Why Use Ordinary Least Squares

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Why Use Ordinary Least Squares?

• The Econometricians Problem Revisited.

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Criteria for a Good Estimator?

• Low cost to compute
• Small residuals
• Unbiasedness
• Efficient/small variance
• Consistency

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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).

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Synthetic.xls

• Theoretical relationship YabX
• Econometric equation YabXe
• Used random number generator to obtain e.
• Assume that e has normal distribution with mean
  zero and constant variance. e is not correlated
  with X. Observations of e are not correlated
  with each other.

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First Sample
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What Happens in Other Samples?

• Because of the disturbance term, Y could vary
  across the same observations of X
• We want to study the properties of the sampling
  distributions of the OLS estmators.
• The other columns of Ys represent different
  samples.
• Monte Carlo simulation

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In Class Work -- OLS

• Calculate regression coefficients
• Intermediate calculations are in sheet CALC
• Results are in Coefficients
• Alternate Method
• Go to Data Analysis in TOOLS menu
• Specify Dependent variable and explanatory
  variable

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What Happens?

• What is the mean of estimated coefficients across
  samples? (Unbiasedness)
• What is the variance? (Efficiency)
• What do you think would happen as we increased
  the number of observations? (Consistency)

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Gauss-Markov Theorem

• OLS is BLUE

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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

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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).

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OLS Estimator of b
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Unbiasedness
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Unbiasedness, cont.
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Alternative Estimator
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Why not use slope estimator?

• It is unbiased
• It is cheap to compute

• It lacks consistency
• It is not the minimum variance linear estimator

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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

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OLS is Consistent
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Efficiency (Proof of Gauss Markov)

• Restrict our attention to the class of linear
  estimators.
• OLS is linear.
• General class of linear estimators
• Consider only unbiased estimators.
• Choose d to minimize variance

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OLS is linear
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General Class of Linear Estimators
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Unbiasedness requires
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Minimize Variance
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Summary

• OLS is BLUE
• Consistency and Unbiasedness require E(e)0 and
  E(Xe)0
• Efficiency requires no serial correlation and
  homoscedastic errors