2038 part 12 asymptotics for the regression model

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Part 12: Asymptotics for the Regression Model General Result for Wald Distance The Wald distance measure: If plim xn = , xn is asymptotically normally distributed with a mean of and variance , and if S n is a consistent estimator of , then the Wald statistic, which is a generalized distance measure between x n converges to a chi- squared variate. ( x n - ) Sn-1 ( x n - )  2[K] ™  21/38
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Part 12: Asymptotics for the Regression Model The F Statistic An application: (Familiar) Suppose b n is the least squares estimator of based on a sample of n observations. No assumption of normality of the disturbances or about nonstochastic regressors is made. The standard F statistic for testing the hypothesis H0: R - q = 0 is F[J, n-K] = [( e*’e* - e’e )/J] / [ e’e / (n-K)] where this is built of two sums of squared residuals. The statistic does not have an F distribution. How can we test the hypothesis? ™  22/38
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Part 12: Asymptotics for the Regression Model JF is a Wald Statistic F[J,n-K] = (1/J)  ( Rbn - q ) [ R s2( XX )-1 R’ ]-1 ( Rbn - q ). Write m = ( Rbn - q ). Under the hypothesis, plim m = 0 . n m  N[0, R (2/n) Q -1 R ’] Estimate the variance with R (s2/n)( X’X /n)-1 R ’] Then, (n m )’ [Est.Var(n m )]-1 (n m ) fits exactly into the apparatus developed earlier. If plim b n = , plim s2 = 2, and the other asymptotic results we developed for least squares hold, then JF[J,n-K]  2[J]. ™  23/38
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Part 12: Asymptotics for the Regression Model Application: Wald Tests read;nobs=27;nvar=10;names= Year, G , Pg, Y , Pnc , Puc , Ppt , Pd , Pn , Ps $ 1960 129.7 .925 6036 1.045 .836 .810 .444 .331 .302 1961 131.3 .914 6113 1.045 .869 .846 .448 .335 .307 1962 137.1 .919 6271 1.041 .948 .874 .457 .338 .314 1963 141.6 .918 6378 1.035 .960 .885 .463 .343 .320 1964 148.8 .914 6727 1.032 1.001 .901 .470 .347 .325 1965 155.9 .949 7027 1.009 .994 .919 .471 .353 .332 1966 164.9 .970 7280 .991 .970 .952 .475 .366 .342 1967 171.0 1.000 7513 1.000 1.000 1.000 .483 .375 .353 1968 183.4 1.014 7728 1.028 1.028 1.046 .501 .390 .368 1969 195.8 1.047 7891 1.044 1.031 1.127 .514 .409 .386 1970 207.4 1.056 8134 1.076 1.043 1.285 .527 .427 .407 1971 218.3 1.063 8322 1.120 1.102 1.377 .547 .442 .431 1972 226.8 1.076 8562 1.110 1.105 1.434 .555 .458 .451 1973 237.9 1.181 9042 1.111 1.176 1.448 .566 .497 .474 1974 225.8 1.599 8867 1.175 1.226 1.480 .604 .572 .513 1975 232.4 1.708 8944 1.276 1.464 1.586 .659 .615 .556 1976 241.7 1.779 9175 1.357 1.679 1.742 .695 .638 .598 1977 249.2 1.882 9381 1.429 1.828 1.824 .727 .671 .648 1978 261.3 1.963 9735 1.538 1.865 1.878 .769 .719 .698 1979 248.9 2.656 9829 1.660 2.010 2.003 .821 .800 .756 1980 226.8 3.691 9722 1.793 2.081 2.516 .892 .894 .839 1981 225.6 4.109 9769 1.902 2.569 3.120 .957 .969 .926 1982 228.8 3.894 9725 1.976 2.964 3.460 1.000 1.000 1.000 1983 239.6 3.764 9930 2.026 3.297 3.626 1.041 1.021 1.062 1984 244.7 3.707 10421 2.085 3.757 3.852 1.038 1.050 1.117 1985 245.8 3.738 10563 2.152 3.797 4.028 1.045 1.075 1.173 1986 269.4 2.921 10780 2.240 3.632 4.264 1.053 1.069 1.224 ™  24/38
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Part 12: Asymptotics for the Regression Model Data Setup Create; G=log(G); Pg=log(PG); y=log(y); pnc=log(pnc); puc=log(puc); ppt=log(ppt); pd=log(pd); pn=log(pn); ps=log(ps); t=year-1960$ Namelist; X=one,y,pg,pnc,puc,ppt,pd,pn,ps,t$ Regress; lhs=g;rhs=X;PrintVC$ ™  25/38
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Part 12: Asymptotics for the Regression Model Regression Model Based on the gasoline data: The regression equation is g =1 + 2y + 3pg + 4pnc + 5puc + 6ppt + 7pd + 8pn + 9ps + 10t +  All variables are logs of the raw variables, so that coefficients are elasticities. The new variable, t, is a time trend, 0,1,…,26, so that 10 is the autonomous yearly proportional growth in G. ™  26/38
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Part 12: Asymptotics for the Regression Model Least Squares Results +----------------------------------------------------+ | Ordinary least squares regression | | LHS=G Mean = 5.308616 | | Standard deviation = .2313508 | | Model size Parameters = 10 | | Degrees of freedom = 17 | | Residuals Sum of squares = .003776938 | | Standard error of e = .01490546 | | Fit R-squared = .9972859 | | Adjusted R-squared = .9958490 | | Model test F[ 9, 17] (prob) = 694.07 (.0000) | | Chi-sq [ 9] (prob) = 159.55 (.0000) | +----------------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant -5.97984140 2.50176400 -2.390 .0287 Y 1.39438363 .27824509 5.011 .0001 9.03448264 PG -.58143705 .06111346 -9.514 .0000 .47679491
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