Lag-wise Confidence band for sample autocorrelation function of AR(1) process Let's say I have a AR(1) process. E.g: $$X_t=\frac{1}{2}X_{t-1}+Z_{t}$$
with $Z_t$ iid student t-distributed with 10 df.
Let's say I have simulated $X_1,X_2 ... X_{1000}$, with $X_{0}=0$.
I want to construct a 95% confidence band for the sample autocorrelation for the first 20 lags. The R function ACF returns the $+/-\frac {1.96}{\sqrt{n=1000}}$. Isn't there a better and more correct way to construct the a CI? 
 A: If you want confidence bands for $\rho_k$ under the assumption that the true model is AR(1), then fit the AR(1) model $x_t = \phi x_{t-1} + w_t$ by maximum likelihood.  This gives you a MLE $\hat\phi$ of $\phi$ and an estimate of the standard error $\widehat{SE\hat\phi}$ based on the observed Fisher information.  Relying on asymptotic normality of $\hat\phi$ an approximate $(1-\alpha)$-confidence interval for $\phi$ is $\hat\phi \pm z_{\alpha/2}\widehat{SE\hat\phi}$.  This means that
$$
P\left(\hat\phi - z_{\alpha/2}\widehat{SE\hat\phi}<\phi<\hat\phi + z_{\alpha/2}\widehat{SE\hat\phi}\right) \approx 1-\alpha.
$$
Thus,
$$
P\left((\hat\phi - z_{\alpha/2}\widehat{SE\hat\phi})^k<\phi^k<\hat\phi + (z_{\alpha/2}\widehat{SE\hat\phi})^k\right) \approx 1-\alpha,
$$
such that
$(\hat\phi \pm z_{\alpha/2}\widehat{SE\hat\phi})^k$ is an approximate confidence interval for the autocorrelation at lag $k$, $\rho_k=\phi^k$.
R example:
x <- arima.sim(n=100,model=list(ar=.8))
model <- arima(x,order=c(1,0,0))
z <- qnorm(.975)
k <- 1:10
confband <- matrix(NA,3,10)
for (i in 1:2)
  confband[i,] <- (model$coef[1] + c(-1,1)[i]*z*sqrt(model$var.coef[1,1]))^k
confband[3,] <- model$coef[1]^k
confband
matplot(k,t(confband),type="l",lty=c(2,2,1),col="black",ylab=expression(rho[k]))


A: After simulating an AR(1) process one possible alternative that I have investigated is to randomly resort the data and compute the acf. Then do this 999 times more ... then analyze the 1000 acf's that were computed and then mark of a central 95% (upper .975 & lower .025) range to provide you with 95% intervals.
