x
x
ECONOMETRICS
x
x
x
CHAPTER
3
Chapter 2:
Estimating the parameters of a linear regression model.
Yi = b1 + b2 Xi + ei
Using OLS
Chapter 3:
Testing hypotheses about the parameters we estimated.
The most common null (default) hypothesis is that a
parameter is equal to zero. We test to see if we can reject
the null hypothesis.
The Classical Linear Regression Model
Assumptions
1. The regression model is linear in the parameters.
Yi = B1 + B2 ln(Xi2) + ui
Yi = B1 + B2 Xi B3 + ui
2. The explanatory variables are uncorrelated with the
disturbance term.
In general, we consider the explanatory variables
fixed. Only the dependent variable is subject
to random disturbances (i.e. stochastic.)
3. The expected value of the disturbance term is zero
for any Xi.
E( u | Xi ) = 0
For any Xi, the disturbances
are just as likely to be
positive as negative
4. The variance of each ui is constant. The model is
homoscedastic.
var( ui ) = σ2
5. There is no correlation between two error terms.
6. The model is correctly specified.
We have included the right variables in the model.
Variances and Stnd Errors of OLS Estimators
We estimate regression coefficients.
These estimators are random variables because their
values change from sample to sample.
1st
sample
QUPS = 511 − 26.2 PUPS + 6.5 PFEDX
2nd
sample
QUPS = 432 − 21.7 PUPS – 1.7 PFEDX
We estimate the variance of the disturbance term in the
population from the residuals in the sample.
From this we estimate the variance of the parameter
estimates :
Yi = b1 + b2 Xi + ei
Estimate var(b1)
Estimate var(b2)
Note: OLS provides parameter estimates that are
unbiased and efficient (minimum variance.)
One Last Assumption
7. The disturbance term is normally distributed.
ui ~ N(0, σ2)
If so, then the estimators are normally distributed.
b1 ~ N(B1, σb12)
b2 ~ N(B2, σb22)
Note: The stnd deviation
of an estimator, σb1,
is usually called the
standard error.
b1
We don’t know the variance of the population disturbance
term, σ2. We only have an estimate, σ2.
b1 ~
N(B1, σb12)
So we can’t use this normal
distribution to test our hypothesis.
But, if we standardize the estimator by σ2, the result
follows the t-distribution (similar to the normal).
b1 – B1
se(b1)
~ tn-2
So we use this t-distribution
to test our hypothesis.