Solved – Lag-wise Confidence band for sample autocorrelation function of AR(1) process

Question

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?

Answer 1

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]))