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%\title{EC771:\ Econometrics \\ Spring 2010}
%Notes Part 3}
%\author{Christopher F\ Baum \and Department of Economics \and Boston College \and % baum@bc.edu}
%\maketitle

\textbf{EC771:\ Econometrics, Spring 2011}

\bigskip \textbf{Fractionally integrated timeseries and \\ARFIMA\ modelling}

%\section{ Fractionally integrated timeseries and ARFIMA\ modelling%
%\protect\footnote{%
This presentation of ARFIMA\ modelling draws heavily from Baum and Wiggins\
(2000). %}

The model of an autoregressive fractionally integrated moving average
process of a timeseries of order $\left( p,\,d,\,q\right) $, denoted by
ARFIMA $\left( p,\,d,\,q\right) $, with mean $\mu $, may be written using
operator notation as 
\begin{equation}
\Phi (L)(1-L)^{d}\left( y_{t}-\mu \right) =\Theta (L)\epsilon
_{t},\;\epsilon _{t}\sim i.i.d.(0,\sigma _{\epsilon }^{2})  \label{one}
\end{equation}
where $L$ is the backward-shift operator, $\Phi (L)$ = 1 - $\phi _{1}L$ - ..
- $\phi _{p}L^{p}$, $\Theta (L)$ = 1 + $\vartheta _{1}L$ + ... + $\vartheta
_{q}L^{q}$, and $(1-L)^{d}$ is the fractional differencing operator defined
by 
\begin{equation}
(1-L)^{d}=\sum\limits_{k=0}^{\infty }\frac{\Gamma (k-d)L^{k}}{\Gamma
(-d)\Gamma (k+1)}  \label{two}
\end{equation}
with $\Gamma $($\cdot $) denoting the gamma (generalized factorial)
function. The parameter $d$ is allowed to assume any real value. The
arbitrary restriction of $d$ to integer values gives rise to the standard
autoregressive integrated moving average (ARIMA) model. The stochastic
process $y_{t}$ is both stationary and invertible if all roots of $\Phi (L)$
and $\Theta (L)$ lie outside the unit circle and $\left| d\right| <0.5$. The
process is nonstationary for $d\geq 0.5$, as it possesses infinite variance,
i.e. see Granger and Joyeux (1980).

Assuming that $d\in\left[ 0,\,0.5\right)$, Hosking (1981) showed that the
autocorrelation function, $\rho (\cdot )$, of an ARFIMA process is
proportional to $k^{2d-1}$ as $k\rightarrow \infty $. Consequently, the
autocorrelations of the ARFIMA process decay hyperbolically to zero as $%
k\rightarrow \infty $ in contrast to the faster, geometric decay of a
stationary ARMA process. For $d\in \left( 0,\/\/0.5\right) $, $%
\sum\nolimits_{j=-n}^{n}\left| \rho (j)\right| $ diverges as $n\rightarrow
\infty $, and the ARFIMA process is said to exhibit long memory, or
long-range positive dependence. The process is said to exhibit intermediate
memory (anti-persistence), or long-range negative dependence, for $d\in
\left( -0.5,\/\/0\right) $. The process exhibits short memory for $d=0$,
corresponding to stationary and invertible ARMA modeling. For $d\in \left[
0.5,\,1\right) $ the process is mean reverting, even though it is not
covariance stationary, as there is no long-run impact of an innovation on
future values of the process.

If a series exhibits long memory, it is neither stationary ($I(0)$) nor is
it a unit root ($I(1)$) process; it is an $I(d)$ process, with $d$ a real
number. A series exhibiting long memory, or persistence, has an
autocorrelation function that damps hyperbolically, more slowly than the
geometric damping exhibited by ``short memory'' (ARMA) processes. Thus, it
may be predictable at long horizons. An excellent survey of long memory
models--which originated in hydrology, and have been widely applied in
economics and finance--is given in Baillie (1996). An up-to-date survey of these
models in finance is available as ``Is there long memory in financial time series?'', Lima, L.\ and 
Xiao, Z., {\it Applied Financial Economics}, 20: 6, 487--500.

%\subsection
{\bf Approaches to estimation of the ARFIMA\ model}

There are two approaches to the estimation of an ARFIMA $\left(
p,\,d,\,q\right) $ model: exact maximum likelihood estimation, as proposed
by Sowell (1992), and semiparametric approaches. Sowell's approach requires
specification of the $p$ and $q$ values, and estimation of the full ARFIMA
model conditional on those choices. This involves all the attendant
difficulties of choosing an appropriate ARMA specification, as well as a
formidable computational task for each combination of $p$ and $q$ to be
evaluated. We first describe semiparametric methods, in which we assume that
the ``short memory'' or ARMA components of the timeseries are relatively
unimportant, so that the long memory parameter $d$ may be estimated without
fully specifying the data generating process.

%\subsection
{\bf Semiparametric estimators for I(d) series}

%\subsubsection
{\it The Lo Modified Rescaled Range estimator\protect\footnote{%
This discussion is drawn from Baum and Room (2000).}}

\texttt{lomodrs} performs Lo's (1991) modified rescaled range (R/S, ``range
over standard deviation'') test for long range dependence of a time series.
The classical R/S statistic, devised by Hurst (1951) and Mandelbrot (1972),
is the range of the partial sums of deviations of a timeseries from its
mean, rescaled by its standard deviation. For a sample of $n$ values $%
\left\{ x_{1},x_{2},\ldots x_{n}\right\} $,

\[
Q_{n}=\frac{1}{s_{n}}\left[ \max_{1\leq k\leq n}\sum_{j=1}^{k}\left( x_{j}-%
\bar{x}_{n}\right) -\min_{1\leq k\leq n}\sum_{j=1}^{k}\left( x_{j}-\bar{x}%
_{n}\right) \right] 
\]

where $s_{n}$ is the maximum likelihood estimator of the standard deviation
of $x.$ The first bracketed term is the maximum of the partial sums of the
first $k$ deviations of $x_{j}$ from the full-sample mean, which is
nonnegative. The second bracketed term is the corresponding minimum, which
is nonpositive. The difference of these two quantities is thus nonnegative,
so that $Q_{n}>0.$ Empirical studies have demonstrated that the R/S
statistic has the ability to detect long-range dependence in the data.

Like many other estimators of long-range dependence, though, the R/S
statistic has been shown to be excessively sensitive to ``short-range
dependence,'' or short memory, features of the data. Lo (1991) shows that a
sizable $AR(1)$ component in the data generating process will seriously bias
the R/S statistic. He modifies the R/S statistic to account for the effect
of short-range dependence by applying a ``Newey-West'' correction (using a
Bartlett window) to derive a consistent estimate of the long-range variance
of the timeseries. For \texttt{maxlag}$>0,$ the denominator of the statistic
is computed as the Newey-West estimate of the long run variance of the
series. If \texttt{maxlag} is set to zero, the test performed is the
classical Hurst-Mandelbrot rescaled-range statistic. Critical values for the
test are taken from Lo, 1991, Table II.

Inference from the modified R/S test for long range dependence is
complementary to that derived from that of other tests for long memory, or
fractional integration in a timeseries, such as \texttt{kpss}, \texttt{%
gphudak}, \texttt{modlpr} and \texttt{roblpr}.

%\subsubsection
{\it The Geweke--Porter-Hudak log periodogram regression estimator}

\texttt{gphudak} performs the Geweke and Porter-Hudak (GPH, 1983)
semiparametric log periodogram regression, often described as the ``GPH
test,'' for long memory (fractional integration) in a timeseries. The GPH
method uses nonparametric methods--a spectral regression estimator--to
evaluate $d$ without explicit specification of the ``short memory'' (ARMA)
parameters of the series. The series is usually differenced so that the
resulting $d$ estimate will fall in the [-0.5, 0.5] interval.

Geweke and Porter-Hudak (1983) proposed a semiparametric procedure to obtain
an estimate of the memory parameter $d$ of a fractionally integrated process 
$X_{t}$ in a model of the form 
\begin{equation}
\left( 1-L\right) ^{d}X_{t}=\epsilon_{t},  \label{arfima}
\end{equation}
where $\epsilon_{t}$ is stationary with zero mean and continuous spectral
density $f_{\epsilon}\left( \lambda \right) >0.$ The estimate $\hat{d}$ is
obtained from the application of ordinary least squares to 
\begin{equation}
\log \left( I_{x}\left( \lambda _{s}\right) \right) =\hat{c}-\hat{d}\log
\left| 1-e^{i\lambda _{s}}\right| ^{2}+residual  \label{lpr}
\end{equation}
computed over the fundamental frequencies \\$\left\{ \lambda _{s}=\frac{2\pi s%
}{n},s=1,...,m,m<n\right\} $. We define \\$\omega _{x}\left( \lambda
_{s}\right) =\frac{1}{\sqrt{2\pi n}}\sum_{t=1}^{n}X_{t}e^{it\lambda _{s}}$
as the discrete Fourier transform (dft) of the timeseries $X_{t}$, $%
I_{x}\left( \lambda _{s}\right) =\omega _{x}\left( \lambda _{s}\right)
\omega _{x}\left( \lambda _{s}\right) ^{*}$ as the periodogram, and $x_{s}=$ 
$\log \left| 1-e^{i\lambda _{s}}\right| .$ Ordinary least squares on (\ref
{lpr})\ yields 
\begin{equation}
\hat{d}=0.5~\frac{\sum_{s=1}^{m}x_{s}\log I_{x}\left( \lambda
_{s}^{{}}\right) }{\sum_{s=1}^{m}x_{s}^{2}}.  \label{gph}
\end{equation}

Various authors have proposed methods for the choice of $m$, the number of
Fourier frequencies included in the regression. The regression slope
estimate is an estimate of the slope of the series' power spectrum in the
vicinity of the zero frequency; if too few ordinates are included, the slope
is calculated from a small sample. If too many are included, medium and
high-frequency components of the spectrum will contaminate the estimate. A
choice of $\sqrt{T}$, or \texttt{power} = 0.5 is often employed. To evaluate
the robustness of the GPH estimate, a range of power values (from
0.40--0.75) is commonly calculated as well. Two estimates of the $d$
coefficient's standard error are commonly employed: the regression standard
error, giving rise to a standard $t$-test, and an asymptotic standard error,
based upon the theoretical variance of the log periodogram of $\frac{\pi ^{2}%
}{6}$. The statistic based upon that standard error has a standard normal
distribution under the null.

\pagebreak
%\subsubsection
{\it The Phillips Modified GPH log periodogram regression estimator
}

\texttt{modlpr} computes a modified form of the GPH estimate of the long
memory parameter, $d$, of a timeseries, proposed by Phillips (1999a, 1999b).
Phillips (1999a) points out that the prior literature on this semiparametric
approach does not address the case of $d=1,$ or a unit root, in (\ref{arfima}%
), despite the broad interest in determining whether a series exhibits
unit-root behavior or long memory behavior, and his work showing that the $%
\hat{d}$ estimate of (\ref{gph}) is inconsistent when $d>1,$ with $\hat{d}$
exhibiting asymptotic bias toward unity. This weakness of the GPH estimator
is solved by Phillips' Modified Log Periodogram Regression estimator, in
which the dependent variable is modified to reflect the distribution of $d$
under the null hypothesis that $d=1$. The estimator gives rise to a test
statistic for $d=1$ which is a standard normal variate under the null.
Phillips suggests that deterministic trends should be removed from the
series before application of the estimator. Accordingly, the routine will
automatically remove a linear trend from the series. This may be suppressed
with the \texttt{notrend} option. The comments above regarding \texttt{power}
apply equally to \texttt{modlpr}.

Phillips' (1999b)\ modification of the GPH\ estimator is based on an exact
representation of the dft in the unit root case. The modification expresses 
\[
\omega _{x}\left( \lambda _{s}\right) =\frac{\omega _{u}\left( \lambda
_{s}\right) }{1-e^{i\lambda _{s}}}-\frac{e^{i\lambda _{s}}}{1-e^{i\lambda
_{s}}}\frac{X_{n}}{\sqrt{2\pi n}} 
\]
and the modified dft as 
\[
\upsilon _{x}\left( \lambda _{s}\right) =\omega _{x}\left( \lambda
_{s}\right) +\frac{e^{i\lambda _{s}}}{1-e^{i\lambda _{s}}}\frac{X_{n}}{\sqrt{%
2\pi n}} 
\]
with associated periodogram ordinates \\$I_{v}\left( \lambda _{s}\right)
=\upsilon _{x}\left( \lambda _{s}\right) \upsilon _{x}\left( \lambda
_{s}\right) ^{*}$ (1999b, p.9). He notes that both $\upsilon _{x}\left(
\lambda _{s}\right) $ and, thus, $I_{v}\left( \lambda _{s}\right) $ are
observable functions of the data. The log-periodogram regression is now the
regression of $\log I_{v}\left( \lambda _{s}\right) $ on $a_{s}=\log \left|
1-e^{i\lambda _{s}}\right| .$ Defining $\bar{a}=m^{-1}\sum_{s=1}^{m}a_{s}$
and $x_{s}=a_{s}-\bar{a},$ the modified estimate of the long-memory
parameter becomes 
\begin{equation}
\tilde{d}=0.5\frac{\sum_{s=1}^{m}x_{s}\log I_{\nu }\left( \lambda
_{s}^{{}}\right) }{\sum_{s=1}^{m}x_{s}^{2}}.  \label{modlpr}
\end{equation}
Phillips proves that, with appropriate assumptions on the distribution of $%
\epsilon _{t},$ the distribution of $\tilde{d}$ follows 
\begin{equation}
\sqrt{m}\left( \tilde{d}-d\right) \rightarrow _{d}N\left( 0,\frac{\pi ^{2}}{%
24}\right) ,  \label{ddist}
\end{equation}
so that $\tilde{d}$ has the same limiting distribution at $d=1$ as does the
GPH estimator in the stationary case so that $\tilde{d}$ is consistent for
values of $d$ around unity. A semiparametric test statistic for a unit root
against a fractional alternative is then based upon the statistic (1999a,
p.10): 
\begin{equation}
z_{d}=\frac{\sqrt{m}\left( \tilde{d}-1\right) }{\pi /\sqrt{24}}
\label{testvs1}
\end{equation}
with critical values from the standard normal distribution. This test is
consistent against both $d<1$ and $d>1$ fractional alternatives.

%\subsubsection
{\it Robinson's Log Periodogram Regression estimator}

\texttt{roblpr} computes the Robinson (1995) multivariate semiparametric
estimate of the long memory (fractional integration) parameters, $d(g)$, of
a set of $G$ timeseries, $y(g)$, $g=1,G$ with $G\geq 1$. When applied to a
set of timeseries, the $d(g)$ parameter for each series is estimated from a
single log-periodogram regression which allows the intercept and slope to
differ for each series. One of the innovations of Robinson's estimator is
that it is not restricted to using a small fraction of the ordinates of the
empirical periodogram of the series: that is, the reasonable values of 
\texttt{power} need not exclude a sizable fraction of the original sample
size. The estimator also allows for the removal of one or more initial
ordinates, and for the averaging of the periodogram over adjacent
frequencies. The rationales for using non-default values of either of these
options are presented in Robinson (1995).

Robinson (1995) proposes an alternative log-periodogram regression estimator
which he claims provides ``modestly superior asymptotic efficiency to $\bar{d%
}\left( 0\right) $'' ($\bar{d}\left( 0\right) $ being the Geweke and
Porter-Hudak estimator) (1995, p.1052). Robinson's formulation of the
log-periodogram regression also allows for the formulation of a multivariate
model, providing justification for tests that different time series share a
common differencing parameter. Normality of the underlying time series is
assumed, but Robinson claims that other conditions underlying his derivation
are milder than those conjectured by GPH.

We present here Robinson's multivariate formulation, which applies to a
single time series as well. Let $X_{t}$ represent a $G-$dimensional vector
with $g^{th}$ element $X_{gt},g=1,...,G$. Assume that $X_{t}$ has a spectral
density matrix $\int_{-\pi }^{\pi }e^{ij\lambda }f\left( \lambda \right)
d\lambda ,$ with $\left( g,h\right) $ element denoted as $f_{gh}\left(
\lambda \right) .$ The $g^{th}$ diagonal element, $f_{gg}\left( \lambda
\right) ,$ is the power spectral density of $X_{gt}.$ For $0<C_{g}<\infty $
and $-\frac{1}{2}<d_{g}<\frac{1}{2},$ assume that $f_{gg}\left( \lambda
\right) \sim C_{g}\lambda ^{-2d_{g}}$ as $\lambda \rightarrow 0+$ for $%
g=1,...,G.$ The periodogram of $X_{gt}$ is then denoted as 
\begin{equation}
I_{g}\left( \lambda \right) =\left( 2\pi n\right) ^{-1}\left|
\sum_{t=1}^{n}X_{gt}e^{it\lambda }\right| ^{2},g=1,...G
\label{multiperiodogram}
\end{equation}
Without averaging the periodogram over adjacent frequencies nor omission of $%
l$ initial frequencies from the regression, we may define $Y_{gk}=\log
I_{g}\left( \lambda _{k}\right) .$ The least squares estimates of $c=\left(
c_{1},...c_{G}\right) ^{\prime }$ and $d=\left( d_{1},...,d_{G}\right)
^{\prime }$ are given by 
\begin{equation}
\left[ 
\begin{array}{c}
\tilde{c} \\ 
\tilde{d}
\end{array}
\right] =vec\left\{ Y^{\prime }Z\left( Z^{\prime }Z\right) ^{-1}\right\} ,
\label{multivarreg}
\end{equation}
where $Z=\left( Z_{1},...Z_{m}\right) ^{\prime },$ $Z_{k}=\left( 1,-2\log
\lambda _{k}\right) ^{\prime }$, $Y=\left( Y_{1},...Y_{G}\right) ,$ and $%
Y_{g}=\left( Y_{g,1},...Y_{g,m}\right) ^{\prime }$ for $m$ periodogram
ordinates. Standard errors for $\tilde{d}_{g}$ and for a test of the
restriction that two or more of the $d_{g}$ are equal may be derived from
the estimated covariance matrix of the least squares coefficients. The
standard errors for the estimated parameters are derived from a pooled
estimate of the variance in the multivariate case, so that their interval
estimates differ from those of their univariate counterparts. Modifications
to this derivation when the frequency-averaging (\texttt{j}) or omission of
initial frequencies (\texttt{l}) options are selected may be found in
Robinson (1995).

%\subsection
{\bf Maximum likelihood estimators of ARFIMA\ models}

The theory and implementation of Sowell's exact maximum likelihood estimator
of the \\ARFIMA(p,d,q) model using Ox is described in Doornik and Ooms
(1999).\pagebreak

%\subsection
{\bf Applications}

Examples of the application of the lomodrs and classical rescaled range
estimators:

Data from Terence Mills' \textit{Econometric Analysis of Financial Time
Series} on returns from the annual S\&P\ 500 index of stock prices,
1871-1997, are analyzed.

{\scriptsize
\begin{verbatim}
. use http://fmwww.bc.edu/ec-p/data/Mills2d/sp500a.dta
. lomodrs sp500ar
 
Lo Modified R/S test for sp500ar
 
Critical values for H0: sp500ar is not long-range dependent
 
90%: [ 0.861, 1.747 ]
95%: [ 0.809, 1.862 ]
99%: [ 0.721, 2.098 ]
 
Test statistic: .780838 (1 lags via Andrews criterion) 
N = 124
 
. lomodrs sp500ar, max(0)
 
Hurst-Mandelbrot Classical R/S test for sp500ar
 
Critical values for H0: sp500ar is not long-range dependent
 
90%: [ 0.861, 1.747 ]
95%: [ 0.809, 1.862 ]
99%: [ 0.721, 2.098 ]
 
Test statistic: .799079 N = 124
 
. lomodrs sp500ar if tin(1946,)
 
Lo Modified R/S test for sp500ar
 
Critical values for H0: sp500ar is not long-range dependent
 
90%: [ 0.861, 1.747 ]
95%: [ 0.809, 1.862 ]
99%: [ 0.721, 2.098 ]
 
Test statistic:  1.08705  (0 lags via Andrews criterion)  
N = 50
\end{verbatim}
}

For the full sample, the null of stationarity may be rejected at 95\% using
either the Lo modified R/S statistic or the classic Hurst-Mandelbrot
statistic. For the postwar data, the null may not be rejected at any level
of significance. Long-range dependence, if present in this series, seems to
be contributed by pre-World War II\ behavior of the stock price series.

\pagebreak
Examples of \texttt{gphudak}, \texttt{modlpr}, and \texttt{roblpr}
estimators:

Data from Terence Mills' \textit{Econometric Analysis of Financial Time
Series} on UK FTA All Share stock returns (ftaret) and dividends (ftadiv)
are analyzed.

 {\tiny 
\begin{verbatim}

 
. use http://fmwww.bc.edu/ec-p/data/Mills2d/fta.dta
 
. tsset
        time variable:  month, 1965m1 to 1995m12
 
. gphudak ftaret,power(0.5 0.6 0.7)
 
GPH estimate of fractional differencing parameter
------------------------------------------------------------------------------
                                                      Asy.
Power   Ords    Est d   StdErr  t(H0: d=0)  P>|t|    StdErr  z(H0: d=0)  P>|z|
------------------------------------------------------------------------------
  .50     20  -.00204  .160313    -0.0127   0.990   .187454    -0.0109   0.991
  .60     35  .228244  .145891     1.5645   0.128   .130206     1.7529   0.080
  .70     64  .141861  .089922     1.5776   0.120   .091267     1.5544   0.120
------------------------------------------------------------------------------
 
 
. modlpr ftaret, power(0.5 0.55:0.8)
 
Modified LPR estimate of fractional differencing parameter
------------------------------------------------------------------------------
Power   Ords      Est d    Std Err  t(H0: d=0)    P>|t|    z(H0: d=1)    P>|z|
------------------------------------------------------------------------------
  .50     19   .0231191    .139872     0.1653     0.870      -6.6401     0.000
  .55     25   .2519889   .1629533     1.5464     0.135      -5.8322     0.000
  .60     34   .2450011   .1359888     1.8016     0.080      -6.8650     0.000
  .65     46   .1024504   .1071614     0.9560     0.344      -9.4928     0.000
  .70     63   .1601207   .0854082     1.8748     0.065     -10.3954     0.000
  .75     84   .1749659     .08113     2.1566     0.034     -11.7915     0.000
  .80    113   .0969439   .0676039     1.4340     0.154     -14.9696     0.000
------------------------------------------------------------------------------
 
. roblpr ftaret
 
Robinson estimates of fractional differencing parameter
-------------------------------------------------------
Power   Ords      Est d    Std Err  t(H0: d=0)    P>|t|
-------------------------------------------------------
  .90    205   .1253645   .0446745     2.8062     0.005
-------------------------------------------------------
 
. roblpr ftap ftadiv
 
Robinson estimates of fractional differencing parameters
Power =  .90                            Ords         =    205
-------------------------------------------------------------
Variable         |     Est d       Std Err      t       P>|t|
-----------------+-------------------------------------------
ftap             |  .8698092      .0163302  53.2640     0.000
ftadiv           |  .8717427      .0163302  53.3824     0.000
-------------------------------------------------------------
Test for equality of d coefficients:   F(1,406) =  .00701   
                                    Prob > F = 0.9333
 
. constraint define 1 ftap=ftadiv
 
. roblpr ftap ftadiv ftaret, c(1)
 
Robinson estimates of fractional differencing parameters
Power =  .90                            Ords         =    205
-------------------------------------------------------------
Variable         |     Est d       Std Err      t       P>|t|
-----------------+-------------------------------------------
ftap             |  .8707759      .0205143  42.4473     0.000
ftadiv           |  .8707759      .0205143  42.4473     0.000
ftaret           |  .1253645      .0290116   4.3212     0.000
-------------------------------------------------------------
Test for equality of d coefficients:   F(1,610) =  440.11   
                                     Prob > F = 0.0000
 

\end{verbatim}
 }

The GPH test, applied to the stock returns series, generates estimates of
the long memory parameter that cannot reject the null at the ten percent
level using the t-test. Phillips' modified LPR, applied to this series,
finds that $d=1$ can be rejected for all powers tested, while $d=0$
(stationarity) may be rejected at the ten percent level for powers 0.6, 0.7,
and 0.75. Robinson's estimate for the returns series alone is quite precise.
Robinson's multivariate test, applied to the price and dividends series,
finds that each series has $d>0$. The test that they share the same $d$
cannot be rejected. Accordingly, the test is applied to all three series
subject to the constraint that price and dividends series have a common $d$,
yielding a more precise estimate of the difference in $d$ parameters between
those series and the stock returns series.

%\begin{thebibliography}{99}
{\bf References}

%\bibitem{and91}  
Andrews, D., 1991. Heteroskedasticity and Autocorrelation
Consistent Covariance Matrix Estimation. \textit{Econometrica}, 59, 817-858.

%\bibitem{baillie}  
Baillie, R. 1996. Long Memory Processes and Fractional
Integration in Econometrics, \textsl{Journal of Econometrics}, 73, 5-59.

%Available from the course homepage.

%\bibitem{lomodrs}  
Baum, Christopher\ F\ and Tairi Room, 2000. The modified
rescaled range test for long memory. Help file for Stata module \texttt{%
lomodrs}, available from SSC-IDEAS\ at \texttt{http://ideas.repec.org}.

%\bibitem{lmstb}  
Baum, Christopher F and Vince Wiggins, 2000. Tests for long
memory in a timeseries. Stata Technical Bulletin 57, available from SSC-IDEAS\ at \texttt{http://ideas.repec.org}.

%\bibitem{Door}  
Doornik, Jurgen A. and Marius Ooms. 1999. A package for
estimating, forecasting and simulating Arfima Models:\ Arfima package 1.0
for Ox. 

%\bibitem{gph}  
Geweke, J. and Porter-Hudak, S. 1983. The Estimation and
Application of Long Memory Time Series Models, \textsl{Journal of Time
Series Analysis}, 221-238.

%\bibitem{grjoy}  
Granger, C. W. J. and R. Joyeux. 1980. An introduction to
long-memory time series models and fractional differencing, \textsl{Journal
of Time Series Analysis}, 1, 15-39.

%\bibitem{hosk}  
Hosking, J. R. M. 1981. Fractional Differencing, \textsl{%
Biometrika}, 68, 165-176.

%\bibitem{hurst}  
Hurst, H., 1951. Long Term Storage Capacity of Reservoirs. 
\textit{Transactions of the American Society of Civil Engineers}, 116,
770-799.

Lima, L.\ and  Xiao, Z., 2010. Is there long memory in financial time series?,  {\it Applied Financial Economics}, 20:6, 487--500.

%\bibitem{lo}  
Lo, Andrew W., 1991. Long-Term Memory in Stock Market Prices. 
\textit{Econometrica}, 59, 1991, 1279-1313.

%\bibitem{mandel}  
Mandelbrot, B., 1972. Statistical Methodology for
Non-Periodic Cycles: From the Covariance to R/S Analysis. \textit{Annals of
Economic and Social Measurement}, 1, 259-290.

%\bibitem{pcb99a}  
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