ARCHTest (Pro.)

	Attachment	Size
	ARCHTest.xlsx

Calculates the p-value of the ARCHⁱ effect test (i.e. the white-noise test for the squared time series).

Syntax

ARCHTest(X, Order, M)

X
is the univariate time series data (a one dimensional array of cells (e.g. rows or columns)).

Order
is the time order of the data series (i.e. whether the first data point corresponds to the earliest or latest 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.

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 hypothesisⁱ.
- $H_{1}$ 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 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.
has an asymptotic chi-square distribution with 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:

Let's consider the sample time series in the spreadsheet attachment on this page.

Sample time series plot for the ARCH effect test

We computed the P-Value for different values of the maximum lags:

P-Value for ARCH effect test in excel Plot of the ARCH effect test P-Values for different lags in excel

Please note the plot of the ARCH test P-value reaches it lowest level (floor) around $\ln(N)$ , and looses its power for larger numbers due to the size of the sample.

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

External Links

Wikipedia - Autoregressive conditional heteroskedasticity