We suggest a rescaled variance type of test for the null hypothesisof stationarity against deterministic and stochastic trends (unitroots). The deterministic trend can be represented as a generalfunction in time (e.g., nonparametric, linear, or polynomialregression, abrupt changes in the mean). Under the null, theasymptotic distribution of the test is derived, and critical valuesare tabulated for a wide class of stationary processes with short,long, or negative dependence structure. A simulation study examinesthe performance of the test in terms of size and power. Theempirical performance of the test is illustrated using the S&P500 data.The authors thank the editor,the referees, and Karim Abadir for helpful comments and AlfredasRačkauskas for drawing our attention to the criterion of Cremersand Kadelka (1986). The first author'swork was supported by the ESRC grants R000238212 and R000239538.The last two authors were supported by a cooperation agreementCNRS/LITHUANIA (4714) and by a bilateral Lithuania-Franceresearch project Gilibert.

This paper proposes a consistent test for the goodness-of-fit ofparametric regression models that overcomes two important problemsof the existing tests, namely, the poor empirical power and sizeperformance of the tests due to the curse of dimensionality and thesubjective choice of parameters such as bandwidths, kernels, andintegrating measures. We overcome these problems by using a residualmarked empirical process based on projections (RMPP). We study theasymptotic null distribution of the test statistic, and we show thatour test is able to detect local alternatives converging to the nullat the parametric rate. It turns out that the asymptotic nulldistribution of the test statistic depends on the data generatingprocess, and so a bootstrap procedure is considered. Our bootstraptest is robust to higher order dependence, in particular toconditional heteroskedasticity. For completeness, we propose a newminimum distance estimator constructed through the same RMPP as inthe testing procedure. Therefore, the new estimator inherits all thegood properties of the new test. We establish the consistency andasymptotic normality of the new minimum distance estimator. Finally,we present some Monte Carlo evidence that our testing procedure canplay a valuable role in econometric regression modeling.The author thanks Carlos Velasco andMiguel A. Delgado for useful comments. The paper has alsobenefited from the comments of two referees and the co-editor.This research was funded by the Spanish Ministry of Educationand Science reference number SEJ2004-04583/ECON and by theUniversidad de Navarra reference number 16037001.

This paper develops a dynamic switching model, with a random walk anda stationary regime, where the time spent in the random walk regimeis endogeneously predetermined. More precisely, we assume that theprocess is recursively defined byYt = μ +Yt−1 +εt, with stochastic probabilityπrw(Yt−1),Yt = μ +εt, with stochastic probability1 −πrw(Yt−1),where (εt) is a strong white noise andπrw is a nondecreasing function.Then, the dynamics of the process(Yt), itsmarginal distribution, and the distribution of the time spent in theunit root regime depend on the pattern of random walk intensityπrw and on the noisedistribution F. Moreover, we study the linksbetween the endogeneous switching regime and the degree ofpersistence of the process(Yt).

Nonstationary integrated time series may be fractionallycointegrated. Here we propose a test for the null hypothesis of nocointegration. It builds on the asymptotically normal Lagrangemultiplier (LM) test against fractional alternatives applied tosingle equation residuals. It is shown that the LM test appliednaively to residuals from a static level regression does not retainasymptotic normality. However, when the LM statistic is employedwith residuals from a regression of differenced variables, then thetest statistic is shown to have a standard normal limitingdistribution. Monte Carlo experiments establish its relevance infinite samples.We thank NorbertChristopeit and three anonymous referees for corrections andvery valuable comments.

The single-index model is one of the most popular semiparametricmodels in applied quantitative sciences. Two new estimation methodshave been proposed recently by Hristache, Juditski, and Spokoiny(2001, Annals ofStatistics 29, 595–623) and Xia, Tong, Li, and Zhu(2002, Journal of the RoyalStatistical Society, Series B 64, 363–410),respectively. However, their asymptotic distributions have not beeninvestigated yet. In this paper, alternative versions for themethods are investigated. Asymptotic distributions of the estimatorsare derived. Efficiency comparisons between the estimation methodsare made.The author is most gratefulto Professor O. Linton and Professor W. Härdle for helpfuldiscussions. Valuable comments from two anonymous reviewers haveimproved the presentation of the paper substantially. Theresearch has been partially supported by NUS research grantR-155-000-048-112, National University of Singapore, Singapore,and the Alexander von Humboldt Foundation, Germany.

This paper extends current theory on the identification andestimation of vector time series models to nonstationary processes.It examines the structure of dynamic simultaneous equations systemsor ARMAX processes that start from a given set of initial conditionsand evolve over a given, possibly infinite, future time horizon. Theanalysis proceeds by deriving the echelon canonical form for suchprocesses. The results are obtained by amalgamating ideas from thetheory of stochastic difference equations with adaptations of theKronecker index theory of dynamic systems. An extension of theseresults to the analysis of unit-root, partially nonstationary(cointegrated) time series models is also presented, leading tostraightforward identification conditions for the error correction,echelon canonical form. An innovations algorithm for the evaluationof the exact Gaussian likelihood is given. The asymptotic propertiesof the approximate Gaussian estimator and the exact maximumlikelihood estimator based upon the algorithm are derived for thecointegrated case. Examples illustrating the theory are discussed,and some experimental evidence is also presented.I thank two referees for insightful comments andhelpful suggestions on the content and presentation of thispaper. I am particularly grateful for the correction of errorsin earlier drafts and reference to the work of B. Hanzon.Financial support under ARC grant DP0343811 is gratefullyacknowledged.

In Wooldridge (2005), I posed a problemwhose solution involved showing that the so-called ignorability oftreatment assumption, commonly used in the treatment effectsliterature, was necessarily violated. In particular, in a settingwith multiple treatment effects and without distributional orfunctional form assumptions, I assumed that treatment was randomizedwith respect to the counterfactual outcomes but not with respect tothe covariates. I applied the law of iterated expectations toprovide a simple proof that ignorability (given the covariates)necessarily fails.

This note analyzes the local asymptotic power properties of a testproposed by Breitung (2000, in B. Baltagi(ed.), Nonstationary Panels, Panel Cointegration, andDynamic Panels). We demonstrate that the Breitung test,like many other tests (including point optimal tests) for panel unitroots in the presence of incidental trends, has nontrivial power inneighborhoods that shrink toward the null hypothesis at the rate ofn−1/4T−1where n and T are thecross-section and time-series dimensions, respectively. This rate isslower than then−1/2T−1rate claimed by Breitung. Simulation evidence documents theusefulness of the asymptotic approximations given here.The authors thank Paolo Paruolo and areferee for comments on an earlier version of the paper.Phillips acknowledges partial support from a Kelly Fellowshipand the NSF under grant SES 04-142254. Perron acknowledgesfinancial support from FQRSC, SSHRC, and MITACS.

Mundlak (1978,Econometrica 46, 69–85) showed that the fixedeffects estimator can be obtained as generalized least squares (GLS)for a panel regression model where the individual effects are randombut are all hopelessly correlated with theregressors. This result was obtained by partitioned inversion aftersubstituting the reduced form expression for the individual effectsas a function of the means of all the regressors.This note shows that Mundlak's result can be obtained usingsystem estimation without using partitionedinversion. System estimation has proved useful for derivingtwo-stage least squares (2SLS) and three-stage least squares (3SLS)counterparts for the random effects panel models by Baltagi (1981, Journal ofEconometrics 17, 189–200). It also has been used forobtaining an alternative derivation of the Hausman tests that isrobust to heteroskedasticity of unknown form (see Arellano, 1993, Journal ofEconometrics 59, 87–97) and more recently, forobtaining generalized method of moments (GMM) estimators for dynamicpanel models (see Arellano and Bover, 1995, Journal of Econometrics 68, 29–51;and Blundell and Bond, 1998,Journal of Econometrics 87, 115–143, to mentiona few). We also show that a necessary and sufficient condition forordinary least squares (OLS) to be equivalent to GLS is satisfiedfor this model.

Applications are now being sought for the eighth A.R. BergstromPrize in Econometrics.