Prelims

Part I: Nonstationarity, Unit Roots, and Fractional Noise

A contribution by Peter C. B. Phillips, not only Joon’s dissertation advisor but also longtime editor of Econometric Theory, appropriately opens the volume on econometric theory and the section on nonstationarity, unit roots, and fractional noise. Specifically, his article “Discrete Fourier Transforms of Fractional Processes With Econometric Applications” presents an exact representation of the discrete Fourier transform in terms of the component data, which he finds to be particularly useful for analyzing the asymptotic behavior of the periodogram when the memory parameter exceeds the threshold for stationarity. He shows that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin as long as the memory parameter is strictly less than unity.

Also studying fractional noise, Xiaohu Wang, Weilin Xiao, and Jun Yu contribute the article “Asymptotic Properties of the Least Squares Estimator in Local to Unity Processes With Fractional Gaussian Noises.” They derive the asymptotic properties of the autoregressive parameter in local to unity processes with errors generated as fractional Gaussian noise with the Hurst parameter over the interval (0,1). The rates of convergence are standard rate-T over the upper half of this interval, but nonstandard and dependent of the Hurst parameter over the lower half. They derive limiting distributions over this interval that are new to the literature except at 1/2.

Critical to ascertaining stationarity or lack thereof are unit root tests. In their contribution, “Powerful Self-normalizing Tests for Stationarity Against the Alternative of a Unit Root,” Uwe Hassler and Mehdi Hosseinkouchack introduce a new and powerful tool to address this well-known problem. Specifically, they propose a family of tests for stationarity against a local unit root that builds on the Karhunen–Loève expansions of the limiting CUSUM process under the null hypothesis and a local alternative. They find that the proposed tests are more powerful than the classic KPSS test.

Also on the topic of testing for unit roots, Kohtaro Hitomi, Keiji Nagai, Yoshihiko Nishiyama, and Junfan Tao contribute “A Sequential Test for a Unit Root in Monitoring a p-th Order Autoregressive Process.” They study unit root tests for autoregressive processes of order p under sequential sampling schemes using stopping times based on the observed Fisher information. They derive the joint limit of the test statistics and the stopping time under the null and local alternatives, which are nonstandard.

Part II: Nonlinearity

As we mentioned, both cointegration and functional coefficients are areas in which Professor Park has made contributions to the literature. Han-Ying Liang, Yu Shen, and Qiying Wang contribute to the volume and this literature with “Functional-coefficient Cointegrating Regression With Endogeneity.” As the title suggests, they explore nonparametric estimation of cointegrating regression models with functional coefficients and where the structural equation errors are serially dependent and the regressor is endogenous. In this context, they show the self-normalized local kernel and local linear estimators to be asymptotically normal.

In “A Specification Test Based on Convolution-type Distribution Function Estimates for Non-linear Autoregressive Processes,” Kun Ho Kim, Hira L. Koul, and Jiwoong Kim develop a test for a parametric specification of the autoregressive function of a given stationary autoregressive time series. Their test is based on the integrated square difference between the empirical distribution function estimate and a convolution-type distribution function estimate of the stationary distribution function obtained from the autoregressive residuals.

Yingqian Lin and Yundong Tu contribute “Transformation Models With Cointegrated and Deterministically Trending Regressors,” which contains important and interesting extensions of the statistical foundation for the nonlinear cointegrated models pioneered by Park and his coauthors. For a general transformation model with a time trend, stationary regressors, and unit root regressors, they estimate the transformation parameter and other model parameters by minimizing the concentrated loss function, and they obtain the asymptotic distributions of the proposed estimators.

The threshold model has been frequently used to model the nonlinearity of time series. Park and Shintani (2016) examine testing issues surrounding threshold effects and unit roots. In “Minimax Risk in Estimating Kink Threshold and Testing Continuity,” Javier Hidalgo, Heejun Lee, Jungyoon Lee, and Myung Hwan Seo derive a risk lower bound in estimating the threshold parameter without knowing whether the threshold regression model is continuous or not. They show that the bound goes to zero as the sample size grows only at the cube root rate. Motivated by this finding, they develop a continuity test for the threshold regression model and a bootstrap to compute its p-values.

Part III: Inference and Prediction Using Models With Trending Series

Articles in the final section of this volume deal with models containing stochastic and/or deterministic trends, as do many of Professor Park’s papers, from his earliest work on cointegration (Park & Phillips, 1988, 1989) and his widely cited CCR paper (Park, 1992) through his more recent work, such as that on estimating stochastic trends in state-space models (Chang et al., 2009).

In the first of these, “Semiparametric Independence Tests Between Two Infinite-order Cointegrated Series,” Chafik Bouhaddioui, Jean-Marie Dufour, and Masaya Takano propose a semiparametric approach for testing independence between two cointegrated vector autoregressive series of infinite order. The residual-based tests allow for computational simplicity and weak assumptions on the form of the underlying process. The authors derive the asymptotic distributions of the test statistics under the null hypothesis and establish consistency of the tests against fixed alternatives of serial cross-correlation of unknown form.

Nikolay Gospodinov, Alex Maynard, and Elena Pesavento contribute “Inference in Conditional Vector Error Correction Models With a Small Signal-to-Noise Ratio,” in which they study vector error correction models when the error correction term is characterized simultaneously by high persistence (near-unit-root behavior) and very small (near zero) variance. The importance of these features lies in the fact that conventional cointegration tests may fail to detect cointegration. The authors develop asymptotic theory for the parameter estimators for unconditional and conditional vector error correction models with these features.

Yixiao Sun, in his contribution entitled “Some Extensions of Asymptotic F and t Theory in Nonstationary Regressions,” extends the asymptotic theory for F- and t-tests to linear regression models where the regressors could contain deterministic trends, unit-root processes, and near-unit-root processes. The tests themselves are implemented in the usual ways, but approximations to the limiting distributions are more accurate than the more commonly used chi-squared and normal approximations.

The last two contributions focus on predictive models with nonstationary series. Ying Zhou, Hsein Kew, and Jiti Gao contribute “Non-stationary Parametric Single-index Predictive Models: Simulation and Empirical Studies.” Their model is designed to handle a wide variety of nonlinear relationships between the regressand and a single-index component containing either the cointegrated predictors or the non-cointegrated predictors. They introduce a new estimation procedure and investigate its finite-sample properties.

We opened the volume with a contribution from Joon’s advisor, so it seems appropriate to close the volume with a contribution from one of his many students. In “Best Linear Prediction in Cointegrated Systems,” Yun-Yeong Kim introduces the best linear predictor with the asymptotic minimum mean squared forecasting error among linear predictors of variables in cointegrated systems with unknown error specification. He suggests a switching predictor that automatically selects the random walk or cointegration model according to the size of the estimated autocorrelation coefficient estimated from the residuals.

We hope you enjoy reading “Essays in Honor of Joon Y. Park: Econometric Theory” and learning about the advances in econometrics made by the authors as much as we have!