Add example for PAC equation (with estimation).

(cherry picked from commit 415a86d1)

examples/pacmodel.mod

+// --+ options: json=compute, stochastic +--
+
+var y x z v;
+
+varexo ex ey ez ;
+
+parameters a_y_1 a_y_2 b_y_1 b_y_2 b_x_1 b_x_2 d_y; // VAR parameters
+
+parameters beta e_c_m c_z_1 c_z_2;               // PAC equation parameters
+
+a_y_1 =  .2;
+a_y_2 =  .3;
+b_y_1 =  .1;
+b_y_2 =  .4;
+b_x_1 = -.1;
+b_x_2 = -.2;
+d_y = .5;
+
+
+beta  =  .9;
+e_c_m =  .1;
+c_z_1 =  .7;
+c_z_2 = -.3;
+
+var_model(model_name=toto, eqtags=['eq:x', 'eq:y']);
+
+pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman);
+
+pac_target_info(pacman);
+  target v;
+  auxname_target_nonstationary vns;
+
+  component y;
+  auxname pv_y_;
+  kind ll;
+
+  component x;
+  growth diff(x(-1));
+  auxname pv_dx_;
+  kind dd;
+
+end;
+
+model;
+
+  [name='eq:y']
+  y = a_y_1*y(-1) + a_y_2*diff(x(-1)) + b_y_1*y(-2) + b_y_2*diff(x(-2)) + ey ;
+
+
+  [name='eq:x']
+  diff(x) = b_x_1*y(-2) + b_x_2*diff(x(-1)) + ex ;
+
+  [name='eq:v']
+  v = x + d_y*y ;  // Composite target, no residuals here only variables defined in the auxiliary VAR model.
+
+  [name='zpac']
+  diff(z) = e_c_m*(pac_target_nonstationary(pacman)-z(-1)) + c_z_1*diff(z(-1))  + c_z_2*diff(z(-2)) + pac_expectation(pacman) + ez;
+
+end;
+
+shocks;
+  var ex = .10;
+  var ey = .15;
+  var ez = .05;
+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');
+
+/*
+**
+** Simulate artificial dataset
+**
+*/
+
+// Set initial conditions to zero.
+initialconditions = dseries(zeros(10, M_.endo_nbr+M_.exo_nbr), 2000Q1, vertcat(M_.endo_names,M_.exo_names));
+
+// Simulate the model for 5000 periods
+TrueData = simul_backward_model(initialconditions, 5000);
+
+/*
+**
+** Estimate PAC equation (using the artificial data)
+**
+*/
+
+
+// Provide initial conditions for the estimated parameters
+clear eparams
+eparams.e_c_m   =  .9;
+eparams.c_z_1   =  .5;
+eparams.c_z_2   =  .2;
+
+edata = TrueData;                // Set the dataset used for estimation
+edata.ez = dseries(NaN, 2000Q1); // Remove residuals for the PAC equation from the database.
+
+pac.estimate.nls('zpac', eparams, edata, 2005Q1:2005Q1+4000, 'fmincon'); // Should produce a table with the estimates (close to the calibration given in lines 21-23)