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

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.本文针对广义自回归条件异方差（GARCH）类模型中遗漏乘性长期成分的检验问题展开研究。在备择假设下，模型包含双成分结构：短期GARCH成分围绕由解释变量动态驱动的平滑时变长期成分波动。本文提出拉格朗日乘子统计量，用于检验“解释变量无解释能力”的原假设。我们推导了该检验统计量的渐近理论，并通过蒙特卡洛模拟考察其有限样本表现。所提检验方法同样适用于收益观测频率高于解释变量的混频场景。通过对标普500（S&P 500）收益数据的实证应用，验证了本文方法的实用性。本文补充材料可在线获取。