Calculates the confidence interval limits (upper/lower) for the autocorrelation function.

## Syntax

**ACFCI**(

**X**,

**Order**,

**K**,

**Method**,

**alpha**,

**upper**)

**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. k=0 (no lag), k=1 (1st lag), etc.). If missing, the default of k=1 is assumed.

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

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

0 | Sample autocorrelation method.(default) |

1 | Periodogram-based estimate. |

2 | Cross-correlation method |

**alpha** is the statistical significance level. If missing, the default of 5% is assumed.

**upper** If true, returns the upper confidence interval limit. Otherwise, returns the lower limit.

Value | Description |
---|---|

0 | return lower limit |

1 | return upper limit |

## 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 ACFCI function calculates the confidence limits as: $$ \hat\rho_k - Z_{\alpha/2}\times \sigma_{\rho_k} \leq \rho_k \leq \hat\rho_k+ Z_{\alpha/2}\times \sigma_{\rho_k} $$ Where:

- $\rho_k$ is the population autocorrelation function.
- $\sigma_{\rho_k}$ is the standard error of the sample autocorrelation.
- $\hat{\rho_{k}}$ is the sample autocorrelation function for lag k.
- $Z\sim N(0,1)$
- $P(\left|Z\right|\geq Z_{\alpha/2}) = \alpha$

- For the case in which the underlying population distribution is normal, the sample autocorrelation also has a normal distribution:

$ \hat \rho_k \sim N(\rho_k,\sigma_{\rho_k}^2)$

Where:

- $\hat \rho_k $ is the sample autocorrelation for lag k.
- $\rho_k $ is the population autocorrelation for lag k.
- $\sigma_{\rho_k}$ is the standard error of the sample autocorrelation for lag k.

- Bartlett proved that the variance of the sample autocorrelation of a stationary normal stochastic process (i.e. independent, identically normal distributed errors) can be formulated as: $$ \sigma_{\rho_k}^2 = \frac{\sum_{j=-\infty}^{\infty}\rho_j^2+\rho_{j+k}\rho_{j-k}-4\rho_j\rho_k\rho_{i-k}+2\rho_j^2\rho_k^2}{T} $$
- Furthermore, the variance of the sample autocorrelation is reformulated: $$ \sigma_{\rho_k}^2 = \frac{1+\sum_{j=1}^{k-1}\hat\rho_j^2}{T} $$ Where:

- $\sigma_{\rho_k}$ is the standard error of the sample autocorrelation for lag k.
- $T$ is the sample data size.
- $\hat\rho_j$ is the sample autocorrelation function for lag j.
- $k$ is the lag order.

## 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) | |

=ACFCI(\$B\$2:\$B\$30,1,1,5%,1) | Upper confidence interval for ACF of order 1 (0.37) | |

=ACFCI(\$B\$2:\$B\$30,1,1,5%,0) | lower confidence interval for ACF of order 2 (-0.37) |

## 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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