Autocorrelation Analysis (ACF, PACF, Ljung-Box)

Autocorrelation analysis estimates how strongly a series correlates with its own past values at each lag (ACF, PACF) and tests jointly for white noise with the Ljung-Box Q statistic.

자기상관 분석은 시계열이 자신의 과거 값과 시차별로 얼마나 상관되는지(ACF·PACF)를 추정하고 Ljung-Box Q 통계량으로 백색잡음 여부를 함께 검정하는 방법이다.

Is there autocorrelation in my monthly sales series, and which lags matter?

월별 매출 시계열에 자기상관이 있는지, 어느 시차가 중요한지 알고 싶어요.

Autocorrelation analysis plots the correlation of the series with itself at each lag and the partial correlation after removing shorter lags. The Ljung-Box test then checks whether the first k autocorrelations are jointly zero, which is the white-noise hypothesis.

When to use Autocorrelation Analysis (ACF, PACF, Ljung-Box)

• You need to identify AR or MA orders before fitting an ARIMA model.
• You want to check whether model residuals are white noise.
• You want to see seasonal dependence at the seasonal lag and its multiples.

When to use something else

• The series has a strong trend or unit root; difference it first or use a unit root test such as Augmented Dickey-Fuller.
• You want the lead-lag relation between two series; use cross-correlation analysis instead.

Assumptions and requirements

• Observations are equally spaced in time.
• The series is weakly stationary (constant mean and variance).
• The ±z/√n bounds assume Gaussian white noise under the null.

What the analysis reports

• ACF and PACF by lag with approximate significance bounds
• Bartlett standard errors for the ACF
• Ljung-Box Q and p-value at lag 10 and at twice the seasonal period
• Count of lags outside the bounds

How to report it

The lag-1 autocorrelation was [value] and the Ljung-Box test over [value] lags gave Q([value]) = [value], p = [value].

Related methods

• Augmented Dickey-Fuller Test
• ARIMA Model
• Cross Correlation Analysis (CCF)

Run Autocorrelation Analysis (ACF, PACF, Ljung-Box) on Quorant