Autocorrelation Function (ACF)

In the previous chapter, we discussed how to decompose time series and ensure they are stationary. Now, we turn our attention to understanding the correlation structure within a stationary time series. Specifically, how is the value of the series at a given time $t$, denoted as $y_t$, related to its past values like $y_{t-1}$, $y_{t-2}$, and so on? The Autocorrelation Function (ACF) is our primary tool for quantifying this relationship.

Understanding Autocorrelation

Autocorrelation simply means "self-correlation". It measures the linear relationship between lagged values of a time series. In simpler terms, it tells us how much the value of the series at time $t$ is correlated with its value $k$ periods ago, at time $t-k$. This lag $k$ can be 1, 2, 3, etc.

The ACF value for a specific lag $k$, often denoted as $\rho_k$, is calculated similarly to the standard correlation coefficient but between $y_t$ and $y_{t-k}$ across all available $t$.

$$\rho_k = \frac{Cov(y_t, y_{t-k})}{Var(y_t)}$$

Where $Cov(y_t, y_{t-k})$ is the covariance between the series and its lagged version, and $Var(y_t)$ is the variance of the series. Since we assume the series is stationary, the variance $Var(y_t)$ is constant over time, and the covariance $Cov(y_t, y_{t-k})$ depends only on the lag $k$, not on the specific time $t$.

The ACF values range from -1 to 1:

• +1: Perfect positive correlation. If the series increases, the value $k$ periods ago also tended to increase.
• -1: Perfect negative correlation. If the series increases, the value $k$ periods ago tended to decrease.
• 0: No linear correlation between the series and its value $k$ periods ago.

By definition, the autocorrelation at lag 0, $\rho_0$, is always 1, as any series is perfectly correlated with itself at no lag.

Calculating and Plotting the ACF

We rarely calculate these values manually. Statistical libraries like statsmodels in Python provide functions to compute and plot the ACF. The standard visualization is a "correlogram" or ACF plot. This plot shows the autocorrelation values $\rho_k$ on the y-axis for different lags $k$ on the x-axis (usually starting from lag 1, though sometimes lag 0 is included).

A critical feature of ACF plots generated by statistical packages is the inclusion of significance boundaries. These are typically represented as a shaded area (often blue). Lags where the autocorrelation bar extends past this boundary are considered statistically significant (usually at the 5% significance level). This suggests that the observed correlation at that lag is unlikely to be due to random chance alone, assuming the true autocorrelation at that lag is zero.

Let's look at an example ACF plot for a stationary time series:

ACF plot showing autocorrelation values for lags 1 through 20. The blue shaded area represents the 95% confidence interval. Bars extending outside this area indicate statistically significant autocorrelation.

Interpreting the ACF Plot

In the plot above:

• Lag 0 shows an autocorrelation of 1 (by definition, though often omitted in plots focusing on model identification).
• Lags 1, 2, 3, and 4 show significant positive autocorrelation, as their bars extend well above the shaded confidence band.
• The autocorrelation gradually decreases as the lag increases. This pattern of slowly decaying ACF is characteristic of certain types of time series processes, like Autoregressive (AR) processes, which we will discuss later.
• After lag 4 or 5, the autocorrelations are generally within the confidence bounds, suggesting they are not statistically significant and might be due to random noise.

Analyzing the ACF plot is a fundamental step in time series analysis. It helps us understand the "memory" of the process. How far back in time do past values significantly influence the current value? The pattern of decay (e.g., sharp cutoff vs. slow decay) in the ACF provides hints about the underlying structure of the data and guides us in selecting appropriate models like Moving Average (MA) or Autoregressive (AR) models. We will explore this connection between ACF patterns and model identification in the section "Interpreting ACF/PACF for Model Selection". For now, focus on understanding what the ACF measures and how to read its plot.