FGNs and FI(d) series can be considered to be true long memory models for reasons outlined above. Other models display the long memory property but are, in fact, short memory series. We consider two classes of such models.
6.4.1 Aggregation Models
In economic and financial applications many time series are aggregates of numerous individual component series. For example, stock market indices, such as the S&P 500, are just a (possibly weighted) sum of a few tens to thousands of individual stocks. Granger and Morris (1976) examined the types of series which resulted from summing a number of AR(p) and MA(q) series. Granger and Morris (1976) considered the AR and MA models to be “. . . intuitively reasonable and could well occur in practice . . . ”. Of particular interest, they reported that the sum of an AR(p1) and an AR(p2) series yielded an ARMA(p1+p2,max(p1, p2)) series.
Subsequently Granger (1980) considered whether the sum
Xt= N
X
j=1
Xt(j)
where Xt(j);j = 1,2, . . . , N are the individual components of an aggregated series
Xt, each of which has short-memory, could exhibit long memory. Granger reported this was the case. Beran (1994, p16) stated “. . . observing long-range dependence in an aggregated series (macro-level) one cannot conclude that this long memory is due to the genuine occurrence of long memory in the individual series (micro-level). It is possible that instead it is induced artificially by aggregation.”
Aggregation models have also appeared in the non-economic literature, albeit indi- rectly. Hare and Mantua (2000) amassed 100 North Pacific climatic and biological time series and used a composite analysis to demonstrate regime shifts in the Pacific Decadal Oscillation (PDO). Rudnick and Davis (2003) responded with a paper criticizing their methodology and demonstrating that composite analysis of red noise series produced
the step-like changes seen in the PDO. Overland et al. (2006) produced a third pa- per which suggested a long-memory model also fitted the data. It would appear all three sets of authors were discussing, in part, a Granger-style long memory aggregation model without being aware of his work.
Cioczek-Georges and Mandelbrot (1995) have studied an alternative aggregation model which exhibits long memory properties. Their model has found application in modeling aggregate traffic on computer networks. Willinger et al. (2003) also studied long memory in computer networks from the point of view of the network protocols which govern how traffic from individual sources is aggregated onto shared data net- works.
6.4.2 Constrained Non-Stationary Models
Klemes (1974) argued that long memory in hydrological time series was a statistical ar- tifact caused by analyzing non-stationary time series with statistical tools which assume stationarity. Often series which display the long memory property are constrained for physical reasons to lie in a bounded range. Beyond that we have no reason to believe that they are stationary. For example, tree ring indices often display long memory (see Chapter 10). For biological reasons there are fixed upper and lower bounds on the rate of tree growth. Long periods of above or below average growth may have biological reasons ultimately rooted in climatic shifts and may not be meaningless fluctuations in growing conditions.
To see how confusion between a non-stationary series and true long memory may arise consider the two ACFs presented in Figure (6.1). The ACF in panel (a) shows no statistically significant autocorrelations as it is the ACF of a series of random numbers drawn from an N(0,1) distribution. The ACF in panel (b) appears to be of the long memory type yet it is the ACF of the same series with a single mean shift at the mid-point.
It is mathematically difficult to handle a general constrained non-stationary model. To make the problem tractable the models which have been studied have often been
0 5 10 15 20 25 0.0 0.4 0.8 Lag ACF (a) 0 5 10 15 20 25 0.0 0.4 0.8 Lag ACF (b)
Figure 6.1: The ACF of two simulated time series. Panel (a) is the ACF of random numbers with no serial correlation. Panel (b) is the ACF of the same data with a mean shift at the mid-point.
of the type where the system has two or more well defined states which it switches between on some irregular basis. Often the models studied are stationary but are used as simplified models of the more general non-stationary series. Thus a number of authors have studied regime switching and hidden Markov models. One of the simplest is due to Granger and Ter¨asvirta (1993) who introduced the regime switching model
wheretis iid N(0, σ2) and sgn(x) = 1 ifx >0 0 ifx= 0 −1 ifx <0.
The process xt is Markovian, with zero mean and is stationary. They observed em- pirically that for infrequent regime changes the ratio log(|ρk|)/logk was constant with increasing k. This was the same theoretical form as for an FI(d) series. In this case the model exhibited statistical long memory in the time domain even though it was Markovian.
Granger and Hyung (2004) showed that the model
xt=µt+t (6.14)
µt=µt−1+qtηt (6.15) where
t= 1, . . . T
µt= mean level of the series at time t
t= a noise variable ηt=N(0, σ2) qt= 0,with probability 1−p 1,with probabilityp
and the assumption that the probability of breaks,p, converged slowly to zero as the sample size increased, i.e.
lim T→∞p→0 and
lim
T→∞T p= non-zero constant yielded an ACF of the long-memory type.
The purpose of the assumption was to ensure that the expected number of breaks,
T p, was bounded from above as T increased to infinity. To preserve the long memory property the number of breaks was required to remain finite as the sample size increased. The assumption made by Granger and Hyung (2004) was the same as that used to obtain the Poisson distribution from the binomial (Miller and Miller, 1999, pp186,187). Further, the model (6.14) and (6.15) is a small variation on the random walk plus noise model (Chan, 2002, p139) often used in the discussion of the Kalman filter. The difference was that the assumption of Granger and Hyung (2004) limited the random walk to a finite number of steps.
A more general model is the structural break model which was defined in Equations (1.1) and (1.2) in Section (1.1).