Testing for an Omitted Multiplicative Long-Term Component in GARCH Models

ABSTRACT

We consider the problem of testing for an omitted multiplicative long-term component in GARCH-type models. Under the alternative, there is a two-component model with a short-term GARCH component that fluctuates around a smoothly time-varying long-term component which is driven by the dynamics of an explanatory variable. We suggest a Lagrange multiplier statistic for testing the null hypothesis that the variable has no explanatory power. We derive the asymptotic theory for our test statistic and investigate its finite sample properties by Monte Carlo simulation. Our test also covers the mixed-frequency case in which the returns are observed at a higher frequency than the explanatory variable. The usefulness of our procedure is illustrated by empirical applications to S&P 500 return data. Supplementary materials for this article are available online.

KEYWORDS:

• GARCH-MIDAS
• LM test
• Long-term volatility
• Mixed-frequency data
• Volatility component models

ACKNOWLEDGMENTS

The authors are most grateful to the editor, Todd Clark, an associate editor, and two anonymous referees for comments that greatly improved their article. The authors thank Torben G. Andersen, Richard Baillie, Ian Dew-Becker, Robert F. Engle, Christian Francq, Eric Ghysels, Tilmann Gneiting, Onno Kleen, Fabian Krüger, Enno Mammen, Rasmus S. Pedersen, Peter Schmidt, Karin Stürmer, Robert Taylor, Timo Teräsvirta, Viktor Todorov, Jeffrey Wooldridge, and Jean-Michel Zakoïan for helpful comments and suggestions.

Notes

1 Throughout the article, we assume that the conditional mean of the returns is zero. For GARCH misspecification testing in the presence of a nonzero conditional mean see Halunga and Orme (2009).

2 Later on, we also consider the one-sided alternative $H_1:{\mathit{}\mathbf{\pi }}_0\ne \mathbf{0},{\mathit{}\mathbf{\pi }}_0\ge \mathbf{0}$ (see Remark 4 in Section 2.3).

3 For α₀ → 0, the GARCH component approaches a constant and all the variation in σ²_{0, t} is due to movements in the long-term component.