Testing for Arch in the Presence of Additive Outliers

SUMMARY
In this paper we investigate the properties of the Lagrange Multiplier [LM] test for autoregressive
conditional heteroscedasticity (ARCH) and generalized ARCH (GARCH) in the presence of additive
outliers (AOs). We show analytically that both the asymptotic size and power are adversely affected if AOs
are neglected: the test rejects the null hypothesis of homoscedasticity too often when it is in fact true. while
the test has difficulty detecting genuine GARCH effects. Several Monte Carlo experiments show that these
phenomena occur in small samples as well. We design and implement a robust test, which has better size and
power properties than the conventional test in the presence of AOs. We apply the tests to a number of US
macroeconomic time series, which illustrates the dangers involved when nonrobust tests for ARCH are
routinely applied as diagnostic tests for misspecification. Copyright ,C1999 John Wiley & Sons, Ltd.

1. INTRODUCTION
It is common practice to subject the residuals of an estimated time series regression model to a
battery of diagnostic tests. One of the tests which is routinely applied is a test for (conditional)
heteroscedasticity. In fact, many econometric programs, such as Evie\r.s, PC-Give and Microjit,
automatically produce such tests as a diagnostic check. The LM test for AutoRegressive
Conditional Heteroscedasticity (ARCH) developed by Engle (1982) undoubtedly is the most
popular one among the available heteroscedasticity tests. When this test is applied to residuals
from an empirical macroeconometric model (for, say, money demand, unemployment, or
output), its significance is often interpreted as a sign of some form of model misspecification, and
not necessarily of conditional heteroscedasticity. Usually the practitioner decides, for example, to
add variables or additional lags to the model. In fact, Lumsdaine and Ng (1997) show that
omitted variables, among other causes, may lead to significant ARCH test statistics.
Alternatively, if the test is used for financial time series, its outcome is often used as a guideline
for subsequent modelling of the conditional variance. ARCH and Generalized ARCH
(GARCH) models, introduced by Engle (1982) and Bollerslev (1986), respectively, are by now
the most widely used models to describe the time-varying volatility observed in many (highfrequency)
financial time series, see Bollerslev et al. (1992, 1994), Bera and Higgins (1993), and
Gourieroux (1997) for recent surveys. Results from the ARCH test may be used to specify the
appropriate orders in a GARCH equation.