文档介绍：Chapter 4 Statistical Properties of the OLS Estimators
Xi’An Institute of Post & munication
Dept of Economic & Management
Prof. Long
yt = household weekly food expenditures
Simple Linear Regression Model
yt = b1 + b2 x t + e t
x t = household weekly e
For a given level of x t, the expected
level of food expenditures will be:
E(yt|x t) = b1 + b2 x t
1. yt = b1 + b2x t + e t
2. E(e t) = 0 <=> E(yt) = b1 + b2x t
3. var(e t) = s 2 = var(yt)
4. cov(e i,e j) = cov(yi,yj) = 0
5. x t c for every observation
6. e t~N(0,s 2) <=> yt~N(b1+ b2x t, s 2)
Assumptions of the Simple
Linear Regression Model
The population parameters b1 and b2 are unknown population constants.
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 sample.
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 least squares estimators.
Estimators are Random Variables
( estimates are not )
The Expected Values of b1 and b2
The least squares formulas (estimators)
in the simple regression case:
b2 =
nSxiyi - Sxi Syi
nSxi2 -(Sxi)
2
2
b1 = y - b2x
where y = Syi / n and x = Sx i / n
()
()
Substitute in yi = b1 + b2xi + e i
to get:
b2 = b2 +
nSxiei - Sxi Sei
nSxi -(Sxi)
2
2
The mean of b2 is:
Eb2 = b2 +
nSxiEei - Sxi SEei
nSxi -(Sxi)
2
2
Since Eei = 0, then Eb2 = b2 .
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.
An Unbiased Estimator
The unbiasedness result on the
previous slide assumes that we
are using the co