## 1. INTRODUCTION

Over the past decade the problem of testing for unit roots in heterogeneous panels has attracted a great of deal attention. See, for example, Bowman D (unpublished 1999), Choi (2001), Hadri (2000), Im *et al.* (1995, 2003), Levin *et al.* (2002), Maddala and Wu (1999), and Shin and Snell (2002). Baltagi and Kao (2000) provide an early review. This literature, however, assumed that the individual time series in the panel were cross-sectionally independently distributed. While it was recognized that this was a rather restrictive assumption, particularly in the context of cross-country (region) regressions, it was thought that cross-sectionally de-meaning the series before application of the panel unit root test could partly deal with the problem (see Im *et al.*, 1995). However, it was clear that cross-section de-meaning could not work in general where pair-wise cross-section covariances of the error terms differed across the individual series. Recognizing this deficiency new panel unit root tests have been proposed in the literature by Chang (2002), Choi (2002), Phillips and Sul (2003), Bai and Ng (2004), Breitung and Das (2005), Choi and Chue (2007), Moon and Perron (2004), and Smith *et al.* (2004).

Chang (2002) proposes a nonlinear instrumental variable approach to deal with the cross-section dependence of a general form and establishes that individual Dickey–Fuller (DF) or the augmented DF (ADF) statistics are asymptotically independent when an integrable function of the lagged dependent variables are used as instruments. From this she concludes that her test is valid for both *T* (the time series dimension) and *N* (the cross-section dimension) are large. However, as shown by Im KS and Pesaran MH (unpublished 2003), her test is valid only if *N* is fixed as *T* → ∞. Using Monte Carlo techniques, Im and Pesaran show that Chang's test is grossly oversized for moderate degrees of cross-section dependence, even for relatively small values of *N*.1

Choi (2002) models the cross-dependence using a two-way error-component model which imposes the same pair-wise error covariances across the different cross-section units. This provides a generalization of the cross-section de-meaning procedure proposed in Im *et al.* (1995) but it can still be restrictive in the context of heterogeneous panels. Smith *et al.* (2004) use bootstrap techniques, and Choi and Chue (2007) employ subsampling techniques to deal with cross-section dependence. Breitung and Das (2005) adopt least-squares and feasible generalized least-squares estimates that are applicable in cases where *T* ≥ *N*. Harris *et al.* (2004) propose a test of joint stationarity (as opposed to unit roots) in panels under cross-section dependence using the sum of lag-*k* sample autocovariances where *k* is taken to be an increasing function of *T*.

Bai and Ng (2004), Moon and Perron (2004), and Phillips and Sul (2003) make use of residual factor models to take account of the cross-section dependence. In the case of a residual one-factor model Phillips and Sul (2003) propose an orthogonalization procedure that asymptotically eliminates the common factors. Similar procedures are used by Bai and Ng (2004) and Moon and Perron (2004) in a more general set-up. Moon and Perron (2004) propose a pooled panel unit root test based on ‘de-factored’ observations and suggest estimating the factor loadings by the principal component method. They derive asymptotic properties of their test under the unit root null and local alternatives, assuming in particular that *N*/*T* → 0, as *N* and *T* → ∞. They show that their proposed test has good asymptotic power properties if the model does not contain deterministic (incidental) trends. In a related paper, Moon HR *et al.* (unpublished 2003) propose a point optimal invariant panel unit root test which is shown to have local power even in the presence of deterministic trends. Bai and Ng (2004) consider a more general set-up and allow for the possibility of unit roots (and cointegration) in the common factors, but continue to assume that *N*/*T* → 0, as *N* and *T* → ∞. To deal with such a possibility they apply the principal component procedure to the first-differenced version of the model, and estimate the factor loadings and the first differences of the common factors. These so-called ‘second generation’ panel unit root tests are reviewed in Breitung and Pesaran (2007) and Choi (2006).

In this paper we adopt a different approach to dealing with the problem of cross-section dependence. Instead of basing the unit root tests on deviations from the estimated factors, we augment the standard DF (or ADF) regressions with the cross-section averages of lagged levels and first-differences of the individual series. Standard panel unit root tests are now based on the simple averages of the individual cross-sectionally augmented ADF statistics (denoted by CADF), or suitable transformations of the associated rejection probabilities. The individual CADF statistics or the rejection probabilities can then be used to develop modified versions of the *t*-bar test proposed by Im *et al.* (IPS), the inverse chi-squared test (or the *P* test) proposed by Maddala and Wu (1999), and the inverse normal test (or the *Z* test) suggested by Choi (2001).2 A truncated version of the test is also considered where the individual CADF statistics are suitably truncated to avoid undue influences of extreme outcomes that could arise when *T* is small (in the region of 10–20).

New asymptotic results are obtained both for the individual CADF statistics, and their simple averages, referred to as the cross-sectionally augmented IPS (*CIPS*) test. The asymptotic null distribution of the individual *CADF*_{i} and the associated statistics are investigated as *N* → ∞ followed by *T* → ∞, as well as jointly with *N* and *T* tending to infinity such that *N*/*T* → *k*, where *k* is a fixed finite non-zero positive constant. It is shown that the *CADF*_{i} statistics are asymptotically similar and do not depend on the factor loadings. But they are asymptotically correlated due to their dependence on the common factor. As a result the standard central limit theorems do not apply to the *CIPS* statistic (or the other combination or meta type tests proposed by Maddala and Wu, and Choi). However, it is shown that the limit distribution of the truncated version of the *CIPS* statistic (denoted by *CIPS**) exists and is free of nuisance parameters. The critical values of *CIPS* and *CIPS** statistics are tabulated for the three main specifications of the deterministics, namely in the case of models without intercepts or trends, models with individual-specific intercepts, and models with incidental linear trends.

The proposed test has the advantage of being simple and intuitive. It is also valid for panels where *N* and *T* are of the same orders of magnitudes, while the tests by Moon and Perron and Bai and Ng require *N*/*T* → 0. But it is based on a one-factor residual model which could be restrictive in some applications. Although it is possible to extend the test to multifactor residual models, such an extension would involve a multivariate panel framework which is outside the scope of the present paper.

The small sample properties of the proposed tests are investigated by Monte Carlo experiments, for a variety of models with incidental deterministics (intercepts as well as linear trends), cross-dependence (low and high) and individual specific residual serial correlation (positive and negative), and sample sizes, *N* and *T* = 10, 20, 30, 50, 100. The simulations show that the cross-sectionally augmented panel unit root tests have satisfactory size and power even for relatively small values of *N* and *T*. This is particularly true of the truncated version of the *CIPS* test and the cross-sectionally augmented version of Choi's inverse normal combination test. These tests show satisfactory size properties even for very small sample sizes, namely when *N* = *T* = 10, and there is a high degree of cross-section dependence with a moderate degree of residual serial correlation. Perhaps not surprisingly, the power of the tests critically depends on the sample sizes *N* and *T*, and on whether the model contains linear time trends. In the case of models with linear time trends power starts to rise with *N* only if *T* is 30 or more. For *T* > 30 the power rises quite rapidly with both *N* and *T*. In their respective simulations Bai and Ng (2004) report Monte Carlo results for *T* = 100 and *N* = 20, 100, Moon and Perron (2004) for *T* = 100, 300, *N* = 10, 20, and Phillips and Sul (2003) for *T* = 50, 100, 200 and *N* = 10, 20, 30. All these studies consider experiments where *T* is much larger than *N*, and hence are difficult to evaluate in relation to our simulation results where *T* could be small relative to *N* and vice versa.3

The plan of the paper is as follows. Section 2 sets out the basic model. Section 3 introduces the cross-sectionally augmented regressions for the individual series for models without residual serial correlations, and derives the null distribution of the CADF statistic. The various CADF-based panel unit root tests (the cross-sectionally augmented versions of the *IPS*, *P* and the *Z* tests) are discussed in Section 4. Section 5 extends the results to the case where the individual specific errors are serially correlated. Small sample performance of the proposed tests are investigated in Section 6 using Monte Carlo experiments. Section 7 provides two empirical applications, and Section 8 concludes the paper.

Notations: *a*_{n} = *O*(*b*_{n}) states the deterministic sequence {*a*_{n}} is at most of order *b*_{n}, **x**_{n} = *O*_{p}(**y**_{n}) states the vector of random variables, **x**_{n}, is at most of order **y**_{n} in probability, and **x**_{n} = *o*_{p}(**y**_{n}) is of smaller order in probability than **y**_{n}. → denotes convergence in quadratic means (q.m.) or mean square errors and ↠ convergence in distribution. All asymptotics are carried out under *N* → ∞, either with a fixed *T*, sequentially, or *jointly* with *T* → ∞. In particular, denotes convergence in distribution (q.m.) with *T* fixed as *N* → ∞, () denotes convergence in distribution (q.m.) for *N* fixed (or when there is no *N*-dependence) as *T* → ∞, denotes sequential convergence with *N* → ∞ first followed by *T* → ∞ (similarly ), denotes joint convergence with *N*, *T* → ∞ jointly such that *N*/*T* → *k*, where *k* is a fixed finite non-zero constant. ∼ denotes asymptotic equivalence in distribution, with , , , , and , similarly defined as , , etc.