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.
The focused information criterion (FIC) was introduced in Claeskens & Hjort (2003) and is based on estimating and comparing the accuracy of model-based estimators for a chosen focus parameter. This focus, say µ, ought to have a clear statistical interpretation across candidate models. For a given candidate model, µ is traditionally expressed as a function of this model's parameters. In general, the focus parameter can be any sufficiently smooth and regular function of the underlying model parameters, or more generally its spectral distribution. This includes quantiles, regression coefficients, a specified lagged correlation, but also various types of predictions and data dependent functions, to name some; see Hermansen & Hjort (2015) for a more complete list and discussion of valid focus parameters for time series models.
Suppose there are candidate models M 1 , . . . , M k , leading to focus parameter estimates µ 1 , . . . , µ k , respectively. The underlying idea leading to the FIC is to estimate the mean squared error (mse) of µ j for each candidate model and then select the model that achieves the smallest value. The mse in question is mse j = E ( µ j -µ true ) 2 = bias( µ j ) 2 + Var µ j , comprising the variance and the squared bias in relation to the true parameter value µ true .
Thus the FIC consists of finding ways of assessing, approximating and then estimating the mse j for each candidate model. The winning model is the one with smallest mse j . How this may be done depends on both the candidate models and the focus parameter, as well as on other characteristics of the underlying situation. The FIC apparatus hence leads to different types of formulae in different setups; see Claeskens & Hjort (2008, Ch. 5 & 6) for a fuller discussion and illustrations of such criteria for selection among parametric models.
Most FIC constructions have been derived by relying on a suitably defined local misspecification framework, see again Claeskens & Hjort (2008, Ch. 5 & 6). In such a framework the true model is assumed to gradually shrink with the sample size, starting from the biggest ‘wide’ model and hitting the simplest ’narrow’ model in the limit. In addition, and all candidate models need to lie between these two model extremes. In the various data settings, such frameworks typically result in squared biases and variances of the same asymptotic order, motivating certain approximation formulae for the mse j in question. In Hermansen & Hjort (2015) such a framework is used to derive FIC machinery for choosing between parametric time series models within broad classes of time series models. See Section 7.5 for some further remarks.
The aim of the present paper is to derive FIC machinery which will justify comparison and selection among both parametric and nonparametric candidate models. The derivation will be somewhat different from that of Claeskens & Hjort (2003) and Hermansen & Hjort (2015) in that we do not rely on a certain local misspecification framework.
We rather take a more direct approach following reasoning similar to the development of Jullum & Hjort (2015), where focused inference and model selection among parametric and nonparametric models are developed for independent observations. By including a nonparametric candidate among the parametric models, we will in particular be able to detect whether our parametric models are off-target. This FIC construction, with a nonparametric alternative, therefore has a built-in insurance mechanism against poorly specified parametric candidates. When one or more parametric models are adequate, such are selected as they typically have lower variance.
Though our methods will be extended to more general setups later, we start our developments with the class of zero-mean stationary Gaussian time series processes. Let {Y t } be such a process. Then the dependency structure, which in such cases determines the entire model, is completely specified by the corresponding covariance function
, defined for all lags k = 0, 1, 2, . . .. Here we will, for mathematical convenience, work with the frequency representation, where the covariance function C(k) can be represented by a unique spectral distribution G such that
provided the corresponding spectral distribution G has a continuous and symmetric density g. See among others Brillinger (1975), Priestley (1981) or Dzhaparidze (1986) for a general introduction to time series modelling in the frequency domain. When necessary, we will write C g to indicate that this is the covariance indexed by the spectral density g. Note also that we can obtain the spectral density as the Fourier transform of the covariance function.
The types of parametric models we will consider are typically the classical autoregressive (AR), moving average (MA) and the mixture (ARMA), all of which have clear and well defined corresponding spectral densities; see e.g. Brockwell & Davis (1991) for an introduction to time series modelling with such
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