Gauss Markov Theorem
&
The Gauss Markov theorem says
that, under certain conditions, the ordinary least squares (OLS) estimator of
the coefficients of a linear regression model is the best linear unbiased estimator
(BLUE)
Consider the simple linear Regression
Model
.png){kind=small}
The estimated model is given by;
.png){kind=small}
Where:
.png)
i. Linearity
The OLS
estimate is linear function of the dependent variable.
.png)
.png)
.png)
.png)
ii. Unbiasedness
The OLS
estimates are unbiased estimators of regression parameters.
.png)
.png)
Hence the OLS estimates are unbiased of their respective parameters.
iii.
Variance of slope is given by:
.png)
.png)
Now variance of intercept is defined as:
.png)
.png)
iii. The variance of the OLS estimates has
minimum variance.
.png)
.png)
.png)
The OLS estimate has
minimum variance.
.png)
.png)
.png)
.png)
Hence, the variance of the OLS estimate is minimum.
.png)
Consider the model
.png)
We know that
.png)
Substitute from eq. 2
& eq. 3 in eq. 1 given below:
.png)
.png)
.png)
.png)
.png)
.png)
- Read More: Sampling distribution of OLS estimators
:
No comments:
Post a Comment