ARCHTest - Test of ARCH Effect

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