The bootstrap in state space models

Coppin, Kim Michelle (2023) The bootstrap in state space models. PhD thesis, Birkbeck, University of London.

Abstract

This dissertation offers some theoretical and practical contributions to the relatively sparse literature on the bootstrap in state space models. In the first chapter, we present the rationale for our research and offer an overview of the structure of the document. In Chapter 2, we introduce a gradient-based approach to the bootstrap which can be applied to heteroskedastic state space models. To the best of our knowledge, there is presently only one bootstrap technique available for models in state space form and it is limited to homoskedastic models. We evaluate the effectiveness of our methodology by analysing the coverage probabilities of different bootstrap confidence intervals and we find that they perform better than their asymptotic equivalents. We go on to use our methodology to modify an existing bootstrap model selection criterion so that it can be utilised with heteroskedastic state space models. Moreover, we introduce a bootstrap procedure that is suitable for homoskedastic state space models, providing an alternative to the current approach. Monte Carlo simulations demonstrate that this new method is significantly faster than the existing technique. In Chapter 3, we apply an approximate bootstrap algorithm to state space models for the purpose of conducting hypothesis tests. We illustrate the advantages of working within the state space framework by highlighting how Kalman filter output can aid in the construction of certain chi-squared test statistics. We then apply the approximate bootstrap approach to actual data and reveal how, in small to moderately sized samples, bootstrap test statistics can lead researchers to different conclusions than those drawn from using standard asymptotic theory. Chapter 4 focuses on the transformation of multilevel models into state space form for the purpose of estimation. We show how the output from the Kalman filter recursions can facilitate the parametric and residual bootstraps in these models. In hierarchical settings, the full information maximum likelihood estimates of the variance elements in the covariance matrix of the vector of random effects are commonly biased downward in small samples. To alleviate this problem, researchers generally use restricted maximum likelihood estimation as it yields less biased estimates. However, we present an alternative solution by employing the parametric and residual bootstraps to correct for this bias in the estimates.

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