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dependent error distributions generated

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Francesco Grechi
Francesco Grechi il 6 Mag 2021
I need to present in a graphic way four examples of dependent error distributions generated using an AR(1) model with different values of the autocorrelation coefficient.
Four further models are used to show the QR behavior with respect to the error independence assumption. In particular, starting from an autoregressive error term of order 1: e_AR(1)_[rho] e_i = ρ * e_(i1) + a_i
where ai N= 0, σ = 1) and using the values ρ = {−0.2, +0.2, 0.5, +0.5} and i have the following models:
model1 y_1 = 1 + 2x + e_AR(1)[ρ=0.2];
model2 y_2 = 1 + 2x + e_AR(1)[ρ=+0.2];
model3 y_3 = 1 + 2x + e_AR(1)[ρ=0.5];
model4 y_4 = 1 + 2x + e_AR(1)[ρ=+0.5].
Below there is my work, i didn't use the function regARIMA('AR',{-0.2},'Variance',1); because i don't really know it
%% INPUT Configures
N = 1e5; % Number of samples
x = -5:0.001:5; % A row vector of equally spaced numbers
X = randn(1,size(x,2));
%% Plot Configuration
Label = 14;
Legend = 12;
%% Main Program
% Define theorical PDF of standard normal distribution
fNorm = @(x) 1/sqrt(2*pi) * exp(-x.^2/2);
% Generate vector of normally distribuited random numbers
% with mean 0 and variance 1
%X = randn(1,N);
Vett_e_AR = [ - 0.2; + 0.2; - 0.5; + 0.5 ];
rnd = normrnd(0,1,1,size(x,2));
e_AR7 = Vett_e_AR(1,1) + rnd(1,1);
e_AR8 = Vett_e_AR(2,1) + rnd(1,1);
e_AR9 = Vett_e_AR(3,1) + rnd(1,1);
e_AR10 = Vett_e_AR(4,1) + rnd(1,1);
for j = 2 : size(Vett_e_AR,1)
R = normrnd(0,1);
e_AR7(j) = Vett_e_AR(1,1) * e_AR7(j-1) + R;
e_AR8(j) = Vett_e_AR(2,1) * e_AR8(j-1) + R;
e_AR9(j) = Vett_e_AR(3,1) * e_AR9(j-1) + R;
e_AR10(j) = Vett_e_AR(4,1) * e_AR10(j-1) + R;
end
for i = 1 : size(rnd,2)
y_7(i) = 1 + 2 * rnd(i) + e_AR7(1,end);
y_8(i) = 1 + 2 * rnd(i) + e_AR8(1,end);
y_9(i) = 1 + 2 * rnd(i) + e_AR9(1,end);
y_10(i) = 1 + 2 * rnd(i) + e_AR10(1,end);
end
fNorm = @(y_7) 1/sqrt(2*pi) * exp(-e_AR7.^2/2);
plot(y_7, fNorm(y_7), 'r-', 'Linewidth', 2);
grid on; axis tight; grid minor; hold on;
fNorm_8 = @(y_8) 1/sqrt(2*pi) * exp(-y_8.^2/2);
plot(x, fNorm_8(x), 'b-', 'Linewidth', 2);
fNorm_9 = @(e_AR9) 1/sqrt(2*pi) * exp(-e_AR9.^2/2);
plot(x, fNorm_9(x), 'g-', 'Linewidth', 2);
fNorm_10 = @(e_AR10) 1/sqrt(2*pi) * exp(-e_AR10.^2/2);
plot(x, fNorm_10(x), 'g-', 'Linewidth', 2);
hold off

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