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I gather from reading various textbooks that the combination of a lagged dependent variable and serially correlated residuals is likely to lead to inconsistent estimators. Would this still be the case if I included an MA(1) term, so that the first order serial correlation is corrected for? If not then what would be the best way to proceed?

It is more likely that you want to use AR(1). But so long as you properly specify the ARMA terms, EViews will get the estimation right.