Grandmont (1985) found that the parameter space of the most classical dynamic models are stratified into an infinite number of subsets supporting an infinite number of different kinds of dynamics, from monotonic stability at one extreme to chaos at the other extreme, and with all forms of multiperiodic dynamics between. The econometric implications of Grandmont¡¯s findings are particularly important, if bifurcation boundaries cross the confidence regions surrounding parameter estimates in policy-relevant models. Stratification of a confidence region into bifurcated subsets seriously damages robustness of dynamical inferences. But Grandmont provided his result with a model in which all policies are Ricardian equivalent, no frictions exist, employment is always full, competition is perfect, and all solutions are Pareto optimal. Hence he was not able to reach conclusions about the policy relevance of his dramatic discovery. As a result, Barnett and He (1999, 2001, 2002) investigated a Keynesian structural model, and found results supporting Grandmont¡¯s conclusions within the parameter space of the Bergstrom-Wymer continuous-time dynamic macroeconometric model of the UK economy. That highly regarded, prototypical Keynesian model was produced from a system of second order differential equations. The model contains frictions through adjustment lags, displays reasonable dynamics fitting the UK economy¡¯s data, and is clearly policy relevant. Criticism of Keynesian structural models by the Lucas critique have motivated development of Euler equations models having policy-invariant deep parameters, which are invariant to policy rule changes. Hence, Barnett and He (2006) chose to continue the investigation of policy-relevant bifurcation by searching the parameter space of the best known of the Euler equations macroeconometric models: the Leeper and Sims (1994) model. The results further confirm Grandmont¡¯s views. Even more recently, interest in policy in some circles has moved to New Keynesian models. As a result, in this paper we explore bifurcation within the class of New Keynesian models. We develop the econometric theory needed to locate bifurcation boundaries in log-linearized New-Keynesian models with Taylor policy rules or inflation-targeting policy rules. Empirical implementation will be the subject of a future paper, in which we shall solve numerically for the location and properties of the bifurcation boundaries and their dependency upon policy-rule parameter settings. Central results needed in this research are our theorems on the existence and location of Hopf bifurcation boundaries in each of the cases that we consider. We provide the proofs of those propositions in this paper. One surprising result from these proofs is the finding that a common setting of a parameter in the future-looking New-Keynesian model can put the model directly onto a Hopf bifurcation boundary. Beginning with Grandmont¡¯s findings with a classical model, we continue to follow the path from the Bergstrom-Wymer policy-relevant Keynesian model, then to the Euler equation macroeconomic models, and now to the New Keynesian models. So far, all of our results suggest that Barnett and He¡¯s initial findings with the path-breaking policy-relevant Bergstrom-Wymer model appear to be generic.
