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Parametric or nonparametric: the FIC approach for stationary time series

Gudmund Hermansen, Nils Lid Hjort, Martin Jullum

TL;DR

A new version of the focused information criterion (FIC) is developed, directly comparing the performance of both parametric and nonparametric time series models, by comparing the mean squared error for estimating a focus parameter under consideration, for each candidate model.

Abstract

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.

Parametric or nonparametric: the FIC approach for stationary time series

TL;DR

A new version of the focused information criterion (FIC) is developed, directly comparing the performance of both parametric and nonparametric time series models, by comparing the mean squared error for estimating a focus parameter under consideration, for each candidate model.

Abstract

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.
Paper Structure (19 sections, 3 theorems, 47 equations, 6 figures)

This paper contains 19 sections, 3 theorems, 47 equations, 6 figures.

Table of Contents

  1. Introduction and summary
  2. Estimation and approximations
  3. Maximum likelihood estimation outside the model
  4. The Whittle approximation
  5. Nonparametric modelling
  6. Parametric versus nonparametric
  7. How to compare parametric and nonparametric models?
  8. Deriving unbiased risk estimates
  9. Models with trends
  10. Average focused information criterion
  11. Performance
  12. FIC under model conditions
  13. FIC in practice
  14. Concluding remarks
  15. Model averaging
  16. ...and 4 more sections

Key Result

Proposition 1

Let $y_{1}, \ldots, y_{n}$ be realisations from a stationary Gaussian time series model with spectral density $g$ assumed to be uniformly bounded away from both zero and infinity. Suppose $|h_0|$ is bounded in $\omega$, that $f_{\theta}$ is two times differentiable with respect to $\theta$, and that where with $J$ and $K$ as defined below (eq:lm_large_sample), and $v_{c} = c_0^{{\rm t}} J(g, f_{\

Figures (6)

  • Figure 1.1: The true spectral density and the raw periodogram from a simulated autoregressive time series of order $4$, with length $n = 100$ and parameters $\rho = (0.2, 0.2, -0.1, -0.2)$ and $\sigma = 1.30$. The shaded regions corresponds to three different focus parameters, namely, the integrated spectrum (or total energy) over that particular region.
  • Figure 1.2: The horizontal lines indicate the true spectral density over the three shaded regions (of the same colour) shown in Figure \ref{['figure:intro:data']}; the three focus parameters $\mu_1, \mu_2$ and $\mu_3$. The corresponding coloured dots show the performance, in terms of the root of the FIC score for the nonparamteric model based on the periodogram (n) and the autoregressive models of order 0--4, where 0 represent the model with independent.
  • Figure 6.1: Relative root-mse for each candidate model fitted to the six focus parameters $\mu_{k} = C(k)$, for $k = 0, \ldots, 5$. The root-mse is computed based on 5000 simulated AR(2) series of length $n = 100$, with $\sigma = 1.0$ and $\rho = (0.7, -0.6)$, For ease of comparison we have scaled the root-mse to the unit interval.
  • Figure 6.2: The five least false covariance functions under the assumption that the true model is an autoregressive model specified by the parameters $\sigma = 1.0$ and $\rho = (0.7, -0.6)$.
  • Figure 6.3: The proportion for which the different criteria selects the model with the theoretical lowest root-mean-squared error. The model-selectors are always nonparametric, FIC, AIC and BIC. The results are based on 5000 simulated series.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Proposition 3
  • proof