Dynamic Models for Multi-Dimensional Time Series
Authors:   , Wiersma Quint 
Published: March 2024

This thesis concerns combining different approaches to introduce a toolbox for econometricians to deal with real-world multi-dimensional and complex time series, furthermore, besides developing the methods we also show their theoretical validity. As some of the techniques developed in this thesis require intricate and demanding numerical algorithms to operate, a lot of attention and time is also devoted to developing efficient codebases and packages to implement the introduced tools. Proper empirical relevance does not only rely on validity and scope of the statistical tool, but also in the ease-of-use for the empirical researcher and practitioner. All in all, this thesis is a collection of three essays that provide flexible tools for handling large dimensional datasets through dimension reduction, sparse modeling, and efficient use of high-dimensional structures in the data.

In Chapter 2 we propose an efficient algorithm for estimating sparse dynamic factor models in a frequentist framework. It optimizes a penalized log-likelihood, which is obtained using the Kalman filter and smoother routines. We show that the penalized estimator has the desirable oracle property, i.e. it is consistent, selects the correct model with probability tending to one, and estimates the truly nonzero parameters as efficiently as if they had been known a priori. A Monte Carlo simulation study showcases the good properties of the estimation procedure and confirms the theoretical results. The methodology is illustrated using monthly exchange rate data with respect to the euro for 34 currencies, spanning the period 2000-2021.

Building upon the sparse dynamic factor model and the results discussed in Chapter 2, we investigate the presence of global and local spatial dependence in the spillover dynamics of the US yield curve in Chapter 3. In this chapter we propose a new unified approach to identifying and estimating spatio-temporal dependence structures in large panels. The model accommodates global cross-sectional dependence due to global dynamic factors as well as local cross-sectional dependence, which may arise from local network structures. A Monte Carlo simulation study illustrates the good performance of the algorithm in terms of determining the presence and magnitude of common factors and local spillover effects. In an empirical application, we investigate monthly US interest rate data on 12 maturities over almost 35 years.

Chapter 4 introduces a new time-varying parameter tensor autoregressive (TAR) framework to model multi-dimensional time series. Our method is able to deal with high-dimensional time series with a natural tensor representation in a structured manner by exploiting the multi-dimensional structure of data using tensor decompositions. This allows us to combine flexibility with interpretability in our modeling approach. Time-variation is build into the tensor decomposition and modeled using a latent dynamic factor model. A Monte Carlo simulation study shows the good small sample performance of our method, even in high-dimensional settings that would have been infeasible in a standard vector autoregressive framework. We apply our method to model sectoral inflation for the five biggest economies in the eurozone, and estimate a sectoral level Phillips curve.

Finally, in Chapter 5 we introduce the Julia software packages developed in combination with Chapters 2,3, and 4. ProximalMethods.jl is a package for non-smooth optimization algorithms based on proximal methods. It provides proximal operator evaluation routines and proximal optimization algorithms, such as a proximal gradient method and alternating direction method of multipliers (ADMM). The package StateSpace.jl is a package for state filtering, smoothing, and parameter estimation for linear Gaussian state space models. Finally, the package TensorAutoregressions.jl is designed to work with tensor-valued time series using the TAR model, introduced in Chapter 4, allowing for either a static or time-varying autoregressive coefficient tensor.