Cauchy Estimators

autoregression, \[y_t = \rho y_{t-1} + u_t\]

Let \(s_t = sign(y_{t-1}) = -1\) if \(y_t-1 < 0\); \(= +1\), otherwise. The Cauchy estimator of \(\rho\) turns out to be just IV estimation, using \(s_t\) as the instrument for \(y_{t-1}\). Importantly, the estimator and the associated \(t\)-test statistic are asymptotically standard normal in distribution, regardless of the value of \(\rho\). In particular, this is the case even if \(\rho = 1\)!

ar1sim <- function(n, phi){
  e <- rnorm(n)
  if(phi<1){
    arima.sim(list(ar=phi), n, innov=e)
  }else{
    cumsum(e)
  }
}
sim <- function(y, rho){
  y_1 <- y[-n]
  y <- y[-1]
  s <- sign(y_1)
  rho_hat <- sum(s*y) / sum(abs(y_1))
  sig <- sqrt(sum((y - rho*y_1)^2)/(n-2))
  se <- sqrt(n)*sig / sum(abs(y_1))
  (rho_hat - rho)/se
}
n <- 1000
nsim <- 10000
phi <- 0.99
tstats <- sapply(1:nsim, function(i)sim(ar1sim(n, phi), phi))
par(mfrow=c(2, 2))
hist(tstats, density=30, breaks=30, prob=TRUE, 
     xlab="x", xlim = c(-4, 4), ylim = c(0, 0.5),
     main="Sampling Distribution of t")
curve(dnorm(x), lwd=2, add=TRUE)
qqnorm(tstats)
plot(ecdf(tstats), main="Empirical Distribution Function")
curve(pnorm(x), 
      col="red", lwd=2, add=TRUE)