MAD/LAD Estimator: The “minimum absolute deviations” or “least absolute deviations” estimator is obtained by solving the following linear programming problem: ()111:sgnLADLADLADiiLADMinyx yxyxβββ−=−−∑∑1i; where all variables are written in deviation form (to avoid that pesky intercept term) and sgn( ) refers to the sign function (just take the signof the terms in parentheses. Thus, if a deviation is (-), then the sgn function returns (-). Plus: an approximate first order condition for this problem illustrates an appealing characteristic. The approximate first order condition is: 1sgnLADx−∑Suppose we change one of our yvalues from a positive number, to an extremely positive number. There is no effect on the estimator as the final term in the first order condition carries only the sign. As long as the sign doesn’t change, there is no effect on the
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