A definition of extended memory is provided, generalizing the ideas of long memory and persistence, based on the properties of forecasts over long horizons. Specification of nonlinear models with variables having extended memory is considered in terms of the balance of an equation and it is suggested that many more types of misspecification can occur than with usual situations and could produce important specification errors. Tests of linearity and standard methods of nonlinear modeling are briefly considered and advice is given on circumstances in which they can be used.