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)).

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), 2=Periodogram-based estimate , 2=Cross corelation).

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 else 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 compute the sample auto-correlation.
8. Although, the ACF estimate using perodogram-based 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 $

Example 1:

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