Econometrics-I-7

# Part 7 estimating the variance of b decomposing m 2 2

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Unformatted text preview: Part 7: Estimating the Variance of b Decomposing M 2 2 2 2 Useful Result: If = ' is the spectral decomposition, then ' (just multiply) = , so . All of the characteristic roots of are 1 or 0. How many of each? trace( ) = trace( ')=trace( = Λ = Λ A C C A C C M M M A C C Λ Λ Λ ' )=trace( ) Trace of a matrix equals the sum of its characteristic roots. Since the roots of are all 1 or 0, its trace is just the number of ones, which is n-K as we saw. C C M Λ Λ &#152;™™™ ™ 8/35 Part 7: Estimating the Variance of b Example: Characteristic Roots of a Correlation Matrix &#152;&#152;™™™ ™ 9/35 Part 7: Estimating the Variance of b Gasoline Data &#152;&#152;™™™ ™ 10/35 Part 7: Estimating the Variance of b X’X and its Roots &#152;&#152;™™™ ™ 11/35 Part 7: Estimating the Variance of b 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 are the square roots of the diagonal elements. &#152;&#152;™™™ ™ 12/35 Part 7: Estimating the Variance of b X’X (X’X)-1 s2(X’X)-1 &#152;&#152;™™™ ™ 13/35 Part 7: Estimating the Variance of b Standard Regression Results---------------------------------------------------------------------- Ordinary least squares regression ........ LHS=G Mean = 226.09444 Standard deviation = 50.59182 Number of observs. = 36 Model size Parameters = 7 Degrees of freedom = 29 Residuals Sum of squares = 778.70227 Standard error of e = 5.18187 <= sqr[778.70227/(36 – 7)] Fit R-squared = .99131 Adjusted R-squared = .98951--------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X--------+------------------------------------------------------------- Constant| -7.73975 49.95915 -.155 .8780 PG| -15.3008*** 2.42171 -6.318 .0000 2.31661 Y| .02365*** .00779 3.037 .0050 9232.86 TREND| 4.14359** 1.91513 2.164 .0389 17.5000 PNC| 15.4387 15.21899 1.014 .3188 1.67078 PUC| -5.63438 5.02666 -1.121 .2715 2.34364 PPT| -12.4378** 5.20697 -2.389 .0236 2.74486--------+------------------------------------------------------------- &#152;&#152;™™ ™ 14/35 Part 7: Estimating the Variance of b 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 ]’ &#152;&#152;™™ ™ 15/35 Part 7: Estimating the Variance of b Bootstrap Application matr;bboot=init(3,21,0.)\$ Store results here name;x=one,y,pg\$ Define X regr;lhs=g;rhs=x\$ Compute b calc;i=0\$ Counter Proc Define procedure regr;lhs=g;rhs=x;quietly\$ … Regression matr;{i=i+1};bboot(*,i)=b\$... Store b(r)matr;{i=i+1};bboot(*,i)=b\$....
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Part 7 Estimating the Variance of b Decomposing M 2 2 2 2...

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