14.1 Introduction


d = 24006 gph (y)
estimates the degree of long-memory of a time series by using a log-periodogram regression
k = 24009 kpss (y)
calculates the KPSS statistics for $ I(0)$ against long-memory alternatives
q = 24012 lo (y{, m})
calculates the Lo Statistics for long-range dependence
t = 24015 lobrob (y{, bdvec})
provides a semiparametric test for $ I(0)$ of a time series
q = 24018 neweywest (y{, m})
calculates the Newey and West heteroskedastic and autocorrelation consistent estimator of the variance
d = 24021 roblm (x{, q, bdvec})
semiparametric average periodogram estimator of the degree of long-memory of a time series
d = 24024 robwhittle (x{, bdvec})
semiparametric Gaussian estimator of the degree of long-memory of a time series, based on the Whittle estimator
k = 24027 rvlm (x{, m})
calculates the rescaled variance test for $ I(0)$ against long-memory alternatives
A stationary stochastic process $ \{Y_t\}$ is called a long-memory process if there exist a real number $ H$ and a finite constant $ C$ such that the autocorrelation function $ \rho(\tau)$ has the following rate of decay:

$\displaystyle \rho(k) \sim C \tau^{2H-2} \quad \textrm{as} \quad \tau \rightarrow \infty.$ (14.1)

The parameter $ H$, called the Hurst exponent, represents the long-memory property of the time series. A long-memory time series is also said fractionally integrated, where the fractional degree of integration $ d$ is related to the parameter $ H$ by the equality $ d = H-1/2$. If $ H \in (1/2,1)$, i.e., $ d \in
(0,1/2)$, the series is said to have long-memory. If $ H>1$, i.e., $ d>1/2$, the series is nonstationary. If $ H\in (0,1/2)$, i.e., $ d\in(-1/2,0)$, the series is called antipersistent.

Equivalently, a long-memory process can be characterized by the behaviour of its spectrum $ f(\lambda_j)$, estimated at the harmonic frequencies $ \lambda_j = {2 \pi j}/{T}$, with $ j =1,\ldots,[T/2]$, near the zero frequency:

$\displaystyle \lim_{\lambda_j \rightarrow 0^+} f(\lambda_j) = C \lambda_j^{-2d}$ (14.2)

where $ C$ is a strictly positive constant. Excellent and exhaustive surveys on long-memory are given in Beran (1994), Robinson (1994a) and Robinson and Henry (1998).

A long-memory process with degree of long-memory $ d$ is said to be integrated of order $ d$ and is denoted by $ I(d)$. The class of long-memory processes generalises the class of integrated processes with integer degree of integration.