A Panel Unit-Root Test with Smooth Breaks and Cross-Sectional Dependence

Transcription

1 A Panel Unit-Root est with Smooth Breaks and Cross-Sectional Dependence Chingnun Lee Institute of Economics, National Sun Yat-sen University, Kaohsiung, aiwan. address: lee Jyh-Lin Wu Institute of Economics, National Sun Yat-sen University, Kaohsiung, aiwan. Department of Economics, National Chung Cheng University, Chia-Yi, aiwan. address: May 26, 212 Corresponding author: Jyh-Lin Wu. address: phone: ext. 5616, fax:

2 A Panel Unit-Root est with Smooth Breaks and Cross-Sectional Dependence Abstract his paper develops a simple panel unit-root test that accommodates cross-sectional dependence among variables and smooth structural changes in deterministic components. he proposed test is the simple average of the individual statistics constructed from the breaks and cross-sectional dependence augmented Dickey-Fuller (BCADF ) regression. Applying both the sequential and joint limit approaches, this paper shows that the asymptotic distribution of the BCADF statistic is free of nuisance parameters as N, go to infinity. We also extend our analysis to the case where shocks are serially correlated. he limiting distribution of the average BCADF statistic is shown to exist and its critical values are tabulated. Monte-Carlo experiments point out that the size and power of the average BCADF statistic are generally good as long as is greater than fifty. he test is then applied to examine the validity of long-run purchasing power parity. Keywords: Panel unit-root test, Fourier approximation, Cross-sectional dependence, Purchasing power parity. JEL classification: C12, C33. 1 Introduction he development of panel unit-root tests had been a hot research topic during the past decade. he first generation panel unit-root tests either neglect cross-sectional dependence or deal with cross dependence by bootstrapping critical values but fail to derive the asymptotic distribution of test statistics (Levin et al., 22; Im et al., 23, IPS; Maddala and Wu, 1999). However, controlling contemporaneous correlation across individuals is important to a panel unit-root test since leaving out cross-sectional dependencies would lead to serious size distortions and power loss (O Connell, 1998). he second generation models allow for cross-sectional dependence and derive the asymptotic distribution of test statistics (Chang, 22; Phillips and Sul, 23; Bai and Ng, 24; Moon and Perron, 24; Smith et al., 24; Breitung and Das, 23; Choi and Chue, 27; Pesaran, 27; Pesaran et al., 29). he above-mentioned articles assume away structural changes in their models. 1

3 Several articles allow panel unit-root tests with structural breaks (Im et al., 25; Carrioni-Silvestre et al., 21; Carrion-i-Silvestre et al., 25). hese tests, however, assume that the variables in the panel are cross-sectionally independent. Recently, Bai and Carrion-i- Silvestre (29) and Im et al. (21, IL) have proposed panel unit-root tests in the presence of multiple structural changes and cross-sectional dependence. Bai and Carrion-i-Silvestre (29) propose a modified Sargan-Bhargava (1983, MSB) test in the panel setting. Although this test is invariant to both mean and trend break parameters, the limiting distribution of the individual MSB (MSB (λ)) test depends on the number of structural breaks. Following the cross-sectionally augmented procedure of Pesaran (27), Im et al. (21) develop an LM-type panel unit-root test to account for possible heterogeneity in both the level and the trend of the series. he IL test is invariant to the nuisance parameters but its limiting distribution depends on the number of trend breaks. However, the number of breaks and break dates are generally unknown in empirical investigation and hence the accuracy of these estimates are crucial to the performance of unit-root tests. It is also difficult to precisely estimate the number and magnitudes of multiple breaks especially when the breaks are of opposite sign (Prodan, 28). Moreover, if asymptotic distribution depends on unknown number and location of breaks then critical values are not readily available in empirical analysis. Instead of adopting dummy variables to capture structural breaks, several articles assumes smoothing breaks in the level or slope of series with a known functional form (Leybourne, et al., 1998; Sollis, et al. 22; Kapetanios, et al. 23; Cerrato, et al., 211). However, the transition function causing smoothing breaks is generally unknown, a priori, in empirical analysis. o take into account smoothing breaks in the deterministic components of a series with an unknown functional form, several papers develop unit-root tests based on Gallant s (1981) flexible Fourier Form. It has been observed that a Fourier approximation to deterministic components can, to any desired degree of accuracy, capture the behavior of a deterministic function of unknown form even if the function itself is not periodic (Gallant, 1981; Davies, 1987; Bierens, 1997; Becker et al., 24, 26; Enders and Lee, 29, 211). One important feature of the Fourier approximation unit-root tests is that it is not necessary to assume that break dates are known a priori. Moreover, Enders and Lee (29, 211) point out that a single-frequency-component Fourier function works better than dummy variable methods regardless of whether the breaks are instantaneous or smooth. Another advantage of the Fourier approximation is that it involves only the determination of the appropriate fre- 2

4 quency component in the model and hence avoids the complication of selecting break dates, the number of breaks and the form of breaks. Because of the above interesting results, Enders and Lee (29, 211) and Rodrigues and aylor (29) develop unit-root tests for time series data by augmenting the conventional OLS and GLS regression to include a singlefrequency-component Fourier function. hey find that their proposed tests are useful since they are robust to a variety of possible break mechanisms in the deterministic trend function of unknown forms and numbers. o allow for both heterogeneous breaks and cross-sectional correlations, this paper generalizes the cross-sectionally augmented ADF (CADF ) regression proposed by Pesaran (27) to incorporate a single-frequency-component Fourier function with heterogeneous amplitudes. he included Fourier function is used to capture the unknown multiple structural breaks (Becker et al., 26; Enders and Lee, 211). We thus construct the individual breaks and cross-sectional dependence augmented ADF (BCADF ) statistics and their simple average. he latter statistic is referred to as the breaks and cross-sectional dependence augmented IP S (BCIP S) statistic. An important advantage of our proposed tests is their simplicity in empirical applications. o analyze the impact of Fourier terms in the BCADF regression in both finite and large sample size ( ), new asymptotic results of BCADF and BCIP S statistics are derived based on the sequential and joint limit approaches, respectively. In the case of serially uncorrelated errors, heorem 1 shows that the asymptotic distribution of BCADF, under a fixed, is not free of nuisance parameters even when N tends to infinity. his is because the inclusion of the Fourier terms in the null process makes the stochastic part of the regressors composed of the terms of different orders in probability under a fixed. heorem 2 shows that the dependence of the asymptotic distribution on nuisance parameters vanishes as both N and sequentially and jointly tend to infinity. his is due to the fact that stochastic trend terms asymptotically dominate Fourier terms as tends to infinity. he Corollary examines the limiting distribution of Pesaran s CADF statistic (that assumes no breaks) in the presence of Fourier form breaks. We show that the asymptotic distribution of the CADF statistic under a fixed includes omitted-variable bias even when N tends to infinity, although the bias vanishes when both N and approach infinity. he limiting distribution of the truncated version of the BCIP S statistic, BCIP S, is also shown to exist. Furthermore, this paper extends the discussion to the case where the residuals are serially correlated. By adding a lagged first-differenced variable and its cross-sectional mean in the regression, heorem 3 3

5 shows that the BCADF statistic, under serially correlated errors, has the same asymptotic distribution as that in heorem 2 when both N, tend to infinity. Although the asymptotic distribution of the BCIP S statistic exists, it is not analytically tractable, this paper therefore tabulates its critical values under different N, and k. he size and power of the BCIP S statistic under different scenarios are then explored by Monte- Carlo simulations. Simulation results indicate that the size and power of our proposed statistic are generally good regardless of N and k as long as is greater than fifty. On the other hand, the size of Pesaran s CIP S test that ignores the Fourier terms in regression is similar to that of BCIP S test when the amplitude of smooth breaks is small. Moreover, the power of the CIP S test is either greater than or equal to that of the BCIP S test in such a case. With large amplitudes of breaks, the size distortion of CIP S statistic is quite serious in many cases even when is as large as 1, but the distortion disappears when is 5,. he finding of substantial finite sample bias is important for empirical investigation with macro or financial data since the number of observations in those researches is generally less than 1. One may be uncertain with the large or small amplitudes of breaks in applied works, our simulation results indicate that applying the BCIP S test is safer than applying the CIP S test when smooth breaks appear in a data generating process (DGP) as long as is greater than 1. Finally, the proposed panel unit-root test is applied to investigate the long-run purchasing power parity (PPP) over the post-bretton Woods period. he remainder of this paper is organized as follows. Section 2 sets out the basic dynamic heterogeneous panel data model with smooth breaks. he cross-sectional dependence across variables is modeled by an unobservable stationary common factor and the smooth breaks in deterministic terms are captured by a single-frequency Fourier function. In Section 3, we derive the null distribution of the individual BCADF statistic with serially uncorrelated errors, discuss the BCADF -based panel unit-root test, and extend our results to the case with serially correlated errors. Section 4 examines the finite-sample properties of the proposed panel unit-root test, BCIP S, by Monte-Carlo simulations. Section 5 provides an empirical application. Finally, Section 6 concludes. 2 he Model Let y it be an observation on the ith cross-sectional unit at time t and suppose that it is generated according to the following simple dynamic linear heterogeneous panel data model 4

6 with an unknown time-dependent deterministic term δ i (t): (1 φ i L)(y it δ i (t) ς i t) = u it, u it = γ i f t + ε it, t = 1,..,, i = 1,.., N, (1) where ς i t is a linear trend, f t is an 1 1 unobserved stationary stochastic common factor, 1 γ i is the associated factor loading reflecting the degree of contemporaneous correlation across individuals and the ε it is an idiosyncratic error and the initial value, y i, is given. Since as shown by Enders and Lee (211) that a single Fourier frequency mimics a variety of breaks in the deterministic component, we begin our analysis with a DGP containing only a Fourier frequency: δ i (t) = ϖ i,k,t = µ i + α i,1 sin(2πkt/ ) + α i,2 cos(2πkt/ ), (2) where k is the frequency component reflecting the number of cycles in the sample period that is assumed to be homogeneous across agents. α i,1 and α i,2 measure the heterogeneous amplitude and displacement of the sinusoidal component of the deterministic term across agents. ϖ i,k,t in (2) captures smooth breaks in deterministic components. Substituting (2) into (1), we obtain 2 y it = β i (y i,t 1 α id t 1 ) + α i d t + γ i f t + ε it, (3) where y it = y it y i,t 1, d t = (1, sin(2πkt/ ), cos(2πkt/ ), t) is a 4 1 vector of deterministic common effect, d t = (, sin(2πkt/ ), cos(2πkt/ ), 1), β i = (1 φ i ), and α i = (µ i, α i,1, α i,2, ς i ). he unit-root hypothesis, φ i = 1, can be expressed as: H : β i =, i (4) against the possibly heterogeneous alternatives, H 1 : β i <, i = 1, 2,..., N 1 ; β i =, i = N 1 + 1, N 1 + 2,..., N. (5) 1 Pesaran et al. (29) extend Pesaran (27) model to contain a multifactor error structure. 2 A restriction embedded in a Fourier function is that the starting and ending value of the function are the same. he introduction of a time trend, ς it, in the function removes the above restriction. As such, changes in the intercept and slope of a deterministic function can be captured by the Fourier approximation. herefore our proposed panel unit-root tests allow for breaks in both the level and trend of the series under investigation. 5

7 Under the above null hypothesis, equation (3) can be solved for y it as follows: y it = ỹ i + α id t + γ i s ft + s it, where α i d t = µ i + α i,1 sin(2πkt/ ) + α i,2 cos(2πkt/ ) + ς i t, s ft = f 1 + f f t, s it = ε i1 + ε i ε it, ỹ i = y i α i d. herefore, under H, y it is composed of a deterministic component with a Fourier element, ỹ i + α i d t, a common stochastic component, s ft I(1), and an idiosyncratic component, s it I(1). We do not assume α i,1 = α i,2 = under the null hypothesis. As such, heterogeneous breaks exist under the null and alternative hypothesis in (4) and (5), respectively, and hence our proposed tests avoid the possibility of spuriously rejecting a unit-root hypothesis (Leybourne et al., 1998 and Enders and Lee, 211). he following assumptions are required for deriving the null distribution of a series-specific unit-root test. Assumption 1 (Idiosyncratic Errors): he idiosyncratic error, ε it, with a zero mean, a constant variance σi 2, and a finite fourth-order moment, is independently distributed across i and t and is independent of f s for all i, t, s. Assumption 2 (Common Factors): he common factor, f t, is serially uncorrelated and has a zero mean, a constant variance σ 2 f, and a finite fourth-order moment. Besides, σ2 f to be unity without loss of generality. is assumed Assumption 3: (Factor Loadings): he factor loading parameter, γ i, satisfying plim 1 N N i=1 γ i = γ, is independently distributed across i and is independent of the idiosyncratic ε it and of the common factor f s for all i, t, s. Assumption 4: (Fourier Amplitude Coefficients): he Fourier amplitude coefficients, α i,1 and α i,2, are nonrandom parameters. Assumptions 1-3 together imply that the composite error, u it, is serially uncorrelated and the case with serially correlated errors will be discussed in Section Breaks and Cross Dependence Augmented Unit-Root ests heorems 1-3 and the Corollary in this section derive the asymptotic distribution of the 6

8 series-specific unit-root test statistic under the null hypothesis in (4). Note that all order results and the proofs of theorems given in the Appendix are derived for the case where d t = (1, sin(2πkt/ ), cos(2πkt/ )), t = 1, 2..,, which implies d t = (, sin(2πkt/ ), cos(2π kt/ )). he asymptotic results for the case where d t = (1, sin(2πkt/ ), cos(2πkt/ ), t) can be derived in a similar manner. 3.1 Unit-root ests with Serially Uncorrelated Errors Although f t in (3) is unobservable, it can be filtered out by taking the cross-sectional average of (3) (Pesaran, 26). By doing so, the common factor f t can be measured by the linear combination of sin(2πkt/ ), cos(2πkt/ ), sin(2πk(t 1)/ ), cos(2πk(t 1)/ ), y t and y t 1, where y t = 1 N N i=1 y it and y t = 1 N N i=1 y i,t 1. Following the argument by Enders and Lee (29, p.14), we remove sin(2πk(t 1)/ ) and cos(2πk(t 1)/ ) to avoid collinearity from the auxiliary regression equation that is equation (3) augmented by crosssectional average as observable proxies for f t. We therefore regress the following breaks and cross dependence augmented Dickey-Fuller equation using OLS: 3 y it = c i, + c i,1 sin(2πkt/ ) + c i,2 cos(2πkt/ ) + c i,3 y t 1 +c i,4 y t + b i y i,t 1 + e it. (6) It is worth noting that sin(2πk(t 1)/ ) and cos(2πk(t 1)/ ) appear in the DGP although they are omitted in equation (6). herefore the above two terms show up under both the unit root null and stationary alternative while studying the properties of the test. he t-statistic of the estimate of b i (ˆb i ) is then applied to examine the unit-root hypothesis and can be expressed as t i (N, ) = y i M zy i, 1 ˆσ i (y i, 1 M, (7) zy i, 1 ) 1/2 where y i = ( y i1, y i2,..., y i ), y i, 1 = (y i, y i1,..., y i, 1 ), ˆσ 2 i = y i M i,z y i 6, M z = I Z(Z Z) 1 Z, M i,z = I G i (G i G i) 1 G i, G i = (Z, y i, 1 ), Z = ( y, τ, Υ 1, Υ 2, y 1 ), τ = (1, 1,..., 1), Υ 1 = ( sin(2πk1/ ), sin(2πk2/ ),..., sin(2πk/ )), Υ 2 = ( cos(2πk1/ ), cos(2πk2/ ),..., cos(2πk/ )), y = ( y 1, y 2,..., y ), y 1 = (y, y 1,..., y 1 ). It should be noted that the exact null distribution of t i (N, ) defined in (7) is affected by nuisance parameters through their effect on the matrix M z and M i,z. In the model of Pesaran 3 See (L5) and (L6) in Lemma 1 in the appendix. 7

9 (27), the effect of the nuisance parameters on t i (N, ) vanishes as N. However, the following theorems show that the asymptotic distribution of t i (N, ) in our model is free of nuisance parameters only when both N and. heorem 1: Let y it be generated based on (3) under the null hypothesis in (4) with the cross-sectional mean of the initial observation y being zero. Suppose that assumptions 1-4 hold and γ, then the null distribution of t i (N, ) given by (7) will not be free of nuisance parameters as N for any fixed > 6. In particular, we have where t i (N, ) N ε i s i, 1 σ 2 i (q i Γ )(Γ Ψ f Γ ) 1 (Γ h i ) J 1 J 2, (8) J 1 = J 2 = Γ = Ξ i = Θ = ( ε i ε i σi 2( 6) (d i Θ )(Θ Ξ i Θ ) 1/2 ) 1 (Θ d i ), (9) 6 ( s ) i, 1 s 1/2 i, 1 σi 2 2 (h i Γ )(Γ Ψ f Γ ) 1 (Γ h i ), (1) γ 1 α1 1 α2 1 1, (11) 1 1/2 α1 1/2 α2 γ [ ] [ ] Ξ (1) i Ξ (2), Ψ i f = Ψ (1) f Ψ (2), (12) f Γ, (13) 1/2 y 1/2 α i,1 1/2 α i,2 γ i 1 8

10 q i = Ξ (1) i = Ξ (2) i = Ψ (2) f = f ε i σ i τ ε i σ i (Υ 1 ε i ) σ i (Υ 2 ε i ) σ i s Υ 1, 1 ε i σ i s Υ 2, 1 ε i σ i s f, 1 ε i σ i s i, 1 f 3/2 f s i, 1 3/2 τ s i, 1 3/2 Υ 1 s i, 1 σ i 1/2 Υ 2 s i, 1 σ i 1/2 s Υ 1, 1 s i, 1 3/2 s Υ 2, 1 s i, 1 3/2 s f, 1 s i, 1 2 s i, 1 s i, 1 2 f s Υ1, 1 τ s Υ1, 1, h i = f s i, 1 σ i 3/2 τ s i, 1 σ i 3/2 (Υ 1 s i, 1) σ i (Υ 2 s i, 1) σ i s Υ 1, 1 s i, 1 σ i 3/2 s Υ 2, 1 s i, 1 σ i 3/2 s f, 1 s i, 1 σ i 2 Ψ f, d i = q i s i, 1 ε i σ i s i, 1 τ s i, 1 Υ 1 s i, 1 Υ 2 s i, 1 s Υ 1, 1 s i, 1 s Υ 2, 1 3/2 σ i 1/2 σ i 1/2 3/2 3/2, Ψ (1) f = f s Υ2, 1 τ s Υ2, 1 Υ 1s Υ1, 1 Υ 1s Υ2, 1 Υ 2s Υ1, 1 Υ 2s Υ2, 1 s Υ 1, 1 s Υ 1, 1 s Υ 1, 1 s Υ 2, 1 s Υ 2, 1 s Υ 1, 1 s Υ 2, 1 s Υ 2, 1 s f, 1 s Υ 1, 1 3/2 3/2 f f, (14) s i, 1 s f, 1 2 f τ f Υ 1 f Υ 2 τ f 1 τ Υ 1 τ Υ 2 Υ 1f Υ 1τ Υ 1Υ 1 Υ 1Υ 2 Υ 2f Υ 2τ Υ 2Υ 1 Υ 2Υ 2 s Υ 1, 1 f s Υ 2, 1 f s f, 1 f 3/2 f s f, 1 3/2 τ s f, 1 3/2 Υ 1 s f, 1 1/2 Υ 2 s f, 1 1/2 s Υ 1, 1 s f, 1 3/2 s Υ 2, 1 s f, 1 3/2 s f, 1 s f, 1 2 s Υ 1, 1 τ s Υ 1, 1 Υ 1 s Υ 1, 1 Υ 2 s Υ 2, 1 τ s Υ 2, 1 Υ 1 s Υ 2, 1 Υ 2 s f, 1 τ s f, 1 Υ 1 s f, 1 Υ 2 3/2 1/2 1/2,,, in which ε i = (ε i1, ε i2,..., ε i ), f = (f 1, f 2,..., f ), s i, 1 = (, s i1, s i2,..., s i, 1 ), s f, 1 = 9

11 (s f, s f1,..., s f, 1 ), s Υ1, 1 = (, s Υ1,1,..., s Υ1, 1), s Υ2, 1 = (, s Υ2,1,..., s Υ2, 1) with s Υ1,t = t j=1 sin(2πkj/ ) and s Υ 2,t = t j=1 cos(2πkj/ ). Proof: See the Appendix. he inclusion of Fourier terms in the null process results in the stochastic part of the regressors, y, y 1 and y i, 1, being composed of the terms of different orders in probability under a fixed. For example (see the Appendix for the detail), y ε i = (α 1 Υ 1ε i ) ( Op( 1/2 )) + (α 2 Υ 2ε i ) ( Op( 1/2 )) +(γ f ε i ) ( Op( 1/2 )) and y 1 ε i = (α 1 s Υ 1, 1 ε i) ( Op( 1/2 )) +(α 2 s Υ 2, 1 ε i) ( O p( 1/2 )) + (γ s f, 1 ε i) ( Op( )). Consequently, the scaling matrix Γ and Θ in (11) and (13), respectively, are non-square and therefore Γ 1 and Θ 1 does not exist. Hence neither can Γ be canceled out in (q i Γ )(Γ Ψ f Γ ) 1 (Γ h i ) and (h i Γ )(Γ Ψ f Γ ) 1 (Γ h i ) nor Θ in (d i Θ )(Θ Ξ i Θ ) 1 (Θ d i ) from matrix inverse production. Hence, unlike the results of Pesaran (27), the null distribution of t i (N, ) given by (7) is not free of nuisance parameters as N for any fixed that is greater than 6 ( > 6). However, in the above example the stochastic trend terms (γ f ε i and γ s f, 1 ε i) asymptotically dominate the other two deterministic trend terms (α 1 Υ 1ε i + α 2 Υ 2ε i and α 1 s Υ 1, 1 ε i + α 2 s Υ 2, 1 ε i) as, resulting in the scaling matrix being a squared one and hence canceling out the effect of nuisance parameters in the distribution. 4 Simulation results in Section 4.2 and in the first panel of able 4 below point out that the effect of nuisance parameters is negligible provided that 3. heorem 2: Suppose y it is generated based on (3) under the null hypothesis in (4) and assumptions 1-4 hold. As N and, t i (N, ) has the same sequential [(N, ) seq ] and joint [(N, ) j ] limit distribution, referred to as the BCADF distribution, given by BCADF if = h if W i(r)dw i (r) q if Ψ 1 f ( ) 1 1/2, (15) W i 2(r)d(r) h if Ψ 1 f h if 4 he result is in contrast to that of West (1988) that includes a nonzero constant in the null unit-root process: y t = α+y t 1+ε t. In such a case, t=1 yt 1εt = t=1 [y+α(t 1)+st 1]εt, where st = t j=1 εj. he deterministic trend term, t=1 α(t 1)εt, asymptotically dominates the stochastic trend term, t=1 st 1εt. See Hamilton (1994, p. 496). 1

12 where Ψ f (4 4) = A = B = D 1 = D 2 = q if = h if = A 2 2 B, (16) B D [ ] (2πk) 2, D = D 1 1 D 2, cos2 (2πkr)dr 1 W f (r)dr ( (2πk) W f (s)ds + 2πk sin(2πkr) [ r W f (s)ds ] ), d(r) (2πk) 2 1 sin2 (2πkr)dr ( (2πk) 2 1 cos(2πkr) [ r W f (s)ds ] ), d(r) ( (2πk)2 cos(2πkr) [ r W f (s)ds ] ) d(r), 1 W f 2(r)dr W i (1) (2πk)σ[W (1) + 2πk sin(2πkr)w (r)d(r)] (2πk) σ[, cos(2πkr)w (r)d(r)] W f (r)dw i (r) (17) 1 W i(r)dr ( (2πk) W (s)ds + 2πk sin(2πkr) [ r ( (2πk) 2 1 cos(2πkr) [ r W (s)ds] d(r) W f (r)w i (r)dr ) W (s)ds] d(r) ). (18) W i (r) and W f (r) are independent standard Brownian motions. For the joint limit distribution to hold it is also required that as (N, ) j, N/ l, where l is a non-zero finite positive constant. Proof: See the Appendix. he asymptotic distribution of t i (N, ) depends only on the frequency component, k, but is invariant to µ i, α i,1, α i,2 and γ i (as long as γ ). Critical values of the individual BCADF statistic can be simulated based on 5, replications of regressing y it on an intercept, sin(2πkt/ ), cos(2πkt/ ), y t, y t 1 and y i,t 1 over the frequency k = 1,..., 5 and the sample t = 1,...,. he individual series were generated based on y it = y i,t

13 f t + ε it for i = 1,..., N, and t = 5, 49,..., 1, 2,..., from y i, 5 =, with f t and ε it as i.i.d. N(, 1). 5 It is interesting to examine the size distortion of Pesaran s (27) CIP S test in the presence of Fourier form breaks. heoretically, Pesaran s test is subject to the problem of the omitted-variable bias when breaks exist. Although the bias vanishes when both N and approach infinity as proved in the following Corollary, the size distortion of Pesaran s t-statistic can be substantial in finite samples when the amplitude of smooth breaks is large as shown in Section 4.1. Corollary: Suppose Assumptions 1-4 hold and y it is generated based on (3) under the null hypothesis in (4). Let t P,B i (N, ) be the statistic for testing the unit-root hypothesis and it is the t- statistic of ˆb i in Pesaran s (27) cross-sectionally augmented Dickey-Fuller regression: y it = c i, +c i,3 y t 1 +c i,4 y t +b i y i,t 1 +e it. hen, t P,B i (N, ) has the following limiting distribution as N : t P,B i (N, ) N ε i s i, 1 σ 2 i (q i Γ P )(ΓP Ψ f Γ P ) 1 (Γ P h i ) J P 1 J P 2 + O p( 1/4 ) + O p ( 1/2 ) J P 1 J P 2 + O p( 1/4 ), where J1 P = J2 P = Γ P = Θ P = ( ) 1/2 ε i ε i σi 2( 4) (d i Θ P )(Θ P Ξ i Θ P ) 1 (Θ P d i ), 4 (19) ( s ) i, 1 s 1/2 i, 1 σi 2 2 (h i, Γ P )(Γ P Ψ f Γ P ) 1 (Γ P h i ), (2) γ 1 α1 1 α2 1, (21) 1/2 α1 1/2 α2 γ Γ P, (22) 1/2 y 1/2 α i,1 1/2 α i,2 γ i 1 and Ψ f, Ξ i, q i, h i and d i and are defined in (12) and (14), respectively. If next to N, 5 he main focus of the paper is the panel test and hence we consider the result from individual test as secondary to the corresponding result for that test. We therefore do not report critical values from individual tests to save space. hese critical values are available from the authors upon request. 12

14 also tends to infinity, then t P,B (N, )seq i (N, ) W i(r)dw i (r) q P ( ) if Ψ P 1 f h P if ( ( ) ) (r)d(r) hp if Ψ P 1 1/2, f h P if W i 2 where q P if = Ψ P f = W i (1), h P 1 W if = f (r)dw i (r) 1 1 W f (r)dr W. 1 f (r)dr W f 2(r)dr W i(r)dr W f (r)w i (r)dr, he expression: W i(r)dw i (r) q P if (Ψ P f ) 1 h P if ( W i 2(r)d(r) hp if (Ψ P f ) 1 h P if ) 1/2 is the limiting distribution of Pesaran s (27) CADF statistic, constructed from the cross-sectionally augmented regression, as N, when there is no break in DGP. Proof: See the Appendix. he limiting distribution of t P,B i (N, ) under a fixed includes the O p ( 1/2 ) and O p ( 1/4 ) terms, which are the finite-sample bias of slope and standard error estimation, respectively, resulting from omitted break terms in Pesaran s cross-sectionally augmented regression. he bias disappears as tend to infinity. It is straightforward to show that under a fixed, O p ( 1/4 ) is positive but O p ( 1/2 ) can be either positive or negative depending on the relative influence of the factor loading (γ i ) and Fourier term parameters (α i,1 and α i,2 ). It is worth noting that a trivial fact from the Corollary is that the finite sample bias of Pesaran s CADF test is generally small when the amplitude of breaks is small. However, Pesaran s CADF test can either under- or over-reject the unit-root hypothesis in finite samples when break amplitudes are large. Our simulation results in Section 4.1 indicate that the size bias of the CIP S test is generally substantial even though is 1 when the amplitude of breaks is large. 3.2 BCADF based Panel Unit-Root ests he null distribution of the individual BCADF statistic is asymptotically independent of nuisance parameters when both N and go to infinity. However, BCADF if and BCADF jf 13

15 are dependently distributed with the same degree of dependence for all i j since they are all nonlinear function of a common process W f. o develop a panel unit-root test, this paper considers the breaks and cross-sectional dependence augmented version of the IP S test as follows: BCIP S(N, ) = 1 N N t i (N, ). i=1 o ensure t i (N, ) having finite moments, we follow Pesaran (27) to construct the truncated version of the BCIP S statistic: BCIP S (N, ) = 1 N N t i (N, ), (23) i=1 where t i (N, ) = t i (N, ), if M 1 < t i (N, ) < M 2, t i (N, ) = M 1, if t i (N, ) M 1, t i (N, ) = M 2, if t i (N, ) M 2, M 1 and M 2 are two positive constants such that P r( M 1 < t i (N, ) < M 2 ) is sufficiently large. Following Pesaran s (27) arguments for the convergence of CIP S (N, ), it can be shown that BCIP S (N, ) converges almost surely to a distribution depending on M 1, M 2 and W f. 6 Although the limiting distribution of BCIP S (N, ) is not analytically tractable, it can be readily simulated using (23). compute Using the normal approximation of t i (N, ), we M 1 = E(BCADF if ) Φ 1 (ɛ/2) V ar(bcadf if ), (24) M 2 = E(BCADF if ) + Φ 1 (1 ɛ/2) V ar(bcadf if ), (25) where Φ 1 is the inverse of the cdf of the normal distribution, and BCADF if is the stochastic limit of t i (N, ) as N and tend to infinity such that N/ l, ( < l < ). After constructing E(BCADF if ) and V ar(bcadf if ) by simulation and setting ɛ = 1 6, we calculate M 1 and M 2 across different k under three different models and report them in able 1. he three models under consideration are the model without intercept and trend, 6 Since the included Fourier terms are deterministic functions which can only affect the conditional expectation of Pesaran s (27) CADF if, i.e. E(CADF 1f W f ), it is appropriate to discuss the convergence of BCIP S (N, ) by following Pesaran s (27) arguments for the convergence of CIP S (N, ). 14

16 with intercept, and with intercept and trend, respectively. his paper simulates M 1 and M 2 across five different k by setting N = 3 and = 5 with 5, replications. < able 1 here > ables 2(a) to 2(c) report 1%, 5% and 1% critical values of BCIP S and BCIP S statistics for three different models across different k, N and. 7 he critical values of the BCIP S statistic may differ from those of the BCIP S statistic when is less than 5 and both critical values are reported with those of the BCIP S statistic in parentheses, otherwise only the critical values of the BCIP S statistic are reported. It is worth noting that if the critical values of the BCIP S and BCIP S statistics are different, then the value of the former statistic is slightly larger than that of the latter. his indicates a slightly rightward shift of the null distribution of the BCIP S statistic relative to that of the BCIP S statistic. Besides, critical values reported in ables 2(a) - 2(c) are also lower than the corresponding values of the CIP S and CIP S statistics in Pesaran (27) that neglects smooth changes in deterministic terms. < ables 2(a), 2(b), (2c) here > 3.3 he Case with Serially Correlated Errors Our discussion in Sections 3.1 and 3.2 can be extended to the case where individual-specific errors are serially correlated. Following Pesaran (27), two different specifications for serially correlated errors are given as follows: u it = ρ i u i,t 1 + η it, η it = γ i f t + ε it ; (26) u it = γ i f t + η it, η it = ρ i η i,t 1 + ε it, (27) where ε it is the idiosyncratic error. o derive the asymptotic distribution of the BCADF statistic, we focus our discussion on the specification of (26). he asymptotic distribution to be derived in this section can be adapted to deal with both specifications in (26) and (27). Replacing u it in (1) with (26) we obtain: y it = α i d t + β i (1 ρ i )(y i,t 1 α id t 1 ) + ρ i (1 + β i )( y i,t 1 α i d t 1 ) +γ i f t + ε it. (28) 7 We also simulate critical values based on the following DGP: y it = y i,t 1 + γ if t + α i,1 sin(2πkt/ ) + α i,2 cos(2πkt/ )+ε it, where γ i i.i.d.u[ 1, 3] and α i,1, α i,2 i.i.d.u[3, 5] and i.i.d.u[ 2, 2], respectively. he critical values in ables 2(a)-2(c) are barely affected by these changes. hese results are available from the authors upon request. 15

17 o test the null hypothesis in (4), this paper estimates the following breaks and cross-sectional dependence augmented ADF regression: y it = c i, + c i,1 sin(2πkt/ ) + c i,2 cos(2πkt/ ) + c i,3 y t 1 + c i,4 y t + c i,5 y i,t 1 +c i,6 y t 1 + b i y i,t 1 + e it. he t-statistic of the estimate of b i (ˆb i ) is then applied to examine the unit-root hypothesis and it can be written as t ρ i (N, ) = y i Mρ zy i, 1 ˆσ i (y i, 1, (29) Mρ zy i, 1 ) 1/2 where ˆσ 2 i = y i Mρ i,z y i 8, M ρ z = I Z ρ (Z ρ Z ρ ) 1 Z ρ, Z ρ = ( y i, 1, y 1, y, τ, Υ 1, Υ 2, y 1 ), M ρ i,z = I G ρ i (G ρ i Gρ i ) 1 G ρ i, Gρ i = (Zρ, y i, 1 ). he exact sampling distribution of t ρ i (N, ) is affected by α i,1, α i,2, γ i and ρ i, but as stated in the following theorem their affection disappears as (N, ). heorem 3: Suppose y it is generated by (28) with ρ i < 1 under the null hypothesis of β i = for all i and assumptions 1-4 hold. Let t ρ i (N, ) be the t-statistic defined in (29). hen, as (N, ), t ρ i (N, ) has the same sequential [(N, ) seq ] and joint [(N, ) j ] limit distribution, given by (15), as obtained under ρ i =. Proof: See the Appendix. Since t ρ i (N, ) in (29) has the same limiting distribution as that of (15), the panel unitroot test, BCIP S (N, ), proposed in the previous section can be applied equally to the case with serially correlated errors. For an AR(p) specification of errors in (26), (1 ρ i,1 L ρ i,p L p )u it = η it, η it = γ i f t + ε it, in which all roots of (1 ρ i,1 z ρ i,p z p ) = lie outside the unit circle. We suggest the following BCADF regressions for the level and trend cases, respectively, 16

18 p y it = c i, + c i,1 sin(2πkt/ ) + c i,2 cos(2πkt/ ) + c i,3 y t 1 + d ij y t j p + δ ij y i,t j + b i y i,t 1 + e it, (3) j=1 y it = c i, + τ t + c i,1 sin(2πkt/ ) + c i,2 cos(2πkt/ ) + c i,3 y t 1 + j= p d ij y t j p + δ ij y i,t j + b i y i,t 1 + e it. (31) j=1 j= 4 Finite Sample Performance he data generating process is given as follows: y it = µ i,1 (1 φ i L)t + (1 φ i L)ϖ i,k,t + φ i y i,t 1 + u it, i = 1, 2,..., N, t = 1, 2,..,, (32) where ϖ i,k,t = µ i + α i,1 sin(2πkt/ ) + α i,2 cos(2πkt/ ); u it = γ i f t + η it, η it = ρ i η i,t 1 + ε it, y, µ i,1, µ i i.i.d.n(, 1). Following Pesaran (27), we set f t i.i.d.n(, 1), ε it i.i.d.n(, σi 2) with σ2 i i.i.d.u[.5, 1.5], ρ i i.i.d.u[.2,.4] and ρ i i.i.d.u[.4,.2] to denote the case of positive and negative residual serial correlations, respectively, γ i i.i.d.u[ 1, 3] and γ i i.i.d.u[,.2] to denote the case of high and low cross-sectional dependence, respectively, and α i,1, α i,2 i.i.d.u[,.2] and α i,1, α i,2 i.i.d.u[1, 2] to denote the case of the low and high amplitudes of breaks, respectively. he specification of residual in (32) differs from (26) that is used to derive heorem 3 and the purpose of this change is to check the robustness of our analysis to an alternative residual serial-correlation specification. Equation (32) degenerates to the intercept case if µ i,1 = for all i and it degenerates to the case with serially uncorrelated errors if ρ i = for all i. Sizes are computed under the null hypothesis of φ i = 1 for all i. Powers are constructed under the alternative hypothesis of φ i i.i.d.u[.85,.95]. he common factor (f t ) was generated independently of ε it and the parameters φ i, µ i, µ i,1, α i,1, α i,2, γ i, ρ i, σ i were also drawn independently of ε it. he tests were one-sided with the nominal size set at 5%, and were conducted for N = 1, 2, 3, 5, 1, 2, = 3, 5, 1, 2, 4 and k = 1, 2, 3. he size and power for each experiment were constructed using 5, replications. Critical values for different combinations of k, N and under three different models, reported in ables 17

19 2(a)-2(c), are adopted to examine the size and power of the BCIP S statistic. 4.1 Size and Power of the CIP S test with Smoothing Breaks in DGP Before discussing the size and power of our proposed test, it would be helpful to discuss the size distortion of Pesaran s (27) CIP S test when smooth breaks exist. o emphasize the role of break amplitudes, size distortions with the high and low amplitudes of breaks under different frequency components (k) are reported. With a small magnitude of breaks, the results from the first panel of able 3(a) indicate that the sizes of the CIP S test are generally close to.5 regardless of N, and k (with a minimum of.43 and a maximum of.65). hese sizes are generally similar to those of the BCIP S statistic as indicated by the second panel of able 3(a). However, the CIP S test can be very conservative when the amplitude of breaks is large as indicated by the third panel of able 3(a). In such a case the size of the CIP S test is generally small even when = 1, but it is close to.5 when = 5,. 8 he above simulation results are consistent with those implied by the Corollary in Section 3.1. able 3(b) reports the powers of the CIP S and BCIP S statistics with low amplitude of breaks and indicates that the power of the CIP S statistic is either greater than or equal to that of the BCIP S statistic, with few exceptions, when 1. 9 It is therefore better to apply the CIP S test with small and medium sample sizes when the amplitude of breaks is small. However, applying our proposed BCIP S test is always helpful when the amplitude of breaks is large since the finite sample bias for the CIP S test is generally serious in such a case even when is as large as 1. Harvey et al. (212) point out that including an irrelevant trend break regressors compromises the power of a fixed-magnitude trend break unit-root test such as Perron and Rodriguez (23). Our results are consistent with theirs. Given that the size distortion of the CIP S test is serious with large break amplitudes, this paper therefore focuses its discussion on the BCIP S test under the case of α i,1, α i,2 i.i.d.u[1, 2]. < ables 3(a), 3(b) here > 4.2 Size and Power under Serially Uncorrelated Errors 8 his paper also considers two different cases for the large amplitude of breaks: α i,1, α i,2 i.i.d.u[ 2, 2] and α i,1, α i,2 i.i.d.u[3, 5]. he sizes of the CIP S test in the former case increase with and are close to.5 when = 5, but the sizes in the latter case vary with with a non-monotonic pattern and are generally close to.5 when = 1,. he above results are available from the authors upon request. 9 Simulating the power of the CIP S statistic under the case of large amplitudes is meaningless since its size distortion is serious in finite samples. 18

20 As a benchmark, we begin by examining the finite-sample size and power of the BCIP S test for various values of k and assume it is known in estimation. he size and power of the BCIP S statistic are similar to those of BCIP S and they are available from the authors upon request. he size of the test with an unknown k is discussed in Section 4.5. he size and power of the BCIP S test for high cross-sectional dependence and large amplitudes of breaks are reported in able 4. he results from the first panel of able 4 indicate that the size of the BCIP S statistic over different N, and k are in general close to.5 (with a minimum of.43 and a maximum of.71) as long as 3. As for the power, the results from the second panel of able 4 indicate that the power is poor when 3. With a sample size of fifty, the power of the BCIP S test is generally high when N > 2 and k > 1. Under a given N, the power of the BCIP S test increases with significantly and it is close to 1. for most cases when > 5. It is interesting to find that, when is very small ( =15), the size of the BCIP S statistic is a little bit high in general even when N is large. Recall that heorem 1 indicates that the BCADF distribution is not free of nuisance parameters in finite even when N. he finding of relatively high sizes, with large N, when is very small ( = 15) agrees with heorem 1. Moreover, the results from the first panel of able 4 indicate that the influence of nuisance parameters on the BCADF distribution is negligible as long as 3. In short, the results from able 4 indicate that, by allowing for high cross dependence and large amplitudes of breaks, the size of the BCIP S test is generally reasonable when 3 and the power of the test is high when > 5. 1 Recall that the results from able 3(a) and 3(b) indicate that the CIP S statistic is favorable when the amplitude of breaks is small regardless of the sample size being small or medium. Moreover, the size of the CIP S test is serious in finite sample with large amplitude of breaks. herefore, the results from ables 3(a), 3(b) and 4 indicate that, with a large sample size ( > 1), both the BCIP S and CIP S tests have same power when the amplitude of breaks is small, but the former achieves better size and power properties than the latter when break amplitudes are large. hese results indicate that applying the BCIP S test is safer than applying the CIP S test regardless of break amplitudes as long as the sample size is greater than 1. < able 4 here > 4.3 Size and Power when Residuals are Serially Correlated 1 Similar results are obtained when the amplitude of breaks is either α i,1, α i,2 i.i.d.u[ 2, 2] or α i,1, α i,2 i.i.d.u[3, 5], which are available from the authors upon request. 19

21 In the case of existing first-order autoregressive errors, we consider the scenarios of positive and negative serial correlations. With a positive residual serial correlation, the sizes of the BCIP S test are close to.5 when k = 1 and they are generally reasonable when k = 2, 3 and > 5 as indicated by the first panel of able 5. With a negative residual serial correlation, the results from the second panel of able 5 point out that the size is generally close to.5 (with a minimum of.41 and a maximum of.7) regardless of N and when k 2. As for the case of k = 3, the size of the BCIP S test is reasonable except for those cases with = 3. In short, the sizes of the BCIP S test, under the case of a positive (negative) residual serial correlation, are generally reasonable when > 5 ( > 3). he results from the third panel of able 5 indicate that the power of the BCIP S test under a positive residual serial correlation is generally high when > 5 regardless of N and k. Similar results are observed when the residual serial correlation is negative as pointed out in the fourth panel of able 5. We also examine the size of BCIP S test under a mis-specified model in which a lagged first-difference of individual series and its cross-sectional mean are neglected in the regression equation. In such a case, the size of BCIP S test is close to. when the residual serial correlation is positive and it is generally grater than.6 when the correlation is negative regardless of N, and k. he above results are available from authors upon request. < able 5 here > 4.4 Size and Power under the Linear rend Model It is beneficial to first examine the performance of our proposed test when DGP includes a linear trend and large amplitudes of smooth breaks in the mean under the case of serially uncorrelated errors. he results from able 6 indicate that the sizes of the BCIP S statistic are generally reasonable and close to.5. Besides, the power of the statistic is generally higher than.7 when > 5. he above results are generally similar to those found in able 4. With a positive residual serial correlation, the results from the first panel of able 7 indicate that the BCIP S test has reasonable sizes when k < 3. In the case of k = 3, the sizes of the BCIP S test are generally reasonable when > 5. As for the case of a negative residual serial correlation, the results from the second panel of able 7 reveal that the size of our proposed test is close to.5 regardless of N, and k. As for the power of the BCIP S test, they are reasonably high in general when > 5 regardless of the residual 2

22 serial correlation being positive or negative as indicated by the third and fourth panels of able < ables 6, 7 here > 4.5 est with k Unknown In the case of serially uncorrelated errors with an unknown k, we first estimate the unknown Fourier frequency component (k) from the data and then employ the estimated k (ˆk) to evaluate the size and power of the BCIP S test. he determination of k is based on the grid-search method suggested by Davis (1987). Following Enders and Lee (29), this paper sets the maximum Fourier frequency component to k max =5 and then estimate (31) with the lag order being zero (p = ) for a given k. After estimating the above equation, we compute the sum of squared residuals (SSR k,i ), i = 1,..., N, k = 1,..., 5 under a given N and. he estimated k is obtained by minimizing the sum of SSR k,i across i: ˆk = min N i=1 SSR k,i. {k} he BCIP S statistic is then calculated as before based on ˆk and its sizes under different N and are reported in able 8. For the intercept case, the results from the first panel of able 8 point out that the sizes of the BCIP S test are generally reasonable and close to.5 regardless of N, and k. As for the linear trend case, the sizes of the BCIP S test are close to.5 when > 3 except for the cases of k=1, 2 and N 2. < ables 8, 9 here > 5 Empirical Application An important characteristic of real exchange rates under the recent float is their long swings (Engel and Hamilton, 199) resulting from common shocks or long-lived bubbles. here are several significant common shocks under the period of recent float, which create long-lived bubbles affecting the nominal exchange rates of many countries. 12 Papell (22) argues that the failure to reject long-run PPP in most of the existing literature could be due to the 11 Our simulation results indicate that the power of the BCIP S test increases with the value of k, which is consistent with that of Enders and Lee (29, able 2). he power of the BCIP S test with k = 1 decreases compared to k = 3 when > 1. One may argue that for large value of k the BCIP S test behave roughly as the CIP S test and hence explained the above power decreases as the cost (in terms of power) of including trigonometric terms. he above argument is not true since large k do not necessarily imply small amplitude of breaks (Hamilton, 1994, p. 161, Figure 6.1.) 12 hese common shocks include the stock market crash of , the UK s secondary banking crisis of , the Latin American debt crisis of 1982, the Plaza Accord of 1985, the US s saving and loan crisis of , the European exchange rate mechanism crisis of , the Mexican crisis of , the East Asian crisis of , the technology bubbles of 2-1, the Argentinean crisis of 21-2, the global financial crisis of 27-1 and the European debt crisis of

23 ignorance of smooth breaks in real exchange rates. Although the conventional literature modeled real exchange rates with a non-trend model, several articles have pointed out the significance of the Balassa-Samuelson effects over the post-bretton Woods period (Bergin et al., 26; Wu and Hu, 29; Paya et al., 23; Sollis, 25). his paper, therefore, applies a trended BCADF regression as indicated by (31) to investigate the validity of PPP. Both monthly and quarterly nominal exchange rates and consumer price indices (CPI) for sixty countries over the post-bretton Woods period are downloaded from the IMF s International Financial Statistics. he sample period starts in January 1974 and ends in April 211. For euro-zone countries, the dollar-based nominal exchange rates after 1999 were constructed by using the euro-dollar rate and the prefixed exchange rates at January 1st, 1999 (Alba and Papell, 27). Australia and New Zealand do not report monthly CPI figures and hence they are converted from quarterly CPI with quadratic-match averages (Molodtsova and Papell, 29). wo different panels are considered in the empirical investigation: P6 including sixty countries from the world and P39 including thirty-nine countries in the panel of P6. o determine the frequency component (k) in the Fourier function, we first allow the frequency component to vary from one to five and then estimate equation (31) for each value of k under a given lag order (p). his paper then constructs the sum of square errors (SSR k,p,i ) for the ith country under a given k and p. he estimate of k (ˆk) under a given lag order, p, is obtained by choosing a specific k that minimizes the sum of SSR k,p,i over i. he maximum lag order (p max ) in (31) is set to 12 for monthly data and 4 for quarterly data. After estimating the frequency component (ˆk) in the Fourier function, the BCIP S statistic is constructed by estimating equation (31) under different lag orders. With monthly data, the results from the first panel of able 9 indicate that the estimated frequency component (ˆk) in the Fourier function is one regardless of the lag order p. his paper then estimates an ADF equation augmented by the Fourier function in which the frequency component is set to one under a given p, and then examines the significance of the cross dependence across estimated residuals by the cross-sectional dependence (CD) test of Pesaran (24). he results from the first panel of able 9 point out that the CD test is significant at conventional levels and the mean of the pair-wise cross-correlation coefficient of estimated residuals is about.17 for the panel of P6. he BCIP S statistic, based on the panel of P6, rejects the unit-root hypothesis at conventional levels for all lags. Besides, the average of the estimated amplitude in the Fourier function, under different lag orders, changes in the interval of (-.3, -.2) for the sine term (Ave(c1)) and (-.13,.48) for the cosine terms (Ave(c2)). However, the 22

24 CIP S statistic rejects the unit-root hypothesis for only five out of twelve lags. As for the panel of P39, the results from the second panel of able 9 indicate significant cross-sectional dependence. he BCIP S statistic rejects the unit-root hypothesis for ten out of twelve lags but the CIP S statistic fails to rejects the hypothesis for all lags. he average estimates of Ave(c1) and Ave(c2) in the Fourier function change in the interval of (-.5, -.28) and (-.41,.14), respectively. 13 As for the quarterly data, the results from the third panel of able 9 again indicate that the estimated frequency component in the Fourier function (ˆk) is one and that Pesaran s (24) CD test reveals significant cross-sectional dependence regardless of the panel being P6 or P39. he BCIP S statistic rejects the unit-root hypothesis at conventional levels for all lags regardless of the panel being P6 or P39. he CIP S statistic rejects the hypothesis for two out of four lags for the panel of P6, but fails to reject the hypothesis for all lags for the panel of P It is interesting to find that the BCIP S statistic reveals stronger evidence of rejecting the unit-root hypothesis than the CP IS statistic. According to our simulation results in Section 4.1, the reason for the above finding should be that the amplitudes of smooth breaks measured by the sine and cosine terms in the Fourier function are large and significant. his paper, therefore, examines if the estimated coefficients of the sine and cosine terms are jointly insignificant. he null hypothesis is α i,1 = α i,2 = for all i N. his paper constructs the average of the F statistic across individuals and refers it to the AF statistic. We then construct the finite sample distribution of AF by a block bootstrap (Kunsch, 1989). 15 he block size is set in accordance with the lag order in the BCADF regression equation. esting the hypothesis of α i,1 = α i,2 = with a F test is appropriate only when the unit-root hypothesis is rejected. We therefore examine the above hypothesis for those lag orders that the BCIP S statistic rejects the unit-root hypothesis. he null hypothesis of α i,1 = α i,2 = for all i is rejected at conventional levels for both panels regardless of the 13 Although many estimates of ave(c1) and ave(c2) are less than or close to.2 in absolute value, one should not argue that these estimates are small simply because we treat the interval of [,.2] as a small amplitude interval in simulations. Whether these estimates are small also depend on other parameters in DGP. Simulation results in Section 4.1 indicate that if estimated amplitudes are small then both BCIP S and CIP S tests have same power properties when > 1. However, our estimated results reveal stronger evidence of rejecting the unit-root null with the BCIP S instead of CIP S test, which indicates that the above estimated break amplitudes are not small. 14 We also select a model across k and p that has the minimum SSR and results reveal that the BCIP S test rejects the unit-root hypothesis but the CIP S test fails to reject the hypothesis regardless of the panel being P6 or P39 and of the data frequency being monthly or quarterly. 15 After generating a bootstrap sample based on the block bootstrap, we estimate the individual BCADF equation and then construct the F (b) i statistic for the hypothesis of α i,1 = α i,2 =. he AF (b) statistic is then obtained by averaging F (b) i, i.e., AF (b) = 1 N times, this paper constructs the finite sample distribution of the AF statistic. N i=1 F (b) i. By repeating the above procedure for b = 1, 23