A stationary unbiased finite sample ARCH-LM test procedure

Abstract

Engle's (1982) Autoregressive Conditional Heteroscedasticity-Lagrange Multiplier (ARCH-LM) test is the undisputed standard test to detect ARCH. In this article, Monte Carlo (MC) simulations are used to demonstrate that the test's statistical size is biased in finite samples. Two complementing remedies to the related problems are proposed. One simple solution is to simulate new unbiased critical values for the ARCH-LM test. A second solution is based on the observation that for econometrics practitioners, detection of ARCH is generally followed by remedial modelling of this time-varying heteroscedasticity by the most general and robust model in the ARCH family; the Generalized ARCH (GARCH(1,1)) model. If the GARCH model's stationarity constraints are violated, as in fact is very often the case, obviously, we can conclude that ARCH-LM's detection of conditional heteroscedasticity has no or limited practical value. Therefore, formulated as a function of whether the GARCH model's stationarity constraints are satisfied or not, an unbiased and more relevant two-stage ARCH-LM test is specified. If the primary objectives of the study are to detect and remedy the problems of conditional heteroscedasticity, or to interpret GARCH parameters, the use of this article's new two-stage procedure, 2-Stage Unbiased ARCH-LM (2S-UARCH-LM), is strongly recommended.

Acknowledgements

I would gratefully like to acknowledge the financial support provided by Sparbankernas Forskningsstiftelse.

Notes

¹ In this article the name ARCH-LM is used, regardless of whether the test's χ²- or F-versions are referred to. ARCH-LM is, despite its small-sample biasedness, the standard test to detect ARCH for all sample sizes (see for instance, Weymark, 1999; Karadaş and Öğünç, 2005; and Hodge, 2006).

² These so-called ‘stationarity constraints’ are presented in Table 1. Notice that these restrictions include both constraints regarding nonstationarity of the GARCH coefficients as well as nonnegativity constraints for these coefficients.

³ See the statistical size of the 2S-ARCH-LM test in Table 4.

⁴ See the unbiased critical values of the UARCH-LM test in Table 5 (that adjusts the size of the ARCH-LM test), and the unbiased critical values of the 2S-UARCH-LM test in Table 5 (that adjusts the size of the 2S-ARCH-LM test).

⁵ Engle's (1982) ARCH-LM test is the undisputed standard test for detection of ARCH. Most of the mentioned alternative tests to ARCH-LM are not substitutes, or directly comparable, to the new approach presented in this article. None of these tests from previous research takes into account whether the parameter constraints are satisfied or not.

⁶ According to, for instance, Lamont et al. (1996) and Gallo and Pacini (1998) ∑(α + β) determines the GARCH model's persistence.

⁷ Unless the QML approach is applied, the covariance matrix estimates will not be consistent which would result in incorrect SEs. The QML correction does not affect the estimated coefficients but it affects only the estimated SEs. Consequently, this adjustment does not affect the executed simulation studies since new simulated critical values are applied. However, when estimating GARCH(1,1) coefficients this correction adjusts the SEs. See, for instance Chan and McAleer (2003).

⁸ 2S-(U)ARCH-LM test stands for 2-Stage (Unbiased) Autoregressive Conditional Heteroscedasticity LM test, since in the first stage the ARCH-LM is executed, and in the next stage we evaluate whether the stationarity constraints are satisfied. Since the 2S-ARCH-LM procedure exhibits severe size-problems, new remedial critical values are simulated for 2S-ARCH-LM (2S-ARCH-LM is simply a replication of the currently most common procedure that is applied by practitioners, and is for practical reasons given a name in this article since it is used as a benchmark model that is compared with the new approach). Besides new critical values, exactly the same procedure is used in the new test. However, since the new critical values make the 2S-ARCH-LM unbiased, the new test procedure is denoted as 2S-Unbiased ARCH-LM or 2S-UARCH-LM.

⁹ In this article, the ARCH-LM test is applied as a benchmark test in order to be able to compare the performance of the new UARCH-LM test.

¹⁰ In this article, the 2S-ARCH-LM test is applied as a benchmark test in order to be able to compare the performance of the new 2S-UARCH-LM test.

¹¹ This is demonstrated in the MC simulations in Table 4.

¹² This is demonstrated in the MC simulations in Table 4.

¹³ Furthermore, observe that the above two-stage approaches do not suffer from so-called mass significance since there are if-statements that evaluate the constraints instead of p-values.

¹⁴ This is demonstrated in the MC simulations in Table 4.

¹⁵ This is demonstrated in the MC simulations in Table 4.

¹⁶ This is an evaluation approach applied in, for instance, Edgerton and Shukur (1999).

¹⁷ The general decision rule in statistical hypothesis testing is that we usually only draw clear conclusions if a test is significant, while if it is insignificant we say that the test is inconclusive. ‘Inconclusive’ does not necessarily imply that we believe in the null hypothesis (at least if we disregard unit root tests), it only implies that we do not have enough statistical evidence to be able to reject the null hypothesis. Consequently, somewhat simplified, power problems may lead to too many inconclusive tests (with essentially no decisions), while statistical size problems lead to too many incorrect decisive conclusions. Since this is the usual practice in statistical hypothesis testing, it is necessary that the size is unbiased.

¹⁸ If, in repeated simulations, the coefficient estimates of the GARCH(1,1) models always satisfy all the stationarity constraints in Table 1, then the ‘percentage share of satisfied stationarity constraints’ is 100 (%). Therefore, optimally, this share should be 100 (%) since we only simulate pure stationary GARCH(1,1) data generating processes with no violated constraints, and then we should, of course, expect the estimated GARCH(1,1) models to obtain coefficient estimates that are stationary. However, if the estimation process does not work optimally this may result in a certain fraction of the estimated GARCH(1,1) models with coefficients that violates the stationarity constraints in Table 1.

¹⁹ Moreover, there are many examples of high ARCH-component values (α) that can be found in Brooks et al. (2001), Kim and Schmidt (1993) and Li et al. (2003).

²⁰ ARCH-LM is the (biased) benchmark test for situations when violations of the stationarity constraints are of no importance, while UARCH-LM is the same test but with new unbiased critical values.

²¹ In some fields of economics 150 observations is a relevant, relatively large, and a fairly common available sample size, such as for instance in macro economics.

²² If one would allow a test to be severely size-biased it is pointless to evaluate the power of this test. In fact it is very easy to construct a test with 100% power if this test always rejects the null hypothesis. The cost would be that the power is misleading due to a deceptive type-1 error, and consequently this would not be a meaningful test.

²³ In this article, this approach is illustrated by the benchmark procedure named 2S-ARCH-LM.

²⁴ Engle's (1982) ARCH-LM test is the undisputed standard test for detection of ARCH.