Econometric analysis Regression and Projection参考.ppt

* * * * * * * * * * * * * * Expected Values of Deviations from the Conditional Mean Observed y will equal E[y|x] + random variation. y = E[y|x] + ? (disturbance) Is there any information about ? in x? That is, does movement in x provide useful information about movement in ?? If so, then we have not fully specified the conditional mean, and this function we are calling ‘E[y|x]’ is not the conditional mean (regression) There may be information about ? in other variables. But, not in x. If E[?|x] ? 0 then it follows that Cov[?,x] ? 0. This violates the (as yet still not fully defined) ‘independence’ assumption Zero Conditional Mean of ε E[?|all data in X] = 0 E[?|X] = 0 is stronger than E[?i | xi] = 0 The second says that knowledge of xi provides no information about the mean of ?i. The first says that no xj provides information about the expected value of ?i, not the ith observation and not any other observation either. “No information” is the same as no correlation. Proof: Cov[X,?] = Cov[X,E[?|X]] = 0 The Difference Between E[ε |X]=0 and E[ε]=0 Conditional Homoscedasticity and Nonautocorrelation Disturbances provide no information about each other, whether in the presence of X or not. Var[?|X] = ?2I. Does this imply that Var[?] = ?2I? Yes: Proof: Var[?] = E[Var[?|X]] + Var[E[?|X]]. Insert the pieces above. What does this mean? It is an additional assumption, part of the model. We’ll change it later. For now, it is a useful simplification Nonrandom (Fixed) Regressors A mathematical convenience. Useful for interpretation of the sampling process, but not a fundamental feature of the linear regression model. Simplifies some proofs, but is without content in the context of the theory and is essentially irrelevant to the results we will obtain. Normal Distribution of ε An assumption of very limited usefulness Used to facilitate finite sample derivations of certain test statistics. Temporary. The Linear