TESTING STRICT STATIONARITY WITH APPLICATIONS TO MACROECONOMIC TIME SERIES

• Author: Yongmiao Hong,Shouyang Wang,Xia Wang
• DOI: http://doi.org/10.1111/iere.12250
• Date: 01 November 2017
• Published date: 01 November 2017

INTERNATIONAL ECONOMIC REVIEW

Vol. 58, No. 4, November 2017

TESTING STRICT STATIONARITY WITH APPLICATIONS

TO MACROECONOMIC TIME SERIES∗

BYYONGMIAO HONG,XIA WANG,AND SHOUYANG WANG1

Cornell University, U.S.A. and Xiamen University,China; Sun Yat-Sen University, China;

Academy of Mathematics and Systems Science,Chinese Academy of Sciences, China and

University of Chinese Academy of Sciences, China

We propose a model-free test for strict stationarity. The idea is to estimate a nonparametric time-varying

characteristic function and compare it with the empirical characteristic function based on the whole sample.

We also propose several derivative tests to check time-invariant moments, weak stationarity, and pth order

stationarity. Monte Carlo studies demonstrate excellent power of our tests. We apply our tests to various

macroeconomic time series and find overwhelming evidence against strict and weak stationarity for both level

and first-differenced series. This suggests that the conventional time series econometric modeling strategies may

have room to be improved by accommodating these time-varying features.

1. INTRODUCTION

A standard modeling strategy in time series econometrics is to first remove a trend from or

take a (log-) difference of an obviously nonstationary economic time series and then employ

a stationary, linear, or nonlinear time series model (e.g., vector autoregression, autoregressive

threshold, and Markov regime switching) for the detrended or differenced series. Another

widespread modeling method for nonstationary economic time series is unit root and cointe-

gration. The popular unit root models assume that the stochastic shocks to the nonstationary

series are stationary, whereas cointegration models assume the individual series are first-order

integrated while their linear combination is stationary (Granger, 1981; Engle and Granger, 1987;

Johansen, 1991). All above modeling strategies rely on the assumption of stationarity for the

differenced series together with constant model parameters.

In time series econometrics, the assumption of stationarity provides a feasible way to infer the

dynamics of a time series by combining realizations over different time periods together. This

assumption simplifies modeling mechanisms and allows for elegant development of time series

econometric methodology. It plays a crucial role in time series inference and forecasting. For

example, many nonparametric and semiparametric estimators, such as kernel and local polyno-

mial estimators, are all constructed based on the implication of the strict stationary assumption

(Pagan and Schwert, 1990; Pagan and Ullah, 1999; Li and Racine, 2006). In addition, strict sta-

tionarity is also an indispensable assumption for the tests involving the entire probability struc-

ture (e.g., Rosenblatt, 1975; Robinson, 1991; Hong and White, 2005; Su and White, 2007, 2008).

Although stationarity is widely assumed in time series analysis, it may not be realistic when

one bases inferences on observations over a long period. A main driving force for economic

∗Manuscript received November 2014; revised May 2016.

1Xia Wang acknowledges financial supports from the National Science Foundation of China (No. 71401160), the

Project of Ministry of Education of Humanities and Social Sciences China (No. 14YJC790120), and Fujian Provincial

Key Laboratory of Statistics (Xiamen University, No. 2016003). We thank the editor Jes´

us Fern´

andez-Villaverde, two

referees, and the participants of 2014 Econometric Society China Meeting, 2015 Econometrics Conference in Honor

of Aman Ullah, and Vanderbilt University for their comments and suggestions. All remaining errors are solely our

own. Please address correspondence to: Xia Wang, Lingnan (University) College, Sun Yat-Sen University, Guangzhou

510275, China. Phone: +86 20 8411 1191. E-mail: wangxia25@mail.sysu.edu.cn.

1227

C

(2017) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1228 HONG,WANG,AND WANG

structural changes is the “shock” induced by institutional changes, such as the changes of

exchange rate systems from a fixed exchange rate mechanism to a floating exchange rate

mechanism or the introduction of the Euro. Changes induced by policy switch, preference

change, and technology progress can also cause structural changes. These changes may make the

characteristics of time series, such as moments and distributions to be time varying. For example,

Stock and Watson (1996) empirically check the stability of 76 univariate autoregressions and

5,700 bivariate relationships constituted by 76 representative U.S. monthly postwar macro-

economic time series and document significant evidence of instability in both univariate and

bivariate autoregressive models. If the stationarity condition fails, inference and forecasting

based on such an assumption may lead to misleading conclusions. Numerous empirical studies

(e.g., Goyal and Welch, 2008; Dangl and Halling, 2012) document overwhelmingly significant

in-sample evidence on predictability of asset returns but rather poor out-of-sample forecasts of

various econometric models, which is likely due to the structural instability of the underlying

economic processes. Obviously, if the underlying time series is nonstationary but we still build

models under the framework of stationarity, the resulting economic analysis and inferences

would be misleading.

Motivated by the important role of stationarity, numerous studies are devoted to test this

fundamental property. Although one can suspect before running a test that the null of station-

arity will be rejected in most cases, it is still necessary to develop some formal econometric

tools to check this assumption rigorously. On the one hand, a formal test procedure may derive

more reliable results than intuitive judgment; on the other hand, when the null of stationarity is

rejected, one may like to gauge possible reasons of rejection, which may provide some ways to

derive the stationary process. For example, if the rejections are due to the smooth and/or abrupt

structural changes of the time series, one may shorten the time periods to get a stationary time

series. There are several notions for stationarity. In the time series literature, a time series {Yt}is

called a strictly stationary process if all its finite-dimensional joint distributions are time invari-

ant, whereas it is called weakly stationary if the first two moments are time invariant. Moreover,

{Yt}is called a pth-order stationary process if its first pth-order joint product moments are time

invariant. Almost all existing tests for stationarity focus on weak stationarity instead of strict

stationarity. More specifically, the existing literature mainly focuses on the unit root process,

which is a special form of nonstationarity. They either test trend stationarity against unit root

(e.g., Kwiatkowski et al., 1992; Bierens, 1993; Bierens and Guo, 1993; Xiao and Lima, 2007)

or test unit root against trend stationarity (e.g., Dickey and Fuller, 1979, 1981; Phillips, 1987;

Phillips and Perron, 1988). Recently, Dette et al. (2011) and Preuß et al. (2013) propose some

model-free tests for weak stationarity by measuring the L2and Kolmogorov–Smirnov type

distances between a time-varying spectral density estimator and a spectral density estimator

based on the whole sample. Although the existing studies have achieved fruitful achievements

on testing weak stationarity, the literature on testing strict stationarity is still deficient. In linear

time series analysis, weak stationarity is a most suitable concept in most cases. However, when

we turn to a nonlinear time series framework, the concept of strict stationarity is more useful, as

the first two moments are insufficient to characterize the full dynamics and nonlinear features

of a time series. With the development and prevalence of nonlinear modeling, strict stationarity

becomes a maintained assumption. Testing strict stationarity is more challenging. To our

knowledge, there are only a few papers that consider testing strict stationarity in the literature.

Kapetanios (2009) tests the strict stationarity assumption by examining the time-invariance

property of a nonparametric marginal density estimator based on a recursive method. Busetti

and Harvey (2010) propose a test for strict stationarity based on a quantile indicator. To execute

their test, one should compute the test statistic for each quantile τ∈(0,1). Recently, Francq

and Zako¨

ıan (2012) propose a test for strict stationarity within the Generalized Autoregressive

Conditional Heteroskedasticity (GARCH) framework.

In this article, we propose a model-free test for strict stationarity of a possibly vector-valued

time series. Assuming that the finite-dimensional distribution of a time series is a smooth

function of time, we estimate the time-varying characteristic function by local smoothing and

TESTING STRICT STATIONARITY 1229

compare it with the empirical characteristic function based on the whole sample. By construc-

tion, our test is most powerful against smooth or evolutionary structural changes in distribution.

This is in contrast to most tests of structural breaks in the econometric literature, which only

focus on abrupt structural breaks. Smooth structural changes are more realistic in reality. As

Hansen (2001) points out, “it may seem unlikely that a structural break could be immediate

and might seem more reasonable to allow a structural change to take a period of time to

take effect.” In addition to smooth structural changes in finite-dimensional distributions of

the time series, our test is also able to detect a finite number of abrupt changes (i.e., struc-

tural breaks) in distribution, with possibly unknown break dates or the number of breaks.

Our test complements the existing tests for strict stationarity and has a number of appealing

features.

First of all, our test is constructed by examining the time invariance property of the finite-

dimensional distribution of a time series instead of only its marginal distribution. Given a

vector valued time series {Yt}and any finite time points (t1,t2,...,tm), we check whether the

joint distribution of (Yt1,Yt2,...,Ytm) is time invariant. A special case of m=1 checks the time

invariant property of the marginal distribution, which is in spirit similar to the null hypothesis

of Kapetanios (2009) and Busetti and Harvey (2010). By choosing mlarger than 1, our test then

checks the time invariance property of a joint distribution and hence could capture a wider

range of deviations from strict stationarity, such as those with a time-varying joint distribution

but time-invariant marginal distributions.

Second, our test could detect a class of local alternatives of smooth distributional changes with

convergence rate T−1/2h−1/4(where his a bandwidth), which depends on neither the dimension

of Ytnor the number of time points checked. Unlike Kapetanios’ (2009) test, which is based on

the marginal density estimator using a recursive method, i.e., estimating the density ft(y)atYt

using the first tobservations, we first take the Fourier transform of a joint finite-dimensional

distribution of the time series and then estimate the time-varying characteristic function by local

smoothing. Thanks to the use of the characteristic function and local regression method, our

test could detect a class of local alternatives with convergence rate T−1/2h−1/4and thus avoids

the potential “curse of dimensionality” problem associated with the dimension of Ytand the

number of time points checked.

Third, unlike Francq and Zako¨

ıan’s (2012) test for strict stationarity, we do not need to impose

any parametric restrictions on the form of alternatives. As a result, our test not only avoids the

misspecification problem but also is powerful in detecting various kinds of nonstationarity, such

as abrupt and smooth changes in various moments, as well as unit root and trend stationarity.

Fourth, by using a multivariate normal weighting function, we avoid the intractable high

dimensional integrations. Unlike Sz´

ekely et al.’s (2007) nonintegrable weighting function, the

normal density function satisfies our regularity conditions, particularly the integrability con-

dition. Simulation studies show that our test with the normal weighting function performs

reasonably well.

Fifth, our test is flexible in gauging possible sources of nonstationarity. By differentiating

our test for strict stationarity with respect to auxiliary parameters at the origin up to various

orders, we obtain a class of derivative tests for the time invariant property of various moments.

The derivative tests could be used to test weak stationarity and, more generally, the pth order

stationarity.

Finally, all the proposed test statistics follow a convenient null asymptotic N(0,1) distribution.

The limiting distributions of most existing tests for stationarity are nonstandard, and one has to

simulate critical values in practice. In contrast, it is easy to implement our tests by comparing

the test statistics with the one-sided N(0,1) critical value zαat significant level α.

In an empirical study, we apply our tests to various macroeconomic time series. We docu-

ment overwhelming evidence of nonstationarity for both level and differenced series. And, our

derivative tests document significant evidence of smoothly changing means and/or variances

for the differenced series. These new findings suggest that the current time series modeling

strategies, namely, modeling nonstationary economic time series by fitting a stationary linear