added geometric lag (Koyck) model example with sim

Author: Bob Carpenter

misc/garch/koyck-sim.R

@@ -0,0 +1,14 @@
+library("rstan");
+T <- 200;
+x <- rnorm(T,0,1);
+alpha <- -1.25;
+beta <- 0.75;
+lambda <- 0.6;
+sigma <- 0.5;
+
+y <- rep(NA,T);
+y[1] <- rnorm(1,alpha + beta * x[1],sigma);
+for (t in 2:T)
+  y[t] <- rnorm(1,alpha + beta * x[t] + lambda * y[t-1], sigma);
+
+fit <- stan(file="koyck.stan", data=c("T","y","x"), iter=500, chains=4, fit=fit);

misc/garch/koyck.stan

@@ -0,0 +1,25 @@
+// geometric lag time-series (Koyck 1951)
+//
+// http://en.wikipedia.org/wiki/Distributed_lag
+
+data {
+  int<lower=0> T;   // number of time points
+  real y[T];        // output at time t
+  real x[T];        // predictor for time t
+}
+parameters {
+  real alpha;                       // intercept
+  real beta;                        // slope
+  real <lower=0, upper=1> lambda;   // lag
+  real <lower=0> sigma;             // noise scale
+}
+model {
+  alpha ~ cauchy(0,5);
+  beta ~ cauchy(0,5);
+  lambda ~ uniform(0,1);
+  sigma ~ cauchy(0,5);
+  for (t in 2:T)
+    y[t] ~ normal(alpha +  beta * x[t] +  lambda * y[t-1],
+                  sigma);
+}
+