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)).

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.

Alpha is the statistical significance of the test (i.e. alpha). If missing or omitted, an alpha value of 5% is assumed.

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 one-side (i.e. one-tail) test, so the computed p-value should be compared with the whole significance level ($\alpha$).

Example 1:

Files Examples

Related Links

• Wikipedia - Autoregressive conditional heteroskedasticity

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