OLS assumptions
and hypothesis testing
1
MADE
What we know about estimation?
• Do we know the true β’s?
– We only know that among linear and unbiased we
have estimators of β (i.e. b’s) that yield lowest
errors
– We also know that b’s are unbiased estimators of
β’s if Gauss-Markov assumptions are fulfilled
• Do we know the residuals of our estimation?
– We only know their estimators and we know that
on average real residuals and estimated ones
should be equal.
2
MADE
What we know about estimation?
 The good news:
– We can use the estimates of residuals to test
whether b’s are what they look or they only seem
to be, because knowing e’s we can tell how
wrong we are in guessing β’s (i.e. what is the
„standard deviation” of our guess)
3
MADE
Properties of OLS - refreshment
1.
2.
3.
4.
4
X’e=0
Fitted and actual values of y are on
average equal
Σe=0 (for a model with a constant)
There is nothing more systematic about y
than already explained by X (fitted y and
residuals are not correlated)
MADE
What Gauss-Markov theorem gives
•
Can we be sure that OLS will always give us the
best possible estimator?
•
If assumptions are fulfilled, OLS is BLUE (meaning
Best Linear Unbiased Estimator)
Assumptions:
1. y=Xβ
2. X is deterministic and exogenous
3. E(ɛi)=0
4. Cov(ɛi,ɛj)=0
5. Var(ɛi)=σ2
•
5
MADE
Hypothesis testing
• α is the assumed confidence level
– it is actually a measure of risking the wrong
conclusion
– it tells you what is the probability of rejecting a
true hypothesis
6
MADE
Hypothesis testing
• For any α, we can define kα,
– it is the critical value of the distribution
– it tells you for what value a predefined 1- α part
of the probability mass is to to the left
• Testing means comparing the estimates
you find with the chosen critical values and
checking whether you are left of right of
them
7
MADE
Probability mass?
• To different chosen values of
8
α
MADE
How do we know the
probability mass function?
• For testing whether our estimators are what we
think they are:
1. we know that E(b) is β
2. we know that var(b) is σ2(X’X)-1 and we
know that s2 is a good estimator of σ2 σ
3. so:
4. and we have:
9
MADE
How do we know the
probability mass function?
• Also, if something has a N or t distribution,
it’s square has a χ2 distribution.
• So our R2 has this distribution, and so do
any tests that will incorporate the squares
of e’s.
10
MADE
11
MADE
How does that work in
practice?
• Example: expenses on housing
expenses=cons + β * income + ε
ln(expenses)=cons + β * ln(income) + ε
12
MADE
How does that work in
practice?
• Residuals
(left for a simple equation, right for logs)
13
MADE
Unemployment, inflation, GDP,
Poland 1995-2003 (quarterly)
14
MADE