Computational Economics 17: 179–201, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
179
Robust Estimation of GARMA Model Parameters
with an Application to Cointegration among
Interest Rates of Industrialized Countries
RAJALAKSHMI RAMACHANDRAN and PAUL BEAUMONT
Department of Economics, Florida State University, Tallahassee, FL 32303, U.S.A.;
E-mail: ramachan@csee.usf.edu and beaumont@scri.fsu.edu
Abstract. We review the literature on long memory ARFIMA and GARMA models and introduce a
new efficient estimator for GARMA models, which we show to be robust. Next we conduct a Monte
Carlo study to demonstate the power of the Dickie–Fuller test when the data are generated from a
stationary GARMA process. We conclude with a brief discussion of cointegration in the context of
GARMA models with an application to international interest rates.
Key words: long memory processes, GARMA models, fractional integration, Dickey–Fuller tests,
comovement, cointegration, international interest rates
1. Introduction
Most economic and financial variables exhibit temporal correlation. The dynamics
of these variables may be studied in either the time domain using Autoregress-
ive Moving Average (ARMA) models or in the frequency domain using spectral
analysis. ARMA models assume that the data process is stationary and that the
autocorrelation function decays exponentially so that there is no strong dependence
between distant observations. The Fourier transform of such an autocorrelation
function produces a spectrum that is finite at all frequencies. Autoregressive Integ-
rated Moving Average (ARIMA) models generalize ARMA models by allowing
the d th difference (where d is a positive integer) of a process to be ARMA. An AR-
IMA process exhibits a special type of nonstationarity in which the autocorrelation
function does not decay at all and the spectrum is unbounded at the origin (zero
frequency). Autoregressive Fractionally Integrated Moving Average (ARFIMA)
models are more general still, allowing the difference parameter d to be fractional.
The autocorrelation function of an ARFIMA process dies off hyperbolically (more
slowly than exponential) and so allows for long-term dependency or long memory
in the process. For some values of d the spectrum is bounded at all frequencies and
for others the spectrum is unbounded at the origin. A model encompassing all these
as special cases is the Generalized Autoregressive Moving Average (GARMA)