The functional central limit theorem and structural change test for the HAR(∞) model

- Some powered process of the hyperbolic-memory HAR model is considered.
- Sufficient conditions for L2-NED of the process are given.
- The FCLT for the process is proved.
- The Limiting distribution of the CUSUM statistics of the process is derived.

In this paper, we study the functional central limit theorem (FCLT) for the infinite-order heterogeneous autoregressive model of realized volatility (HAR(∞)). We prove under proper assumptions that the process is L2-NED and then obeys the FCLT. As an application of the FCLT, we derive the limit distribution of the CUSUM statistics to detect the structural change of the model.