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Economics 20 - Prof. Anderson * Unbiasedness of OLS (cont) Economics 20 - Prof. Anderson * Unbiasedness Summary The OLS estimates of b1 and b0 are unbiased Proof of unbiasedness depends on our 4 assumptions – if any assumption fails, then OLS is not necessarily unbiased Remember unbiasedness is a description of the estimator – in a given sample we may be “near” or “far” from the true parameter Economics 20 - Prof. Anderson * Variance of the OLS Estimators Now we know that the sampling distribution of our estimate is centered around the true parameter Want to think about how spread out this distribution is Much easier to think about this variance under an additional assumption, so Assume Var(u|x) = s2 (Homoskedasticity) Economics 20 - Prof. Anderson * Variance of OLS (cont) Var(u|x) = E(u2|x)-[E(u|x)]2 E(u|x) = 0, so s2 = E(u2|x) = E(u2) = Var(u) Thus s2 is also the unconditional variance, called the error variance s, the square root of the error variance is called the standard deviation of the error Can say: E(y|x)=b0 + b1x and Var(y|x) = s2 Economics 20 - Prof. Anderson * . . x1 x2 Homoskedastic Case E(y|x) = b0 + b1x y f(y|x) Economics 20 - Prof. Anderson * . x x1 x2 y f(y|x) Heteroskedastic Case x3 . . E(y|x) = b0 + b1x Economics 20 - Prof. Anderson * Variance of OLS (cont) Economics 20 - Prof. Anderson * Variance of OLS Summary The larger the error variance, s2, the larger the variance of the slope estimate The larger the variability in the xi, the smaller the variance of the slope estimate As a result, a larger sample size should decrease the variance of the slope estimate Problem that the error variance is unknown Economics 20 - Prof. Anderson * Estimating the Error Variance We don’t know what the error variance, s2, is, because we don’t observe the errors, ui What we observe are the residuals, ?i We can use the residuals to form an estimate of the error variance Economics 20 - Prof. Anderson * Error Variance Estimate (cont) Economics 20
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