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).


ARCHTest(X, Order, M, Return_type, $\alpha$)

is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
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).
is the maximum number of lags included in the ARCH effect test. If omitted, the default value of log(T) is assumed.
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.
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 a one-side (i.e., one-tail) test, so the computed p-value should be compared with the whole significance level ($\alpha$).

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