Parametric or nonparametric: the FIC approach for stationary time series

We seek to narrow the gap between parametric and nonparametric modelling of stationary time series processes. The approach is inspired by recent advances in focused inference and model selection techniques. The paper generalises and extends recent work by developing a new version of the focused information criterion (FIC), directly comparing the performance of parametric time series models with a nonparametric alternative. For a pre-specified focused parameter, for which scrutiny is considered valuable, this is achieved by comparing the mean squared error of the model-based estimators of this quantity. In particular, this yields FIC formulae for covariances or correlations at specified lags, for the probability of reaching a threshold, etc. Suitable weighted average versions, the AFIC, also lead to model selection strategies for finding the best model for the purpose of estimating e.g.~a sequence of correlations.

💡 Research Summary

The paper tackles a long‑standing dilemma in time‑series analysis: when to rely on a parametric model (e.g., ARMA, GARCH) and when a non‑parametric alternative is preferable. Traditional information criteria such as AIC or BIC evaluate models on a global fit basis and do not take into account the specific inferential goal of the analyst. To bridge this gap, the authors develop a Focused Information Criterion (FIC) that directly compares the mean‑squared error (MSE) of estimators for a pre‑specified “focused parameter” – for example, a covariance at a particular lag, a correlation, or the probability that the series exceeds a threshold.

The methodology proceeds as follows. The analyst first defines one or more focused parameters, θ_f. For each candidate model, a parametric estimator (\hat θ_{f}^{(i)}) is obtained (e.g., using the fitted ARMA coefficients). In parallel, a non‑parametric estimator (\hat θ_{f}^{NP}) is constructed using kernel‑based spectral density estimation, local polynomial regression, or block‑bootstrap techniques. The MSE of each estimator is then decomposed into bias² plus variance. The FIC for the i‑th parametric model is defined as

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