(4.1a) (4.1b)
where y = Syi / n and. x = Sxi / n 6
Substitute in yi = b1 + b2xi + ei
4.7
to get:
b2 =
b2
+
nSxiei - Sxi Sei nSxi2-(Sxi) 2
The mean of b2 is:
Eb2 =
the variance of the estimator b2 is:
var(b2) =
s2
S(xi - x)2
b2 is a function of the yi values but
they are called estimators of b1 and b2
which are random variables because they are different fr.om sample to 4
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 prop. erties of the 5
Since the distribution of b2 is centered at b2 ,we say that b2 is an unbiased e. stimator of b2. 8
Wrong Model Specification 4.9
The unbiasedness result on the previous slide assumes that we are using the correct model.
4.6
The Expected Values of b1 and b2
The least squares formulas (estimators) in the simple regression nSxi22 -(Sxi) 2
b1 = y - b2x
If the model is of the wrong form
or is missing important variables,
then Eei = 0, then Eb2 = b2 .
.
9
4.10
Unbiased Estimator of the Intercept
In a similar manner, the estimator b1 of the intercept or constant term can be
b2
+
nSxiEei - Sxi SEei nSxi2-(Sxi) 2
Since Eei = 0, then Eb2 = b2 .
.
7
An Unbiased Estimator
4.8
The result Eb2 = b2 means that the distribution of b2 is centered at b2.
Chapter 4 Statistical Properties of the OLS
Estimators
Xi’An Institute of Post & Telecommunication Dept of Economic & Management Prof. Long
.
1
Simple Linear Regression Model 4.2
shown to be an unbiased estimator of b1
when the model is correctly specified.
Eb1 = b1
.
10
4.11
Equivalent expressions for b2:
b2
=
S(xi - x )(yi S(xi - x )2
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,
y
)
(4.3a)
Expand and multiply top and bottom by n:
b2 =
nSxiyi - Sxi Syi nSxi2-(S.xi) 2
(4.3b)
11
Variance of b2
4.12
Given that both yi and ei have variance s2,
E(yt|xt) =
b1
+
.
b2
xt
2
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)
4. cov(ei,ej) = cov(yi,yj) = 0 5. xt c for every observation
6. et~N(0,s 2) <=.> yt~N(b1+ b2x3 t,
The population parameters b1 and b2 4.4 are unknown population constants.