Finite-sample size distortion of the AESTAR unit root test: GARCH, corrected variance–covariance matrix estimators and adjusted critical values

ABSTRACT

The behaviour of the asymmetric exponential smooth transition autoregressive (AESTAR) unit root test, which allows for asymmetric and nonlinear reversion to equilibrium, is examined in the presence of generalized autoregressive conditional heteroscedasticity (GARCH). It is found that while the test is relatively robust in the presence of ‘low volatility’ GARCH processes, it exhibits substantial size distortion when large values of the volatility parameter are considered. Attempted resolution via the routine application of heteroscedasticity consistent (or ‘corrected’) covariance matrix estimators (HCCMEs) is shown to result in overwhelming size distortion due to their impact upon the finite-sample distribution of the underlying test statistic. However, application of a corrected HCCME, in combination with critical values derived specifically under its use, results in the control of test size. Analogous results for the Dickey–Fuller (1979) test are presented to permit comparison with a test considering linear, symmetric adjustment. It is found that the AESTAR test is subject to far greater distortion than its linear, symmetric alternative. In summary, the results indicate that caution must be exercised when applying the AESTAR test to macroeconomic and financial time series, particularly if routine application of corrected covariance matrix estimators occurs.

KEYWORDS:

• GARCH
• unit root tests
• nonlinearity
• asymmetry
• consistent-threshold estimation
• size distortion

JEL CLASSIFICATION:

• C12
• C15
• C58

Notes

¹ The importance and prominence of GARCH in financial series are illustrated by the studies of Andersen and Bollerslev (1998) and Engle and Patton (2001).

² Further details on the derivation of the testing equation, Equation 5, can be obtained from Sollis (2009). The analysis provided is similar to that of Kapetanios, Snell, and Shin (2003) for an ESTAR unit root test, while similar AESTAR models have been considered by Anderson (1997) and Siliverstovs (2005).

³ In this paper the less empirically realistic, or relevant, cases of degenerate GARCH (φ) and integrated GARCH (φ₁ + φ) are not considered.