Long Memory Properties and Complex Systems
On the other hand , the Local Whittle Estimator depends also on the periodogram , as defined right above , but consists of a semiparametric estimate , which is carried out by the minimization of an objective function , as in Künsch ( 1987 ):
where :
• is any admissible value
•
• , being H the Hurst exponent .
Keeping that in mind , both estimators are applied on generated time series , which are result of the simulation of each model explained . Hence ,
for
, the time series displays long memory properties .
Nonetheless , it is also important to notice that despite the fact that ARI- MA processes can be seen as truncated fractionally integrated processes ; the results of both tests do not suggest fractional difference coefficients where they do not exist .
About the first part of the affirmation above , if a fractionally integrated process can be written as : obviously , simple autoregressive processes can be seen as their respective truncations of order p .
Moreover , in order to obtain an interesting proof in terms of the statistical test performances with no further mathematical abstraction , a Monte Carlo-based test was carried out , with 100 simulations for each time series specification with 2,000 observations .
Only first-order processes were tested , since none of the second-order autoregressive models suggested any spurious fractional difference coefficient . The same occurred for third-order processes and so on .
So , basically , it was simulated AR ( 1 ) processes with the firstorder term ranging from to b , and the results are shown in Figure 1 , in terms of the suggested fractional difference coefficient . In addition , it is important to keep in mind that an AR ( 1 ) process ( without constant term ) can be represented as a model of the form :
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