Volume 6, Issue 1

Let H be an infinite-dimensional real separable Hilbert space. Given an unknown mapping M:H (r)H that can only be observed with noise, we consider two modified Robbins-Monro procedures to estimate the zero point ?o ( H of M. These procedures work in appropriate finite dimensional sub-spaces of growing dimension. Almost-sure convergence, functional central limit theorem (hence asymptotic normality), law of iterated logarithm (hence almost-sure loglog rate of convergence), and mean rate of convergence are obtained for Hilbert space-valued mixingale, (-dependent error processes.

In this paper we construct a model of stock market, interest rate and output interaction which is a generalization of the well known 1981 model of Blanchard. We allow for imperfect substitutability between stocks and bonds in the asset market and for lagged portfolio adjustment. The reaction of agents to changes in the stock market is dependent on the state of the economy. We analyze the dynamics of the model and its local stability properties. A discretization in terms of observable variables is derived. Some empirical results for U.S. output, stock price and interest rate data are presented using nonlinear least square estimates. We perform some stochastic simulations of the estimated non-linear model, obtaining distributions of the key economic quantities, their autocorrelation structure and financial statistics which are compared with historical data and RBC models. In addition, following Mittnik and Zadrozny (1993) a VAR with confidence bands for historical data is estimated and cumulative impulse-response functions compared to the model's impulse response functions. We find that the model captures a number of features of the data.Acknowledgements: Willi Semmler wants to acknowledge financial support from the CEPA of the New School University and the Ministry of Science and Technology of the State of Northrhine-Westfalia, Germany. Carl Chiarella acknowledges support from Australia Research Council grant number: A79802872.

The purpose of this paper is to show how the stability properties of non-linear dynamic models may be characterized and studied, where the degree of stability is defined by the effects of exogenous shocks on the evolution of the observed stochastic system. This type of stability concept is frequently of interest in economics, e.g., in real business cycle theory.We argue that smooth Lyapunov exponents can be used to measure the degree of stability of a stochastic dynamic model. It is emphasized that the stability properties of the model should be considered when the volatility of the variable modelled is of interest. When a parametric model is fitted to observed data, an estimator of the largest smooth Lyapunov exponent is presented which is consistent and asymptotically normal. The small sample properties of this estimator are examined in a Monte Carlo study. Finally, we illustrate how the presented framework can be used to study the degree of stability and the volatility of an exchange rate.

Several authors have suggested that, instead of being unit root processes, some macro variables may actually be stationary around nonlinear deterministic trends (Perron, 1989, 1990, Bierens, 1997). This paper investigates this for four variables in a standard money demand specification, using Canadian data. Evidence is first presented that the null of unit root with drift (constant, linear, or nonlinear) can be rejected in favor of nonlinear trend stationarity for the variables. Then, Bierens' (2000) nonlinear co-trending test finds two common nonlinear trends among the variables. The trends are consistent with a standard money demand relationship. All unit root and co-trending test conclusions are based on size and power results from Monte Carlo simulations as well as on asymptotic critical values. The paper concludes with a discussion of how the observed nonlinear trending and co-trending might arise in a theoretical model, and with implications for further empirical tests.