Commodity Modeling and Pricing: Methods for Analyzing Resource Market Behavior

CHAPTER 1

Indirect Inference and Long Memory A New Truncated-Series Estimation Method

Armand Sadler, Jean-Baptiste Lesourd and Vêlayoudom Marimoutou

INTRODUCTION

Long-memory processes are an important and even fundamental advance in time-series modeling. More precisely, the so-called autoregressive fractionally integrated moving average (ARFIMA) model has been introduced by Granger and Joyeux (1980) and Hosking (1981). It is a generalization of the ARIMA model, which is a short memory process, by allowing the differencing parameter d to take any real value. The goal of this specification is to capture parsimoniously long-run multipliers that decay very slowly, which amounts to modeling long memories in a time series. ARFIMA processes, however, are associated with hyperbolically decaying autocorrelations, impulse response weights, and spectral density function exploding at zero frequency. As noted by Brockwell et al. (1998), while a long memory process can always be approximated by an ARMA(p, q) process, the orders p and q required to achieve a good approximation may be so large as to make parameter estimation extremely difficult. In any case, this approximation is not possible with small samples.

ARFIMA processes are defined as follows in their canonical form:

where d ∈ (-0.5, 0.5) is the fractional difference operator and μ can be any deterministic function of time. If μ is zero, ...