Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF)

If today's value is high, does that help predict tomorrow's value? What about the value two periods later? The autocorrelation function (ACF) measures the overall linear association at each separation. The partial autocorrelation function (PACF) asks what remains at a separation after the intermediate lags have been accounted for linearly.

These questions differ because association can travel through a chain. In an AR(1), $X_{t-2}$ helps predict $X_{t-1}$, and $X_{t-1}$ helps predict $X_t$. The lag-2 ACF can therefore be nonzero even when $X_{t-2}$ adds no further linear information once $X_{t-1}$ is included.

This distinction is useful for proposing dynamic models, but it must be applied to the right series. A trend, seasonal mean, or break can generate strong sample correlation without a stationary AR mechanism. First examine the time plot and the mean structure; then use ACF and PACF to study the remaining lag dependence. Fit and diagnose candidate models rather than selecting an order from one conspicuous spike.

Worked calculation: AR(1) dependence

For a stationary AR(1),

$$ X_t=0.7X_{t-1}+\varepsilon_t, $$

the theoretical autocorrelation is

$$ \rho(h)=0.7^{|h|}. $$

The lag-2 association is $0.7^2=0.49$ because one period of persistence is followed by another. At lag 3 it is $0.7^3=0.343$. None of those population ACF values is exactly zero. The PACF, however, is 0.7 at lag 1 and zero afterward: the first lag already accounts for the linear chain. A sample correlogram estimates these patterns with noise and will not reproduce their exact values.

Two summaries of lagged dependence

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)

For a weakly stationary series with positive variance, lag-$k$ autocorrelation is the covariance of $X_t$ and $X_{t-k}$ divided by their common variance. The normalization makes it independent of the measurement units. A value near 1 indicates a strong same-direction linear relationship across that lag; a value near $-1$ indicates strong opposite-direction association. The definition is:

$$ \rho_k = \frac{\gamma_k}{\gamma_0} $$

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:

$$ r_k = \frac{\sum_{t=1}^{N-k} (x_t - \bar{x})(x_{t+k} - \bar{x})}{\sum_{t=1}^{N} (x_t - \bar{x})^2} $$

Here:

• $\bar{x}$ is the sample mean of the series.
• $N$ is the number of observations.
• $x_t$ is the observed value at time $t$.

The displayed sample formula uses one overall sample mean and the same lag-0 denominator for all lags. Its numerator uses fewer pairs as $k$ grows. It can be calculated for a nonconstant nonstationary record, but interpreting it as an estimate of one time-invariant population lag function requires stationarity. An exactly constant record has zero denominator, so its ordinary ACF is undefined.

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 $n$.

Positive cross-time covariance makes the sample mean more variable than an independent-sample calculation would suggest. The exact variance formula includes each lag covariance, weighted by how many observation pairs have that separation. In practice, distant lag estimates are noisy, so a bandwidth $m$ truncates the sum and the weights taper it. The following estimator approximates the variance of the mean rather than the marginal variance of an observation:

$$ \hat v_n =\frac{1}{n} \left[ \hat\gamma(0) +2\sum_{h=1}^{m} \left(1-\frac{h}{m+1}\right)\hat\gamma(h) \right], $$

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 covariance matrix $W$ describes how estimates at different lags vary together. In a dependent process, these sampling errors can be correlated, so several correlogram spikes should be interpreted as one estimated pattern. The approximation also requires appropriate moment and dependence conditions; weak stationarity alone does not guarantee an asymptotic normal distribution.

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:

1. 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.
2. Repeated peaks at regular intervals can indicate seasonal dependence.
3. 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:

$$ \pm \frac{1.96}{\sqrt{n}} $$

The rough band concerns one autocorrelation under an appropriate white-noise approximation. Inspecting many lags creates many opportunities for a chance crossing. The errors at different lags can also be related. A few isolated spikes are therefore less persuasive than a coherent pattern, such as repeated seasonal peaks or sustained slow decay, especially in a short record.

For some short-memory processes, Bartlett-type approximations give a larger sampling variance at later lags:

$$ \text{Var}(r_k) \approx \frac{1}{n} \left(1 + 2 \sum_{j=1}^{k-1} \rho_j^2\right) $$

Earlier nonzero correlations affect the sampling variation of later estimates. That is why a single white-noise band cannot be treated as an exact confidence region for every dependent process. Distinguish a rough screening band on a plot from a model-specific uncertainty calculation or a joint residual test. Each uses different assumptions and answers a different question.

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: comparing sample ACFs

The simulation contrasts three mechanisms. A random walk retains every shock in its level, producing broad persistence. A seasonal signal repeats at a known period, producing correlation at related separations. An MA(1) shares shocks only between adjacent observations, producing a finite population covariance range. Plotting the generated paths before their ACFs helps connect the lag patterns to the mechanism rather than treating the plots as memorized shapes.

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)

To understand the lag-2 PACF, predict $X_t$ from $X_{t-1}$ and separately predict $X_{t-2}$ from $X_{t-1}$. Correlate the two remaining prediction errors. This removes the linear association explained by the intervening observation. At higher lags, project out all the intermediate lags in the same way. The result measures additional linear association, not necessarily conditional independence or direct causation.

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

Each step of the Durbin-Levinson recursion adds one older lag to the prediction equation. The numerator below measures the correlation left after the previous coefficients have explained what they can. The denominator rescales by the remaining prediction-error variance. The newly added coefficient $\phi_{kk}$ is the PACF at lag $k$; the earlier coefficients are then revised because the predictor set has changed. Start with $\phi_{11}=\rho_1$. For $k\ge2$:

$$ \phi_{kk} = \frac{\rho_k - \sum_{j=1}^{k-1} \phi_{k-1,j} \rho_{k-j}}{1 - \sum_{j=1}^{k-1} \phi_{k-1,j} \rho_j} $$

and the intermediate coefficients $\phi_{kj}$ for $j<k$ are updated using

$$ \phi_{kj} = \phi_{k-1,j} - \phi_{kk} \phi_{k-1,k-j} $$

Plotting the PACF

A PACF plot displays the estimated partial autocorrelation at each lag. Because it removes the linear contribution of intermediate lags, it is particularly useful for proposing the order of an autoregressive component.

For an ideal AR($p$) process, the population PACF is zero beyond lag $p$. In a finite sample, the estimated values do not become exactly zero, so the practical pattern is a set of notable early lags followed by values consistent with sampling variation. By contrast, MA and ARMA processes generally have PACFs that tail off rather than cut off cleanly.

As with the ACF, these patterns suggest candidate models; they do not prove a model order on their own.

Python example: comparing sample PACFs

This example simulates AR, MA, and ARMA processes and compares their PACFs. Reading the time-series plots first makes it easier to connect each process with its lag-domain pattern.

import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.arima_process import ArmaProcess
from statsmodels.graphics.tsaplots import plot_pacf

# Example 1: Simulating an AR(2) process
np.random.seed(42)
ar2 = np.array([1, -0.75, 0.25])  # AR(2) coefficients (X_t = 0.75*X_t-1 - 0.25*X_t-2 + noise)
ma0 = np.array([1])  # No MA component
AR_process = ArmaProcess(ar2, ma0)
ar_series = AR_process.generate_sample(nsample=1000)

# Example 2: Simulating a Moving Average (MA) process
ma1 = np.array([1, 0.5])  # MA(1) coefficients
MA_process = ArmaProcess([1], ma1)
ma_series = MA_process.generate_sample(nsample=1000)

# Example 3: Simulating an ARMA(1,1) process
ar1 = np.array([1, -0.5])  # AR(1): X_t = 0.5 X_{t-1} + noise
ma1 = np.array([1, 0.5])  # MA(1): current noise + 0.5 previous noise
ARMA_process = ArmaProcess(ar1, ma1)
arma_series = ARMA_process.generate_sample(nsample=1000)

# Plotting the time series
plt.figure(figsize=(12, 8))
plt.subplot(3, 1, 1)
plt.plot(ar_series, label="AR(2) Process")
plt.title('AR(2) Process')
plt.grid(True)

plt.subplot(3, 1, 2)
plt.plot(ma_series, label="MA(1) Process")
plt.title('MA(1) Process')
plt.grid(True)

plt.subplot(3, 1, 3)
plt.plot(arma_series, label="ARMA(1,1) Process")
plt.title('ARMA(1,1) Process')
plt.grid(True)

plt.tight_layout()
plt.show()

# Plotting PACF for each time series
plt.figure(figsize=(12, 8))

# PACF for the AR(2) process
plt.subplot(3, 1, 1)
plot_pacf(ar_series, lags=30, ax=plt.gca())
plt.title('PACF of AR(2) Process')

# PACF for the MA(1) process
plt.subplot(3, 1, 2)
plot_pacf(ma_series, lags=30, ax=plt.gca())
plt.title('PACF of MA(1) Process')

# PACF for the ARMA(1,1) process
plt.subplot(3, 1, 3)
plot_pacf(arma_series, lags=30, ax=plt.gca())
plt.title('PACF of ARMA(1,1) Process')

plt.tight_layout()
plt.show()

The generated series provide the time-domain context for the PACF comparison.

The PACF plots then show how the direct lag relationships differ across the three models.

For the AR(2) process, the population PACF cuts off after lag 2. The MA(1) and ARMA(1,1) processes instead have PACFs that tail off, although the exact finite-sample shapes depend on the parameters and simulated data.

Comparing ACF and PACF

The ACF captures both direct and indirect linear dependence across lags, whereas the PACF removes the linear contribution of intermediate lags. This distinction leads to the familiar identification patterns:

• for an AR($p$) process, the ACF tails off while the PACF cuts off after lag $p$;
• for an MA($q$) process, the ACF cuts off after lag $q$ while the PACF tails off;
• for an ARMA process, both functions usually tail off.

The following figure summarizes these identification patterns visually.

Example: ACF and PACF for AR(1) Process

Consider the autoregressive process of order 1, denoted AR(1):

$$ X_t = \phi X_{t-1} + \epsilon_t $$

where $\epsilon_t$ is white noise.

ACF for AR(1)

The autocorrelation function for an AR(1) process is:

$$ \rho_k = \phi^k $$

For $|\phi|<1$, the autocorrelation decays geometrically with the lag, producing a gradual tail-off in the ACF plot. If $\phi<0$, the signs alternate while the magnitude still decays.

PACF for AR(1)

In AR(1), once the previous value has been used, every older observation contributes only through information already carried by that previous value. Its extra linear contribution is therefore zero. The ordinary lag-2 correlation still equals $\phi^2$, so zero PACF does not mean zero pairwise correlation. This is the mechanism behind the characteristic first-lag PACF cutoff.

Visualization of ACF and PACF

The following mock series illustrates short-term dependence consistent with an AR-type process. The point of the figure is the contrast between a gradually decaying ACF and a PACF that is concentrated at the first lag.

Left Plot: Autocorrelation Function (ACF)

• The ACF equals 1 at lag 0 because the series is perfectly correlated with itself.
• The ACF then decays gradually while remaining positive over several lags, which is consistent with autoregressive persistence.
• The shaded region is a reference band for sampling uncertainty. Spikes outside it indicate evidence of nonzero autocorrelation at those lags, but the overall decay pattern is more informative than any single crossing.

Right Plot: Partial Autocorrelation Function (PACF)

• The PACF shows a sharp drop after lag 1, which is the characteristic population pattern of an AR(1) process.
• Taken together with the tailing ACF, the dominant first-lag PACF suggests AR(1) as a candidate model to estimate and validate.

Connecting ACF and PACF to AR and MA models

Autoregressive (AR) and moving-average (MA) models produce different population ACF and PACF patterns. Use those patterns to propose plausible orders, then estimate and check the candidates. The model equation explains why a cutoff occurs; a noisy sample plot provides evidence about that equation rather than an automatic identification rule.

Autoregressive (AR) model

An AR($p$) model expresses the current value as a linear function of its own previous values plus a new innovation:

$$ y_t=c+\phi_1y_{t-1}+\phi_2y_{t-2}+\dots+\phi_py_{t-p}+\varepsilon_t. $$

The order $p$ is the number of lagged values included. In the ideal stationary population, the PACF is zero after lag $p$, while the ACF tails off. This makes the PACF especially useful for proposing an AR order, although estimation and diagnostics are still required.

Moving Average (MA) Model

An MA($q$) model expresses the current value in terms of the current innovation and a finite number of previous innovations:

$$ y_t=\mu+\varepsilon_t+\theta_1\varepsilon_{t-1}+\theta_2\varepsilon_{t-2}+\dots+\theta_q\varepsilon_{t-q}. $$

The order $q$ is the number of past shocks retained in the model. In the ideal population, the ACF is zero after lag $q$, while the PACF tails off. The following figure illustrates that finite shock duration in the MA representation.

Visual comparison across common patterns

The following comparison places several common patterns side by side: AR(1), AR(2), MA(1), MA(2), a deterministic linear trend, a constant series, a noisy sine wave, and white noise.

Use the chart as a visual reference rather than a lookup rule. In particular, the ordinary ACF is not defined for an exactly constant series because its variance is zero, and non-stationary trend or seasonal examples should not be interpreted with stationary ARMA cutoff rules.

From a sample calculation to a defensible model proposal

A proposed lag interpretation should survive reasonable treatment of the mean structure. If removing an evident deterministic trend eliminates a broad ACF tail, the original tail was not good evidence for a high-order stationary AR model. If peaks remain at seasonal separations, investigate seasonal dependence or missing calendar variables. The numerical calculations below make the sample convention explicit before those patterns are interpreted.

Sample ACF calculation

For observations $(1,2,4,3)$, the sample mean is $\bar x=2.5$, giving centered values

$$ z=(-1.5,-0.5,1.5,0.5). $$

Using $n$ as the denominator for this autocovariance convention,

$$ \hat\gamma(0)=\frac{2.25+0.25+2.25+0.25}{4}=1.25, $$

and

$$ \hat\gamma(1)=\frac{(-1.5)(-0.5)+(-0.5)(1.5)+(1.5)(0.5)}4 =0.1875. $$

Thus

$$ \hat\rho(1)=\frac{0.1875}{1.25}=0.15. $$

The ratio is $0.1875/1.25=0.15$. The numerator retains the signed products of adjacent centered values; the denominator gives their common variance scale. A value of 0.15 is a small positive sample association in this example, but four observations are far too few to identify a dynamic model from that number alone. Changing the ordering changes the numerator even though the values and denominator remain the same.

AR(1) pattern

For

$$ X_t=0.7X_{t-1}+\varepsilon_t, $$

the theoretical ACF is

$$ \rho(h)=0.7^{|h|}. $$

The first values are $1$, $0.7$, $0.49$, $0.343$, and so on. The PACF is $0.7$ at lag 1 and zero at later lags in the population. The following figure shows how this persistence appears as a decaying lag pattern.

A finite sample will show nonzero later values, so use uncertainty bands and model checks rather than expecting an exact theoretical cutoff.

PACF as a conditional relationship

The lag-2 PACF measures the relationship between $X_t$ and $X_{t-2}$ after linearly removing the contribution of the intermediate lag $X_{t-1}$. It is not an ordinary pairwise correlation.

For an AR(2),

$$ X_t=\phi_1X_{t-1}+\phi_2X_{t-2}+\varepsilon_t, $$

the population PACF is zero after lag 2. The worked AR(2) figure below shows the contrast between a tailing ACF and a PACF whose direct dependence is concentrated in the first two lags.

For an MA process, the pattern is reversed: the ACF has a finite cutoff while the PACF generally tails off.

Approximate significance bands

Under a white-noise approximation, an individual sample autocorrelation is often compared with

$$ \pm\frac{1.96}{\sqrt n}. $$

At $n=100$, the rough threshold is $\pm0.196$. Under the reference approximation, about 5% of individual lag estimates cross a 95% band by chance. These are not simultaneous bounds for the entire correlogram. Fitting parameters, inspecting large lags, or using a strongly dependent process changes the relevant uncertainty. Use the band to screen a pattern, then test or fit the specific dynamic hypothesis being proposed.

Identification is iterative

A practical identification loop is:

1. remove or model trend and seasonality;
2. calculate ACF/PACF of the transformed series;
3. propose a small ARMA candidate set;
4. estimate candidates;
5. inspect residual ACF/PACF and squared residuals;
6. backtest the candidates.

The cutoff rules are therefore a starting point rather than a proof. Near-unit roots, small samples, outliers, missing values, and structural changes can all mimic familiar patterns.

Seasonal autocorrelation

For a seasonal period $s$, inspect lags $s,2s,3s$ and the neighboring lags. A spike at lag 12 in monthly data can arise from several different mechanisms:

• deterministic seasonal means;
• seasonal AR dependence;
• seasonal MA effects;
• calendar aggregation;
• an unmodeled annual predictor.

A repeated seasonal mean can create lag-12 correlation even when the remaining errors are uncorrelated. A seasonal AR term instead models dependence after the mean is specified. A calendar predictor can explain a known recurring mechanism, while seasonal differencing changes the target to year-over-year increments. These operations are not interchangeable responses to the same spike; choose the one that matches the source and forecasting use of the pattern.

Identification table

These patterns describe ideal population behavior. Use them to propose a small set of candidates, then confirm the choice through estimation, residual diagnostics, and forecast validation.

A runnable companion is the ACF and PACF notebook.

Formula reference