## 2.9 White noise

Time series that show no autocorrelation are called “white noise”. Figure 2.15 gives an example of a white noise series.

```
set.seed(30)
y <- ts(rnorm(50))
autoplot(y) + ggtitle("White noise")
```

`ggAcf(y)`

For white noise series, we expect each autocorrelation to be close to zero. Of course, they will not be exactly equal to zero as there is some random variation. For a white noise series, we expect 95% of the spikes in the ACF to lie within \(\pm 2/\sqrt{T}\) where \(T\) is the length of the time series. It is common to plot these bounds on a graph of the ACF (the blue dashed lines above). If one or more large spikes are outside these bounds, or if substantially more than 5% of spikes are outside these bounds, then the series is probably not white noise.

In this example, \(T=50\) and so the bounds are at \(\pm 2/\sqrt{50} = \pm 0.28\). All of the autocorrelation coefficients lie within these limits, confirming that the data are white noise.