3.4 Autocorrelation

Autocorrelation Definition¶

As discussed in Chapter 2, covariance, and by extension autocovariance, has some undesirable characteristics. For this reason, in time series analysis we often prefer to work with the autocorrelation, which is a unitless quantity bounded by $[-1,1]$.

For any finite variance process, the autocorrelation is simply the autocovariance normalized by the the variances.

When considering stationary time series, we can employ the simplifications

and

As such, a stationary time series has autocorrelation given by

Note that $\rho(h)=\rho(-h)$ and $\rho(0)=1$.

Autocorrelation is Positive Semidefinite¶

The autocorrelation matrix $\mathbf{P}$ is related to the autocovariance matrix $\boldsymbol{\Gamma}$ by

$\gamma(0)$ is the variance, and hence cannot be negative, so $\mathbf{v}^T\mathbf{P}\mathbf{v}$ will only differ from $\mathbf{v}^T\boldsymbol{\Gamma}\mathbf{v}$ by a positive constant term given by $\frac{1}{\gamma(0)}$. As a result, since $\boldsymbol{\Gamma}$ is positive semidefinite, $\mathbf{P}$ must also be as well.

Autocorrelation Plots¶

It is often instructive to plot the autocorrelation of a time series. statsmodels has a nice built-in function to create plots:

from statsmodels.graphics.tsaplots import plot_acf
plot_acf(my_time_series, lags=20)

where the parameter lags controls the maximum value of $h$. lags=20 is generally a good starting choice for most scenarios, though for longer patterns such as hourly data with a 24-hour period you may want to set lags to something like 72 or 96.

plot_acf has other arguments that we will discuss once we’ve covered more theory. For now, let’s take a look at a couple of different autocorrelation plots to get a feeling for how they behave.

Figure 1 shows two significant lags (plus the value of 1 at $h=0$), and falls to statistical insignificance thereafter as denoted by the blue shading. Later on we will see that this is a hallmark characteristic of a moving average process.

Figure 2 demonstrates an exponentially decaying autocorrelation that remains statistically significant even at $h=20$. In later chapters we will see that this behavior corresponds to an autoregressive process.