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

- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g., #N/A) at either end.
- The ARCH effect applies the white-noise test on the time series squared: $$y_t=x_t^2$$
- 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.

- 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.

- $Q^*(m)$ 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.

- This is a 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

- Hamilton, J .D.; Time Series Analysis, Princeton University Press (1994), ISBN 0-691-04289-6.
- Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740.

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