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variables are chosen according to some pattern, care must be taken to choose data points such that none of the p + 1 vectors Xi is a linear combination of the others. 4.6d. Let us verify that b + t*d 3 = A* - IC*, as stated in the subsection on justification of the algorithm. We want to show that A*b + t*A*d J = c*. The components of this vector equation are x;b + t*x;d J = Yi (i = 1,2,4) and xkb + t*x k d 3 = Yk' The last equation follows from the definition of t*. From the fact that AA -I is the identity matrix, we see that x;d J = 0 for i = 1,2,4. Now note that x;b = Yi because Ab = c. 4.6e. The vector PLAD of least-absolute-deviations estimates has a variance-covariance matrix approximately equal to T 2( X' X) - I, where T is described in Section 4.4 and X is the matrix of explanatory variables with a vector of l's as the first column. Similarly, as seen in Note 3.7, the vector PLS of least-squares estimates has variance-covariance matrix 0"2(X' X)-I. Therefore the standar~ deviation of jJj, LAD is approximately T /0" times the standard deviation of f3 j , LS' 4.7a. Three different types of LAD tests were proposed by Koenker and Bassett (1982). The test in Section 4.7 is their likelihood ratio test. To describe the other two tests, partition the parameter vector into two parts 131 = (130,131"'" f3q) and 132 = (f3 q + I " ' " f3 p)' and let XI and X 2 be the matrices of corresponding explanatory variables. We want to test whether 132 is the zero vector. Let be its LAD estimate in the full model. The Wald test statistic is jJ~V2 IjJ2' where V2 is an estimate of COV(jJ2)' namely f 2 W, where W = (X 1X 2 - X 1X I(X;X I )-IX;X 2)-I. When p = 1 and q = 0, this is the square of the test statistic in Section 4.4. Let u be the vector of signs of the LAD residuals in the reduced model with 132 = O. The Lagrange multiplier test statistic is u' X 2WX 1 This test avoids the problem of estimating T. u. Koenker and Bassett (1982) showed that the three tests are asymptotically equivalent. For infinitely large samples, they have the same power and their p-values can be calculated by assuming the test statistic has a chi-squared distribution with p - q degrees of freedom. Dielman and Pfaffenberger (1990) applied the three tests to simulated data following a simple linear regression model for sample sizes n = 20, 40 and 100. For n = 20, the Lagrange multiplier test was more powerful than the other two tests when the error distribution was heavy-tailed, but the likelihood ratio test had somewhat greater power when the error distribution was normal. For n = 40 and n = 100, the likelihood ratio test tended to have greater power.
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4.7h. The modified calculation of the p-value based on G is recommended by Schrader and McKean (1987). 4.8. In S-PLUS, the function that performs LAD regression is IIftt. It uses the Barrodale-Roberts (1974) algorithm, as does ROBSYS and the IMSL
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subroutine RLLA V. This is basically the same algorithm as the one described in Section 4.6, but Barrodale and Roberts have devised refinements to speed up the algorithm.
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Additional Reading. For more on theory and computation for LAD regression, see the book by Bloomfield and Steiger (1983) and the conference proceedings edited by Dodge (1987a, 1987b, 1992). For a review article, see Dielman and Pfaffenberger (1982).
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