Pnc 29476979 25797920 1143 2690 28100132 puc 20153591

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PNC -.29476979 .25797920 -1.143 .2690 .28100132 PUC -.20153591 .07415599 -2.718 .0146 .40523616 PPT .08050720 .08706712 .925 .3681 .47071442 PD 1.50606609 .29745626 5.063 .0001 -.44279509 PN .99947385 .27032812 3.697 .0018 -.58532943 PS -.81789420 .46197918 -1.770 .0946 -.62272267 T -.01251291 .01263559 -.990 .3359 13.0000000 ™  27/38
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Part 12: Asymptotics for the Regression Model Covariance Matrix ™  28/38
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Part 12: Asymptotics for the Regression Model Linear Hypothesis H0: Aggregate price variables are not significant determinants of gasoline consumption H0: β7 = β8 = β9 = 0 H1: At least one is nonzero ™  29/38 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 , = 0 0 0 0 0 0 0 0 0 1 0 0           R - q = 0 β R q
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Part 12: Asymptotics for the Regression Model Wald Test Matrix ; R = [0,0,0,0,0,0,1,0,0,0/                        0,0,0,0,0,0,0,1,0,0/                        0,0,0,0,0,0,0,0,1,0]            ;  q = [0 / 0 / 0 ] $ Matrix ; m = R*b - q ; Vm = R*Varb*R'             ; List ; Wald = m'<Vm>m $ Matrix WALD     has  1 rows and  1 columns.                1         +--------------        1|   66.91506 ™  30/38
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Part 12: Asymptotics for the Regression Model Restricted Regression Compare Sums of Squares Regress; lhs=g;rhs=X; cls:b(7)=0,b(8)=0,b(9)=0$ +----------------------------------------------------+ | Linearly restricted regression | | Ordinary least squares regression | | LHS=G Mean = 5.308616 | | Standard deviation = .2313508 | | Residuals Sum of squares = .01864365 | .00377694 | Standard error of e = .3053166E-01 | | Fit R-squared = .9866028 | .9972859 without restrictions | Adjusted R-squared = .9825836 | | Model test F[ 6, 20] (prob) = 245.47 (.0000) | | Restrictns. F[ 3, 17] (prob) = 22.31 (.0000) | Note: J(=3)*F = Chi-Squared = 66.915 from before | Not using OLS or no constant. Rsqd & F may be < 0. | | Note, with restrictions imposed, Rsqd may be < 0. | +----------------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant -4.46504223 4.77789711 -.935 .3631 Y 1.05851456 .55196204 1.918 .0721 9.03448264 PG -.15852276 .05008100 -3.165 .0057 .47679491 PNC .21765564 .18336687 1.187 .2516 .28100132 PUC -.24298315 .10328032 -2.353 .0309 .40523616 PPT -.12617610 .10436708 -1.209 .2432 .47071442 PD .000000 ...... (Fixed Parameter) ....... -.44279509 PN .222045D-15 ...... (Fixed Parameter) ....... -.58532943 PS -.444089D-15 ...... (Fixed Parameter) ....... -.62272267 T .02944666 .02126600 1.385 .1841 13.0000000 ™  31/38
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Part 12: Asymptotics for the Regression Model Nonlinear Restrictions I am interested in testing the hypothesis that certain ratios of elasticities are equal. In particular, 1 = 4/5 - 7/8 = 0 2 = 4/5 - 9/8 = 0 ™  32/38
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Part 12: Asymptotics for the Regression Model Setting Up the Wald Statistic To do the Wald test, I first need to estimate the asymptotic covariance matrix for the sample estimates of 1 and 2. After estimating the regression by least squares, the estimates are f1 = b4/b5 - b7/b8 f2 = b4/b5 - b9/b8. Then, using the delta method, I will estimate the asymptotic variances of f1 and f2 and the asymptotic covariance of f1 and f2. For this, write f1 = f1( b ), that is a function of the entire 101 coefficient vector. Then, I compute the 110 derivative vectors, d 1 = f1( b )/ b and d2 = f2( b )/ b These vectors are 1 2 3 4 5 6 7 8 9 10 d 1 = 0, 0, 0, 1/b5, -b4/b52, 0, -1/b8, b7/b82, 0, 0 d 2 = 0, 0, 0, 1/b5, -b4/b52, 0, 0, b9/b82, -1/b8, 0 ™  33/38
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Part 12: Asymptotics for the Regression Model Wald Statistics Then, D = the 210 matrix with first row d 1 and second row d 2. The estimator of the asymptotic covariance matrix of [f1,f2] (a 21 column vector) is V = D  s2 ( XX )-1  D. Finally, the Wald test of the
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