Generalized M-estimates (minimum contrast estimates) and their asymptotically equivalent approximate versions are considered. A relatively simple condition is found which is equivalent with consistency of all approximate M-estimates under wide assumptions about the model. This condition is applied in several directions. (i) A more easily verifiable condition equivalent with consistency of all approximate M-estimates is derived and illustrated on models with stationary and ergodic observations. (ii) A condition sufficient for inconsistency of all approximate M-estimates is obtained and illustrated on models with i.i.d. observations. (iii) A simple necessary and sufficient condition for consistency of all approximate M-estimates in linear regression with i.i.d. errors is found. This condition is weaker than sufficient conditions for consistency of M-estimators known from the literature. A linear regression example is presented where the M-estimate is consistent and an approximate M-estimate is inconsistent.
