The autocorrelation function measures the unit-free linear association between a time series and lagged versions of itself. The partial autocorrelation function asks a narrower question: how much direct linear relationship remains at a given lag after the intervening lags have been accounted for.
Together, ACF and PACF reveal persistence and help propose AR and MA orders, but their sample patterns are noisy and can also be distorted by trend, seasonality, or structural change. They are most useful as identification tools when interpreted with stationarity, sampling uncertainty, residual diagnostics, and fitted-model comparisons.
Worked calculation: AR(1) dependence
For a stationary AR(1),
$$ X_t=0.7X_{t-1}+\varepsilon_t, $$
the theoretical autocorrelation is
$$ \rho(h)=0.7^{|h|}. $$
Therefore $\rho(1)=0.7$, $\rho(2)=0.49$, and $\rho(3)=0.343$. The ACF tails off geometrically rather than becoming exactly zero. The PACF is $0.7$ at lag 1 and zero at later lags in the population, which is the ideal identification pattern for an AR(1). Finite samples only approximate these values.
ACF and PACF summarize dependence across time lags and are useful for model identification and forecasting. They describe related but different aspects of that dependence:
- The ACF measures the correlation between observations separated by a given lag.
- The PACF isolates the direct linear relationship at a lag after accounting for the intermediate lags.
Autocorrelation Function (ACF)
The Autocorrelation Function (ACF) measures the correlation between a time series and its lagged values. It summarizes how strongly observations separated by $k$ periods move together. Persistent or repeating ACF patterns can signal serial dependence, non-stationarity, or seasonality, although the ACF alone does not identify their cause. The autocorrelation at lag $k$, denoted $\rho_k$, is defined as:
$$ \rho_k = \frac{\gamma_k}{\gamma_0} $$
where:
- $\gamma_k$ is the autocovariance at lag $k$;
- $\gamma_0$ is the variance of the series, or equivalently the autocovariance at lag 0.
Autocovariance Function
The autocovariance at lag $k$ measures how observations separated by $k$ periods vary together. For a weakly stationary series with mean $\mu$, it is:
$$ \gamma_k = \text{Cov}(X_t, X_{t+k}) = \mathbb{E}[(X_t - \mu)(X_{t+k} - \mu)] $$
where $\mu$ is the constant mean of the series and $\mathbb{E}$ denotes expectation. The following figure gives a geometric view of how observations separated by a lag contribute to autocovariance.
Autocorrelation Coefficient
The autocorrelation coefficient at lag $k$ normalizes $\gamma_k$ by the variance $\gamma_0$. This makes it dimensionless and bounded between $-1$ and $1$, so values can be compared across lags and series:
$$ \rho_k = \frac{\gamma_k}{\gamma_0} = \frac{\mathbb{E}[(X_t - \mu)(X_{t+k} - \mu)]}{\mathbb{E}[(X_t - \mu)^2]} $$
Sample Autocorrelation Function
In practice, the population ACF is unknown and is estimated from the observed series. One common sample autocorrelation coefficient at lag $k$ is:
$$ r_k = \frac{\sum_{t=1}^{N-k} (x_t - \bar{x})(x_{t+k} - \bar{x})}{\sum_{t=1}^{N} (x_t - \bar{x})^2} $$
Where:
- $\bar{x}$ is the sample mean of the series.
- $N$ is the number of observations.
- $x_t$ is the observed value at time $t$.
The sample ACF can be computed for any series, but the standard interpretation of its lag pattern is most useful when the series is approximately stationary.
Sampling Properties (Large Samples)
For a weakly stationary series with mean $\mu$ and autocovariance $\gamma(h)$:
- $E(\bar{X}_n)=\mu$.
- The exact variance of the sample mean is
$$ \text{Var}(\bar{X}n) =\frac{1}{n}\sum{h=-(n-1)}^{n-1} \left(1-\frac{|h|}{n}\right)\gamma(h) $$
- Under suitable weak-dependence conditions, $\bar{X}_n$ is approximately normal for large $n$.
A practical confidence interval therefore needs an estimate of the variance of $\bar X_n$. Using a truncated, weighted autocovariance estimate with bandwidth $m$ gives
$$ \hat v_n =\frac{1}{n} \left[ \hat\gamma(0) +2\sum_{h=1}^{m} \left(1-\frac{h}{m+1}\right)\hat\gamma(h) \right], $$
and an approximate $(1-\alpha)$ confidence interval is
$$ \bar X_n\pm z_{1-\alpha/2}\sqrt{\hat v_n}. $$
For a fixed set of lags in a stationary linear process, the vector of sample autocorrelations also has an approximate large-sample normal distribution:
$$ \hat{\rho} = (\hat{\rho}(1),\dots,\hat{\rho}(k))^\top \approx \mathcal{N}\left(\rho,\frac{W}{n}\right). $$
A Bartlett-type expression for the entries of the asymptotic covariance matrix is
$$ W_{ij} = \sum_{m=1}^{\infty} {\rho(m+i)+\rho(m-i)-2\rho(i)\rho(m)} {\rho(m+j)+\rho(m-j)-2\rho(j)\rho(m)}. $$
The main practical point is that sampling errors are correlated across lags, so uncertainty should be interpreted as a joint pattern rather than as a collection of independent tests.
At large lags, fewer observation pairs contribute to each estimate, so the sample ACF becomes increasingly noisy. Rules such as limiting plots to roughly $n/4$ lags can be useful for display, but they are heuristics rather than requirements.
Plotting the ACF
An ACF plot, or correlogram, displays the sample autocorrelation at each lag. It helps reveal persistence, repeating seasonal structure, and cutoff patterns that may suggest candidate time-series models.
Useful questions include:
- Does the series resemble white noise, with most autocorrelations near zero?
- How quickly does dependence decay as the lag increases?
- Are there repeating spikes at seasonal lags?
- Does the ACF show a cutoff pattern consistent with a low-order MA model?
Key points for interpretation are:
- A slow decay across many lags can indicate strong persistence or non-stationarity, including a trend, but it does not by itself establish the cause.
- Repeated peaks at regular intervals can indicate seasonal dependence.
- A sharp cutoff after lag $q$ is the ideal population pattern for an MA($q$) process, in which the current value depends on the current shock and a finite number of past shocks.
For a large white-noise sample, an approximate 95% reference band is:
$$ \pm \frac{1.96}{\sqrt{n}} $$
Spikes outside these bounds are evidence against zero autocorrelation at an individual lag, but several lags are being inspected at once, so isolated crossings should be interpreted cautiously.
For some short-memory processes, Bartlett-type approximations give a larger sampling variance at later lags:
$$ \text{Var}(r_k) \approx \frac{1}{n} \left(1 + 2 \sum_{j=1}^{k-1} \rho_j^2\right) $$
This approximation illustrates why uncertainty can widen when earlier lags are correlated. In model diagnostics, reference bands are best used together with the full residual pattern and formal checks rather than as a mechanical lag-by-lag decision rule.
The following synthetic AR(1) example shows the gradual ACF decay expected from autoregressive persistence.
For comparison, an ARMA(1,1) process usually has no clean cutoff in either function; both ACF and PACF tend to tail off.
Python Example
The following example generates three contrasting series—a random walk with drift, a seasonal signal, and an MA(1) process—and compares their ACFs. The purpose is to connect visible time-domain behavior with the corresponding lag-correlation pattern.
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.graphics.tsaplots import plot_acf
# code for simulating time series with trend and seasonality
np.random.seed(42)
N = 1000
# Example 1: Time Series with a stronger trend (Random Walk)
trend_series = np.cumsum(np.random.normal(1, 1, N)) # Random walk simulating a trend with positive drift
# Example 2: Time Series with clearer seasonality (less noise)
seasonal_series = np.sin(np.linspace(0, 20 * np.pi, N)) # A sine wave to emphasize seasonality
# Moving Average Process (MA(1))
noise = np.random.normal(0, 1, N)
ma_series = np.zeros(N)
ma_series[0] = noise[0]
for i in range(1, N):
ma_series[i] = noise[i] + 0.5 * noise[i - 1] # MA(1): current noise plus weighted previous noise
# Plotting the time series
plt.figure(figsize=(12, 8))
plt.subplot(3, 1, 1)
plt.plot(trend_series, label="Time Series with Trend")
plt.title('Time Series with Trend')
plt.grid(True)
plt.subplot(3, 1, 2)
plt.plot(seasonal_series, label="Time Series with Seasonality")
plt.title('Time Series with Seasonality')
plt.grid(True)
plt.subplot(3, 1, 3)
plt.plot(ma_series, label="Moving Average (MA(1)) Process")
plt.title('Moving Average (MA(1)) Process')
plt.grid(True)
plt.tight_layout()
plt.show()
# Plotting ACF for each time series
plt.figure(figsize=(12, 8))
# ACF for the time series with trend
plt.subplot(3, 1, 1)
plot_acf(trend_series, lags=50, ax=plt.gca())
plt.title('ACF of Time Series with Trend')
# ACF for the time series with seasonality
plt.subplot(3, 1, 2)
plot_acf(seasonal_series, lags=50, ax=plt.gca())
plt.title('ACF of Time Series with Seasonality')
# ACF for the MA(1) process
plt.subplot(3, 1, 3)
plot_acf(ma_series, lags=50, ax=plt.gca())
plt.title('ACF of Moving Average (MA(1)) Process')
plt.tight_layout()
plt.show()The first figure shows the three generated series themselves, which provides the context needed before interpreting their ACFs.
The corresponding ACF plots make those structures visible in lag space.
The random walk has a slowly decaying ACF because it is non-stationary and highly persistent. The seasonal series produces a repeating correlation pattern, while the MA(1) series has the characteristic population cutoff after lag 1, subject to sampling noise in a finite sample.
Partial Autocorrelation Function (PACF)
The Partial Autocorrelation Function (PACF) measures the linear relationship between observations $k$ periods apart after removing the linear effects of the intervening lags. It is especially useful for identifying autoregressive order.
The PACF at lag $k$, often denoted $\phi_{kk}$, is the coefficient on the $k$th lag when $X_t$ is linearly projected on $X_{t-1},\ldots,X_{t-k}$. Equivalently, it is the correlation between $X_t$ and $X_{t-k}$ after the intermediate lags have been accounted for.
Yule-Walker Equations
The Yule-Walker equations connect the autocovariances of a stationary AR($p$) process to its AR coefficients. If
$$ X_t=\phi_1X_{t-1}+\cdots+\phi_pX_{t-p}+\varepsilon_t, $$
then
$$ \gamma_k=\sum_{j=1}^{p}\phi_j\gamma_{k-j}, $$
for positive lags $k$, with $\gamma_{-h}=\gamma_h$. Solving finite Yule-Walker systems of increasing order produces coefficients $\phi_{k1},\ldots,\phi_{kk}$; the final coefficient $\phi_{kk}$ is the PACF at lag $k$.
Recursive Calculation of PACF
The Durbin-Levinson recursion calculates these coefficients efficiently. It starts with $\phi_{11}=\rho_1$. For $k\geq2$,
$$ \phi_{kk} = \frac{\rho_k - \sum_{j=1}^{k-1} \phi_{k-1,j} \rho_{k-j}}{1 - \sum_{j=1}^{k-1} \phi_{k-1,j} \rho_j} $$
and the intermediate coefficients $\phi_{kj}$ for $j