%% "A property of the Hodrick Prescott filter and its application," Neslihan Sakarya and Robert M. de Jong (2018)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% This file allows you to compare Theorem 1 representation of the HP filter with the HP filter code written by Ivailo Izvorski, the Department of Economics,  Yale University.
% The link for the code: https://dge.repec.org/codes/izvorski/hpfilter.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% After you run the program, the following results are displayed in the command window:

% (2)A (Tx3) table that allows you to compare 3 series (in order)
%    (i) The cyclical component (c) of a process (y) that we generate. The cyclical component is calculated by using Izvorski's code; [labeled in the table as "c series"]
%   (ii) The cyclical component of y that is calculated by using Theorem 1 of Sakarya and de Jong (2018) [labeled in the table as "c1 series"]
%  (iii) The difference between the two cyclical components [labeled in the table as "Difference"] 
%    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Explanation of the codes in this file:
% [1] In this program, we first generate a process denoted by y. The HP filter
% is going to be applied to y. We offer a selection of y processes such as
% a standard normal variable, a linear trend series, a quadratic trend
% series, and a deterministic exponential trend series.
% 
% [2] In the next step, we calculate the cyclical component of y, which is
% denoted as c. The calculation of c is based on the HP filter code written by Ivailo Izvorski, the Department of Economics,  Yale University.
% The link for the code: https://dge.repec.org/codes/izvorski/hpfilter.m
%
% [3] To verify the result of Theorem 1, we create a (Tx1) vector that is denoted as tilde_y. Note that tilde_y corresponds to {\tilde{y}_{Tt}} series in Theorem 1 of Sakarya and de Jong (2018).
%
% [4] We calculate the trend of tilde_y that is denoted as tilde_tau. Note
% that tilde_tau corresponds to
% \hat{\tau}_{Tt}(\tilde{y}_{T1},\tilde{y}_{T2},...,\tilde{y}_{TT}) series in Theorem 1 of Sakarya and de Jong (2018).
%
% [5] Lastly, we calculate the cyclical component of y by using the result of Theorem 1. The cyclical component that is calculated by using the representation in Theorem 1 is denoted as c1. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all;
close all;
clc;

T=10;          %The number of observations
lmb=1600;       %The smoothing parameter value

%% [1]: Generate a series called y

y=randn(T,1);       %y is a standard normal random variable.

% y=ones(T,1);      %y is a linear trend.
% y=cumsum(y);

% y=ones(T,1);      %y is a quadratic trend.
% y=cumsum(y).^2;
 
% y=zeros(T,1);     %y is a deterministic exponential trend.
% 
% for t=1:T;
%     y(t)=exp(t);
% end

%% [2]: Calculate the cyclical component by using the HP filter code written by Ivailo Izvorski

[tau,c]=hpfilter(y,lmb);


%% [3]: Generate \tilde{y}_{Tt} series by using the formula in Theorem 1 of Sakarya and de Jong (2018)

tilde_y=zeros(T,1);

for t=3:T-2;
    
    tilde_y(t)=y(t+2)-4*y(t+1)+6*y(t)-4*y(t-1)+y(t-2);
end


tilde_y(1)=y(3)-2*y(2)+y(1);
tilde_y(2)=(y(4)-2*y(3)+y(2))-2*(y(3)-2*y(2)+y(1));
tilde_y(T-1)=(y(T-1)-2*y(T-2)+y(T-3))-2*(y(T)-2*y(T-1)+y(T-2));
tilde_y(T)=y(T)-2*y(T-1)+y(T-2);

%% [4]: Calculate the trend of \tilde{y}_{Tt} series

[tilde_tau,tilde_c]=hpfilter(tilde_y,lmb);


%% [5]: Calculate the cyclical component of y by using the representation in Theorem 1 of Sakarya and de Jong (2018)

c1=lmb*tilde_tau;

%% Result:

R=[c, c1, c-c1];

format long

disp('  c series                c1 series             Difference')
disp(R)