added section to manual and example models/sim for ARMA models

Author: Bob Carpenter

misc/moving-avg/arma11-sim.R

@@ -0,0 +1,16 @@
+mu <- -1.25;
+sigma <- 0.75;
+theta <- 0.5;
+phi <- 0.2;
+
+T <- 1000;
+
+err <- rnorm(T,0,sigma);
+nu <- rep(0,T);
+y <- rep(0,T);
+y[1] <- err[1] + mu + phi * mu;
+for (t in 2:T)
+  y[t] <- err[t] + (mu + phi * y[t-1] + theta * err[t-1]);
+
+library('rstan')
+# fit <- stan('arma11.stan', data=list(T=T, y=y), iter=2000, chains=4);

misc/moving-avg/arma11.stan

@@ -0,0 +1,39 @@
+data {
+  int<lower=1> T;       // number of observations
+  real y[T];            // observed outputs
+}
+parameters {
+  real mu;              // mean term
+  real phi;             // autoregression coeff
+  real theta;           // moving avg coeff
+  real<lower=0> sigma;  // noise scale
+}
+model {
+  vector[T] nu;         // prediction for time t
+  vector[T] err;        // error for time t
+  nu[1] <- mu + phi * mu;   // assume err[0] == 0
+  err[1] <- y[1] - nu[1];
+  for (t in 2:T) {
+    nu[t] <- mu + phi * y[t-1] + theta * err[t-1];
+    err[t] <- y[t] - nu[t];
+  }
+
+  // priors
+  mu ~ normal(0,10);
+  phi ~ normal(0,2);
+  theta ~ normal(0,2);
+  sigma ~ cauchy(0,5);
+
+  // likelihood
+  err ~ normal(0,sigma);
+}
+
+// alternative encoding:
+//
+// model {
+//   err <- 0;
+//   for (t in 2:T) {
+//     err <- y[t-1] - (mu + phi * y[t-1] + theta * err[t-1]);
+//     err ~ normal(0,sigma);
+//   }
+