Essays on nonstationary econometrics - LIU, Yanbo (Institutional Knowledge at Singapore Management University, 2020)
My dissertation consists of three essays that contribute new theoretical results to robust inference procedures and machine learning algorithms in nonstationary models. Chapter 2 compares OLS and GLS in autoregressions with integrated noise terms. Grenander and Rosenblatt (2008) gave sufficient conditions for the asymptotic equivalence of GLS and OLS in deterministic trend extraction. However when extending to univariate autoregression model yt = ρnyt−1 + ut , ρn = 1 + c nα , ut = ut−1 + t , and t is one iid disturbance term with zero expectation and σ 2 variance, the asymptotic equivalence no longer holds. Under the mildly explosive (c > 0, α ∈ (0, 1)) and pure explosive (c > 0, α = 0) cases, the limiting distributions of OLS and GLS estimates are identical as standard Cauchy distribution, and the OLS estimate has a slower convergence rate. Under the mildly stationary (c < 0, α ∈ (0, 1)) case, the limiting distribution of OLS is degenerate centered at −c, while the GLS estimate is Gaussian distributed. Under the local to unity (α = 1) case, when c ≥ c ∗ , the mean and variance of the asymptotic distribution of the OLS estimate are smaller than the GLS estimate, showing the efficiency gains in OLS. Chapter 3 proposes novel mechanisms for identifying explosive bubbles in panel autoregressions with a latent group structure. Two post-classification panel data approaches are employed to test the explosiveness in time-series data. The first approach applies a recursive k-means clustering algorithm to explosive panel autoregressions. The second approach uses a modified k-means clustering algorithm for mixed-root panel autoregressions. We establish the uniform consistency of both clustering algorithms. The abovementioned k-means procedures achieve the oracle properties so that the post-classification estimators are asymptotically equivalent to the infeasible estimators that use the true group identities. Two right-tailed t-statistics, based on post-classification estimators, are introduced to detect explosiveness. A panel recursive procedure is proposed to estimate the origination date of explosiveness. The asymptotic theory is available for concentration inequalities, clustering algorithms, and right-tailed t-tests based on mixed-root panels. Extensive Monte Carlo simulations provide strong evidence that the proposed panel approaches lead to substantial power gains compared with the time-series approach. Chapter 4 explores predictive regression models with stochastic unit root (STUR) components and robust inference procedures that encompass a wide class of persistent and time-varying stochastically nonstationary regressors. The paper extends the mechanism of endogenously generated instrumentation known as IVX, showing that these methods remain valid for short- and long-horizon predictive regressions in which the predictors have STUR and local STUR (LSTUR) generating mechanisms. Both mean regression and quantile regression methods are considered. The asymptotic distributions of the IVX estimators are new compared to previous work but again lead to pivotal limit distributions for Wald testing procedures that remain robust for both single and multiple regressors with various degrees of persistence and stochastic and fixed local departures from unity. Numerical experiments corroborate the asymptotic theory, and IVX testing shows good power and size control. The new methods are illustrated in an empirical application to evaluate the predictive capability of economic fundamentals in forecasting excess returns in the Dow Jones industrial average index.