# ARCHTest - Test of ARCH Effect

Calculates the p-value of the ARCH effect test (i.e., the white-noise test for the squared time series).

## Syntax

ARCHTest(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 included in the ARCH effect 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 ARCH effect applies the white-noise test on the time series squared: $$y_t=x_t^2$$
4. The test hypothesis for the ARCH effect: $$H_{o}: \rho_{1}=\rho_{2}=...=\rho_{m}=0$$ $$H_{1}: \exists \rho_{k}\neq 0$$ $$1\leq k \leq m$$< Where:
• $H_{o}$ is the null hypothesis.
• $H_{1}$ is the alternate hypothesis.
• $\rho$ is the population autocorrelation function for the squared time series (i.e., $y_t=x_t^2$).
• $m$ is the maximum number of lags included in the ARCH effect test.
5. The Ljung-Box modified $Q^*$ statistic is computed as: $$Q^*(m)=T(T+2)\sum_{j=1}^{m}\frac{\hat\rho_{j}^2}{T-l}$$ Where:
• $m$ is the maximum number of lags included in the ARCH effect test.
• $\hat{\rho_j}$ is the sample autocorrelation at lag $j$ for the squared time series.
• $T$ is the number of non-missing values in the data sample.
6. $Q^*(m)$ has an asymptotic chi-square distribution with $m$ degrees of freedom and can be used to test the null hypothesis that the time series has an ARCH effect. $$Q^*(m) \sim \chi_{\nu=m}^2()$$ Where:
• $\chi_{\nu}^2()$ is the Chi-square probability distribution function.
• $\nu$ is the degrees of freedom for the Chi-square distribution.
7. This is a one-side (i.e., one-tail) test, so the computed p-value should be compared with the whole significance level ($\alpha$).