Introduction to
Econometrics
Ekki Syamsulhakim Undergraduate Program Department of Economics
Sampling Variance of the
OLS estimator
•
We know that when is not biased
•
The variance of can be computed
using the formula:
Estimator Variance, Perfect
Estimator Variance, Perfect
Estimator Variance, Perfect
Estimator Variance, Perfect
Estimator Variance, Perfect
Estimator Variance, Perfect
Estimator Variance, Perfect
Estimator Variance, Perfect
Inference
•
We assume that
unobserved error
is
normally distributed in the population
Hypothesis Testing
•
t-test or (later) F-test (individual
coefcient vs overall model tests)
•
two sided vs one sided test
– Your hypothesis
– check the theory
– our research question
•
2 methods
– t-stat method
Hypothesis Testing
•
The long steps:
– State the null and alternative hypothesis
– Choose the level of signifcance
– For t-test method: observe t-statistics
and compute t-critical
– For p-value method: compute p-value
– State the decision rule
Regression
reg rent room sqrm if rent<4000000 & sqrm<3000 & room<30
Compute t-crit
t-crit (a, df=n-k-1=11043-2-1=11040) = 1.960179
Rejection criteria:
Reject H0 if |t-stat |> |t-crit|
Conclusion:
Since our |t-stat| > |t-crit| or 22.57> 1.960179, we reject H0.
Conclusion:
Since our t-stat > t-crit (22.57 > 1.960179) we reject H0.
Therefore we have sufcient evidence that number of room has an impact on rent
Rejection criteria:
Reject H0 if p-value < Conclusion:
Since our p-value=0.0000… is less than =0.05, we reject H0.
Therefore we have sufcient evidence that number of room has an impact on rent
•
One sided t-test (ex: t-stat
app)
• As number of room increases, it is sensible to
think that the rent also increases (probably based on theory)
• We can (should) use 1 tail test
– We must compute new t-critical as the output of STATA /
One sided t-test (ex: t-stat
app)
Compute t-crit for 1 sided
t-crit (2a, df=n-k-1=11043-2-1=11040) = 1.645 (positive side)
Rejection criteria:
Reject H0 if |t-stat |> |t-crit|
Conclusion:
Since our |t-stat| > |t-crit| or 22.57> 1.645, we reject H0.
One sided t-test (ex: p-value
app)
• As number of room increases, it is sensible
to think that the rent also increases
• We can (should) use 1 tail test
One sided t-test (ex: p-value
Because we are doing 1 tail test, P-value given by Econometric Software must be divided by 2;
Hence calculated =0.0000…
Example:
Therefore we have sufcient evidence that number of room has a positive
impact on rent
Testing Other Hypotheses About
•
Consider a simple model relating the
annual number of crimes on college
campuses (crime) to student
enrollment (enroll)
•
This is a constant elasticity model,
where is the elasticity of crime with
respect to enrollment
Testing Other Hypotheses
About
• It is not much use to test H0: , as we
expect the total number of crimes to increase as the size of the campus increases
• A more interesting hypothesis to test
would be that the elasticity of crime with respect to enrollment is one
H0 :
– This means that a 1% increase in enrollment
Testing Other Hypotheses
About
•
A noteworthy alternative is
H
1:
,
which implies that a 1% increase in
enrollment increases campus crime
by
more than
1%
•
If , then, in a relative sense—not just
an absolute sense—crime is more of
a problem on larger campuses.
Testing Other Hypotheses
About
•
The estimated elasticity of crime with
respect to enroll, 1.27, is in the
direction of the alternative .
•
But
is there enough evidence
to
conclude that ?
Testing Other Hypotheses
About
•
if the null is stated as H
0:
•
where is our hypothesized value of ,
then the appropriate t statistic is
•
The usual t statistic is obtained when
.
Testing Other Hypotheses
About
•
The correct t statistic is
•
The one-sided 5% critical value for a
t distribution with df is about 1.66
•
So we clearly reject in favor of at
the 5% level
F-test (F-stat approach)
H0: b1=b2=0 all coefcients are zero (or: all independent
variables do not afect dependent variables; or: room and sqrm do not afect rent)
HA: At least one of bi is NOT zero (or: at least one independent
variable is NOT zero)
F-stat = 434.33
F-crit (a=0.05,k=2,n-k-1=26)2.99 Because F-stat > F crit, reject H0
Conclusion: we have sufcient evidence that at least one of our independent variable is useful in explaining house rent
F-test (p-value approach)
H
0:
b
1=
b
2=0 all coefcients are zero
H
A: At least one of
b
iis zero
Using p-value approach, we can see that our p-value for F-test is 0.000… which is less than our (default) a=0.05 Hence, reject H0
Joint / Multiple hypothesis
test
•
We often test hypotheses involving
more than one of the population
parameters.
– test a single hypothesis involving more
than one of the .
– test multiple hypotheses (multiple linear
restrictions – the F-test)
Testing Multiple Linear Restrictions:
The
F -
Test
•
We begin with the leading case of
testing whether a set of independent
variables has no partial efect on a
dependent variable
– we want to test whether a group of
variables has no efect on the dependent variable.
– the null hypothesis is that a set of variables
Testing Multiple Linear Restrictions:
The
F -
Test
• consider the following model that explains major
league baseball players’ salaries:
(4.28)
salary is the 1993 total salary, years is years in the
league, gamesyr is average games played per year,
bavg is career batting average (for example, bavg = 250), hrunsyr is home runs per year, and rbisyr is
runs batted in per year.
Testing Multiple Linear Restrictions:
The
F -
Test
• Suppose we want to test the null
hypothesis that, once years in the league and games per year have been controlled for, the statistics measuring performance
—bavg, hrunsyr, and rbisyr—have no
efect on salary.
• Essentially, the null hypothesis states that
Testing Multiple Linear Restrictions:
The
F -
Test
• In terms of the parameters of the model, the null hypothesis is stated as
(4.29)
The null (4.29) constitutes three exclusion restrictions:
If (4.29) is true, then bavg, hrunsyr, and rbisyr have no efect on log(salary), after years and gamesyr have
been controlled for, and therefore should be excluded
from the model.
Testing Multiple Linear Restrictions:
The
F -
Test
• What should be the alternative to (4.29)? If what we have in mind is that “performance statistics matter, even after controlling for years in the league and games per year,”
then the appropriate alternative is simply
is not true
The alternative (4.30) holds if at least one of or is diferent from zero. (Any or all could be diferent from zero.)
Testing Multiple Linear Restrictions:
The
F -
Test
• The steps to be done:
1. Conduct a regression for the unrestricted model (in the example above, the model with all performance variables included)
• Note the SSR and R2
2. Conduct a regression for the restricted model (in the example above, the model with none of the
performance variables included)
• Note the SSR and R2
3. Calculate the F-Statistic, that is
Where is numerator degree of freedom = and is called the denominator degree of freedom =
Testing Multiple Linear Restrictions:
The
F -
Test
• The outcome of the joint test may seem
surprising in light of the insignifcant t
-statistics for the three variables.
• What is happening is that the two
variables hrunsyr and rbisyr are highly correlated, and this multicollinearity
makes it difcult to uncover the partial
The
R
-Squared form of the
F
Statistic
• It is often more convenient to have a form of the
F statistic that can be computed using the R -squareds from the restricted and unrestricted models.
• One reason for this is that the R-squared is
always between zero and one, whereas the SSRs can be very large depending on the unit of
measurement of y, making the calculation based
