ACF - Autocorrelation Function

Calculates the sample autocorrelation function (ACF) of a stationary time series.

Syntax

X

is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).

Order

is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).

Order	Description
1	Ascending (the first data point corresponds to the earliest date) (default).
0	Descending (the first data point corresponds to the latest date).

K

is the lag order (e.g., 0=no lag, 1=1st lag, etc.). If missing, the default lag order of zero (i.e., Lag=0) is assumed.

Method

is the calculation method for estimating the autocorrelation function (0 = Sample Autocorrelation (Default), 1 = Periodogram-based estimate, 2 = Cross-correlation).

Value	Method
0	Sample autocorrelation method (default).
1	Periodogram-based estimate. method.
2	Cross-correlation method.

Remarks

1. The time series is homogeneous or equally spaced.
2. The time series may include missing values (e.g., #N/A) at either end.
3. The lag order (k) must be less than the time series size, or an error value (#VALUE!) is returned.
4. The ACF values are bound between -1 and 1, inclusive.
5. Method 1: The sample autocorrelation is computed as:$$\hat{\rho}(h)=\frac{\sum_{k=h}^T{(y_{k}-\bar y)(y_{k-h}-\bar y)}}{\sum_{k=1}^T(y_{k}-\bar y)^2}$$Where:
   
   • $y_{t}$ is the value of the time series at time $t$.
   • $h$ is the lag order.
   • $T$ is the number of non-missing values in the time series data.
   • $\bar y$ is the sample average/mean of the time series.
     $\bar y=\frac{\sum_{i=1}^{N} y_i}{N}$
6. Note that we subtract the full sample mean $\bar y$.
7. Method 2: Periodogram estimate.
   In this method, we compute the spectral density (periodogram is an estimator) of the sample data set and use it to calculate the sample auto-correlation.
8. Although the periodogram-based ACF estimate is usually biased, it generally exhibits a smaller standard error.
9. Method 3: cross-correlation method:$$\rho(h)=\frac{\sum_{i=1}^{N-h}(y_i-\bar y)\times (y_{i+h}-\bar y_h)}{\sqrt{\sum_{i=1}^{N-h}(y_i-\bar y)^2 \times\sum_{j=h}^N (y_j-\bar y_h)^2}}$$Where:
   
   • $y_{t}$ is the value of the time series at time $t$.
   • $h$ is the lag order.
   • $\bar y=\frac{\sum_{i=1}^{N-h} y_i}{N-h}$
   • $\bar y_h=\frac{\sum_{i=h}^N y_i}{N-h}$
   • $T$ is the number of non-missing values in the time series data.
   • $\bar y$ is the sample average/mean of the time series.
10. Special Cases:
    
    • By definition, $\hat{\rho}(0) \equiv 1.0$.

Files Examples

Related Links

• Wikipedia - Autocorrelation function (ACF).

References

• D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906.
• James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
• Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.
• Box, Jenkins, and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848.
• Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568.