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// --+ options: json=compute, transform_unary_ops, stochastic +--
var x1 x2 x1bar x2bar z ;
varexo ex1 ex2 ex1bar ex2bar ez ;
parameters a_x1_0 a_x1_1 a_x1_2 a_x1_x2_1 a_x1_x2_2
a_x2_0 a_x2_1 a_x2_2 a_x2_x1_1 a_x2_x1_2
e_c_m c_z_1 c_z_2 gamma beta ;
a_x1_0 = -.9;
a_x1_1 = .4;
a_x1_2 = .3;
a_x1_x2_1 = .1;
a_x1_x2_2 = .2;
a_x2_0 = -.9;
a_x2_1 = .2;
a_x2_2 = -.1;
a_x2_x1_1 = -.1;
a_x2_x1_2 = .2;
beta = .1;
e_c_m = .1;
c_z_1 = .07;
c_z_2 = -.3;
gamma = .7;
trend_component_model(model_name=toto, eqtags=['eq:x1', 'eq:x2', 'eq:x1bar', 'eq:x2bar'], targets=['eq:x1bar', 'eq:x2bar']);
pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman);
model;
[name='eq:x1']
diff(diff(x1)) = a_x1_0*(diff(x1(-1))-diff(x1bar(-1))) + a_x1_1*diff(diff(x1(-1))) + a_x1_2*diff(diff(x1(-2))) + a_x1_x2_1*diff(log(x2(-1))) + a_x1_x2_2*diff(log(x2(-2))) + ex1;
[name='eq:x2']
diff(log(x2)) = a_x2_0*(log(x2(-1))-log(x2bar(-1))) + a_x2_1*diff(diff(x1(-1))) + a_x2_2*diff(diff(x1(-2))) + a_x2_x1_1*diff(log(x2(-1))) + a_x2_x1_2*diff(log(x2(-2))) + ex2;
[name='eq:x1bar']
diff(x1bar) = diff(x1bar(-1)) + ex1bar;
[name='eq:x2bar']
log(x2bar) = log(x2bar(-1)) + ex2bar;
[name='eq:pac']
diff(z) = gamma*(e_c_m*(x1(-1)-z(-1)) + c_z_1*diff(z(-1)) + c_z_2*diff(z(-2)) + pac_expectation(pacman)) + (1-gamma)*ez;
end;
shocks;
var ex1 = 1;
var ex2 = 1;
var ex1bar = 1;
var ex2bar = 1;
var ez = 1;
end;
// Initialize the PAC model (build the Companion VAR representation for the auxiliary model).
pac.initialize('pacman');
// Update the parameters of the PAC expectation model (h0 and h1 vectors).
pac.update.expectation('pacman');
// Set initial conditions to zero for non logged variables, and one for logged variables
init = zeros(10, M_.endo_nbr+M_.exo_nbr);
init(:,[2,4]) = ones(10,2);
initialconditions = dseries(init, 2000Q1, vertcat(M_.endo_names,M_.exo_names));
// Simulate the model for 500 periods
TrueData = simul_backward_model(initialconditions, 500);
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