function [fX,AA] = bpass(X,pl,pu,root,drift,ifilt,nfix,thet)

%

%	MATLAB COMMAND FOR DEFAULT FILTER:  fX = bpass(X,pl,pu)

%

%	Required Inputs:

%  	X		- matrix of data (T x # variables) with variables in columns

%		pl		- minimum period of oscillation of desired component 

%		pu		- maximum period of oscillation of desired component (2<=pl<pu<infinity).

%

%  Output:

%		fX 	- matrix (T x nvars) containing filtered data 

%

%	Examples:  

%   Quarterly data: pl=6, pu=32 returns component with periods between 1.5 and 8 yrs.

%	 Monthly data  : pl=2, pu=24 returns component with all periods less than 2 yrs.

%

%  Note:  When feasible, we recommend dropping 2 years of data from the beginning 

%		and end of the filtered data series.  These data are relatively poorly estimated.

%

%	Optional Inputs:  root,drift,ifilt,nfix,thet.  THESE INPUTS ARE NOT REQUIRED.

%

%	Additional Output:  AA - T x T matrix which premultiplies a T x Nvars data matrix 

%				to compute the band-pass filtered data.  (Note: data should have any drift

%				term or time trend removed before applying AA.)

%

%=======================================================================================

%		The next discussion is only for users who wish to use optimal band pass filters 

%		other than the default filter recommended in Christiano and Fitzgerald (1999). 

%=======================================================================================

%

%  Version Date: 6/10/99 (Please notify Terry Fitzgerald at

%			tj.fitzgerald@clev.frb.org or (216)579-2293 if you encounter bugs).  

%

%  This is a matlab function that filters time series data using  

%  approximations to the band pass filter as discussed in the paper

%  "The Band Pass Filter" by Lawrence J. Christiano and Terry J. Fitzgerald (1999).



%	Several band-pass approximation strategies can be selected in this 

%  program.  To explain how to select these alternatives, we rely upon the notation 

%  developed in the paper.  The default setting used above returns the filtered data fX 

%  associated with the unrestricted optimal filter assuming a Random Walk with drift

%  time series process.  

%

%	Alternative filters require selecting the type of filter (asymetric, symmetric, or 

%  fixed length symmetric) and the underlying time series assumption (random walk, IID, 

%  or a user provided MA process with or without a unit root).  The fixed length 

%  symmetric filter advocated by Baxter and King(1999), and the trigonometric regression

%	approach, can also be chosen.  

%

%  Optional Inputs:

%		root		= 0	no unit root in time series						(default: root = 1);

%					= 1	unit root in time series

%		drift		= 0	no drift or time trend in time series			(default: drift = 1);

%					= 1	time series is Random Walk with possibly a nonzero drift

%					 		or time series is stationary about a linear time trend.

%		ifilt 	= 0	Asymmetric Filter 									(default: ifilt = 0);

%					= 1	Symmetric Filter

%					= 2	Fixed Length Symmetric Filter

%					= 3 	Baxter-King Fixed Length Symmetric Filter 

%					= 4	Trigonometric Regression Filter

%		nfix	= sets fixed length (used w/ ifilt=2 or 3)				(default: nfix=-1).

%		thet	= MA coefficients for time series model 					(default: thet=1)

%				if root=1 :	x(t) = mu + x(t-1) + thet(1)e(t) + thet(2)e(t-1) + ...

%				if root=0 :	x(t) = thet(1)e(t) + thet(2)e(t-1) + ...

%						Examples:

%									root=1, thet=1 	Random Walk filter

%									root=0, thet=1		IID filter

%

%  The following function calls are permitted:

%		fX = bpass(X,pl,pu)										: optimal Random Walk filter

%		fX = bpass(X,pl,pu,root)								: choose unit root option

%		fX = bpass(X,pl,pu,root,drift)						: choose drift/trend option

%		fX = bpass(X,pl,pu,root,drift,ifilt)				: choose type of filter

%		fX = bpass(X,pl,pu,root,drift,ifilt,nfix)			: choose fixed filter length

%		fX = bpass(X,pl,pu,root,drift,ifilt,nfix,thet)	: choose MA time series model

%

%  Notes:  

%	1) Users wishing to use a fixed length symmetric filter (ifilt = 2 or 3) must

%     also supply the lead/lag length of the filter (nfix).

%	2) The filtered data matrix (fX)	associated with some filters include zeros at the 

%		beginning and end of the data set.  The filter is not available for these points.

%		You should truncate these points from your analysis.  Examples include

%		fixed length filters and filters for user supplied time series with length(thet)>1.  

%		The default filter returns all nonzero data.

%

if nargin < 4

   root = 1;

   drift = 1;

   ifilt = 0;

   nfix = -1;

   thet = 1;

elseif nargin < 5

   drift = 1;

   ifilt = 0;

   nfix = -1;

   thet = 1;

elseif nargin < 6

   ifilt = 0;

   nfix = -1;

   thet = 1;

elseif nargin < 7

   nfix = -1;

   thet = 1;

elseif nargin < 8

   thet = 1;

end



nq=length(thet)-1;

b=2*pi/pl;

a=2*pi/pu;



[T,nvars] = size(X);



if T < 5

   warning(' (bpass): T < 5')

end

if T < 2*nq+1

   error(' (bpass): T must be at least 2*q+1');

end

if pu <= pl

   error(' (bpass): pu must be larger than pl')

end

if pl < 2

   warning(' (bpass): pl less than 2 , reset to 2')

   pl = 2;

end

if root ~= 0 & root ~= 1

   error(' (bpass): root must be 0 or 1')

end

if drift<0 | drift > 1

   error(' (bpass): drift must be 0 or 1')

end

if ifilt<0 | ifilt > 4

   error(' (bpass): ifilt must be 0, 1, 2, 3, or 3')

end

if (ifilt == 2 | ifilt == 3) & nfix < 1

   error(' (bpass): fixed lag length must be >= 1')

end

if ifilt == 2 & nfix < nq

   error(' (bpass): fixed lag length must be >= q')

end

if (ifilt == 2 | ifilt ==3) & nfix >= T/2

      error(' (bpass): fixed lag length must be < T/2')

end

if (ifilt == 4 & T-2*floor(T/2) ~= 0)

   error(' (bpass): trigonometric regressions only available for even T')

end



[m1,m2]=size(thet);

if m1 > m2 

   th=thet;

else

   th=thet';

end

%   compute g(thet)

%   [g(1) g(2) .... g(2*nq+1)] correspond to [c(q),c(q-1),...,c(1),

%                                        c(0),c(1),...,c(q-1),c(q)]

%   cc = [c(0),c(1),...,c(q)]

thp=flipud(th);

g=conv(th,thp);

cc = g(nq+1:2*nq+1);

%   compute "ideal" Bs

j = 1:2*T;

B = [(b-a)/pi (sin(j*b)-sin(j*a))./(j*pi)]';

%    compute R using closed form integral solution

R = zeros(T,1);

if nq > 0

   R0 = B(1)*cc(1) + 2*B(2:nq+1)'*cc(2:nq+1);

   R(1) = pi*R0;

   for i = 2:T

      dj = Bge(i-2,nq,B,cc);

      R(i) = R(i-1) - dj;

   end

else

   R0 = B(1)*cc(1);

   R(1) = pi*R0;

   for j = 2:T

      dj = 2*pi*B(j-1)*cc(1);

      R(j) = R(j-1) - dj;

   end

end

%

%======================================================================

%

AA = zeros(T,T);

%

%----------------------------------

%  asymmetric filter

%----------------------------------

%

if ifilt == 0

   if nq==0

      for i = 1:T

         AA(i,i:T) = B(1:T-i+1)';

         if root == 1;

            AA(i,T) = R(T+1-i)/(2*pi);

         end

      end

      AA(1,1) = AA(T,T);

      %  Use symmetry to construct bottom 'half' of AA

      AA = AA + flipud(fliplr(triu(AA,1)));

   else

      % CONSTRUCT THE A MATRIX size T x T

      A = Abuild(T,nq,g,root);

      Ainv = inv(A);

      % CONSTRUCT THE d MATRIX size T x 1 

      for np = 0:ceil(T/2-1)

         d = zeros(T,1);

         ii = 0;

        	for jj = np-root:-1:np+1+root-T

           	ii = ii+1;

	         d(ii) = Bge(jj,nq,B,cc);

   	   end

         if root == 1

            d(T-1) = R(T-np);

         end

         %  COMPUTE Bhat = inv(A)*d

         Bhat = Ainv*d;

         AA(np+1,:) = Bhat';

      end

      %  Use symmetry to construct bottom 'half' of AA

      AA(ceil(T/2)+1:T,:) = flipud(fliplr(AA(1:floor(T/2),:)));

   end

end   

%

%-------------------------------------

%  symmetric filter

%-------------------------------------

%

if ifilt == 1

   if nq==0

      for i = 2:ceil(T/2)

         np = i-1;

         AA(i,i:i+np) = B(1:1+np)';

         if root == 1;

            AA(i,i+np) = R(np+1)/(2*pi);

         end

         AA(i,i-1:-1:i-np) = AA(i,i+1:i+np);

      end

      %  Use symmetry to construct bottom 'half' of AA

      AA(ceil(T/2)+1:T,:) = flipud(fliplr(AA(1:floor(T/2),:)));

   else

      for np = nq:ceil(T/2-1)

         nf = np;

         nn = 2*np+1;

      % CONSTRUCT THE A MATRIX size nn x nn

       	A = Abuild(nn,nq,g,root);

         Ainv = inv(A);

      % CONSTRUCT THE d MATRIX size nn x 1

         d = zeros(nn,1);

         ii = 0;

         for jj = np-root:-1:-nf+root

  	         ii = ii+1;

     	      d(ii) = Bge(jj,nq,B,cc);

         end

         if root == 1

            d(nn-1) = R(nf+1);

			end            

      %  COMPUTE Bhat = inv(A)*d

         Bhat = Ainv*d;

         AA(np+1,1:2*np+1) = Bhat';

      end

      %  Use symmetry to construct bottom 'half' of AA

      AA(ceil(T/2)+1:T,:) = flipud(fliplr(AA(1:floor(T/2),:)));

   end

end

%

%----------------------------------

%  fixed length symmetric filter

%----------------------------------

%

if ifilt == 2

   if nq==0

      bb = zeros(2*nfix+1,1);

      bb(nfix+1:2*nfix+1) = B(1:nfix+1);

      bb(nfix:-1:1) = B(2:nfix+1);

      if root == 1

         bb(2*nfix+1) = R(nfix+1)/(2*pi);

         bb(1) = R(nfix+1)/(2*pi);

      end

      for i = nfix+1:T-nfix

         AA(i,i-nfix:i+nfix) = bb';

      end

   else

      nn = 2*nfix+1;

      % CONSTRUCT THE A MATRIX size nn x nn

     	A = Abuild(nn,nq,g,root);

      Ainv = inv(A);

      % CONSTRUCT THE d MATRIX size nn x 1

      d = zeros(nn,1);

      ii = 0;

      for jj = nfix-root:-1:-nfix+root

 	      ii = ii+1;

	  	   d(ii) = Bge(jj,nq,B,cc);

      end

      if root == 1

     	   d(nn-1) = R(nn-nfix);

      end

      %  COMPUTE Bhat = inv(A)*d

      Bhat = Ainv*d;

      for ii = nfix+1:T-nfix

         AA(ii,ii-nfix:ii+nfix) = Bhat';

      end

   end

end

%

%----------------------------------

%  Baxter-King filter

%----------------------------------

%

if ifilt == 3

   bb = zeros(2*nfix+1,1);

   bb(nfix+1:2*nfix+1) = B(1:nfix+1);

   bb(nfix:-1:1) = B(2:nfix+1);

   bb = bb - sum(bb)./(2*nfix+1);

   for i = nfix+1:T-nfix

      AA(i,i-nfix:i+nfix) = bb';

   end

end

%

%----------------------------------

%  Trigonometric Regression filter

%----------------------------------

%

if ifilt == 4

	jj = 1:T./2;

	% find frequencies in desired band omitting T/2;

	jj = find(T./pu<=jj & jj<=T./pl & jj<T./2);

	if isempty(jj)

	   error(' (bpass): frequency band is empty in trigonometric regression');

	end

	om = 2*pi*jj./T;

	if pl > 2

  		for t = 1:T

   		for k = T:-1:1

      		l = t-k;

      		tmp = sum(cos(om*l));

      		AA(t,k) = tmp;

   		end

  		end

	else

  		for t = 1:T

   		for k = T:-1:1

      		l = t-k;

      		tmp = sum(cos(om*l));

      		tmp2 = ( cos(pi*(t-l)).*cos(pi*t) )./ 2;

      		AA(t,k) = tmp + tmp2;

   		end

  		end

	end

	AA = AA*2/T;

end

%

%======================================================================

%

%  check that sum of all filters equal 0 if assuming unit root

%

if root == 1

   tst = max(abs(sum(AA')));

   if tst > 1.e-09 & root ~= 0

      warning(' (bpass): Bhat does not sum to 0 ')

      tst   

   end

end

%

%======================================================================

%

%	compute filtered time series using selected filter matrix AA

%

if drift > 0;

   X = undrift(X);

end

fX = AA*X;

%

%======================================================================

%				Functions

%======================================================================

%

function y = Bge(jj,nq,B,cc)

%

%  closed form solution for integral of B(z)g(z)(1/z)^j  (eqn 16)

%     nq > 0, jj >= 0

%     if nq = 0, y = 2*pi*B(absj+1)*cc(1);

%

absj =abs(jj);

if absj >= nq

   dj = B(absj+1)*cc(1) + B(absj+2:absj+nq+1)'*cc(2:nq+1);

   dj = dj + flipud(B(absj-nq+1:absj))'*cc(2:nq+1);

elseif absj >= 1

   dj = B(absj+1)*cc(1) + B(absj+2:absj+nq+1)'*cc(2:nq+1);

   dj = dj + flipud(B(1:absj))'*cc(2:absj+1);

   dj = dj + B(2:nq-absj+1)'*cc(absj+2:nq+1);

else

   dj = B(absj+1)*cc(1) + 2*B(2:nq+1)'*cc(2:nq+1);

end

y = 2*pi*dj;

%

%-----------------------------------------------------------------------

%

function A = Abuild(nn,nq,g,root)

%

%  builds the nn x nn A matrix in A.12

%   if root == 1 (unit root)

%      Abig is used to construct all but the last 2 rows of the A matrix

%   elseif root == 0 (no unit root)

%      Abig is used to construct the entire A matrix

%

if root == 1

	Abig=zeros(nn-2,nn+2*(nq-1));

	for j = 1:nn-2      

	   Abig(j,[j:j+2*nq]) = g'; 

	end

	A = Abig(:,[nq:nn+nq-1]);

	%   construct A(-f)

	Q = -ones(nn-1,nn);

	Q = tril(Q);

	F = zeros(1,nn-1);

	F(nn-1-nq:nn-1) = g(1:nq+1);

	A(nn-1,:) = F*Q;

	%    construct last row of A

   A(nn,:) = ones(1,nn);

else

	Abig=zeros(nn,nn+2*(nq-1));

	for j = 1:nn      

   	Abig(j,[j:j+2*nq]) = g'; 

	end

	A = Abig(:,[nq+1:nn+nq]);

end   

%    multiply A by 2*pi

A = 2*pi*A;

%

%-----------------------------------------------------------------------

%

function xun = undrift(x)

%

%  This function removes the drift or a linear time trend from a time series using the formula

%			drift = (x(T) - x(1)) / (T-1).

%

%  Input:  x - data matrix x where columns represent different variables, x is (T x # variables). 

%  Output: xun - data matrix same size as x with a different drift/trend removed from each variable.

%

[T,nvars] = size(x);

xun = zeros(T,nvars);

dd = [0:T-1]';

for ivar = 1:nvars

   drift = (x(T,ivar)-x(1,ivar)) / (T-1);

   xun(:,ivar) = x(:,ivar) - dd*drift;

end



