Likelihood Inference for Possibly Non-Stationary Processes via Adaptive Overdifferencing

We make an observation that facilitates exact likelihood-based inference for the parameters of the popular ARFIMA model without requiring stationarity by allowing the upper bound d¯ for the memory parameter d to exceed 0.5: estimating the parameters of a single non-stationary ARFIMA model is equivalent to estimating the parameters of a sequence of stationary ARFIMA models. This allows for the use of existing methods for evaluating the likelihood for an invertible and stationary ARFIMA model. This enables improved inference because many standard methods perform poorly when estimates are close to the boundary of the parameter space. It also allows us to leverage the wealth of likelihood approximations that have been introduced for estimating the parameters of a stationary process. We explore how estimation of the memory parameter d depends on the upper bound d¯ and introduce adaptive procedures for choosing d¯. We show via simulation how our adaptive procedures estimate the memory parameter well, relative to existing alternatives, when the true value is as large as 2.5.

我们发现了一个关键现象，该现象可推动对常用ARFIMA模型（Autoregressive Fractionally Integrated Moving Average）参数的精确似然推断——无需要求平稳性，只需允许记忆参数d的上界d̄超过0.5即可：估计单个非平稳ARFIMA模型的参数等价于估计一系列平稳ARFIMA模型的参数。这使得我们能够利用现有方法评估可逆且平稳的ARFIMA模型的似然值，进而提升推断性能（因为当估计值接近参数空间边界时，许多标准方法表现不佳）。此外，这还能让我们充分利用已提出的大量用于平稳过程参数估计的似然近似方法。我们探究了记忆参数d的估计对上限d̄的依赖关系，并提出了选择d̄的自适应方法。通过仿真实验，我们证明了当d的真实值高达2.5时，与现有替代方法相比，我们的自适应方法对记忆参数的估计效果更优。