GARCH Toolbox™

archtest

Run Engle's hypothesis test to detect presence of ARCH/GARCH effects

Syntax

[H,pValue,ARCHstat,CriticalValue] = archtest(Residuals,Lags,Alpha)

Description

[H,pValue,ARCHstat,CriticalValue] = archtest(Residuals,Lags,Alpha) tests the null hypothesis that a time series of sample residuals consists of independent identically distributed (i.i.d.) Gaussian disturbances; that is, that no ARCH effects exist.

Given sample residuals obtained from a curve fit (for example, a regression model), archtest tests for the presence of Mth order ARCH effects. It does so by regressing the squared residuals on a constant and the lagged values of the previous M squared residuals.

Under the null hypothesis, the asymptotic test statistic, T(R²), where:

T is the number of squared residuals included in the regression.
R² is the sample multiple correlation coefficient.

is asymptotically chi-square distributed with M degrees of freedom.

When testing for ARCH effects, a GARCH(P,Q) process is locally equivalent to an ARCH(P+Q) process.

Input Arguments

`Residuals`	Time-series column vector of sample residuals obtained from a curve fit, which `archtest` examines for the presence of ARCH effects. The last row contains the most recent observation.
`Lags`	Vector of positive integers indicating the lags of the squared sample residuals included in the ARCH test statistic. If specified, each lag should be less than the length of `Residuals`. If `Lags = []` or is unspecified, the default is `1` lag (that is, first-order ARCH).
`Alpha`	Significance levels of the hypothesis test. `Alpha` can be a scalar applied to all lags in `Lags`, or a vector of significance levels the same length as `Lags`. If `Alpha = []` or is unspecified, the default is `0.05`. For all elements, α of `Alpha`, 0 < α < 1.

Output Arguments

`H`	Boolean decision vector. `0` indicates acceptance of the null hypothesis that no ARCH effects exist; that is, there is homoscedasticity at the corresponding element of `Lags`. `1` indicates rejection of the null hypothesis. The length of `H` is the same as the length of `Lags`.
`pValue`	Vector of p-values (significance levels) at which `archtest` rejects the null hypothesis of no ARCH effects at each lag in `Lags`.
`ARCHstat`	Vector of ARCH test statistics for each lag in `Lags`.
`CriticalValue`	Vector of critical values of the chi-square distribution for comparison with the corresponding element of `ARCHstat`.

Examples

Example 1

Create a time-series column vector of 100 (synthetic) residuals, then test for the first, second, and fourth order ARCH effects at the 10 percent significance level:

randn('state', 0)                % Start from a known state.
residuals     = randn(100, 1);   % 100 Gaussian deviates ~ N(0, 1)
[H, P, Stat, CV] = archtest(residuals, [1 2 4]', 0.10);
[H, P, Stat, CV]

ans =

         0    0.3925    0.7312    2.7055
         0    0.5061    1.3621    4.6052
         0    0.7895    1.7065    7.7794

Example 2

See Example: Analysis and Estimation Using the Default Model for another example.

References

Box, G.E.P., G.M. Jenkins, and G.C. Reinsel, Time Series Analysis: Forecasting and Control, Third edition, Prentice Hall, 1994.

Engle, Robert, "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Vol. 50, 1982, pp. 987-1007.

Gourieroux, C., ARCH Models and Financial Applications, Springer-Verlag, 1997.

Hamilton, J.D., Time Series Analysis, Princeton University Press, 1994.

aicbic

autocorr