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 one-side (i.e. one-tail) test, so the computed p-value should be compared with the whole significance level ($\alpha$).
Examples
Example 1:
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Formula | Description (Result) |
---|---|
=ARCHTest($B$2:$B$30,1) | p-value of ARCH effect test (0.5663) |
Files Examples
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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