ACF and PACF — Identifying ARIMA Orders

Statistics

Reading the Signature of Time Series Dependence

ACF and PACF plots reveal the autocorrelation structure needed to specify ARIMA models. The ACF shows total correlation at each lag, while the PACF isolates direct correlation after removing intermediate effects.

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The cutoff patterns in these plots tell you exactly which lags matter.

The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) are essential tools for identifying the orders of ARIMA models.

DfAutocorrelation (ACF)

The correlation between a time series and its lagged version. The ACF at lag $k$ measures the linear dependence between $Y_t$ and $Y_{t-k}$ .

Autocorrelation Function

\rho(k) = \frac{\gamma(k)}{\gamma(0)}

Here,

$\rho(k)$ =Autocorrelation at lag k
$\gamma(k)$ =Autocovariance at lag k: $\text{Cov}(Y_t, Y_{t-k})$
$\gamma(0)$ =Variance of the series

Sample ACF

The sample ACF is estimated as: $\hat{\rho}(k) = \frac{\sum_{t=k+1}^{T}(Y_t - \bar{Y})(Y_{t-k} - \bar{Y})}{\sum_{t=1}^{T}(Y_t - \bar{Y})^2}$ . For a white noise series, $\hat{\rho}(k) \approx 0 \pm \frac{2}{\sqrt{T}}$ .

Partial Autocorrelation (PACF)

The PACF measures the correlation between $Y_t$ and $Y_{t-k}$ after removing the linear effect of intermediate lags.

DfPartial Autocorrelation

The PACF at lag $k$ is the coefficient $\phi_{kk}$ in the regression:

Y_t = \phi_{1}Y_{t-1} + \phi_{2}Y_{t-2} + \cdots + \phi_{k}Y_{t-k} + \varepsilon_t

Identification Rules

The ACF and PACF patterns help determine the orders (p, d, q) of an ARIMA(p,d,q) model:

| Model | ACF Pattern | PACF Pattern |

|-------|------------|--------------|

| AR(p) | Tails off (exponential/sinusoidal decay) | Cuts off after lag p |

| MA(q) | Cuts off after lag q | Tails off |

| ARMA(p,q) | Tails off | Tails off |

Quick Guide

AR: Look at PACF for the order (where it cuts off)
MA: Look at ACF for the order (where it cuts off)
ARMA: Both tail off — use information criteria (AIC/BIC)

Confidence Bands

For a white noise series, approximately 95% of sample ACF values should fall within $\pm \frac{2}{\sqrt{T}}$ .

Significance Threshold

\text{Bounds} = \pm \frac{2}{\sqrt{T}}

Here,

$T$ =Length of the time series

Values outside these bounds are considered statistically significant at the 5% level.

Worked Examples

AR(1) Process

AR(1): $Y_t = 0.7Y_{t-1} + \varepsilon_t$

ACF: Exponential decay: $\rho(k) = 0.7^k$
PACF: Spike at lag 1 ( $\phi_{11} \approx 0.7$ ), then cuts off

This pattern clearly identifies an AR(1) model.

MA(1) Process

MA(1): $Y_t = \varepsilon_t + 0.5\varepsilon_{t-1}$

ACF: Spike at lag 1 ( $\rho(1) = \frac{0.5}{1+0.5^2} \approx 0.4$ ), then cuts off
PACF: Exponential decay

This pattern clearly identifies an MA(1) model.

Python Implementation


import numpy as np

import matplotlib.pyplot as plt

from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

from statsmodels.tsa.arima_process import ArmaProcess



np.random.seed(42)



# AR(1) process

ar_params = np.array([1, -0.7])

ma_params = np.array([1])

ar1 = ArmaProcess(ar_params, ma_params)

y_ar1 = ar1.generate_sample(n=200)



# MA(1) process

ar_params_ma = np.array([1])

ma_params = np.array([1, 0.5])

ma1 = ArmaProcess(ar_params_ma, ma_params)

y_ma1 = ma1.generate_sample(n=200)



fig, axes = plt.subplots(2, 2, figsize=(12, 8))



plot_acf(y_ar1, ax=axes[0, 0], lags=20)

axes[0, 0].set_title('ACF - AR(1)')



plot_pacf(y_ar1, ax=axes[0, 1], lags=20)

axes[0, 1].set_title('PACF - AR(1)')



plot_acf(y_ma1, ax=axes[1, 0], lags=20)

axes[1, 0].set_title('ACF - MA(1)')



plot_pacf(y_ma1, ax=axes[1, 1], lags=20)

axes[1, 1].set_title('PACF - MA(1)')



plt.tight_layout()

plt.show()

Partial Autocorrelation via Durbin-Levinson

Durbin-Levinson Recursion

\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)}

Here,

$\phi_{kk}$ =PACF at lag k
$\rho(k)$ =ACF at lag k
$\phi_{k-1,j}$ =Previous order coefficients

Key Takeaways

Summary: ACF and PACF

ACF measures total correlation between $Y_t$ and $Y_{t-k}$
PACF measures direct correlation after removing intermediate effects
AR(p): PACF cuts off at lag p; ACF tails off
MA(q): ACF cuts off at lag q; PACF tails off
ARMA(p,q): Both ACF and PACF tail off
Use significance bounds $\pm 2/\sqrt{T}$ to identify significant lags
Combine ACF/PACF analysis with AIC/BIC for model selection

ACF and PACF — Identifying ARIMA Orders

ACF and PACF — Identifying ARIMA Orders

Reading the Signature of Time Series Dependence

DfAutocorrelation (ACF)

Autocorrelation Function

Partial Autocorrelation (PACF)

DfPartial Autocorrelation

Identification Rules

Confidence Bands

Significance Threshold

Worked Examples

AR(1) Process

AR(1): $Y_t = 0.7Y_{t-1} + \varepsilon_t$

MA(1) Process

MA(1): $Y_t = \varepsilon_t + 0.5\varepsilon_{t-1}$

Python Implementation

Partial Autocorrelation via Durbin-Levinson

Durbin-Levinson Recursion

Key Takeaways

Summary: ACF and PACF

Related Topics

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