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

## 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, it returns the upper confidence interval limit. Otherwise, it 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 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.

## 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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