The Population Linear Regression Model general notation

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The Population Linear Regression Model general
notation
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Population Regression Line
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The OLS estimator solves
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Sample Regression Equation
chosen in sample not chosen in sample
Y
estimated error for X3 (residual)
Observed Value of Y for X3
Predicted Value of Y for X3
Estimated slope
estimated Intercept
X
X3
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California Test Score/Class Size data

• Interpretations

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Predicted values residuals
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OLS regression STATA output
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Measures of Fit(Section 4.3)
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The Standard Error of the Regression (SER)
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• Technical note why divide by (n-2) instead of
  (n-1)?
• When n is large, it makes negligible difference
  whether n, n-1, or n-2 are used (although
  conventional formula uses n-2 whre there is one
  X)

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Example of the R2 and the SER
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The Least Squares Assumptions (SW Section 4.4)
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The Least Squares Assumptions
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Least squares assumption 1 E(uX x) 0.
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Least squares assumption 1, ctd.
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Least squares assumption 2 (Xi,Yi), i 1,,n
are i.i.d.
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Least squares assumption 3 Large outliers are
rare Technical statement E(X4) lt ? and E(Y4) lt ?
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OLS can be sensitive to an outlier
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The Sampling Distribution of
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The mean and variance of the sampling
distribution of
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Now we can calculate E( ) and var( )
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Next calculate var( )
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What is the sampling distribution of ?
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Large-n approximation to the distribution of
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The larger the variance of X, the smaller the
variance of
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The larger the variance of X, the smaller the
variance of
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Summary of the sampling distribution of
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