Some tests of dynamic specification for a single equation

It may often be the case that we wish to explore the possibility that the errors on a structural equation are ARMA in form, as modelled in the equation (2). If we ignore the implicit constraints on the coefficients of f(L) which follow from fr(L) = p(L)a (L), we still retain a valid stochastic model, but (i) the estimates of a and p will be relatively inefficient, (ii) the dynamic structure of f(L) will be unnecessarily complicated compared with a (L), and (iii) prior constraints on a (L) when applied to f(L) may yield inconsistent estimates. A traditional approach to estimating the model defined by (3) would be to estimate (1) with errors generated by a relatively simple stochastic process of type (2), possibly using such estimates to test various restrictions on the equation (1). Then tests would be made for the adequacy of the formulation (2), and so on.