Albert Bergstrom

CONTINUOUS TIME STOCHASTIC MODELS AND ISSUES OF AGGREGATION OVER TIME

Publisher Summary This chapter describes statistical methods that are applicable to a class of continuous time stochastic models and discusses the theoretical foundations of these methods. An important feature of the class of models is that such models allow for the incorporation of a priori restrictions, such as those derived from economic theory, through the structural parameters of the continuous time system. They are used to represent a dynamic system of causal relations in which each variable adjusts continuously in response to the stimulus provided by other variables, and the adjustment relations involve the basic structural parameters in some optimization theory. These structural parameters are estimated from a sample comprising a sequence of discrete observations of the variables that are a mixture of stock variables and flow variables. In this way, it is possible to take advantage of the a priori restrictions derived from economic theory without making the unrealistic assumption that the economy moves in discrete jumps among successive positions of temporary equilibrium. On the theoretical side an important and relatively unexplored field of research is in the development of methods of estimation for systems of nonlinear stochastic differential equations. In some cases, it is possible to derive the exact likelihood function in terms of the discrete observations generated by a system of nonlinear stochastic differential equations. But, more generally, approximate methods are relied on, possibly involving the use of numerical solutions to the nonlinear differential equations system.

Continuous time econometric modelling

Part 1 Theoretical models of cyclical growth: a model of technical progress, the production function and cyclical growth monetary phenomena and economic growth - a synthesis of neoclassical and Keynsian theories. Part 2 Estimation, forecasting and control: nonrecursive models as discrete approximations to systems of stochastic differential equations Gaussian estimation of structural parameters in higher order continuous time dynamic models the estimation of parameters in nonstationary higher order continuous time dynamic models the estimation of open higher order continuous time dynamic models with mixed stock and flow data hypothesis testing in continuous time econometric models optimal forecasting of discrete stock and flow data generated by a higher order continuous time system optimal control in wide-sense stationary continous time stochastic models. Part 3 Applications: a model of disequilibrium neoclassical growth and its application to the United Kingdom, C.R.Wymer monetary, fiscal, and exchange rate policy in a continuous time econometric model of the United Kingdom Gaussian estimation of a continuous time model of demand for consumer durable goods with applications to demand in the United Kingdom, 1973-84, M.J.Chambers.

The History of Continuous-Time Econometric Models

Although it is only during the last decade that continuous-time models have been extensively used in applied econometric work, the development of statistical methods applicable to such models commenced over 40 years ago. The first significant contribution to the problem of estimating the parameters of continuous-time stochastic models from discrete data was made by the British statistician Bartlett (1946) only three years after the pioneer contribution of Haavelmo (1943) on simultaneous-equations models. Moreover, by this time the fundamental mathematical theory of continuous-time stochastic models was already well-developed, major contributions having been made by some of the leading mathematicians of the twentieth century, including Einstein, Wiener and Kolmogorov.

THE ESTIMATION OF PARAMETERS IN NONSTATIONARY HIGHER-ORDER CONTINUOUS-TIME DYNAMIC MODELS

This paper is concerned with derivation of a new efficient algorithm for computing the exact Gaussian likelihood for structural parameters in nonstationary higher-order continuous-time dynamic models and with its application in the estimation of these parameters. The algorithm completely avoids the computation of the covariance matrix of the observations and is applicable to a system of any order with mixed stock and flow data. It is used as the basis for an iterative procedure in which the structural parameters and the initial state vector are estimated alternately.

The Estimation of Open Higher-Order Continuous Time Dynamic Models with Mixed Stock and Flow Data

This article extends recent work on the Gaussian or quasi-maximum likelihood estimation of the parameters of a closed higher-order continuous time dynamic model by introducing exogenous variables into the model The method presented yields exact maximum likelihood estimates when the innovations are Gaussian and the exogenous variables are polynomials in time of degree not exceeding two, and it can be expected to yield very good estimates under more general conditions. It is applicable, in principle, to a system of any order with mixed stock and iow data. The precise formulas for its implementation are derived, in this article, for a second-order system in which both the endog-enous and exogenous variables are a mixture of stock and flow variables.

Gaussian estimation of a second order continuous time macroeconometric model of the UK

Abstract This paper describes the first application to an economy-wide macroeconometric model of recently developed methods for the exact Gaussian estimation of higher order continuous time dynamic models. The new model is formulated as a system of second order differential equations, thus providing a much richer dynamic specification than the predominantly first order continuous time macroeconometric models developed during the last 15 years. It also makes intensive use of economic theory to obtain a parsimonious parametrization, is designed in such a way as to permit a rigorous mathematical investigation of its steady state and asymptotic stability properties, and makes systematic use of the assumption of long-run rational expectations. In addition to the exact Gaussian estimates of the structural parameters, the paper includes the first set of continuous lag distributions derived from estimates that take account of the exact restrictions on the distribution of the discrete data implied by a continuous time model. It also includes the first estimates of the coefficient matrices of the exact discrete model, in its VARMAX form, satisfied by the discrete stock and flow data generated by a higher order continuous time dynamic model.

Gaussian Estimation of Mixed-Order Continuous-Time Dynamic Models with Unobservable Stochastic Trends from Mixed Stock and Flow Data

This paper develops an algorithm for the exact Gaussian estimation of a mixed-order continuous-time dynamic model, with unobservable stochastic trends, from a sample of mixed stock and flow data. Its application yields exact maximum likelihood estimates when the innovations are Brownian motion and either the model is closed or the exogenous variables are polynomials in time of degree not exceeding two, and it can be expected to yield very good estimates under much more general circumstances. The paper includes detailed formulae for the implementation of the algorithm, when the model comprises a mixture of first- and second-order differential equations and both the endogenous and exogenous variables are a mixture of stocks and flows.

Monetary and fiscal policy in a second-order continuous time macroeconometric model of the United Kingdom

Abstract This paper is concerned with the application to monetary and fiscal policy analysis of a recently developed second-order continuous time macroeconometric model of the United Kingdom. It deals with the effect on stability of both simple ad hoc policy feedbacks and more sophisticated feedbacks derived by using optimal control theory. The latter are designed to minimize an infinite horizon quadratic cost function involving deviations of output, the exchange rate, and the monetary and fiscal policy instruments from their steady state paths and, also, the rates of change of the policy instruments.

Optimal control in wide-sense stationary continuous-time stochastic models

Abstract This paper provides a rigorous treatment of the optimal control problem for a continuous-time stochastic model, with an infinite-horizon quadratic cost function, under weaker assumptions concerning the white-noise innovations than have been made in the literature on this subject. Since it is not assumed that the innovations are generated by Brownian motion, or even that the sample paths of the state variables are integrable, the Ito calculus is inapplicable. The solution depends on a more recently developed theory of stochastic differential equations, based on random measure theory, and on the ergodic theorem for a wide-sense stationary process.

The Estimation of Nonparametric Functions in a Hilbert Space

This paper is concerned with the estimation of a nonlinear regression function which is not assumed to belong to a prespecified parametric family of functions. An orthogonal series estimator is proposed, and Hilbert space methods are used in the derivation of its properties and the proof of several convergence theorems. One of the main objectives of the paper is to provide the theoretical basis for a practical stopping rule which can be used for determining the number of Fourier coefficients to be estimated from a given sample.