# 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), 2=Periodogram-based estimate , 2=Cross corelation).

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 normal 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 ($\alpha/2$).

## 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\$30,1,2) Autocorrelation of order 2 (-0.008)
=ACFTest(\$B\$2:\$B\$30,1,2,0) p-value of ACF(2) test when ACF(2) = 0 (0.483)