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Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information

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 augmented Dickey–Fuller (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 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

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. 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 See, for example, Bowman D (unpublished 1999), Choi (2001), Hadri (2000), Im et al.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 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 [RIGHTWARDS ARROW] ∞. 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 TN. 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 [RIGHTWARDS ARROW] 0, as N and T [RIGHTWARDS ARROW] ∞. 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 [RIGHTWARDS ARROW] 0, as N and T [RIGHTWARDS ARROW] ∞. 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 CADFi and the associated equation image statistics are investigated as N [RIGHTWARDS ARROW] ∞ followed by T [RIGHTWARDS ARROW] ∞, as well as jointly with N and T tending to infinity such that N/T [RIGHTWARDS ARROW] k, where k is a fixed finite non-zero positive constant. It is shown that the CADFi 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 [RIGHTWARDS ARROW] 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: an = O(bn) states the deterministic sequence {an} is at most of order bn, xn = Op(yn) states the vector of random variables, xn, is at most of order yn in probability, and xn = op(yn) is of smaller order in probability than yn. [RIGHTWARDS ARROW] denotes convergence in quadratic means (q.m.) or mean square errors and ↠ convergence in distribution. All asymptotics are carried out under N [RIGHTWARDS ARROW] ∞, either with a fixed T, sequentially, or jointly with T [RIGHTWARDS ARROW] ∞. In particular, equation image denotes convergence in distribution (q.m.) with T fixed as N [RIGHTWARDS ARROW] ∞, equation image (equation image) denotes convergence in distribution (q.m.) for N fixed (or when there is no N-dependence) as T [RIGHTWARDS ARROW] ∞, equation image denotes sequential convergence with N [RIGHTWARDS ARROW] ∞ first followed by T [RIGHTWARDS ARROW] ∞ (similarly equation image), equation image denotes joint convergence with N, T [RIGHTWARDS ARROW] ∞ jointly such that N/T [RIGHTWARDS ARROW] k, where k is a fixed finite non-zero constant. ∼ denotes asymptotic equivalence in distribution, with equation image, equation image, equation image, equation image, and equation image, similarly defined as equation image, equation image, etc.

2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information

Let yit be the observation on the ith cross-section unit at time t and suppose that it is generated according to the simple dynamic linear heterogeneous panel data model

  • equation image(1)

where initial value, yi0, has a given density function with a finite mean and variance, and the error term, uit, has the single-factor structure

  • equation image(2)

in which ft is the unobserved common effect, and εit is the individual-specific (idiosyncratic) error.

It is convenient to write (1) and (2) as

  • equation image(3)

where αi = (1 − ϕii, βi = − (1 − ϕi) and Δyit = yityi, t−1. The unit root hypothesis of interest, ϕi = 1, can now be expressed as

  • equation image(4)

against the possibly heterogeneous alternatives,

  • equation image(5)

We shall assume that N1/N, the fraction of the individual processes that are stationary, is non-zero and tends to the fixed value δ such that 0 < δ≤ 1 as N[RIGHTWARDS ARROW]∞. 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, εit, i = 1, 2, …, N, t = 1, 2, …, T, are independently distributed both acrossiandt, have mean zero, varianceequation image, and finite fourth-order moment.

Assumption 2The common factor,ft, is serially uncorrelated with mean zero and a constant variance,equation image, and finite fourth-order moment. Without loss of generalityequation imagewill be set equal to unity.

Assumption 3 εit, ft, and γiare independently distributed for alli.

The cross-section independence of εit (across i) is standard in one-factor 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, uit, is serially uncorrelated. Both of these assumptions can be relaxed. It is possible to relax Assumption 2 by allowing the common factor, ft, 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[RIGHTWARDS ARROW]∞, jointly.4 Alternatively, one could consider serial correlation in both ft and εit 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 ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information

Let equation image and suppose that equation image for a fixed N and as N[RIGHTWARDS ARROW]∞. Then following the line of reasoning in Pesaran (2006), the common factor ft can be proxied by the cross-section mean of yit, namely equation image, and its lagged value(s), ȳt−1, ȳt−2, … for N sufficiently large. In the simple case where uit is serially uncorrelated, it turns out that ȳt and ȳt−1 (or equivalently ȳt−1 and Δȳt) are sufficient for asymptotically filtering out the effects of the unobserved common factor, ft. We shall therefore base our test of the unit root hypothesis, (4), on the t-ratio of the OLS estimate of bi(i) in the following cross-sectionally augmented DF (CADF) regression:

  • equation image(6)

Denoting this t-ratio by ti(N, T) we have

  • equation image(7)

where

  • equation image(8)
  • equation image(9)
  • equation image(10)
  • equation image(11)

and

  • equation image(12)

Under βi = 0 we have Δyit = γift + εit and hence

  • equation image(13)
  • equation image(14)
  • equation image(15)
  • equation image(16)

where εi = (εi1, εi2, …, εiT)′, f =(f1, f2, …, fT)′, equation image, equation image, yi0 is a given initial value (fixed or random), equation image, si,−1 = (0, si1, …, si, T−1)′, sf,−1 = (sf0, sf1, …, sf, T−1)′, −1 = (0, 1, …, T−1) with equation image, and equation image, for t = 1, 2, ., and equation image. Using (15) to eliminate f from (13), and noting that by assumption equation image, we have

  • equation image(17)

where

  • equation image(18)

Therefore,

  • equation image(19)

where equation image

  • equation image(20)

and equation image. Therefore,

  • equation image(21)

where υi = ξii∼(0, IT).

Similarly, using (16) to eliminate sf,−1 from (14) we obtain

  • equation image(22)

and hence

  • equation image(23)

where equation image. It is easily seen that equation image is the random walk associated with υi.

Using (11), (21) and (23) in (7) we have

  • equation image(24)

Hence, the exact null distribution of ti(N, T) will depend on the nuisance parameters only through their effects on w and Mi, w. But, as shown in the Appendix, this dependence vanishes as N[RIGHTWARDS ARROW]∞, irrespective of whether T is fixed or tends to infinity jointly with N. In the case where T is fixed, to ensure that ti(N, T) does not depend on the nuisance parameters the effect of the initial cross-section mean, ȳ0, must also be eliminated. This can be achieved by applying the test to the deviations yitȳ0. The following theorems provide a formal statement of these results.

Theorem 3.1Suppose the seriesyit, fori = 1, 2, …, Nandt = 1, 2, …, T, are generated under(4)according to(3)and by construction ȳ0(the cross-section mean of the initial observations) is set to zero. Then under Assumptions 1 and 2 the null distribution ofti(N, T) given by(24)will be free of nuisance parameters asN[RIGHTWARDS ARROW]for any fixedT > 4. In particular, we have (in quadratic mean)

  • equation image(25)

where

  • equation image

εiiandfare independently distributed as (0, IT), si = si,−1 + εi, andsf = sf,−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 (εi, f).

Theorem 3.2Letyitbe defined by(3)and consider the statisticsti(N, T) defined by(7). Suppose that Assumptions 1–3 hold andequation imagetends to a finite non-zero limit asN[RIGHTWARDS ARROW]∞, then under(4)and asNandT[RIGHTWARDS ARROW]∞, ti(N, T) has the same sequential (N[RIGHTWARDS ARROW]∞, T[RIGHTWARDS ARROW]∞) and joint [(N, T)j[RIGHTWARDS ARROW]∞] limit distributions, referred to as cross-sectionally augmented Dickey–Fuller (CADF) distribution given by

  • equation image(26)

where

  • equation image(27)

and

  • equation image(28)

withWi(r) andWf(r) being independent standard Brownian motions. For the joint limit distribution to hold it is also required that as (N, T)j[RIGHTWARDS ARROW]∞, N/T[RIGHTWARDS ARROW]k, wherekis a non-zero, finite positive constant.

Remark 3.1The critical values ofCADFifcan be computed by stochastic simulation assuming that

  • equation image

Remark 3.2From(26)it is clear that

  • equation image(29)

whereG(.) is a general nonlinear function common for alli. Therefore,CADFifandCADFjfare dependently distributed, with the same degree of dependence for allij.

Remark 3.3The random variablesCADF1f, CADF2f, …, CADFNfform an exchangeable sequence. This follows from the fact that under Assumption 3, conditional onWfthe random variables {CADFif, 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 non-stationary 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 cross-section average,ȳt, 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 cross-dependence problem. But following Pesaran (2006) weighted averages could also be used instead, for any given set of fixed weights,wi, such thatequation imageasN[RIGHTWARDS ARROW]∞.

Remark 3.6Our assumption thatequation image, with a non-zero limit asN[RIGHTWARDS ARROW]∞, might be viewed by some as restrictive. Underequation image, for example, the possibility of cross-section independence is ruled out. However, it is instructive to recall from(15)that in the case whereequation image, we haveequation image, and therefore Δȳtwould tend to zero asN[RIGHTWARDS ARROW]∞, andȳtto 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 whereequation image, asN[RIGHTWARDS ARROW]∞, could be of theoretical interest and is worth further consideration.

Remark 3.7As noted earlier, the asymptotic distribution ofti(N, T), given by(26), continues to hold if Assumption 2 is relaxed andftis allowed to follow the general linear stationary process:

  • equation image

whereequation image, equation imageis 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), withWf(r) replaced byequation image(see, for example, Hamilton, 1994, pp. 504–505). But it is readily verified that the term ψ(1) cancels out from the asymptotic limit ofti(N, T) asNandT[RIGHTWARDS ARROW]∞.

3.1. Critical Values of the Individual CADF Test

The computation of the critical values were based on 50,000 CADF regressions of Δy1t on y1, t−1, ȳt−1, Δȳt, 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 yit = yi, t−1 + ft + εit, for i = 1, 2, …, N, and t = − 50, − 49, …, 1, 2, …, T from yi,−50 = 0, with ft and εit 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 cross-sectionally augmented Dickey–Fuller distribution (Case I: no intercept and no trend)
T/N101520305070100200
  1. Notes:

  2. The calculations are carried out for 50,000 replications based on the OLS regression of Δyit on yi, t−1, ȳt−1 and Δȳt, where equation image. CADFi refers to the OLS t-ratio of the coefficient of yi, t−1.

  3. The critical values for the truncated version of the test statistics are indicated in parentheses if they differ from the non-truncated ones.

1%(CADFi)
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%(CADFi)
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%(CADFi)
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 cross-sectionally augmented Dickey–Fuller distribution (Case II: intercept only)
T/N101520305070100200
  1. Notes:

  2. The calculations are carried out for 50,000 replications based on the OLS regression of Δyit on an intercept, yi, t−1, ȳt−1 and Δȳt, where equation image. CADFi refers to the OLS t-ratio of the coefficient of yi, t−1.

  3. The critical values for the truncated version of the test statistics are indicated in parentheses if they differ from the non-truncated ones.

1%(CADFi)
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%(CADFi)
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%(CADFi)
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 cross-sectionally augmented Dickey–Fuller distribution (Case III: intercept and trend)
T/N101520305070100200
  1. Notes:

  2. The calculations are carried out for 50,000 replications based on the OLS regression of Δyit on an intercept, trend, yi, t−1, ȳt−1 and Δȳt, where equation image. CADFi refers to the OLS t-ratio of the coefficient of yi, t−1.

  3. The critical values for the truncated version of the test statistics are indicated in parentheses if they differ from the non-truncated ones.

1%(CADFi)
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%(CADFi)
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%(CADFi)
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 pair-wise dependence of the CADFif statistics across i. We simulated the simple pair-wise correlation coefficient, corr(CADFif, CADFjf), 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 non-zero. 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 equation image, 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

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. 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 cross-sectionally independent errors can also be applied to the present more general case. One possibility would be to consider a cross-sectionally augmented version of the IPS test based on

  • equation image(30)

where ti(N, T) is the cross-sectionally augmented Dickey–Fuller statistic for the ith cross-section unit given by the t-ratio of the coefficient of yi, t−1 in the CADF regression defined by (6). One could also consider combining the p-values of the individual tests as proposed by Maddala and Wu (1999) and Choi (2001). Examples are the inverse chi-squared (or Fisher) test statistic defined by

  • equation image(31)

where piT is the p-value corresponding to the unit root test of the ith individual cross-section unit. Another possibility would be to use the inverse normal test statistic defined by

  • equation image(32)

Here we focus on the t-bar version of the panel unit root test (30) and consider the mean deviations

  • equation image(33)

where CADFif is the stochastic limit of ti(N, T) as N and T tend to infinity such that N/T[RIGHTWARDS ARROW]k(0 < k < ∞). See (26). It seems reasonable to expect that D(N, T) = op(1) for N and T sufficiently large. This conjecture would clearly hold in the case where for each i, ti(N, T) in (33) have finite moments for all N and T above some finite threshold values, say N0, and T0. However, such moment conditions are difficult to establish even under cross-section independence (see IPS).

One possible method of dealing with these technical difficulties would be to base the t-bar 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 second-order 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 ti(N, T), say equation image, where

  • equation image(34)

where K1 and K2 are positive constants that are sufficiently large so that Pr[−K1 < ti(N, T)< K2] is sufficiently large, say in excess of 0.9999. Using the normal approximation of ti(N, T) as a crude benchmark we would have equation image, and equation image, where ε is a sufficiently small positive constant. Using simulated values of E(CADFif) and var(CADFif), and setting ε = 1 × 10−6, we obtain:

  • equation image

The associated truncated panel unit root test is now given by

  • equation image(35)

Since by construction all moments of equation image exist it then follows that conditional on Wf

  • equation image(36)

where equation image is given by

  • equation image(37)

and CADFif is defined by (26) in Theorem 3.2. The distributions of the average CADF statistic or its truncated counterpart, equation image, are non-standard even for sufficiently large N. This is due to the dependence of the individual CADFif variates on the common process Wf 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 cross-section independence where a standardized version of equation image was shown to be normally distributed for N sufficiently large. Nevertheless, it is possible to show that CADF* converges in distribution as N[RIGHTWARDS ARROW]∞, without any need for further normalization. Recall that CADFif = G(Wi, Wf), i = 1, 2, …, N, where W1, W2, …., WN and Wf are i.i.d. Brownian motions. Similarly, equation image defined by (37) will be a nonlinear function of Wi and Wf and hence, conditional on Wf, equation image will be independently distributed across i. Therefore, since by construction equation image, it follows that, conditional on Wf,

  • equation image(38)

where π1 = Pr(CADFif ≤ − K1|Wf) and π2 = Pr(CADFifK2|Wf). This result simplifies further if we could also establish that E|CADFif|< ∞, a property that we conjecture to be true. By letting K1 and K2[RIGHTWARDS ARROW]∞, and noting that in this case π2K2 − π1K1[RIGHTWARDS ARROW]0, we have equation image.

The above results establish that the CADF* converges almost surely to a distribution which depends on K1, K2 and Wf. 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(CADFi, CADFj) the greater one would expect the density to depart from normality.

We carried out the same analysis for the non-truncated 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 cross-sectionally augmented Dickey–Fuller distribution (Case I: no intercept and no trend)
T/N101520305070100200
  1. Note: CADF statistic is computed as the simple average of the individual-specific CADFi statistics. See notes to Table I(a).

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 cross-sectionally augmented Dickey–Fuller distribution (Case II: intercept only)
T/N101520305070100200
  1. Note: CADF statistic is computed as the simple average of the individual-specific CADFi statistics. See notes to Table I(b).

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 cross-sectionally augmented Dickey–Fuller distribution (Case III: intercept and trend)
T/N101520305070100200
  1. Note: CADF statistic is computed as the simple average of the individual-specific CADFi statistics. See notes to Table I(c).

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 cross-sectionally augmented versions of these statistics, where the rejection probabilities, piT, are computed using the CADF regressions, (6), will be denoted by CP(N, T) and CZ(N, T). Note that in the presence of cross-section 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.

5. CASE OF SERIALLY CORRELATED ERRORS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information

The CIPS testing procedure can be readily extended to the case where, in addition to the cross-dependence, the individual-specific error terms are also serially correlated.

5.1. Alternative Residual Serial Correlation Models with Cross-Dependence

The residual serial correlation can be modelled in a number of different ways: directly via the idiosyncractic components, through the common effects, or a mixture of the two. To simplify the exposition we shall confine our analysis to stationary first-order autoregressive processes and consider three general types of specifications.7 The case where the residual serial correlation is confined to the common effects has already been discussed in Remark 3.7. When the idiosyncratic components as well as the common effects are serially correlated we have

  • equation image(39)

where

  • equation image(40)

and equation image. In conjunction with (3) this would yield the following ADF regression:

  • equation image(41)

with cross-sectionally dependent errors. Serial correlation in the common factor does not pose additional difficulties, and as before these common factors can be proxied by ȳt and ȳt−1.

A third possibility would be to model the residual serial correlation first as

  • equation image(42)

and then allow for the cross-section dependence by assuming a one-factor model for the residuals

  • equation image(43)

Under this specification we have

  • equation image(44)

5.2. Individual-Specific CADF Statistics for the Serially Correlated Case

All three models yield the same ADF regressions, but with different error specifications and parameter heterogeneity. The asymptotic theory to be developed in this section can be adapted to deal with all three specifications, but to save space here we focus on the third specification given by (44). We shall also confine our attention to the case where the autoregressive coefficients, ρi, are homogeneous across i, but shall consider the implications of relaxing this assumption using Monte Carlo simulations. The mathematical details become much more complicated if ρi is allowed to differ across i.

To deal with the unobserved common effects, ft, we first note that in this case under the unit root hypothesis we have (using (44) with βi = 0 and ρi = ρ)

  • equation image

and

  • equation image

Hence, for sufficiently large N, and under our assumption that equation image tends to a non-zero limit as N[RIGHTWARDS ARROW]∞, the common effects can be proxied by a linear combination of Δȳt and Δȳt−1. In addition, the DF regressions must be augmented for residual serial correlation and the lagged levels of the cross-section means of the processes, namely Δyi, t−1 and ȳt−1. Accordingly, we propose running the following CADF regressions which are augmented to asymptotically filter out the effects of both cross-section and time dependence patterns in the residuals:

  • equation image

where i = (Δyi,−1, Δȳ, Δȳ−1, τT, ȳ−1) is a T × 5 matrix of observations defined in Section 3. The individual CADF statistics are given by

  • equation image(45)

where equation image, i = ITi(ii)−1i, Mi, w = ITGi(GiGi)−1Gi, and Gi = (yi,−1, i).

To establish the asymptotic invariance of ti(N, T) to the coefficients of the common effects, γi, we first note that under βi = 0

  • equation image(46)

or

  • equation image(47)

and

  • equation image(48)

where

  • equation image(49)
  • equation image(50)

and L is a one-period lag operator. Using these results we now have the following generalizations of (17) and (22):

  • equation image(51)

and

  • equation image(52)

which if used in (45) yields (under βi = 0)

  • equation image(53)

where as before equation image, ωi is defined by (20), and

  • equation image

The elements of szi,−1 and z,−1 are defined in (50).

The exact sample distribution of ti(N, T) depends on δi, equation image and ρ, but as stated in the following theorem this dependence vanishes for N and T[RIGHTWARDS ARROW]∞, such that N/T[RIGHTWARDS ARROW]k, where k is a finite, non-zero positive constant.

Theorem 5.1Letyitbe defined by(44)withi| = |ρ|< 1, and consider the statisticti(N, T) defined by(45). Suppose that Assumptions 1–3 hold andequation imagetends to a finite non-zero limit asN[RIGHTWARDS ARROW]∞, then under(4)and asNandT[RIGHTWARDS ARROW]∞, ti(N, T) has the same sequential (N[RIGHTWARDS ARROW]∞, T[RIGHTWARDS ARROW]∞) and joint [(N, T)j[RIGHTWARDS ARROW]∞] limit distributions given by(26), obtained under ρ = 0.

For a proof see Section A.3 in the Appendix.

This theorem establishes that ADF regression results in the case of pure time series models also apply to cross-sectionally augmented regressions. Although our proof assumes a first-order error process, the approach readily extends to higher-order processes. For example, for an AR(p) error specification the relevant individual CADF statistics will be given by the OLS t-ratio of bi in the following pth order cross-section/time-series augmented regression:

  • equation image(54)

This testing procedure also readily extend to models containing linear trends.

5.3. Panel Unit Root Tests for Panels with Serially Correlated Errors

It is now relatively easy to construct panel unit root tests that simultaneously take account of cross-section dependence and residual serial correlation. Once again we focus on the truncated version of the CIPS statistic given by (35), with equation image computed using the cross-section/time-series augmented regression, (54), subject to the truncation scheme defined by (34).8 Using Theorem 5.1 and noting that the result of the theorem applies equally to the truncated version of the CADF statistics, we have equation image. Hence equation image, and CIPS* in the case of serially correlated errors has the same limit distribution as (38) obtained under ρi = 0 and the critical values reported in Tables II(a)–II(c) also apply equally to the serially correlated case.

6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information

Initially we shall consider dynamic panels with fixed effects and cross-section dependence, but without residual serial correlation or linear trends. The data-generating process (DGP) in this case is given by

  • equation image(55)

where

  • equation image(56)

and

  • equation image(57)

We shall consider two levels of cross-section dependence where we generate γi∼i.i.d.U[0, 0.20] as an example of ‘low cross-section dependence’, and γi∼i.i.d.U[−1, 3] to represent the case of ‘high cross-section dependence’. The average pair-wise cross correlation coefficient of uit and ujt 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 εit were generated as

  • equation image(58)

with ρi∼i.i.d.U[0.2, 0.4], as an example of positive residual serial correlations, and ρi∼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, ρi, 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 yit were generated as follows:

  • equation image

with μi∼i.i.d.U[0.0, 0.02] and δi∼i.i.d.U[0.0, 0.02]. This ensures that yit has the same average trend properties under the null and the alternative hypotheses. The errors, uit, were generated according to (56), (57) and (58) for different values of ρi as set out above.

Size and power of the tests were computed under the null ϕi = 1 for all i, and the heterogeneous alternatives ϕi∼i.i.d.U[0.85, 0.95], using 1000 replications per experiment.9 The tests were one-sided 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, μi, δi, ϕi, ρi, equation image, and γi were generated independently of the errors, eitit) and ft, with ft also generated independently of eitit).

6.1. Size Distortion of the Standard Panel Unit Root Tests

Before reporting the results for the proposed cross-sectionally augmented tests, it would be helpful first to examine the extent to which the size of the standard panel unit root tests (that assume cross-section independence) are distorted in the presence of cross-section dependence. Table III reports the empirical sizes of the IPS, truncated IPS, the inverse chi-squared (P), and the inverse normal (Z) tests when the DGP is subject to cross-section 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 t-bar statistic defined by (see IPS, 2003):

  • equation image(59)

where equation image, and tiT is the t-ratio of the estimated coefficient of yi, t−1 in the OLS regression of Δyit on an intercept and yi, t−1. The truncated version of the IPS test uses the same formula as above but replaces tiT with the individually truncated statistic, equation image, defined by (34) with K1 = 6.19, and K2 = 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 equation image and the equation image tests.

Table III. Size of panel unit root tests that do not allow for cross-sectional dependence: no serial correlation, low and high cross-section dependence, intercept case
NTestLow cross-section dependence THigh cross-section dependence T
1020305010010203050100
  1. Notes:

  2. This table reports the size of various test statistics defined in the paper. The underlying data is generated by yit = (1 − ϕii + ϕiyi, t−1 + uit, i = 1, 2, .., N, t = − 51, − 50, …, T, and uit = γift + εit where we generate γi∼i.i.d.U[0, 0.2] for low cross-section dependence, and γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and equation image with σi2∼i.i.d.U[0.5, 1.5]. The test statistics are computed using various regression specifications as specified in the text for the case with intercept only and no lag augmentation. The size (under the null ϕi = 1) of the tests are computed at the 5% nominal level. ‘DF’ refers to the Dickey–Fuller regression with an intercept defined as Δyit = ai + biyi, t−1 + uit, i = 1, 2, .., N, t = 1, 2, …, T. The IPS test statistic is defined as equation image, where ti is the OLS t-ratio of bi in the ‘DF’ regression defined above, is the simple average of these t-ratios. The IPS* is the truncated IPS statistic computed by applying the truncation procedure similar to that used in (33) to the individual DF (or ADF) statistics. For P-test and Z-test, the normal and empirical approximations distinguish the use of standard normal critical values and empirical critical values obtained by stochastic simulations. Simulation results reported in this and the subsequent tables are based on 1000 replications.

10IPS0.0410.0460.0560.0510.0460.0930.1170.1220.1110.114
IPS*0.0400.0460.0560.0510.0460.0950.1180.1220.1110.114
P-test (DF) 
 Normal approximation0.1860.1050.0920.0740.0630.2070.1590.1550.1260.124
 Empirical distribution0.0300.0420.0480.0480.0410.0540.0820.1080.0920.095
Z-test (DF) 
 Normal approximation0.0780.0550.0660.0570.0480.1390.1350.1350.1220.122
 Empirical distribution0.0390.0430.0560.0520.0430.0940.1180.1230.1140.119
20IPS0.0430.0540.0450.0480.0510.1790.2180.2130.2150.246
IPS*0.0450.0540.0450.0480.0510.1820.2190.2130.2150.246
P-test (DF) 
 Normal approximation0.2270.1230.0840.0830.0710.2420.2160.1780.1880.207
 Empirical distribution0.0350.0530.0410.0490.0510.1040.1480.1340.1570.191
Z-test (DF) 
 Normal approximation0.0670.0580.0470.0560.0520.2140.2270.2170.2210.243
 Empirical distribution0.0430.0510.0390.0520.0520.1780.2130.2040.2160.243
30IPS0.0370.0350.0520.0470.0600.2080.2340.2360.2560.283
IPS*0.0380.0340.0520.0470.0600.2100.2340.2360.2560.283
P-test (DF) 
 Normal approximation0.2420.1190.1030.0710.0670.2990.2370.2220.2210.231
 Empirical distribution0.0270.0390.0560.0470.0510.1250.1570.1630.1800.208
Z-test (DF) 
 Normal approximation0.0660.0420.0560.0450.0520.2510.2450.2390.2530.277
 Empirical distribution0.0420.0350.0500.0420.0520.2140.2340.2370.2490.277
50IPS0.0460.0320.0500.0470.0390.2850.3140.3180.3340.370
IPS*0.0470.0320.0500.0470.0390.2910.3140.3180.3340.370
P-test (DF) 
 Normal approximation0.3140.1450.1280.0890.0590.3270.2740.2710.2770.291
 Empirical distribution0.0400.0320.0530.0540.0440.1680.2140.2230.2460.271
Z-test (DF) 
 Normal approximation0.0750.0350.0520.0470.0410.3070.3140.3220.3300.354
 Empirical distribution0.0430.0320.0500.0480.0420.2850.3050.3160.3300.362
100IPS0.0350.0670.0590.0570.0590.3300.3820.3720.3830.406
IPS*0.0360.0670.0590.0570.0590.3360.3830.3720.3830.406
P-test (DF) 
 Normal approximation0.4090.1810.1200.0900.0910.3790.3400.3060.3100.325
 Empirical distribution0.0260.0490.0400.0480.0540.2180.2590.2630.2890.314
Z-test (DF) 
 Normal approximation0.0520.0620.0410.0480.0490.3450.3670.3540.3760.394
 Empirical distribution0.0380.0620.0440.0490.0520.3300.3670.3550.3780.394

As to be expected, the extent of over-rejection of the tests very much depends on the degree of cross-section dependence. Under the low cross-section dependence the different version of the IPS and the Z tests perform reasonably well. The standard P test tends to over-reject for small values of T, but the normal approximation begins to work as T is increased. Overall, when the cross-section 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 cross-section scenario all the tests tend to over-reject, often by a substantial amount. Clearly, the standard panel unit root tests that do not allow for cross-section dependence can be seriously biased if the degree of cross-section dependence is sufficiently large. Therefore, in what follows we shall focus on the high cross-section dependence scenario.

6.2. Size and Power: Case of Serially Uncorrelated Errors

The tests to be considered are the cross-sectionally augmented IPS test, CIPS(N, T), and its truncated version, CIPS*(N, T), and the cross-sectionally augmented versions of the inverse chi-squared 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 individual-specific rejection probabilities by stochastic simulations. In particular, the cross-sectionally augmented inverse chi-squared test statistic is given by

  • equation image(60)

where the rejection probabilities are computed as

  • equation image(61)

where ����(s) is the sth random draw from the distribution of ����, defined by (26), and S is the number of replications used to compute equation image, 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,

  • equation image(62)

To avoid very extreme values the rejection probabilities were truncated to lie in the range [0.000001, 0.999999].

The size and power characteristics of these tests are summarized in Table IV for the high cross-section dependence scenario. There is no evidence of size distortions in the case of the CIPS, CIPS*, and the versions of the CP, and CZ tests (denoted by equation image, and equation image) that use the correct critical values. Using normal approximations for the CP and CZ tests does not work and results in substantial size distortions.12 In terms of power all the tests perform similarly well, except for the equation image test which is generally dominated by the other three. But none of the tests exhibit much power when T = 10, irrespective of the size of N. Only when T is increased to 20 and beyond can one begin to see the benefit of increasing N on the power of the tests. Finally, in the present simple case of serially uncorrelated residuals little seems to be gained by the truncation procedure.

Table IV. Size and power of cross-sectionally augmented panel unit root tests: no serial correlation, high cross-section dependence, intercept case
NTestSize TPower T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = (1 − ϕii + ϕiyi, t−1 + uit, i = 1, 2, .., N, t = − 51, − 50, …, T, and uit = γift + εit where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and equation image with equation image. Size (under the null ϕi = 1) and Power (under the heterogeneous alternatives ϕi∼i.i.d.U[0.85, 0.95]) are computed at the 5% nominal level based on the cross-section augmented Dickey–Fuller regression (CADF): Δyit = ai + biyi, t−1 + ciȳt−1 + di0Δȳt + uit, i = 1, 2, .., N, t = 1, 2, …, T, where equation image. The CIPS test statistic is defined as equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regression. Similarly, truncated CIPS statistic is computed with ti(N, T) replaced by equation image, defined by (33). The equation imageP and equation imageZ tests are based on the statistics defined by (62) and (64), respectively, using their corresponding empirical critical values.

10CIPS0.0430.0480.0570.0520.0630.0620.1150.1920.3820.958
CIPS*0.0490.0480.0570.0520.0630.0680.1150.1920.3820.958
equation imageP0.0360.0380.0600.0520.0560.0480.0950.1410.3000.890
equation imageZ0.0500.0490.0550.0500.0600.0660.1140.1880.3810.957
20CIPS0.0340.0620.0550.0640.0570.0710.1140.2430.6881.00
CIPS*0.0410.0620.0550.0640.0570.0710.1140.2430.6881.00
equation imageP0.0280.0570.0560.0450.0570.0480.0870.1750.5201.00
equation imageZ0.0370.0630.0570.0630.0570.0720.1130.2370.6871.00
30CIPS0.0460.0540.0540.0430.0450.0580.1220.2310.6741.00
CIPS*0.0450.0540.0540.0430.0450.0670.1220.2310.6741.00
equation imageP0.0290.0530.0490.0440.0410.0430.0850.1810.5361.00
equation imageZ0.0470.0560.0520.0420.0440.0680.1210.2290.6671.00
50CIPS0.0390.0450.0480.0460.0510.0610.1460.2980.8491.00
CIPS*0.0400.0450.0480.0460.0510.0690.1450.2980.8491.00
equation imageP0.0260.0310.0520.0530.0520.0450.1210.2280.7081.00
equation imageZ0.0410.0460.0490.0460.0520.0650.1460.2980.8431.00
100CIPS0.0370.0490.0460.0610.0440.0550.1890.3790.9631.00
CIPS*0.0410.0490.0460.0610.0440.0600.1880.3790.9631.00
equation imageP0.0260.0490.0440.0640.0400.0350.1410.2920.8871.00
equation imageZ0.0400.0490.0480.0600.0440.0600.1890.3770.9601.00

6.3. Size and Power: Case of Serially Correlated Errors

In the case of models with serially correlated errors, the cross-sectionally augmented tests (CIPS, CIPS*, equation image, and equation image) 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 yit and ȳt, 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 cross-section 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 under-rejections for positive residual serial correlation, and substantial over-rejections in the case of negative residual serial correlations. But the test sizes stabilize at around 5% when the CADF regressions are augmented with Δyi, 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 equation image test, and to a lesser extent the CIPS test, are over-sized. 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.

Table V(a). Size of cross-sectionally augmented panel unit root tests: positive serial correlation, high cross-section dependence, intercept case
NTestCADF(0) TCADF(1) T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = (1 − ϕii + ϕiyi, t−1 + uit, i = 1, 2, .., N, t = − 51, − 50, …, T, and uit = γift + εit where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and εit = ρiεi, t−1 + eit, where equation image with equation image and ρi∼i.i.d.U[0.2, 0.4]. Size (under the null ϕi = 1) is computed at the 5% nominal level based on cross-section augmented Dickey–Fuller regressions: Δyit = ai + biyi, t−1 + ciȳt−1 + di0Δȳt + uit, i = 1, 2, .., N, t = 1, 2, …, T, where equation image, and the CADF(1) regressions: Δyit = ai + biyi, t−1 + ciȳt−1 + di0Δȳt + di1Δȳt−1 + δi1Δyi, t−1 + uit. The CIPS test statistic is defined by equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regressions. Similarly, truncated CIPS statistic is computed with ti(N, T) replaced by equation image as defined by (33). The equation imageP and equation imageZ tests are based on the statistics defined by (62) and (64), respectively, using their corresponding empirical critical values.

10CIPS0.0050.0120.0050.0050.0000.0900.0620.0510.0480.051
CIPS*0.0050.0110.0050.0050.0000.0550.0620.0510.0480.051
equation imageP0.0150.0150.0060.0130.0020.1220.0770.0620.0540.048
equation imageZ0.0050.0110.0040.0050.0000.0560.0650.0510.0500.050
20CIPS0.0070.0000.0000.0000.0000.0980.0560.0470.0420.053
CIPS*0.0050.0000.0000.0000.0000.0620.0540.0470.0420.053
equation imageP0.0080.0030.0040.0010.0010.1480.0770.0630.0650.066
equation imageZ0.0050.0000.0000.0000.0000.0600.0540.0480.0430.053
30CIPS0.0040.0000.0000.0000.0000.0830.0330.0490.0370.047
CIPS*0.0010.0000.0000.0000.0000.0520.0330.0490.0370.047
equation imageP0.0100.0030.0030.0000.0010.1770.0530.0680.0510.048
equation imageZ0.0010.0000.0000.0000.0000.0520.0340.0490.0380.048
50CIPS0.0000.0000.0000.0000.0000.0720.0390.0350.0280.047
CIPS*0.0000.0000.0000.0000.0000.0400.0390.0350.0280.047
equation imageP0.0050.0010.0000.0000.0000.1630.0690.0630.0440.051
equation imageZ0.0000.0000.0000.0000.0000.0390.0390.0370.0290.046
100CIPS0.0000.0000.0000.0000.0000.0690.0370.0290.0370.040
CIPS*0.0000.0000.0000.0000.0000.0260.0370.0290.0370.040
equation imageP0.0000.0000.0000.0000.0000.1940.0800.0560.0530.055
equation imageZ0.0000.0000.0000.0000.0000.0270.0390.0310.0390.041
Table V(b). Power of cross-sectionally augmented panel unit root tests: positive serial correlation, high cross-section dependence, intercept case
NTestCADF(0) TCADF(1) T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = (1 − ϕii + ϕiyi, t−1 + uit, i = 1, 2, .., N, t = − 51, − 50, …, T, and uit = γift + εit where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and εit = ρiεi, t−1 + eit, where equation image with equation image and ρi∼i.i.d.U[0.2, 0.4]. Power, under the heterogeneous alternatives ϕi∼i.i.d.U[0.85, 0.95], is computed at the 5% nominal level based on the cross-section augmented Dickey–Fuller regressions: Δyit = ai + biyi, t−1 + ciȳt−1 + di0Δȳt + uit, i = 1, 2, .., N, t = 1, 2, …, T, where equation image, and the CADF(1) regressions: Δyit = ai + biyi, t−1 + ciȳt−1 + di0Δȳt + di1Δȳt−1 + δi1Δyi, t−1 + uit. The CIPS test statistic is given by equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regressions. Similarly, truncated CIPS statistic is computed with ti(N, T) replaced by equation image, defined by (33). The equation imageP and equation imageZ tests are based on the statistics defined by (62) and (64), respectively, using their corresponding empirical critical values.

10CIPS0.0150.0100.0130.0110.1010.1310.0940.1090.2690.873
CIPS*0.0180.0100.0130.0110.1010.1000.0930.1090.2690.873
equation imageP0.0200.0190.0170.0110.0520.1610.1050.1110.2220.778
equation imageZ0.0170.0100.0130.0110.0940.1000.0940.1110.2680.868
20CIPS0.0060.0020.0030.0040.1430.1250.0750.1290.4120.992
CIPS*0.0070.0020.0030.0040.1430.0800.0740.1290.4120.992
equation imageP0.0100.0060.0000.0020.0400.1830.0930.1240.3300.973
equation imageZ0.0070.0010.0030.0040.1330.0810.0750.1290.4030.992
30CIPS0.0020.0000.0020.0000.2130.1260.0720.1710.4200.999
CIPS*0.0030.0000.0020.0000.2130.0650.0710.1710.4200.999
equation imageP0.0050.0010.0030.0020.1120.2110.1220.1730.3580.990
equation imageZ0.0030.0000.0020.0000.2040.0640.0750.1720.4160.999
50CIPS0.0000.0000.0000.0000.3480.1130.0890.1600.5951.00
CIPS*0.0000.0000.0000.0000.3480.0650.0890.1600.5951.00
equation imageP0.0030.0010.0010.0000.1290.2220.1190.1750.4771.00
equation imageZ0.0000.0000.0000.0000.3320.0650.0940.1620.5861.00
100CIPS0.0010.0000.0000.0000.3980.0900.0880.1870.7131.00
CIPS*0.0010.0000.0000.0000.3980.0440.0870.1880.7131.00
equation imageP0.0010.0000.0000.0000.1490.2350.1670.2130.6411.00
equation imageZ0.0010.0000.0000.0000.3710.0470.0910.1930.7131.00
Table VI(a). Size of cross-sectionally augmented panel unit root tests: negative serial correlation, high cross-section dependence, intercept case
NTestCADF(0) TCADF(1) T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = (1 − ϕii + ϕiyi, t−1 + uit, i = 1, 2, …, N, t = − 51, − 50, …, T, and uit = γift + εit where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and εit = ρiεi, t−1 + eit, where equation image with equation image and ρi∼i.i.d.U[−0.4, − 0.2]. Size (under the null ϕi = 1) is computed at the 5% nominal level based on the cross-section augmented Dickey–Fuller regressions: Δyit = ai + biyi, t−1 + ciȳt−1 + di0Δȳt + uit, i = 1, 2, …, N, t = 1, 2, …, T, where equation image, and the CADF(1) regressions: Δyit = ai + biyi, t−1 + ciȳt−1 + di0Δȳt + di1Δȳt−1 + δi1Δyi, t−1 + uit. The CIPS test statistic is defined by equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regressions. Similarly, truncated CIPS statistic is computed with ti(N, T) replaced by equation image as defined by (33). The equation imageP and equation imageZ tests are based on the statistics defined by (62) and (64), respectively, using their corresponding empirical critical values.

10CIPS0.2370.4360.4980.5310.5790.1010.0650.0570.0440.049
CIPS*0.2580.4360.4980.5310.5790.0740.0650.0570.0440.049
equation imageP0.1850.4030.5010.5420.6020.1170.0680.0520.0540.051
equation imageZ0.2550.4370.5010.5320.5820.0720.0650.0560.0450.050
20CIPS0.3650.5950.6460.7240.7460.1060.0540.0510.0460.056
CIPS*0.3720.5960.6460.7240.7460.0760.0530.0510.0460.056
equation imageP0.2980.5820.6740.7510.7980.1550.0610.0590.0480.059
equation imageZ0.3750.5930.6460.7240.7470.0780.0530.0500.0450.055
30CIPS0.4230.6940.7480.8080.8220.1030.0460.0560.0340.046
CIPS*0.4450.6940.7480.8070.8220.0650.0460.0560.0340.046
equation imageP0.3460.7080.7790.8470.8810.1330.0480.0600.0450.052
equation imageZ0.4440.6950.7510.8060.8250.0640.0460.0560.0340.047
50CIPS0.4990.7950.8540.8860.9190.1030.0420.0410.0300.047
CIPS*0.5130.7950.8540.8860.9190.0680.0420.0410.0300.047
equation imageP0.3870.8160.8870.9280.9550.1650.0510.0460.0320.044
equation imageZ0.5080.7950.8570.8900.9200.0670.0450.0410.0280.047
100CIPS0.5870.8350.8940.9410.9670.0940.0440.0400.0450.045
CIPS*0.6020.8340.8940.9410.9660.0510.0450.0400.0450.045
equation imageP0.5200.8980.9450.9690.9860.1730.0600.0460.0470.053
equation imageZ0.6010.8370.8980.9410.9670.0520.0450.0410.0440.044
Table VI(b). Power of cross-sectionally augmented panel unit root tests: negative serial correlation, high cross-section dependence, intercept case
NTestCADF(0) TCADF(1) T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = (1 − ϕii + ϕiyi, t−1 + uit, i = 1, 2, …, N, t = − 51, − 50, …, T, and uit = γift + εit, where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and εit = ρiεi, t−1 + eit, where equation image with equation image and ρi∼i.i.d.U[−0.4, − 0.2]. Power, under the heterogeneous alternatives ϕi∼i.i.d.U[0.85, 0.95], is computed at the 5% nominal level based on the cross-section augmented Dickey–Fuller regressions: Δyit = ai + biyi, t−1 + ciȳt−1 + di0Δȳt + uit, i = 1, 2, .., N, t = 1, 2, …, T, where equation image, and the CADF(1) regressions: Δyit = ai + biyi, t−1 + ciȳt−1 + di0Δȳt + di1Δȳt−1 + δi1Δyi, t−1 + uit. The CIPS test statistic is given by equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regressions. Similarly, truncated CIPS statistic is computed with ti(N, T) replaced by equation image, defined by (33). The equation imageP and equation imageZ tests are based on the statistics defined by (62) and (64), respectively, using their corresponding empirical critical values.

10CIPS0.2720.6170.8200.9801.000.1300.0960.1340.3440.938
CIPS*0.2920.6180.8190.9801.000.0940.0960.1340.3440.938
equation imageP0.1830.5400.7500.9601.000.1460.0810.1070.2660.857
equation imageZ0.2890.6170.8170.9801.000.0920.0960.1340.3370.937
20CIPS0.4180.8640.9730.9991.000.1440.0880.1600.5360.999
CIPS*0.4420.8620.9730.9991.000.0980.0880.1600.5360.999
equation imageP0.3190.8070.9530.9991.000.1620.0880.1210.4030.993
equation imageZ0.4430.8610.9740.9991.000.0930.0900.1580.5280.999
30CIPS0.4530.8820.9901.001.000.1330.0930.1870.5791.00
CIPS*0.4750.8830.9901.001.000.0880.0920.1870.5791.00
equation imageP0.3470.8310.9751.001.000.1840.0970.1630.4480.998
equation imageZ0.4760.8820.9901.001.000.0890.0910.1860.5721.00
50CIPS0.6170.9841.0001.001.000.1290.1070.2080.7511.00
CIPS*0.6340.9841.0001.001.000.0900.1070.2080.7511.00
equation imageP0.5120.9570.9991.001.000.1860.1040.1780.6071.00
equation imageZ0.6350.9841.0001.001.000.0910.1080.2090.7451.00
100CIPS0.6650.9981.001.001.000.1210.1080.2540.8801.00
CIPS*0.6850.9981.001.001.000.0720.1070.2540.8801.00
equation imageP0.5650.9951.001.001.000.2180.1290.2250.7761.00
equation imageZ0.6800.9981.001.001.000.0740.1100.2550.8751.00

Focusing on CIPS* and equation image 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

Size and power of CIPS, CIPS*, equation image, and equation image tests in the case of models with linear deterministic trends are summarized in Tables VII–IX. Table VII gives the results for models without residual serial correlation and shows that all the various tests continue to have sizes very close to the nominal value of 5%. However, as to be expected, the inclusion of linear trends in the CADF regressions come at the cost of a lower power. We now need T to be 30 or more before power begins to increase with N. For example, when T = 20 the power of the tests stays around 7% irrespective of the value of N. But when T = 50 the power of the CIPS test rises from 18% to 62% as N is increased from 10 to 100. Once again the equation image test is dominated by the other three tests, which have very similar power characteristics.

Table VII. Size and power of cross-sectionally augmented panel unit root tests: no serial correlation, high cross-section dependence, the linear trend case
NTestSize TPower T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = αi + di(1 − ϕi)t + ϕiyi, t−1 + uit, i = 1, 2, …, N, t = − 51, − 50, …, T, with uit = γift + εit, where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. αi and di∼i.i.d.U[0, 0.02], ft∼i.i.d.N(0, 1), and equation image with equation image. Size (under the null ϕi = 1) and power (under the heterogeneous alternative ϕi∼i.i.d.U[0.85, 0.95]) are computed at the 5% nominal level based cross-section augmented Dickey–Fuller regressions with linear trends: Δyit = ai0 + ai1t + biyi, t−1 + ciȳt−1 + di0Δȳt + uit, i = 1, 2, …, N, t = 1, 2, …, T, where equation image. The CIPS test statistic is defined as equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regression. Similarly, the truncated CIPS statistic (CIPS*) is computed with ti(N, T) replaced by equation image, defined by (33). The equation imageP and equation imageZ tests are defined by (62) and (64) respectively using their corresponding empirical critical values.

10CIPS0.0380.0590.0620.0590.0490.0400.0700.0780.1830.692
CIPS*0.0450.0580.0620.0590.0490.0540.0690.0790.1830.692
equation imageP0.0320.0470.0590.0580.0460.0380.0670.0750.1430.577
equation imageZ0.0470.0580.0630.0600.0460.0570.0700.0760.1870.697
20CIPS0.0320.0480.0610.0520.0410.0420.0700.0930.2700.935
CIPS*0.0500.0480.0610.0510.0410.0520.0700.0950.2700.935
equation imageP0.0310.0440.0600.0510.0380.0380.0560.0860.2090.866
equation imageZ0.0500.0490.0600.0530.0410.0530.0700.0920.2690.937
30CIPS0.0280.0600.0520.0490.0470.0320.0810.0970.3320.987
CIPS*0.0390.0600.0530.0490.0470.0380.0820.0970.3320.987
equation imageP0.0250.0540.0470.0480.0500.0220.0590.0810.2240.954
equation imageZ0.0400.0630.0530.0510.0470.0400.0820.1000.3320.987
50CIPS0.0200.0580.0430.0480.0580.0250.0750.1330.4671.00
CIPS*0.0280.0580.0440.0480.0580.0340.0750.1330.4671.00
equation imageP0.0080.0480.0490.0460.0550.0170.0590.0990.3250.996
equation imageZ0.0280.0570.0440.0470.0570.0340.0750.1320.4601.00
100CIPS0.0160.0470.0580.0580.0450.0340.0740.1370.6211.00
CIPS*0.0290.0470.0580.0580.0450.0430.0760.1370.6211.00
equation imageP0.0090.0360.0520.0480.0410.0200.0580.1060.4781.00
equation imageZ0.0290.0450.0560.0600.0440.0420.0760.1350.6171.00
Table VIII(a). Size of cross-sectionally augmented panel unit root tests: positive serial correlation, high cross-section dependence, the linear trend case
NTestCADF(0) TCADF(1) T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = αi + di(1 − ϕi)t + ϕiyi, t−1 + uit, i = 1, 2, …, N, t = − 51, − 50, …, T, and uit = γift + εit where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and εit = ρiεi, t−1 + eit, where equation image with equation image and ρi∼i.i.d.U[0.2, 0.4]. Size (under the null ϕi = 1) is computed at the 5% nominal level based on the cross-section augmented Dickey–Fuller regressions: Δyit = ai0 + ai1t + biyi, t−1 + ciȳt−1 + di0Δȳt + uit, i = 1, 2, …, N, t = 1, 2, …, T, where equation image, and the CADF(1) regressions: Δyit = ai0 + ai1t + biyi, t−1 + di0Δȳt + ciȳt−1 + δi1Δyi, t−1 + di1Δȳt−1 + uit. The CIPS test statistic is defined by equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regressions. Similarly, the truncated CIPS statistic is computed with ti(N, T) replaced by equation image as defined by (33). The equation imageP and equation imageZ tests are based on the statistics defined by (62) and (64), respectively, using their corresponding empirical critical values.

10CIPS0.0100.0020.0010.0000.0000.1810.0540.0390.0580.044
CIPS*0.0120.0020.0010.0000.0000.0800.0520.0380.0580.044
equation imageP0.0120.0060.0010.0020.0000.1990.0750.0540.0610.040
equation imageZ0.0120.0020.0010.0000.0000.0830.0520.0390.0620.046
20CIPS0.0040.0010.0000.0000.0000.1980.0480.0380.0550.050
CIPS*0.0050.0010.0000.0000.0000.0690.0470.0390.0550.050
equation imageP0.0070.0030.0000.0000.0000.2070.0950.0560.0650.053
equation imageZ0.0050.0010.0000.0000.0000.0680.0480.0380.0550.049
30CIPS0.0030.0000.0000.0000.0000.2260.0490.0480.0490.054
CIPS*0.0010.0000.0000.0000.0000.0530.0480.0480.0490.054
equation imageP0.0040.0000.0000.0000.0000.2680.0840.0690.0590.064
equation imageZ0.0010.0000.0000.0000.0000.0540.0500.0480.0500.053
50CIPS0.0000.0000.0000.0000.0000.2610.0490.0520.0410.039
CIPS*0.0000.0000.0000.0000.0000.0520.0490.0520.0410.039
equation imageP0.0000.0000.0000.0000.0000.3130.1060.0800.0570.050
equation imageZ0.0000.0000.0000.0000.0000.0560.0500.0530.0430.041
100CIPS0.0000.0000.0000.0000.0000.2830.0480.0340.0470.033
CIPS*0.0020.0000.0000.0000.0000.0380.0420.0340.0470.033
equation imageP0.0000.0000.0000.0000.0000.3690.1050.0760.0600.046
equation imageZ0.0020.0000.0000.0000.0000.0390.0460.0340.0490.035
Table VIII(b). Power of cross-sectionally augmented panel unit root tests: positive serial correlation, high cross-section dependence, the linear trend case
NTestCADF(0) TCADF(1) T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = αi + di(1 − ϕi)t + ϕiyi, t−1 + uit, i = 1, 2, …, N, t = − 51, − 50, …, T, and uit = γift + εit, where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and εit = ρiεi, t−1 + eit, where equation image with equation image and ρi∼i.i.d.U[0.2, 0.4]. Power, under the heterogeneous alternative ϕi∼i.i.d.U[0.85, 0.95], is computed at the 5% nominal level based cross-section augmented Dickey–Fuller regressions with linear trends: Δyit = ai0 + ai1t + biyi, t−1 + ciȳt−1 + + di0Δȳt + uit, i = 1, 2, …, N, t = 1, 2, …, T, where equation image, and the CADF(1) regressions: Δyit = ai0 + ai1t + biyi, t−1 + di0Δȳt + ciȳt−1 + δi1Δyi, t−1 + di1Δȳt−1 + uit. The CIPS test statistic is defined by equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regressions. Similarly, truncated CIPS statistic is computed with ti(N, T) replaced by equation image as defined by (33). The equation imageP and equation imageZ tests are based on the statistics defined by (62) and (64), respectively, using their corresponding empirical critical values.

10CIPS0.0050.0030.0020.0040.0100.1770.0640.0740.1310.521
CIPS*0.0060.0030.0020.0040.0100.0780.0620.0740.1310.521
equation imageP0.0090.0060.0010.0060.0020.1970.0840.0800.1180.417
equation imageZ0.0060.0030.0020.0040.0100.0790.0610.0730.1320.523
20CIPS0.0010.0010.0010.0010.0070.2050.0660.0690.1740.816
CIPS*0.0020.0010.0010.0010.0070.0700.0660.0690.1740.816
equation imageP0.0020.0030.0010.0010.0040.2380.1050.0760.1570.709
equation imageZ0.0020.0010.0010.0010.0070.0710.0660.0690.1730.814
30CIPS0.0040.0000.0000.0000.0030.2060.0560.0790.2010.921
CIPS*0.0040.0000.0000.0000.0030.0500.0560.0780.2010.921
equation imageP0.0020.0000.0000.0000.0020.2450.0920.0960.1820.817
equation imageZ0.0050.0000.0000.0000.0030.0490.0560.0800.2010.922
50CIPS0.0010.0000.0000.0000.0000.2550.0410.0730.2230.988
CIPS*0.0010.0000.0000.0000.0000.0370.0390.0720.2230.988
equation imageP0.0010.0000.0000.0000.0000.3190.1030.1120.1980.957
equation imageZ0.0010.0000.0000.0000.0000.0400.0390.0720.2230.990
100CIPS0.0000.0000.0000.0000.0010.2740.0440.0940.2901.00
CIPS*0.0000.0000.0000.0000.0010.0460.0410.0940.2901.00
equation imageP0.0010.0000.0000.0000.0010.3610.0950.1080.2870.998
equation imageZ0.0000.0000.0000.0000.0010.0470.0450.0960.2891.00
Table IX(a). Size of cross-sectionally augmented panel unit root tests: negative serial correlation, high cross-section dependence, the linear trend case
NTestCADF(0) TCADF(1) T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = αi + di(1 − ϕi)t + ϕiyi, t−1 + uit, i = 1, 2, …, N, t = − 51, − 50, …, T, and uit = γift + εit, where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and εit = ρiεi, t−1 + eit, where equation image with equation image and ρi∼i.i.d.U[−0.4, − 0.2]. Size (under the null ϕi = 1) is computed at the 5% nominal level based on the cross-section augmneted Dickey–Fuller regressions: Δyit = ai0 + ai1t + biyi, t−1 + ciȳt−1 + di0Δȳt + uit, i = 1, 2, …, N, t = 1, 2, …, T, where equation image, and the CADF(1) regressions: Δyit = ai0 + ai1t + biyi, t−1 + di0Δȳt + ciȳt−1 + δi1Δyi, t−1 + di1Δȳt−1 + uit. The CIPS test statistic is defined by equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regressions. Similarly, the truncated CIPS statistic is computed with ti(N, T) replaced by equation image as defined by (33). The equation imageP and equation imageZ tests are based on the statitics defined by (62) and (64), respectively, using their corresponding empirical critical values.

10CIPS0.2020.6160.7090.8050.8280.1920.0600.0390.0590.040
CIPS*0.2500.6140.7090.8050.8280.0920.0590.0390.0590.040
equation imageP0.1660.5570.6900.7890.8220.1900.0660.0450.0620.037
equation imageZ0.2530.6100.7070.8040.8260.0910.0600.0390.0590.042
20CIPS0.3370.7970.9040.9480.9640.2270.0530.0350.0570.051
CIPS*0.3860.7950.9030.9480.9650.1040.0560.0350.0570.051
equation imageP0.2490.7530.8900.9550.9700.2320.0810.0560.0590.055
equation imageZ0.3880.7940.9050.9480.9630.1050.0510.0360.0590.050
30CIPS0.3590.8950.9670.9830.9940.2660.0520.0590.0620.059
CIPS*0.4330.8920.9670.9830.9940.0680.0510.0590.0620.059
equation imageP0.2840.8660.9710.9790.9950.2540.0660.0610.0520.067
equation imageZ0.4300.8940.9670.9810.9940.0670.0520.0610.0620.061
50CIPS0.4200.9650.9870.9980.9990.2930.0560.0570.0490.048
CIPS*0.4940.9650.9880.9980.9990.0840.0550.0570.0490.048
equation imageP0.3340.9520.9880.9980.9990.3090.0640.0560.0520.044
equation imageZ0.4910.9650.9860.9970.9990.0840.0560.0570.0500.049
100CIPS0.5520.9951.0001.0001.0000.3160.0580.0480.0530.040
CIPS*0.6100.9951.0001.0001.0000.0880.0590.0480.0530.040
equation imageP0.4710.9931.0001.0001.0000.3680.0790.0630.0520.037
equation imageZ0.6070.9941.0001.0001.0000.0870.0600.0470.0540.040
Table IX(b). Power of cross-sectionally augmented panel unit root tests: negative serial correlation, high cross-section dependence, the linear trend case
NTestCADF(0) TCADF(1) T
1020305010010203050100
  1. Notes: The underlying data is generated by yit = αi + di(1 − ϕi)t + ϕiyi, t−1 + uit, i = 1, 2, …, N, t = − 51, − 50, …, T, and uit = γift + εit where we generate γi∼i.i.d.U[−1, 3] for high cross-section dependence. μi and ft∼i.i.d.N(0, 1), and εit = ρiεi, t−1 + eit, where equation image with equation image and ρi∼i.i.d.U[−0.4, − 0.2]. Power, under the heterogeneous alternative ϕi∼i.i.d.U[0.85, 0.95], is computed at the 5% nominal level based cross-section augmented Dickey–Fuller regressions with linear trends: Δyit = ai0 + ai1t + biyi, t−1 + ciȳt−1 + di0Δȳt + uit, i = 1, 2, …, N, t = 1, 2, …, T, where equation image, and the CADF(1) regressions: Δyit = ai0 + ai1t + biyi, t−1 + di0Δȳt + ciȳt−1 + δi1Δyi, t−1 + di1Δȳt−1 + uit. The CIPS test statistic is defined by equation image, where ti(N, T) is the OLS t-ratio of bi in the above CADF regressions. Similarly, the truncated CIPS statistic is computed with ti(N, T) replaced by equation image as defined by (33). The equation imageP and equation imageZ tests are based on the statistics defined by (62) and (64), respectively, using their corresponding empirical critical values.

10CIPS0.2170.6860.8360.9691.000.1980.0720.0830.1510.640
CIPS*0.2770.6840.8360.9691.000.0960.0710.0830.1510.640
equation imageP0.1850.6310.8060.9491.000.1750.0640.0730.1270.512
equation imageZ0.2810.6860.8360.9691.000.0960.0700.0840.1520.639
20CIPS0.3380.8820.9770.9981.000.2430.0710.0710.2080.919
CIPS*0.4190.8840.9770.9981.000.0830.0700.0710.2080.919
equation imageP0.2740.8270.9660.9991.000.2370.0860.0790.1720.832
equation imageZ0.4170.8840.9780.9981.000.0850.0690.0710.2070.916
30CIPS0.3910.9610.9971.001.000.2460.0640.0890.2560.977
CIPS*0.4630.9630.9971.001.000.0840.0640.0880.2560.977
equation imageP0.3220.9390.9931.001.000.2470.0790.0800.1970.908
equation imageZ0.4590.9630.9971.001.000.0820.0650.0890.2550.977
50CIPS0.4490.9810.9991.001.000.2900.0660.0870.3171.00
CIPS*0.5230.9810.9991.001.000.0780.0660.0870.3171.00
equation imageP0.3440.9630.9991.001.000.3030.0890.0850.2550.995
equation imageZ0.5230.9800.9991.001.000.0820.0650.0880.3181.00
100CIPS0.6000.9991.001.001.000.3110.0610.1030.4071.00
CIPS*0.6540.9991.001.001.000.0880.0630.1030.4071.00
equation imageP0.5280.9991.001.001.000.3560.0800.1010.3521.00
equation imageZ0.6500.9991.001.001.000.0870.0620.1030.4071.00

The results for the linear trend case combined with residual serial correlation are presented in Tables VIII and IX. Table VIII(a) and VIII(b) give the results for positive residual correlations, and Tables IX(a) and IX(b) summarize the results for negative residual serial correlations. The sizes of the CIPS* and equation image tests continue to be satisfactory, even for T = 10, once the augmentation for the residual serial correlation is implemented (see the results under ACDF(1)). In contrast, the CIPS and equation image tests are grossly over-sized when T = 10. Note that in the case of CADF(1) regressions with linear trends the number of parameters being estimated is 7, and with only 3 degrees of freedom remaining the non-truncated individual CADFi(1) statistics might not have moments, which could be the reason why the CIPS test breaks down. The application of the truncation procedure fixes the lack of the moment problem and renders the truncated CIPS test valid even when the degrees of freedom of the underlying CADF regressions is as low as 3. Similarly, the equation image statistic overcomes the problem of extreme values by using the inverse probability transformation and by the fact that the rejection probabilities used in the construction of equation image are truncated to avoid very extreme values. See (62).

7. EMPIRICAL APPLICATIONS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information

To illustrate the applicability of the proposed test to micro as well as macro panels here we consider (i) an international macro data set composed of 17 real exchange rate series observed over 100 quarters, and (ii) a micro panel data on real earnings of 181 households from the PSID data observed over 20 years. These data sets are chosen as representing panels with T > N and N > T, respectively. Based on the Monte Carlo results one would expect the CIPS test to be applicable to both types of panels, while the test developed by Moon and Perron (2004) is likely to be valid only in the case of panels where T > N. With this in mind we shall also report the results from the application of the Moon and Perron (MP) test to these data sets.

7.1. Real Exchange Rates

Panel unit root tests have been used in the literature primarily to test the purchasing power parity (PPP) hypothesis.14 These applications have been particularly important considering the relative lack of power of unit root tests applied to single series. Reliance on long time series covering 60 or more years of data in order to enhance the power of single-series unit root tests have also been problematic due to changes in exchange rate regimes and the incidence of structural breaks within the same regime. However, as originally emphasized by O'Connell (1998) panel unit root tests can also lead to spurious results (spuriously favouring the PPP hypothesis) if there are significant degrees of positive error cross-section dependence and this is ignored. This is confirmed by the Monte Carlo results summarized in Table III, but only if the degree of error cross-section dependence is sufficiently high. Application of panel unit root tests that allow for cross-section dependence is, therefore, desirable once it is established that the panel is subject to a significant degree of error cross-section dependence. In cases where cross-section dependence is not sufficiently high, loss of power might result if panel unit root tests that allow for cross-section dependence are used. Therefore, before an appropriate choice of a panel unit root test is made it is important that some evidence on the degree of residual cross-section dependence is provided.

In this subsection we consider two panels of quarterly real exchange rates from 17 OECD countries. The first panel covers the period 1974Q1–1998Q4 (T = 100), and the second shorter panel covers the period 1988Q1–1998Q4 (T = 44), which has been recently analyzed by Smith et al. (2004), on the grounds that the latter is less likely to be subject to structural breaks.

Log real exchange rates are computed as yit = sit + pustpit, where sit is the log of the nominal exchange rate of country ith currency in terms of US dollar, pust and pit are logarithms of consumer price indices in the USA and country i, respectively.15 Here we also take the opportunity of correcting an error in the computation of the real exchange rate used in the panel unit root tests reported in Smith et al. (2004, p. 165). These authors base their tests on sitpust + pit, which would have been correct (apart from a sign) if the nominal exchange rate had been defined as US dollars per unit of country ith currency. But as Smith et al. state and the data deposited on the JAE Archive establishes, the nominal exchange rates used are units (fractions) of foreign currency in one US dollar.16

As the first stage in our analysis we estimated individual ADF(p) regressions (without cross-section augmentations) for p = 1, 2, 3 and 4, for the two sample periods 1974Q1–1998Q4 and 1988Q1–1998Q4, and computed pair-wise cross-section correlation coefficients of the residuals from these regressions (namely equation image). The simple average of these correlation coefficients across all the (17 × 18)/2 = 153 pairs, equation image, together with the associated cross-section dependence (CD) test statistics proposed in Pesaran (2004), are given in Table X.17 The average cross-section error correlation coefficients is around 0.60, and the CD statistics are highly significant. This result is robust to the choice of p and the sample period and is in line with the findings of O'Connell (1998) and others.

Table X. Cross-Section correlations of the errors in the ADF(p) regressions of real exchange rates across countries (N = 17)
 1974Q1–1998Q4 (T = 100)1988Q1–1998Q4 (T = 44)
p = 1p = 2p = 3p = 4p = 1p = 2p = 3p = 4
equation image0.6000.5990.5980.5860.5990.6090.6090.589
CD70.0169.8669.7468.3046.3547.1447.1545.54

The panel unit root test statistics based on ADF(p) and CADF(p) regressions are summarized in Table XI. The IPS statistic is the standardized t-bar test of Im et al. (2003) defined by (59), and the CIPS statistic is the cross-section average of the t-ratio of the OLS coefficient of yi, t−1 in the CADF regressions (54). Under the unit root hypothesis and no cross-section dependence, IPS is asymptotically distributed as N(0, 1), and therefore on the basis of the IPS statistics in Table XI it would be concluded that the unit root hypothesis is rejected for p ≥ 2, with the probability of rejection being particularly high for p = 4.18 But, due to the large and significant degree of cross-section dependence in real exchange rates documented in Table X, this conclusion might not be safe and we should be considering the CIPS test that allows for cross-section dependence. At the 5% significance level the critical value of the CIPS statistic for N = 17 and T in the range of 30–100 is around − 2.22. (see Table II(b)). Therefore, according to the CIPS test the null of unit root cannot be rejected at the 5% level irrespective of the value of p.19 Therefore, the apparent support obtained for the PPP hypothesis using the IPS test could be spurious.

Table XI. IPS and CIPS Test statistics for the real exchange rates
Tests1974Q1–1998Q4 (T = 100)1988Q1–1998Q4 (T = 44)
p = 1p = 2p = 3p = 4p = 1p = 2p = 3p = 4
  1. Note: All statistics are based on univariate AR(p) specifications in the level of the variables with p ≤ 4 including an intercept term only and the underlying ADF and CADF regressions are estimated on the same sample period, namely, 1974Q1–1998Q4 and 1988Q1–1998Q4 for panels A and B, respectively.

IPS0.353−2.006−1.903−3.5480.361−2.400−1.997−4.183
CIPS−1.694−2.072−1.961−2.1540.9791.5631.522−1.788

The above conclusion is in line with the test results reported by Harris et al. (2004) for a similar data set using an altogether different procedure that tests the null hypothesis of the real exchange rates being stationary. A similar conclusion is also reached by Choi and Chue (2007) for the G7 countries using subsampling techniques. Turning to the bootstrap procedure advanced in Smith et al. (2004), in Table XII we report the p-values of three out of the five tests considered by these authors, namely *, Max*, and WS*, which are the bootstrap versions of the ADF test, the Max test of Leybourne (1995), and the weighted symmetric (WS) test of Pantula et al. (1994) that allow for cross-section dependence using bootstrap blocks of m = 30 or 100.20 These test results and the conclusion one obtains from them concerning the validity of the PPP hypothesis critically depend on the ADF version of the test used in the bootstrap procedure. The * version of the test (which is based on the same underlying statistic used in the CIPS test) does not reject the unit root hypothesis for values of p ≤ 3. For p = 4 the * test rejects the null at 7% and 5% levels for the sample periods 1974Q1–1998Q4 and 1988Q1–1998Q4, respectively. In contrast, the Max and the WS tests reject the unit root hypothesis at 6% or less so long as p ≥ 2. Overall, the test results are inconclusive and further analysis of the small sample properties of the unit root tests based on the Max and the WS variants of the ADF statistics would seem desirable. It would also be interesting to investigate the small-sample properties of CIPS type tests based on Max and WS statistics computed using CADF regressions.

Table XII. p-values for bootstrap panel data unit root tests applied to 17 OECD quarterly real exchange rates
  1. Note: All statistics are based on univariate AR(p) specifications in the level of the variables with p ≤ 4 including an intercept term only and the regressions are estimated on the same sample period, namely, 1974Q1–1998Q4 and 1988Q1–1998Q4 for panels A and B, respectively. Estimates were obtained using 5000 replications.

Panel A: 1974Q1–1998Q4 (T = 100)
Block size statisticp = 1 mp = 2 mp = 3 mp = 4 m
30100301003010030100
*0.5830.5830.2030.2190.2230.2320.0660.069
Max*0.3240.3240.0550.0600.0570.0640.0080.007
WS*0.2540.2540.0420.0440.0420.0500.0060.006
Panel B: 1988Q1–1998Q4 (T = 44)
Block size statisticp = 1 mp = 2 mp = 3 mp = 4 m
30100301003010030100
*0.5810.5810.1770.1710.2230.2270.0400.046
Max*0.2330.2330.0280.0310.0400.0370.0030.004
WS*0.1500.1500.0220.0240.0280.0250.0030.004

7.2. Real Earnings

In their analysis of variance dynamics of real earnings Meghir and Pistaferri (2004) impose a unit root on log real earnings of households in the PSID, without providing any empirical evidence in its support. Here we address the validity of their procedure. These authors consider households with male heads aged 25–55 with at least 9 years of usable earnings data. In our application we follow them, but in view of the Monte Carlo results on the power properties of the CIPS test we further restrict the set of households to those with 22 years of usable earnings so that we end with T = 20 when estimating ADF(p) or CADF(p) regressions with p = 1. In fact, for models with linear trends one would even need a larger T for the CIPS test to have reasonable power even for N large. We also follow Meghir and Pistaferri (2004) and group the households by their educational backgrounds into high school dropouts (HSD, those with less than 12 grades of schooling), high school graduates (HSG, those with at least a high school diploma, but no college degree), and college graduates (CLG, those with a college degree or more). Although our proposed test allows for heterogeneity of income dynamics across households, it is not necessarily true that all household groupings would be subject to the same degree of cross-section dependence.

The CD statistics for all the households (N = 181), and the three subgroups (NHSD = 36, NHSG = 87, and NCLG = 58), together with the associated IPS and CIPS statistics, are given in Table XIII. The CD test is statistically significant for the sample as a whole, and for two of the three educational groups. It is not statistically significant in the case of the high school dropouts. This outcome is robust to the choice of p and does not depend on whether a linear trend is included in the earnings equations. Based on the IPS test, the unit root hypothesis is rejected for the whole sample and in the case of all the three subgroups, irrespective of whether a linear trend is included in the individual earning equations. But once the CIPS test is considered the test outcomes are mixed. The unit root hypothesis is rejected for the sample as a whole, but not for all the subgroups.21 This could be partly due to lack of power of the CIPS test in small samples as compared to the IPS test. It is therefore important that the CIPS test is used when there is significant evidence of cross-section dependence. This is readily seen in the case of the test results for the HSD group, where the unit root hypothesis is rejected by IPS but not if CIPS is used in the case of models with linear trends.

Table XIII. Panel unit root tests applied to log real earnings of households in PSID data (T = 20)
Panel A: With intercept
Household groupingsAllHSDHSGCLG
N = 181N = 36N = 87N = 58
p = 0
 CD11.620.604.746.90
 IPS−15.11−8.68−10.82−6.61
 CIPS−3.03−2.67−3.03−3.44
p = 1
 CD9.730.574.314.44
 IPS−7.34−4.83−5.64−2.25
 CIPS−2.30−2.00−2.22−2.68
Panel B With intercept and trend
Household groupingsAllHSDHSGCLG
N = 181N = 36N = 87N = 58
p = 0    
 CD10.551.033.384.06
 IPS−17.73−7.30−11.99−10.89
 CIPS−3.34−3.02−3.32−3.66
p = 1    
 CD10.811.193.714.41
 IPS−7.43−3.00−5.05−4.57
 CIPS−2.58−2.30−2.51−2.85

7.3. Test Results Based on Moon–Perron Procedure

So far our empirical analyses assume that the error cross-section dependence can be approximated by a single factor model. It is therefore important that the factor structure of the data sets being analysed is investigated and the robustness of the conclusions to the number of factors is examined. To this end we first applied the factor selection procedure advanced in Bai and Ng (2002), using their information criteria, PCi and ICi, i = 1, 2, 3, to the standardized first differences of the series, assuming the maximum number of factors, kmax, to be 1, 4 and 6. For the real exchange rate series all the six selection criteria choose the maximum number of factors assumed, for both sample periods, 1974Q1–1998Q4 and 1988Q1–1998Q4. Not surprisingly, the Bai and Ng selection procedure does not seem to have discriminatory power in this application where N = 17 and T is relatively small. On this also see Bai and Ng (2002, p. 203).

The application of Bai and Ng selection criteria to the real earnings data tended to be more informative, but still quite sensitive to the choice of the information criteria and the assumed maximum number of factors. For example, when kmax = 8, PC1PC3 selected 6–8 factors, while IC1 and IC2 selected no factors at all, and IC3 selected 8 factors in the case of the subsamples, HSD and CLG, and 0 factors for the pooled sample and the HSG subsample.

In view of the considerable uncertainty that surrounds the choice of factors, at least in the case of our data sets, in the applications of the Moon–Perron unit root tests we decided to compute their test statistics, equation image and equation image, for different values of k (the number of factors) in the range 1–4.22 The results are summarized Table XIV. In these applications the choice of the number of factors does not seem to play an important role and the test outcomes are the same for different values of k. Also in the case of the real exchange rate series the results are in line with the tests based on the CIPS statistic and the unit root hypothesis cannot be rejected.

Table XIV. Moon and Perron panel unit root tests for different numbers of factors k = 1, 2, 3, 4 (the intercept case)
Number of factorsk = 1k = 2k = 3k = 4
  1. Notes: Under the unit root hypothesis the Moon–Perron statistics are standard normal for large T. Long-run variance used in the construction of equation imageandequation image is computed using the Andrews and Monahan (1992) estimator.

17 OECD real exchange rate
1974Q1–1998Q4 (T = 103)
 equation image test−0.047−0.044−0.0170.015
 equation image test−0.407−0.447−0.2130.210
1988Q1–1998Q4 (T = 47)
 equation image test0.0040.0110.0130.022
 equation image test0.0520.1520.2370.512
Log real earnings (T = 21)
Pooled (N = 181)
 equation image test0.1090.1130.1190.133
 equation image test1.9252.0932.2192.599
HSD (high school dropouts, N = 36)
 equation image test−0.006−0.0010.0080.017
 equation image test−0.082−0.0190.1300.319
HSG (high school graduates, N = 87)
 equation image test0.0890.1090.1100.108
 equation image test2.0112.6142.6472.582
CLG (higher college graduates, N = 58)
 equation image test0.0980.0940.0940.096
 equation image test2.0781.9492.0682.303

The application of the MP tests to the real earnings series, however, is very sensitive to the choice of test statistic. The two tests yield similar outcomes only in the case of the subsample HSD (high school dropouts). For the other groupings only the equation image test rejects the null hypothesis. Focusing on the equation image test (also favoured by MP), we note that these test results are in accordance with the outcomes of the CIPS tests discussed above, and the imposition of a unit root on log real earnings does not seem justified.

8. CONCLUDING REMARKS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information

This paper presents a new and simple procedure for testing unit roots in dynamic panels subject to (possibly) cross-sectionally dependent as well as serially correlated errors. The procedure involves augmenting the standard ADF regressions for the individual series with current and lagged cross-section 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 cross-section averages to perform a similar task in dealing with the cross-dependence 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 cross-dependence 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 cross-section averages formed over subgroups, such as regions, sectors or industries. Another worthwhile extension would be to consider cross-section 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 small-sample power properties.

In their analysis Bai and Ng (2004) also consider the possibility of unit root in the common factors. However, under their set-up 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):

  • equation image

where ft is the common factor, γi the associated factor loadings, and vit the idiosyncractic component assumed independently distributed of ft. The unit root properties of yit is determined by the maximum order of integration of the two series ft and vit. Hence, yit will be I(1) if either vit and/or ft contain a unit root. Averaging across i and letting N[RIGHTWARDS ARROW]∞, for each t, t[RIGHTWARDS ARROW]0, if vit is stationary, and t[RIGHTWARDS ARROW]c, where c is a fixed constant if vit is I(1). Therefore, a unit root in ft may be tested by testing the presence of a unit root in ȳt 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 ϕi = 1, the order of integration of yit changes from being I(1) if ft is stationary, to I(2) if ft is I(1). Therefore, in our set-up it makes sense not to allow ft to have a unit root. The models advanced here and the static factor model used by Bai and Ng serve different purposes.

APPENDIX: MATHEMATICAL PROOFS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information

A.1. SOME PRELIMINARY ORDER RESULTS

Recall from (18) and (20) that equation image, where equation image. Also equation image where

  • equation image(A.1)

Hence

  • equation image(A.2)
  • equation image(A.3)
  • equation image(A.4)
  • equation image(A.5)
  • equation image(A.6)
  • equation image(A.7)
  • equation image(A.8)

Now using results in Pesaran (2006, Appendix) it is easily seen that

  • equation image(A.9)
  • equation image(A.10)
  • equation image(A.11)
  • equation image(A.12)

Also using results in Fuller (1996, p. 547) and carrying out similar derivations, we also have

  • equation image(A.13)
  • equation image(A.14)
  • equation image(A.15)
  • equation image(A.16)
  • equation image(A.17)
  • equation image(A.18)
  • equation image(A.19)
  • equation image(A.20)
  • equation image(A.21)

A.2. ASYMPTOTIC DISTRIBUTION OF Ti(N, T): SERIALLY UNCORRELATED CASE

The equation imagei(N, T) statistic defined by (24) may be written as

  • equation image(A.22)

where υi∼(0, IT), and equation image is defined by (A.1), which is the standardized random walk associated to υi.

First consider the numerator of ti(N, T), and note that

  • equation image(A.23)

where

  • equation image(A.24)

Also

  • equation image(A.25)
  • equation image(A.26)

Using (15) and (16) we have

  • equation image(A.27)
  • equation image(A.28)
  • equation image(A.29)
  • equation image(A.30)
  • equation image(A.31)
  • equation image(A.32)
  • equation image(A.33)
  • equation image(A.34)
  • equation image(A.35)

Similarly, apart from equation image and T−1(υiυi), the remaining terms in the denominator of ti(N, T) may also be written in terms of the above expressions.

A.2.1. T Fixed and N [RIGHTWARDS ARROW]

In this case

  • equation image

where equation image. Hence, for a fixed T those elements in ti(N, T) that involve equation image and −1 will converge to zero in mean square errors as N[RIGHTWARDS ARROW]∞. Assuming also that the series, yit, are in the form of deviations from the cross-sectional mean of the initial observations so that ȳ0 = 0, using the above results for a fixed T and as N[RIGHTWARDS ARROW]∞ we have (in mean square errors)

  • equation image

where

  • equation image

γ* is the limit of equation image as N[RIGHTWARDS ARROW]∞,

  • equation image

Similarly,

  • equation image

where

  • equation image

Using these results in (A.22) we finally obtain

  • equation image(A.36)

which is free of nuisance parameters and its probability distribution can be simulated for any given value of T > 4. Recall that ft and εiti are independently distributed as i.i.d.(0, 1).

A.2.2. Sequential Asymptotic: N[RIGHTWARDS ARROW]∞ then T[RIGHTWARDS ARROW]

First, using familiar results from the unit root literature as T[RIGHTWARDS ARROW]∞, we have (see, for example, Hamilton, 1994, p. 486)

  • equation image

where Wi(r) and Wf(r) are independently distributed standard Brownian motions defined on [0, 1]. Similarly,

  • equation image

where Wf(r) and Wfi(r) are also distributed as standard Brownian motions. Finally, it is easily seen that

  • equation image

Using these results in (A.36) as T[RIGHTWARDS ARROW]∞, we obtain the following sequential limit distribution:

  • equation image(A.37)

where

  • equation image(A.38)
  • equation image(A.39)
A.2.3. Joint Asymptotics

Using the results (A.9)–(A.21) it is easily seen that all the terms in (A.2)–(A.8) that contain the cross-section means, equation image and −1, converge in quadratic means to zero as N,T[RIGHTWARDS ARROW]∞, jointly so long as equation image. This latter condition is satisfied if N/T[RIGHTWARDS ARROW]k, where k is a fixed finite non-zero positive constant. Under this condition we also have equation image. It therefore follows that the asymptotic result, (A.37), also holds under joint asymptotics so long as equation image. In particular

  • equation image(A.40)

where ti(N, T) is defined by (7).

A.3. ASYMPTOTIC DISTRIBUTION OF Ti(N, T): SERIALLY CORRELATED CASE

Consider first the numerator of (A.22) and note that

  • equation image

where now

  • equation image

The elements of szi,−1 and z,−1 are defined by (49) and (50), and can be written in terms of general first-difference stationary processes. Recall also that equation image.

Using the results set out above, together with the familiar results on stationary first-difference processes summarized, for example, in Proposition 17.3 of Hamilton (1994), the following limits can now be established under joint asymptotics (with N and T[RIGHTWARDS ARROW]∞, such that N/T[RIGHTWARDS ARROW]k, ∞> k > 0):

  • equation image

where Λf, ψif, and κif, are defined by (A.38) and (A.39), γ* ≠ 0 is the limit of equation image as N[RIGHTWARDS ARROW]∞, Wi(r) and Wf(r) are independent standard Brownian motions, and

  • equation image

Similarly,

  • equation image

where it is also easily seen that

  • equation image

and equation image. Using the above results in (A.22) we now have (|ρ|< 1)

  • equation image

which reduces to (A.40), the asymptotic limit distribution of the CADF obtained in the case of the serially uncorrelated errors.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information

I would like to thank Soren Johansen and Chris Rogers for helpful discussions with respect to the analysis of exchangeable processes, and Jörg Breitung, Roger Moon, Benoit Perron, Ron Smith, four anonymous referees, and the editor (Badi Baltagi) for constructive comments. I am also particularly grateful to Yongcheol Shin for carrying out some preliminary computations that were instrumental in helping me form some of the ideas developed formally in the paper. Excellent research assistance by Mutita Akusuwan is also gratefully acknowledged, who carried out all the Monte Carlo computations reported in this paper. Computations of the empirical applications in Section 7 were carried out by Vanessa Smith and Takashi Yamagata.

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A SIMPLE DYNAMIC PANEL WITH CROSS-SECTION DEPENDENCE
  5. 3. UNIT ROOT TESTS FOR ONE-FACTOR RESIDUAL MODELS WITH SERIALLY UNCORRELATED ERRORS
  6. 4. CADF PANEL UNIT ROOT TESTS
  7. 5. CASE OF SERIALLY CORRELATED ERRORS
  8. 6. SMALL-SAMPLE PERFORMANCE: MONTE CARLO EVIDENCE
  9. 7. EMPIRICAL APPLICATIONS
  10. 8. CONCLUDING REMARKS
  11. APPENDIX: MATHEMATICAL PROOFS
  12. Acknowledgements
  13. REFERENCES
  14. Supporting Information
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  • 1

    Following Maddala and Wu (1999) bootstrap techniques have also been utilized to deal with cross-section dependence in panel unit root tests. See, for example, Smith et al. (2004) and Chang (2004). These procedures are also valid if N is fixed as T [RIGHTWARDS ARROW] ∞.

  • 2

    Clearly, it is also possible to construct the CADF test based on recent modifications of the ADF test proposed in the literature, for example, the ADF-GLS test of Elliott et al. (1996), the weighted symmetric ADF (WS-ADF) test of Fuller and Park (1995) and Fuller (1996, Section 10.1.3), or the Max-ADF test of Leybourne (1995). The use of the latter two modifications of ADF statistics in the IPS panel unit root test have been recently considered by Smith et al. (2004), who report significant gain in power as compared to the IPS test based on standard ADF statistics.

  • 3

    A comparative study of the small sample properties of the panel unit root tests proposed in this paper with those advanced by Moon and Perron (2004) and Bai and Ng (2004) is provided by Gengenbach et al. (2005) and Kapetanios (2007).

  • 4

    I am grateful to one of the referees for bringing this point to my attention.

  • 5

    See, for example, Theorem 1.2.2 in Taylor et al. (1985, p. 13).

  • 6

    To save space the figures of the simulated densities are not included but are available upon request.

  • 7

    The analysis can be readily extended to higher-order processes.

  • 8

    The order of augmentation, p, can be estimated using model selection criteria such as Akaike or Schwartz applied to the underlying time series specification, namely (54).

  • 9

    Under the alternative hypothesis μi are drawn as μi∼i.i.d.U[0, 0.02].

  • 10

    In calculation of P and Z statistics the rejection probabilities, piT, are truncated to lie in the range [0.000001, 0.999999], in order to avoid very extreme values affecting these test statistics. This, in effect, is a kind of truncation, similar to the truncated version of the IPS statistic.

  • 11

    See (31), (32), and the notes to Table III for further details.

  • 12

    To save space the normal approximation results are not reported.

  • 13

    These Monte Carlo results are in line with the theoretical findings of Moon HR et al. (2007), who show that local power of panel unit root tests is in N−1/2T−1 neighborhood of the null in the case of models that contain intercepts only and N−1/4T−1 for models with linear trends.

  • 14

    Panel unit root tests have also been applied to test the convergence of log output per capita across countries. But, as argued in Pesaran (2007), unit root tests applied to per capita output series either individually or in their panel forms are not informative about within- or cross-country convergence.

  • 15

    Further details together with the data are available from the Journal of Applied Econometrics Data Archive (http://qed.econ.queensu.ca/jae/).

  • 16

    This error was identified by Vanessa Smith (one of the authors of the Smith et al. paper) in the process of replicating and extending the test results for inclusion in the current paper.

  • 17

    Specifically, equation image, and equation image.

  • 18

    The same conclusion would be reached if p is selected for each series by a model selection criteria such as AIC.

  • 19

    Identical results are obtained using the truncated version, the CIPS* statistic. Also similar results are obtained using the average probability versions of the CIPS test defined by (31) and (32).

  • 20

    For details of the bootstrap procedure used see Smith et al. (2004). I am grateful to Vanessa Smith for carrying out the computations.

  • 21

    The critical values for the CIPS tests are given in Tables III(a)–III(c). For example, for T = 20 and N = 181 the 5% critical value of the CIPS test in the case of models with an intercept is − 2.04, and for models with an intercept and a linear time trend it is − 2.55.

  • 22

    I am grateful to Takashi Yamagata for carrying out the computations and to Benoit Perron for kindly providing us with his codes.