diff -u -w -u -w -r1.6 timeseries.tex
--- doc/tex/timeseries.tex	27 Feb 2006 18:29:51 -0000	1.6
+++ doc/tex/timeseries.tex	2 Mar 2006 16:37:45 -0000
@@ -291,30 +291,63 @@
 \subsection{The KPSS test}
 \label{sec:KPSStest}
 
+The KPSS test (Kwiatkowski, Phillips, Schmidt and Shin (1992)) is a
+unit root test in which the null hypothesis is opposite to that in the
+ADF test: under the null, the series subjected to the test is
+stationary, against the alternative that it is $I(1)$.
+
+The basic intuition behind this test statistic is very simple: if
+$y_t$ can be written as $y_t = \mu + u_t$, where $u_t$ is some
+zero-mean stationary process, then not only the sample average of the
+$y_t$'s is a consistent estimator of $\mu$, but the long-run variance
+of $u_t$ is a well-defined, finite number. Neither of these properties
+hold under the alternative.
+
+The test itself is based on the following statistic:
 \begin{equation}
   \label{eq:KPSStest}
   \eta = \frac{\sum_{i=1}^T S_t^2 }{ T^2 \bar{\sigma}^2 }
 \end{equation}
-where $S_t = \sum_{s=1}^t e_s$ and $\bar{\sigma}^2$ is an
-estimate of the long-run variance of $e_t = (y_t - \bar{y})$.
-\begin{itemize}
-\item $H_0$: $y_t$ is I(0);
-\item one-sided test: $H_0$ is rejected if $\eta$ is ``big'';
-\item extension to deterministic trend ($e_t$ are the residuals from
-  an OLS regression of $y_t$ on a constant and a linear trend);
-\item nonstandard distribution --- we provide 90\%, 95\%, 97.5\% and
-  99\% quantiles.
-\end{itemize}
-Syntax:
+where $S_t = \sum_{s=1}^t e_s$ and $\bar{\sigma}^2$ is an estimate of
+the long-run variance of $e_t = (y_t - \bar{y})$. Under the null, this
+statistic has a well-defined (nonstandard) asymptiotic distribution,
+which is free of nuisance parameters and has been tabulated by
+simulation. Under the alternative, the statistic diverges.
+
+As a consequence, it is possible to construce a one-sided test based
+on $\eta$, where $H_0$ is rejected if $\eta$ is bigger than the
+appropriate critical value; \app{gretl} provides the 90\%, 95\%,
+97.5\% and 99\% quantiles.
+
+Usage example:
 \begin{code}
-  kpss n x1
+  kpss n y
 \end{code}
-\begin{itemize}
-\item \verb|--trend| option;
-\item \verb|n| is used for estimating $\bar{\sigma}^2$; in the GUI,
-  default is the integer part of $4 \left( \frac{T}{100}
-  \right)^{1/4}$.
-\end{itemize}
+where \verb|n| is an integer used for estimating $\bar{\sigma}^2$ by
+the formula 
+\[
+  \bar{\sigma}^2 = \sum_{i=-m}^m \left( 1 - \frac{|i|}{m+1} \right) \hat{\gamma}_i
+\]
+and $\hat{\gamma}_k$ is the empirical $k$-th order empirical
+autocovariance of $e_t$. For this estimator to be consistent, the
+bandwidth $m$ must be large enough to accommodate the short-run
+persistence of $e_t$, but not too large compared to the sample size
+$T$.  In the GUI interface of \app{gretl}, this value defaults to the
+integer part of $4 \left( \frac{T}{100} \right)^{1/4}$.
+
+The above concept can be generalized to the case where $y_t$ is
+thought to be stationary around a deterministic trend. In this case,
+formula (\ref{eq:KPSStest}) remains unchanged, but the series $e_t$ is
+defined as the residuals from an OLS regression of $y_t$ on a constant
+and a linear trend. This second form of the test is obtaining by
+appending the \verb|--trend| option to the \cmd{kpss} command:
+\begin{code}
+  kpss n y --trend
+\end{code}
+Note that in this case the asymptotic distribution of the test is
+different and the critical values reported by \app{gretl} change
+accordingly.
+
 
 \subsection{The Johansen tests}
 \label{sec:Joh-test}
@@ -363,7 +396,6 @@
 \item Critical values by Doornik
 \end{itemize}
 
-
 \section{ARCH and GARCH}
 \label{sec:arch-garch}
 
