Abstract
This paper re-examines the limiting distribution of the conventional Pearson sample cross- correlation coefficients (?ˆ??) when testing for the uncorrelatedness between two time series (??,? and ??,?), of which the integrated orders are not equal to each other. We first demonstrate the invalidation of √??ˆ?? → ?(0,1) and of conventional liming distribution for its resulting Lagrangian multiplier (LM) correlation test statistics of Breusch and Pagan (1980) when the integrated orders of two time series are imbalanced. We then suggest to reconstruct this ?ˆ?? by using the AR-filtered residuals from two integrated order-imbalanced time series as well as the AR-filtering version of LM test. The mathematical theorems justify the standard normal distribution followed by the new built Pearson sample correlation coefficient, i.e., √??ˆ??,?? → ?(0,1) and then show a chi-squared distribution with ?(? − 1)/2 degrees of freedom of its corresponding LM test. Extending the Hong(1996) cross-correlation tests, we further propose two easy-to-implement lagged correlation tests for two integrated order-imbalanced processes being uncorrelated via AR approximations. Our simulations confirm the limiting distribution of the conventional ?ˆ?? fails to follow the standard normal distribution as two time series displaying different integrated orders in contrast to the traditional understanding of ?ˆ??, as well as demonstrate that in finite samples, the significant improvement of the newly built Pearson sample correlation coefficient in terms of the size and promising power performances for the new proposed correlation tests. More importantly, our new methodology could be treated as a simple correlation test which avoids the spurious correlation and inaccurate results caused by the two order-imbalanced series analyzed in previous literature. Finally, a newly constructed crises-detecting index and a revisiting study of risk-return trade-offs provide the usefulness of our methodology.