Forecasting Financial Times Series with Generalized Long Memory Processes

L. Ferrara1, D. Guégan2

Abstract : In this paper, we present a parametric modelling of financial time series by using long memory k-factor Gegenbauer processes suggested by Gray et al. (1989) and known in the statistical literature as generalized long memory processes. Using forecasting performances of this kind of processes and recent results on estimation theory we show how on an application to the monthly time series of exchange rates for the French franc against the US dollar, we can improve the efficiency of the long-term forecasting process.

Key words : Financial series, forecasting, generalized long memory processes.

1

Université Paris 13, CNRS UMR 7539 - RATP, Département Commercial, LAC A73, 54 Quai de la Rapée, 75599 Paris Cedex 12, France. e-mail: [email protected] 2 Université de Reims, UPESA 6056 - CREST, Timbre J340, 3 av. Pierre Larousse 92245 Malakoff Cedex, France. email: [email protected]

1. Introduction Long-range dependence between observations of a time series is an evidence in diverse fields of applied statistics. For instance, the pionner work of Hurst (1951) stems from problems dealing with hydrology, and later many authors analyzed river flows through long memory processes, see for instance Noakes et al. (1988) or Ooms and Franses (1998). More recently, researchers founded evidence of persistence in telecommunication data (Taqqu et al. (1997) or in urban transport data (Ferrara and Guégan (1999(a))). However, most long memory applications concern financial time series. Without being exhaustive, empirical evidence of long memory has been found in time series of inflation rates (Delgado and Robinson (1994), Hassler and Wolters (1995), Baillie et al. (1996), Franses and Ooms (1997)), stock market prices (Cheung and Lai (1995), Barkoulas and Baum (1996), Willinger et al. (1999)) or exchange rates (Cheung (1993), Bisaglia and Guégan (1998), Velasco (1999)). Thus, several studies have shown the efficiency of long memory processes to analyze financial time series. Moreover, it seems that long memory processes constitute an adequate natural tool to provide long-term forecasts, in comparison with short memory processes, although no study really showed it up to now. Recall that a stationary process is said to be long memory if it exhibits a slowly decaying autocorrelation function (ACF), denoted ρ(n), approximated as follows : ρ(n)∼ cρ(n)n2d-1, as n tends to infinity, where cρ(n) is a slowly varying function at infinity and d is the long memory parameter such that 0 0, for i=1,…, k, then (Xt){t∈Z} is a long memory process.

If the k-factor Gegenbauer process (Xt){t∈Z} is invertible (see Proposition A.1(1)), then, from equation (3), we can formally write: k

X t = ∏ (1 − 2ν i B + B 2 ) − di ε t ,

(13)

i =1

and from equation (1) and (13), Giraitis and Leipus (1995) derive the infinite movingaverage representation of the k-factor Gegenbauer process (Xt){t∈Z} :

X t = ∑ψ j (d ,ν ) B j ε t ,

(14)

j≥0

where:

ψ j (d ,ν ) =

∑C

l1 0 ≤l1 ,...,l k ≤ j , l1 +...+ l k = j

(d 1 ,ν 1 )...C lk (d k ,ν k ) ,

(15)

where (Cj(d, ν)){j ∈ Z} are the Gegenbauer polynomials previously defined. Moreover, it is worthwhile to note that Giraitis and Leipus (1995) give the asymptotical expansion of the weights (ψj(d,ν)){j∈Z}.

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