(4.3a)
Expand and multiply top and bottom by n:
b2
=
nSxiyi - Sxi Syi nSxi2-(Sxi) 2
(4.3b)
Variance of b2
4.12
Given that both yi and ei have variance s2,
the variance of the estimator b2 is:
4. cov(ei,ej) = cov(yi,yj) = 0 5. xt c for every observation
6. et~N(0,s 2) <=> yt~N(b1+ b2xt,
The population parameters b1 and b2 4.4 are unknown population constants.
b2
+
nSxiEei - Sxi SEei nSxi2-(Sxi) 2
Since Eei = 0, then Eb2 = b2 .
An Unbiased Estimator
4.8
The result Eb2 = b2 means that the distribution of b2 is centered at b2.
4.6
The Expected Values of b1 and b2
The least squares formulas (estimators) in the simple regression case:
b2 =
nSxiyi - Sxi Syi nSxi22 -(Sxi) 2
b1 = y - b2x
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 )
shown to be an unbiased estimator of b1
when the model is correctly specified.
Eb1 = b1
4.11
Equivalent expressions for b2:
b2
=
S(xi - x )(yi S(xi - x )2
y
)
Since the distribution of b2 is centered at b2 ,we say that b2 is an unbiased estimator of b2.
Wrong Model Specification 4.9
The unbiasedness result on the previous slide assumes that we are using the correct model.
(4.1a) (4.1b)
where y = Syi yi = b1 + b2xi + ei
4.7
to get:
b2 =
b2
+
nSxiei - Sxi Sei nSxi2-(Sxi) 2
The mean of b2 is:
Eb2 =
• 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
yt = b1 + b2 xt + et yt = household weekly food expenditures xt = household weekly income
For a given level of xt, the expected
level of food expenditures will be:
The formulas that produce the
sample estimates b1 and called the estimators of
b2 are
b1 and
b2.
When b0 and b1 are used to represent the formulas rather than specific values,
E(yt|xt) = b1 + b2 xt
Assumptions of the Simple 4.3 Linear Regression Model
1. yt = b1 + b2xt + et 2. E(et) = 0 <=> E(yt) = b1 +
b2x t
3. var(et) = s 2 = var(yt)
Chapter 4 Statistical Properties of the OLS
Estimators
Xi’An Institute of Post & Telecommunication Dept of Economic & Management Prof. Long
Simple Linear Regression Model 4.2
var(b2) =
s2
S(xi - x)2
b2 is a function of the yi values but var(b2) does not involve yi directly.
If the model is of the wrong form or is missing important variables,
then Eei = 0, then Eb2 = b2 .
4.10
Unbiased Estimator of the Intercept
In a similar manner, the estimator b1 of the intercept or constant term can be