Stimulated by ideas of Hannan (1980), convergence results are proved for misspecified MA(1) models for pseudolinear regression (PLR---also referred to as AML and RML(1)) and RML2 under monitoring. RML2 requires monitoring to ensure invertibility of the estimates. Conditions are presented that guarantee convergence under a simple, implementable monitoring scheme. Examples are presented where PLR converges, but to a value that is suboptimal (under mean square prediction error) whereas RML2 converges to the optimal value. These seem to be the first rigorous recursive estimation results for misspecfied MA models. Finally, we indicate how these results could be generalized.