Summary
 Top of page
 Summary
 1 Introduction
 2 The Models
 3 Test Statistics
 4 Panel Data Cointegration Tests
 5 Monte Carlo Simulations
 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
 7 Conclusions
 Acknowledgements
 References
 Supporting Information
The power of standard panel cointegration statistics may be affected by misspecification errors if structural breaks in the parameters generating the process are not considered. In addition, the presence of crosssection dependence among the panel units can distort the empirical size of the statistics. We therefore design a testing procedure that allows for both structural breaks and crosssection dependence when testing the null hypothesis of no cointegration. The paper proposes test statistics that can be used when one or both features are present. We illustrate our proposal by analysing the passthrough of import prices on a sample of European countries. Copyright © 2013 John Wiley & Sons, Ltd.
1 Introduction
 Top of page
 Summary
 1 Introduction
 2 The Models
 3 Test Statistics
 4 Panel Data Cointegration Tests
 5 Monte Carlo Simulations
 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
 7 Conclusions
 Acknowledgements
 References
 Supporting Information
The literature on panel data econometrics with integrated data has experienced rapid development since the 1990s. The driving force behind the popularity of the use of the panel data techniques is the idea that the power of tests for unit roots and cointegration might be increased by combining the information that comes from the crosssection and the time dimensions, especially when the time dimension is restricted by the lack of availability of long series of reliable time series data. As a result, new statistics to assess the stochastic properties of panel datasets have appeared in the literature; see Breitung and Pesaran (2008) and Banerjee and Wagner (2009) for overviews of the field.
The issue of structural instability has received relatively recent attention in the panel data cointegration framework; see, for example, Kao and Chiang (2000), Banerjee and CarrioniSilvestre (2004), Westerlund (2006) and Gutierrez (2010). One important feature to consider from a practical point of view is the crosssection dependence of the units of the panel. The main characteristic that is shared by the papers mentioned above is that they assume that individuals are crosssection independent. In this paper we contribute to the panel cointegration literature by extending the analysis in Banerjee and CarrioniSilvestre (2004) to allow for the presence of both structural breaks and crosssection dependence, where dependence is modelled using a factor model as in Bai and Ng (2004). Our approach also covers, as special cases, the situations where there are no structural breaks in the panel and/or the crosssection units are independent.
The paper in the literature that is closest to our analysis is by Westerlund and Edgerton (2008), who also consider the case of structural breaks in panels with crosssection dependence. However, there are three distinct aspects in which our papers differ. First, we begin by considering a slightly more restrictive version of Westerlund and Edgerton's (2008) models to allow for the presence of only a level term in the deterministic part of the stochastic processes (without a trend). Second, we allow for the factors generating the crosssection dependence to be integrated stochastic processes. Finally, we allow for possible breaks in the trends generating the processes.
For the specifications which do not allow for changes in trend, our results are completely general to allow for homogeneous or heterogeneous (multiple) breaks in the levels and cointegrating vectors of the processes. However, for the specification in which trend breaks are present, we are able to allow only for multiple homogeneous breaks in trend across the units where the break dates are known. The reasons for this limitation are both theoretical and practical and become clear in the actual statements of the theorems. The difficulty essentially lies in the dependence of the critical values of the tests on the location of the break dates when trend breaks are present.
The paper is organized as follows. Section 2 presents the models, while Section 3 designs statistics for the null hypothesis of no cointegration allowing for crosssection dependence. Section 4 proposes panel cointegration tests under different specifications of the models. Section 5 focuses on the finitesample properties of the statistics that have been proposed. Section 6 provides an empirical illustration of the use of our tests using data on exchange rate passthrough. The issue of the degree of exchange rate passthrough is an important focus of investigation in the macroeconomics literature, although much of the testing has been undertaken under severely restrictive assumptions. Section 7 concludes with some remarks. Details of the data, proofs and additional tables are collected in a companion appendix, available online as supporting information and also upon request.
2 The Models
 Top of page
 Summary
 1 Introduction
 2 The Models
 3 Test Statistics
 4 Panel Data Cointegration Tests
 5 Monte Carlo Simulations
 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
 7 Conclusions
 Acknowledgements
 References
 Supporting Information
Let Y_{i,t} = (y_{i,t}, x′_{i,t})′ be an (m × 1) vector of nonstationary stochastic process whose elements are individually I(1). Further, let us specify the datagenerating process (DGP) in structural form as
 (1)
 (2)
 (3)
 (4)
 (5)
 (6)
i = 1, …, N, t = 1, …, T, where , , and . The general functional form for the deterministic term D_{i,t} is given by
 (7)
where DU_{i, j,t} = 1 and for and 0 otherwise, with denoting the timing of the jth break, j = 1, …, m_{i}, for the ith unit, i = 1, …, N, , Λ being a closed subset of (0, 1).1 Note also that the cointegrating vector in equation (1) is specified as a function of time so that
 (8)
with the convention that and , where denoting the jth time of the break, j = 1, …, n_{i}, for the ith unit, i = 1, …, N, .
We begin by assuming throughout this section that both the number and positions of the structural breaks in each unit are known a priori. This includes the special case where there are no structural breaks in some or all of the units of the panel. In Section 2.2 we show how the assumption of known structural breaks may be relaxed in some cases.
The combination of the specifications given by equations (7) and (8) define the six different models that are covered in this paper:
 Model 1: no linear trend– β_{i} = γ_{i,j} = 0 ∀ i, j in (7) and κ_{i} = 0 ∀ i in (5)–and stable cointegrating vector– δ_{i,j} = δ_{i} ∀ j in equation (8).
 Model 2: stable trend– β_{i} ≠ 0 ∀ i and γ_{i,j} = 0 ∀ i, j in (7)–and stable cointegrating vector– δ_{i,j} = δ_{i} ∀ j in equation (8).
 Model 3: changes in level and trend– β_{i} ≠ γ_{i,j} ≠ 0 ∀ i, j in (7)–and stable cointegrating vector– δ_{i,j} = δ_{i} ∀ j in equation (8).
 Model 4: no linear trend– β_{i} = γ_{i,j} = 0 ∀ i, j in (7) and κ_{i} = 0 ∀ i in equation (5)–but the presence of multiple structural breaks affects both the level and the cointegrating vector of the model.
 Model 5: stable trend– β_{i} ≠ 0 ∀ i and γ_{i,j} = 0 ∀ i, j in (7)–with the presence of multiple structural breaks affects both the level and the cointegrating vector of the model.
 Model 6: changes in the level, trend and in the cointegrating vector. No constraints are imposed on the parameters of equations (7) and (8).
Remarks.  Note that our framework is general enough to allow for multiple structural breaks affecting either the deterministic component or the cointegrating vector. Further, the number and position of these structural breaks do not need to be the same. Thus it would be possible that the number of structural breaks affecting the deterministic component of the model (m_{i}) is different from the number of structural breaks affecting the cointegrating vector (n_{i}). Moreover, even in the case that m_{i} = n_{i}, the break dates do not need to be located at the same date, i.e. . We define the break fraction vector for each unit for the most general situation as .2 From an empirical point of view, it is worth pointing out that, although certainly possible in principle, differing structural breaks for the deterministic component and cointegrating vector are infeasible in practice unless the number of breaks is small.
 Also note that it is not a restriction of our framework to specify the breaks in level and trend as happening at the same time . As we show below, the densities of the statistics are invariant to the location of the shifts in level, and these shifts in level can therefore be specified to occur at times distinct from the timings of the shifts in trend. This does not mean that we can ignore the presence of level shifts if adverse effects on the empirical power of the test statistics are to be avoided. Our model can be reformulated to separate shifts in levels from shifts in trend and shifts in the cointegrating vectors. Since there is no gain in generality of the results from this reformulation, we proceed to operate within the simpler formulation.
 It should be understood that our framework can be also applied for those cases where there are no structural breaks. This situation can be found if we impose κ_{i} = 0 in equation (5), β_{i} = θ_{i,j} = γ_{i,j} = 0 ∀ i, j in equation (7) and δ_{i,j} = δ_{i} ∀ j in equation (8), which gives rise to the socalled ‘constant term case’. Further, it is also possible to include a time trend with no structural breaks in the model just by specifying β_{i} ≠ 0 and θ_{i,j} = γ_{i,j} = 0 ∀ i, j in equation (7) and δ_{i,j} = δ_{i} ∀ j in equation (8). This model defines the socalled ‘time trend case’.
 The component designed as F_{t} denotes an (r × 1) vector containing the common factors, with π_{i} the vector of loadings. Our model specification considers the case where the stochastic regressors are assumed to be either crosssection independent–impose ς_{i} = 0 in equation (5)–or crosssection dependent with dependence driven by a set of common factors G_{t}–consider ς_{i} ≠ 0 in (5). Throughout this paper we assume that the set of common factors affecting x_{i} are different from those affecting y_{i}–i.e. F_{t} or transformations of F_{t}–although it is possible to use the approach in Bai and CarrioniSilvestre (2013) to consider the case where some of the factors in G_{t} are also in F_{t} (or transformations of F_{t}).
 Despite the presence of the operator (IL) in equation (3), F_{t} does not have to be I(1). In fact, F_{t} can be I(0), I(1) or a combination of both, depending on the rank of C(1). If C(1) = 0, then F_{t} is I(0). If C(1) is of full rank, then each component of F_{t} is I(1). If C(1) ≠ 0, but not full rank, then some components of F_{t} are I(1) and some are I(0).
 The presence of cointegration among Y_{i,t} = (y_{i,t}, x′_{i,t})′ requires F_{t} to be I(0). However, allowing F_{t} to be I(1) is also relevant from an empirical point of view since, in this case, F_{t} might be capturing effects from outside the model that are not included in Y_{i,t}. Then, cointegration among the elements in Y_{i,t} up to the inclusion of I(1) factors is possible, which will imply e_{i,t} to be I(0).3 As noted previously in the Introduction, this is a generalization of the Westerlund and Edgerton (2008) framework.
Our analysis is based on the same set of assumptions as in Bai and Ng (2004) and Bai and CarrioniSilvestre (2013). Let M < ∞ be a generic positive number, not depending on T and N. Further, ‖A‖ = trace(A′A)^{1/2}. Then
Assumption A. (i) For nonrandom π_{i}, ‖π_{i}‖ ≤ M; for random π_{i}, E‖π_{i}‖^{4} ≤ M, (ii) , a (r × r) positive definite matrix.
Assumption C. (i) for each i, , Eε_{i,t}^{8} ≤ M, , ; (ii) E(ε_{i, t} ε_{j, t}) = τ_{i, j} with for all j; (iii) , for every (t,s).
Assumption D. The errors ε_{i,t}, w_{t}, v_{i,t}, ϖ_{t} and the loadings π_{i} and ς_{i} are mutually independent groups.
Assumption E. E ‖F_{0}‖ ≤ M, and for every i = 1, …, N, E e_{i,0} ≤ M and E ‖x_{i,0}‖ ≤ M.
Assumption G. (i) E(e_{i,t}v_{i,t}) = 0 when stochastic regressors are assumed to be strictly exogenous; or (ii) E(e_{i,t}v_{i,t}) = Δx′_{i,t}A_{i}(L) + ξ_{i,t}, with A_{i}(L) being a (υ_{i} × 1) vector of lags and leads polynomials of finite orders and , when stochastic regressors are nonstrictly exogenous.
Assumption A ensures that the factor loadings are identifiable. Assumption B establishes the conditions on the short and longrun variance of ΔF_{t} –i.e. the shortrun variance matrix is positive definite and the longrun variance matrix may have reduced rank in order to accommodate stationary linear combinations of I(1) factors. Assumption C(i) allows for some weak serial correlation in (1 − ρ_{i}L)e_{i,t}, whereas Assumptions C(ii) and C(iii) allow for weak crosssection correlation. Assumption D imposes mutual independence among the factors, loadings, idiosyncratic residuals and stochastic regressors x_{i,t}. Assumption E defines the initial conditions. Assumption F establishes conditions on the first differences of the stochastic regressors. Finally, Assumption G defines two situations depending on whether the stochastic regressors are strictly exogenous regressors or endogenous. This distinction is important here, because in the common factor framework the limiting distributions of the statistics do not depend on the number of stochastic regressors if strict exogeneity holds. However, this is no longer true when correlation between e_{i,t} and v_{i,s} is allowed and modifications need to be introduced to account for endogenous regressors. Here we suggest using the dynamic ordinary least squares (DOLS) estimation method in Stock and Watson (1993) to account for endogeneity, where we assume that the number of leads and lags is fixed as in Stock and Watson (1993); see Bai and CarrioniSilvestre (2013). For ease of exposition, in what follows we assume strictly exogenous stochastic regressors, although the derivation for the more general case can be found in the companion appendix.
3 Test Statistics
 Top of page
 Summary
 1 Introduction
 2 The Models
 3 Test Statistics
 4 Panel Data Cointegration Tests
 5 Monte Carlo Simulations
 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
 7 Conclusions
 Acknowledgements
 References
 Supporting Information
The cointegration analysis that is proposed in this paper requires us to assess the order of integration of the idiosyncratic component and the number of global stochastic trends. Before doing so, we first show that it is possible to write the model in terms of a common factor representation similar to the one given in Bai and Ng (2004) and Bai and CarrioniSilvestre (2013), and proceed then with the estimation of the common factor and idiosyncratic components following their approach. First, we write the model in first difference:
 (9)
where ; ‘.*’ denotes the elementbyelement product of each column of Δx_{i} with the vector DU_{i,j}, j = 1, …, n_{i}; and .
The first of the two steps is to take the orthogonal projections with respect to the deterministic components:
where and the dotted variables denote that they have been projected against ΔD_{i} . Second, we can take the orthogonal projections with respect to :
where . Note that in this expression we have , which involves projecting the first difference of the common factors using a unitspecific projection matrix. This violates the standard definition of a common factor model, since this component is no longer common. However, we can rewrite the model as a common factor model as follows:
 (10)
where , f = ΔF and . Note that now equation (10) has a component that is common to all units and an idiosyncratic component, which gives a standard common factor representation where the elements and entering in z_{i} are asymptotically negligible for testing for panel cointegration in this paper; see Assumption D above and Bai and CarrioniSilvestre (2013, Remark 3.2 and Lemma A.1). This result is valid for all the models except Models 3 and 6 since , j = 1, …, m_{i}, have a negligible effect in the limit.
We can recover the idiosyncratic disturbance terms through cumulation, i.e. , and test the unit root hypothesis (α_{i,0} = 0) using the augmented Dickey–Fuller (ADF) type regression equation:
 (12)
The null hypothesis of a unit root can be tested using the pseudo t ratio , j = c, τ, γ, for testing α_{i,0} = 0 in eqution (12). Here the models that do not include a time trend–i.e. Models 1 and 4–are denoted by c; models that include a linear time trend with stable trend–i.e. Models 2 and 5–are denoted τ; and, finally, γ refers to the models with a time trend with changing trend–i.e. Models 3 and 6.
When r = 1 we can use an ADFtype equation to analyse the order of integration of F_{t}. However, in this case we need to proceed in two steps. In the first step we regress on the deterministic specification. In the second step we estimate the ADF regression equation using the detrended common factor .
If r > 1 we should use one of the two statistics proposed in Bai and Ng (2004) to fix the number of common stochastic trends (r_{1}) . As before, let denote the detrended common factors. Start with q = r and proceed in three stages; we reproduce these steps here for completeness:
 Let be the q eigenvectors associated with the q largest eigenvalues of .
 Let , from which we can define two statistics:
 Let K(j) = 1 − j/(J + 1), j = 0, 1, 2, …, J:
 Let be the residuals from estimating a firstorder VAR in , and let
 Let .
 Define for the case of no change in the trend and for the case of changes in the trend.
 For p fixed that does not depend on N and T:
 Estimate a VAR of order p in to obtain . Filter by to get .
 Let be the smallest eigenvalue of
 Define the statistic for the case of no change in the trend and for the case of changes in the trend.
 If H_{0} : r_{1} = q is rejected, set q = q − 1 and return to the first step. Otherwise, and stop.
The following theorem consolidates the main results concerning these statistics.
Theorem 1. Let {y_{i,t}} the stochastic process with DGP given by equations (1)–(5). The following results hold as N, T ∞.
 Let k_{i} be the order of autoregression chosen such that k_{i} ∞ and . Under the null hypothesis that ρ_{i} = 1 in equation (4)

1(a) Models 1 and 4:

1(b) Models 2 and 5:

1(c) Models 3 and 6:
where W_{i}(s) denotes a standard Brownian motion, V_{i}(s) = W_{i}(s) − sW_{i}(1), and V_{i}(b_{j}) = W_{i}(b_{j}) − b_{j}W_{i}(1), with b_{j} = (s − λ_{j − 1})/(λ_{j} − λ_{j − 1}) so that 0 < b_{j} < 1, and with and .  Let k be the order of autoregression chosen such that k ∞ and k^{3}/min[N,T] 0. When r = 1, under the null hypothesis that F_{t} has a unit root and no change in trend: where for Models 1 and 4 and for Models 2 and 5, with W_{w}(s) a standard Brownian motion. When changes in the trend are allowed, i.e. for Models 3 and 6: where , M_{χ} = I − χ(χ′χ)^{− 1}χ′, is a Brownian motion projected onto the subspace generated by χ_{s} = (1, s, dt_{1}(s, λ_{1}), …, dt_{m}(s, λ_{m})), where dt_{j}(s, λ_{j}) = (s − λ_{j}) for s − λ_{j} > 0 and 0 otherwise, j = 1, 2, …, m, and [⋅]_{s} denotes the sth element of the matrix between brackets.
 When r > 1, let W_{q}(s) be a q vector of standard Brownian motion and and the detrended counterparts; see statement 2 of this theorem. For the models that do not include change in the trend, let be the smallest eigenvalues of For Models 3 and 6, let be the smallest eigenvalues of
 Let J be the truncation lag of the Bartlett kernel, chosen such that J ∞ and . Then, under the null hypothesis that F_{t} has q stochastic trends, and .
 Under the null hypothesis that F_{t} has q stochastic trends with a finite VAR representation and a VAR(p) is estimated with , and .
Proof. See the companion appendix.
Remarks.  The limiting distributions of the statistics do not depend on the stochastic regressors because these are assumed to be orthogonal to the factors and strictly exogenous to the idiosyncratic errors.4 This is also the reason why the presence of common factors G_{t} in equation (5) does not disrupt any of our results while allowing for crosssectional dependence among the stochastic regressors.
 Except for Models 3 and 6, the presence of multiple structural breaks does not affect the limiting distributions in Theorem 1. For Models 3 and 6 the limiting distributions do depend on the number and position of the structural breaks. The limitations introduced on our procedure by this feature are also mentioned in Sections 'Known Breaks Case' and 'Unknown Structural Break Dates Case' below.
 Note that in the particular case of no structural breaks valid cointegration test statistics can be computed that have the same limiting distributions in Theorem 1.5
 The limiting distributions for and derived in (1a) and (2) of Theorem 1 correspond to the standard Dickey–Fuller distributions.
 Finally, the limiting distribution of the statistic for the one structural break case can be found in Perron (1989), i.e. Model C in Perron (1989). The limiting distributions of the MQ test in (3) with no change in trend can be found in Bai and Ng (2004), while the corresponding distributions for a single known break date in trend, , are reported in Table 1. The asymptotic critical values reported in Table 1 depend both on the number of stochastic common trends and on the break fraction. It is worth mentioning that we only provide critical values for the case of only one (known) structural break in trend, although critical values for multiple changes in trend can be easily computed.6
 When the number of common factors is not known, it can be estimated using the panel Bayesian information criterion (BIC) as suggested in Bai and Ng (2004) and Bai and CarrioniSilvestre (2009), which considers the presence of structural breaks.
Table 1. Asymptotic critical values for the MQ(q,λ) testsr  λ = 0.1  λ = 0.2  λ = 0.3 

1%  5%  10%  1%  5%  10%  1%  5%  10% 


1  −32.163  −23.629  −19.865  −34.858  −26.091  −22.144  −36.123  −27.562  −23.619 
2  −43.372  −34.321  −30.056  −46.436  −37.139  −32.688  −46.773  −37.778  −33.492 
3  −53.648  −44.378  −39.748  −55.828  −46.232  −41.766  −57.136  −47.511  −42.775 
4  −63.359  −53.470  −48.595  −65.206  −55.582  −50.645  −65.570  −55.883  −51.370 
5  −73.691  −62.796  −57.434  −74.601  −64.165  −59.199  −75.573  −64.731  −59.919 
6  −81.346  −71.238  −65.663  −83.575  −72.562  −67.309  −83.921  −73.247  −67.908 
 λ = 0.4  λ = 0.5  λ = 0.6 
r  1%  5%  10%  1%  5%  10%  1%  5%  10% 
1  −36.635  −28.147  −24.140  −36.775  −28.226  −24.419  −36.805  −28.178  −24.176 
2  −47.134  −38.391  −34.282  −48.148  −38.907  −34.553  −47.611  −38.587  −34.246 
3  −57.176  −47.642  −43.088  −56.753  −47.715  −43.333  −57.230  −47.865  −43.200 
4  −67.481  −56.958  −52.039  −65.752  −56.418  −51.708  −67.094  −56.599  −51.785 
5  −75.603  −65.386  −60.204  −75.378  −65.302  −60.251  −75.182  −64.986  −60.057 
6  −84.718  −73.703  −68.372  −83.902  −73.746  −68.222  −84.059  −73.136  −67.973 
 λ = 0.7  λ = 0.8  λ = 0.9 
r  1%  5%  10%  1%  5%  10%  1%  5%  10% 
1  −36.302  −27.751  −23.890  −35.249  −26.722  −22.713  −32.918  −24.712  −20.896 
2  −47.383  −38.223  −34.045  −46.572  −37.227  −33.085  −43.959  −35.248  −31.190 
3  −56.908  −47.282  −42.693  −55.960  −46.442  −41.998  −54.568  −45.183  −40.623 
4  −66.869  −56.270  −51.337  −65.833  −55.750  −50.890  −63.920  −53.985  −49.399 
5  −75.074  −64.828  −59.867  −74.046  −64.430  −59.290  −74.177  −63.063  −57.839 
6  −85.434  −73.646  −68.332  −83.244  −72.857  −67.721  −82.664  −71.518  −66.449 
4 Panel Data Cointegration Tests
 Top of page
 Summary
 1 Introduction
 2 The Models
 3 Test Statistics
 4 Panel Data Cointegration Tests
 5 Monte Carlo Simulations
 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
 7 Conclusions
 Acknowledgements
 References
 Supporting Information
The individual statistics for the idiosyncratic disturbance terms can be pooled to define panel data cointegration tests. Different proposals can be designed depending, first, on whether the break dates are known or unknown and, second, on the degree of heterogeneity that is desired and allowed in the setup. At this stage, a comment on the interpretation of the outcome of the panel data test statistics should be made. Thus, in our setup, the null hypothesis of no panel cointegration implies that there is no cointegration in all units, whereas, depending on the power of the testing procedure, the null hypothesis could be rejected if there are some units for which cointegration holds. Therefore, the rejection of the null hypothesis does not necessarily imply that there is cointegration in all units; see also Pesaran (2012).
4.1 Known Breaks Case
The panel data cointegration test that is proposed is based on the sum of the individual ADF (SADF) cointegration statistics:
 (13)
where λ = (λ′_{1}, λ′_{2}, …, λ′_{N})′ for j = c, τ and λ = (λ_{1}, λ_{2}, …,λ_{m})′ for j = γ. The limit distribution of the statistic in equation (13) is given in the following theorem.
Theorem 2. Let {y_{i,t}} be the stochastic process with DGP given by equations (1)–(5), and suppose that e_{i,t} is independent across i. Denote by and the mean and variance, respectively, of the vector Brownian motion functionals defined in Theorem 1. Let k_{i} be the order of autoregression in equation (12) chosen such that k_{i} ∞ and . Then, under the null hypothesis that ρ_{i} = 1 ∀ i in equation (4), the distribution of the standardized statistic SADF_{j}(λ), j = c, τ, γ, converges, as N, T ∞ with N/T 0, to
 (14)
where and for j = c, τ.
Remarks.  As in Pedroni (2004), in order to prove Theorem 2 we require only the assumption of finite second moments of the random variables characterized as Brownian motion functionals, which will allow us to apply the Lindberg–Levy central limit theorem as N ∞. We have computed the moments of the limiting distribution of the statistics by means of Monte Carlo simulation–using 1000 steps to approximate the standard Brownian motion and 100,000 replications–which has produced and . Note also that these densities do not depend upon the location of the breaks.
 Following on from (1) above, for j = c, τ, our analysis can be carried out for specifications where the structural breaks are either homogeneous (common for all units) or heterogeneous.
 For Models 3 and 6, the mean and the variance depend on the number and position of the common structural breaks. Table 2 reports the simulated values of these moments for the one and two structural breaks cases for different values of the break fraction vector.7
4.2 Unknown Structural Break Dates Case
The developments above have all been based on the assumption that the break dates are known. However, there might be situations where this assumption cannot be imposed. Our discussion here distinguishes between two different situations, depending on whether the structural breaks are assumed to be homogeneous or heterogeneous. As noted previously, for the unknown breaks case, our results extend to covering the specifications that do not include changes in the trend. Models 3 and 6 are thus excluded from our discussion.
4.2.1 Heterogeneous Break Dates
First, if break dates (across the units) are allowed to be individual specific (heterogeneous), the break dates for each unit can be estimated by minimizing the sum of square residuals following the proposal in Bai and Perron (1998) and Bai and CarrioniSilvestre (2009). Thus, from equation (9), we define Δy_{i,t} = ΔD_{i,t} + ΔX′_{i,t}δ_{i} + Δu_{i,t}, where Δu_{i,t} = ΔF′_{t}π_{i} + Δe_{i,t}, and proceed to estimate the break dates, ΔD_{i,t} and δ_{i} using ordinary least squares (OLS). Since ΔF has zero mean, the factors can be embedded in the residual term of the regression.8 Conditional on these initial estimates, define and estimate the factors and loadings using principal components on the model . The estimated factors and loadings are denoted as and . Then, define and get an updated estimate of the break dates, ΔD_{i,t} and δ_{i}. Using these updated estimates, define from which the factors and the loadings are estimated again, giving . This strategy naturally leads to an iterative estimation procedure of the break dates, common factors and loadings until convergence is achieved—convergence is taken to occur when the improvement in the sum of squared residuals across all equations is smaller than a given error tolerance. The standardized test statistic is then constructed as in equation (14) using the estimated break dates. Since the moments do not depend on the location of the breaks, the same corrections can be used for the cases where the (heterogeneous) breaks are known or unknown. Consequently, the limiting distributions given in Theorem 2 hold in this case.
4.2.2 Homogeneous Break Dates
It is possible that investigators would want to impose common (homogeneous) structural breaks, affecting all units of the panel at the same time but with different magnitudes. In this case, we can compute the Z_{j}(λ), j = c, τ, statistic for the break dates, where the break dates are the same for each unit, using the idiosyncratic disturbance terms. The statistic used to test the null hypothesis of noncointegration for the idiosyncratic disturbance term is given by
 (15)
where . The limiting distribution of , j = c, τ, is given in the following theorem.
Theorem 3. Let {y_{i,t}} be the stochastic process with DGP given by equations (1)–(5), and suppose that e_{i,t} is independent across i. Let k_{i} be the order of autoregression in equation (12) chosen such that k_{i} ∞ and . Then, under the null hypothesis that ρ_{i} = 1 ∀ i in equation (4), the distribution of the , j = c, τ, test in ( 15) converges, as N, T ∞ with N/T 0, to
The proof follows from noting that under the null hypothesis , j = c, τ, is the infimum of a sequence of perfectly correlated random variables Z_{j}(λ) that are asymptotically standard normal. The perfect correlation arises from the fact that the distributions of the statistics under the null hypothesis do not depend on λ. In this case Embrechts et al. (1997, p. 210) and Dolado et al. (2005) show that the infimum is also standard normal. Finally, Table 3 provides the critical values for equation (15) for the one break case obtained by simulation for different values of T and N = 100, which confirm the validity of the limiting result.9
Table 3. Critical values for the , j = c, τ, statisticsT  1%  2.5%  5%  10% 

Constant with or without level shifts 
( test statistic) 
50  −2.926  −2.517  −2.219  −1.901 
100  −2.824  −2.402  −2.113  −1.759 
250  −2.560  −2.250  −1.985  −1.619 
Time trend with or without level shifts 
( test statistic) 
50  −2.900  −2.537  −2.120  −1.822 
100  −2.924  −2.538  −2.240  −1.835 
250  −2.619  −2.269  −1.931  −1.506 
5 Monte Carlo Simulations
 Top of page
 Summary
 1 Introduction
 2 The Models
 3 Test Statistics
 4 Panel Data Cointegration Tests
 5 Monte Carlo Simulations
 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
 7 Conclusions
 Acknowledgements
 References
 Supporting Information
Consider the DGP given by a bivariate system:
where π_{i, j} ∼ i. i. d. N(1,1), (w′_{t},ε_{i,t},v_{i,t})′, with w_{t} = (w_{1,t}, …,w_{r,t})′, follow a mutually i.i.d. standard multivariate Normal distribution for ∀ i, j, i ≠ j, and ∀ t, s, t ≠ s. The parameters that define the deterministic component and the cointegrating vector are μ_{i} = 1, β_{i} = 0.3, θ_{i} = 3, δ_{i,1} = 1 ∀ i and, when there is a change in the cointegrating vector, δ_{i,2} = 3 ∀ i, otherwise δ_{i,2} = 0 ∀ i. In this paper we consider two different situations depending on the number of common factors, i.e. r = {1,3}, and specify three values for the autoregressive parameters φ = {0.8,0.9,1} and ρ_{i} = {0.95,0.99,1} ∀ i. Note that these values allow us to analyse both the empirical size and power of the statistics. The importance of the common factors is controlled through the specification of . The number of common factors is estimated using the panel BIC information criterion in Bai and Ng (2004) with r_{max} = 6 as the maximum number of factors. In the computation of the ADFtype regression equation we select the order of the autoregressive correction using the tsig criterion in Ng and Perron (1995) with a maximum of k_{max} = 4ceil[min[N,T]/100]^{1/4} lags. We consider N = 40 units and T = {50,100,250} time observations. Finally, the nominal size is set at the 5% level and 1000 replications are used throughout the section.
The simulation results for size and power for the case with no breaks (with one or more factors) are close to those reported by Bai and Ng (2004) and are therefore not included in the paper; they are available upon request. From these results it may be seen that the empirical size of the pooled idiosyncratic tratio statistic and the ADF statistic of the common factor–when there is only one factor in the DGP–is close to the nominal size. As expected, the power of the tests increases as the autoregressive parameter moves away from unity. These results do not change regardless of the deterministic specification and the number of common factors.10
Let us turn now to the results for the case where there is one unknown structural break. In order to save space, we only report simulations for Models 2 and 5 with λ_{i} = 0.5 ∀ i, where the common break date is estimated as described in Section 'Homogeneous Break Dates'—results for the other specifications that have been considered in the paper are available upon request.
Table 4. Empirical size and power of the Z^{*} and tests for Models 2 and 5 with one common factor  φ  T  Panel A: Model 2  Panel B: Model 5 

ρ_{i} = 1  ρ_{i} = 0.99  ρ_{i} = 0.95  ρ_{i} = 1  ρ_{i} = 0.99  ρ_{i} = 0.95 

           

0.5  1  50  0.049  0.057  0.074  0.006  0.410  0.039  0.091  0.059  0.604  0.001  0.843  0.002 
0.5  1  100  0.042  0.054  0.095  0.042  0.978  0.056  0.065  0.045  0.727  0.001  1  0.003 
0.5  1  250  0.050  0.059  0.420  0.047  1  0.050  0.084  0.054  0.971  0.001  1  0.032 
0.5  0.9  50  0.046  0.081  0.091  0.088  0.389  0.030  0.082  0.097  0.569  0.001  0.849  0.001 
0.5  0.9  100  0.034  0.194  0.082  0.140  0.986  0.183  0.052  0.173  0.749  0.001  1  0.018 
0.5  0.9  250  0.044  0.758  0.446  0.709  1  0.776  0.066  0.727  0.979  0.001  1  0.546 
0.5  0.8  50  0.044  0.220  0.082  0.142  0.398  0.139  0.083  0.201  0.556  0.001  0.848  0.006 
0.5  0.8  100  0.048  0.580  0.085  0.480  0.98  0.500  0.058  0.549  0.719  0.001  0.999  0.045 
0.5  0.8  250  0.058  0.994  0.440  0.987  1  0.999  0.069  0.999  0.974  0.001  1  0.675 
1  1  50  0.065  0.061  0.085  0.006  0.418  0.049  0.105  0.076  0.610  0.001  0.847  0.001 
1  1  100  0.043  0.045  0.075  0.061  0.98  0.060  0.083  0.051  0.735  0.001  0.999  0.005 
1  1  250  0.049  0.043  0.436  0.047  1  0.054  0.081  0.050  0.975  0.001  1  0.039 
1  0.9  50  0.044  0.087  0.085  0.065  0.401  0.104  0.088  0.092  0.586  0.002  0.821  0.007 
1  0.9  100  0.051  0.190  0.085  0.200  0.972  0.171  0.056  0.209  0.724  0.001  1  0.016 
1  0.9  250  0.044  0.773  0.409  0.742  1  0.811  0.076  0.750  0.971  0.002  1  0.519 
1  0.8  50  0.059  0.221  0.073  0.180  0.361  0.180  0.074  0.207  0.608  0.001  0.825  0.004 
1  0.8  100  0.040  0.602  0.067  0.561  0.972  0.588  0.063  0.583  0.712  0.001  1  0.039 
1  0.8  250  0.053  0.999  0.428  1  1  0.999  0.076  0.998  0.980  0.002  1  0.631 
10  1  50  0.078  0.039  0.112  0.041  0.381  0.033  0.168  0.024  0.543  0.001  0.730  0.001 
10  1  100  0.054  0.049  0.089  0.050  0.931  0.045  0.119  0.034  0.675  0.001  0.991  0.002 
10  1  250  0.055  0.038  0.398  0.052  1  0.049  0.105  0.046  0.968  0.001  1  0.019 
10  0.9  50  0.062  0.055  0.094  0.048  0.355  0.036  0.144  0.038  0.498  0.001  0.731  0.002 
10  0.9  100  0.052  0.178  0.074  0.174  0.957  0.194  0.070  0.148  0.685  0.001  0.994  0.007 
10  0.9  250  0.048  0.823  0.430  0.785  1  0.793  0.122  0.783  0.962  0.002  1  0.376 
10  0.8  50  0.061  0.143  0.105  0.078  0.339  0.139  0.146  0.070  0.471  0.001  0.715  0.002 
10  0.8  100  0.043  0.525  0.084  0.552  0.922  0.552  0.077  0.495  0.628  0.001  0.988  0.020 
10  0.8  250  0.048  1  0.402  0.993  1  0.985  0.100  0.984  0.972  0.002  1  0.401 
6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
 Top of page
 Summary
 1 Introduction
 2 The Models
 3 Test Statistics
 4 Panel Data Cointegration Tests
 5 Monte Carlo Simulations
 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
 7 Conclusions
 Acknowledgements
 References
 Supporting Information
Campa and GonzálezMínguez (2006) (henceforth CM) and Campa et al. (2005) (henceforth CGM) have investigated the issue of exchange rate pass through (ERPT) of foreign to domestic prices. Studies of ERPT have been conducted both for the USA and for countries of the euro area to analyse the importance of institutional arrangements in generating responses to exchange rate institutions and changes. All the theories contain a steadystate relationship in the levels of a measure of import unit values (in domestic currency), the exchange rate (relating the domestic to the numeraire currency) and a measure of foreign prices (unit values in the numeraire currency, typically US dollars); this long run is routinely disregarded in most of the empirical implementations. There is a substantial consensus in the literature that these time series are integrated variables. Therefore, one way of defining the long run is in the sense of Engle and Granger (1987), where the long run is given by the cointegrating relationship. When the time dimension is relatively short, a panel approach to testing for cointegration may be used.
6.1 Exchange Rate PassThrough into Import Prices
By definition, import prices for any type of goods j, , are a transformation of export prices of a country's trading partners, , using the bilateral exchange rate, ER_{t}, i.e., and dropping superscript j for clarity, we have MP_{t} = ER_{t} ⋅ XP_{t}; in logarithms (depicted in lower case), mp_{t} = er_{t} + xp_{t}. The export price consists of the exporters marginal cost and a markup, i.e., XP_{t} = FMC_{t} ⋅ FMKUP_{t}. After suitable substitutions we have
 (16)
Markups in an industry are assumed to consist of a component specific to the type of good, independent of the exchange rate and a reaction to exchange rate movements:
 (17)
It is also important to consider the effects working through the marginal cost. These are a function of demand conditions in the importing country (y_{t}), marginal costs of production (labour wages, fw_{t}) in the exporting country, and the commodity prices denominated in foreign currency (fcp_{t}):
 (18)
Substituting equations (18) and (17) into equation (16), we have
where the coefficient b on the exchange rate er_{t} is the passthrough elasticity. In the CGM ‘integrated world market’ specification, c_{0} ⋅ y_{t} + c_{1} ⋅ fw_{t} + c_{3} ⋅ fcp_{t}, independent of the exchange rate, is called the opportunity cost of allocating those same goods to other customers and is reflected in the world price of the product fp_{t} in the world currency (here taken to be the US dollar).11 Thus the final equation can be rewritten as follows:
 (19)
which gives the longrun relation between the import price, exchange rate and a measure of foreign price. Note that equation (19) also includes a time trend to capture the trending nature of the variables involved in the model.
6.2 Data
A sample of data for the euro area at a sectoral level from Eurostat is used with a time span of 1995–2005. The construction of the variables follows CM; more details are given in the companion appendix. The indicator we use for import prices, the index of import unit values (IUV), has a series of caveats associated with their use but we are constrained in our investigations by the quality of the publicly available data.
There are a number of reasons why we expect there may be a change in the longrun ERPT within our sample. Firstly, on 1 January 1999 11 European countries fixed their exchange rates by adopting the euro.12 This constituted a change in monetary policy, especially for countries that previously had less credible policy regimes. This is more noticeable in countries with previously rather less successful monetary policy. The perceived stabilization of monetary policy may well have induced the producers to change their pricing strategies and would thus be expected to have an influence on the longrun ERPT.
Moreover, the adoption of a common currency changed the competitive conditions, by increasing the share of goods denominated in the (new) domestic currency. Virtually all the currencies were depreciating against the US dollar in the period 1995–2000, especially since 1996, but after a short period of a stable euro dollar exchange rate the euro started appreciating from 2000/2001 until the end of our sample. Thus both the anticipation and introduction of the common currency, and the differential behaviour with respect to the US dollar, may be expected to be events that cause breaks in the cointegrating relationship. These should be revealed in terms of altering, in equation (19), either the mean level of the markup (a) or the passthrough (b) (or both). The direction of the changes will depend inter alia upon the changes in competition induced by the alteration in monetary conditions and the reactions of the participants in the market to these changes, for example by generating more or less pricing to market depending upon the nature of the industry. We keep this in mind in discussing the empirical results which follow.
6.3 Empirical Results
There are essentially three ways of proceeding in order to construct panels from the datasets: (i) creating country panels of industry crosssections; (ii) industry panels with country crosssections; and (iii) a pooled panel in which every country and industry combination constitutes a separate unit. In search of the existence of a cointegrating relationship in the series we try to maximize the dimensions of our panel, and thus will focus on (iii). In this empirical illustration we have decided to use the longest panel dataset in order to satisfy the condition that the time dimension of the panel is larger than the crosssection dimension (N/T is smaller compared to the case where the crosssection dimension is maximized). This implies excluding Austria, Finland and Portugal for the dataset and starting from January 1995 until March 2005 (T = 123) with seven countries and nine sectors for each country (N = 63).
6.3.1 Evidence of Parameter Instability
Before presenting the results based on the panel data procedure proposed in this paper, we report the results of an initial unitbyunit analysis to motivate the need to allow for the inclusion of structural breaks in the cointegration analysis. First, we estimate the relationship given in equation (19) and compute the Engle–Granger ADF tratio statistic to test the null hypothesis of no cointegration against the alternative hypothesis of cointegration. The evidence obtained by the use of this statistic leads to rejection of the null hypothesis of no cointegration at the 5% level of significance in 33% of units.13 Second, we compute the ADF tratio statistic in Gregory and Hansen (1996) allowing for one structural break that affects the level of equation (19); i.e. the C/T model specification in Gregory and Hansen (1996). In this case, the rejection of the null hypothesis of no cointegration at the 5% level of significance increases up to 76% of units, which indicates that consideration of solely one structural break can change the conclusions of the cointegration analysis considerably.14 Finally, we also compute the instability supF and the meanF test statistics in Hansen (1992). The results indicate that the null hypothesis of parameter stability is rejected at the 5% level of significance in 13.3% of cases using the supF statistic. The evidence against the null hypothesis of stability increases if we use the meanF statistic, where the percentages of rejection is 34%.15 This initial evidence therefore indicates that there are serious grounds for accounting for structural instability when performing the cointegration analysis if meaningful conclusions are to be obtained.
6.3.2 Cointegration Analysis
In this empirical application we are dealing with variables that show trending behaviour. This implies that the models that are to be estimated for the dataset need to include a time trend; i.e. Models 1 and 4 are then excluded. Further, we wish to illustrate the results that are obtained when the break dates are assumed not to be known a priori, which leads us to exclude Models 3 and 6 since these specifications require the break dates to be known and common to all units. Therefore, from the six different model specifications that we have proposed in the paper, we are left to considering only the ones given by Models 2 and 5.
As for the number of structural breaks, we cover three different situations: (i) the no break case; (ii) the one structural break case; and (iii) the two structural breaks case. For those specifications where both the parameters of the deterministic component and the cointegrating vector change, we consider that the structural breaks affect both components at the same time, i.e. m_{i} = n_{i} with ∀ i, j. We consider that allowing for up to two structural breaks is enough due to the restrictions imposed by the length of the time series and in order to avoid criticisms relating to data mining. Next, as shown in Figure 1 in the companion appendix for the case of one structural break (lefthand panel), the estimated break dates indicate that the assumption of a homogeneous break seems not to be adequate for our data. A similar picture is obtained when allowing for two structural breaks (Figure 1 in the companion appendix, righthand panel) and we therefore focus on the heterogeneous structural break case. Finally, throughout this section the maximum number of factors allowed is r_{max} = 12 and we have used the panel BIC in Bai and Ng (2004) to estimate the number of common factors .
Modelling choices for investigators using other datasets will depend upon specific imperatives imposed by the data such as those described above. This means that the number of modelling options available to them will perforce be narrowed from the six original models to a much smaller subset, leading to a considerable simplification of the modelling framework.
Since the limiting distribution of the panel data statistics requires the individual statistics to be crosssection independent, we have proceeded to test the null hypothesis of crosssection independence of e_{i,t} with the WCD test statistic in Pesaran (2013) computed for for different values of the autoregressive correction (k).16 Table 5 reveals that the null hypothesis of crosssection independence of the idiosyncratic disturbance terms is clearly rejected at the 5% level of significance for the specifications that do not account for structural breaks and for Model 2 with two structural breaks, regardless of the value of k. Therefore, in these cases, inference using the panel data cointegration test statistic that is proposed in this paper cannot be undertaken and the values of the panel cointegration test statistic are not reported. For Model 2 with one structural break we do not reject the null hypothesis of independence for most cases—rejections are found only for lag lengths k = 2 and k > 7. Finally, the null hypothesis is not rejected for Model 5, regardless of the value of k and the number of breaks.
Table 5. Pesaran's WCD statistic to test the null hypothesis of crosssection independence on the idiosyncratic residuals and panel cointegration test statisticsPesaran's WCD test statistic 

k  No structural breaks  One structural break  Two structural breaks 

Model 2  Model 5  Model 2  Model 5 

0  2.029  0.923  0.318  1.793  0.740 
1  2.591  1.769  0.342  2.201  0.684 
2  3.071  1.896  0.434  2.537  0.662 
3  3.014  1.772  0.388  2.317  0.656 
4  2.613  1.582  0.142  2.289  0.298 
5  2.737  1.650  0.346  2.394  0.409 
6  2.506  1.591  0.464  2.234  0.429 
7  2.405  1.603  0.743  2.270  0.609 
8  2.583  1.830  0.876  2.480  0.616 
9  2.614  1.888  0.760  2.538  0.429 
10  2.692  2.049  0.630  2.551  0.370 
11  2.351  1.905  0.482  2.408  0.427 
12  2.643  1.921  0.818  2.549  0.754 
Panel cointegration analysis 
% Individual rejections at the 5% level of sig.  38.1%  36.5%  34.9%  27.0%  36.5% 
Panel data test statistic  —  −3.011  −3.410  —  −4.130 
 10  12  11  10  11 
 1  1  3  1  5 
 1  1  3  3  3 
The same information is presented in a slightly different way in Figure 2 in the companion appendix, where the ratio of the average correlation coefficients pre and post defactoring are plotted for different choices of k for the one and two structural breaks cases. The sharp drops of this ratio are evident for Models 2 and 5, an indication that the common factor model does indeed capture successfully the crosssection dependence among the units of the panel.
The individual pseudo tratio statistics for testing α_{i,0} = 0 in equation (12) have been computed with the order of the autoregressive correction (k) selected using the tsig criterion in Ng and Perron (1995) with k_{max} = 12ceil[min[N,T]/100]^{1/4}. As can be seen from Table 5, the null hypothesis of spurious regression is rejected in a percentage range that goes from 28.6 to 38.1, depending on the model specification used.
This low evidence against the null hypothesis of no cointegration might be due to the fact that we are basing the statistical inference on individual information, so that panel data statistics may be expected to improve statistical inference by combining information from the time and crosssection dimensions. Based on this individual information, the panel cointegration test statistics for Model 2 with one structural break and Model 5 with one and two structural breaks reported in Table 5 lead to the same qualitative conclusion, i.e. the null hypothesis of panel spurious regression is rejected at the 5% level of significance. It is worth noting that the procedure detects the presence of at least one nonstationary common factor (r_{1}).17 Therefore, we can conclude that the observable variables do not cointegrate by themselves alone; i.e. we need to consider the presence of nonstationary common factors to obtain a longrun relationship.
This result shows the usefulness of the approach described in the paper, since the need for common factors in the analysis is often to capture misspecification in the model—misspecification that is to some extent to be expected if we proxy for economic variables that are difficult to measure in practice. For instance, in our illustration, and based on standard procedures used in the empirical literature on exchange rate passthrough, we proxy variables such as the markup or the world price in order to estimate the passthrough equation. An alternative interpretation of the need for nonstationary common factors might be to model the common driving trends in the empirical model such as the foreign price of commodities in dollars or the exchange rate with respect to the dollar which introduces dependence across the units of the panel.
Figure 1 in the companion appendix reports the years of the estimated break dates for the one structural break model specifications—63 estimated break dates for each of the model specifications, in the left panel—and for the two structural breaks case—63 first and second estimated break dates for each of the model specifications, in the right panel. As can be seen, the estimated break dates are mostly placed around the introduction of the euro in 1999 and, to a lesser extent, around 2002 to reflect entry (and anticipated entry) into the common currency area, as well as the impact of the originally depreciating and then appreciating euro with respect to the dollar.
However, it should be borne in mind that these estimated break dates do not necessarily lead to consistent estimates of the break dates. As described above, estimation of these break dates has been carried out using the model in equation (1) in first differences. This implies estimating a model with I(0) variables where the structural breaks are captured by impulse dummies (Model 2) or by impulse dummies and changing coefficients for the first difference of the stochastic regressors (Model 5). For Model 5, the use of the minimization of the sum of squared residuals leads to consistent estimates of the break fractions; see Bai and Perron (1998). The problem appears for Model 2, where the consistent estimation of the break fractions requires the magnitudes of the structural breaks to be large; see CarrioniSilvestre et al. (2009). Therefore, unless the magnitudes of the structural breaks in Model 2 are large enough to ensure that the effect of the structural breaks does not vanish asymptotically, some caution should be exercised when examining the estimated break dates obtained from Model 2.
6.3.3 LongRun PassThrough Cointegration Relationship
Consistent estimates of the cointegrating vector are obtained from the OLS estimation of the model:
 (20)
with x_{i,t} = (er_{i,t},fp_{i,t})′, where the definition of the deterministic component and whether the cointegration vector changes depends on the model under consideration. Note that equation (20) includes the common factors that have been estimated when implementing the test of panel cointegration with common factors. Therefore, and provided that evidence of panel cointegration has been found, equation (20) defines a cointegration relationship that can be estimated unit by unit.
Provided the estimated common factors are consistent estimates of the (rotated) unobservable common factors, they may be used in equation (20) as if they were known; see Bai and Ng (2004). Furthermore, the estimation of the break dates unit by unit can be achieved by minimizing the sum of squared residuals of the cointegrating relationship in equation (20), provided that this estimation procedure renders consistent estimates of the break fractions for both Model 2 and 5 specifications; see CarrioniSilvestre and Sansó (2006), Arai and Kurozumi (2007) and Kejriwal and Perron (2010).
Based on this approach, we have estimated equation (20) for Model 2 allowing for one structural break and Model 5 allowing for one and two structural breaks. In order to select among these three options, and following the spirit of Westerlund and Edgerton (2008), we have computed the mean squared error (MSE) for each unit and option as a way of obtaining an indication of the fit to the data provided by each of the options. Figure 3 in the companion appendix (left panel) reports the estimated density functions of the MSE statistic, showing that the MSE density of Model 5 with two structural breaks is more concentrated around zero, which makes this specification our preferred one.18 We henceforth focus on Model 5 with two structural breaks.
Figure 3 in the companion appendix (right panel) presents the densities of the first and second estimated break dates (in years) associated with the OLS estimation of equation (20) for Model 5 with two structural breaks. Similar to what has been mentioned above, the estimated break dates are mostly placed around the introduction of the euro in 1999—note that the densities of the estimated break dates superpose around 1999—and also around 1996 and 2002. Figure 4 in the companion appendix summarizes the estimated coefficients of the cointegrating vector for the three regimes that define the two estimated break dates. One interesting feature that is obtained when looking at the densities of the estimated coefficients is that the degree of the passthrough increases through time—i.e. the densities move to the right as we move across regimes—potentially reflecting a lower importance of pricingtomarket behaviour by importers in the economy.
Finally, our analysis has proceeded assuming that all units define a cointegration relationship. However, and as stated above, the rejection of the null hypothesis of no panel cointegration does not necessarily mean that all the units of the panel are cointegrated. In this regard, the information drawn from the unitbyunit analysis can be helpful in assessing the robustness of our conclusions. To this end, we have taken into account only the units for which the null hypothesis of no cointegration is rejected in the unitbyunit analysis. As can be seen, Figures 5 and 6 in the companion appendix indicate that the same qualitative conclusions are found.
7 Conclusions
 Top of page
 Summary
 1 Introduction
 2 The Models
 3 Test Statistics
 4 Panel Data Cointegration Tests
 5 Monte Carlo Simulations
 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
 7 Conclusions
 Acknowledgements
 References
 Supporting Information
Starting from the observation that structural breaks, if not accounted for, have a harmful effect on the size and power properties of panel tests for cointegration, we have established a general testing framework within which to allow for such breaks. Building on our own earlier work and on related papers in the literature, we use a common factor framework allied to specifications that accommodate changes in level, trend and cointegrating coefficients to develop tests for cointegration that consider not only the presence of breaks but also the existence of crosssection dependence among the units of the panel. Loosely speaking, these may be termed the third generation of panel cointegration tests to distinguish them from the firstgeneration tests, which disregarded crosssection dependence, and the secondgeneration tests, which disregarded structural breaks.
We are able to show a number of interesting results. First, for the cases where the breaks are assumed to occur only in the level component and in the cointegrating coefficients in the units, we derive tests and critical values that control not only for multiple breaks in each unit of the panel but also permit heterogeneity of break locations across the units. Within this setting we show how to work with cases not only where the break dates are assumed to be known a priori but also where these may be unknown (and thus themselves subject to a process of discovery before the testing for cointegration is undertaken).
We show that the analysis is complicated considerably if allowing for breaks in trend. In this case the restrictions imposed by the assumption of a factor model to model dependence mean that the testing for breaks can be undertaken successfully only for cases where these breaks are imposed homogeneously across the units of the panel. Multiple breaks are still permissible, only the set of breaks must be temporally the same (although of differing magnitudes) across the units. Our analysis so far has considered only the case where these homogeneous breaks are assumed to be known, although work is in hand on the difficult task of extending these results to the unknown breaks in trend case.
Our theoretical analysis is illustrated through Monte Carlo simulations, from which we conclude that the statistics show good performance once the procedures have included the structural breaks. An empirical example based on looking at exchange rate passthrough in the euro area helps to illustrate the value of our proposal.