# 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 ($\alpha$).