| Summary: | This dissertation addresses various issues related to statistical inference in the
context of parameter time-variation. The problem is considered within general regression
models as well as in the context of methods for forecast evaluation.
The first chapter develops a theory of evolutionary spectra for heteroskedasticityand
autocorrelation-robust (HAR) inference when the data may not satisfy secondorder
stationarity. We introduce a class of nonstationary stochastic processes that
have a time-varying spectral representation and presents a new positive semidefinite
heteroskedasticity- and autocorrelation consistent (HAC) estimator. We obtain an
optimal HAC estimator under the mean-squared error (MSE) criterion and show its
consistency. We propose a data-dependent procedure based on a “plug-in” approach
that determines the bandwidth parameters for given kernels and a given sample size.
The second chapter develops a continuous record asymptotic framework to build
inference methods for the date of a structural change in a linear regression model.
We impose very mild regularity conditions on an underlying continuous-time model assumed to generate the data. We consider the least-squares estimate of the break
date and establish consistency and convergence rate. We provide a limit theory for
shrinking magnitudes of shifts and locally increasing variances.
The third chapter develops a novel continuous-time asymptotic framework for
inference on whether the predictive ability of a given forecast model remains stable
over time. As the sampling interval between observations shrinks to zero the
sequence of forecast losses is approximated by a continuous-time stochastic process
possessing certain pathwise properties. We consider an hypotheses testing problem
based on the local properties of the continuous-time limit counterpart of the sequence
of losses.
The fourth chapter develops a class of Generalized Laplace (GL) inference methods
for the change-point dates in a linear time series regression model with multiple
structural changes. The GL estimator is defined by an integration rather than
optimization-based method and relies on the least-squares criterion function. On the
theoretical side, depending on some smoothing parameter, the class of GL estimators
exhibits a dual limiting distribution; namely, the classical shrinkage asymptotic
distribution of the least-squares estimator, or a Bayes-type asymptotic distribution.
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