# ACFTest - Autocorrelation Test

Calculates the p-value of the statistical test for the population autocorrelation function.

## Syntax

ACFTest(X, Order, k, Method, $\rho$, Return_type, $\alpha$)

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 one (i.e., Lag = 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.
$\rho$
is the assumed autocorrelation function value. If missing, the default of zero is assumed.
Return_type
is a switch to select the return output (1 = P-Value (default), 2 = Test Stats, 3 = Critical Value.
Method Description
1 P-Value.
2 Test Statistics (e.g., Z-score).
3 Critical Value.
$\alpha$
is the statistical significance of the test (i.e., alpha). If missing or omitted, an alpha value of 5% is assumed.

## 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 test hypothesis for the population autocorrelation: $$H_{o}: \rho_{k}=\rho_o$$ $$H_{1}: \rho_{k}\neq a$$ Where:
• $H_{o}$ is the null hypothesis.
• $H_{1}$ is the alternate hypothesis.
• $\hat \rho_o$ is the assumed population autocorrelation function for lag $k$.
• $k$ is the lag order.
5. Assuming a normally distributed population, the sample autocorrelation 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 deviation of the sample autocorrelation function for lag $k$.
6. The variance of the sample autocorrelation is computed as: $$\sigma_{\rho_k}^2 = \frac{1+\sum_{j=1}^{k-1}\hat\rho_j^2}{T}$$ 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.
7. This is a two-sides (i.e., two-tails) test, so the computed p-value should be compared with half of the significance level ($\alpha/2$).