Estimating the Variance of the Least Squares Estimator参考.ppt

* * * * * * * * * * Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 7. Estimating the Variance of the Least Squares Estimator Context The true variance of b is ?2E[(X?X)-1] We consider how to use the sample data to estimate this matrix. The ultimate objectives are to form interval estimates for regression slopes and to test hypotheses about them. Both require estimates of the variability of the distribution. We then examine a factor which affects how large this variance is, multicollinearity. Estimating ?2 Using the residuals instead of the disturbances: The natural estimator: e?e/n as a sample surrogate for ???/n Imperfect observation of ?i = ei + (? - b)?xi Downward bias of e?e/n. We obtain the result E[e?e] = (n-K)?2 Expectation of e’e Method 1: Estimating σ2 The unbiased estimator is s2 = e?e/(n-K). “Degrees of freedom.” Therefore, the unbiased estimator is s2 = e?e/(n-K) = ??M?/(n-K). Method 2: Some Matrix Algebra Decomposing M Example: Characteristic Roots of a Correlation Matrix Var[b|X] Estimating the Covariance Matrix for b|X The true covariance matrix is ?2 (X’X)-1 The natural estimator is s2(X’X)-1 “Standard errors” of the individual coefficients. How does the conditional variance ?2(X’X)-1 differ from the unconditional one, ?2E[(X’X)-1]? Regression Results X’X (X’X)-1 s2(X’X)-1 Bootstrapping Some assumptions that underlie it - the sampling mechanism Method: 1. Estimate using full sample: -- b 2. Repeat R times: Draw n observations from the n, with replacement Estimate ? with b(r). 3. Estimate variance with V = (1/R)?r [b(r) - b][b(r) - b]’ Bootstrap Application -- matr;bboot=init(3,21,0.)$ -- name;x=one,y,pg$ -- regr;lhs=g;rhs=x$ -- calc;i=0$ -- proc -- regr;lhs=g;rhs=x$ -- matr;{i=i+1};bboot(*,i)=b$ -- endproc -- exec;n=20;bootstrap=b$ -- matr;list;bboot $ +--------+--------------+----------------+--------+--------+----------+ |Variabl