Supplement to “reconstructing past Temperatures from Natural Proxies and Estimated Climate Forcings Using Short- and Long-memory

A.1. Background on long memory models. As mentioned in the introduction, long-memory estimation is typically difficult both in theory and in practice; fundamental stochastic analysis research on this question is ongoing. In discrete time series, the long-memory autoregressive moving average (ARMA) and autoregressive conditional heteroskedasticity (ARCH) models that are popular in financial econometrics (see Granger and Joyeux (1980) and Baillie, Bollerslev and Mikkelsen (1996) for variations of ARMA and ARCH), cannot be adapted to our Bayesian context. These models are complex and are not explicitly specified in terms of distributions; they often rely on long data series or high-frequency in-fill data. It can be argued that the only models for which the discrete-time parametric memory-length estimation is straightforward and reliable in finite samples are those that come from continuous-time underlying models which exhibit self-similarity (identical distribution at all time scales), of which the typical example is fractional Brownian motion (fBm). Pioneering work in this direction was performed in Coeurjolly (2001), and further details and developments were provided in Chronopoulou, Viens and Tudor (2009). Long-memory structures which are slightly more complex than those studied therein are already difficult to estimate. For instance, the so-called fractional Ornstein-Uhlenbeck (fOU) process has a well-known estimator for its drift parameter in continuous time (Kleptsyna and Le Breton, 2002), and with discrete data for increasing horizon asymptotics (Tudor and Viens, 2007), but the question of estimating its long-memory parameter remains elusive. When in-fill asymptotics can be attained in the data this can be accomplished via path regularity in continuous time (see Istas and Lang (1997) and its application in Brouste and Iacus (2012)). Neither this, nor the use of increasing-horizon asymptotics in Tudor and Viens (2007) are applicable in our paleoclimatology situation, because our observation frequency is not high enough and our calibration period is too short.