06Econometrics_Statistics_Basic_1-8.ppt

Basic Econometrics
Introduction:
What is Econometrics?
2

Introduction
 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)
3

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)
4

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)
5

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
6

7
Econometrics
Economic
Theory
Mathematical
Economics
Economic
Statistics
Mathematic
Statistics

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
8

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
9

10
Econometrics
Economic
Theory
Mathematical
Economics
Economic
Statistics
Mathematic
Statistics

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”
11

Introduction
(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
12

Introduction
(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
13

Introduction
(4) Obtaining Data
(See Table 1.1, page 6)
Y= Personal consumption
expenditure
X= Gross Domestic Product
all in Billion US Dollars
14

Introduction
(4) Obtaining Data
May 2004 Prof.VuThieu 15
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

Introduction
(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

Introduction
(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)

Introduction
(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

Introduction
(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

Introduction
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

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

Basic Econometrics
Chapter 1:
THE NATURE OF REGRESSION ANALYSIS

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”)

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”

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)

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

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)

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”)

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)

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)

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

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)

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

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

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)

• 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 )

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 ($)

• 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)

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

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

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

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:

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

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
--------------------

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

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)

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

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

• 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.

• 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.

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

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:

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

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 - ]

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

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

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

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

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

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

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

3-5. The coefficient of determination r2: A
measure of “Goodness of fit”
 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

3-5. The coefficient of determination r2: A
measure of “Goodness of fit”
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)

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

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

• 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

• 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)

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

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)

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]

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

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

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

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

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

•(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

•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)

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)

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)

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

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 > *

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)

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)

• 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

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 =>

• 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

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

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 ]

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

•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

• 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

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:

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)

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

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

• 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

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

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

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.

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

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, but

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.

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

(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.

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

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

Chapter 6
REGRESSION MODELS
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

Chapter 6
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

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

Chapter 6
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)

Chapter 6
REGRESSION MODELS
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)

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

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.

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)

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)

 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.

 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.

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

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) */

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)

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

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 Ui:: 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

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

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)

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

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

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

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

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 thuswhen number of X variables increases
R2
bar increases less than R2 and R2
bar can be negative

• 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 thuswhen number of X variables increases
R2
bar increases less than R2 and R2
bar can be negative

• 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

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)

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

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)

Basic Econometrics
Chapter 8
The Problem of Inference

Chapter 8
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

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

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

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)

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

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

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).

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;

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)]
(8.6.4)
(see appendix)

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

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)

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

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

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)

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

Chapter 8
8-8. Comparing two regressions: Testing for structural stability
of regression models
+ 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

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;

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.
+ Apply for the data in Table 8.8

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;

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 the
coefficient of Z2 is statistically significant, by the
usual t-test

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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