Computes the p-value of the statistical portmanteau test (i.e. whether any of a group of autocorrelations of a time series are different from zero).

## Syntax

**WNTest**(**X**, **Order**, **M**, **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). - M
- is the maximum number of lags to include in the test. If omitted, the default value of log(T) 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 test hypothesis for white-noise: $$H_{o}: \rho_{1}=\rho_{2}=...=\rho_{m}=0$$ $$H_{1}: \exists \rho_{k}\neq 0$$ $$1\leq k \leq m$$ Where:
- $H_{o}$ is the null hypothesis.
- $H_{1}$ is the alternate hypothesis.
- $\rho_k$ is the population autocorrelation function for lag $k$.
- $m$ is the maximum number of lags included in the white-noise test.

- The Ljung-Box modified $Q^*(m)$ statistic is computed as: $$Q^* =T(T+2)\sum_{j=1}^{m}\frac{\hat\rho_{j}^2}{T-l}$$ Where:
- $m$ is the maximum number of lags included in the test.
- $\hat\rho_j$ is the sample autocorrelation at lag $j$.
- $T$ is the number of non-missing values in the data sample.

- The Ljung-Box modified $Q^*$ statistic has an asymptotic chi-square distribution with $m$ degrees of freedom and can be used to test the null hypothesis that the time series is not serially correlated. $$Q^*(m) \sim \chi_{\nu=m}^2()$$ Where:
- $\chi_{\nu}^2()$ is the Chi-square probability distribution function.
- $\nu$ is the degrees of freedom for the Chi-square distribution.

- The Ljung-Box test is a suitable test for all sample sizes including small ones.
- This is one-side (i.e., one-tail) test, so the computed p-value should be compared with the whole significance level ($\alpha$).

## 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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