Abstract
 Top of page
 Abstract
 1. INTRODUCTION
 2. A SIMPLE DYNAMIC PANEL WITH CROSSSECTION DEPENDENCE
 3. UNIT ROOT TESTS FOR ONEFACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
 4. CADF PANEL UNIT ROOT TESTS
 5. CASE OF SERIALLY CORRELATED ERRORS
 6. SMALLSAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
 7. EMPIRICAL APPLICATIONS
 8. CONCLUDING REMARKS
 APPENDIX: MATHEMATICAL PROOFS
 Acknowledgements
 REFERENCES
 Supporting Information
A number of panel unit root tests that allow for crosssection dependence have been proposed in the literature that use orthogonalization type procedures to asymptotically eliminate the crossdependence 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 augmented Dickey–Fuller (ADF) regressions are augmented with the crosssection averages of lagged levels and firstdifferences of the individual series. New asymptotic results are obtained both for the individual crosssectionally augmented ADF (CADF) statistics and for 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. Copyright © 2007 John Wiley & Sons, Ltd.
1. INTRODUCTION
 Top of page
 Abstract
 1. INTRODUCTION
 2. A SIMPLE DYNAMIC PANEL WITH CROSSSECTION DEPENDENCE
 3. UNIT ROOT TESTS FOR ONEFACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
 4. CADF PANEL UNIT ROOT TESTS
 5. CASE OF SERIALLY CORRELATED ERRORS
 6. SMALLSAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
 7. EMPIRICAL APPLICATIONS
 8. CONCLUDING REMARKS
 APPENDIX: MATHEMATICAL PROOFS
 Acknowledgements
 REFERENCES
 Supporting Information
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 See, for example, Bowman D (unpublished 1999), Choi (2001), Hadri (2000), Im et al.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 crosssectionally independently distributed. While it was recognized that this was a rather restrictive assumption, particularly in the context of crosscountry (region) regressions, it was thought that crosssectionally demeaning 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 crosssection demeaning could not work in general where pairwise crosssection 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 crosssection 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 crosssection 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 crosssection dependence, even for relatively small values of N.1
Choi (2002) models the crossdependence using a twoway errorcomponent model which imposes the same pairwise error covariances across the different crosssection units. This provides a generalization of the crosssection demeaning 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 crosssection dependence. Breitung and Das (2005) adopt leastsquares and feasible generalized leastsquares 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 crosssection dependence using the sum of lagk sample autocovariances where k is taken to be an increasing function of T.
In this paper we adopt a different approach to dealing with the problem of crosssection dependence. Instead of basing the unit root tests on deviations from the estimated factors, we augment the standard DF (or ADF) regressions with the crosssection averages of lagged levels and firstdifferences of the individual series. Standard panel unit root tests are now based on the simple averages of the individual crosssectionally 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 tbar test proposed by Im et al. (IPS), the inverse chisquared 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).
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 onefactor 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), crossdependence (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 crosssectionally 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 crosssectionally 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 crosssection 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 crosssectionally augmented regressions for the individual series for models without residual serial correlations, and derives the null distribution of the CADF statistic. The various CADFbased panel unit root tests (the crosssectionally 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 Ndependence) 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 nonzero constant. ∼ denotes asymptotic equivalence in distribution, with , , , , and , similarly defined as , , etc.
2. A SIMPLE DYNAMIC PANEL WITH CROSSSECTION DEPENDENCE
 Top of page
 Abstract
 1. INTRODUCTION
 2. A SIMPLE DYNAMIC PANEL WITH CROSSSECTION DEPENDENCE
 3. UNIT ROOT TESTS FOR ONEFACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
 4. CADF PANEL UNIT ROOT TESTS
 5. CASE OF SERIALLY CORRELATED ERRORS
 6. SMALLSAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
 7. EMPIRICAL APPLICATIONS
 8. CONCLUDING REMARKS
 APPENDIX: MATHEMATICAL PROOFS
 Acknowledgements
 REFERENCES
 Supporting Information
Let y_{it} be the observation on the ith crosssection unit at time t and suppose that it is generated according to the simple dynamic linear heterogeneous panel data model
 (1)
where initial value, y_{i0}, has a given density function with a finite mean and variance, and the error term, u_{it}, has the singlefactor structure
 (2)
in which f_{t} is the unobserved common effect, and ε_{it} is the individualspecific (idiosyncratic) error.
It is convenient to write (1) and (2) as
 (3)
where α_{i} = (1 − ϕ_{i})μ_{i}, β_{i} = − (1 − ϕ_{i}) and Δy_{it} = y_{it} − y_{i, t−1}. The unit root hypothesis of interest, ϕ_{i} = 1, can now be expressed as
 (4)
against the possibly heterogeneous alternatives,
 (5)
We shall assume that N_{1}/N, the fraction of the individual processes that are stationary, is nonzero and tends to the fixed value δ such that 0 < δ≤ 1 as N∞. As noted in Im et al. (2003) this condition is necessary for the consistency of the panel unit root tests. We shall also make the following assumptions:
Assumption 1The idiosyncratic shocks, ε_{it}, i = 1, 2, …, N, t = 1, 2, …, T, are independently distributed both acrossiandt, have mean zero, variance, and finite fourthorder moment.
Assumption 2The common factor,f_{t}, is serially uncorrelated with mean zero and a constant variance,, and finite fourthorder moment. Without loss of generalitywill be set equal to unity.
Assumption 3 ε_{it}, f_{t}, and γ_{i}are independently distributed for alli.
The crosssection independence of ε_{it} (across i) is standard in onefactor models, although its validity in more general settings may require specification of more than one common factor in (2). Assumptions 1 and 2 together imply that the composite error, u_{it}, is serially uncorrelated. Both of these assumptions can be relaxed. It is possible to relax Assumption 2 by allowing the common factor, f_{t}, to follow a general stationary process. As we shall see, this does not affect the asymptotic distribution of the proposed test as N and T∞, jointly.4 Alternatively, one could consider serial correlation in both f_{t} and ε_{it} processes. In that case the effects of residual serial correlation need to be taken into account explicitly in the derivation of the proposed test statistic. This problem will be considered in Section 5.
3. UNIT ROOT TESTS FOR ONEFACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
 Top of page
 Abstract
 1. INTRODUCTION
 2. A SIMPLE DYNAMIC PANEL WITH CROSSSECTION DEPENDENCE
 3. UNIT ROOT TESTS FOR ONEFACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
 4. CADF PANEL UNIT ROOT TESTS
 5. CASE OF SERIALLY CORRELATED ERRORS
 6. SMALLSAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
 7. EMPIRICAL APPLICATIONS
 8. CONCLUDING REMARKS
 APPENDIX: MATHEMATICAL PROOFS
 Acknowledgements
 REFERENCES
 Supporting Information
Let and suppose that for a fixed N and as N∞. Then following the line of reasoning in Pesaran (2006), the common factor f_{t} can be proxied by the crosssection mean of y_{it}, namely , and its lagged value(s), ȳ_{t−1}, ȳ_{t−2}, … for N sufficiently large. In the simple case where u_{it} is serially uncorrelated, it turns out that ȳ_{t} and ȳ_{t−1} (or equivalently ȳ_{t−1} and Δȳ_{t}) are sufficient for asymptotically filtering out the effects of the unobserved common factor, f_{t}. We shall therefore base our test of the unit root hypothesis, (4), on the tratio of the OLS estimate of b_{i}(b̂_{i}) in the following crosssectionally augmented DF (CADF) regression:
 (6)
Denoting this tratio by t_{i}(N, T) we have
 (7)
where
 (8)
 (9)
 (10)
 (11)
and
 (12)
Under β_{i} = 0 we have Δy_{it} = γ_{i}f_{t} + ε_{it} and hence
 (13)
 (14)
 (15)
 (16)
where ε_{i} = (ε_{i1}, ε_{i2}, …, ε_{iT})′, f =(f_{1}, f_{2}, …, f_{T})′, , , y_{i0} is a given initial value (fixed or random), , s_{i,−1} = (0, s_{i1}, …, s_{i, T−1})′, s_{f,−1} = (s_{f0}, s_{f1}, …, s_{f, T−1})′, s̄_{−1} = (0, s̄_{1}, …, s̄_{T−1}) with , and , for t = 1, 2, ., and . Using (15) to eliminate f from (13), and noting that by assumption , we have
 (17)
where
 (18)
Therefore,
 (19)
where
 (20)
and . Therefore,
 (21)
where υ_{i} = ξ_{i}/ω_{i}∼(0, I_{T}).
Similarly, using (16) to eliminate s_{f,−1} from (14) we obtain
 (22)
and hence
 (23)
where . It is easily seen that is the random walk associated with υ_{i}.
Hence, the exact null distribution of t_{i}(N, T) will depend on the nuisance parameters only through their effects on M̄_{w} and M_{i, w}. But, as shown in the Appendix, this dependence vanishes as N∞, irrespective of whether T is fixed or tends to infinity jointly with N. In the case where T is fixed, to ensure that t_{i}(N, T) does not depend on the nuisance parameters the effect of the initial crosssection mean, ȳ_{0}, must also be eliminated. This can be achieved by applying the test to the deviations y_{it} − ȳ_{0}. The following theorems provide a formal statement of these results.
Theorem 3.1Suppose the seriesy_{it}, fori = 1, 2, …, Nandt = 1, 2, …, T, are generated under(4)according to(3)and by construction ȳ_{0}(the crosssection mean of the initial observations) is set to zero. Then under Assumptions 1 and 2 the null distribution oft_{i}(N, T) given by(24)will be free of nuisance parameters asN∞ for any fixedT > 4. In particular, we have (in quadratic mean)
 (25)
where
ε_{i}/σ_{i}andfare independently distributed as (0, I_{T}), s_{i} = s_{i,−1} + ε_{i}, ands_{f} = s_{f,−1} + f.
The critical values of the CADF test can be computed by stochastic simulation for any fixed T > 4, and for given distributional assumptions for the random variables (ε_{i}, f).
Theorem 3.2Lety_{it}be defined by(3)and consider the statisticst_{i}(N, T) defined by(7). Suppose that Assumptions 1–3 hold andtends to a finite nonzero limit asN∞, then under(4)and asNandT∞, t_{i}(N, T) has the same sequential (N∞, T∞) and joint [(N, T)_{j}∞] limit distributions, referred to as crosssectionally augmented Dickey–Fuller (CADF) distribution given by
 (26)
where
 (27)
and
 (28)
withW_{i}(r) andW_{f}(r) being independent standard Brownian motions. For the joint limit distribution to hold it is also required that as (N, T)_{j}∞, N/Tk, wherekis a nonzero, finite positive constant.
Remark 3.1The critical values ofCADF_{if}can be computed by stochastic simulation assuming that
Remark 3.2From(26)it is clear that
 (29)
whereG(.) is a general nonlinear function common for alli. Therefore,CADF_{if}andCADF_{jf}are dependently distributed, with the same degree of dependence for alli ≠ j.
Remark 3.3The random variablesCADF_{1f}, CADF_{2f}, …, CADF_{Nf}form an exchangeable sequence. This follows from the fact that under Assumption 3, conditional onW_{f}the random variables {CADF_{if}, i = 1, 2, …, N} are identically and independently distributed.5
Remark 3.4The idea of augmenting ADF regressions with covariates has been considered by Hansen (1995), where stationary covariates are added to the ADF regression with the aim of increasing the power of the unit root test. In our application the covariates are nonstationary as well as being endogenous for a finiteN. As in Hansen (1995) it is possible to gain power by application of the CADF procedure to a single time series when information on the crosssection average,ȳ_{t}, is available. Of course, different critical values would now apply, and must be computed using the CADF distribution given by(26). Also see Section3.1.
Remark 3.5In the above analysis we have opted for simple averages (i.e.,ȳ) in dealing with the crossdependence problem. But following Pesaran (2006) weighted averages could also be used instead, for any given set of fixed weights,w_{i}, such thatasN∞.
Remark 3.6Our assumption that, with a nonzero limit asN∞, might be viewed by some as restrictive. Under, for example, the possibility of crosssection independence is ruled out. However, it is instructive to recall from(15)that in the case where, we have, and therefore Δȳ_{t}would tend to zero asN∞, andȳ_{t}to a fixed constant for allt. In most economic and financial panels of interest this does not seem to be a very likely outcome. However, the case where, asN∞, could be of theoretical interest and is worth further consideration.
Remark 3.7As noted earlier, the asymptotic distribution oft_{i}(N, T), given by(26), continues to hold if Assumption 2 is relaxed andf_{t}is allowed to follow the general linear stationary process:
where, is an i.i.d. sequence with mean zero, a unit variance, and finite fourth moment. In this case all the various terms in(25)tend to the same limits as those in(26), withW_{f}(r) replaced by(see, for example, Hamilton, 1994, pp. 504–505). But it is readily verified that the term ψ(1) cancels out from the asymptotic limit oft_{i}(N, T) asNandT∞.
3.1. Critical Values of the Individual CADF Test
The computation of the critical values were based on 50,000 CADF regressions of Δy_{1t} on y_{1, t−1}, ȳ_{t−1}, Δȳ_{t}, over the sample t = 1, 2, …, T; including appropriate deterministics, namely without any (Case I), with an intercept (Case II), and with an intercept and a linear trend (Case III). The individual series were generated as y_{it} = y_{i, t−1} + f_{t} + ε_{it}, for i = 1, 2, …, N, and t = − 50, − 49, …, 1, 2, …, T from y_{i,−50} = 0, with f_{t} and ε_{it} as i.i.d. N(0, 1). Critical values of the individual CADF distribution for values of T and N in the range of 10–200 for the three standard cases are provided in Tables I(a) to I(c). Each Table gives the 1%, 5%, and 10% critical values.
Table I(a). Critical values of individual crosssectionally augmented Dickey–Fuller distribution (Case I: no intercept and no trend)T/N  10  15  20  30  50  70  100  200 


1%(CADF_{i}) 
10  −4.23  −4.26  −4.21  −4.25  −4.22  −4.25  −4.31  −4.28 
15  −3.69  −3.72  −3.67  −3.71  −3.75  −3.75  −3.70  −3.65 
20  −3.53  −3.52  −3.48  −3.51  −3.55  −3.51  −3.52  −3.53 
30  −3.40  −3.43  −3.38  −3.39  −3.41  −3.41  −3.40  −3.40 
50  −3.33  −3.32  −3.33  −3.31  −3.31  −3.35  −3.33  −3.30 
70  −3.28  −3.24  −3.29  −3.25  −3.28  −3.30  −3.28  −3.31 
100  −3.26  −3.27  −3.25  −3.28  −3.30  −3.26  −3.25  −3.25 
200  −3.21  −3.22  −3.25  −3.24  −3.24  −3.24  −3.25  −3.24 
5%(CADF_{i}) 
10  −2.95  −2.92  −2.91  −2.91  −2.93  −2.94  −2.96  −2.94 
15  −2.75  −2.75  −2.75  −2.76  −2.77  −2.77  −2.75  −2.74 
20  −2.69  −2.71  −2.70  −2.69  −2.71  −2.69  −2.68  −2.69 
30  −2.66  −2.65  −2.66  −2.66  −2.66  −2.66  −2.65  −2.66 
50  −2.63  −2.64  −2.63  −2.63  −2.61  −2.63  −2.62  −2.62 
70  −2.60  −2.61  −2.61  −2.60  −2.61  −2.62  −2.62  −2.62 
100  −2.60  −2.61  −2.61  −2.61  −2.60  −2.62  −2.60  −2.61 
200  −2.60  −2.61  −2.60  −2.60  −2.59  −2.60  −2.60  −2.60 
10%(CADF_{i}) 
10  −2.39  −2.39  −2.37  −2.38  −2.39  −2.38  −2.39  −2.39 
15  −2.31  −2.32  −2.31  −2.32  −2.32  −2.32  −2.32  −2.32 
20  −2.29  −2.31  −2.30  −2.30  −2.30  −2.30  −2.30  −2.29 
30  −2.29  −2.28  −2.28  −2.28  −2.28  −2.28  −2.27  −2.29 
50  −2.28  −2.28  −2.26  −2.28  −2.27  −2.28  −2.26  −2.27 
70  −2.26  −2.26  −2.28  −2.26  −2.27  −2.26  −2.27  −2.27 
100  −2.26  −2.26  −2.27  −2.27  −2.26  −2.27  −2.26  −2.26 
200  −2.26  −2.27  −2.26  −2.27  −2.26  −2.27  −2.27  −2.26 
Table I(b). Critical values of individual crosssectionally augmented Dickey–Fuller distribution (Case II: intercept only)T/N  10  15  20  30  50  70  100  200 


1%(CADF_{i}) 
10  −5.75  −5.73  −5.78  −5.73  −5.71  −5.72  −5.89  −5.72 
15  −4.65  −4.65  −4.62  −4.68  −4.66  −4.64  −4.69  −4.61 
20  −4.35  −4.34  −4.32  −4.35  −4.35  −4.33  −4.36  −4.34 
30  −4.11  −4.12  −4.11  −4.12  −4.11  −4.12  −4.11  −4.09 
50  −3.94  −4.00  −3.99  −3.97  −3.95  −3.99  −3.96  −3.96 
70  −3.92  −3.90  −3.91  −3.92  −3.94  −3.93  −3.91  −3.94 
100  −3.88  −3.86  −3.87  −3.90  −3.86  −3.85  −3.85  −3.89 
200  −3.81  −3.83  −3.84  −3.84  −3.83  −3.85  −3.83  −3.84 
5%(CADF_{i}) 
10  −3.93  −3.96  −3.94  −3.97  −3.94  −3.93  −3.96  −3.99 
15  −3.53  −3.57  −3.54  −3.55  −3.55  −3.55  −3.57  −3.55 
20  −3.43  −3.43  −3.42  −3.43  −3.43  −3.42  −3.44  −3.43 
30  −3.36  −3.36  −3.34  −3.34  −3.34  −3.34  −3.33  −3.34 
50  −3.29  −3.30  −3.28  −3.27  −3.27  −3.28  −3.28  −3.28 
70  −3.26  −3.26  −3.27  −3.27  −3.27  −3.28  −3.26  −3.29 
100  −3.24  −3.25  −3.24  −3.27  −3.26  −3.24  −3.24  −3.24 
200  −3.22  −3.23  −3.23  −3.24  −3.24  −3.23  −3.24  −3.22 
10%(CADF_{i}) 
10  −3.26  −3.27  −3.24  −3.26  −3.25  −3.25  −3.28  −3.27 
15  −3.06  −3.08  −3.06  −3.07  −3.07  −3.07  −3.07  −3.06 
20  −3.00  −3.02  −3.01  −3.01  −3.01  −3.00  −3.02  −3.01 
30  −2.97  −2.98  −2.96  −2.97  −2.97  −2.97  −2.95  −2.97 
50  −2.94  −2.95  −2.94  −2.93  −2.94  −2.94  −2.93  −2.94 
70  −2.93  −2.94  −2.94  −2.94  −2.93  −2.94  −2.93  −2.94 
100  −2.92  −2.92  −2.92  −2.93  −2.93  −2.92  −2.91  −2.92 
200  −2.91  −2.92  −2.91  −2.92  −2.92  −2.91  −2.92  −2.91 
Table I(c). Critical values of individual crosssectionally augmented Dickey–Fuller distribution (Case III: intercept and trend)T/N  10  15  20  30  50  70  100  200 


1%(CADF_{i}) 
10  −7.49  −7.67  −7.50  −7.64  −7.69  −7.44  −7.40  −7.51 
 (−6.40)  (−6.40)  (−6.40)  (−6.40)  (−6.40)  (−6.40)  (−6.40)  (−6.40) 
15  −5.44  −5.46  −5.40  −5.50  −5.48  −5.42  −5.49  −5.41 
20  −4.97  −4.98  −4.96  −4.97  −5.01  −5.00  −5.02  −4.95 
30  −4.67  −4.67  −4.68  −4.69  −4.69  −4.64  −4.68  −4.68 
50  −4.49  −4.51  −4.52  −4.51  −4.47  −4.46  −4.48  −4.47 
70  −4.41  −4.41  −4.39  −4.41  −4.41  −4.41  −4.40  −4.42 
100  −4.35  −4.35  −4.35  −4.34  −4.37  −4.35  −4.35  −4.35 
200  −4.28  −4.32  −4.32  −4.30  −4.32  −4.28  −4.30  −4.31 
5%(CADF_{i}) 
10  −4.89  −4.93  −4.89  −4.87  −4.91  −4.90  −4.88  −4.88 
15  −4.17  −4.17  −4.14  −4.18  −4.17  −4.19  −4.19  −4.17 
20  −3.99  −3.99  −4.00  −4.01  −4.01  −4.00  −4.01  −4.01 
30  −3.87  −3.88  −3.87  −3.88  −3.87  −3.86  −3.87  −3.87 
50  −3.78  −3.79  −3.79  −3.80  −3.78  −3.78  −3.79  −3.79 
70  −3.76  −3.75  −3.76  −3.75  −3.76  −3.76  −3.77  −3.78 
100  −3.72  −3.74  −3.74  −3.74  −3.74  −3.73  −3.73  −3.74 
200  −3.69  −3.71  −3.71  −3.71  −3.72  −3.72  −3.72  −3.71 
10%(CADF_{i}) 
10  −4.00  −4.00  −3.99  −4.00  −4.02  −3.99  −4.01  −4.02 
15  −3.64  −3.63  −3.62  −3.65  −3.63  −3.63  −3.65  −3.64 
20  −3.55  −3.54  −3.55  −3.56  −3.56  −3.55  −3.56  −3.56 
30  −3.49  −3.49  −3.49  −3.49  −3.49  −3.49  −3.48  −3.49 
50  −3.44  −3.44  −3.44  −3.45  −3.44  −3.43  −3.45  −3.45 
70  −3.43  −3.43  −3.43  −3.43  −3.43  −3.44  −3.42  −3.44 
100  −3.41  −3.42  −3.42  −3.43  −3.42  −3.42  −3.41  −3.42 
200  −3.39  −3.39  −3.41  −3.40  −3.41  −3.41  −3.41  −3.41 
The CADF distribution turns out to be more skewed to the left as compared to the standard DF distribution. Another interesting aspect of the CADF distribution, which becomes important when the test is used in a panel data context, is the pairwise dependence of the CADF_{if} statistics across i. We simulated the simple pairwise correlation coefficient, corr(CADF_{if}, CADF_{jf}), for different values of N and T and found it be remarkably stable: around 0.03 for the intercept case and in the range of 0.01–0.02 for the linear trend case, both quite small but nonzero. Finally, the CADF distribution, like the standard DF distribution, departs from normality in two important respects: it has a substantially negative mean and its standard deviation is less than unity, although not by a large amount. However, the distribution of a standardized version of the CADF statistic, defined by , looks remarkably like a standard normal distribution. The skewness and (kurtosis − 3) coefficients of the standardized CADF distributions are 0.17 and 0.27 for the intercept case, and 0.06 and 0.29 for the linear trend case. They are quite small, although statistically significant.
4. CADF PANEL UNIT ROOT TESTS
 Top of page
 Abstract
 1. INTRODUCTION
 2. A SIMPLE DYNAMIC PANEL WITH CROSSSECTION DEPENDENCE
 3. UNIT ROOT TESTS FOR ONEFACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
 4. CADF PANEL UNIT ROOT TESTS
 5. CASE OF SERIALLY CORRELATED ERRORS
 6. SMALLSAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
 7. EMPIRICAL APPLICATIONS
 8. CONCLUDING REMARKS
 APPENDIX: MATHEMATICAL PROOFS
 Acknowledgements
 REFERENCES
 Supporting Information
Given that the null distribution of the individual CADF statistics are asymptotically independent of the nuisance parameters, the various panel unit root tests developed in the literature for the case of the crosssectionally independent errors can also be applied to the present more general case. One possibility would be to consider a crosssectionally augmented version of the IPS test based on
 (30)
where t_{i}(N, T) is the crosssectionally augmented Dickey–Fuller statistic for the ith crosssection unit given by the tratio of the coefficient of y_{i, t−1} in the CADF regression defined by (6). One could also consider combining the pvalues of the individual tests as proposed by Maddala and Wu (1999) and Choi (2001). Examples are the inverse chisquared (or Fisher) test statistic defined by
 (31)
where p_{iT} is the pvalue corresponding to the unit root test of the ith individual crosssection unit. Another possibility would be to use the inverse normal test statistic defined by
 (32)
Here we focus on the tbar version of the panel unit root test (30) and consider the mean deviations
 (33)
where CADF_{if} is the stochastic limit of t_{i}(N, T) as N and T tend to infinity such that N/Tk(0 < k < ∞). See (26). It seems reasonable to expect that D(N, T) = o_{p}(1) for N and T sufficiently large. This conjecture would clearly hold in the case where for each i, t_{i}(N, T) in (33) have finite moments for all N and T above some finite threshold values, say N_{0}, and T_{0}. However, such moment conditions are difficult to establish even under crosssection independence (see IPS).
One possible method of dealing with these technical difficulties would be to base the tbar test on a suitably truncated version of the CADF statistics. The simulation results discussed in Section 3.1 suggest that the standardized version of these statistics are very close to being standard normal with finite first and secondorder moments. Therefore, for the purpose of the panel unit root test it would be equally valid to base the test on an average of the truncated versions of t_{i}(N, T), say , where
 (34)
where K_{1} and K_{2} are positive constants that are sufficiently large so that Pr[−K_{1} < t_{i}(N, T)< K_{2}] is sufficiently large, say in excess of 0.9999. Using the normal approximation of t_{i}(N, T) as a crude benchmark we would have , and , where ε is a sufficiently small positive constant. Using simulated values of E(CADF_{if}) and var(CADF_{if}), and setting ε = 1 × 10^{−6}, we obtain:
The associated truncated panel unit root test is now given by
 (35)
Since by construction all moments of exist it then follows that conditional on W_{f}
 (36)
where is given by
 (37)
and CADF_{if} is defined by (26) in Theorem 3.2. The distributions of the average CADF statistic or its truncated counterpart, , are nonstandard even for sufficiently large N. This is due to the dependence of the individual CADF_{if} variates on the common process W_{f} which invalidates the application of the standard central limit theorems to CADF or CADF*, and is in contrast to the results obtained by IPS under crosssection independence where a standardized version of was shown to be normally distributed for N sufficiently large. Nevertheless, it is possible to show that CADF* converges in distribution as N∞, without any need for further normalization. Recall that CADF_{if} = G(W_{i}, W_{f}), i = 1, 2, …, N, where W_{1}, W_{2}, …., W_{N} and W_{f} are i.i.d. Brownian motions. Similarly, defined by (37) will be a nonlinear function of W_{i} and W_{f} and hence, conditional on W_{f}, will be independently distributed across i. Therefore, since by construction , it follows that, conditional on W_{f},
 (38)
where π_{1} = Pr(CADF_{if} ≤ − K_{1}W_{f}) and π_{2} = Pr(CADF_{if} ≥ K_{2}W_{f}). This result simplifies further if we could also establish that ECADF_{if}< ∞, a property that we conjecture to be true. By letting K_{1} and K_{2}∞, and noting that in this case π_{2}K_{2} − π_{1}K_{1}0, we have .
The above results establish that the CADF* converges almost surely to a distribution which depends on K_{1}, K_{2} and W_{f}. This distribution is not analytically tractable, but can be readily simulated using (35). We simulated the distribution of CIPS* for the three cases setting N = 100, T = 500, and using 50,000 replications. All the three densities show marked departures from normality, although the extent of the departure depends on the nature of the deterministic included in the model.6 The density in the case of the model without any deterministics is bimodal and shows the greatest degree of departure from normality. For the other two cases the densities are unimodal but are highly skewed. The density for the model with a linear trend is closest to being normal. The larger the value of corr(CADF_{i}, CADF_{j}) the greater one would expect the density to depart from normality.
We carried out the same analysis for the nontruncated version, using CIPS defined by (30), and obtained identical results. The finite sample distributions of CIPS*(N, T) and CIPS(N, T) differ only for very small values of T and are indistinguishable for T > 20. The comparative small sample performances of the CIPS and the CIPS* tests will be considered in Section 6.
The 1%, 5% and 10% critical values of CIPS and CIPS* tests are given in Tables II(a)–II(c) for the three cases I–III, respectively. In most cases the critical values for the two versions of the CIPS test are identical and only one value is reported. In cases where the two critical values differ the truncated version is included in parentheses.
Table II(a). Critical values of average of individual crosssectionally augmented Dickey–Fuller distribution (Case I: no intercept and no trend)T/N  10  15  20  30  50  70  100  200 


1%(CADF) 
10  −2.16  −2.02  −1.93  −1.85  −1.78  −1.74  −1.71  −1.70 
 (−2.14)  (−2.00)  (−1.91)  (−1.84)  (−1.77)  (−1.73)   (−1.69) 
15  −2.03  −1.91  −1.84  −1.77  −1.71  −1.68  −1.66  −1.63 
20  −2.00  −1.89  −1.83  −1.76  −1.70  −1.67  −1.65  −1.62 
30  −1.98  −1.87  −1.80  −1.74  −1.69  −1.67  −1.64  −1.61 
50  −1.97  −1.86  −1.80  −1.74  −1.69  −1.66  −1.63  −1.61 
70  −1.95  −1.86  −1.80  −1.74  −1.68  −1.66  −1.63  −1.61 
100  −1.94  −1.85  −1.79  −1.74  −1.68  −1.65  −1.63  −1.61 
200  −1.95  −1.85  −1.79  −1.73  −1.68  −1.65  −1.63  −1.61 
5%(CADF) 
10  −1.80  −1.71  −1.67  −1.61  −1.58  −1.56  −1.54  −1.53 
 (−1.79)   (−1.66)   (−1.57)  (−1.55)  (−1.53)  (−1.52) 
15  −1.74  −1.67  −1.63  −1.58  −1.55  −1.53  −1.52  −1.51 
20  −1.72  −1.65  −1.62  −1.58  −1.54  −1.53  −1.52  −1.50 
30  −1.72  −1.65  −1.61  −1.57  −1.55  −1.54  −1.52  −1.50 
50  −1.72  −1.64  −1.61  −1.57  −1.54  −1.53  −1.52  −1.51 
70  −1.71  −1.65  −1.61  −1.57  −1.54  −1.53  −1.52  −1.51 
100  −1.71  −1.64  −1.61  −1.57  −1.54  −1.53  −1.52  −1.51 
200  −1.71  −1.65  −1.61  −1.57  −1.54  −1.53  −1.52  −1.51 
10%(CADF) 
10  −1.61  −1.56  −1.52  −1.49  −1.46  −1.45  −1.44  −1.43 
 (−1.55)   (−1.48)   (−1.43)  
15  −1.58  −1.53  −1.50  −1.48  −1.45  −1.44  −1.44  −1.43 
20  −1.58  −1.52  −1.50  −1.47  −1.45  −1.45  −1.44  −1.43 
30  −1.57  −1.53  −1.50  −1.47  −1.46  −1.45  −1.44  −1.43 
50  −1.58  −1.52  −1.50  −1.47  −1.45  −1.45  −1.44  −1.43 
70  −1.57  −1.52  −1.50  −1.47  −1.46  −1.45  −1.44  −1.43 
100  −1.56  −1.52  −1.50  −1.48  −1.46  −1.45  −1.44  −1.43 
200  −1.57  −1.53  −1.50  −1.47  −1.45  −1.45  −1.44  −1.43 
Table II(b). Critical values of average of individual crosssectionally augmented Dickey–Fuller distribution (Case II: intercept only)T/N  10  15  20  30  50  70  100  200 


1%(CADF) 
10  −2.97  −2.76  −2.64  −2.51  −2.41  −2.37  −2.33  −2.28 
 (−2.85)  (−2.66)  (−2.56)  (−2.44)  (−2.36)  (−2.32)  (−2.29)  (−2.25) 
15  −2.66  −2.52  −2.45  −2.34  −2.26  −2.23  −2.19  −2.16 
20  −2.60  −2.47  −2.40  −2.32  −2.25  −2.20  −2.18  −2.14 
30  −2.57  −2.45  −2.38  −2.30  −2.23  −2.19  −2.17  −2.14 
50  −2.55  −2.44  −2.36  −2.30  −2.23  −2.20  −2.17  −2.14 
70  −2.54  −2.43  −2.36  −2.30  −2.23  −2.20  −2.17  −2.14 
100  −2.53  −2.42  −2.36  −2.30  −2.23  −2.20  −2.18  −2.15 
200  −2.53  −2.43  −2.36  −2.30  −2.23  −2.21  −2.18  −2.15 
5%(CADF) 
10  −2.52  −2.40  −2.33  −2.25  −2.19  −2.16  −2.14  −2.10 
 (−2.47)  (−2.35)  (−2.29)  (−2.22)  (−2.16)  (−2.13)  (−2.11)  (−2.08) 
15  −2.37  −2.28  −2.22  −2.17  −2.11  −2.09  −2.07  −2.04 
20  −2.34  −2.26  −2.21  −2.15  −2.11  −2.08  −2.07  −2.04 
30  −2.33  −2.25  −2.20  −2.15  −2.11  −2.08  −2.07  −2.05 
50  −2.33  −2.25  −2.20  −2.16  −2.11  −2.10  −2.08  −2.06 
70  −2.33  −2.25  −2.20  −2.15  −2.12  −2.10  −2.08  −2.06 
100  −2.32  −2.25  −2.20  −2.16  −2.12  −2.10  −2.08  −2.07 
200  −2.32  −2.25  −2.20  −2.16  −2.12  −2.10  −2.08  −2.07 
10%(CADF) 
10  −2.31  −2.22  −2.18  −2.12  −2.07  −2.05  −2.03  −2.01 
 (−2.28)  (−2.20)  (−2.15)  (−2.10)  (−2.05)  (−2.03)  (−2.01)  (−1.99) 
15  −2.22  −2.16  −2.11  −2.07  −2.03  −2.01  −2.00  −1.98 
20  −2.21  −2.14  −2.10  −2.07  −2.03  −2.01  −2.00  −1.99 
30  −2.21  −2.14  −2.11  −2.07  −2.04  −2.02  −2.01  −2.00 
50  −2.21  −2.14  −2.11  −2.08  −2.05  −2.03  −2.02  −2.01 
70  −2.21  −2.15  −2.11  −2.08  −2.05  −2.03  −2.02  −2.01 
100  −2.21  −2.15  −2.11  −2.08  −2.05  −2.03  −2.03  −2.02 
200  −2.21  −2.15  −2.11  −2.08  −2.05  −2.04  −2.03  −2.02 
Table II(c). Critical values of average of individual crosssectionally augmented Dickey–Fuller distribution (Case III: intercept and trend)T/N  10  15  20  30  50  70  100  200 


1%(CADF) 
10  −3.88  −3.61  −3.46  −3.30  −3.15  −3.10  −3.05  −2.98 
 (−3.51)  (−3.31)  (−3.20)  (−3.10)  (−3.00)  (−2.96)  (−2.93)  (−2.88) 
15  −3.24  −3.09  −3.00  −2.89  −2.81  −2.77  −2.74  −2.71 
 (−3.21)  (−3.07)  (−2.98)  (−2.88)  (−2.80)  (−2.76)   (−2.70) 
20  −3.15  −3.01  −2.92  −2.83  −2.76  −2.72  −2.70  −2.65 
30  −3.10  −2.96  −2.88  −2.81  −2.73  −2.69  −2.66  −2.63 
50  −3.06  −2.93  −2.85  −2.78  −2.72  −2.68  −2.65  −2.62 
70  −3.04  −2.93  −2.85  −2.78  −2.71  −2.68  −2.65  −2.62 
100  −3.03  −2.92  −2.85  −2.77  −2.71  −2.68  −2.65  −2.62 
200  −3.03  −2.91  −2.85  −2.77  −2.71  −2.67  −2.65  −2.62 
5%(CADF) 
10  −3.27  −3.11  −3.02  −2.94  −2.86  −2.82  −2.79  −2.75 
 (−3.10)  (−2.97)  (−2.89)  (−2.82)  (−2.75)  (−2.73)  (−2.70)  (−2.67) 
15  −2.93  −2.83  −2.77  −2.70  −2.64  −2.62  −2.60  −2.57 
 (−2.92)  (−2.82)  (−2.76)  (−2.69)   (−2.59)  
20  −2.88  −2.78  −2.73  −2.67  −2.62  −2.59  −2.57  −2.55 
30  −2.86  −2.76  −2.72  −2.66  −2.61  −2.58  −2.56  −2.54 
50  −2.84  −2.76  −2.71  −2.65  −2.60  −2.58  −2.56  −2.54 
70  −2.83  −2.76  −2.70  −2.65  −2.61  −2.58  −2.57  −2.54 
100  −2.83  −2.75  −2.70  −2.65  −2.61  −2.59  −2.56  −2.55 
200  −2.83  −2.75  −2.70  −2.65  −2.61  −2.59  −2.57  −2.55 
10%(CADF) 
10  −2.98  −2.89  −2.82  −2.76  −2.71  −2.68  −2.66  −2.63 
 (−2.87)  (−2.78)  (−2.73)  (−2.67)  (−2.63)  (−2.60)  (−2.58)  (−2.56) 
15  −2.76  −2.69  −2.65  −2.60  −2.56  −2.54  −2.52  −2.50 
 (−2.68)  (−2.64)  (−2.59)  (−2.55)  (−2.53)  (−2.51)  
20  −2.74  −2.67  −2.63  −2.58  −2.54  −2.53  −2.51  −2.49 
30  −2.73  −2.66  −2.63  −2.58  −2.54  −2.52  −2.51  −2.49 
50  −2.73  −2.66  −2.63  −2.58  −2.55  −2.53  −2.51  −2.50 
70  −2.72  −2.66  −2.62  −2.58  −2.55  −2.53  −2.52  −2.50 
100  −2.72  −2.66  −2.63  −2.59  −2.55  −2.53  −2.52  −2.50 
200  −2.73  −2.66  −2.63  −2.59  −2.55  −2.54  −2.52  −2.51 
Similar arguments also apply to the other forms of the panel unit root tests given by (31) and (32). The crosssectionally augmented versions of these statistics, where the rejection probabilities, p_{iT}, are computed using the CADF regressions, (6), will be denoted by CP(N, T) and CZ(N, T). Note that in the presence of crosssection dependence these statistics are no longer asymptotically normally distributed and their critical values must be obtained by stochastic simulations. The 1%, 5% and 10% critical values of CP(N, T) and CZ(N, T) are computed by Mutita Akusuwan for all pairs of N, T = 10, 15, 20, 30, 50, 70, 100, 200, and are available from the author on request.
6. SMALLSAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
 Top of page
 Abstract
 1. INTRODUCTION
 2. A SIMPLE DYNAMIC PANEL WITH CROSSSECTION DEPENDENCE
 3. UNIT ROOT TESTS FOR ONEFACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
 4. CADF PANEL UNIT ROOT TESTS
 5. CASE OF SERIALLY CORRELATED ERRORS
 6. SMALLSAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
 7. EMPIRICAL APPLICATIONS
 8. CONCLUDING REMARKS
 APPENDIX: MATHEMATICAL PROOFS
 Acknowledgements
 REFERENCES
 Supporting Information
Initially we shall consider dynamic panels with fixed effects and crosssection dependence, but without residual serial correlation or linear trends. The datagenerating process (DGP) in this case is given by
 (55)
where
 (56)
and
 (57)
We shall consider two levels of crosssection dependence where we generate γ_{i}∼i.i.d.U[0, 0.20] as an example of ‘low crosssection dependence’, and γ_{i}∼i.i.d.U[−1, 3] to represent the case of ‘high crosssection dependence’. The average pairwise cross correlation coefficient of u_{it} and u_{jt} under these two scenarios are 1% and 50%, respectively, and cover a wide range of values applicable in practice.
To examine the impact of the residual serial correlation on the proposed tests we considered a number of experiments where the errors ε_{it} were generated as
 (58)
with ρ_{i}∼i.i.d.U[0.2, 0.4], as an example of positive residual serial correlations, and ρ_{i}∼i.i.d.U[−0.4, − 0.2], as an example of negative residual serial correlations. This yields the augmented ADF model given by (41). This DGP differs from model (44) that underlies the theoretical derivations in Section 5, and is intended to check the robustness of our analysis to alternative residual serial correlation models. It also allows the residual serial correlation coefficients, ρ_{i}, to differ across i, thus providing an opportunity to check the robustness of our results to such heterogeneities.
In a third set of experiments we allow for deterministic trends in the DGP and the CADF regressions. For this case y_{it} were generated as follows:
with μ_{i}∼i.i.d.U[0.0, 0.02] and δ_{i}∼i.i.d.U[0.0, 0.02]. This ensures that y_{it} has the same average trend properties under the null and the alternative hypotheses. The errors, u_{it}, were generated according to (56), (57) and (58) for different values of ρ_{i} as set out above.
Size and power of the tests were computed under the null ϕ_{i} = 1 for all i, and the heterogeneous alternatives ϕ_{i}∼i.i.d.U[0.85, 0.95], using 1000 replications per experiment.9 The tests were onesided with the nominal size set at 5%, and were conducted for all combinations of N and T = 10, 20, 30, 50, 100. All the parameters, μ_{i}, δ_{i}, ϕ_{i}, ρ_{i}, , and γ_{i} were generated independently of the errors, e_{it}(ε_{it}) and f_{t}, with f_{t} also generated independently of e_{it}(ε_{it}).
6.1. Size Distortion of the Standard Panel Unit Root Tests
Before reporting the results for the proposed crosssectionally augmented tests, it would be helpful first to examine the extent to which the size of the standard panel unit root tests (that assume crosssection independence) are distorted in the presence of crosssection dependence. Table III reports the empirical sizes of the IPS, truncated IPS, the inverse chisquared (P), and the inverse normal (Z) tests when the DGP is subject to crosssection dependence with serially uncorrelated errors as defined by (55) and (56).10 All these tests are based on simple DF regressions and utilize the individual DF statistics, or the associated rejection probabilities.11 The IPS statistic is the familiar standardized tbar statistic defined by (see IPS, 2003):
 (59)
where , and t_{iT} is the tratio of the estimated coefficient of y_{i, t−1} in the OLS regression of Δy_{it} on an intercept and y_{i, t−1}. The truncated version of the IPS test uses the same formula as above but replaces t_{iT} with the individually truncated statistic, , defined by (34) with K_{1} = 6.19, and K_{2} = 2.61. In the case of P and Z tests we report two sets of results: one set based on normal approximations as originally proposed by Maddala and Wu (1999) and Choi (2001), and another set based on empirical critical values obtained from the simulated distribution of these statistics under the null hypothesis. We refer to the latter versions of these tests as and the tests.
Table III. Size of panel unit root tests that do not allow for crosssectional dependence: no serial correlation, low and high crosssection dependence, intercept caseN  Test  Low crosssection dependence T  High crosssection dependence T 

10  20  30  50  100  10  20  30  50  100 


10  IPS  0.041  0.046  0.056  0.051  0.046  0.093  0.117  0.122  0.111  0.114 
IPS*  0.040  0.046  0.056  0.051  0.046  0.095  0.118  0.122  0.111  0.114 
Ptest (DF)  
Normal approximation  0.186  0.105  0.092  0.074  0.063  0.207  0.159  0.155  0.126  0.124 
Empirical distribution  0.030  0.042  0.048  0.048  0.041  0.054  0.082  0.108  0.092  0.095 
Ztest (DF)  
Normal approximation  0.078  0.055  0.066  0.057  0.048  0.139  0.135  0.135  0.122  0.122 
Empirical distribution  0.039  0.043  0.056  0.052  0.043  0.094  0.118  0.123  0.114  0.119 
20  IPS  0.043  0.054  0.045  0.048  0.051  0.179  0.218  0.213  0.215  0.246 
IPS*  0.045  0.054  0.045  0.048  0.051  0.182  0.219  0.213  0.215  0.246 
Ptest (DF)  
Normal approximation  0.227  0.123  0.084  0.083  0.071  0.242  0.216  0.178  0.188  0.207 
Empirical distribution  0.035  0.053  0.041  0.049  0.051  0.104  0.148  0.134  0.157  0.191 
Ztest (DF)  
Normal approximation  0.067  0.058  0.047  0.056  0.052  0.214  0.227  0.217  0.221  0.243 
Empirical distribution  0.043  0.051  0.039  0.052  0.052  0.178  0.213  0.204  0.216  0.243 
30  IPS  0.037  0.035  0.052  0.047  0.060  0.208  0.234  0.236  0.256  0.283 
IPS*  0.038  0.034  0.052  0.047  0.060  0.210  0.234  0.236  0.256  0.283 
Ptest (DF)  
Normal approximation  0.242  0.119  0.103  0.071  0.067  0.299  0.237  0.222  0.221  0.231 
Empirical distribution  0.027  0.039  0.056  0.047  0.051  0.125  0.157  0.163  0.180  0.208 
Ztest (DF)  
Normal approximation  0.066  0.042  0.056  0.045  0.052  0.251  0.245  0.239  0.253  0.277 
Empirical distribution  0.042  0.035  0.050  0.042  0.052  0.214  0.234  0.237  0.249  0.277 
50  IPS  0.046  0.032  0.050  0.047  0.039  0.285  0.314  0.318  0.334  0.370 
IPS*  0.047  0.032  0.050  0.047  0.039  0.291  0.314  0.318  0.334  0.370 
Ptest (DF)  
Normal approximation  0.314  0.145  0.128  0.089  0.059  0.327  0.274  0.271  0.277  0.291 
Empirical distribution  0.040  0.032  0.053  0.054  0.044  0.168  0.214  0.223  0.246  0.271 
Ztest (DF)  
Normal approximation  0.075  0.035  0.052  0.047  0.041  0.307  0.314  0.322  0.330  0.354 
Empirical distribution  0.043  0.032  0.050  0.048  0.042  0.285  0.305  0.316  0.330  0.362 
100  IPS  0.035  0.067  0.059  0.057  0.059  0.330  0.382  0.372  0.383  0.406 
IPS*  0.036  0.067  0.059  0.057  0.059  0.336  0.383  0.372  0.383  0.406 
Ptest (DF)  
Normal approximation  0.409  0.181  0.120  0.090  0.091  0.379  0.340  0.306  0.310  0.325 
Empirical distribution  0.026  0.049  0.040  0.048  0.054  0.218  0.259  0.263  0.289  0.314 
Ztest (DF)  
Normal approximation  0.052  0.062  0.041  0.048  0.049  0.345  0.367  0.354  0.376  0.394 
Empirical distribution  0.038  0.062  0.044  0.049  0.052  0.330  0.367  0.355  0.378  0.394 
As to be expected, the extent of overrejection of the tests very much depends on the degree of crosssection dependence. Under the low crosssection dependence the different version of the IPS and the Z tests perform reasonably well. The standard P test tends to overreject for small values of T, but the normal approximation begins to work as T is increased. Overall, when the crosssection dependence is low all tests (possibly except for the P test) have the correct size, which is in line with the results in the literature, reported, for example, by Choi (2001). But in the case of high crosssection scenario all the tests tend to overreject, often by a substantial amount. Clearly, the standard panel unit root tests that do not allow for crosssection dependence can be seriously biased if the degree of crosssection dependence is sufficiently large. Therefore, in what follows we shall focus on the high crosssection dependence scenario.
6.2. Size and Power: Case of Serially Uncorrelated Errors
The tests to be considered are the crosssectionally augmented IPS test, CIPS(N, T), and its truncated version, CIPS*(N, T), and the crosssectionally augmented versions of the inverse chisquared and the inverse normal tests, denoted by CP(N, T) and CZ(N, T), respectively. The CIPS and CIPS* statistics are defined by (30) and (35), respectively. The computation of the CP and CZ statistics require the estimation of individualspecific rejection probabilities by stochastic simulations. In particular, the crosssectionally augmented inverse chisquared test statistic is given by
 (60)
where the rejection probabilities are computed as
 (61)
where ����ℱ^{(s)} is the sth random draw from the distribution of ����ℱ, defined by (26), and S is the number of replications used to compute , which we also set equal to 50,000. I[A] is the indicator function that takes the value of 1 when A > 0, and 0 otherwise. Similarly,
 (62)
To avoid very extreme values the rejection probabilities were truncated to lie in the range [0.000001, 0.999999].
6.3. Size and Power: Case of Serially Correlated Errors
In the case of models with serially correlated errors, the crosssectionally augmented tests (CIPS, CIPS*, , and ) are computed both for the basic CADF regressions without time series augmentation (which we denote by CADF(0)), and the CADF regressions are augmented with lagged changes of y_{it} and ȳ_{t}, as in (54), which we refer to by CADF(p), where p is the order of the time series augmentation. We computed the tests for p = 0, 1, and focused on the high crosssection dependence scenario. The size and power results for the experiments with positive residual serial correlation are summarized in Tables V(a) and V(b), and the ones for negative residual serial correlation are given in Tables VI(a) and VI(b). As to be expected, significant size distortions will be present if CADF regressions are not augmented to account for the time series dependence. There are substantial underrejections for positive residual serial correlation, and substantial overrejections in the case of negative residual serial correlations. But the test sizes stabilize at around 5% when the CADF regressions are augmented with Δy_{i, t−1}. Irrespective of whether the residual serial correlations are positive or negative, the tests based on CADF(1) regressions tend to have the correct size. There is, however, some evidence that for small T (less than 20 in these experiments) the test, and to a lesser extent the CIPS test, are oversized. But the truncated version of the CIPS does not seem to suffer from this problem even for T as small as 10. The truncation of the extreme individual CADF statistics seem to have paid out in the present application where T is very small relative to the number of parameters of the underlying CADF(1) regressions.
Focusing on CIPS* and we note from Tables V(b) and VI(b) that both tests have very similar power properties. Neither of the tests seem to have any power for T = 10 or less, and as in the serially uncorrelated case the power does not rise with N if T is too small. However, with T = 20 or higher the power of both tests begins to rise quite rapidly with N.13 The tests tend to show higher power for negative as compared to positive residual serial correlations. Finally, there is very little to choose between the two tests, although as noted earlier the CIPS* statistic is much simpler to compute.
6.4. Size and Power: Models with Linear Trends and Serially Correlated Errors
8. CONCLUDING REMARKS
 Top of page
 Abstract
 1. INTRODUCTION
 2. A SIMPLE DYNAMIC PANEL WITH CROSSSECTION DEPENDENCE
 3. UNIT ROOT TESTS FOR ONEFACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
 4. CADF PANEL UNIT ROOT TESTS
 5. CASE OF SERIALLY CORRELATED ERRORS
 6. SMALLSAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
 7. EMPIRICAL APPLICATIONS
 8. CONCLUDING REMARKS
 APPENDIX: MATHEMATICAL PROOFS
 Acknowledgements
 REFERENCES
 Supporting Information
This paper presents a new and simple procedure for testing unit roots in dynamic panels subject to (possibly) crosssectionally dependent as well as serially correlated errors. The procedure involves augmenting the standard ADF regressions for the individual series with current and lagged crosssection averages of all the series in the panel. This is a natural extension of the DF approach to dealing with residual serial correlation where lagged changes of the series are used to filter out the time series dependence when T is sufficiently large. Here we propose to use crosssection averages to perform a similar task in dealing with the crossdependence problem. Our approach should be seen as providing a simple alternative to the orthogonalization type procedures advanced in the literature by Bai and Ng (2004), Moon and Perron (2004), and Phillips and Sul (2003). Although we have provided extensive simulation results in support of our proposed tests, further simulation experiments are needed to shed light on the relative merits of the various panel unit roots that are now available in the literature.
Our analysis and testing approach can also be extended in a number of directions. One obvious generalization is to allow for a richer pattern of crossdependence by including additional common factors in the model. This is likely to pose additional technical difficulties, but can be dealt with by augmenting the individual ADF regressions with additional crosssection averages formed over subgroups, such as regions, sectors or industries. Another worthwhile extension would be to consider crosssection augmented versions of unit root tests due to Elliott et al. (1996), Fuller and Park (1995), and Leybourne (1995). Such tests are likely to have better smallsample power properties.
In their analysis Bai and Ng (2004) also consider the possibility of unit root in the common factors. However, under their setup the unit root properties of the common factor(s) and the idiosyncractic component of the individual series are unrelated. As a result they are able to carry out separate unit root tests in the common and the idiosyncractic components. The specification used by Bai and Ng is given by the static factor model (assuming one factor for ease of comparison):
where f_{t} is the common factor, γ_{i} the associated factor loadings, and v_{it} the idiosyncractic component assumed independently distributed of f_{t}. The unit root properties of y_{it} is determined by the maximum order of integration of the two series f_{t} and v_{it}. Hence, y_{it} will be I(1) if either v_{it} and/or f_{t} contain a unit root. Averaging across i and letting N∞, for each t, v̄_{t}0, if v_{it} is stationary, and v̄_{t}c, where c is a fixed constant if v_{it} is I(1). Therefore, a unit root in f_{t} may be tested by testing the presence of a unit root in ȳ_{t} independently of whether the idiosyncractic components are I(0) or I(1). By contrast, in our specifications, the common factor is introduced to model cross section dependence of the stationary components. As a result when testing ϕ_{i} = 1, the order of integration of y_{it} changes from being I(1) if f_{t} is stationary, to I(2) if f_{t} is I(1). Therefore, in our setup it makes sense not to allow f_{t} to have a unit root. The models advanced here and the static factor model used by Bai and Ng serve different purposes.