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

- 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 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.

- 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 $k$.

- 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.

- This is a two-sides (i.e., two-tails) test, so the computed p-value should be compared with half of the significance level ($\alpha/2$).

## Files Examples

## Related Links

## 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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