Counting Process Generated by Boundary-crossing Events. Theory and Statistical Applications

In this paper, we introduce and analyze a new class of stochastic process, named Boundary Crossing Counting Process. We show how to assess its finite-sample probability distribution using first exit time distributions. We discuss and calibrate three different methods for estimating this probability distribution and show how to use this distribution for non-parametric statistical specification testing or model validation. We apply this tool to the problem of unit root testing and to the problem of modeling financial data. First, we find that the BCC-test is less powerful than specific parametric unit-root tests, yet more powerful than another non-parametric test based on probability integral transform. Next, we learn that the BCC-test is relatively powerful in differentiating between commonly used GARCH type of models when the model under the null hypothesis and the model under the alternative hypothesis differ in the distribution of the error term. Finally, we show that the BCC-test often rejects the null hypothesis that the data generating process for major American and Australian stock indexes is well represented by the GARCH(1,1) model. The BCC-test cannot reject the GJR model for any specification, which is additional supportive evidence for the presence and importance of asymmetry in financial markets.