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 ($\alpha$).

## Examples

**Example 1: **

A | B | |
---|---|---|

1 | Date | Data |

2 | January 10, 2008 | -0.30 |

3 | January 11, 2008 | -1.28 |

4 | January 12, 2008 | 0.24 |

5 | January 13, 2008 | 1.28 |

6 | January 14, 2008 | 1.20 |

7 | January 15, 2008 | 1.73 |

8 | January 16, 2008 | -2.18 |

9 | January 17, 2008 | -0.23 |

10 | January 18, 2008 | 1.10 |

11 | January 19, 2008 | -1.09 |

12 | January 20, 2008 | -0.69 |

13 | January 21, 2008 | -1.69 |

14 | January 22, 2008 | -1.85 |

15 | January 23, 2008 | -0.98 |

16 | January 24, 2008 | -0.77 |

17 | January 25, 2008 | -0.30 |

18 | January 26, 2008 | -1.28 |

19 | January 27, 2008 | 0.24 |

20 | January 28, 2008 | 1.28 |

21 | January 29, 2008 | 1.20 |

22 | January 30, 2008 | 1.73 |

23 | January 31, 2008 | -2.18 |

24 | February 1, 2008 | -0.23 |

25 | February 2, 2008 | 1.10 |

26 | February 3, 2008 | -1.09 |

27 | February 4, 2008 | -0.69 |

28 | February 5, 2008 | -1.69 |

29 | February 6, 2008 | -1.85 |

30 | February 7, 2008 | -0.98 |

Formula | Description (Result) | |
---|---|---|

=WNTest(\$B\$2:\$B\$30,1) | p-value of white-noise test (0.5995) |

## Files Examples

## 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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