计量经济学(英文版).

Chapter 4 Statistical Properties of the OLS Estimators
Xi’An Institute of Post & Telecommunication Dept of Economic & Management Prof. Long
Simple Linear Regression Model y t = b1 + b 2 x t + e t
b1 + b2 x t
Assumptions of the Simple Linear Regression Model yt = b1 + b2x t + e t 2. E(e t) = 0 <=> E(yt) = b1 + b2x t
1.
3. var(e t)
4.3
=
4.
5.
cov(e i,e j)
x t c for every observation
= cov(yi,yj)
s 2 = var(yt)
= 0
6.
e t~N(0,s 2) <=> yt~N(b1+ b2x t,
The population parameters b1 and b2 are unknown population constants.
4.2
yt = household weekly food expenditures
x t = household weekly income
For a given level of x t, the expected level of food expenditures will be: E(yt|x t) =
Eb1 = b1
4.11
Equivalent expressions for b2:
S(xi - x )(yi - y ) b2 = 2 S(xi - x )
(4.3a)
Expand and by n:
b2 =
nSxiyi - Sxi Syi nSxi -(Sxi)
2 2
(4.3b)
Variance of b2
4.12
Given that both yi and ei have variance s 2, the variance of the estimator b2 is:
var(b2) =
S(x i - x)
s2
2
4.4
The formulas that produce the sample estimates b1 and b2 are called the estimators of b1 and
b2.
When b0 and b1 are used to represent the formulas rather than specific values, they are called estimators of b1 and b2 which are random variables because they are different from sample to
4.5
Estimators are Random Variables ( estimates are not )
• If the least squares estimators b0 and b1 are random variables, then what are their means, variances, covariances and probability distributions? • Compare the properties of alternative estimators to the properties of the
b1 = y - b2x
where
y = Syi / n and x = Sx i / n
Substitute in to get:
yi = b1 + b2xi + e i
4.7
nSxiei - Sxi Sei b2 = b2 + 2 2 nSxi -(Sxi)
The mean of b2 is:
4.9
4.10
Unbiased Estimator of the Intercept
In a similar manner, the estimator b1 of the intercept or constant term can be shown to be an unbiased estimator of b1 when the model is correctly specified.
4.6
The Expected Values of b1 and b2
The least squares formulas (estimators) in the simple regression case:
b2 =
nSxiyi - Sxi Syi
2 2 nSxi
-(Sxi)
2
(4.1a) (4.1b)
nSxiEei - Sxi SEei Eb2 = b2 + 2 2 nSxi -(Sxi)
Eei = 0, then Eb2 = b2 .
Since
An Unbiased Estimator
4.8
The result Eb2 = b2 means that the distribution of b2 is centered at b2.
Since the distribution of b2
is centered at b2 ,we say that b2 is an unbiased estimator of b2.
Wrong Model Specification
The unbiasedness result on the previous slide assumes that we are using the correct model. If the model is of the wrong form or is missing important variables, then Eei = 0, then Eb2 = b2 .