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## Applied Regression

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### Efficient/small variance. Consistent. Lecture Outline, continued ... The error term has constant variance (no heteroskedasticity) ... – PowerPoint PPT presentation

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Title: Applied Regression

1
Applied Regression
• Feb 1- Feb 6, 2006

2
Lecture Outline
• The Econometricians Problem revisited
• The disturbance term
• Samples of Y given X
• Choose an estimator of beta
• Properties of a Good Estimator
• Low cost
• Unbiased
• Efficient/small variance
• Consistent

3
Lecture Outline, continued
• The Ordinary Least Squares Estimator
• Theoretical Equation
• OLS chooses a,b,g to minimize sum of the squared
residuals.
• Derivation of OLS estimators
• Classical Assumptions
• Gauss Markov Theorem

4
The Econometricians Problem Revisited
• Theoretical relationship YabX
• Econometric equation YabXe
• Synthetic.xls Used random number generator to
obtain e. Because of the disturbance term, Y
could vary across the same observations of X
• Mock Data Estimate b
• We want to study the properties of the sampling
distributions of estimator for b.

5
Properties of Good Estimator
• Low cost
• Unbiased
• Efficient/small variance
• Consistent

6
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8
How Does OLS Work?
• OLS chooses a,bx,by to minimize the sum of the
squared residuals

9
The OLS Estimated Coefficients
10
Classical Assumptions
• The regression model is linear in the
coefficients and the error term.
• The error term has zero population mean
• All explanatory variables are uncorrelated with
the error term
• Observations of the error term are uncorrelated
with each other (no serial correlation)
• The error term has constant variance (no
heteroskedasticity)
• No explanatory variable is a perfect linear
function of the other explanatory variables (no
perfect multicollinearity)
• The error term is normally distributed.
(optional).

11
OLS Estimator of b
12
Unbiasedness
13
Unbiasedness, cont.
14
Alternative Estimator
15
Consistency
• What happens as the size of the sample approaches
the population?
• If X and e are not correlated (independent) and
Var (X)gt0, OLS estimator gets closer to its true
value.
• Slope estimator doesnt depend on T. So it can
not be consistent

16
OLS is Consistent
17
Gauss-Markov Theorem
• OLS is BLUE

18
Gauss Markov Theorem
• Under the Classical Assumptions, the OLS
estimator of bk is the minimum variance estimator
from among the set of all linear unbiased
estimators for bk, for k1,,K

19
Flow Chart of Proof of Gauss-Markov Theorem
Observe That
So the best linearly unbiased estimator is
Identify Restrictions to Insure Unbiasedness
Choose di to minimize variance
20
OLS is linear
21
General Class of Linear Estimators
22
Unbiasedness requires
23
Minimize Variance
24
Summary
• OLS is BLUE
• Consistency and Unbiasedness require E(e)0 and
E(Xe)0
• Efficiency requires no serial correlation and
homoscedastic errors