Multivariate Fractional Integration Tests allowing for Conditional Heteroskedasticity with an Application to Return Volatility and Trading Volume

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 2 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 crosssectional 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 over-sizing problems seen with extant tests when conditional heteroskedasticity is present in the data. We report an empirical case study for a sample of major U.S. stocks investigating the order of fractional integration in trading volume and different measures of volatility in returns, including realized 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 realized variance are used.