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# OLSStatsNotes - Notes Statistical Properties of the Linear...

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Statistical Properties of the Linear Model Charlie Gibbons University of California, Berkeley Economics 140 Fall 2011 Outline 1 Assumptions 2 Unbiasedness 3 Calculating ˆ σ 2 4 Variances of coefficients 5 R 2 6 Evaluating Residuals 7 Variance of predicted values Assumptions Let’s make the following assumptions: y i = β 0 + β 1 x i + i (linear model) E [ i | X ] = 0 (exogeneity) x i takes on at least two different values in the sample We need these for unbiasedness . We’ll add an additional assumption for calculating standard errors ( i.e. , variances of parameter estimates): Var( i | X ) = σ 2 i (homoskedasticity) Cov( i , j | X ) = 0 i 6 = j (independence/no serial correlation) Notes Notes Notes

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Unbiasedness Suppose that we are interested in a univariate regression: y i = β 0 + β 1 x i + i . Our estimates of the coefficients are ˆ β 1 = d Cov( y , x ) d Var( x ) = ˆ ρ xy ˆ σ y ˆ σ x ˆ β 0 = ¯ y - ˆ β 1 ¯ x . These estimates are unbiased i.e. , their expectation is equal to their true values. Unbiasedness in small samples What if you are trying to estimate β 0 and β 1 with only three observations (each with different x values)? Are your estimates still unbiased? Yes. The distribution of the estimates that you get has a mean of the true value, but the variance of the estimate around the true value is very wide and decreases as you add more observations. Calculating ˆ σ 2 We need to estimate the variance of y given x : σ 2 Var( y i | x i ) = E h ( y i - E [ y i | x i ]) 2 | x i i .
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OLSStatsNotes - Notes Statistical Properties of the Linear...

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