# WNTest - White-Noise Test

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

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 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.
4. 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.
5. 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.
6. The Ljung-Box test is a suitable test for all sample sizes including small ones.
7. 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)