Tài liệu MULTIPLE LINEAR REGRESSION MODEL Introduction and Estimation ppt

Fulbright Economics Teaching Program Analytical Methods Lecture notes 7
Lecture 7
MULTIPLE LINEAR REGRESSION MODEL
Introduction and Estimation
1) Introduction to the multiple linear regression model
The simple linear regression model cannot explain everything.
So far, we have considered the simple linear regression model. In both theory and
practice, there are many cases in which a given economic variable cannot be
explained by such the simple regression model. We can offer the following
examples : -
Quantity demanded depends on price, income, and the prices of other
goods, etc.
Recall consumer behaviour theory.
Q
D
= f(P, I, P
s
, P
c
, Market size,P
f
(expected price), T (preference))
Output depends on price, primary inputs, intermediate inputs, technology,
etc. Recall production function theory : - Q
S
=f(K,L, TECH)
Nguyen Trong Hoai 1 12/20/13
Fulbright Economics Teaching Program Analytical Methods Lecture notes 7
the growth rate of an economy depends on investment, labour,
technological change, etc. Recall the total factor productivity theory : -
Wages depends on education, experience, gender, age, etc.

House prices depends on size, the number of bedrooms and bathroom, etc.
Nguyen Trong Hoai 2 12/20/13
Fulbright Economics Teaching Program Analytical Methods Lecture notes 7
Household expenditure on food depends on the household size, the
income, the location, etc.
National children mortality rates depends on the income per capita,
eduction, etc.
Nguyen Trong Hoai 3 12/20/13
Fulbright Economics Teaching Program Analytical Methods Lecture notes 7
The demand for money depends on the rate of interest, the price, the GDP
in the economy; etc.

When we collect data of some economic variables (called dependent
variables) and its determinants (called explanatory variables), studies of
separate influences (direct or net) of the various factors on an economic
variable can be explained by the multiple regression model.
2) Data requirement
Some data is expressed in terms of a spreadsheet as above mentioned.
3) Population Regression Function-PRF
Study the model : -
iK33221

εββββ
+++++=
Kiiii
XXXY 
PRF
[ ] [ ]
sX' | E sX' | E
iK33221

εββββ
+++++=
Kiiii
XXXY 

The
β
-coefficients are called the partial regression coefficients and each one has
the following interpretation : -
Nguyen Trong Hoai 4 12/20/13
Fulbright Economics Teaching Program Analytical Methods Lecture notes 7
[ ]
k
i

sX' | Y
β
=
∂
∂
k
X
E
4) Important assumptions of the multiple linear regression model
PRF consists of two components : a controlled part and a stochastic part
(stochastic disturbance - random disturbance). ε
i
is a random variable and follows
normal a distribution ε
i

≈ N(0, σ
2
), X’s are controlled variables or given variables.
Since Y
i
is the sum of such two parts, Y
i
is also a random variable.
4.1 OLS assumptions in a simple regression model are interpreted in a
multiple regression model : these assumptions relate to stochastic disturbance
(ε
i
)
a) Mean value of ε
i
is zero => E(ε
i
| X’s) = 0
b) No serial correlation (autocorrelation) => cov(εi, εj| X’s ) = 0 vôùi i # j
c) Homoscedasticity => var(ε
i
) = σ
2
d) Random disturbance has no correlation with Xs => cov(ε
i
, X
ki
) = 0 (k: number
of explanatory variables in the model)
e) No error model specification

4.2 Additional assumptions of OLS for multiple regression model
Regressors do not perfectly satisfy any linear relationship (perfect multi-
collinearity). That is, there is no set of coefficients for which the following
expression is always true : -
0 1
K3322
=++++
Kiii
XXX
λλλ

We will explain this condition clearly by the two-explanatory variable (two
regressor) model. We temporarily accept this assumption.
5) Sample Regression Function-SRF
We address the estimation problem by specifying the sample regression function
(SRF) : -

ˆ

ˆ

ˆ

ˆ

ˆ
K33221 Kiiii
XXXY
ββββ
++++=


The residuals are defined in just the same way as they were defined in the simple
regression framework : -
ii
Y
ˆ
- Y
=
i
e
6) Ordinary Least Squares Estimators - OLS
By definition, we can invoke the ordinary least squares principle to choose the
estimators of partial regression coefficients.
Nguyen Trong Hoai 5 12/20/13