# 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).

## 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

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 ($\alpha$).

## Examples

Example 1:

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Date Data
January 10, 2008 -2.827
January 11, 2008 -0.947
January 12, 2008 -0.877
January 14, 2008 1.209
January 13, 2008 -1.669
January 15, 2008 0.835
January 16, 2008 -0.266
January 17, 2008 1.361
January 18, 2008 -0.343
January 19, 2008 0.475
January 20, 2008 -1.153
January 21, 2008 1.144
January 22, 2008 -1.070
January 23, 2008 -1.491
January 24, 2008 0.686
January 25, 2008 0.975
January 26, 2008 -1.316
January 27, 2008 0.125
January 28, 2008 0.712
January 29, 2008 -1.530
January 30, 2008 0.918
January 31, 2008 0.365
February 1, 2008 -0.997
February 2, 2008 -0.360
February 3, 2008 1.347
February 4, 2008 -1.339
February 5, 2008 0.481
February 6, 2008 -1.270
February 7, 2008 1.710
February 8, 2008 -0.125
February 9, 2008 -0.940

Formula Description (Result)
=ARCHTest($B$2:$B$30,1) p-value of ARCH effect test (0.5663)