SEARCH

SEARCH BY CITATION

Summary

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. 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 cross-section 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 cross-section 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 pass-through of import prices on a sample of European countries. Copyright © 2013 John Wiley & Sons, Ltd.

1 Introduction

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. 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 cross-section 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 Carrion-i-Silvestre (2004), Westerlund (2006) and Gutierrez (2010). One important feature to consider from a practical point of view is the cross-section 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 cross-section independent. In this paper we contribute to the panel cointegration literature by extending the analysis in Banerjee and Carrion-i-Silvestre (2004) to allow for the presence of both structural breaks and cross-section 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 cross-section 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 cross-section 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 cross-section 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 cross-section dependence. Section 4 proposes panel cointegration tests under different specifications of the models. Section 5 focuses on the finite-sample 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 pass-through. The issue of the degree of exchange rate pass-through 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

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. Supporting Information

Let Yi,t = (yi,t, xi,t)′ be an (m × 1) vector of non-stationary stochastic process whose elements are individually I(1). Further, let us specify the data-generating process (DGP) in structural form as

  • display math(1)
  • display math(2)
  • display math(3)
  • display math(4)
  • display math(5)
  • display math(6)

i = 1, …, N, t = 1, …, T, where inline image, inline image, inline image and inline image. The general functional form for the deterministic term Di,t is given by

  • display math(7)

where DUi, j,t = 1 and inline image for inline image and 0 otherwise, with inline image denoting the timing of the jth break, j = 1, …, mi, for the ith unit, i = 1, …, N, inline image, 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

  • display math(8)

with the convention that inline image and inline image, where inline image denoting the jth time of the break, j = 1, …, ni, for the ith unit, i = 1, …, N, inline image.

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– βi = γi,j = 0 ∀ i, j in (7) and κi = 0 ∀ i in (5)–and stable cointegrating vector– δi,j = δi ∀ j in equation (8).
  • Model 2: stable trend– βi ≠ 0 ∀ i and γi,j = 0 ∀ i, j in (7)–and stable cointegrating vector– δi,j = δi ∀ j in equation (8).
  • Model 3: changes in level and trend– βi ≠ γi,j ≠ 0 ∀ i, j in (7)–and stable cointegrating vector– δi,j = δi ∀ j in equation (8).
  • Model 4: no linear trend– βi = γi,j = 0 ∀ i, j in (7) and κi = 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– βi ≠ 0 ∀ i and γi,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.
  1. 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 (mi) is different from the number of structural breaks affecting the cointegrating vector (ni). Moreover, even in the case that mi = ni, the break dates do not need to be located at the same date, i.e. inline image. We define the break fraction vector for each unit for the most general situation as inline image.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.
  2. 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 inline image. 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.
  3. 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 κi = 0 in equation (5), βi = θi,j = γi,j = 0 ∀ i, j in equation (7) and δi,j = δi ∀ j in equation (8), which gives rise to the so-called ‘constant term case’. Further, it is also possible to include a time trend with no structural breaks in the model just by specifying βi ≠ 0 and θi,j = γi,j = 0 ∀ i, j in equation (7) and δi,j = δi ∀ j in equation (8). This model defines the so-called ‘time trend case’.
  4. The component designed as Ft denotes an (r × 1) vector containing the common factors, with πi the vector of loadings. Our model specification considers the case where the stochastic regressors are assumed to be either cross-section independent–impose ςi = 0 in equation (5)–or cross-section dependent with dependence driven by a set of common factors Gt–consider ςi ≠ 0 in (5). Throughout this paper we assume that the set of common factors affecting xi are different from those affecting yi–i.e. Ft or transformations of Ft–although it is possible to use the approach in Bai and Carrion-i-Silvestre (2013) to consider the case where some of the factors in Gt are also in Ft (or transformations of Ft).
  5. Despite the presence of the operator (I-L) in equation (3), Ft does not have to be I(1). In fact, Ft can be I(0), I(1) or a combination of both, depending on the rank of C(1). If C(1) = 0, then Ft is I(0). If C(1) is of full rank, then each component of Ft is I(1). If C(1) ≠ 0, but not full rank, then some components of Ft are I(1) and some are I(0).
  6. The presence of cointegration among Yi,t = (yi,t, xi,t)′ requires Ft to be I(0). However, allowing Ft to be I(1) is also relevant from an empirical point of view since, in this case, Ft might be capturing effects from outside the model that are not included in Yi,t. Then, cointegration among the elements in Yi,t up to the inclusion of I(1) factors is possible, which will imply ei,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 Carrion-i-Silvestre (2013). Let M <  be a generic positive number, not depending on T and N. Further, ‖A‖ = trace(AA)1/2. Then

Assumption A. (i) For non-random πi, ‖πi‖ ≤ M; for random πi, Eπi4 ≤ M, (ii) inline image, a (r × r) positive definite matrix.

Assumption B. (i) wt ∼ i. i. d. (0,inline imagew), Ewt4 ≤ M; (ii) inline image; (iii) inline image; and (iv) C(1) has rank r1, 0 ≤ r1 ≤ r.

Assumption C. (i) for each i, inline image, E|εi,t|8 ≤ M, inline image, inline image; (ii) E(εi, tεj, t) = τi, j with inline image for all j; (iii) inline image, for every (t,s).

Assumption D. The errors εi,t, wt, vi,t, ϖt and the loadings πi and ςi are mutually independent groups.

Assumption E. EF0‖ ≤ M, and for every i = 1, …, N, E |ei,0| ≤ M and Exi,0‖ ≤ M.

Assumption F. (i) inline image, Evi,t4 ≤ M, ϖt  ∼ i. i. d. (0,inline imageϖ), Eϖt4 ≤ M; (ii) inline image; (iii) inline image, inline image; and (iv) Γ(1) and Ξi(1) have full rank.

Assumption G. (i) E(ei,t|vi,t) = 0 when stochastic regressors are assumed to be strictly exogenous; or (ii) E(ei,t|vi,t) = Δxi,tAi(L) + ξi,t, with Ai(L) being a (υi × 1) vector of lags and leads polynomials of finite orders and inline image, when stochastic regressors are non-strictly exogenous.

Assumption A ensures that the factor loadings are identifiable. Assumption B establishes the conditions on the short- and long-run variance of ΔFt –i.e. the short-run variance matrix is positive definite and the long-run 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 − ρiL)ei,t, whereas Assumptions C(ii) and C(iii) allow for weak cross-section correlation. Assumption D imposes mutual independence among the factors, loadings, idiosyncratic residuals and stochastic regressors xi,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 ei,t and vi,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 Carrion-i-Silvestre (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

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. 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 Carrion-i-Silvestre (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:

  • display math(9)

where inline image; ‘.*’ denotes the element-by-element product of each column of Δxi with the vector DUi,j, j = 1, …, ni; and inline image.

From equation (9) we could take the orthogonal projection of the model with respect to ΔDi and ΔXi defining the projection matrix inline image —the superscript d in inline image indicates that there are deterministic elements, i.e. inline image for the most general specification, where inline image for inline image and 0 elsewhere, j = 1, …, mi —and obtain a standard common factor model representation. In order to show this more easily, we proceed by following an equivalent two-step approach for expositional purposes. Let us begin by considering the model specifications without changes in the slope of the trend (Models 1, 2, 4 and 5). We will then address the models that allow for these changes (Models 3 and 6).

The first of the two steps is to take the orthogonal projections with respect to the deterministic components:

  • display math

where inline image and the dotted variables denote that they have been projected against ΔDi . Second, we can take the orthogonal projections with respect to inline image:

  • display math

where inline image. Note that in this expression we have inline image, which involves projecting the first difference of the common factors using a unit-specific 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:

  • display math(10)

where inline image, f = ΔF and inline image. 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 inline image and inline image entering in zi are asymptotically negligible for testing for panel cointegration in this paper; see Assumption D above and Bai and Carrion-i-Silvestre (2013, Remark 3.2 and Lemma A.1). This result is valid for all the models except Models 3 and 6 since inline image, j = 1, …, mi, have a negligible effect in the limit.

For Models 3 and 6 the presence of DUi,j, j = 1, …, mi, makes the terms in inline image and inline image not negligible in the limit, so that in this case we need to assume common break dates, i.e. λi = λ ∀ i so that inline image ∀ i. Under this further assumption equation (10) again represents a standard common factor model with inline image, f = MΔDΔF and inline image.

The estimation of the common factors and factor loadings can be done as in Bai and Ng (2004) and Bai and Carrion-i-Silvestre (2013) using principal components. Specifically, the estimated principal components inline image are inline image times the r eigenvectors corresponding to the first r largest eigenvalues of the (T − 1) × (T − 1) matrix inline image, where inline image and given that the number of common factors is known. Under the normalization inline image, the estimated loading matrix is inline image. Therefore, the estimated residuals are defined as

  • display math(11)

We can recover the idiosyncratic disturbance terms through cumulation, i.e. inline image, and test the unit root hypothesis (αi,0 = 0) using the augmented Dickey–Fuller (ADF) type regression equation:

  • display math(12)

The null hypothesis of a unit root can be tested using the pseudo t -ratio inline image, j = c, τ, γ, for testing αi,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 ADF-type equation to analyse the order of integration of Ft. However, in this case we need to proceed in two steps. In the first step we regress inline image on the deterministic specification. In the second step we estimate the ADF regression equation using the detrended common factor inline image.

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 (r1) . As before, let inline image denote the detrended common factors. Start with q = r and proceed in three stages; we reproduce these steps here for completeness:

  1. Let inline image be the q eigenvectors associated with the q largest eigenvalues of inline image.
  2. Let inline image, from which we can define two statistics:
    • Let K(j) = 1 − j/(J + 1), j = 0, 1, 2, …, J:
      1. Let inline image be the residuals from estimating a first-order VAR in inline image, and let
        • display math
      2. Let inline image.
      3. Define inline image for the case of no change in the trend and inline image for the case of changes in the trend.
    • For p fixed that does not depend on N and T:
      1. Estimate a VAR of order p in inline image to obtain inline image. Filter inline image by inline image to get inline image.
      2. Let inline image be the smallest eigenvalue of
        • display math
      3. Define the statistic inline image for the case of no change in the trend and inline image for the case of changes in the trend.
  3. If H0 : r1 = q is rejected, set q = q − 1 and return to the first step. Otherwise, inline image and stop.

The following theorem consolidates the main results concerning these statistics.

Theorem 1. Let {yi,t} the stochastic process with DGP given by equations (1)(5). The following results hold as N, T[RIGHTWARDS ARROW].

  1. Let ki be the order of autoregression chosen such that ki[RIGHTWARDS ARROW] and inline image. Under the null hypothesis that ρi = 1 in equation (4)
    • 1(a) Models 1 and 4: inline image

    • 1(b) Models 2 and 5: inline image

    • 1(c) Models 3 and 6: inline image

    where Wi(s) denotes a standard Brownian motion, Vi(s) = Wi(s) − sWi(1), and Vi(bj) = Wi(bj) − bjWi(1), with bj = (s − λj − 1)/(λj − λj − 1) so that 0 < bj < 1, and inline image with inline image and inline image.
  2. Let k be the order of autoregression chosen such that k[RIGHTWARDS ARROW] and k3/min[N,T] [RIGHTWARDS ARROW] 0. When r = 1, under the null hypothesis that Ft has a unit root and no change in trend:
    • display math
    where inline image for Models 1 and 4 and inline image for Models 2 and 5, with Ww(s) a standard Brownian motion. When changes in the trend are allowed, i.e. for Models 3 and 6:
    • display math
    where inline image, Mχ = I − χ(χχ)− 1χ′, is a Brownian motion projected onto the subspace generated by χs = (1, s, dt1(s, λ1), …, dtm(s, λm)), where dtj(s, λj) = (s − λj) for s − λj > 0 and 0 otherwise, j = 1, 2, …, m, and [⋅]s denotes the sth element of the matrix between brackets.
  3. When r > 1, let Wq(s) be a q vector of standard Brownian motion and inline image and inline image the detrended counterparts; see statement 2 of this theorem. For the models that do not include change in the trend, let inline image be the smallest eigenvalues of
    • display math
    For Models 3 and 6, let inline image be the smallest eigenvalues of
    • display math
    • Let J be the truncation lag of the Bartlett kernel, chosen such that J[RIGHTWARDS ARROW] and inline image. Then, under the null hypothesis that Ft has q stochastic trends, inline image and inline image.
    • Under the null hypothesis that Ft has q stochastic trends with a finite VAR inline image representation and a VAR(p) is estimated with inline image, inline image and inline image.

Proof. See the companion appendix.

Remarks.
  1. 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 Gt in equation (5) does not disrupt any of our results while allowing for cross-sectional dependence among the stochastic regressors.
  2. 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.
  3. 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
  4. The limiting distributions for inline image and inline image derived in (1a) and (2) of Theorem 1 correspond to the standard Dickey–Fuller distributions.
  5. Finally, the limiting distribution of the inline image 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, inline image, 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
  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 Carrion-i-Silvestre (2009), which considers the presence of structural breaks.
Table 1. Asymptotic critical values for the MQ(q,λ) tests
rλ = 0.1λ = 0.2λ = 0.3
1%5%10%1%5%10%1%5%10%
  1. Note: 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.

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 = 0.4λ = 0.5λ = 0.6
r1%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 = 0.7λ = 0.8λ = 0.9
r1%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

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. 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 set-up. At this stage, a comment on the interpretation of the outcome of the panel data test statistics should be made. Thus, in our set-up, 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:

  • display math(13)

where λ = (λ1, λ2, …, λN)′ for j = c, τ and λ = (λ1, λ2, …,λm)′ for j = γ. The limit distribution of the statistic in equation (13) is given in the following theorem.

Theorem 2. Let {yi,t} be the stochastic process with DGP given by equations (1)(5), and suppose that ei,t is independent across i. Denote by inline image and inline image the mean and variance, respectively, of the vector Brownian motion functionals inline image defined in Theorem 1. Let ki be the order of autoregression in equation (12) chosen such that ki[RIGHTWARDS ARROW] and inline image. Then, under the null hypothesis that ρi = 1 ∀ i in equation (4), the distribution of the standardized statistic SADFj(λ), j = c, τ, γ, converges, as N, T[RIGHTWARDS ARROW] with N/T[RIGHTWARDS ARROW] 0, to

  • display math(14)

where inline image and inline image for j = c, τ.

Remarks.
  1. 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[RIGHTWARDS ARROW]. 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 inline image and inline image. Note also that these densities do not depend upon the location of the breaks.
  2. 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.
  3. 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
Table 2. Moments for the Zγ(λ) for Models 3 and 6
One structural break
λ0.10.20.30.40.50.60.70.80.9
  1. Note: 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. For the two structural breaks case, we report the value of the mean on the upper triangle of the matrix, and the value of the variance on the lower triangle of the matrix. For example, for (λ1,λ2) = (0.3,0.7), inline image and inline image.

inline image−1.676−1.809−1.902−1.950−1.972−1.953−1.900−1.799−1.691
inline image0.3890.3920.3710.3380.3390.3460.3650.3900.392
Two structural breaks
inline image on the upper triangle, inline image on the lower triangle
λ1 \ λ20.10.20.30.40.50.60.70.80.9
0.1 −1.855−1.994−2.093−2.143−2.139−2.097−1.998−1.853
0.20.445 −1.995−2.146−2.238−2.270−2.238−2.142−1.998
0.30.4390.436 −2.094−2.237−2.309−2.315−2.236−2.098
0.40.4010.4160.406 −2.144−2.270−2.315−2.263−2.141
0.50.3740.3730.3740.376 −2.144−2.241−2.240−2.144
0.60.3730.3520.3350.3520.373 −2.094−2.145−2.093
0.70.4020.3710.3370.3380.3740.405 −1.996−1.995
0.80.4400.4150.3720.3520.3710.4160.438 −1.856
0.90.4420.4400.4040.3770.3730.4020.4410.444 

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 Carrion-i-Silvestre (2009). Thus, from equation (9), we define Δyi,t = ΔDi,t + ΔXi,tδi + Δui,t, where Δui,t = ΔFtπi + Δei,t, and proceed to estimate the break dates, ΔDi,t and δi 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 inline image and estimate the factors and loadings using principal components on the model inline image . The estimated factors and loadings are denoted as inline image and inline image. Then, define inline image and get an updated estimate of the break dates, ΔDi,t and δi. Using these updated estimates, define inline image from which the factors and the loadings are estimated again, giving inline image. 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 Zj(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 non-cointegration for the idiosyncratic disturbance term is given by

  • display math(15)

where inline image. The limiting distribution of inline image, j = c, τ, is given in the following theorem.

Theorem 3. Let {yi,t} be the stochastic process with DGP given by equations (1)(5), and suppose that ei,t is independent across i. Let ki be the order of autoregression in equation (12) chosen such that ki[RIGHTWARDS ARROW] and inline image. Then, under the null hypothesis that ρi = 1 ∀ i in equation (4), the distribution of the inline image, j = c, τ, test in ( 15) converges, as N, T[RIGHTWARDS ARROW] with N/T[RIGHTWARDS ARROW] 0, to

  • display math

The proof follows from noting that under the null hypothesis inline image , j = c, τ, is the infimum of a sequence of perfectly correlated random variables Zj(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 inline image, j = c, τ, statistics
T1%2.5%5%10%
Constant with or without level shifts
(inline image 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
(inline image 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

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. Supporting Information

Consider the DGP given by a bivariate system:

  • display math

where πi, j ∼ i. i. d. N(1,1), (wt,εi,t,vi,t)′, with wt = (w1,t, …,wr,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 μi = 1, βi = 0.3, θi = 3, δi,1 = 1 ∀ i and, when there is a change in the cointegrating vector, δi,2 = 3 ∀ i, otherwise δi,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 ρi = {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 inline image. The number of common factors is estimated using the panel BIC information criterion in Bai and Ng (2004) with rmax = 6 as the maximum number of factors. In the computation of the ADF-type regression equation we select the order of the autoregressive correction using the t-sig criterion in Ng and Perron (1995) with a maximum of kmax = 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 t-ratio 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 λi = 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 offers results for the empirical size and power when there is only one common factor for Model 2 (Panel A of the table) and Model 5 (Panel B of the table). Looking at these results we can conclude that for Model 2 the empirical size of the inline image test is close to the nominal one, while there is some over-rejection for the inline image test when inline image. For Model 2, the empirical power is close to the nominal size for ρi = 0.99, which is to be expected given that the alternative is close to the null hypothesis. It is worth noting that decent power results can be obtained even for this value of the autoregressive parameter if the sample size is large (T = 250). Things are different for Model 5, for which the inline image test shows high power in all cases. These features are reproduced when ρi = 0.95, but with higher values for the empirical power, although the tendency of this test to over-reject must be kept in mind.

Table 4. Empirical size and power of the Z* and inline image tests for Models 2 and 5 with one common factor
inline imageφTPanel A: Model 2Panel B: Model 5
ρi = 1ρi = 0.99ρi = 0.95ρi = 1ρi = 0.99ρi = 0.95
inline imageinline imageinline imageinline imageinline imageinline imageinline imageinline imageinline imageinline imageinline imageinline image
0.51500.0490.0570.0740.0060.4100.0390.0910.0590.6040.0010.8430.002
0.511000.0420.0540.0950.0420.9780.0560.0650.0450.7270.00110.003
0.512500.0500.0590.4200.04710.0500.0840.0540.9710.00110.032
0.50.9500.0460.0810.0910.0880.3890.0300.0820.0970.5690.0010.8490.001
0.50.91000.0340.1940.0820.1400.9860.1830.0520.1730.7490.00110.018
0.50.92500.0440.7580.4460.70910.7760.0660.7270.9790.00110.546
0.50.8500.0440.2200.0820.1420.3980.1390.0830.2010.5560.0010.8480.006
0.50.81000.0480.5800.0850.4800.980.5000.0580.5490.7190.0010.9990.045
0.50.82500.0580.9940.4400.98710.9990.0690.9990.9740.00110.675
11500.0650.0610.0850.0060.4180.0490.1050.0760.6100.0010.8470.001
111000.0430.0450.0750.0610.980.0600.0830.0510.7350.0010.9990.005
112500.0490.0430.4360.04710.0540.0810.0500.9750.00110.039
10.9500.0440.0870.0850.0650.4010.1040.0880.0920.5860.0020.8210.007
10.91000.0510.1900.0850.2000.9720.1710.0560.2090.7240.00110.016
10.92500.0440.7730.4090.74210.8110.0760.7500.9710.00210.519
10.8500.0590.2210.0730.1800.3610.1800.0740.2070.6080.0010.8250.004
10.81000.0400.6020.0670.5610.9720.5880.0630.5830.7120.00110.039
10.82500.0530.9990.428110.9990.0760.9980.9800.00210.631
101500.0780.0390.1120.0410.3810.0330.1680.0240.5430.0010.7300.001
1011000.0540.0490.0890.0500.9310.0450.1190.0340.6750.0010.9910.002
1012500.0550.0380.3980.05210.0490.1050.0460.9680.00110.019
100.9500.0620.0550.0940.0480.3550.0360.1440.0380.4980.0010.7310.002
100.91000.0520.1780.0740.1740.9570.1940.0700.1480.6850.0010.9940.007
100.92500.0480.8230.4300.78510.7930.1220.7830.9620.00210.376
100.8500.0610.1430.1050.0780.3390.1390.1460.0700.4710.0010.7150.002
100.81000.0430.5250.0840.5520.9220.5520.0770.4950.6280.0010.9880.020
100.82500.04810.4020.99310.9850.1000.9840.9720.00210.401

Turning now to the inline image statistic. In general, we can see that we require T to be large for the statistic to show good properties in terms of empirical size and power. This is not surprising, since this statistic relies on only one source of information, i.e. the one coming from the time dimension, whereas the inline image and inline image tests pool across the N dimension also.

When there are three common factors, the empirical size of the inline image test is close to the nominal one, regardless of the parameter combination that is used; owing to space constraints, we do not report the results, although they are available in Tables VI–VIII of the companion appendix. This does not happen for the inline image test, since we observe over-rejection when the importance of the common factors is large inline image. The empirical power of the inline image and inline image tests mimics the behaviour shown for the one common factor case.

Regarding the inline image test for Model 2, we can see that the statistic selects the correct number of common stochastic trends (r1) when T is large and inline image is small, but in general it tends to detect more non-stationary common stochastic trends for T small and inline image –the bandwidth for the Bartlett spectral window is set as J = 4ceil[min[N,T]/100]1/4. This behaviour is more pronounced for Model 5 than for Model 2. Thus, in the case of Model 5, we can see that the statistic selects the correct number of stochastic trends when T is large, inline image is small and ρi = 1, but it tends to detect more non-stationary common stochastic trends for T small, inline image and ρi = 1. In general, we can observe that the inline image test detects more non-stationary common stochastic trends than exist when the idiosyncratic component is I(0) regardless of the value of φ.

To sum up, the simulations that have been conducted in this section reveal that the inline image and inline image test statistics have good properties in terms of empirical size and power, but the inline image test tends to overestimate the number of common stochastic trends.

6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. Supporting Information

Campa and González-Mínguez (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 steady-state 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 Pass-Through into Import Prices

By definition, import prices for any type of goods j, inline image, are a transformation of export prices of a country's trading partners, inline image , using the bilateral exchange rate, ERt, i.e., and dropping superscript j for clarity, we have MPt = ERt ⋅ XPt; in logarithms (depicted in lower case), mpt = ert + xpt. The export price consists of the exporters marginal cost and a mark-up, i.e., XPt = FMCt ⋅ FMKUPt. After suitable substitutions we have

  • display math(16)

Mark-ups 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:

  • display math(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 (yt), marginal costs of production (labour wages, fwt) in the exporting country, and the commodity prices denominated in foreign currency (fcpt):

  • display math(18)

Substituting equations (18) and (17) into equation (16), we have

  • display math

where the coefficient b on the exchange rate ert is the pass-through elasticity. In the CGM ‘integrated world market’ specification, c0 ⋅ yt + c1 ⋅ fwt + c3 ⋅ fcpt, 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 fpt in the world currency (here taken to be the US dollar).11 Thus the final equation can be rewritten as follows:

  • display math(19)

which gives the long-run 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 long-run 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 long-run 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 mark-up (a) or the pass-through (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 cross-sections; (ii) industry panels with country cross-sections; 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 cross-section dimension (N/T is smaller compared to the case where the cross-section 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 unit-by-unit 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 t-ratio 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 t-ratio 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 sup-F and the mean-F 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 sup-F statistic. The evidence against the null hypothesis of stability increases if we use the mean-F 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. mi = ni with inline image ∀ 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 (left-hand 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, right-hand panel) and we therefore focus on the heterogeneous structural break case. Finally, throughout this section the maximum number of factors allowed is rmax = 12 and we have used the panel BIC in Bai and Ng (2004) to estimate the number of common factors inline image.

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 cross-section independent, we have proceeded to test the null hypothesis of cross-section independence of ei,t with the WCD test statistic in Pesaran (2013) computed for inline image for different values of the autoregressive correction (k).16 Table 5 reveals that the null hypothesis of cross-section 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 cross-section independence on the idiosyncratic residuals and panel cointegration test statistics
Pesaran's WCD test statistic
kNo structural breaksOne structural breakTwo structural breaks
Model 2Model 5Model 2Model 5
02.0290.9230.3181.7930.740
12.5911.7690.3422.2010.684
23.0711.8960.4342.5370.662
33.0141.7720.3882.3170.656
42.6131.5820.1422.2890.298
52.7371.6500.3462.3940.409
62.5061.5910.4642.2340.429
72.4051.6030.7432.2700.609
82.5831.8300.8762.4800.616
92.6141.8880.7602.5380.429
102.6922.0490.6302.5510.370
112.3511.9050.4822.4080.427
122.6431.9210.8182.5490.754
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
inline image1012111011
inline image11315
inline image11333

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 de-factoring 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 cross-section dependence among the units of the panel.

The individual pseudo t-ratio statistics inline image for testing αi,0 = 0 in equation (12) have been computed with the order of the autoregressive correction (k) selected using the t-sig criterion in Ng and Perron (1995) with kmax = 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 cross-section 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 non-stationary common factor (r1).17 Therefore, we can conclude that the observable variables do not cointegrate by themselves alone; i.e. we need to consider the presence of non-stationary common factors to obtain a long-run 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 pass-through, we proxy variables such as the mark-up or the world price in order to estimate the pass-through equation. An alternative interpretation of the need for non-stationary 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 Carrion-i-Silvestre 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 Long-Run Pass-Through Cointegration Relationship

Consistent estimates of the cointegrating vector are obtained from the OLS estimation of the model:

  • display math(20)

with xi,t = (eri,t,fpi,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 Carrion-i-Silvestre 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 pass-through increases through time—i.e. the densities move to the right as we move across regimes—potentially reflecting a lower importance of pricing-to-market 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 unit-by-unit 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 unit-by-unit analysis. As can be seen, Figures 5 and 6 in the companion appendix indicate that the same qualitative conclusions are found.

7 Conclusions

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. 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 cross-section 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 first-generation tests, which disregarded cross-section dependence, and the second-generation 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 pass-through in the euro area helps to illustrate the value of our proposal.

Acknowledgements

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. Supporting Information

We thank the editor and referees for helpful comments and suggestions. We also thank participants at seminars in Copenhagen, Ente Einaudi (Rome), City University (London), Maastricht University, University of Algarve (Portugal) and the Tinbergen Institute (Rotterdam) for their comments. Carrion-i-Silvestre acknowledges financial support from the Generalitat de Catalunya under BE-2004 grant, and from the Spanish Ministerio de Ciencia e Innovación under grant ECO2011-30260-C03-03.

  1. 1

    For instance, Gregory and Hansen (1996) follow the previous literature and define Λ = [0.15,0.85].

  2. 2

    For Models 1–3 the break fraction vector is defined by inline image. Similarly, for Models 4–6 where the same breaks affect both the deterministic component and the cointegrating vector, it is sufficient to define inline image.

  3. 3

    Note that in this case the common factors will be accounting for misspecification errors in the model, due, for instance, to the omission of relevant stochastic regressors.

  4. 4

    As noted above, in the case where strict exogeneity does not hold, we recommend the use of DOLS estimation. Bai and Carrion-i-Silvestre (2013) follow the approach in Bai (2009) and consider the case where the regressors are not orthogonal to the factors. A derivation of the proofs using DOLS is given in the companion appendix.

  5. 5

    In this case, it is only required to impose the constraint that θi,j = γi,j = δi,j = 0 ∀ i, j in the model specification and proceed as described above.

  6. 6

    A computer program is available from the authors to simulate the critical values for multiple structural breaks.

  7. 7

    A computer program is available from the authors to simulate the critical values for multiple structural breaks.

  8. 8

    If ΔF does has have zero mean, we can always redefine the constant term of the model so that it captures the mean of the common factor component; see footnote 8 in Bai and Carrion-i-Silvestre (2009).

  9. 9

    A computer program is available from the authors to simulate the critical values for multiple structural breaks.

  10. 10

    We also have results for the case where the break point is known. The tests on the idiosyncratic disturbance terms show good properties in terms of empirical size and power. The ADF statistic on the common factor, when there is one single common factor, shows the right size although, as expected, it has low power when the autoregressive parameter is close to unity and the sample size is small. Our results for three factors are comparable to the one factor case.

  11. 11

    The integrated market hypothesis in CM is based on the assumption that there exists a single world market for each good. Therefore, regardless of the origin of the product, on the world market it has one world price. This price constitutes the opportunity cost of selling to a local market. Thus, in the CM set-up for the integrated market and, consequently, in ours, it proxies for the foreign price. The currency denomination does not in fact matter, as long as the exchange rate for the local currency is taken vis-à-vis this ‘world’ currency. In the CM case the extra-euro area imports denominated in US dollars are taken as a proxy for the world price.

  12. 12

    Greece failed to fulfil the Maastricht Treaty criteria, and therefore joined 2 years later, effective 1 January 2001.

  13. 13

    The order of the autoregressive correction that is used to compute the ADF test statistic is selected using the t-sig criterion in Ng and Perron (1995) with a maximum of kmax = 12ceil[T/100]1/4 lags.

  14. 14

    The order of the autoregressive correction that is required to compute the ADF t-ratio statistic is selected as described for the Engle–Granger test statistic.

  15. 15

    If the level of significance is set at 10% the percentages of rejection of the null hypothesis of parameter stability are 28% for the sup-F test, and 47% for the mean-F test statistic.

  16. 16

    The WCD test statistic is distributed as a standard normal under the null hypothesis of cross-section independence; see Pesaran (2013) for further details on the computation of the statistic. We use two-sided inference to test the null hypothesis, so that the critical values are ±1.96 at the 5% level of significance.

  17. 17

    For the non-parametric (NP) version of the MQ test statistic, the bandwidth for the Bartlett spectral window is set as J = 4ceil[min[N,T]/100]1/4. For the parametric (P) version, the order of the VAR(p) model has been selected using the BIC information criterion using a maximum of pmax = 4ceil[T/100]1/4 lags.

  18. 18

    The quartiles are 10-2 times 0.1069, 0.2315 and 0.3684 (Model 2, one structural break), 0.1553, 0.2808 and 0.4934 (Model 5, one structural break), and 0.1122, 0.2165 and 0.3384 (Model 5, two structural breaks).

References

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. Supporting Information
  • Arai Y, Kurozumi E. 2007. Testing for the null hypothesis of cointegration with a structural break. Econometric Reviews 26: 705739.
  • Bai J. 2009. Panel data models with interactive effects. Econometrica 77: 12291279.
  • Bai J, Carrion-i-Silvestre J. 2009. Structural changes, common stochastic trends, and unit roots in panel data. Review of Economic Studies 76: 471501.
    Direct Link:
  • Bai J, Carrion-i-Silvestre J. 2013. Testing panel cointegration with unobservable dynamic common factors that are correlated with the regressors. Econometrics Journal 16: 222249.
  • Bai J, Ng S. 2004. A PANIC attack on unit roots and cointegration. Econometrica 72: 11271177.
  • Bai J, Ng S. 2009. Panel unit root tests with cross-section dependence: a further investigation. Econometric Theory 26: 10881114.
  • Bai, J, Perron P. 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66: 4778.
  • Banerjee A, Carrion-i-Silvestre J. 2004. Breaking panel data cointegration. Working paper. Available: http://www.cass.city.ac.uk/conferences/cfl/index.html [29 July 2013].
  • Banerjee A, Wagner M. 2009. Panel methods to test for unit roots and cointegration. In Palgrave Handbook of Econometrics, Vol. 2: Applied Econometrics, Mills TC, Patterson K (eds). Palgrave Macmillan: Basingstoke, UK.
  • Breitung J, Pesaran MH. 2008. Unit roots and cointegration in panels. In The Econometrics of Panel Data, Matyas L, Sevestre P (eds). Kluwer Academic: Dordrecht.
  • Campa JM, González-Mínguez JM. 2006. Difference in exchange rate pass-through in the euro area. European Economic Review 50: 121145.
  • Campa JM, Goldberg LS, González-Mínguez JM. 2005. Exchange-rate pass-through to import prices in the euro area. NBER working paper 11632.
  • Carrion-i-Silvestre J, Sansó A. 2006. Testing the null of cointegration with structural breaks. Oxford Bulletin of Economics and Statistics 68: 623646.
  • Carrion-i-Silvestre J, Kim D, Perron P. 2009. GLS-based unit root tests with multiple structural breaks under both the null and the alternative hypotheses. Econometric Theory 25: 17541792.
  • Dolado JJ, Gonzalo J, Mayoral M. 2005. What is what? A simple test of long-memory vs. structural breaks in the time domain. Working paper, Open Access Publications from Universidad Carlos III de Madrid.
  • Embrechts P, Klüppelberg C, Mikosch T. 1997. Modelling Extremal Events. Springer: Berlin.
  • Engle RF, Granger CWJ. 1987. Cointegration and error correction: representation, estimation, and testing. Econometrica 55: 251276.
  • Gregory AW, Hansen BE. 1996. Residual-based tests for cointegration in models with regime shifts. Journal of Econometrics 70: 99126.
  • Gutierrez L. 2010. Simple tests for cointegration in panels with structural breaks. Applied Economics Letters 17: 197200.
  • Hansen BE. 1992. Tests for parameter instability in regressions with I(1) processes. Journal of Business and Economic Statistics 10: 321335.
  • Kao C, Chiang MH. 2000. Testing for structural change of a cointegrated regression. Mimeo, Department of Economics, Syracuse University.
  • Kejriwal M, Perron P. 2010. Testing for multiple structural changes in cointegrated regression models. Journal of Business and Economic Statistics 28: 503522.
  • Ng S, Perron P. 1995. Unit root tests in ARMA models with data dependent methods for the selection of the truncation lag. Journal of the American Statistical Association 90: 268281.
  • Pedroni P. 2004. Panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis. Econometric Theory 20: 597625.
  • Perron P. 1989. The great crash, the oil price shock and the unit root hypothesis. Econometrica 57: 13611401.
  • Pesaran MH. 2012. On the interpretation of panel unit root tests. Economics Letters 116: 545546.
  • Pesaran MH. 2013. Testing weak cross-sectional dependence in large panels. Econometric Reviews (forthcoming).
  • Stock JH, Watson MW. 1993. A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 64: 783820.
  • Westerlund J. 2006. Testing for panel cointegration with multiple structural breaks. Oxford Bulletin of Economics and Statistics 68: 101132.
  • Westerlund J, Edgerton DL. 2008. A simple test for cointegration in dependent panels with structural breaks. Oxford Bulletin of Economics and Statistics 70: 665704.
  • Westerlund J, Larsson R. 2009. A note on the pooling of individual PANIC unit root tests. Econometric Theory 25: 18511868.

Supporting Information

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 The Models
  5. 3 Test Statistics
  6. 4 Panel Data Cointegration Tests
  7. 5 Monte Carlo Simulations
  8. 6 Empirical Illustration: Exchange Rate Pass Through in the Euro Area
  9. 7 Conclusions
  10. Acknowledgements
  11. References
  12. Supporting Information

The JAE Data Archive directory is available at http://qed.econ.queensu.ca/jae/datasets/banerjee003/

FilenameFormatSizeDescription
jae2348-sup-0001-appendices.pdfPDF document258KSupporting Information

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.