Pnc 29476979 25797920 1143 2690 28100132 puc 20153591

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
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Part 12: Asymptotics for the Regression Model
Covariance Matrix
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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
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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

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
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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
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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
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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
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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