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:
|
|
| 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
Related Links
- P.A.P Moran; Testing of Significance of Correlation between Time Series, page 397
; Biometrica, 1947 - Wikipedia - Confidence Interval
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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