A Unit Root Test Using a Fourier Series to Approximate Smooth Breaks*

We develop a unit‐root test based on a simple variant of Gallant's (1981) flexible Fourier form. The test relies on the fact that a series with several smooth structural breaks can often be approximated using the low frequency components of a Fourier expansion. Hence, it is possible to test for a unit root without having to model the precise form of the break. Our unit‐root test employing Fourier approximation has good size and power for the types of breaks often used in economic analysis. The appropriate use of the test is illustrated using several interest rate spreads.     

Does real wage converge in China?

This study adopts flexible Fourier unit root test proposed by Enders and Lee (Oxf Bull Econ Stat 74(4):574, 2012) to explore real wage convergence in China in the process of market-oriented reform. We find that our approximation has higher power to detect U-shaped and smooth breaks than linear method if the true data generating process of relative real wage is in fact a stationary non-liner process. The empirical results show that property of stationarity of relative real wage varies across different regions. Specifically, stationarity prevails in eastern and western regions, implying the regional development strategies launched there enhance labor migration, commodity trade and thus wage convergence. In contrast, almost all provinces in northeastern and central regions show non-stationarity, indicating the strategies there are less effective in promoting wage convergence. These results have practical policy implications for China’s regional development and income equality.     

Panel Unit‐root Tests for Cross‐sectionally Correlated Panels: A Monte Carlo Comparison*

This paper deals with the finite‐sample performance of a set of unit‐root tests for cross‐correlated panels. Most of the available macroeconomic time series cover short time periods. The lack of information, in terms of time observations, implies that univariate tests are not powerful enough to reject the null of a unit‐root while panel tests, by exploiting the large number of cross‐sectional units, have been shown to be a promising way of increasing the power of unit‐root tests. We investigate the finite sample properties of recently proposed panel unit‐root tests for cross‐sectionally correlated panels. Specifically, the size and power of Choi's [Econometric Theory and Practice: Frontiers of Analysis and Applied Research: Essays in Honor of Peter C. B. Phillips, Cambridge University Press, Cambridge (2001)], Bai and Ng's [Econometrica (2004), Vol. 72, p. 1127], Moon and Perron's [Journal of Econometrics (2004), Vol. 122, p. 81], and Phillips and Sul's [Econometrics Journal (2003), Vol. 6, p. 217] tests are analysed by a Monte Carlo simulation study. In synthesis, Moon and Perron's tests show good size and power for different values of T and N, and model specifications. Focusing on Bai and Ng's procedure, the simulation study highlights that the pooled Dickey–Fuller generalized least squares test provides higher power than the pooled augmented Dickey–Fuller test for the analysis of non‐stationary properties of the idiosyncratic components. Choi's tests are strongly oversized when the common factor influences the cross‐sectional units heterogeneously.     

Exact Skewness–Kurtosis Tests for Multivariate Normality and Goodness‐of‐Fit in Multivariate Regressions with Application to Asset Pricing Models*

We study the problem of testing the error distribution in a multivariate linear regression (MLR) model. The tests are functions of appropriately standardized multivariate least squares residuals whose distribution is invariant to the unknown cross‐equation error covariance matrix. Empirical multivariate skewness and kurtosis criteria are then compared with a simulation‐based estimate of their expected value under the hypothesized distribution. Special cases considered include testing multivariate normal and stable error distributions. In the Gaussian case, finite‐sample versions of the standard multivariate skewness and kurtosis tests are derived. To do this, we exploit simple, double and multi‐stage Monte Carlo test methods. For non‐Gaussian distribution families involving nuisance parameters, confidence sets are derived for the nuisance parameters and the error distribution. The tests are applied to an asset pricing model with observable risk‐free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over 5‐year subperiods from 1926 to 1995.     

Testing for unit roots in the presence of uncertainty over both the trend and initial condition

In this paper we provide a joint treatment of two major problems that surround testing for a unit root in practice: uncertainty as to whether or not a linear deterministic trend is present in the data, and uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. We suggest decision rules based on the union of rejections of four standard unit root tests (OLS and quasi-differenced demeaned and detrended ADF unit root tests), along with information regarding the magnitude of the trend and initial condition, to allow simultaneously for both trend and initial condition uncertainty.     

Further Evidence on the Statistical Properties of Real GNP*

The well‐known lack of power of unit‐root tests has often been attributed to the short length of macroeconomic variables and also to data‐generating processes (DGPs) departing from the I(1)–I(0) models. This paper shows that by using long spans of annual real gross national product (GNP) and GNP per capita (133 years), high power can be achieved, leading to the rejection of both the unit‐root and the trend‐stationary hypothesis. More flexible representations are then considered, namely, processes containing structural breaks (SB) and fractional orders of integration (FI). Economic justification for the presence of these features in GNP is provided. It is shown that both FI and SB formulations are in general preferred to the autoregressive integrated moving average (ARIMA) [I(1) or I(0)] formulations. As a novelty in this literature, new techniques are applied to discriminate between FI and SB. It turns out that the FI specification is preferred, implying that GNP and GNP per capita are non‐stationary, highly persistent but mean‐reverting series. Finally, it is shown that the results are robust when breaks in the deterministic component are allowed for in the FI model. Some macroeconomic implications of these findings are also discussed.     

New unit root asymptotics in the presence of deterministic trends

Recent work by Phillips (Econometrica 66 (1998) 1299) has shown that stochastic trends can be validly represented in empirical regressions in terms of deterministic functions of time. These representations offer an alternative mechanism for modelling stochastic trends. It is shown here that the alternate representations affect the asymptotics of all commonly used unit root tests in the presence of trends. In particular, the critical values of unit root tests diverge when the number of deterministic regressors K→∞ as the sample size n→∞. When they are appropriately recentered and standardized, unit root limit distributions are shown to be normal as K→∞.     

Evidence on purchasing power parity from univariate models: the case of smooth transition trend‐stationarity

Recent research has found that trend‐break unit root tests derived from univariate linear models do not support the hypothesis of long‐run purchasing power parity (PPP) for US dollar real exchange rates. In this paper univariate smooth transition models are utilized to develop unit root tests that allow under the alternative hypothesis for stationarity around a gradually changing deterministic trend function. These tests reveal statistically significant evidence against the null hypothesis of a unit root for the real exchange rates of a number of countries against the US dollar. However, restrictions consistent with long‐run PPP are rejected for some of the countries for which a rejection of the unit root hypothesis is obtained. Copyright © 2005 John Wiley & Sons, Ltd.     

Asymmetry and nonstationarity for a seasonal time series model

Tests for symmetry and seasonal unit roots are developed for an extended model of Hylleberg et al. (1990. Seasonal integration and cointegration. Journal Econometrics 44, 215–238.) which can represent both partial seasonal unit roots and threshold effects. Methods based on ordinary least squares (OLS) estimation and instrumental variable (IV) estimation are proposed and compared. For adjusting mean functions, ordinary mean adjustment and recursive mean adjustment are both considered. Several tests are constructed from various combination of estimation schemes and mean adjustment schemes. Among the tests, the tests based on IV-estimation are recommended because they have very simple limiting null distributions and have finite sample power properties comparable to those based on the OLSE. The recommended tests are applied to a US unemployment rate data set and find evidences for both nonstationarities associated with zero frequency and threshold effects.     

Dickey–Fuller Type of Tests against Nonlinear Dynamic Models*

In this paper, we introduce several test statistics testing the null hypothesis of a random walk (with or without drift) against models that accommodate a smooth nonlinear shift in the level, the dynamic structure and the trend. We derive analytical limiting distributions for all the tests. The power performance of the tests is compared with that of the unit‐root tests by Phillips and Perron [Biometrika (1988), Vol. 75, pp. 335–346], and Leybourne, Newbold and Vougas [Journal of Time Series Analysis (1998), Vol. 19, pp. 83–97]. In the presence of a gradual change in the deterministics and in the dynamics, our tests are superior in terms of power.     

Practical Considerations for Choosing Between Tobit and SCLS or CLAD Estimators for Censored Regression Models with an Application to Charitable Giving*

Practical considerations for choosing between Tobit, symmetrically censored least squares (SCLS) and censored least absolute deviations (CLAD) estimators are offered. Practical considerations deal with when a Hausman test is better than a conditional moment test for judging the severity of a misspecification, the need to bootstrap the sampling distributions of the Hausman tests, what to look for in a graphical examination of the residuals and the limited value of SCLS. The practical considerations are applied to a model of the intergenerational transmission of charitable giving using new data from the Panel Study of Income Dynamics (PSID). The paper shows how to use relative distribution methods to calculate CLAD‐based marginal effects on the observable dependent variable.     

Levels,Differences and ECMs – Principles for Improved Econometric Forecasting*

Unit‐root testing can be a preliminary step in model development, an intermediate step, or an end in itself. Some researchers have questioned the value of any unit‐root and cointegration testing, arguing that restrictions based on theory are at least as effective. Such confusion is unsatisfactory. Needed is a set of principles that limit and define the role of the tacit knowledge of the model builders. In a forecasting context, we enumerate the various possible model selection strategies and, based on simulation and empirical evidence, recommend using these tests to improve the specification of an initial general vector autoregression model.     

HEGY Tests in the Presence of Moving Averages*

We analyze the asymptotic distributions associated with the seasonal unit root tests of Hylleberg et al. (1990) for quarterly data when the innovations follow a moving average process. Although both the t‐ and F‐type tests suffer from scale and shift effects compared with the presumed null distributions when a fixed order of autoregressive augmentation is applied, these effects disappear when the order of augmentation is sufficiently large. However, as found by Burridge and Taylor (2001) for the autoregressive case, individual t‐ratio tests at the semi‐annual frequency are not pivotal even with high orders of augmentation, although the corresponding joint F‐type statistic is pivotal. Monte Carlo simulations verify the importance of the order of augmentation for finite samples generated by seasonally integrated moving average processes.     

Detection of Structural Change in the Long‐run Persistence in a Univariate Time Series*

In this paper, we investigate a test for structural change in the long‐run persistence in a univariate time series. Our model has a unit root with no structural change under the null hypothesis, while under the alternative it changes from a unit‐root process to a stationary one or vice versa. We propose a Lagrange multiplier‐type test, a test with the quasi‐differencing method, and ‘demeaned versions’ of these tests. We find that the demeaned versions of these tests have better finite‐sample properties, although they are not necessarily superior in asymptotics to the other tests.     

Prewhitening Bias in HAC Estimation*

Heteroskedasticity and autocorrelation consistent (HAC) estimation commonly involves the use of prewhitening filters based on simple autoregressive models. In such applications, small sample bias in the estimation of autoregressive coefficients is transmitted to the recolouring filter, leading to HAC variance estimates that can be badly biased. The present paper provides an analysis of these issues using asymptotic expansions and simulations. The approach we recommend involves the use of recursive demeaning procedures that mitigate the effects of small‐sample autoregressive bias. Moreover, a commonly used restriction rule on the prewhitening estimates (that first‐order autoregressive coefficient estimates, or largest eigenvalues, >0.97 be replaced by 0.97) adversely interferes with the power of unit‐root and [ Kwiatkowski, Phillips, Schmidt and Shin (1992) Journal of Econometrics, Vol. 54, pp. 159–178] (KPSS) tests. We provide a new boundary condition rule that improves the size and power properties of these tests. Some illustrations of the effects of these adjustments on the size and power of KPSS testing are given. Using prewhitened HAC estimates and the new boundary condition rule, the KPSS test is consistent, in contrast to KPSS testing that uses conventional prewhitened HAC estimates [ Lee, J. S. (1996) Economics Letters, Vol. 51, pp. 131–137].     

Testing for a unit root in a random coefficient panel data model

This paper proposes new unit root tests in the context of a random autoregressive coefficient panel data model, in which the null of a unit root corresponds to the joint restriction that the autoregressive coefficient has unit mean and zero variance. The asymptotic distributions of the test statistics are derived and simulation results are provided to suggest that they perform very well in small samples.     

考虑结构突变的单位根检验程序研究——基于“新息异常值模型”的Perron检验分析

Perron检验是一种考虑结构突变的单位根检验方法,检验统计量的分布依赖于数据生成过程中所包含的确定性趋势和所选取的检验回归式;而在实证分析中真实的数据生成过程是未知的,这使得单位根检验缺乏必要依据,因而探寻科学有效的单位根检验程序是受到广泛关注的问题。基于此,本文在"IO模型"分析框架下,依据Perron检验提出了一套考虑结构突变的单位根检验程序,并通过蒙特卡洛模拟分析了该程序在有限样本情形下的表现。本研究完善了带有结构突变的单位根检验理论,为实证分析提供了有益的建议和参考。     

Power of Tests for Unit Roots in the Presence of a Linear Trend*

Dickey and Fuller [Econometrica (1981) Vol. 49, pp. 1057–1072] suggested unit‐root tests for an autoregressive model with a linear trend conditional on an initial observation. TPower of tests for unit roots in the presence of a linear trendightly different model with a random initial value in which nuisance parameters can easily be eliminated by an invariant reduction of the model. We show that invariance arguments can also be used when comparing power within a conditional model. In the context of the conditional model, the Dickey–Fuller test is shown to be more stringent than a number of unit‐root tests motivated by models with random initial value. The power of the Dickey–Fuller test can be improved by making assumptions to the initial value. The practitioner therefore has to trade‐off robustness and power, as assumptions about initial values are hard to test, but can give more power.     

Testing the Null of Cointegration with Structural Breaks*

We propose a Lagrange Multiplier‐type statistic to test the null hypothesis of cointegration allowing for the possibility of a structural break, in both the deterministic and the cointegration vectors. Our proposal focuses on the presence of endogenous regressors. The test complements the usual non‐cointegration tests so as to obtain stronger evidence of cointegration. We consider the cases of known and unknown dates of the break. In the latter case, we show that minimizing the Sum of Squared Residuals results in a super‐consistent estimator of the break fraction. Finally, the behaviour of the tests is studied through Monte Carlo experiments.     

Taking a new contour: A novel approach to panel unit root tests

The paper introduces a novel approach to testing for unit roots in panels, which takes a new contour that is drawn along the line given by the equi-squared-sum instead of the traditional one given by the equi-sample-size. We show in the paper that the distributions of the unit root tests are asymptotically normal along the new contour under both the null and the local-to-unity alternatives. Subsequently, we demonstrate that this startling finding may be exploited constructively to invent tools and methodologies for effective inferences in panel unit root models. Simulations show that our approach works quite well in finite samples.