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Dec 5th, 2016

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% params
randn('seed',2016)
T = 100; % length of time series
Tf = 12; % forecast horizon
mu = 0; % unc. mean
rho = 0.95; % autocorrelation
sigma = 0.01; % st. dev. of shock
% simulate series
X = zeros(1,T);
Y = zeros(1,T);
X(1) = 1;
Y(1) = mu + (sigma/sqrt(1-rho^2)) * randn();
for t=2:T
X(t) = t;
Y(t) = (1-rho)*mu + rho*Y(t-1) + sigma*randn();
end
% compute forecasts
Xf = cell(1,T);
Yf = cell(1,T);
for t=1:T
Xf{t} = zeros(1,Tf+1);
Yf{t} = zeros(1,Tf+1);
Xf{t}(1) = t;
Yf{t}(1) = Y(t);
for i=2:(Tf+1)
Xf{t}(i) = t+i-1;
Yf{t}(i) = (1-rho)*mu + rho*Yf{t}(i-1);
end
end
% make chart
figure
plot(X, Y, 'color','blue')
xlim([0,T+Tf+1])
line([0,T+Tf+1],[mu,mu],'color','black','linestyle','--')
for t=1:T
line(Xf{t},Yf{t},'color','red','linestyle',':')
end

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% params
randn('seed',2016)
T = 100;      % length of time series
Tf = 12;      % forecast horizon
mu = 0;       % unc. mean
rho = 0.95;   % autocorrelation
sigma = 0.01; % st. dev. of shock

% simulate series
X = zeros(1,T);
Y = zeros(1,T);
X(1) = 1;
Y(1) = mu + (sigma/sqrt(1-rho^2)) * randn();
for t=2:T
	X(t) = t;
	Y(t) = (1-rho)*mu + rho*Y(t-1) + sigma*randn();
end

% compute forecasts
Xf = cell(1,T);
Yf = cell(1,T);
for t=1:T
	Xf{t} = zeros(1,Tf+1);
	Yf{t} = zeros(1,Tf+1);
	Xf{t}(1) = t;
	Yf{t}(1) = Y(t);
	for i=2:(Tf+1)
		Xf{t}(i) = t+i-1;
		Yf{t}(i) = (1-rho)*mu + rho*Yf{t}(i-1);
	end
end

% make chart
figure
plot(X, Y, 'color','blue')
xlim([0,T+Tf+1])
line([0,T+Tf+1],[mu,mu],'color','black','linestyle','--')
for t=1:T
	line(Xf{t},Yf{t},'color','red','linestyle',':')
end