Finite Sample Properties of the Least Squares Estimator参考.ppt

* * * * * * * * * * * * * * * * * * 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 $ +--------+--------------+----------------+--------+--------+----------+ |Variable| Coefficient | Standard Error |t-ratio |P[|T|t]| Mean of X| +--------+--------------+----------------+--------+--------+----------+ Constant| 7 6 1.277 .2104 Y | 12.955 .0000 9232.86111 PG | -8 1 -6.558 .0000 2Completed 20 bootstrap iterations. +------------------------------------------+ | Results of bootstrap estimation of model.| | Model has been reestimated 20 times. | | Means shown below are the means of the | | bootstrap estimates. Coefficients shown | | below are the original estimates based | | on the full sample. | | bootstrap samples have 36 observations.| +------------------------------------------+ +--------+--------------+----------------+--------+--------+----------+ |Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|z]| Mean of X| +--------+--------------+----------------+--------+--------+----------+ B001 | 7 6 1.184 .2365