Ols introduction

Econometrics II
Part I
The Classical Linear
Regression Model (CLRM) and the OLS Estimator
17/02/2010

Notation y1 y

y2 yn column vector containing the n sample observations on the dependent variable y.

(1)

x 1k xk 

x 2k x nk

column vector containing the n sample observations on the independent variable x k , with k  1, 2, . . . , K.

(2)

x1 x2

xK

X

n  K data matrix containing the n sample observations on the K independent variables. Usually the vector x 1 is assumed to be a column of 1s (constant).

(3)

1


2
n

column vector containing the n disturbances.

(4)

Assumptions of the CLRM
Assumption 1: linearity
Observed data are generated by the following linear model
′
y i   1 x i1   2 x i2 . . .  K x iK   i  x i    i i  1, 2, . . . , n.

(5)

The K unknown parameters of the model can be collected in a column vector, 
1


2

(6)

K and the model can be rewritten in compact form: y  x 1  1  x 2  2 . . . x K  K   y  X  

(7)

Assumption 2: the strict exogeneity assumption
The expected value of each disturbance,  i conditional on all observations is zero
E i |X  0

(8)

i  1, 2, . . . , n
In compact form:
E|X  0

(9)

First implication of strict exogeneity
The unconditional mean is also zero. In fact, by the Law of Total
Expectations:
(10)
E i   E X E i |X  0 i  1, 2, . . . , n

Second implication of strict exogeneity
The regressors are orthogonal to the error term for all observations: E i x jk   0 i, j  1, 2, . . . , n; k  1, 2, . . . , K

(11)

Third implication of strict exogeneity
The orthogonality conditions are equivalent to zero-correlation conditions Covx jk ,  i   Ex jk  i  − Ex jk E i   0

(12)

Assumption 3: no (perfect) multicollinearity
The rank of the n  K matrix, X, is K with probability 1.
This implies that X has full column rank; the