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OLS assumptions and hypothesis testing

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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.

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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)

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Properties of OLS - refreshment

1. X’e=02. Fitted and actual values of y are on

average equal3. Σe=0 (for a model with a constant)4. There is nothing more systematic about

y than already explained by X (fitted y and residuals are not correlated)

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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=Xβ2. X is deterministic and exogenous3. E(ɛi)=0

4. Cov(ɛi,ɛj)=0

5. Var(ɛi)=σ2

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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

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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

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Probability mass?

• To different chosen values of α

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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:

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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.

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How does that work in practice?

• Example: expenses on housingexpenses=cons + β * income + ε

ln(expenses)=cons + β * ln(income) + ε

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How does that work in practice?

• Residuals (left for a simple equation, right for logs)

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Unemployment, inflation, GDP, Poland 1995-2003 (quarterly)