FiniteSamplePropertiesoftheLeastSquaresEstimator文件材料(二).ppt

* * * * Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 6. Finite Sample Properties of the Least Squares Estimator Deriving the Properties So, b = a parameter vector + a linear combination of the disturbances, each times a vector. Therefore, b is a vector of random variables. We analyze it as such. The assumption of nonstochastic regressors. How it is used at this point. We do the analysis conditional on an X, then show that results do not depend on the particular X in hand, so the result must be general – i.e., independent of X. Properties of the LS Estimator Expected value and the property of unbiasedness. E[b|X] = ? = E[b]. Prove this result. A Crucial Result About Specification: y = X1?1 + X2?2 + ? Two sets of variables. What if the regression is computed without the second set of variables? What is the expectation of the short regression estimator? b1 = (X1?X1)-1X1?y The Left Out Variable Formula (This is a VVIR!) E[b1] = ?1 + (X1?X1)-1X1?X2?2 The (truly) short regression estimator is biased. Application: Quantity = ?1Price + ?2Income + ? If you regress Quantity on Price and leave out Income. What do you get? (Application below) The Extra Variable Formula A Second Crucial Result About Specification: y = X1?1 + X2?2 + ? but ?2 really is 0. Two sets of variables. One is superfluous. What if the regression is computed with it anyway? The Extra Variable Formula: (This is a VIR!) E[b1.2| ?2 = 0] = ?1 The long regression estimator in a short regression is unbiased.) Extra variables in a model do not induce biases. Why not just include them, then? Well pursue this later. Application: Left out Variable Leave out Income. What do you get? E[b1] = ?1 + ?2 In time series data, ?1 0, ?2 0 (usually) Cov[Price,Income] 0 in time series data. So, the short regression will ov