Robustness and local linearisation in economic models

The Journal of Finance, 1991

SSRN Electronic Journal, 2015

We examine the inter-linkages between financial factors and real economic activity. We review the main theoretical approaches that allow financial frictions to be embedded into general equilibrium models. We outline, from a policy perspective, the most recent empirical papers focusing on the propagation of exogenous shocks to the economy, with a particular emphasis on works dealing with time variation of parameters and other types of nonlinearities. We then present an application to the analysis of the changing transmission of financial shocks in the euro area. Results show that the effects of a financial shock are time-varying and contingent on the state of the economy. They are of negligible importance in normal times but they greatly matter in conditions of stress.

Journal of Monetary Economics, 1988

In an attempt to resolve the controversies that exist within the field of economics regarding nonlinearity, chaos, and bifurcation, we investigate the relevancy to these controversies of a controlled competition among nonparametric econometric tests for nonlinearity and chaos, and we also report on our results with experiments using parametric macroeconomic models to investigate the implications of bifurcation for macroeconomic policy. These experiments are part of an ongoing research project. What we find so far is that existing views on nonlinearity, chaos, and bifurcation in economics are based upon oversimplified views that currently neither can be confirmed nor contradicted with empirical results that are now available. Since these issues are deep and difficult, considerably more research is needed before any serious conclusions on the subject can be stated with confidence. This fact is particularly true regarding the relevancy of nonlinearity, chaos, and bifurcation for macroeconomic stabilization policy.

Chaos, Solitons & Fractals, 2001

In this paper a discrete-time economic model is considered where the savings are proportional to income and the investment demand depends on the dierence between the current income and its exogenously assumed equilibrium level, through a nonlinear S-shaped increasing function. The model can be ultimately reduced to a two-dimensional discrete dynamical system in income and capital, whose time evolution is``driven'' by a family of two-dimensional maps of triangular type. These particular two-dimensional maps have the peculiarity that one of their components (the one driving the income evolution in the model at study) appears to be uncoupled from the other, i.e., an independent one-dimensional map. The structure of such maps allows one to completely understand the forward dynamics, i.e., the asymptotic dynamic behavior, starting from the properties of the associated one-dimensional map (a bimodal one in our model). The equilibrium points of the map are determined, and the in¯uence of the main parameters (such as the propensity to save and the ®rms' speed of adjustment to the excess demand) on the local stability of the equilibria is studied. More important, the paper analyzes how changes in the parameters' values modify both the asymptotic dynamics of the system and the structure of the basins of the dierent and often coexisting attractors in the phase-plane. Finally, a particular``global'' (homoclinic) bifurcation is illustrated, occurring for suciently high values of the ®rms' adjustment parameter and causing the switching from a situation of bi-stability (coexistence of two stable equilibria, or attracting sets of dierent nature) to a regime characterized by wide chaotic oscillations of income and capital around their exogenously assumed equilibrium levels. Ó : S 0 9 6 0 -0 7 7 9 ( 0 0 ) 0 0 0 5 5 -2

Journal of Monetary Economics, 1988

An empirical assessmez? of a linear-stochastic perspective for Canadian macroecono series is presented. The methods used are based on the mathematics of 'chaos'. Present evidence suggests that low-order deterministic chaos does not provide a satisfactory characterization of the data. The absence of significant nonlinear structure for the investment and unemployment series is of particular note in light of past findings with American data. The degree to which the use of a time trend can impose a pseudo-structure on the data is illhlstrated. *This research was partly supported by a grant from the Research Excellence Program of University of Guelph. We would like to thank William Brock, Roger Farmer, Chera Sayers, Jose Scheinkman for helpful discussions ab chaos. ne suggestions of an anonymous referee were quite useful. Our i~tellectu~ debt to acknowledged. Any deficiencies remain our responsibility.

Studies in Nonlinear Dynamics & Econometrics, 2000

Many monetary and fiscal policy measures have aimed at mitigating the effects of the financial market meltdown that started in the U.S. subprime sector in 2008 and has subsequently spread world wide as a great recession. Slowly some recovery appears to be on the horizon, yet it is worthwhile exploring the fragility and potentially destabilizing feedbacks of advanced macroeconomies in the context of a framework that builds on the ideas of Keynes and Tobin. This framework stresses the fragilities and destabilizing feedback mechanisms that are potential features of all major markets-those for goods, labor, and financial assets. We use a Tobin macroeconomic portfolio approach and the interaction of heterogeneous agents on the financial market to characterize the potential for financial market instability. Though the study of the latter has been undertaken in many partial models, we focus here on the interconnectedness of all three markets. Furthermore, we study what potential labor market, fiscal and monetary policies can have in stabilizing unstable macroeconomies. In order to study this problem we introduce, besides money, long term bonds and equity into the asset market. We in particular propose a countercyclical monetary policy that sells assets in the boom and purchases them in recessions. Modern stability analysis is brought to bear to demonstrate the stabilizing effects of the suggested policies. The policies suggested here could help the Fed in its search for an appropriate exit strategy after its massive intervention in the financial market. * The authors acknowledge the comments and suggestions from the audience at the conference in honour of Professor Giancarlo Gandolfo. The usual caveat applies. the financial market), and the period we have just lived through could easily occur 2

Structural Change and Economic Dynamics, 2000

This paper introduces and discusses an heuristic model meant to clarify why and how economic instability may play a crucial role in a modern sophisticated monetary economy. In this model economic instability is specified in terms of structural instability rather than in the usual terms of dynamic instability. This different view of instability implies a different approach to the analysis of the dynamic behaviour of the economic system and of its structural changes. In particular, the qualitative changes in the economic behaviour of the economic system are seen not as purely exogenous as in the received view but as essentially endogenous. This approach is applied to the analysis of financial crises and of their impact on the fluctuations of a sophisticated monetary economy. The crucial variable, the degree of financial fragility of the economic units, is specified in terms of structural instability, and this implies that, beyond certain thresholds of its value, the qualitative characteristics of their dynamic behaviour change radically in such a way to produce cyclical, though fairly irregular, fluctuations. The interplay between these microeconomic fluctuations is sufficient to produce cyclical macroeconomic fluctuations whose characteristics and implications for policy are briefly examined.

The Pakistan Development Review, 1994

Recently there has been an increased interest in the theory of chaos by macroeconomists and fmancial economists. Originating in the natural sciences, applications of the theory have spread through various fields including brain research, optics, metereology, and economics. The attractiveness of chaotic dynamics is its ability to generate large movements which appear to be random, with greater frequency than linear models. Two of the most striking features of any macro-economic data are its random-like appearance and its seemingly cyclical character. Cycles in economic data have often been noticed, from short-run business cycles, to 50 years Kodratiev waves. There have been many attempts to explain them, e.g. Lucas (1975), who argues that random shocks combined with various lags can give rise to phenomena which have the appearance of cycles, and Samuelson (1939) who uses the familiar multiplier accelerator model. The advantage of using non-linear difference (or differential) equation...

Studies in Nonlinear Dynamics & Econometrics, 1996

The possibility of cycles and chaos arising from nonlinear dynamics in economics emerged in the literature in the 1980s, and it came as a surprise. 1 The possibility of deterministic cycles in economic models had been noted before, for example in the well-known multiplier-accelerator models, but not in equilibrium models with complete markets, no frictions, and full intertemporal arbitrage. 2 The reason for the surprise was understandable: deterministic fluctuations in equilibrium models involve predictable changes in relative prices which should be ruled out by intertemporal arbitrage. In models of overlapping generations, however, finite lives can restrict complete arbitrage over time. As a result, some people thought, and still think, that cycles that are shorter than the agents' postulated lifespans would not be possible in equilibrium models, and therefore are irrelevant for business-cycle analysis. This view is clearly wrong, and of course ignores the extensive literature on cycles and chaos in optimal growth models with infinitely lived agents. In such models deterministic cycles in relative prices occur easily, but the amplitudes of the cycles remain within bounds of the discount rate. 3 It is not difficult to show in the context of multisector growth models, say with Cobb-Douglas production functions, that for any positive discount rate there is a large class of technologies for which cycles occur. (See Benhabib and Rustichini [1990].) Getting chaos, however, is harder. Recent works by Sorger (1992), by Mitra (1995), and by Nishimura and Yano (1995) give lower bounds for the discount rate, below which chaos is ruled out for one-sector models of optimal growth. Yet even in that context, going to a multisector framework may considerably lower the bounds on the discount rate thus far established. A second reason for the attention that chaotic dynamics received in the economics literature regards prediction. The common wisdom has been that economic fluctuations are driven by exogenous shocks. Chaotic dynamics not only supplied an alternative explanation for at least some part of economic fluctuations, but also provided an excuse for economists' difficulties with forecasting. Sometimes, however, an important feature of chaotic dynamics that makes forecasting difficult, namely, sensitive dependence on initial conditions, is used in a cavalier way to explain short-run dynamics, forgetting that the effect of sensitive dependence becomes significant only after some periods, but not in the very short run. When it became obvious that very-standard equilibrium models could easily generate cycles and chaos, the attention in the literature naturally turned to the empirical plausibility of such dynamics. The most interesting approach, inspired by developments in natural sciences and mathematics, was also atheoretical, and reminiscent of VAR methods of time series. 4 The idea was to try to infer whether a particular economic time series was generated by a deterministic, low (at most four-or five-) dimensional system that was chaotic, or whether it came from a simple (linear) stochastic system. It is not difficult to see that such inferences are hard to make when the time-series data is short, as is the case with most economic series, with the exception of financial data. It is not surprising, then, that many applications of this approach are in the area of finance, but even there, where we have very high-frequency data, it is hard to pick up fluctuations that may occur at lower

2006

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.

Unpublished Ph. D. dissertation, …, 1995

We have benefitted from helpful comments from presentations of earlier versions of this paper at seminars and at meetings of the Society for Nonlinear Dynamics and Econometrics, the Western Economic Association and the Eastern Finance Association. Parts of this paper are based ...

Journal of Economic Behavior & Organization, 2006

This paper proposes a simple prototype model that describes the complex dynamics of a sophisticated monetary economy. The interaction between the current and intertemporal financial constraints of economic units brings about irregular fluctuations at the micro and macro levels. By means of qualitative dynamic analysis and numerical simulations, we reformulate in more operational terms, and extend in a number of new directions, the model suggested recently by one of the authors (Vercelli, 2000) to study the interaction between financial fragility, modelled in terms of structural instability, and dynamically unstable financial fluctuations.

2013

Following Mulligan and Sala-i-Martin (1993) we study a general class of endogenous growth models formalized as a non linear autonomous three-dimensional differential system. We consider the abstract model. By using the Shilnikov Theorem statements, we determine the parameters space in which the condition for the existence of a homoclinic Shilnikov orbit and Smale horseshoe chaos are true. The Lucas model (1998) can be considered as an application of the general result. The series expression of the homoclinic orbit is derived by the undetermined coefficient method. We show the optimality for the solutions path based on the Shilnikov Theorem. Some economic implications of this analysis are discussed.

2010

The impact of increasing leverage in the economy produces hyperreaction of market participants to variations of their revenues. If the income of banks decreases, they mass-reduce their lendings; if corporations sales drop, and due to existing debt they cannot adjust their liquidities by further borrowings, then they must immediately reduce their expenses, lay off staff, and cancel investments. This hyperreaction produces a bifurcation mechanism, and eventually a strong dynamical instability in capital markets, commonly called systemic risk. In this article, we show that this instability can be monitored by measuring the highest eigenvalue of a matrix of elasticities. These elasticities measure the reaction of each sector of the economy to a drop in its revenues from another sector. This highest eigenvalue-the spectral radius-of the elasticity matrix, can be used as an early indicator of market instability and potential crisis. Grandmont (1985) and subsequent research showed the possibility that the "invisible hand" of markets become chaotic, opening the door to uncontrolled swings. Our contribution is to provide an actual way of measuring how close to chaos the market is. Estimating elasticities and actually generating the indicators of instability will be the topic of forthcoming research.