Continuous Record Asymptotics for Change‐Point Models

In the context of a linear regression model with a single break point, we develop a continuous record asymptotic framework to build inference methods for the break date. We have T$$ T $$ observations with a sampling frequency h$$ h $$ over a fixed‐time horizon 0,N,$$ \left[0,N\right], $$ and let T→∞$$ T\to \infty $$ with h↓0$$ h\downarrow 0 $$ while keeping the time span N$$ N $$ fixed. We consider the least‐squares estimate of the break date and establish consistency and convergence rate. We provide a limit theory for shrinking magnitudes of shifts and locally increasing variances. The asymptotic distribution corresponds to the location of the extremum of a function of the quadratic variation of the regressors and of a Gaussian‐centered martingale process over a certain time interval. We can account for the asymmetric informational content provided by the pre‐ and post‐break regimes and show how the location of the break and shift magnitude are key ingredients in shaping the distribution. We consider a feasible version based on plug‐in estimates, which provides a very good approximation to the finite sample distribution. We use the concept of the Highest Density Region to construct confidence sets. Overall, our method is reliable and delivers accurate coverage probabilities and the relatively short average length of the confidence sets. Importantly, it does so irrespective of the size of the break.