Econometrics
Applied Econometrics
Macroeconomics

Econometrics
Applied Econometrics
Macroeconomics

(Oral) May 1999, Econometrics (with distinction) and Macroeconomics
(Written) May 1998, Microeconomics and Macroeconomics

  Modeling and Testing Nonlinearity with Nonstationary Time Series

Professor Peter C.B. Phillips
Professor Donald W.K. Andrews
Professor Guido Kuersteiner

May 2004

M.Phil., Economics, Yale University, May 2001
M.A., Economics, Yale University, November 1999
B.A., summa cum laude, Economics, Seoul National University, February 1996

Dissertation Fellowship, Yale University, 2002
Cowles Foundation Prize, Yale University, 1999, 2000
Japan-IMF Scholarship for Advanced Studies, 1998-1999
Scholarship for Honors, Seoul National University, 1990-1991, 1994-1995
Dan-Am Scholarship, Seoul National University, 1991, 1994-1995

Teaching Assistant, Econometrics II (Graduate course), Yale University, Spring 2001, Spring 2002
Teaching Assistant, Introduction to Probability and Statistics (Undergraduate course), Yale University, Fall 2003
Teaching Assistant, Econometrics and Data Analysis (Undergraduate course), Yale University, Fall 1999
Teaching Assistant, Intermediate Macroeconomics (Undergraduate course), Yale University, Spring 2000, Fall
   2001
Teaching Assistant, Introductory Macroeconomics (Undergraduate course), Yale University, Fall 2000
Teaching Assistant, Applied Econometrics (Graduate course), Seoul National University, Fall 1996
Teaching Assistant, Advanced Econometrics (Graduate course), Seoul National University, Spring 1997
Teaching Assistant, Econometrics (Undergraduate course), Seoul National University, Spring 1996

Summer Intern at the International Monetary Fund (EU1 department, division of European policy). Research
     assistant to Zenon Kontolemius and Kevin Ross for inflation rate forecasting in EURO area, Summer 2000
Research Assistant to Professor Hak Pyo and Professor Joon Park, Seoul National University, for a project on
     forecasting electricity demand in Korea, 1997
Research Assistant at the Institute of World Economy at Seoul National University, 1997
Research Assistant to Professor Hak Pyo, Seoul National University, for a project on constructing the annual
     capital stock series of each industry in Korea, 1996

"Testing Linearity in Cointegrating Relations: Application to PPP," mimeo, Yale University, 2003 [Job Market Paper]

"Spurious Regression of Another Kind: Stationary Series with Structural Breaks," mimeo, Yale University, 2002

"Implications of Nonlinear Processes for Linear Regression Analysis: ARCH and Bilinear Models," Work in progress

"Testing Linear Cointegration against Both Nonlinear Cointegration and No Cointegration," Work in progress

"PPP, Reconsidering Linear Specification: European Countries Case," Work in progress

Econometric Theory

Economic time series are often believed to exhibit nonlinear behavior. Economists usually formulate this nonlinearity in two ways: 1) by building nonlinear dynamics into the time series model; or 2) by allowing for nonlinear relationships among different linear time series. While numerous nonlinear models (and almost as many specification tests) have been developed and applied to analyzing economic time series, most applications have been confined to the analysis of stationary time series, and little is known about the properties of nonlinear, nonstationary time series. This dissertation seeks to fill this gap in the literature by investigating certain issues of nonlinearity in nonstationary time series. In particular, we focus on the implications of nonlinear processes for linear regression analysis and the problems of testing (non) linear specification between nonstationary time series.

We first investigate nonlinear cointegrating relationships from a model specification perspective. Despite the fact that many economic theories contain nonlinear elements, the linear model has been the predominant approach to formulate most of the testable implications from these theories. This is partly because linear models exhibit relatively clear and simple statistical analyses but also because they can often provide reasonable approximations for nonlinear models of stationary time series. Unlike mean-reverting stationary processes, nonstationary time series tend to wander randomly or even diverge, so that such a linear local approximation to a nonlinear model produces a high risk of faulty inference from the misspecified model, especially in the case of misspecified long-run relationships. Despite this shortcoming of the linear model of nonstationary time series, the importance of correct specification of cointegrating relations seems to have been under-appreciated in the literature.

Accordingly, we develop and present a linearity test that can be applied to cointegrating relations. Considering the possibility of nonlinearity in cointegrating relations gives rise to three possible cases: linear cointegration, nonlinear cointegration, and absence of cointegration. Since existing cointegration tests do not effectively distinguish between linear and nonlinear cointegration, linear cointegration analysis requires a further specification test to address the issue. Existing linearity tests, however, fail to provide guidance concerning the type of relationship between nonstationary time series as well. In the absence of dual specification tests, economists have frequently used the existing cointegration tests as linear cointegration tests. We consider two widely used linearity tests, based on the general approximation theory: the regression error specification test (RESET) and neural network (NN) test. We show that these tests, when applied to nonstationary time series, follow asymptotic noncentral chi² distributions and the noncentrality brings severe size distortions. These tests arise from estimated nonlinear cointegration regressions and involve evaluating the sample moments of nonlinearly transformed time series. We show that nonstationarity introduces two bias terms in the limit distribution of the sample covariance, and these terms are the main source of the big size distortions, leading to noncentral chi² distributions. By providing the appropriate corrections for these asymptotic bias terms, we present modified tests and prove that these tests follow asymptotic central chi² distribution under linearity. We also show that these modified tests have power against not only nonlinear cointegration but also the case of no cointegration. Therefore, these modified tests can be used as omnibus tests for both cointegration and linearity. Simulation results show that the modified test has correct size asymptotically and has good power against many nonlinear models as well as no cointegration cases, confirming our analytic results.

The modified test is applied in an empirical study to access evidence to support the linear specification of Purchasing Power Parity (PPP). PPP is a clear, intuitively appealing economic hypothesis but it has had little empirical support so far. Without considering nonlinear cointegration, this lacking empirical evidence invalidates the PPP hypothesis, while this may be simply due to misspecified cointegration model. We consider two groups of countries: US, Canada and Japan in the first group and selected European countries in the second group. Existing cointegration tests (ADF and PP tests) yield mixed results, so that the linear cointegration model is not clearly supported. Existing linearity tests cannot confirm linear cointegration specification either. However, when we apply our modified test to the absolute and relative forms of PPP (using both CPI and PPI), the test rejects linear cointegration in favor of nonlinear cointegration for most cases. Test results show that, in general, the PPP relationship with PPI tends to be closer to a linear model than the one using CPI, and the pair of countries with lower trade barriers show their PPP relationships closer to be linear, for example, US-Canada vs. US-Japan and Germany-Spain vs. Germany-UK.

The second major issue investigated is how nonlinearity in time series manifests itself when linear regressions are employed. We start with one of the simplest but widely analyzed nonlinear processes: a stationary time series around a deterministic trend with breaks. Spurious regression arises in the linear regression of this process on a set of independent I(1) processes. Most existing spurious regression results apply to regressions among nonstationary integrated processes. We illustrate that nonlinearity constitutes another significant source of (spurious regression) failure in linear regression analysis. The asymptotics of regression statistics such as t-ratios, F statistics, and R squared are shown to be the same as in typical spurious regression. We investigate the mathematical representation behind these spurious regressions, as in Phillips (1996, 1998). Because of the Gibbs phenomenon, we develop another mathematical representation (in place of Fourier series) of an integrated process using Gegenbauer polynomials. Based on this representation, we derive a Wiener process version of the Weierstrass theory that extends Phillips’s results to the class of nonperiodic processes with the possibility of break(s). This result illuminates that there exists a clear mathematical relationship behind this type of spurious regression and the regression statistics correctly reflect its existence.

In addition, we explore the case of linear regression between nonlinear time series. Two, more elaborate nonlinear processes are considered in particular: ARCH (nonlinear in variance) and bilinear (nonlinear in mean) models. We examine the effects of nonlinearity on linear regression in terms of the coefficient estimate and its t-ratio. First, we show that adding nonlinearity to the two independent I(1) processes tends to reduce the chance of spurious linear correlation, thereby extending Subba Rao & Terdik's (2000) simulation results on bilinear processes. Additionally, we find another effect of nonlinearity working in the opposite direction that increases the likelihood of spurious linear correlation between two independent processes. We give an analytic explanation to these correcting and corrupting effects of nonlinearity in relation to the degree of nonlinearity of the process.