Trace trace trace λ λ a c c a c c m m m a c c λ λ

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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 ™    8/35
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Part 7: Estimating the Variance of  b Example: Characteristic Roots of a Correlation Matrix ™    9/35
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Part 7: Estimating the Variance of  b Gasoline Data ™    10/35
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Part 7: Estimating the Variance of  b X’X and its Roots ™    11/35
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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. ™    12/35
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Part 7: Estimating the Variance of  b X’X (X’X)-1 s2(X’X)-1 ™    13/35
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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 --------+------------------------------------------------------------- ™    14/35
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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 ]’ ™    15/35
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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$... Store b(r) Endproc Ends procedure exec;n=20;bootstrap=b$ 20 bootstrap reps matr;list;bboot' $ Display results ™    16/35
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Part 7: Estimating the Variance of  b --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -79.7535*** 8.67255 -9.196 .0000 Y| .03692*** .00132 28.022 .0000 9232.86 PG| -15.1224*** 1.88034 -8.042 .0000 2.31661 --------+------------------------------------------------------------- Completed 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| -79.7535*** 8.35512 -9.545 .0000 -79.5329 B002| .03692*** .00133 27.773 .0000 .03682 B003| -15.1224*** 2.03503 -7.431 .0000 -14.7654 --------+------------------------------------------------------------- Results of Bootstrap Procedure ™    17/35
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Part 7: Estimating the Variance of  b Bootstrap Replications Full sample result Bootstrapped sample results ™    18/35
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Part 7: Estimating the Variance of  b OLS vs. Least Absolute Deviations ---------------------------------------------------------------------- Least absolute deviations estimator ...............
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