06 Properties of OLS Estimators

06 Properties of OLS Estimators - Economics 140A Properties...

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Economics 140A Properties of OLS Estimators As discussed last time, we begin with the population regression Y t = & 0 + 1 X t + U t : One estimator, which we introduced in our previous discussion, is the ordinary least squares (OLS) estimator. Below we determine the properties of the OLS es- timator of the coe¢ cients in the population regression model. The OLS estimator is linear and unbiased. After verifying these facts, we determine the variance of the OLS estimator. Linear Estimators A linear estimator is de&ned to be a linear function of the dependent vari- able. (A nonlinear function would be P n t =1 c t Y 2 t .) The OLS estimators are linear functions of the dependent variable. For B 1 : B 1 = P n t =1 X t X n Y t Y n ± P n t =1 X t X n ± 2 = P n t =1 X t Y t X n P n t =1 Y t P n t =1 X t X n ± 2 = n X t =1 C t Y t ; with C t = X t X n P n t =1 ( X t X n ) 2 . For B 0 : B 0 = Y n B 1 X n = 1 n n X t =1 Y t X n n X t =1 C t Y t = n X t =1 ² 1 n X n C t ³ Y t ± n X t =1 D
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06 Properties of OLS Estimators - Economics 140A Properties...

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