GARCH Toolbox

archtest

Engle's hypothesis test for the presence of ARCH/GARCH effects

Syntax

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

Arguments

Residuals	Time series vector of sample residuals obtained from a curve fit, which archtest examines for the presence of ARCH effects. The last element contains the most recent observation.
Lags	(optional) Vector of positive integers indicating the lags of the squared sample residuals included in the ARCH test statistic. If specified, each lag should be significantly less than the length of Residuals. If Lags = [] or is not specified, the default is 1 lag (i.e., first order ARCH).
Alpha	(optional) Significance level(s) 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 not specified, the default is 0.05. For all elements, , of Alpha, 0 < < 1.

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, i.e., no ARCH effects exist.

Given sample residuals obtained from a curve fit (e.g., a regression model), archtest tests for the presence of Mth order ARCH effects 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 and 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.

H	Boolean decision vector. 0 indicates acceptance of the null hypothesis that no ARCH effects exist, i.e., there is homoskedasticity 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.

Example

Create a vector of 100 (synthetic) residuals, then test for the 1st, 2nd, and 4th 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
  

See Also
lbqtest

References

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

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

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

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

aicbic

autocorr