The State Space Model (SSM) encompasses the class of multivariate linear models, in
particular, regression models with fixed, time-varying and random parameters, time series models, unobserved components models and combinations thereof. The well-known
Kalman Filter (KF) provides a unifying tool for conducting statistical inferences with
the SSM.
A major practical problem with the KF concerns its initialization when either the
initial state or the regression parameter (or both) in the SSM are diffuse. In these situations, it is common practice to either apply the KF to a transformation of the data which
is functionally independent of the diffuse parameters or else initialize the KF with an arbitrarily large error covariance matrix. However neither approach is entirely satisfactory.
The data transformation required in the first approach can be computationally tedious
and furthermore it may not preserve the state space structure. The second approach is
theoretically and numerically unsound. Recently however, De Jong (1991) has developed
an extension of the KF, called the Diffuse Kalman Filter (DKF) to handle these diffuse
situations. The DKF does not require any data transformation.
The thesis contributes further to the theoretical and computational aspects of con
ducting statistical inferences using the DKF. First, we demonstrate the appropriate initialization of the DKF for the important class of time-invariant SSM’s. This result is
useful for maximum likelihood statistical inference with the SSM. Second, we derive and
compare alternative pseudo-likelihoods for the diffuse SSM. We uncover some interesting
characteristics of the DKF and the diffuse likelihood with the class of ARMA models.
Third, we propose an efficient implementation of the DKF, labelled the collapsed DKF (CDKF). The latter is derived upon sweeping out some columns of the pertinent matrices
in the DKF after an initial number of iterations. The CDKF coincides with the KF in
the absence of regression effects in the SSM. We demonstrate that in general the CDKF
is superior in practicality and performance to alternative algorithms proposed in the literature. Fourth, we consider maximum likelihood estimation in the SSM using an EM
(Expectation-Maximization) approach. Through a judicious choice of the complete data,
we develop an CDKF-EM algorithm which does not require the evaluation of lag one
state error covariance matrices for the most common estimation exercise required for the
SSM, namely the estimation of the covariance matrices of the disturbances in the SSM.
Last we explore the topic of diagnostic testing in the SSM. We discuss and illustrate the
recursive generation of residuals and the usefulness of the latters in pinpointing likely
outliers and points of structural change.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:BVAU.2429/3318 |
| Date | 05 1900 |
| Creators | Chu-Chun-Lin, Singfat |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Relation | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
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