A number of panel unit root tests that allow for cross section dependence have been proposed in the literature that use orthogonalization type procedures to asymptotically eliminate the cross dependence of the series before standard panel unit root tests are applied to the transformed series. In this paper we propose a simple alternative where the standard ADF regressions are augmented with the cross section averages of lagged levels and first-differences of the individual series. New asymptotic results are obtained both for the individual cross sectionally augmented ADF (CADF) statistics, and their simple averages. It is shown that the individual CADF statistics are asymptotically similar and do not depend on the factor loadings. The limit distribution of the average CADF statistic is shown to exist and its critical values are tabulated. Small sample properties of the proposed test are investigated by Monte Carlo experiments. The proposed test is applied to a panel of 17 OECD real exchange rate series as well as to log real earnings of households in the PSID data.

This paper provides a review of the literature on unit roots and cointegration in panels where the time dimension (T), and the cross section dimension (N) are relatively large. It distinguishes between the …rst generation tests developed on the assumption of the cross section independence, and the second generation tests that allow, in a variety of forms and degrees, the dependence that might prevail across the di¤erent units in the panel. In the analysis of cointegration the hypothesis testing and estimation problems are further complicated by the possibility of cross section cointegration which could arise if the unit roots in the di¤erent cross section units are due to common random walk components.

This paper re-examines the panel unit root tests proposed by Chang (2002). She establishes asymptotic independence of the t-statistics when integrable functions of lagged dependent variable are used as instruments even if the original series are cross sectionally dependent. From this rather remarkable result she claims that her non-linear instrumental variable (NIV) panel unit root test is valid under general error cross correlations for any N (the cross section dimension) as T (the time dimension of the panel) tends to infinity. We show that her claim is valid only if N ln T / √ T → 0, as N and T → ∞, and this condition is unlikely to hold in practice, unless N is very small. The favourable simulation results reported by Chang are largely due to her particular choice of the error correlation matrix, which results in weak cross section dependence. Also, the asymptotic independence property of the t-statistics disappears when Chang’s modified instruments are used. Using a common factor model with a sizeable degree of cross section correlations, we are able to show that Chang’s NIV panel unit root test suffers from gross size distortions, even when N is small relative to T (for example N =5, T =100). JEL Classification: C12, C15, C22, C23. Key Words: Non-linear Instrumental Variable (NIV) Panel unit root tests, Cross-section dependence, Finite sample properties. We would like to thank Michael Binder, George Kapetanios, and Junsoo Lee for helpful discussions and Mutita Akusuwan for computing the results reported in Table 5 of this paper. We are also grateful to Yoosoon Chang for providing us with her Gauss program. 1 1

In this paper we establish the strong approximations for the nonlinear transformations of integrated time series. Both the asymptotically homogeneous and integrable transformations are considered, and the explicit rates for the convergence to their limit distributions are obtained under mild regularity conditions that are satisfied by virtually all nonlinear models used in practical applications. The first order asymptotics are also derived under the conditions that are significantly weaker than those required by earlier works.

An IV approach, using as instruments nonlinear transformations of the lagged levels, is explored to test for unit roots in panels with general dependency and heterogeneity across cross-sectional units. We allow not only for the cross-sectional dependencies of innovations, but also for the presence of cointegration across crosssectional levels. Unbalanced panels and panels with di¤ering individual shortrun dynamics and cross-sectionally related dynamics are also permitted. Panels with such cross-sectional dependencies and heterogeneities appear to be quite commonly observed in practical applications. Yet none of the currently available tests can be used to test for unit roots in such general panels. We also more carefully formulate the unit root hypothesis in panels. In particular, using order statistics we make it possible to test for and against the presence of unit roots in some of the individual units for a given panel. The individual IV t-ratios, which are the bases of our tests, are asymptotically normally distributed and cross-sectionally independent. Therefore, the critical values of the order statistics as well as the usual averaged statistic can be easily obtained from simple elementary probability computations. We show via a set of simulations that our tests work well, while other existing tests fail to perform properly. As an illustration, our tests are applied to some of the data sets that were used in earlier studies.

The competitive environment hypothesis revisited: Nonlinearity,

This paper analyzes nonlinear cointegrating regressions as have been recently analyzed in a paper by Park and Phillips in Econometrica. I analyze the consequences of removing Park and Phillips' exogeneity assumption, which for the special case of a linear model would imply the asymptotic validity of the least squares estimator for linear cointegrating regressions. For the linear model, the unlikeliness of such an exogeneity assumption to hold in practice has inspired the "fully modified" technique, the "leads and lags" technique, and Park's "canonical regressions". In this paper, a "fully modified" type technique is proposed for nonlinear cointegrating regressions. The mathematical tool for proving this result is a new so-called "convergence to stochastic integrals" result. This result is proven for objects that are summations of a stationary random variable times an asymptotically homogeneous function of an integrated process.

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This paper presents the nonlinear IV methodology as an effective inferential basis for nonstationary panels. The nonlinear IV method resolves the inferential difficulties in testing for unit roots arising from the intrinsic heterogeneities and cross-dependencies of panel models. Individual units are allowed to be dependent through correlations among innovations, interrelatedness of short-run dynamics and/or cross-sectional cointegrations. If based on the instrumental variables that are nonlinear transformations of the lagged levels, the usual IV estimation of the augmented Dickey-Fuller type regressions yields asymptotically normal unit root tests for panels with general dependencies and heterogeneities. Moreover, the nonlinear IV estimation allows for the use of covariates to further increase power, and order statistics to test for more flexible forms of hypotheses, which are especially important in heterogeneous panels.

Traditionally economic panels had large number of cross-section units and relatively few time periods and econometric methods were developed for such ‘large N small T ’data. More recently panels with observations for a large numbers of time periods have become available on cross-section units like …rms, industries, regions or countries. These notes explore the econometric methods developed for such ‘large N large T ’ data. Such data allow more explicit treatment of (a) heterogeneity across units (b) dynamics, including the treatment of unit roots and cointegration and (c) cross-section dependence arising from spatial interactions or unobserved