Local powers of least-squares-based test for panel fractional Ornstein–Uhlenbeck process

In recent years, significant advancements have been made in the field of identifying financial asset price bubbles, particularly through the development of time-series unit-root tests featuring fractionally integrated errors and panel unit-root tests. This study introduces an innovative approach for assessing the sign of the persistence parameter ($\alpha$) within a panel fractional Ornstein-Uhlenbeck process, based on the least squares estimator of $\alpha$. This method incorporates three distinct test statistics based on the Hurst parameter ($H$), which can take values in the range of $(1/2,1)$, be equal to $1/2$, or fall within the interval of $(0,1/2)$. The null hypothesis corresponds to $\alpha =0$. Based on a panel of continuous records of observations, the null asymptotic distributions are obtained when the time span ($T$) is fixed and the number of cross sections ($N$) goes to infinity. The power function of the tests is obtained under the local alternative where $\alpha$ is close to zero in the order of $1/\left(T\sqrt{N}\right)$. This alternative covers the departure from the unit root hypothesis from the explosive side, enabling the calculation of lower power in bubble tests. The hypothesis testing problem and the local power function are also considered when a panel of discrete-sampled observations is available under a sequential limit, that is, the sampling interval shrinks to zero followed by the $N$ goes to infinity.