Autocorrelation

For an \( n\times 1 \) stationary time series \( \boldsymbol x \), the sample autocovariance function is

\[ \hat{\gamma}_x\left(h\right) = \frac{1}{n}\sum_{t = 1}^{n - h}\left(x_{t + h} - \hat{\mu}_x\right)\left(x_t - \hat{\mu}_x\right) \]

The sample autocorrelation function is

\[ \hat{\rho}_x\left(h\right) = \hat{\gamma}_x\left(h\right)/\hat{\gamma}_x\left(0\right) \]

We'll rely a lot on this result:

If \( \boldsymbol x \) white noise and \( v\sim\mathcal{N}\left(0, 1\right) \) \( \implies \) \( \hat{\rho}_x\left(h\right) \approx v/\sqrt{n} \)

Look at a sample ACF of \( n=10 \) white noise.

Keep in mind that \( \hat{\rho}_x\left(h\right) \boldsymbol \approx v/\sqrt{n} \) - we want to think carefully about when this is a good approximation.

Let's simulate \( \hat{\rho}_x\left(1\right) \) for a \( n=10 \) white noise time series.

sim <- 10000
acf.1 <- unlist(lapply(1:sim, function(x) {
  x <- rnorm(n)
  x.acf <- acf(x, plot = FALSE, lag.max = 1)
  return(x.acf$acf[2, 1, 1])
}))

What if we look at bigger values of \( h \)?

Let's simulate \( \hat{\rho}_x\left(9\right) \) for a \( n=10 \) white noise time series.

sim <- 10000
acf.9 <- unlist(lapply(1:sim, function(x) {
  x <- rnorm(n)
  x.acf <- acf(x, plot = FALSE, lag.max = 9)
  return(x.acf$acf[10, 1, 1])
}))

What if we have a longer time series, e.g. \( n=100 \)

Look at the lag-1 sample autocorrelation again over repeated samples.

sim <- 10000
acf.1 <- unlist(lapply(1:sim, function(x) {
  x <- rnorm(n)
  x.acf <- acf(x, plot = FALSE, lag.max = 1)
  return(x.acf$acf[2, 1, 1])
}))