## 2.9 White noise

Time series that show no autocorrelation are called **white noise**. Figure 2.22 gives an example of a white noise series.

```
set.seed(30)
<- tsibble(sample = 1:50, wn = rnorm(50), index = sample)
y %>% autoplot(wn) + labs(title = "White noise", y = "") y
```

```
%>%
y ACF(wn) %>%
autoplot() + labs(title = "White noise")
```

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.