Non-Robust Dynamic Inferences from Macroeconometric Models: Bifurcation Stratification of Confidence Regions

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.Bifurcation, inference, dynamic general equilibrium, Pareto optimality, Hopf bifurcation, Euler equations, New Keynesian macroeconometrics, Bergstrom-Wymer model.

Non-Robust Dynamic Inferences from Macroeconometric Models: Bifurcation Stratification of Confidence Regions

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 many 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. 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. 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.Bifurcation, Hopf bifurcation, Euler equations, New Keynesian macroeconometrics, Bergstrom-Wymer model

Suppression of Limit Cycle Oscillations using the Nonlinear Tuned Vibration Absorber

The objective of the present study is to mitigate, or even completely eliminate, the limit cycle oscillations in mechanical systems using a passive nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA). An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is not imposed a priori, as it is the case for most existing nonlinear absorbers. The NLTVA parameters are determined analytically using stability and bifurcation analyses, and the resulting design is validated using numerical continuation. The proposed developments are illustrated using a Van der Pol-Duffing primary system

Empirical assessment of bifurcation regions within new Keynesian models

As is well known in systems theory, the parameter space of most dynamic models is stratified into subsets, each of which supports a different kind of dynamic solution. Since we do not know the parameters with certainty, knowledge of the location of the bifurcation boundaries is of fundamental importance. Without knowledge of the location of such boundaries, there is no way to know whether the confidence region about the parameters’ point estimates might be crossed by one or more such boundaries. If there are intersections between bifurcation boundaries and a confidence region, the resulting stratification of the confidence region damages inference robustness about dynamics, when such dynamical inferences are produced by the usual simulations at the point estimates only. Recently, interest in policy in some circles has moved to New Keynesian models, which have become common in monetary policy formulations. As a result, we explore bifurcations within the class of New Keynesian models. We study different specifications of monetary policy rules within the New Keynesian functional structure. In initial research in this area, Barnett and Duzhak (2008) found a New Keynesian Hopf bifurcation boundary, with the setting of the policy parameters influencing the existence and location of the bifurcation boundary. Hopf bifurcation is the most commonly encountered type of bifurcation boundary found among economic models, since the existence of a Hopf bifurcation boundary is accompanied by regular oscillations within a neighborhood of the bifurcation boundary. Now, following a more extensive and systematic search of the parameter space, we also find the existence of Period Doubling (flip) bifurcation boundaries in the class of models. Central results in this research are our theorems on the existence and location of Hopf bifurcation boundaries in each of the considered cases. We also solve numerically for the location and properties of the Period Doubling bifurcation boundaries and their dependence upon policy-rule parameter settings.Bifurcation; dynamic general equilibrium; Hopf bifurcation; flip bifurcation; period doubling bifurcation; robustness; New Keynesian macroeconometrics; Taylor rule; inflation targeting

Empirical Assessment of Bifurcation Regions within New Keynesian Models

As is well known in systems theory, the parameter space of most dynamic models is stratified into subsets, each of which supports a different kind of dynamic solution. Since we do not know the parameters with certainty, knowledge of the location of the bifurcation boundaries is of fundamental importance. Without knowledge of the location of such boundaries, there is no way to know whether the confidence region about the parameters’ point estimates might be crossed by one or more such boundaries. If there are intersections between bifurcation boundaries and a confidence region, the resulting stratification of the confidence region damages inference robustness about dynamics, when such dynamical inferences are produced by the usual simulations at the point estimates only. Recently, interest in policy in some circles has moved to New Keynesian models, which have become common in monetary policy formulations. As a result, we explore bifurcations within the class of New Keynesian models. We study different specifications of monetary policy rules within the New Keynesian functional structure. In initial research in this area, Barnett and Duzhak (2008) found a New Keynesian Hopf bifurcation boundary, with the setting of the policy parameters influencing the existence and location of the bifurcation boundary. Hopf bifurcation is the most commonly encountered type of bifurcation boundary found among economic models, since the existence of a Hopf bifurcation boundary is accompanied by regular oscillations within a neighborhood of the bifurcation boundary. Now, following a more extensive and systematic search of the parameter space, we also find the existence of Period Doubling (flip) bifurcation boundaries in the class of models. Central results in this research are our theorems on the existence and location of Hopf bifurcation boundaries in each of the considered cases. We also solve numerically for the location and properties of the Period Doubling bifurcation boundaries and their dependence upon policy-rule parameter settings.Bifurcation, dynamic general equilibrium, Hopf bifurcation, flip bifurcation, period doubling bifurcation, robustness, New Keynesian macroeconometrics, Taylor rule, inflation targeting.

Non-Robust Dynamic Inferences from Macroeconometric Models: Bifurcation Stratification of Confidence Regions

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 policy-relevant Bergstrom-Wymer model appear to be generic.Bifurcation; inference; dynamic general equilibrium; Pareto optimality; Hopf bifurcation; Euler equations; New Keynesian macroeconometrics; Bergstrom-Wymer model