  We consider a general nonlinear regression problem where the predictors
contain measurement error. It has been recently discovered that several
well-known dimension reduction methods, such as OLS, SIR and pHd, can be
performed on the surrogate regression problem to produce consistent estimates
for the original regression problem involving the unobserved true predictor. In
this paper we establish a general invariance law between the surrogate and the
original dimension reduction spaces, which implies that, at least at the
population level, the two dimension reduction problems are in fact equivalent.
Consequently we can apply all existing dimension reduction methods to
measurement error regression problems. The equivalence holds exactly for
multivariate normal predictors, and approximately for arbitrary predictors. We
also characterize the rate of convergence for the surrogate dimension reduction
estimators. Finally, we apply several dimension reduction methods to real and
simulated data sets involving measurement error to compare their performances.
