

    \filetitle{cumsumk}{Cumulative sum with a k-period leap}{tseries/cumsumk}

	\paragraph{Syntax}
 
 \begin{verbatim}
 Y = cumsumk(X,K,RHO,RANGE)
 Y = cumsumk(X,K,RHO)
 Y = cumsumk(X,K)
 Y = cumsumk(X)
 \end{verbatim}
 \paragraph{Input arguments}
 
 \begin{itemize}
 \item
   \texttt{X} {[} tseries {]} - Input data.
 \item
   \texttt{K} {[} numeric {]} - Number of periods that will be leapt the
   cumulative sum will be taken; if not specified, \texttt{K} is chosen
   to match the frequency of the input data (e.g. \texttt{K = -4} for
   quarterly data), or \texttt{K = -1} for indeterminate frequency.
 \item
   \texttt{RHO} {[} numeric {]} - Autoregressive coefficient; if not
   specified, \texttt{RHO = 1}.
 \item
   \texttt{RANGE} {[} numeric {]} - Range on which the cumulative sum
   will be computed and the output series returned.
 \end{itemize}
 \paragraph{Output arguments}
 
 \begin{itemize}
 \item
   \texttt{Y} {[} tseries {]} - Output data constructed as described
   below.
 \end{itemize}
 \paragraph{Options}
 
 \begin{itemize}
 \item
   \texttt{'log='} {[} \texttt{true} \textbar{} \emph{\texttt{false}} {]}
   - Logarithmise the input data before, and de-logarithmise the output
   data back after, running \texttt{x12}.
 \end{itemize}
 \paragraph{Description}
 
 If \texttt{K \textless{} 0}, the first \texttt{K} observations in the
 output series \texttt{Y} are copied from \texttt{X}, and the new
 observations are given recursively by
 
 \begin{verbatim}
 Y{t} = RHO*Y{t-K} + X{t}.
 \end{verbatim}
 If \texttt{K \textgreater{} 0}, the last \texttt{K} observations in the
 output series \texttt{Y} are copied from \texttt{X}, and the new
 observations are given recursively by
 
 \begin{verbatim}
 Y{t} = RHO*Y{t+K} + X{t},
 \end{verbatim}
 going backwards in time.
 
 If \texttt{K == 0}, the input data are returned.
 
 \paragraph{Example}
 
 Construct random data with seasonal pattern, and run X12 to seasonally
 adjust these series.
 
 \begin{verbatim}
 x = tseries(qq(1990,1):qq(2020,4),@randn);
 x1 = cumsumk(x,-4,1);
 x2 = cumsumk(x,-4,0.7);
 x1sa = x12(x1);
 x2sa = x12(x2);
 \end{verbatim}
 The new series \texttt{x1} will be a unit-root process while \texttt{x2}
 will be stationary. Note that the command on the second line could be
 replaced with \texttt{x1 = cumsumk(x)}.


