The authors are most grateful to the discussants who have put in substantial work in understanding the details of our analysis, and have come up with many interesting extensions of our work. In what follows, we will refer to the different discussion contributions as follows: MF Madonna (2022) PS Poppick and Stein (2022) SR Schmidt and Rodríguez (2022) MK McKinnon (2022) TP Thorne (2022)We will deal with some of the common threads first, and leave remaining individual points to later.
TREND MODELINGEvery statistical analysis must make a variety of modeling choices, each of which may be debatable. Our choice of B-splines to model the trend was perhaps the most frequent point of disagreement between us and the discussants. While the concept of trend is not always well defined in the literature, we often define trend to be continuous changes over longer scales that can be modeled independently from the rest of the series. SR suggest using a dynamic linear model (essentially a locally linear trend model with dependent errors) and show that this yields similar answers to our B-spline approach. This is perhaps not surprising considering the relationship between splines and random walk/stochastic differential equations (see, e.g.,