations:
Apply rank condition to all the equations.
Solution:
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Advanced Econometrics
Equation
1
-1
2
0
3
0
4
1
0
-1
0
1
0
0
-1
0
-1
0
0
0
0
1
0
0
Consider equation 1.
[
|
|
|
|
]
|
|
|
|
|
|
Hence equation 1 is unidentified.
Consider equation 2.
[
|
|
|
|
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|
]
|
|
|
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Advanced Econometrics
Hence equation 2 is identified.
Consider equation 3.
[
|
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|
]
|
|
|
|
|
|
Hence equation 3 is identified.
Consider equation 4.
[
|
|
|
|
]
Hence equation 4 is also identified.
Example: Consider the following system of equations
Determine the system of equation is exactly, Over and
unidentified by using:
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Advanced Econometrics
a) Rank condition
b) Order condition
Solution:
a) Rank condition
Equation
1
2
3
4
1
0
1
0
0
0
0
1
1
0
0
0
0
0
0
0
Consider equation 1.
[
|
|
|
|
]
|
|
Hence equation 1 is unidentified.
Consider equation 2.
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Advanced Econometrics
[
|
|
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|
]
|
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|
|
|
Hence equation 2 is also unidentified.
Consider equation 3.
[
|
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|
]
|
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|
|
|
|
Hence equation 3 is also unidentified.
Consider equation 4.
[
|
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|
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|
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]
|
|
Hence equation 4 is identified.
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Advanced Econometrics
b) Order condition
M = number of endogenous variables in a system of
equations.
K = number of exogenous variables in a system of equations.
i.e. (
K=3
)
i.e. (
)
m = number of endogenous variables in a given equations.
For equation 1:
m=3
i.e. (
)
For equation 2:
m=2
i.e. (
)
i.e. (
)
For equation 3:
m=2
For equation 4:
m=3
i.e. (
)
k = number of exogenous variables in a given equation.
For equation 1:
k=1
i.e. (
)
For equation 2:
k=2
i.e. (
For equation 3:
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Advanced Econometrics
k=2
i.e. (
)
For equation 4:
k=1
i.e. (
)
Equation
Result
1
Identified
2
Identified
3
Identified
4
Identified
Thus by order condition all the equations are identified but by
rank condition only equation 4 is identified.
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Advanced Econometrics
11.4: Exercise
i.
ii.
iii.
iv.
Discuss the problem of identification.
Explain the rank condition of identification.
Briefly discuss the procedure of order condition of
identification.
Check the identifiability of the following model:
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