  For a partial structural change in a linear regression model with a single
break, we develop a continuous record asymptotic framework to build inference
methods for the break date. We have T observations with a sampling frequency h
over a fixed time horizon [0, N] , and let T with h 0 while keeping the time
span N fixed. We impose very mild regularity conditions on an underlying
continuous-time model assumed to generate the data. 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 Highest
Density Region to construct confidence sets. Overall, our method is reliable
and delivers accurate coverage probabilities and relatively short average
length of the confidence sets. Importantly, it does so irrespective of the size
of the break.
