ECONOMETRICS

Chrispin Mphuka

University of Zambia

January 2011

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 1 / 15

Assumptions

Generic form of the linear regression model : population regression h

y = f (x1 , x2 , ..., xk ) + ε
= x1 β1 + x2 β2 + ...xk βk + ε

Classical regression model is a set of joint distributions satisfying

Assumptions 1.1 - 1.4

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 2 / 15

Classical Linear Regression Assumptions
Linearity Assunption.

yi = xi 1 β1 + xi 2 β2 + ...xik βk + εi (i = 1, 2, 3, ..., n) (1)

0
where β s are unkown parameters to be estimated, and εi is the
unobserved error term with certain properties to be speci…ed below

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 3 / 15

Classical Linear Regression Assumptions
Linearity Assunption.

yi = xi 1 β1 + xi 2 β2 + ...xik βk + εi (i = 1, 2, 3, ..., n) (1)

0
where β s are unkown parameters to be estimated, and εi is the
unobserved error term with certain properties to be speci…ed below
the RHS: xi 1 β1 + xi 2 β2 + ...xik βk is called the regression function

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 3 / 15

Classical Linear Regression Assumptions
Linearity Assunption.

yi = xi 1 β1 + xi 2 β2 + ...xik βk + εi (i = 1, 2, 3, ..., n) (1)

0
where β s are unkown parameters to be estimated, and εi is the
unobserved error term with certain properties to be speci…ed below
the RHS: xi 1 β1 + xi 2 β2 + ...xik βk is called the regression function
β0 s are regression coe¢ cientst that represent marginal and separate
e¤ects of regressors

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 3 / 15

Classical Linear Regression Assumptions
Linearity Assunption.

yi = xi 1 β1 + xi 2 β2 + ...xik βk + εi (i = 1, 2, 3, ..., n) (1)

0
where β s are unkown parameters to be estimated, and εi is the
unobserved error term with certain properties to be speci…ed below
the RHS: xi 1 β1 + xi 2 β2 + ...xik βk is called the regression function
β0 s are regression coe¢ cientst that represent marginal and separate
e¤ects of regressors
e.g , β2 represents the the change in the dependent variable when the
second regressor increases by 1 unit holding constant other variables.
i.e ∂yi /∂xi 2 = β2

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 3 / 15

Classical Linear Regression Assumptions
Linearity Assunption.

yi = xi 1 β1 + xi 2 β2 + ...xik βk + εi (i = 1, 2, 3, ..., n) (1)

0
where β s are unkown parameters to be estimated, and εi is the
unobserved error term with certain properties to be speci…ed below
the RHS: xi 1 β1 + xi 2 β2 + ...xik βk is called the regression function
β0 s are regression coe¢ cientst that represent marginal and separate
e¤ects of regressors
e.g , β2 represents the the change in the dependent variable when the
second regressor increases by 1 unit holding constant other variables.
i.e ∂yi /∂xi 2 = β2
Linearity implies that the marginal e¤ects do not depend on the level of
regressors

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 3 / 15

Classical Linear Regression Assumptions
Linearity Assunption.

yi = xi 1 β1 + xi 2 β2 + ...xik βk + εi (i = 1, 2, 3, ..., n) (1)

0
where β s are unkown parameters to be estimated, and εi is the
unobserved error term with certain properties to be speci…ed below
the RHS: xi 1 β1 + xi 2 β2 + ...xik βk is called the regression function
β0 s are regression coe¢ cientst that represent marginal and separate
e¤ects of regressors
e.g , β2 represents the the change in the dependent variable when the
second regressor increases by 1 unit holding constant other variables.
i.e ∂yi /∂xi 2 = β2
Linearity implies that the marginal e¤ects do not depend on the level of
regressors
Example: (Wage Equation)

log(WAGEi ) = β1 + β2 Si + β3 TENUREi + β4 EXPRi + εi

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 3 / 15

Classical Linear Regression Assumptions

2 3 2 3
xi 1 β1
6 xi 2 7 6 β2 7
6 7 6 7
6 . 7 6 . 7
Xi = 6
6
7 ,
7 β =6
6
7
7
6 . 7 6 . 7
4 . 5 4 . 5
xik βk

thus Xi0 β =xi 1 β1 + xi 2 β2 + ...xik βk and

yi = Xi0 β + εi , (i = 1, 2, ..., n)

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 4 / 15

Classical Linear Regression Assumptions

Also de…ned

2 3 2 3 2 3 2 3
y1 ε1 X10 x11 . . . x1K
6 . 7 6 . 7 6 . 7 6 . 7
6 7 6 7 6 7 6 7
y =6
6 . 7,
7 ε=6
6 . 7,
7 X =6
6 . 7=6
7 6 . 7
7
4 . 5 4 . 5 4 . 5 4 . . . . 5
yn εn Xn0 xn1 . . . xnK

Therefore assumption1.1 can be written as:

y = Xβ + ε

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 5 / 15

Classical Linear Regression Assumptions

Assumption 1.2 (Strict Exogeneity)

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 6 / 15

Classical Linear Regression Assumptions

Assumption 1.2 (Strict Exogeneity)

E ( εi jX) = 0 (i = 1, 2, ..., n)

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 6 / 15

Classical Linear Regression Assumptions

Assumption 1.2 (Strict Exogeneity)

E ( εi jX) = 0 (i = 1, 2, ..., n)
di¤erently put implies E (εi jX1 , ..., Xn ) = 0 (i = 1, 2, ..., n)

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 6 / 15

Classical Linear Regression Assumptions

Assumption 1.2 (Strict Exogeneity)

E ( εi jX) = 0 (i = 1, 2, ..., n)
di¤erently put implies E (εi jX1 , ..., Xn ) = 0 (i = 1, 2, ..., n)
Implications of Strict Exogeneighty

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 6 / 15

Classical Linear Regression Assumptions

Assumption 1.2 (Strict Exogeneity)

E ( εi jX) = 0 (i = 1, 2, ..., n)
di¤erently put implies E (εi jX1 , ..., Xn ) = 0 (i = 1, 2, ..., n)
Implications of Strict Exogeneighty
The unconditional mean if the error term is 0 i.e E ( εi ) = 0
(i = 1, 2, ..., n)

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 6 / 15

Classical Linear Regression Assumptions

Assumption 1.2 (Strict Exogeneity)

E ( εi jX) = 0 (i = 1, 2, ..., n)
di¤erently put implies E (εi jX1 , ..., Xn ) = 0 (i = 1, 2, ..., n)
Implications of Strict Exogeneighty
The unconditional mean if the error term is 0 i.e E (εi ) = 0
(i = 1, 2, ..., n)
If cross moments E (xy ) = 0 =) x is orthogonal to y. Under strict
exogeneity the regressors are orthogonal to the error terms.
i.e., E xjk εi = 0 (j, k = 1, ..., n; k = 1, ..., K )

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 6 / 15

Classical Linear Regression Assumptions

Assumption 1.2 (Strict Exogeneity)

E ( εi jX) = 0 (i = 1, 2, ..., n)
di¤erently put implies E (εi jX1 , ..., Xn ) = 0 (i = 1, 2, ..., n)
Implications of Strict Exogeneighty
The unconditional mean if the error term is 0 i.e E (εi ) = 0
(i = 1, 2, ..., n)
If cross moments E (xy ) = 0 =) x is orthogonal to y. Under strict
exogeneity the regressors are orthogonal to the error terms.
i.e., E xjk εi = 0 (j, k = 1, ..., n; k = 1, ..., K )
Strict exogeneity resquires that the regressors be orthorgonal not only
to the error term from the same observation but also to the error
term from other observations.

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 6 / 15

Classical Linear Regression Assumptions

Assumption 1.3: No multicollinearity

- the rank of the nXK data matrix, X, is K with probability 1.
-implies X is a full column rank
-implies that n > k i.e., there must be at least as many observations
as variables.

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 7 / 15

Classical Linear Regression Assumptions

Assumption 1.3: No multicollinearity

- the rank of the nXK data matrix, X, is K with probability 1.
-implies X is a full column rank
-implies that n > k i.e., there must be at least as many observations
as variables.
Assumption 1.4: Spherical error variance
- (homoskedasticity) E ε2i jX = σ2 > 0 (i = 1, 2, ..., n)
2
Var ε2i jX = E ε2i jX E (εi jX) = E ε2i jX , since E (εi jX) = 0
by assumption

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 7 / 15

Classical Linear Regression Assumptions

Assumption 1.3: No multicollinearity

- the rank of the nXK data matrix, X, is K with probability 1.
-implies X is a full column rank
-implies that n > k i.e., there must be at least as many observations
as variables.
Assumption 1.4: Spherical error variance
- (homoskedasticity) E ε2i jX = σ2 > 0 (i = 1, 2, ..., n)
2
Var ε2i jX = E ε2i jX E (εi jX) = E ε2i jX , since E (εi jX) = 0
by assumption
The homoskedaisticity assumption says that the conditional second
moment is a constant
-(no correlation between errors) E (εi εj jX) = 0
(i = 1, 2, ..., n; i 6= j )

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 7 / 15

Classical Linear Regression Assumptions

Assumption 1.3: No multicollinearity

- the rank of the nXK data matrix, X, is K with probability 1.
-implies X is a full column rank
-implies that n > k i.e., there must be at least as many observations
as variables.
Assumption 1.4: Spherical error variance
- (homoskedasticity) E ε2i jX = σ2 > 0 (i = 1, 2, ..., n)
2
Var ε2i jX = E ε2i jX E (εi jX) = E ε2i jX , since E (εi jX) = 0
by assumption
The homoskedaisticity assumption says that the conditional second
moment is a constant
-(no correlation between errors) E (εi εj jX) = 0
(i = 1, 2, ..., n; i 6= j )
The two assumptions combined =) E (εε0 jX) = σ2 In or equivalently
Var (εε0 jX) = σ2 In
Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 7 / 15
Classical Linear Regression Assumptions

Random Sample
-Assume (yi ,Xi ) is a random sample =) (yi ,Xi ) is iid accros
distributions

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 8 / 15

Classical Linear Regression Assumptions

Random Sample
-Assume (yi ,Xi ) is a random sample =) (yi ,Xi ) is iid accros
distributions
Then
-E (εi jX) = E (εi jxi )
-E ε2i jX = E ε2i jxi

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 8 / 15

Classical Linear Regression Assumptions

Random Sample
-Assume (yi ,Xi ) is a random sample =) (yi ,Xi ) is iid accros
distributions
Then
-E (εi jX) = E (εi jxi )
-E ε2i jX = E ε2i jxi
and E εi εj jX = E (εi jxi ) E εj jxj (i 6 = j )

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 8 / 15

Algebra of Least Squares

Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 9 / 15

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Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 10 / 15

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Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 11 / 15

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Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 12 / 15

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Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 13 / 15

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Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 14 / 15

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Lecture 1: Classical Regression Model Notes

2011-01-25 The Classical Multiple Linear Regression Model

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Chrispin Mphuka (Institute) Lecture 1: Classical Regression Model 01/2010 15 / 15