A Mirror Image Invariance for M-Estimators

Abstract

MUCH RESEARCH INTO THE PERFORMANCE of econometric estimators utilizes Monte Carlo experiments, either as a primary tool or as a check on the accuracy of first or higher order asymptotic approximations. It is important to design efficient experiments and to extract all the information they provide. But it is also important to avoid constructing experiments which give an overly optimistic view of the performance of an estimator or test. This note gives a theorem concerning choice of covariate distribution which is relevant to both these considerations. It is shown that, when a subset of covariates, X, has a symmetric conditional distribution given the remaining covariates, Z, a mirror image invariance applies to M-estimators (Huber (1977)) of the coefficients that measure the impact of X on the conditional distribution of the response variates, namely that reversal of the signs of the coefficients causes the marginal2 sampling distributions of their M-estimators to be reflected around the origin while the marginal sampling distributions of estimators of other parameters are unaffected. It follows that, when covariate distributions are symmetric, results for one half of the parameter space can be deduced from results for the other half. For example, if bias(b)( 3) and MSE b)(P) are respectively the bias and mean squared error (assumed for the moment to exist) of an estimator ,3 when the true value of ,3 is b, then bias(b)(13)= -bias(b)(!3) and MSE(b(3) =MSE( b(P). It follows that bias(O)(13)=0 and, more generally, the result implies that sampling distributions are symmetric when coefficients associated with symmetrically distributed covariates are zero. This typically removes the (n- 1/2) term from the Edgeworth expansion of the estimator and associated t statistics. Therefore choosing a symmetric covariate distribution in a Monte Carlo experiment may lead to a false view of the accuracy of first order approximations to distributions of certain estimators and test statistics. The theorem highlights the potential sensitivity of the results of Monte Carlo experiments to the choice of covariate distribution. The mirror image result and the associated symmetry result are unusual because they do not require strong assumptions about the distribution of the response variate, in contrast to the symmetry results described by Andrews (1986) and the mirror image invariance given by Cryer, Nankervis, and Savin (1989). Apart from the symmetry of the conditional distribution of a subset of covariates, the main requirements are that the coefficients of interest, /3, satisfy two related single index restrictions, namely that they