ECON
Econometrics-I-12

Hypothesis that 0 is carried out by using the chi

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hypothesis that = 0 is carried out by using the chi-squared statistic W = ( f-0)V-1(f-0) . This is a chi-squared statistic with 2 degrees of freedom. The critical value from the chi- squared table is 5.99, so if my sample chi-squared statistic is greater than 5.99, I reject the hypothesis. ™  34/38

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Part 12: Asymptotics for the Regression Model Wald Test In the example below, to make this a little simpler, I computed the 10 variable regression, then extracted the 51 subvector of the coefficient vector c = (b4,b5,b7,b8,b9) and its associated part of the 1010 covariance matrix. Then, I manipulated this smaller set of values. ™  35/38
Part 12: Asymptotics for the Regression Model Application of the Wald Statistic ? Extract subvector and submatrix for the test matrix;list ; c =[b(4)/b(5)/b(7)/b(8)/b(9)]\$ matrix;list ; vc=[varb(4,4)/ varb(5,4),varb(5,5)/ varb(7,4),varb(7,5),varb(7,7)/ varb(8,4),varb(8,5),varb(8,7),varb(8,8)/ varb(9,4),varb(9,5),varb(9,7),varb(9,8),varb(9,9)]\$ ? Compute derivatives calc ;list ; g11=1/c(2); g12=-c(1)*g11*g11; g13=-1/c(4); g14=c(3)*g13*g13 ; g15=0 ; g21=g11 ; g22=g12 ; g23=0 ; g24=c(5)/c(4)^2 ; g25=-1/c(4)\$ ? Move derivatives to matrix matrix;list; dfdc=[g11,g12,g13,g14,g15 / g21,g22,g23,g24,g25]\$ ? Compute functions, then move to matrix and compute Wald statistic calc;list ; f1=c(1)/c(2) - c(3)/c(4) ; f2=c(1)/c(2) - c(5)/c(4) \$ matrix ; list; f = [f1/f2]\$ matrix ; list; vf=dfdc * vc * dfdc' \$ matrix ; list ; wald = f' * <vf> * f\$ (This is all automated in the WALD command.) ™  36/38

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Part 12: Asymptotics for the Regression Model Computations Matrix C is 5 rows by 1 columns. 1 1 -0.2948 -0.2015 1.506 0.9995 -0.8179 Matrix VC is 5 rows by 5 columns. 1 2 3 4 5 1 0.6655E-01 0.9479E-02 -0.4070E-01 0.4182E-01 -0.9888E-01 2 0.9479E-02 0.5499E-02 -0.9155E-02 0.1355E-01 -0.2270E-01 3 -0.4070E-01 -0.9155E-02 0.8848E-01 -0.2673E-01 0.3145E-01 4 0.4182E-01 0.1355E-01 -0.2673E-01 0.7308E-01 -0.1038 5 -0.9888E-01 -0.2270E-01 0.3145E-01 -0.1038 0.2134 G11 = -4.96184 G12 = 7.25755 G13= -1.00054 G14 = 1.50770 G15 = 0.000000 G21 = -4.96184 G22 = 7.25755 G23 = 0 G24 = -0.818753 G25 = -1.00054 DFDC=[G11,G12,G13,G14,G15/G21,G22,G23,G24,G25] Matrix DFDC is 2 rows by 5 columns. 1 2 3 4 5 1 -4.962 7.258 -1.001 1.508 0.0000 2 -4.962 7.258 0.0000 -0.8188 -1.001 F1= -0.442126E-01 F2= 2.28098 F=[F1/F2] VF=DFDC*VC*DFDC' Matrix VF is 2 rows by 2 columns. 1 2 1 0.9804 0.7846 2 0.7846 0.8648 WALD Matrix Result is 1 rows by 1 columns. 1 1 22.65 ™  37/38
Part 12: Asymptotics for the Regression Model Noninvariance of the Wald Test   38/38
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