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.

Partial Autocorrelation (PACF)

The PACF measures the correlation between and after removing the linear effect of intermediate lags.

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 |

Confidence Bands

For a white noise series, approximately 95% of sample ACF values should fall within .

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

Worked Examples

AR(1) Process

MA(1) Process

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()

ACF and PACF — Identifying ARIMA Orders

ACF and PACF — Identifying ARIMA Orders

Reading the Signature of Time Series Dependence

Partial Autocorrelation (PACF)

Identification Rules

Confidence Bands

Worked Examples

AR(1) Process

MA(1) Process

Python Implementation

Partial Autocorrelation via Durbin-Levinson

Key Takeaways

Related Topics

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