We introduce a new joint test for the order of fractional integration of a multivariate fractionally integrated vector autoregressive (FIVAR) time series based on applying the Lagrange multiplier principle to a feasible generalised least squares estimate of the FIVAR model obtained under the null hypothesis. A key feature of the test we propose is that it is constructed using a heteroskedasticity-robust estimate of the variance matrix. As a result, the test has a standard χ² limiting null distribution under considerably weaker conditions on the innovations than are permitted in the extant literature. Specifically, we allow the innovations driving the FIVAR model to follow a vector martingale difference sequence allowing for both serial and cross-sectional dependence in the conditional second-order moments. We also do not constrain the order of fractional integration of each element of the series to lie in a particular region, thereby allowing for both stationary and non-stationary dynamics, nor do we assume any particular distribution for the innovations. A Monte Carlo study demonstrates that our proposed tests avoid the large oversizing problems seen with extant tests when conditional heteroskedasticity is present in the data. We report an empirical case study for a sample of major US stocks investigating the order of fractional integration in trading volume and different measures of volatility in returns, including realised variance. Our results suggest that both return volatility and trading volume are fractionally integrated, but with the former generally found to be more persistent (having a higher fractional exponent) than the latter, when more reliable proxies for volatility such as the range or realised variance are used.

中文翻译：

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多变量分数集成测试允许条件异方差性与应用程序返回波动率和交易量

我们基于将拉格朗日乘数原理应用于在原假设下获得的 FIVAR 模型的可行广义最小二乘估计，引入了一种新的联合检验，用于对多元分数积分向量自回归 (FIVAR) 时间序列的分数积分阶次进行检验。我们提出的测试的一个关键特征是它是使用方差矩阵的异方差稳健估计构建的。因此，检验具有标准的χ ²在比现有文献中允许的更弱的创新条件下限制零分布。具体来说，我们允许驱动 FIVAR 模型的创新遵循向量鞅差分序列，允许条件二阶矩中的序列和横截面相关性。我们也不限制序列中每个元素的分数积分顺序位于特定区域，从而允许静态和非静态动态，我们也不假设创新的任何特定分布。蒙特卡罗研究表明，当数据中存在条件异方差时，我们提出的测试避免了现有测试中出现的大尺寸过大问题。我们报告了一个主要美国股票样本的实证案例研究，调查了交易量的分数积分顺序和回报波动的不同度量，包括已实现的方差。我们的结果表明，回报波动率和交易量都是分数集成的，但当使用更可靠的波动率指标（例如范围或已实现方差）时，前者通常比后者更持久（具有更高的分数指数） .