A stochastic process is if it needs to be differentiated
times
in order to become
. We shall test for
against fractional
alternatives by using more formal definitions.
In a first approach, we define a stochastic process as
if the normalized partial sums follow a particular
distribution. We only require is the existence of a consistent
estimator of the variance for normalizing the partial sums. The
tests presented here make use of the Newey and West (1987)
heteroskedastic and autocorrelation consistent (HAC) estimator of the
variance, defined as
The quantlet
neweywest
computes the Newey and West (1987)
estimator of the variance of a unidimensional process. Its syntax is:
sigma = neweywest(y{, q})where the input parameters are:
In the following example the HAC variance of the first 2000 observations of the 20 minutes spaced sample of Deutschmark-Dollar FX is computed.
library("times") y = read("dmus58.dat") y = y[1:2000] q = 5|10|25|50 sigma = neweywest(y,q) q~sigma
Contents of _tmp [1,] 5 0.0047841 [2,] 10 0.008743 [3,] 25 0.020468 [4,] 50 0.039466
The first test for long-memory was devised by the hydrologist Hurst (1951) for the design of an optimal reservoir for the Nile river, of which flow regimes were persistent. Although Mandelbrot (1975) gave a formal justification for the use of this test, Lo (1991) demonstrated that this statistic was not robust to short range dependence, and proposed the following one:
![]() |
(14.4) |
This statistic is extremely sensitive to the order of truncation
but there is no statistical criteria for
choosing
in the framework of this statistic. Andrews (1991) rule
gives mixed results. If
is too small, this estimator does not
account for the autocorrelation of the process, while if
is too
large, it accounts for any form of autocorrelation and the power of
this test tends to its size. Given that the power of a useful test
should be greater than its size, this statistic is not very helpful.
For that reason, Teverovsky, Taqqu and Willinger (1999) suggest to use this
statistic with other tests.
Since there is no data driven guidance for the choice of this parameter, we
consider the default values for = 5, 10, 25, 50.
XploRe
users have the option to provide their own vector of truncation lags.
Let's consider again the series of absolute returns on the 20 minutes spaced Deutschmark-Dollar FX rates.
library("times") y = read("dmus58.dat") ar = abs(tdiff(y[1:2000])) lostat = lo(ar) lostat
Contents of lostat [1,] " Order Statistic" [2,] "__________________ " [3,] "" [4,] " 5 2.0012 *" [5,] " 10 1.8741 *" [6,] " 25 1.7490 " [7,] " 50 1.6839 "This result illustrates the issue of the choice of the bandwidth parameter q. For q = 5 and 10, we reject the null hypothesis of no long-memory. However, when q = 25 or 50, this null hypothesis is accepted, as the power of this test is too low for these levels of truncation orders.
Equivalently, we can test for against fractional alternatives
by using the KPSS test Kwiatkowski, Phillips, Schmidt, and Shin (1992), as Lee and Schmidt (1996)
have shown that this test has a power equivalent to Lo's statistic
against long-memory processes. The two KPSS statistics, denoted by
and
, are respectively based on the residuals of
two regression models: on an intercept and a trend
, and on a
constant
. If we denote by
the partial sums
, where
are the residuals of these
regressions, the KPSS statistic is defined by:
The quantlet
kpss
computes both statistics. The default
bandwidths, denoted by
,
and
are the one given in
Kwiatkowski, Phillips, Schmidt, and Shin (1992). We evaluate both tests on the series of
absolute returns ar as follows:
library("times") y = read("dmus58.dat") ar = abs(tdiff(y[1:2000])) kpsstest = kpss(ar) kpsstest
Thus, XploRe returns the following table:
Contents of kpsstest [1,] " Order Test Statistic Crit. Value " [2,] "_________________________________________ " [3,] "" [4,] " L0 = 0 const 1.8259 * 0.4630" [5,] " L4 = 8 const 1.2637 * 0.4630" [6,] " L12= 25 const 1.0483 * 0.4630" [7,] " L0 = 0 trend 0.0882 0.1460" [8,] " L4 = 8 trend 0.0641 0.1460" [9,] " L12= 25 trend 0.0577 0.1460"
Giraitis, Kokoszka and Leipus (1998) have proposed a centering of the
KPSS statistic based on the partial sum of the deviations from the
mean. They called it a rescaled variance test as its expression
given by
![]() |
(14.6) |
![]() |
(14.7) |
We evaluate the statistic with the quantlet
rvlm
which has
the following syntax:
vstest = rvlm(ary{, q})where
library("times") y = read("dmus58.dat") ar = abs(tdiff(y[1:2000])) vstest = rvlm(ar) vstest
Contents of vstest [1,] " Order Statistic Crit. Value " [2,] "_________________________________" [3,] "" [4,] " 0 0.3305 * 0.1869" [5,] " 8 0.2287 * 0.1869" [6,] " 25 0.1897 * 0.1869"
Lobato and Robinson (1998) nonparametric test for against
is also based on the
approximation (14.2) of the spectrum of a long-memory process.
In the univariate case, the
statistic is equal to:
The quantlet
lobrob
evaluates the Lobato-Robinson test. Its
syntax is as follows:
l = lobrob(ary{, m})where
library("times") y = read("dmus58.dat") ar = abs(tdiff(y[1:2000])) l = lobrob(ar) l
Contents of l [1,] "Bandwidth Statistic " [2,] "_____________________ " [3,] "" [4,] " 334 -4.4571"
In the next case, we provide a vector of bandwidths m, and evalutate this statistic for all the elements of m. The sequence of instructions:
library("times") y = read("dmus58.dat") ar = abs(tdiff(y[1:2000])) m = #(100,150,200) l = lobrob(ar,m) l
Contents of l [1,] "Bandwidth Statistic " [2,] "_____________________ " [3,] "" [4,] " 100 -1.7989" [5,] " 150 -2.9072" [6,] " 200 -3.3308"