Model Selection and Statistical Inference in Time Series Analysis

Time series analysis concerns the study of data collected sequentially over time, with the dual objectives of understanding its dependence structure and generating reliable forecasts. Model selection and statistical inference lie at the heart of this discipline, guiding the choice among candidate families—such as autoregressive–moving average, state-space and volatility models—and providing measures of uncertainty for parameter estimates. A delicate trade-off between complexity and generalisation must be managed: overly simple models fail to capture key dynamics, while excessively flexible ones risk poor out-of-sample performance. To address this, practitioners draw on information criteria, cross-validation, penalised likelihood and Bayesian frameworks, often enriched by machine-learning techniques for regularisation and ensemble modelling. Rigorous inferential procedures underpin confidence intervals and hypothesis tests, even in the presence of non-stationarity, structural breaks or exogenous covariates. Recent innovations have emphasised high-dimensional systems, nonlinear dynamics and robust methods for heavy-tailed or heteroskedastic data. Applications span financial risk management, climate modelling, epidemiology and engineering, where accurate model choice and parameter estimation are essential for real-time monitoring and policy decision making.

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Model Selection and Statistical Inference in Time Series Analysis publication trend

The graph below shows the total number of articles in model selection and statistical inference in time series analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Autoregressive (AR) model: A representation in which each value is expressed as a linear combination of its own past observations.

Moving average (MA) model: A structure that models current observations as a linear function of past disturbance terms.

Generalised autoregressive conditional heteroskedasticity (GARCH): A class of models in which current variance depends on past squared errors and past variances, capturing time-varying volatility.

Quasi-maximum likelihood estimator (QMLE): An estimation technique that maximises an assumed likelihood function, providing consistent results even under mild misspecification.

Information criterion: A scalar measure, such as Akaike’s or Bayesian information criterion, that balances goodness of fit against model complexity for comparative selection.

Stationarity: A property of a time series whose statistical moments (mean, variance, autocorrelation) remain invariant over time shifts.

References

1. Inference and model selection in general causal time series with exogenous covariates. Electronic Journal of Statistics (2022).
2. Scale-Invariant and consistent Bayesian information criterion for order selection in linear regression models. Signal Processing (2022).
3. Consistent model selection criteria and goodness-of-fit test for common time series models. Electronic Journal of Statistics (2020).