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ACFCI - ACF Confidence Interval Function

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Jacquie Nesbitt
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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

  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 an error value (#VALUE!) is returned.
  4. 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 $k$.
    • $Z\sim N(0,1)$.
    • $P(\left|Z\right|\geq Z_{\alpha/2}) = \alpha$.
  5. 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 $k$.
  6. 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: $$ \sigma_{\rho_k}^2 = \frac{\sum_{j=-\infty}^{\infty}\rho_j^2+\rho_{j+k}\rho_{j-k}-4\rho_j\rho_k\rho_{i-k}+2\rho_j^2\rho_k^2}{T} $$
  7. 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.

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