Econometrics-I-12

Part 12 asymptotics for the regression model setting

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Unformatted text preview: 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 101 coefficient vector. Then, I compute the 110 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, d 2 = 0, 0, 0, 1/b5, -b4/b52, 0, 0, b9/b82, -1/b8, &#152;&#152;&#152;&#152;™ ™ 33/38 Part 12: Asymptotics for the Regression Model Wald Statistics Then, D = the 210 matrix with first row d 1 and second row d 2. The estimator of the asymptotic covariance matrix of [f1,f2] (a 21 column vector) is V = D s2 ( XX )-1 D. Finally, the Wald test of the 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. &#152;&#152;&#152;&#152; ™ 34/38 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 51 subvector of the coefficient vector c = (b4,b5,b7,b8,b9) and its associated part of the 1010 covariance matrix. Then, I manipulated this smaller set of values. &#152;&#152;&#152;&#152; &#152;™ 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.) &#152;&#152;&#152;&#152; &#152;™ 36/38 Part 12: Asymptotics for the Regression Model Computations Matrix C is 5 rows by 1 columns....
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Part 12 Asymptotics for the Regression Model Setting Up the...

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