Linear Models and Econometrics Chapter 4 Econometrics.ppt

1
Chapter 4
Econometrics
Dr. A. PHILIP AROKIADOSS
Assistant Professor
Department of Statistics
St. Joseph’s College (Autonomous)
Tiruchirappalli-620 002.

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Basic Econometrics
Introduction:
What is
Econometrics?

3
Introduction
What is Econometrics?
 Definition 1: Economic Measurement
 Definition 2: Application of the
mathematical statistics to economic data
in order to lend empirical support to the
economic mathematical models and
obtain numerical results (Gerhard Tintner,
1968)

4
Introduction
 Definition 3: The quantitative
analysis of actual economic phenomena
based on concurrent development of
theory and observation, related by
appropriate methods of inference
(P.A.Samuelson, T.C.Koopmans and
J.R.N.Stone, 1954)

5
Introduction
 Definition 4: The social science
which applies economics, mathematics
and statistical inference to the analysis of
economic phenomena (By Arthur S.
Goldberger, 1964)
 Definition 5: The empirical
determination of economic laws (By H.
Theil, 1971)

6
Introduction
 Definition 6: A conjunction of
economic theory and actual
measurements, using the theory and
technique of statistical inference as a
bridge pier (By T.Haavelmo, 1944)
 And the others

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Econometrics
Economic
Theory
Mathematical
Economics
Economic
Statistics
Mathematic
Statistics

8
Introduction
Why a separate discipline?
 Economic theory makes statements
that are mostly qualitative in nature,
while econometrics gives empirical
content to most economic theory
 Mathematical economics is to
express economic theory in
mathematical form without empirical
verification of the theory, while
econometrics is mainly interested in the
later

9
Introduction
Why a separate discipline?
 Economic Statistics is mainly
concerned with collecting, processing and
presenting economic data. It does not
being concerned with using the collected
data to test economic theories
 Mathematical statistics provides
many of tools for economic studies, but
econometrics supplies the later with
many special methods of quantitative
analysis based on economic data

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Econometrics
Economic
Theory
Mathematical
Economics
Economic
Statistics
Mathematic
Statistics

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Introduction
Methodology of
Econometrics
(1) Statement of theory or
hypothesis:
Keynes stated: ”Consumption
increases as income increases, but
not as much as the increase in
income”. It means that “The
marginal propensity to consume
(MPC) for a unit change in
income is grater than zero but less
than unit”

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Introduction
Methodology of
Econometrics
(2) Specification of the
mathematical model of the
theory
Y = ß1+ ß2X ; 0 < ß2< 1
Y= consumption expenditure
X= income
ß1 and ß2 are parameters; ß1 is
intercept, and ß2 is slope coefficients

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Introduction
Methodology of
Econometrics
(3) Specification of the
econometric model of the
theory
Y = ß1+ ß2X + u ; 0 < ß2< 1;
Y = consumption expenditure;
X = income;
ß1 and ß2 are parameters; ß1is
intercept and ß2 is slope coefficients;
u is disturbance term or error term. It
is a random or stochastic variable

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Introduction
Methodology of
Econometrics
(4) Obtaining Data
(See Table 1.1, page 6)
Y= Personal consumption
expenditure
X= Gross Domestic Product
all in Billion US Dollars

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Introduction
Methodology of
Econometrics
(4) Obtaining Data
Year X Y
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
2447.1
2476.9
2503.7
2619.4
2746.1
2865.8
2969.1
3052.2
3162.4
3223.3
3260.4
3240.8
3776.3
3843.1
3760.3
3906.6
4148.5
4279.8
4404.5
4539.9
4718.6
4838.0
4877.5
4821.0

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Introduction
Methodology of
Econometrics
(5) Estimating the Econometric
Model
Y^ = - 231.8 + 0.7194 X (1.3.3)
MPC was about 0.72 and it means
that for the sample period when real
income increases 1 USD, led (on
average) real consumption expenditure
increases of about 72 cents
Note: A hat symbol (^) above one
variable will signify an estimator of the
relevant population value

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Introduction
Methodology of Econometrics
(6) Hypothesis Testing
Are the estimates accord with the
expectations of the theory that is being
tested? Is MPC < 1 statistically? If so,
it may support Keynes’ theory.
Confirmation or refutation of
economic theories based on
sample evidence is object of Statistical
Inference (hypothesis testing)

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Introduction
Methodology of
Econometrics
(7) Forecasting or Prediction
 With given future value(s) of X, what
is the future value(s) of Y?
 GDP=$6000Bill in 1994, what is the
forecast consumption expenditure?
 Y^= - 231.8+0.7196(6000) = 4084.6
 Income Multiplier M = 1/(1 – MPC)
(=3.57). decrease (increase) of $1 in
investment will eventually lead to
$3.57 decrease (increase) in income

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Introduction
Methodology of
Econometrics
(8) Using model for control or
policy purposes
Y=4000= -231.8+0.7194 X  X  5882
MPC = 0.72, an income of $5882 Bill
will produce an expenditure of $4000
Bill. By fiscal and monetary policy,
Government can manipulate the
control variable X to get the desired
level of target variable Y

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Introduction
Methodology of Econometrics
Figure 1.4: Anatomy of economic
modelling
• 1) Economic Theory
• 2) Mathematical Model of Theory
• 3) Econometric Model of Theory
• 4) Data
• 5) Estimation of Econometric Model
• 6) Hypothesis Testing
• 7) Forecasting or Prediction
• 8) Using the Model for control or policy
purposes

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Economic Theory
Mathematic Model Econometric Model Data Collection
Estimation
Hypothesis Testing
Forecasting
Application
in control or
policy
studies

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Basic Econometrics
Chapter 1:
THE NATURE OF
REGRESSION
ANALYSIS

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1-1. Historical origin of the term
“Regression”
 The term REGRESSION was
introduced by Francis Galton
 Tendency for tall parents to have tall
children and for short parents to have
short children, but the average height
of children born from parents of a
given height tended to move (or
regress) toward the average height in
the population as a whole (F. Galton,
“Family Likeness in Stature”)

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1-1. Historical origin of the term
“Regression”
 Galton’s Law was confirmed by Karl
Pearson: The average height of sons of
a group of tall fathers < their fathers’
height. And the average height of sons
of a group of short fathers > their
fathers’ height. Thus “regressing” tall
and short sons alike toward the average
height of all men. (K. Pearson and A.
Lee, “On the law of Inheritance”)
 By the words of Galton, this was
“Regression to mediocrity”

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1-2. Modern Interpretation of
Regression Analysis
 The modern way in interpretation of
Regression: Regression Analysis is
concerned with the study of the
dependence of one variable (The
Dependent Variable), on one or more
other variable(s) (The Explanatory
Variable), with a view to estimating
and/or predicting the (population)
mean or average value of the former in
term of the known or fixed (in
repeated sampling) values of the latter.
 Examples: (pages 16-19)

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Dependent Variable Y; Explanatory Variable Xs
1. Y = Son’s Height; X = Father’s Height
2. Y = Height of boys; X = Age of boys
3. Y = Personal Consumption Expenditure
X = Personal Disposable Income
4. Y = Demand; X = Price
5. Y = Rate of Change of Wages
X = Unemployment Rate
6. Y = Money/Income; X = Inflation Rate
7. Y = % Change in Demand; X = % Change in the
advertising budget
8. Y = Crop yield; Xs = temperature, rainfall, sunshine,
fertilizer

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1-3. Statistical vs.
Deterministic Relationships
 In regression analysis we are
concerned with STATISTICAL
DEPENDENCE among variables (not
Functional or Deterministic), we
essentially deal with RANDOM or
STOCHASTIC variables (with the
probability distributions)

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1-4. Regression vs. Causation:
Regression does not necessarily imply
causation. A statistical relationship
cannot logically imply causation. “A
statistical relationship, however strong
and however suggestive, can never
establish causal connection: our ideas
of causation must come from outside
statistics, ultimately from some theory
or other” (M.G. Kendal and A. Stuart,
“The Advanced Theory of Statistics”)

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1-5. Regression vs.
Correlation
 Correlation Analysis: the primary objective
is to measure the strength or degree of
linear association between two variables
(both are assumed to be random)
 Regression Analysis: we try to estimate or
predict the average value of one variable
(dependent, and assumed to be stochastic)
on the basis of the fixed values of other
variables (independent, and non-stochastic)

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1-6. Terminology and Notation
Dependent Variable

Explained Variable

Predictand

Regressand

Response

Endogenous
Explanatory
Variable(s)

Independent
Variable(s)

Predictor(s)

Regressor(s)

Stimulus or control
variable(s)

Exogenous(es)

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1-7. The Nature and Sources
of Data for Econometric
Analysis
1) Types of Data :
 Time series data;
 Cross-sectional data;
 Pooled data
2) The Sources of Data
3) The Accuracy of Data

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1-8. Summary and Conclusions
1) The key idea behind regression
analysis is the statistic dependence of
one variable on one or more other
variable(s)
2) The objective of regression analysis is
to estimate and/or predict the mean or
average value of the dependent
variable on basis of known (or fixed)
values of explanatory variable(s)

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3) The success of regression depends on
the available and appropriate data
4) The researcher should clearly state the
sources of the data used in the analysis,
their definitions, their methods of
collection, any gaps or omissions and
any revisions in the data

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Basic Econometrics
Chapter 2:
TWO-VARIABLE
REGRESSION ANALYSIS:
Some basic Ideas

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2-1. A Hypothetical Example
 Total population: 60 families
 Y=Weekly family consumption expenditure
 X=Weekly disposable family income
 60 families were divided into 10 groups of
approximately the same income level
(80, 100, 120, 140, 160, 180, 200, 220, 240, 260)

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 Table 2-1 gives the conditional distribution
of Y on the given values of X
 Table 2-2 gives the conditional probabilities
of Y: p(YX)
 Conditional Mean
(or Expectation): E(YX=Xi )

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X
Y
80 100 120 140 160 180 200 220 240 260
Weekly
family
consumption
expenditure
Y ($)
55 65 79 80 102 110 120 135 137 150
60 70 84 93 107 115 136 137 145 152
65 74 90 95 110 120 140 140 155 175
70 80 94 103 116 130 144 152 165 178
75 85 98 108 118 135 145 157 175 180
-- 88 -- 113 125 140 -- 160 189 185
-- -- -- 115 -- -- -- 162 -- 191
Total 325 462 445 707 678 750 685 1043 966 1211
Mean 65 77 89 101 113 125 137 149 161 173
Table 2-2: Weekly family income X ($), and consumption Y ($)

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 Figure 2-1 shows the population
regression line (curve). It is the
regression of Y on X
 Population regression curve is the
locus of the conditional means or
expectations of the dependent variable
for the fixed values of the explanatory
variable X (Fig.2-2)

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2-2. The concepts of population
regression function (PRF)
 E(YX=Xi ) = f(Xi) is Population
Regression Function (PRF) or
Population Regression (PR)
 In the case of linear function we have
linear population regression function (or
equation or model)
E(YX=Xi ) = f(Xi) = ß1 + ß2Xi

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2-2. The concepts of population
regression function (PRF)
E(YX=Xi ) = f(Xi) = ß1 + ß2Xi
 ß1 and ß2 are regression coefficients, ß1is
intercept and ß2 is slope coefficient
 Linearity in the Variables
 Linearity in the Parameters

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2-4. Stochastic Specification of PRF
 Ui = Y - E(YX=Xi ) or Yi = E(YX=Xi ) + Ui
 Ui = Stochastic disturbance or stochastic
error term. It is nonsystematic component
 Component E(YX=Xi ) is systematic or
deterministic. It is the mean consumption
expenditure of all the families with the same
level of income
 The assumption that the regression line
passes through the conditional means of Y
implies that E(UiXi ) = 0

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2-5. The Significance of the Stochastic
Disturbance Term
 Ui = Stochastic Disturbance Term is a
surrogate for all variables that are
omitted from the model but they
collectively affect Y
 Many reasons why not include such
variables into the model as follows:

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2-5. The Significance of the Stochastic
Disturbance Term
Why not include as many as variable into
the model (or the reasons for using ui)
+ Vagueness of theory
+ Unavailability of Data
+ Core Variables vs. Peripheral Variables
+ Intrinsic randomness in human behavior
+ Poor proxy variables
+ Principle of parsimony
+ Wrong functional form

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2-6. The Sample Regression
Function (SRF)
Table 2-4: A random
sample from the
population
Y X
------------------
70 80
65 100
90 120
95 140
110 160
115 180
120 200
140 220
155 240
150 260
------------------
Table 2-5: Another random
sample from the population
Y X
-------------------
55 80
88 100
90 120
80 140
118 160
120 180
145 200
135 220
145 240
175 260
--------------------

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SRF1
SRF2
Weekly Consumption
Expenditure (Y)
Weekly Income (X)

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Function (SRF)
 Fig.2-3: SRF1 and SRF 2
 Yî = ^1 + ^2Xi (2.6.1)
 Yî = estimator of E(YXi)
 ^1 = estimator of 1
 ^2 = estimator of 2
 Estimate = A particular numerical value
obtained by the estimator in an application
 SRF in stochastic form: Yi= ^1 + ^2Xi + uî
or Yi= Yî + uî (2.6.3)

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Function (SRF)
 Primary objective in regression analysis is
to estimate the PRF Yi= 1 + 2Xi + ui on
the basis of the SRF Yi= ^1 + ^2Xi + ei
and how to construct SRF so that ^1 close
to 1 and ^2 close to 2 as much as
possible

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Function (SRF)
 Population Regression Function PRF
 Linearity in the parameters
 Stochastic PRF
 Stochastic Disturbance Term ui plays a
critical role in estimating the PRF
 Sample of observations from
population
 Stochastic Sample Regression Function
SRF used to estimate the PRF

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 The key concept underlying regression
analysis is the concept of the
population regression function (PRF).
 This book deals with linear PRFs:
linear in the unknown parameters.
They may or may not linear in the
variables.

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 For empirical purposes, it is the stochastic
PRF that matters. The stochastic
disturbance term ui plays a critical role in
estimating the PRF.
 The PRF is an idealized concept, since in
practice one rarely has access to the entire
population of interest. Generally, one has a
sample of observations from population
and use the stochastic sample regression
(SRF) to estimate the PRF.

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Basic Econometrics
Chapter 3:
TWO-VARIABLE
REGRESSION MODEL:
The problem of Estimation

52
3-1. The method of ordinary
least square (OLS)
 Least-square criterion:
 Minimizing U^2
i = (Yi – Y^i) 2
= (Yi- ^1 - ^2X)2 (3.1.2)
 Normal Equation and solving it
for ^1 and ^2 = Least-square
estimators [See (3.1.6)(3.1.7)]
 Numerical and statistical
properties of OLS are as follows:

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3-1. The method of ordinary least
square (OLS)
 OLS estimators are expressed solely in terms of
observable quantities. They are point estimators
 The sample regression line passes through
sample means of X and Y
 The mean value of the estimated Y^ is equal to
the mean value of the actual Y: E(Y) = E(Y^)
 The mean value of the residuals Uî is zero:
E(uî )=0
 uî are uncorrelated with the predicted Yî and
with Xi : That are uîYî = 0; uîXi = 0

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3-2. The assumptions underlying
the method of least squares
 Ass 1: Linear regression model
(in parameters)
 Ass 2: X values are fixed in repeated
sampling
 Ass 3: Zero mean value of ui : E(uiXi)=0
 Ass 4: Homoscedasticity or equal
variance of ui : Var (uiXi) = 2
[VS. Heteroscedasticity]
 Ass 5: No autocorrelation between the
disturbances: Cov(ui,ujXi,Xj ) = 0
with i # j [VS. Correlation, + or - ]

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3-2. The assumptions underlying
the method of least squares
 Ass 6: Zero covariance between ui and Xi
Cov(ui, Xi) = E(ui, Xi) = 0
 Ass 7: The number of observations n must be
greater than the number of parameters
to be estimated
 Ass 8: Variability in X values. They must
not all be the same
 Ass 9: The regression model is correctly
specified
 Ass 10: There is no perfect multicollinearity
between Xs

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3-3. Precision or standard errors of
least-squares estimates
 In statistics the precision of an
estimate is measured by its standard
error (SE)
 var( ^2) = 2 / x2
i (3.3.1)
 se(^2) =  Var(^2) (3.3.2)
 var( ^1) = 2 X2
i / n x2
i (3.3.3)
 se(^1) =  Var(^1) (3.3.4)
 ^ 2 = u^2
i / (n - 2) (3.3.5)
 ^ =  ^ 2 is standard error of the
estimate

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3-3. Precision or standard errors of
least-squares estimates
 Features of the variance:
+ var( ^2) is proportional to 2 and
inversely proportional to x2
i
+ var( ^1) is proportional to 2 and
X2
i but inversely proportional to x2
i
and the sample size n.
+ cov ( ^1 , ^2) = - var( ^2) shows
the independence between ^1 and ^2
X

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3-4. Properties of least-squares
estimators: The Gauss-Markov Theorem
 An OLS estimator is said to be BLUE if :
+ It is linear, that is, a linear function of a
random variable, such as the dependent
variable Y in the regression model
+ It is unbiased , that is, its average or expected
value, E(^2), is equal to the true value 2
+ It has minimum variance in the class of all
such linear unbiased estimators
An unbiased estimator with the least variance is
known as an efficient estimator

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3-4. Properties of least-squares
estimators: The Gauss-Markov
Theorem
 Gauss- Markov Theorem:
Given the assumptions of the
classical linear regression model,
the least-squares estimators, in
class of unbiased linear estimators,
have minimum variance, that is,
they are BLUE

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3-5. The coefficient of determination
r2: A measure of “Goodness of fit”
 Yi = i + i or
 Yi - = i - i + i or
 yi = i + i (Note: = )
Squaring on both side and summing =>
  yi
2 = 2 x2
i +  2
i ; or
 TSS = ESS + RSS
Y
Y
Ŷ
Ŷ Ŷ
Ŷ
Û
Û
Û
ŷ Û
2
β̂
2
β̂

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3-5. The coefficient of determination r2:
A measure of “Goodness of fit”
 TSS =  yi
2 = Total Sum of Squares
 ESS =  Y^ i
2 = ^2
2 x2
i =
Explained Sum of Squares
 RSS =  u^2
I = Residual Sum of
Squares
ESS RSS
1 = -------- + -------- ; or
TSS TSS
RSS RSS
1 = r2 + ------- ; or r2 = 1 - -------
TSS TSS

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 r2 = ESS/TSS
is coefficient of determination, it measures
the proportion or percentage of the total
variation in Y explained by the regression
Model
 0  r2  1;
 r =  r2 is sample correlation coefficient
 Some properties of r

63
3-6. A numerical Example (pages 80-83)
3-7. Illustrative Examples (pages 83-85)
3-8. Coffee demand Function
3-9. Monte Carlo Experiments (page 85)
3-10. Summary and conclusions (pages
86-87)

64
Basic Econometrics
Chapter 4:
THE NORMALITY
ASSUMPTION:
Classical Normal Linear
Regression Model
(CNLRM)

65
4-2.The normality assumption
 CNLR assumes that each u i is distributed
normally u i  N(0, 2) with:
Mean = E(u i) = 0 Ass 3
Variance = E(u2
i) = 2 Ass 4
Cov(u i , u j ) = E(u i , u j) = 0 (i#j) Ass 5
 Note: For two normally distributed variables,
the zero covariance or correlation means
independence of them, so u i and u j are not only
uncorrelated but also independently distributed.
Therefore u i  NID(0, 2) is Normal and
Independently Distributed

66
 Why the normality assumption?
(1) With a few exceptions, the distribution of sum
of a large number of independent and
identically distributed random variables tends
to a normal distribution as the number of
such variables increases indefinitely
(2) If the number of variables is not very large or
they are not strictly independent, their sum
may still be normally distributed

67
 Why the normality assumption?
(3) Under the normality assumption
for ui , the OLS estimators ^1 and
^2 are also normally distributed
(4) The normal distribution is a
comparatively simple distribution
involving only two parameters
(mean and variance)

68
4-3. Properties of OLS
estimators under the normality
assumption
 With the normality assumption the
OLS estimators ^1 , ^2 and ^2 have
the following properties:
1. They are unbiased
2. They have minimum variance.
Combined 1 and 2, they are efficient
estimators
3. Consistency, that is, as the sample size
increases indefinitely, the estimators
converge to their true population
values

69
4-3. Properties of OLS estimators
under the normality assumption
4. ^1 is normally distributed 
N(1, ^1
2)
And Z = (^1- 1)/ ^1 is  N(0,1)
5. ^2 is normally distributed N(2 ,^2
2)
And Z = (^2- 2)/ ^2 is  N(0,1)
6. (n-2) ^2/ 2 is distributed as the
2
(n-2)

70
4-3. Properties of OLS estimators
under the normality assumption
7. ^1 and ^2 are distributed
independently of ^2. They have
minimum variance in the entire class
of unbiased estimators, whether linear
or not. They are best unbiased
estimators (BUE)
8. Let ui is  N(0, 2 ) then Yi is 
N[E(Yi); Var(Yi)] = N[1+ 2X i ; 2]

71
Some last points of chapter 4
4-4. The method of Maximum likelihood (ML)
 ML is point estimation method with some
stronger theoretical properties than OLS
(Appendix 4.A on pages 110-114)
The estimators of coefficients ’s by OLS and ML are
 identical. They are true estimators of the ’s
 (ML estimator of 2) = u^i
2/n (is biased
estimator)
 (OLS estimator of 2) = u^i
2/n-2 (is unbiased
estimator)
 When sample size (n) gets larger the two estimators
tend to be equal

72
Some last points of chapter 4
4-5. Probability distributions related
to the Normal Distribution: The t, 2,
and F distributions
See section (4.5) on pages 107-108
with 8 theorems and Appendix A, on
pages 755-776
See 10 conclusions on pages 109-110

73
Basic Econometrics
Chapter 5:
TWO-VARIABLE
REGRESSION:
Interval Estimation
and Hypothesis Testing

74
Chapter 5 TWO-VARIABLE REGRESSION:
Interval Estimation and Hypothesis Testing
5-1. Statistical Prerequisites
 See Appendix A with key concepts
such as probability, probability
distributions, Type I Error, Type II
Error,level of significance, power of a
statistic test, and confidence interval

75
5-2. Interval estimation: Some basic Ideas
 How “close” is, say, ^2 to 2 ?
Pr (^2 -   2  ^2 + ) = 1 -  (5.2.1)
 Random interval ^2 -   2  ^2 + 
if exits, it known as confidence interval
 ^2 -  is lower confidence limit
 ^2 +  is upper confidence limit

76
 (1 - ) is confidence coefficient,
 0 <  < 1 is significance level
 Equation (5.2.1) does not mean that the Pr of
2 lying between the given limits is (1 - ), but
the Pr of constructing an interval that contains
2 is (1 - )
 (^2 -  , ^2 + ) is random interval

77
 In repeated sampling, the intervals will
enclose, in (1 - )*100 of the cases, the true
value of the parameters
 For a specific sample, can not say that the
probability is (1 - ) that a given fixed interval
includes the true 2
 If the sampling or probability distributions of
the estimators are known, one can make
confidence interval statement like (5.2.1)

78
5-3. Confidence Intervals for Regression
Coefficients
 Z= (^2 - 2)/se(^2) = (^2 - 2) x2
i / ~N(0,1)
(5.3.1)
We did not know  and have to use ^ instead, so:
 t= (^2 - 2)/se(^2) = (^2 - 2) x2
i /^ ~ t(n-2)
(5.3.2)
 => Interval for 2
Pr [ -t /2  t  t /2] = 1-  (5.3.3)

79
5-3. Confidence Intervals for Regression
Coefficients
 Or confidence interval for 2 is
Pr [^2-t /2se(^2)  2  ^2+t /2se(^2)] = 1- 
(5.3.5)
 Confidence Interval for 1
Pr [^1-t /2se(^1)  1  ^1+t /2se(^1)] = 1- 
(5.3.7)

80
5-4. Confidence Intervals for 2
Pr [(n-2)^2/ 2
/2  2 (n-2)^2/ 2
1- /2] = 1- 
(5.4.3)
 The interpretation of this interval is: If we
establish (1- ) confidence limits on 2 and
if we maintain a priori that these limits
will include true 2, we shall be right in
the long run (1- ) percent of the time

81
5-5. Hypothesis Testing: General Comments
 The stated hypothesis is known as the
null hypothesis: Ho
 The Ho is tested against and alternative
hypothesis: H1
5-6. Hypothesis Testing: The confidence
interval approach
One-sided or one-tail Test
H0: 2  * versus H1: 2 > *

82
Two-sided or two-tail Test
H0: 2 = * versus H1: 2 # *
^2 - t /2se(^2)  2  ^2 + t /2se(^2) values
of 2 lying in this interval are plausible under Ho
with 100*(1- )% confidence.
 If 2 lies in this region we do not reject Ho (the
finding is statistically insignificant)
 If 2 falls outside this interval, we reject Ho (the
finding is statistically significant)

83
5-7. Hypothesis Testing:
The test of significance approach
A test of significance is a procedure by
which sample results are used to verify the
truth or falsity of a null hypothesis
 Testing the significance of regression
coefficient: The t-test
Pr [^2-t /2se(^2)  2  ^2+t /2se(^2)]= 1-

(5.7.2)

84
 5-7. Hypothesis Testing: The test of
significance approach
 Table 5-1: Decision Rule for t-test of significance
Type of
Hypothesis
H0 H1 Reject H0
if
Two-tail 2 = 2* 2 # 2* |t| > t/2,df
Right-tail 2  2* 2 > 2* t > t,df
Left-tail 2 2* 2 < 2* t < - t,df

85
Testing the significance of 2 : The 2 Test
Under the Normality assumption we have:
^2
2 = (n-2) ------- ~ 2
(n-2) (5.4.1)
2
From (5.4.2) and (5.4.3) on page 520 =>

86
 Table 5-2: A summary of the 2 Test
H0 H1 Reject H0 if
2 = 2
0 2 > 2
0 Df.(^2)/ 2
0 > 2
,df
2 = 2
0 2 < 2
0 Df.(^2)/ 2
0 < 2
(1-),df
2 = 2
0 2 # 2
0 Df.(^2)/ 2
0 > 2
/2,df
or < 2
(1-/2), df

87
Some practical aspects
1) The meaning of “Accepting” or
“Rejecting” a Hypothesis
2) The Null Hypothesis and the Rule of
Thumb
3) Forming the Null and Alternative
Hypotheses
4) Choosing , the Level of Significance

88
Some practical aspects
5) The Exact Level of Significance:
The p-Value [See page 132]
6) Statistical Significance versus
Practical Significance
7) The Choice between Confidence-
Interval and Test-of-Significance
Approaches to Hypothesis Testing
[Warning: Read carefully pages 117-134 ]

89
5-9. Regression Analysis and Analysis
of Variance
 TSS = ESS + RSS
 F=[MSS of ESS]/[MSS of RSS] =
= 2^2 xi
2/ ^2 (5.9.1)
 If ui are normally distributed; H0: 2 = 0
then F follows the F distribution with 1
and n-2 degree of freedom

90
 5-9. Regression Analysis and
Analysis of Variance
 F provides a test statistic to test the
null hypothesis that true 2 is zero by
compare this F ratio with the F-critical
obtained from F tables at the chosen
level of significance, or obtain the p-
value of the computed F statistic to
make decision

91
 5-9. Regression Analysis and Analysis
of Variance
 Table 5-3. ANOVA for two-variable regression model
Source of
Variation
Sum of square ( SS) Degree of
Freedom -
(Df)
Mean sum of
square ( MSS)
ESS (due to
regression)
yî
2 = 2^2 xi
2 1 2^2 xi
2
RSS (due to
residuals)
uî
2 n-2 uî
2 /(n-2)=^2
TSS y i
2 n-1

92
5-10. Application of Regression
Analysis: Problem of Prediction
 By the data of Table 3-2, we obtained the
sample regression (3.6.2) :
Y^i = 24.4545 + 0.5091Xi , where
Y^i is the estimator of true E(Yi)
 There are two kinds of prediction as
follows:

93
5-10. Application of Regression
Analysis: Problem of Prediction
 Mean prediction: Prediction of the
conditional mean value of Y
corresponding to a chosen X, say X0, that
is the point on the population regression
line itself (see pages 137-138 for details)
 Individual prediction: Prediction of an
individual Y value corresponding to X0
(see pages 138-139 for details)

94
5-11. Reporting the results of
regression analysis
 An illustration:
Y^I= 24.4545 + 0.5091Xi (5.1.1)
Se = (6.4138) (0.0357) r2= 0.9621
t = (3.8128) (14.2405) df= 8
P = (0.002517) (0.000000289) F1,2=2202.87

95
5-12. Evaluating the results of regression
analysis:
 Normality Test: The Chi-Square (2)
Goodness of fit Test
2
N-1-k =  (Oi – Ei)2/Ei (5.12.1)
Oi is observed residuals (u^i) in interval i
Ei is expected residuals in interval i
N is number of classes or groups; k is number of
parameters to be estimated. If p-value of
obtaining 2
N-1-k is high (or 2
N-1-k is small) =>
The Normality Hypothesis can not be rejected

96
analysis:
 Normality Test: The Chi-Square (2)
Goodness of fit Test
H0: ui is normally distributed
H1: ui is un-normally distributed
Calculated-2
N-1-k =  (Oi – Ei)2/Ei (5.12.1)
Decision rule:
Calculated-2
N-1-k > Critical-2
N-1-k then H0 can
be rejected

97
analysis:
The Jarque-Bera (JB) test of normality
This test first computes the Skewness (S)
and Kurtosis (K) and uses the following
statistic:
JB = n [S2/6 + (K-3)2/24] (5.12.2)
Mean= xbar = xi/n ; SD2 = (xi-xbar)2/(n-1)
S=m3/m2
3/2 ; K=m4/m2
2 ; mk= (xi-xbar)k/n

98
5-12. (Continued)
Under the null hypothesis H0 that the
residuals are normally distributed Jarque
and Bera show that in large sample
(asymptotically) the JB statistic given in
(5.12.12) follows the Chi-Square
distribution with 2 df. If the p-value of the
computed Chi-Square statistic in an
application is sufficiently low, one can
reject the hypothesis that the residuals
are normally distributed. But if p-value is
reasonable high, one does not reject the

99
1. Estimation and Hypothesis testing
constitute the two main branches of
classical statistics
2. Hypothesis testing answers this question:
Is a given finding compatible with a stated
hypothesis or not?
3. There are two mutually complementary
approaches to answering the preceding
question: Confidence interval and test of
significance.

100
4. Confidence-interval approach has a
specified probability of including
within its limits the true value of the
unknown parameter. If the null-
hypothesized value lies in the
confidence interval, H0 is not
rejected, whereas if it lies outside this
interval, H0 can be rejected

101
5. Significance test procedure develops a test
statistic which follows a well-defined
probability distribution (like normal, t, F, or
Chi-square). Once a test statistic is
computed, its p-value can be easily
obtained.
The p-value The p-value of a test is the
lowest significance level, at which we
would reject H0. It gives exact probability of
obtaining the estimated test statistic under
H0. If p-value is small, one can reject H0,

102
6. Type I error is the error of rejecting a
true hypothesis. Type II error is the
error of accepting a false hypothesis.
In practice, one should be careful in
fixing the level of significance , the
probability of committing a type I error
(at arbitrary values such as 1%, 5%,
10%). It is better to quote the p-value
of the test statistic.

103
7. This chapter introduced the normality
test to find out whether ui follows the
normal distribution. Since in small
samples, the t, F,and Chi-square tests
require the normality assumption, it is
important that this assumption be
checked formally

104
(ended)
8. If the model is deemed practically
adequate, it may be used for
forecasting purposes. But should not
go too far out of the sample range of
the regressor values. Otherwise,
forecasting errors can increase
dramatically.

105
Basic Econometrics
Chapter 6
EXTENSIONS OF THE
TWO-VARIABLE LINEAR
REGRESSION MODEL

106
Chapter 6
EXTENSIONS OF THE TWO-VARIABLE
LINEAR REGRESSION MODELS
6-1. Regression through the origin
 The SRF form of regression:
 Yi = ^2X i + u^ i (6.1.5)
 Comparison two types of regressions:
* Regression through-origin model and
* Regression with intercept

107
Chapter 6
Comparison two types of regressions:
^2 = XiYi/X2
i (6.1.6) O
^2 = xiyi/x2
i (3.1.6) I
var(^2) = 2/ X2
i (6.1.7) O
var(^2) = 2/ x2
i (3.3.1) I
^2 = (u^i)2/(n-1) (6.1.8) O
^2 = (u^i)2/(n-2) (3.3.5) I

108
Chapter 6
EXTENSIONS OF THE TWO-VARIABLE LINEAR
REGRESSION MODELS
 r2 for regression through-origin model
Raw r2 = (XiYi)2 /X2
i Y2
i (6.1.9)
 Note: Without very strong a priory expectation, well
advise is sticking to the conventional, intercept-
present model. If intercept equals to zero
statistically, for practical purposes we have a
regression through the origin. If in fact there is an
intercept in the model but we insist on fitting a
regression through the origin, we would be
committing a specification error

109
Chapter 6
 Illustrative Examples:
1) Capital Asset Pricing Model - CAPM (page 156)
2) Market Model (page 157)
3) The Characteristic Line of Portfolio Theory
(page 159)

110
Chapter 6
EXTENSIONS OF THE TWO-VARIABLE LINEAR
REGRESSION MODELS
6-2. Scaling and units of measurement
 Let Yi = ^1 + ^2Xi + u^ i (6.2.1)
 Define Y*i=w 1 Y i and X*i=w 2 X i then:
 *^2 = (w1/w2) ^2 (6.2.15)
 *^1 = w1^1 (6.2.16)
 *^2
= w1
2^2 (6.2.17)
 Var(*^1) = w2
1 Var(^1) (6.2.18)
 Var(*^2) = (w1/w2)2 Var(^2) (6.2.19)
 r2
xy = r2
x*y* (6.2.20)

111
Chapter 6
6-2. Scaling and units of measurement
 From one scale of measurement, one can derive the results
based on another scale of measurement. If w1= w2 the
intercept and standard error are both multiplied by w1. If
w2=1 and scale of Y changed by w1, then all coefficients and
standard errors are all multiplied by w1. If w1=1 and scale of
X changed by w2, then only slope coefficient and its standard
error are multiplied by 1/w2. Transformation from (Y,X) to
(Y*,X*) scale does not affect the properties of OLS
Estimators
 A numerical example: (pages 161, 163-165)

112
6-3. Functional form of regression model
 The log-linear model
 Semi-log model
 Reciprocal model

113
6-4. How to measure elasticity
The log-linear model
 Exponential regression model:
 Yi= 1Xi
2 e u
i (6.4.1)
By taking log to the base e of both side:
 lnYi = ln1 +2lnXi + ui , by setting ln1 =  =>
 lnYi =  +2lnXi + ui (6.4.3)
(log-log, or double-log, or log-linear model)
This can be estimated by OLS by letting
 Y*i =  +2X*i + ui , where Y*i=lnYi, X*i=lnXi ;
2 measures the ELASTICITY of Y respect to X, that is,
percentage change in Y for a given (small) percentage
change in X.

114
6-4. How to measure elasticity
The log-linear model
The elasticity E of a variable Y with
respect to variable X is defined as:
E=dY/dX=(% change in Y)/(% change in X)
~ [(Y/Y) x 100] / [(X/X) x100]=
= (Y/X)x (X/Y) = slope x (X/Y)
 An illustrative example: The coffee
demand function (pages 167-168)

115
6-5. Semi-log model:
Log-lin and Lin-log Models
 How to measure the growth rate: The log-lin model
 Y t = Y0 (1+r) t (6.5.1)
 lnYt = lnY0 + t ln(1+r) (6.5.2)
 lnYt = 1 + 2t , called constant growth model (6.5.5)
where 1 = lnY0 ; 2 = ln(1+r)
 lnYt = 1 + 2t + ui (6.5.6)
 It is Semi-log model, or log-lin model. The slope
coefficient measures the constant proportional or
relative change in Y for a given absolute change in the
value of the regressor (t)
 2 = (Relative change in regressand)/(Absolute change
in regressor) (6.5.7)

116
 Instantaneous Vs. compound rate of growth
 2 is instantaneous rate of growth
 antilog(2) – 1 is compound rate of growth
The linear trend model
 Yt = 1 + 2t + ut (6.5.9)
 If 2 > 0, there is an upward trend in Y
 If 2 < 0, there is an downward trend in Y
 Note: (i) Cannot compare the r2 values of
models (6.5.5) and (6.5.9) because the
regressands in the two models are different,
(ii) Such models may be appropriate only if a
time series is stationary.

117
 The lin-log model:
 Yi = 1 +2lnXi + ui (6.5.11)
 2 = (Change in Y) / Change in lnX =
(Change in Y)/(Relative change in X) ~
(Y)/(X/X) (6.5.12)
 or Y = 2 (X/X) (6.5.13)
 That is, the absolute change in Y equal
to 2 times the relative change in X.

118
6-6. Reciprocal Models:
The reciprocal model:
 Yi = 1 + 2( 1/Xi ) + ui (6.5.14)
 As X increases definitely, the term
2( 1/Xi ) approaches to zero and Yi
approaches the limiting or asymptotic value
1 (See figure 6.5 in page 174)
 An Illustrative example: The Phillips Curve
for the United Kingdom 1950-1966

119
6-7. Summary of Functional Forms
Table 6.5 (page 178)
Model Equation Slope =
dY/dX
Elasticity =
(dY/dX).(X/Y)
Linear Y = 1 + 2 X 2 2(X/Y) */
Log-linear
(log-log)
lnY = 1 + 2 lnX 2 (Y/X) 2
Log-lin lnY = 1 + 2 X 2 (Y) 2 X */
Lin-log Y = 1 + 2 lnX 2(1/X) 2 (1/Y) */
Reciprocal Y = 1 + 2 (1/X) - 2(1/X2) - 2 (1/XY) */

120
6-7. Summary of Functional Forms
 Note: */ indicates that the elasticity
coefficient is variable, depending on the
value taken by X or Y or both. when no X
and Y values are specified, in practice, very
often these elasticities are measured at the
mean values E(X) and E(Y).
-----------------------------------------------
6-8. A note on the stochastic error term
6-9. Summary and conclusions
(pages 179-180)

121
Basic Econometrics
Chapter 7
MULTIPLE REGRESSION
ANALYSIS:
The Problem of Estimation

122
7-1. The three-Variable Model:
Notation and Assumptions
 Yi = ß1+ ß2X2i + ß3X3i + u i (7.1.1)
 ß2 , ß3 are partial regression coefficients
 With the following assumptions:
+ Zero mean value of U i:: E(u i|X2i,X3i) = 0. i (7.1.2)
+ No serial correlation: Cov(ui,uj) = 0, i # j (7.1.3)
+ Homoscedasticity: Var(u i) = 2 (7.1.4)
+ Cov(ui,X2i) = Cov(ui,X3i) = 0 (7.1.5)
+ No specification bias or model correct specified (7.1.6)
+ No exact collinearity between X variables (7.1.7)
(no multicollinearity in the cases of more explanatory
vars. If there is linear relationship exits, X vars. Are said
to be linearly dependent)
+ Model is linear in parameters

123
7-2. Interpretation of Multiple
Regression
 E(Yi| X2i ,X3i) = ß1+ ß2X2i + ß3X3i (7.2.1)
 (7.2.1) gives conditional mean or
expected value of Y conditional upon
the given or fixed value of the X2 and
X3

124
7-3. The meaning of partial
regression coefficients
 Yi= ß1+ ß2X2i + ß3X3 +….+ ßsXs+ ui
 ßk measures the change in the mean
value of Y per unit change in Xk,
holding the rest explanatory variables
constant. It gives the “direct” effect of
unit change in Xk on the E(Yi), net of Xj
(j # k)
 How to control the “true” effect of a
unit change in Xk on Y? (read pages
195-197)

125
7-4. OLS and ML estimation of the
partial regression coefficients
 This section (pages 197-201) provides:
1. The OLS estimators in the case of three-
variable regression
Yi= ß1+ ß2X2i + ß3X3+ ui
2. Variances and standard errors of OLS
estimators
3. 8 properties of OLS estimators (pp 199-201)
4. Understanding on ML estimators

126
7-5. The multiple coefficient of
determination R2 and the multiple
coefficient of correlation R
 This section provides:
1. Definition of R2 in the context of multiple
regression like r2 in the case of two-variable
regression
2. R = R2 is the coefficient of multiple
regression, it measures the degree of
association between Y and all the explanatory
variables jointly
3. Variance of a partial regression coefficient
Var(ß^k) = 2/ x2
k (1/(1-R2
k)) (7.5.6)
Where ß^k is the partial regression coefficient
of regressor Xk and R2
k is the R2 in the
regression of Xk on the rest regressors

127
7-6. Example 7.1: The
expectations-augmented Philips
Curve for the US (1970-1982)
 This section provides an
illustration for the ideas
introduced in the chapter
 Regression Model (7.6.1)
 Data set is in Table 7.1

128
7-7. Simple regression in the
context of multiple regression:
Introduction to specification bias
 This section provides an
understanding on “ Simple
regression in the context of
multiple regression”. It will cause
the specification bias which will be
discussed in Chapter 13

129
7-8. R2 and the Adjusted-R2
 R2 is a non-decreasing function of the number of
explanatory variables. An additional X variable will not
decrease R2
R2= ESS/TSS = 1- RSS/TSS = 1-u^2
I / y^2
i (7.8.1)
 This will make the wrong direction by adding more
irrelevant variables into the regression and give an idea
for an adjusted-R2 (R bar) by taking account of degree of
freedom
 R2
bar= 1- [ u^2
I /(n-k)] / [y^2
i /(n-1) ] , or (7.8.2)
R2
bar= 1- ^2 / S2
Y (S2
Y is sample variance of Y)
K= number of parameters including intercept term
– By substituting (7.8.1) into (7.8.2) we get
R2
bar = 1- (1-R2) (n-1)/(n- k) (7.8.4)
– For k > 1, R2
bar < R2 thus when number of X variables
increases R2
bar increases less than R2 and R2
bar can be
negative

130
 R2 is a non-decreasing function of the number of
explanatory variables. An additional X variable will not
decrease R2
R2= ESS/TSS = 1- RSS/TSS = 1-u^2
I / y^2
i (7.8.1)
 This will make the wrong direction by adding more
irrelevant variables into the regression and give an idea
for an adjusted-R2 (R bar) by taking account of degree of
freedom
 R2
bar= 1- [ u^2
I /(n-k)] / [y^2
i /(n-1) ] , or (7.8.2)
R2
bar= 1- ^2 / S2
Y (S2
Y is sample variance of Y)
K= number of parameters including intercept term
– By substituting (7.8.1) into (7.8.2) we get
R2
bar = 1- (1-R2) (n-1)/(n- k) (7.8.4)
– For k > 1, R2
bar < R2 thus when number of X variables
increases R2
bar increases less than R2 and R2
bar can be
negative

131
 Comparing Two R2 Values:
To compare, the size n and the dependent variable must
be the same
 Example 7-2: Coffee Demand Function Revisited (page
210)
 The “game” of maximizing adjusted-R2:
Choosing the model that gives the highest R2
bar may be
dangerous, for in regression our objective is not for that
but for obtaining the dependable estimates of the true
population regression coefficients and draw statistical
inferences about them
 Should be more concerned about the logical or
theoretical relevance of the explanatory variables to the
dependent variable and their statistical significance

132
7-9. Partial Correlation
Coefficients
 This section provides:
1. Explanation of simple and partial
correlation coefficients
2. Interpretation of simple and partial
correlation coefficients
(pages 211-214)

133
7-10. Example 7.3: The Cobb-
Douglas Production function
More on functional form
 Yi = 1X2
2i X3
3ieU
i (7.10.1)
By log-transform of this model:
 lnYi = ln1 + 2ln X2i + 3ln X3i +
Ui = 0 + 2ln X2i + 3ln X3i
+ Ui
(7.10.2)
Data set is in Table 7.3
Report of results is in page 216

134
7-11 Polynomial Regression
Models
 Yi = 0 + 1 Xi + 2 X2
i +…+ k Xk
i + Ui
(7.11.3)
 Example 7.4: Estimating the Total Cost
Function
 Data set is in Table 7.4
 Empirical results is in page 221
--------------------------------------------------------------
 7-12. Summary and Conclusions
(page 221)

135
Basic Econometrics
Chapter 8
MULTIPLE REGRESSION
ANALYSIS:
The Problem of Inference

136
Chapter 8
MULTIPLE REGRESSION ANALYSIS:
8-3. Hypothesis testing in multiple regression:
Testing hypotheses about an individual partial regression
coefficient
Testing the overall significance of the estimated multiple
regression model, that is, finding out if all the partial slope
coefficients are simultaneously equal to zero
Testing that two or more coefficients are equal to one
another
Testing that the partial regression coefficients satisfy
certain restrictions
Testing the stability of the estimated regression model
over time or in different cross-sectional units
Testing the functional form of regression models

137
Chapter 8
8-4. Hypothesis testing about individual partial
regression coefficients
With the assumption that u i ~ N(0,2) we can
use t-test to test a hypothesis about any
individual partial regression coefficient.
H0: 2 = 0
H1: 2  0
If the computed t value > critical t value at the
chosen level of significance, we may reject the
null hypothesis; otherwise, we may not reject it

138
Chapter 8
8-5. Testing the overall significance of a multiple
regression: The F-Test
For Yi = 1 + 2X2i + 3X3i + ........+ kXki + ui
 To test the hypothesis H0: 2 =3 =....= k= 0 (all
slope coefficients are simultaneously zero) versus H1: Not at
all slope coefficients are simultaneously zero,
compute
F=(ESS/df)/(RSS/df)=(ESS/(k-1))/(RSS/(n-k)) (8.5.7)
(k = total number of parameters to be estimated
including intercept)
 If F > F critical = F(k-1,n-k), reject H0
 Otherwise you do not reject it

139
Chapter 8
regression
 Alternatively, if the p-value of F obtained from
(8.5.7) is sufficiently low, one can reject H0
 An important relationship between R2 and F:
F=(ESS/(k-1))/(RSS/(n-k)) or
R2/(k-1)
F = ---------------- (8.5.1)
(1-R2) / (n-k)
( see prove on page 249)

140
Chapter 8
regression in terms of R2
For Yi = 1 + 2X2i + 3X3i + ........+ kXki + ui
 To test the hypothesis H0: 2 = 3 = .....= k = 0
(all slope coefficients are simultaneously zero)
versus H1: Not at all slope coefficients are
simultaneously zero, compute
 F = [R2/(k-1)] / [(1-R2) / (n-k)] (8.5.13) (k = total
number of parameters to be estimated including
intercept)
 If F > F critical = F , (k-1,n-k), reject H0

141
Chapter 8
regression
Alternatively, if the p-value of F obtained from
(8.5.13) is sufficiently low, one can reject H0
The “Incremental” or “Marginal”
contribution of an explanatory variable:
Let X is the new (additional) term in the
right hand of a regression. Under the usual
assumption of the normality of ui and the
HO:  = 0, it can be shown that the following
F ratio will follow the F distribution with
respectively degree of freedom

142
Chapter 8
regression
[R2
new - R2
old] / Df1
F com = ---------------------- (8.5.18)
[1 - R2
new] / Df2
Where Df1 = number of new regressors
Df2 = n – number of parameters in the
new model
R2
new is standing for coefficient of determination of the
new regression (by adding X);
R2
old is standing for coefficient of determination of the old
regression (before adding X).

143
Chapter 8
regression
Decision Rule:
If F com > F , Df1 , Df2 one can reject the Ho that  =
0 and conclude that the addition of X to the model
significantly increases ESS and hence the R2 value
 When to Add a New Variable? If |t| of coefficient
of X > 1 (or F= t 2 of that variable exceeds 1)
 When to Add a Group of Variables? If adding a
group of variables to the model will give F value
greater than 1;

144
Chapter 8
8-6. Testing the equality of two regression coefficients
Yi = 1 + 2X2i + 3X3i + 4X4i + ui (8.6.1)
Test the hypotheses:
H0: 3 = 4 or 3 - 4 = 0 (8.6.2)
H1: 3  4 or 3 - 4  0
Under the classical assumption it can be shown:
t = [(^3 - ^4) – (3 - 4)] / se(^3 - ^4)
follows the t distribution with (n-4) df because (8.6.1)
is a four-variable model or, more generally, with (n-k)
df. where k is the total number of parameters
estimated, including intercept term.
se(^3 - ^4) =  [var((^3) + var( ^4) – 2cov(^3, ^4)]

145
Chapter 8
t = (^3 - ^4) /  [var((^3) + var( ^4) – 2cov(^3, ^4)]
(8.6.5)
Steps for testing:
1. Estimate ^3 and ^4
2. Compute se(^3 - ^4) through (8.6.4)
3. Obtain t- ratio from (8.6.5) with H0: 3 = 4
4. If t-computed > t-critical at designated level of
significance for given df, then reject H0. Otherwise do
not reject it. Alternatively, if the p-value of t statistic
from (8.6.5) is reasonable low, one can reject H0.
 Example 8.2: The cubic cost function revisited

146
Chapter 8
8-7. Restricted least square: Testing linear
equality restrictions
Yi = 1X 2
2i X 3
3i eu
i (7.10.1) and (8.7.1)
Y = output
X2 = labor input
X3 = capital input
In the log-form:
lnYi = 0 + 2lnX2i + 3lnX3i + ui (8.7.2)
with the constant return to scale: 2 + 3 = 1
(8.7.3)

147
Chapter 8
8-7. Restricted least square: Testing linear
equality restrictions
How to test (8.7.3)
 The t Test approach (unrestricted): test of the
hypothesis H0: 2 + 3 = 1 can be conducted by t- test:
t = [(^2 + ^3) – (2 + 3)] / se(^2 - ^3) (8.7.4)
 The F Test approach (restricted least square -RLS):
Using, say, 2 = 1-3 and substitute it into (8.7.2) we get:
ln(Yi /X2i) = 0 + 3 ln(X3i /X2i) + ui (8.7.8). Where (Yi /X2i)
is output/labor ratio, and (X3i / X2i) is capital/labor ratio

148
Chapter 8
8-7. Restricted least square: Testing linear equality
restrictions
u^2
UR=RSSUR of unrestricted regression (8.7.2)
and  u^2
R = RSSR of restricted regression (8.7.7),
m = number of linear restrictions,
k = number of parameters in the unrestricted regression,
n = number of observations.
R2
UR and R2
R are R2 values obtained from unrestricted and
restricted regressions respectively. Then
F=[(RSSR – RSSUR)/m]/[RSSUR/(n-k)] =
= [(R2
UR – R2
R) / m] / [1 – R2
UR / (n-k)] (8.7.10)
follows F distribution with m, (n-k) df.
Decision rule: If F > F m, n-k , reject H0: 2 + 3 = 1

149
Chapter 8
8-7. Restricted least square: Testing linear equality
restrictions
 Note: R2
UR  R2
R (8.7.11)
 and  u^2
UR   u^2
R (8.7.12)
 Example 8.3: The Cobb-Douglas Production
function for Taiwanese Agricultural Sector,
1958-1972. (pages 259-260). Data in Table 7.3
(page 216)
 General F Testing (page 260)
 Example 8.4: The demand for chicken in the US,
1960-1982. Data in exercise 7.23 (page 228)

150
Chapter 8
8-8. Comparing two regressions: Testing for structural
stability of regression models
Table 8.8: Personal savings and income data, UK, 1946-
1963 (millions of pounds)
Savings function:
 Reconstruction period:
Y t = 1+ 2X t + U1t (t = 1,2,...,n1)
 Post-Reconstruction period:
Y t = 1 + 2X t + U2t (t = 1,2,...,n2)
Where Y is personal savings, X is personal income, the
us are disturbance terms in the two equations and n1, n2
are the number of observations in the two period

151
Chapter 8
+ The structural change may mean that the two
intercept are different, or the two slopes are different,
or both are different, or any other suitable
combination of the parameters. If there is no structural
change we can combine all the n1, n2 and just estimate
one savings function as:
Y t = l1 + l2X t + Ut (t = 1,2,...,n1, 1,....n2). (8.8.3)
How do we find out whether there is a structural
change in the savings-income relationship between the
two period? A popular test is Chow-Test, it is simply
the F Test discussed earlier
HO: i = i i Vs H1: i that i  i

152
Chapter 8
+ The assumptions underlying the Chow test
u1t and u2t ~ N(0,s2), two error terms are normally
distributed with the same variance
u1t and u2t are independently distributed
Step 1: Estimate (8.8.3), get RSS, say, S1 with df =
(n1+n2 – k); k is number of parameters estimated )
Step 2: Estimate (8.8.1) and (8.8.2) individually
and get their RSS, say, S2 and S3 , with df = (n1 –
k) and (n2-k) respectively. Call S4 = S2+S3; with df
= (n1+n2 – 2k)
Step 3: S5 = S1 – S4;

153
Chapter 8
Step 4: Given the assumptions of the Chow Test,
it can be show that
F = [S5 / k] / [S4 / (n1+n2 – 2k)] (8.8.4)
follows the F distribution with Df = (k, n1+n2 – 2k)
Decision Rule: If F computed by (8.8.4) > F-
critical at the chosen level of significance a =>
reject the hypothesis that the regression (8.8.1)
and (8.8.2) are the same, or reject the hypothesis
of structural stability; One can use p-value of the
F obtained from (8.8.4) to reject H0 if p-value low
reasonably.

154
Chapter 8
8-9. Testing the functional form of
regression:
Choosing between linear and log-linear
regression models: MWD Test
(MacKinnon, White and Davidson)
H0: Linear Model Y is a linear function of
regressors, the Xs;
H1: Log-linear Model Y is a linear function
of logs of regressors, the lnXs;

155
Chapter 8
8-9. Testing the functional form of regression:
Step 1: Estimate the linear model and obtain the
estimated Y values. Call them Yf (i.e.,Y^). Take
lnYf.
Step 2: Estimate the log-linear model and obtain
the estimated lnY values, call them lnf (i.e., ln^Y )
Step 3: Obtain Z1 = (lnYf – lnf)
Step 4: Regress Y on Xs and Z1. Reject H0 if the
coefficient of Z1 is statistically significant, by the
usual t - test
Step 5: Obtain Z2 = antilog of (lnf – Yf)
Step 6: Regress lnY on lnXs and Z2. Reject H1 if

156
Chapter 8
Example 8.5: The demand for Roses (page 266-
267). Data in exercise 7.20 (page 225)
8-10. Prediction with multiple regression
Follow the section 5-10 and the illustration in
pages 267-268 by using data set in the Table 8.1
(page 241)
8-11. The troika of hypothesis tests: The
likelihood ratio (LR), Wald (W) and Lagarange
Multiplier (LM) Tests

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