2026 Theses Doctoral

Essays on Robust Time Series Estimation and Forecasting under Outliers and Fat Tails

Kim, Hyoseok

We examine the estimation and forecasting of time series models for macroeconomic analysis in the presence of outliers and fat tails. Chapter 1 considers the decomposition of a time series into a trend, a cycle, and outliers, motivated by the importance of identifying slow-moving components in macroeconomic data. Classic approaches to trend extraction include those of Baxter and King [1], Beveridge and Nelson [2], and Hodrick and Prescott [3]. More recently, Muller and Watson [4] introduced the concept of the low-frequency trend, which is defined as the fitted value from a linear regression of a time series on a small set of low-frequency cosine functions.

While the ordinary least squares estimator is commonly used, the COVID-19 pandemic highlighted the need for estimation methods that are robust to extreme observations. Accordingly, we propose an outlier-robust low-frequency trend based on an L1-penalized least squares regression model. Following Gannaz [5], we show that the outlier-robust estimator of the regression coefficients is equivalent to the Huber M-estimator. Building on Wu [6], we derive sufficient conditions for the asymptotic normality of our estimator. As an empirical application, we decompose the levels and growth rates of U.S. real GDP and real consumption per capita into outlier-robust low-frequency trends, cycles, and outliers. In addition, we interpret the outliers in light of major historical episodes, such as the Great Depression, the World Wars, and pandemics.

Chapter 2 extends the penalized regression framework to a multivariate setting by considering vector autoregressive (VAR) models, which are widely used for macroeconomic forecasting and structural analysis. The COVID-19 pandemic has posed significant challenges for forecasting in the post-pandemic era, highlighting the need for outlier-robust forecasting models. Accordingly, we propose an outlier-robust VAR model that decomposes the reduced-form error term into a regular component and an outlier component. The VAR coefficients are estimated by imposing an L1 penalty on the outlier component, which yields an outlier-robust estimator that is equivalent to the multivariate Huber M-estimator. We then establish the asymptotic normality of the proposed outlier-robust estimator. To assess its practical relevance for improving forecast accuracy in the post-COVID period, we estimate a 5-variable VAR model using monthly data from March 1959 to June 2024 obtained from the FRED-MD database. The results show that the out-of-sample forecasts for the post-COVID period from our outlier-robust VAR are more accurate than those from the standard VAR and the outlier-augmented Bayesian VARs proposed by Carriero et al. [7].

Chapter 3 examines the performance of Bayesian shrinkage and variable-selection models in high-dimensional macroeconomic forecasting. We consider two linear models, the Bayesian Lasso [8] and the spike-and-slab [9], and a tree-based model, the Bayesian Additive Regression Trees (BART) [10]. The Bayesian Lasso model imposes a shrinkage prior on the regression coefficients, leading to a dense representation, whereas the spike-and-slab model adopts a mixture prior that explicitly allows for a sparse representation. The BART model is a non-linear, tree-based model that is flexible enough to approximate a wide range of functional forms. These models are evaluated based on their ability to forecast the one-month-ahead growth rate of U.S. industrial production using 120 monthly variables from the FRED-MD database. Among these models, the BART model exhibits the best in-sample predictive performance due to its flexible non-parametric structure. However, the linear models outperform the BART model in terms of out-of-sample forecasting. Furthermore, allowing for Student-t errors improves out-of-sample forecast performance, highlighting the importance of accounting for fat tails in macroeconomic data. These findings provide empirical guidance for model selection in the presence of model uncertainty and non-Gaussian errors.