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

## Syntax

**ACF**(

**X**,

**Order**,

**K**,

**Method**)

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

Value | Method |
---|---|

0 | Sample autocorrelation method.(default) |

1 | Periodogram-based estimate. method |

2 | Cross-correlation method |

## Remarks

- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g. #N/A) at either end.
- The lag order (k) must be less than the time series size, or else an error value (#VALUE!) is returned.
- The ACF values are bound between -1 and 1, inclusive.
- 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.

$\bar y=\frac{\sum_{i=1}^{N} y_i}{N}$

- Note that we subtract the full sample mean $\bar y$
- 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. - Although, the ACF estimate using perodogram-based is usually biased, it generally exhibits a smaller standard error.
- 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.

- Special Cases:

- By definition, $\hat{\rho}(0) \equiv 1.0 $

## Examples

**Example 1: **

A | B | |
---|---|---|

1 | Date | Data |

2 | 1/1/2008 | #N/A |

3 | 1/2/2008 | -1.28 |

4 | 1/3/2008 | 0.24 |

5 | 1/4/2008 | 1.28 |

6 | 1/5/2008 | 1.20 |

7 | 1/6/2008 | 1.73 |

8 | 1/7/2008 | -2.18 |

9 | 1/8/2008 | -0.23 |

10 | 1/9/2008 | 1.10 |

11 | 1/10/2008 | -1.09 |

12 | 1/11/2008 | -0.69 |

13 | 1/12/2008 | -1.69 |

14 | 1/13/2008 | -1.85 |

15 | 1/14/2008 | -0.98 |

16 | 1/15/2008 | -0.77 |

17 | 1/16/2008 | -0.30 |

18 | 1/17/2008 | -1.28 |

19 | 1/18/2008 | 0.24 |

20 | 1/19/2008 | 1.28 |

21 | 1/20/2008 | 1.20 |

22 | 1/21/2008 | 1.73 |

23 | 1/22/2008 | -2.18 |

24 | 1/23/2008 | -0.23 |

25 | 1/24/2008 | 1.10 |

26 | 1/25/2008 | -1.09 |

27 | 1/26/2008 | -0.69 |

28 | 1/27/2008 | -1.69 |

29 | 1/28/2008 | -1.85 |

30 | 1/29/2008 | -0.98 |

Formula | Description (Result) | |
---|---|---|

=ACF(\$B\$2:\$B\$30,1,1) | Autocorrelation of order 1 (0.235) | |

=ACF(\$B\$2:\$B\$30,1,2) | Autocorrelation of order 2 (-0.008) | |

=ACF(\$B\$2:\$B\$30,1,3) | Autocorrelation of order 3 (0.054) |

## Files Examples

## 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

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