Marcus J. Chambers

Long Memory and Aggregation in Macroeconomic Time Series

This paper explores the interaction between long memory and aggregation. Results are derived which link the (possibly fractional) orders of integration of the aggregate series with those of the underlying series when the aggregation is either cross-sectional or temporal (in discrete or continuous time). These results provide empirically testable hypotheses that are examined using six U.K. macroeconomic series. A semiparametric method is found to be broadly consistent with the implications of the theory but fully parametric ARFIMA models show considerable variation. Each series appears to be integrated of an order of between one and one-and-a-half.

A Theory of Commodity Price Fluctuations

This paper studies the price fluctuations of storable commodities that are traded in open markets and are subject to random shocks to demand or, more particularly, to supply. It relaxes the common assumption that the shocks are identically and independently distributed in favor of temporally dependent and periodic disturbances. The existence of a unique stationary rational expectations equilibrium is demonstrated for each of the models analyzed, and testable implications of the models are derived. An illustrative empirical investigation is then undertaken for the model with periodic disturbances using monthly time-series observations for seven commodities over the period 1960-93.

Granger Causality and the Sampling of Economic Processes

This paper provides a discussion of the developments in econometric modelling that are designed to deal with the problem of spurious Granger causality relationships that can arise from temporal aggregation.We outline the distortional e ects of using discrete time models that explicitly depend on the unit of time and outline a remedy of constructing timeinvariant discrete time models via a structural continuous time model.In an application to testing for money-income causality, we demonstrate the importance of incorporating exact temporal aggregation restrictions on the discrete time data.We do this by conducting causality tests in discrete time models that: (a) impose the temporal aggregation restrictions exactly; (b) impose the temporal aggregation restrictions approximately; and (c) do not impose these restrictions at all.

The estimation of continuous parameter long-memory time series models

A class of univariate fractional ARIMA models with a continuous time parameter is developed for the purpose of modeling long-memory time series. The spectral density of discretely observed data is derived for both point observations (stock variables) and integral observations (flow variables). A frequency domain maximum likelihood method is proposed for estimating the longmemory parameter and is shown to be consistent and asymptotically normally distributed, and some issues associated with the computation of the spectral density are explored.

Discrete time representation of stationary and non-stationary continuous time systems

This paper derives the formulae for an exact discrete time representation corresponding to a system of higher-order stochastic differential equations. The formulae are applicable in stationary, non-stationary and explosive systems and for data observed as a mixture of both stock and flow variables. Expressions are also provided for an explicit moving average representation of the disturbance vector in the discrete time model, which can be used, under the assumption of white noise continuous time disturbances, to derive formulae for the computation of the exact Gaussian likelihood function.

Forecasting with the almost ideal demand system: evidence from some alternative dynamic specifications

The almost ideal demand system is used as a representation of long run demands in discrete time and continuous time error correction models to produce forecasts of budget shares beyond the sample period. The estimated models are subjected to a battery of tests, and an analysis of the forecasts indicates that continuous time adjustment mechanisms, based around fully modified estimates of the long run preference parameters, provide a remarkably accurate method of forecasting budget shares.

Forecasting with demand systems: A comparative study

Six systems of consumer demand equations are considered using seasonally adjusted quarterly U.K. data. The homogeneity and symmetry restrictions are tested where appropriate and are only convincingly rejected by one model. The estimated models are used to generate forecasts using fourteen post-sample observations and a simple habit formation model is found to be superior. Each system is tested for parameter constancy over the forecast period and only the dynamic models pass this test. The results highlight the importance of dynamics in demand systems but suggest that simpler structures may be more appropriate for forecasting. The use of small-sample adjusted test statistics can have a dramatic impact on inferences drawn.

Cointegration and Sampling Frequency

This paper analyses the effects of sampling frequency on the properties of spectral regression estimators of cointegrating parameters. Large sample asymptotic properties are derived under three scenarios concerning the span of data and sampling frequency, each scenario depending on whether span or frequency (or both) tends to infinity. The limiting distributions are shown to be different in each case. Furthermore, the asymptotic efficiency of the estimators obtained with a fixed sampling frequency is compared with that obtained with a continuous record of data, and it is shown that the only inefficiencies arise with respect to stock variables. Some simulation results and an empirical illustration are also provided.

THE ASYMPTOTIC EFFICIENCY OF COINTEGRATION ESTIMATORS UNDER TEMPORAL AGGREGATION

This paper examines the effects of temporal aggregation on the asymptotic variances of estimators in cointegrated systems. Two important findings are obtained. First, estimators based on flow data alone are more efficient than when the data are all stocks or a mixture of stocks and flows. Second, estimators based on flow data are as efficient as when the data are recorded continuously. A method of improving efficiency with stock variables is also proposed, and an empirical illustration of the method is provided in the context of long-run money demand regressions.I thank Roy Bailey, Rex Bergstrom, Roderick McCrorie, a co-editor, and two anonymous referees for helpful comments. I also thank Katsumi Shimotsu for help with some data issues. None of these individuals are implicated, however, in any possible shortcomings of this paper. The financial support provided by the ESRC under grant R000221818 is gratefully acknowledged.

The simulation of random vector time series with given spectrum

A method is proposed for generating multivariate time series which are required to satisfy a given spectral density function, which extends previous work on univariate time series. The performance of the method is assessed in a small simulation exercise for a bivariate long memory model and is found to perform well.