Language Reference

CALL LAV (rc, xr, a, b <, x0> <, opt> ) ;

This subroutine is supported only by the IML procedure.

The LAV subroutine performs linear least absolute value regression by solving the norm minimization problem.

The LAV subroutine returns the following values:

The input arguments to the LAV subroutine are as follows:

Missing values are not permitted in the a or b argument. The argument is ignored if it contains any missing values. Missing values in the opt argument cause the default value to be used.

The LAV subroutine is designed for solving the unconstrained linear norm minimization problem,

for m equations with n (unknown) parameters . This is equivalent to estimating the unknown parameter vector, x, by least absolute value regression in the model

where b is the vector of n observations, A is the design matrix, and is a random error term.

An algorithm by Madsen and Nielsen (1993) is used, which can be faster for large values of m and n than the Barrodale and Roberts (1974) algorithm. The current version of the algorithm assumes that A has full column rank. Also, constraints cannot be imposed on the parameters in this version.

The norm minimization problem is more difficult to solve than the least squares ( norm) minimization problem because the objective function of the norm problem is not continuously differentiable (the first derivative has jumps). A function that is continuous but not continuously differentiable is called nonsmooth. By using PROC NLP and the nonlinear optimization subroutines, you can obtain the estimates in linear and nonlinear norm estimation (even subject to linear or nonlinear constraints) as long as the number of parameters, n, is small. Using the nonlinear optimization subroutines, there are two ways to solve the nonlinear norm, , problem:

Both approaches are successful only for a small number of parameters and good initial estimates. If you cannot supply good initial estimates, the optimal results of the corresponding nonlinear least squares ( norm) estimation can provide fairly good initial estimates.

Gonin and Money (1989) show that the nonlinear norm estimation problem

can be reformulated as a linear optimization problem with nonlinear constraints in the following ways.

For linear functions , , you obtain linearly constrained linear optimization problems, for which the number of variables and constraints is on the order of the number of observations, m. The advantage that the algorithm by Madsen and Nielsen (1993) has over the Barrodale and Roberts (1974) algorithm is that its computational cost increases only linearly with m, and it can be faster for large values of m.

In addition to computing an optimal solution that minimizes , you can also compute approximate standard errors and the approximate covariance matrix of . The standard errors can be used to compute confidence limits.

The following example is the same one used for illustrating the LAV subroutine by Lee and Gentle (1986). A and b are as follows:

The following statements specify the matrix A, the vector b, and the options vector opt. The options vector specifies that all output is printed (opt[2]=5), that the asymptotic standard errors and covariance matrix are computed based on the McKean-Schrader estimate of the variance of the median (opt[3]=0), and that the convergence test be performed (opt[4]=1).

a = { 0,  1, -1, -1,  2,  2 };
m = nrow(a);
a = j(m, 1, 1.) || a;
b = { 1,  2,  1, -1,  2,  4 };

opt= { . 5  0 1 };
call lav(rc, xr, a, b, , opt);

The first part of the output is shown in Figure 216. This output displays the least squares solution, which is used as the starting point. The estimates of the largest and smallest nonzero eigenvalues of give only an idea of the magnitude of these values, and they can be very crude approximations.

The second part of the printed output shows the iteration history. It is shown in Figure 217.

The third part of the output is shown in Figure 218. This output displays the norm solution (first row) together with asymptotic standard errors (second row) and the asymptotic covariance matrix of parameter estimates. The ASEs are the square roots of the diagonal elements of this covariance matrix.

The last part of the printed output shows the predicted values and residuals, as in Lee and Gentle (1986). It is shown in Figure 219.