[VR,Zk,Zhk]=vrt(x,k)

function [VR,Zk,Zhk]=vrt(x,k)
% 
% Syntax: [vr,zk,zhk]=vrt(x,k)
%
% Calculates the Variance Ratio Test (VR Test) of a time series x, with
% and without the heteroskedasticity assumption.
% input  : x is a vector of time serie (observed prices)
%          k is a scalar 
% output : VR is the value of the VR Test
%          Zk is the homoscedastic statistic of the variance ratio
%          Zhk is the heteroscedastic statistic of the variance ratio
% *****************************************************************
% Reference:
% Lo A, MacKinley AC (1989): "The size and power of the variance ratio
% test in finite samples". Journal of Econometrics 40:203-238.
%
% Elaborated by : BEN HASSEN Anis 
% "Institut Suprieur de Gestion de Tunis" (ISG Tunis)
% University of Tunis
% 41, rue de la Libert - Cit Bouchoucha - C.P. : 2000 Le Bardo
% Tunisia
% University e-mail: http://www.isg.rnu.tn/
% Personal e-mail: [e-mail]
% _________________________________________________________________
% January 14, 2004.
%
rt1=diff(log(x)); % one period rate of return
rtk=rkp(rt1,k); % k period rate of return
moy=mean(rt1);
v=var(rt1);
T=length(rt1);
m=k*(T-k+1)*(1-k/T);
VR=1/m*sum((rtk-k*moy).^2)/v; % Variance ratio statistic
Zk=sqrt(T)*(VR-1)*(2*(2*k-1)*(k-1)/(3*k))^(-.5); % homoscedastic statistic
j=1:k-1;
vec1=(2/k*(k-j)).^2;
rst=(rt1-moy).^2;
aux=zeros(1,k-1);
for i=1:k-1,
   aux(i)=rst(i+1:T)'*rst(1:T-i);
end
vec2=aux/((T-1)*v)^2;
Zhk=(VR-1)*(vec1*vec2')^(-.5); % heteroscedastic statistic