Local Power of Panel Unit Root Tests Allowing for. Structural Breaks

Transcription

1 Local Power o Panel Unit Root Tests Allowing or Structural Breaks Yiannis Karavias a, and Elias Tzavalis b Abstract The asymptotic local power o least squares based ixed-t panel unit root tests allowing or a structural break in their individual eects and/or incidental trends o the AR panel data model is studied. Limiting distributions o these tests are derived under a sequence o local alternatives and analytic expressions show how their means and variances are unctions o the break date and the time dimension o the panel. The considered tests have non-trivial local power in a /2 neighborhood o unity when the panel data model includes individual intercepts. For panel data models with incidental trends, the power o the tests becomes trivial in this neighborhood. However, this problem does not always appear i the tests allow or serial correlation in the error term and completely vanishes in the presence o cross section correlation. These results show that ixed-t tests have very dierent theoretical properties than their large-t counterparts. Monte Carlo experiments demonstrate the useulness o the asymptotic theory in small samples. JEL classiication: C22, C23 Keywords: Cross section correlation, strong actors, incidental trends, bias correction, ixed T a :School o Economics and Granger Centre or Time Series Econometrics, University o ottingham. University Park, ottingham G7 2RD, UK. Tel: ioannis.karavias@nottingham.ac.uk. b :Department o Economics, Athens University o Economics & Business. etzavalis@aueb.gr :Corresponding author The authors would like to thank the Editor Esandiar Maasoumi, an Associate Editor, two anonynous reerees, Tassos Magdalinos, Robert Taylor and the participants o the 8th International Conerence on Computational and Financial Econometrics or their helpul comments.

2 Introduction There is recently growing interest in developing panel data unit root tests allowing or a break in their deterministic components, namely in their individual eects and/or individual linear trends see, Carrion-i-Silvestre et al. 2005, Harris et al. 2005, Karavias and Tzavalis 204a, 204b, Chan and Pauwels 20, Bai and Carrion-i-Silvestre 2009, Hadri et al. 202 and Pauwels et al As is aptly noted by Perron 989 in the single timeseries literature, not accounting or a break point in the level and/or deterministic trend o economic series can lead to a unit root test which can hardly reject the null hypothesis o unit root rom its alternative o stationarity. Panel unit root tests suer rom this problem too. But, despite the many panel unit root tests proposed in the literature, to our knowledge, there has been no attempt o studying the behaviour o these tests theoretically. This paper constistutes the irst work in this direction. It investigates the power properties o ixed-t panel unit root tests that allow or structural breaks. These tests are appropriate or panels with ew time series observations and many cross-section units, oten met in practice see, e.g., Baltagi The asymptotic theory employed considers the time dimension T as ixed and the cross section one as going to ininity. In particular, the ocus is in the asymptotic local power o two tests proposed in Karavias and Tzavalis 204a and in Karavias and Tzavalis 204b. The irst test generalizes the Harris and Tzavalis tests 999 to allow or a common break, and will be henceorth denoted as HT. The second test denoted as KT allows, in addition to structural breaks, or serial correlation in the error term o the individual series o the panel. One o the contributions o this paper is that it extends these tests to the case o cross sectionally dependent errors, so that ote that a version o the KT test or the case o no structural breaks has been suggested by Kruiniger and Tzavalis 2002, and Moon and Peron 2004 or the case that T is large. 2

3 we can study their power properties in a more general setting. Both the above tests are based on the within groups estimator o the autoregressive coei cient o the AR panel data model, which is just the least squares LS estimator on the transormed AR model. This transormation is necessary or the removal o the individual deterministic terms and the initial conditions o the series, but it renders the LS estimator inconsistent as it induces correlation between the lagged dependent variable and the error term. The HT and KT tests correct or this inconsistency o the LS estimator or simplicity, we will also use the term bias in dierent ways. The HT test corrects or the bias o both the numerator and denominator, while the KT test corrects only or the bias o the numerator. This bias correction is undamentally dierent in the ixed-t and large-t settings. As we show in the paper, it is the main source o the distinct and superior behaviour o the ixed-t tests over the large-t ones, or short panels. In the large-t setting, Moon and Perron 2004 show how the bias o the numerator is a unction o a long run variance, which they estimate using kernel estimators that, as they note, have bad small sample properties. The KT test however corrects or the bias using a ixed T non-parametric estimator based on the covariance matrix estimation method o Abowd and Card 989 and Arellano 990, This method is consistent across the dimension o the panel and has good small sample properties. The paper makes a number o contributions into the literature o panel data unit root tests, which have practical implications. First, it shows that, or the standard panel data model with IID errors and individual intercepts, the HT test has higher asymptotic local power than the KT test. This can be attributed to the act that the HT test does not require a consistent estimator o the variance o the error term, compared to the KT test. The HT test is invariant to this nuisance parameter, as it adjusts the LS estimator or its 3

4 inconsistency o both its numerator and numerator. Second, as with panel unit root tests that do not allow or a break, the HT and KT tests have trivial asymptotic local power i incidental trends are included in the deterministic components o the AR panel data model. The allowance or a break in the deterministic components o this panel data model does not save these tests rom this problem. Third, when short term serial correlation o arbitrary orm is permitted, the KT test can increase its power and, or the panel data model with incidental trends, it has non-trivial asymptotic local power. This is important because large-t panel unit root tests have trivial power in the natural /2 T neighbourhood o unity, when incidental trends are present see Moon et al This rise o the power can be attributed to the interaction between the serial correlation parameters and the ixed T non-parametric estimator o the bias. Finally, the paper extends the two tests or the case that the error term has a strong actor structure. It is shown that both the HT and KT tests have good power properties or the panel data model with individual intercepts. However, or the model with incidental trends, only the KT test is ound to have non-trivial power. This inding is in sharp contrast with the large-t case o the KT test, which has trivial local power or panels with a large cross section dimension. It is shown that this power comes rom the way the LS estimator is bias corrected which is dierent than in the large-t case. The above results are conirmed through a Monte Carlo experiment. This exercise also provides small sample results on the power perormance o the tests and shows the useulness o the asymptotic approximation. The paper is organized as ollows: Section 2 presents the assumptions on the data generating process required by the HT and KT tests. Section 3 derives the limiting distributions o the tests or IID errors. For the KT test allowing or serial correlation eects, this is done in Section 4. Section 5 considers the case o cross section dependence. Section 6 carries 4

5 out the Monte Carlo exercise. Section 7 concludes the paper. All proos are given in the appendix. 2 Models and Assumptions Consider the ollowing AR dynamic panel data models allowing or a common structural break in their deterministic components individual eects and/or individual linear trends at time point λ, or all individual units o the panel i: M: y i = a i e + a 2 i e 2 + ζ i, i =, 2,..,, M2: y i = a i e + a 2 i e 2 + β i τ + β 2 i τ 2 + ζ i, i =,..., where ζ i = ϕζ i, + u i, ϕ, ], y i = y i,,..., y i,t and y i, = y i,0,..., y i,t are T vectors, u i = u i,,..., u i,t is the T vector o error terms u i,t, a i and β i denote the individual eects and slope coei - cients o the linear incidental trends o the panel. In particular, a i is deined as a i = a i i t T 0 and a i = a 2 i i t > T 0, while e and e 2 are T -column vectors deined as ollows: e t = i t T 0 and 0 otherwise, and e 2 t = i t > T 0 and 0 otherwise. Slope coei cients β i are deined as β i = β i i t T 0 and β i = β 2 i i t > T 0, while τ and τ 2 are T -column vectors deined as ollows: τ t = t i t T 0, and zero otherwise, and τ 2 t = t i t > T 0, and zero otherwise. Throughout the paper, we will denote the set o possible dates that the break can occurs with I λ and the break raction with λ = T 0 /T, i.e. λ I λ. 5

6 The above models nest in the same ramework both the null hypothesis o unit roots in ϕ, i.e., ϕ =, and its alternative o stationarity, ϕ <. They can be written in a non-linear orm as ollows: y i = ϕy i, + ϕa i e + a 2 i e 2 + u i, i =, 2,..., and y i = ϕy i, + ϕβ i e + ϕβ 2 i e 2 + ϕa i e + a 2 i e 2 + ϕβ i τ + β 2 i τ 2 + u i, respectively. The within group least squares LS known also as least squares dummy variables LSDV estimator o autoregressive coei cient ϕ o the models can be written as ollows: ˆϕ λ = y i, Q λ y i, y i, Q λ y i, i= i= where Q λ is the T T within transormation annihilator matrix o the individual series o the panel y i,t. Q λ is deined as Q λ = I X λ X λ X λ X λ, where X λ = e, e 2 or model M and X λ = e, e 2, τ, τ 2 or model M2. I denotes the T T identity matrix. The within transormation o the data wipes o the individual eects and/or incidental trends o the panel, as well as its initial conditions y i,0, but it results in an inconsistent estimator because it induces correlation between the transormed error and the transormed lagged dependent variable. Thus, ixed-t panel unit root tests based on it must rely on a correction o estimator ˆϕ λ or its inconsistency asymptotic bias see, e.g., Harris and Tzavalis 999, To study the asymptotic local power o these tests, deine the autoregressive coei cient ϕ as ϕ = c/. Then, the hypotheses o interest become H 0 : c = 0 and H a : c > 0, 6

7 where c is the local to unity parameter. The limiting distributions o the tests based on LSDV estimator ˆϕ λ will be derived under the sequence o local alternatives ϕ, by making the ollowing general assumptions: Assumption A: a {u i }, i {, 2,..., }, constitutes a sequence o T independent random vectors with means Eu i = 0 and all mixed 4+δ moments are inite. a2 Eu i u i = Γ i with γ i,ts = Eu i,t u i,s = 0 or t < s and s = t + p i +,..., T, where p i denotes the order o serial correlation or each i and p i p max. a3 Deine Γ = / i= Γ i or which it holds that lim Γ Γ i = lim i= Γ i Γi = 0 and also assume that matrix Γ = lim Γ is positive deinite. a4 The error terms u i,t is independent o a i, a 2 i V ary i,0 < +. Assumption B: p max = [T/2 2] or model M and T 2 p max = 3 i T is even and T 0 = T/2, min{t 0 2, T T 0 2} otherwise and y i,0, or all i, and or model M2, where [.] denotes the greatest integer unction. Assumption C: b β i and β 2 i are sequences o independent random variables with inite 4+δ moments. They are also independent rom u i. b2 lim maxeβ j2 i / β j2 = 0, where β j2 = / i= Eβj2 i or j =, 2. Also, β j2 = lim βj2 is inite. Assumption D: The break raction λ I λ = {2/T, 3/T,..., T /T } or model M and λ I λ = {2/T, 3/T,..., T 2/T } or model M2. Assumption A enables us to derive the limiting distribution o the ixed-t panel data unit root tests o Karavias and Tzavalis 204a or Γ i = σ 2 i I. These tests denoted as HT extend those o Harris and Tzavalis 999 or the case o a common break in the deterministic 7

8 components o models M and M2. It also allows the derivation o this limiting distribution or Karavias and Tzavalis 204b ixed-t panel data unit root test denoted as KT, which allows or a structural break under heteroscedasticity and serial correlation o error terms u i,t, or both models M and M2. Condition a states that u i is mean zero and that, element-wise, all possible 4 + δ moments o it are inite. Condition a2 allows u i,t to have dierent types o heteroscedasticity and serial correlation across the cross section units and the time dimension o the panel. However, there is a common bound to the order o serial correlation see also Assumption B. I Γ i = σ 2 I or all i, then a2 is consistent with the assumption o Karavias and Tzavalis 204a panel data unit root tests, considering the simpler case o u i IID0, σ 2 I. Condition a3 states that no individual variance o u i,t is big enough to dominate the rest. This is an assumption required by the Lindeberg-Feller CLT. Finally, condition a4 imposes independence between the parameters o the series o the panel y i,t and the innovations. It also implies that V ary i,0 < + which is consistent with assumptions like constant, random and mean stationary initial conditions y i,0. Covariance stationarity o y i,0, implying V ary i,0 = σ 2 / ϕ 2 see Kruiniger 2008 and Madsen 200 is not considered. This is because, as is also aptly noted by Moon et al. 2007, this assumption implies that V ary i,0 when ϕ, which means that the variance o the initial condition increases with the number o cross-section units. This is not meaningul or cross-section data sets. Assumption B determines the maximum allowable order o serial correlation because o its interaction with the structural break. We chose to restrict the order o serial correlation and let the break date ree, rather than the opposite. Finally, Assumption C is relevant only or the case o the KT test or model M2. Conditions c and c2 guarantee that 8

9 β i and β 2 i, which appear in the estimator o the bias correction, obey the Lindeberg- Feller CLT. Assumption D determines the possible break points. An advantage o the HT and KT tests is that the trimming o the sample depends on the deterministic speciication o the panel data models M and M2, and it is less severe than that assumed by single time series unit root tests allowing or breaks, i.e. or M, I λ = {2/T, 3/T,..., T /T } and, thereore, only the irst and last dates are trimmed out as opposed to the {0.5, 0.85} interval, advocated in Andrews 993. To study the asymptotic local power o the tests, we will rely on the slope parameter, denoted as k, o local power unctions o the orm Φz a + ck, where Φ is the standard normal cumulative distribution unction and z a denotes the α-level percentile. Since Φ is strictly monotonic, a larger k means greater power or the same value o c. I k is positive, then the tests will have non-trivial power. I it is zero, they will have trivial power, which is equal to a, and, inally, i c < 0 they will be biased. 3 The limiting distribution o the tests i u i IID0, σ 2 I This section presents the limiting distribution o the HT and KT test statistics under the sequence o local alternatives ϕ = c/ when u i IID0, σ 2 I. This is a special case o Assumption A where p max = 0. The assumption o normality is made only or convenience, because in this case we can calculate the analytic ormula o the HT test statistic variance. For ease o exposition, it is also assumed that β j i are IID. This is a special case 9

10 o Assumption C. As mentioned beore, the HT test corrects both the numerator and the denominator o the LS estimator ˆϕ λ or its inconsistency, while the KT test corrects only the numerator o ˆϕ λ. This enables the KT test to be easily extended to allow or more general dependence structures o error term u i. 3. Model M For model M, the HT test allowing or a break is based on the ollowing statistic: Z λ HT = V λ /2 HT ˆϕ λ Bλ, where Bλ = p limˆϕ λ = trλ Q λ /trλ Q λ Λ is the inconsistency o LS estimator ˆϕ λ under null hypothesis H 0 : c = 0, where Λ is a T T dimension matrix having unities at its lower than its main diagonals and zeroes elsewhere. V λ HT = 2trAλ2 HT /trλ Q λ Λ 2, with A λ HT = /2Λ Q λ + Q λ Λ BλΛ Q λ Λ, is the variance o the limiting distribution o the corrected or its inconsistency LS estimator ˆϕ λ, i.e. ˆϕ λ Bλ. The KT test is based on the ollowing statistic: KT = V ˆδλ λ /2 KT ˆϕ λ Z λ ˆb λ ˆδλ, where ˆb λ /ˆδ λ ˆσ 2 trλ Q λ / i= y i, Q λ y i, is a consistent estimator o the bias o ˆϕ λ based on a consistent estimator o the bias o the numerator, namely ˆσ 2 trλ Q λ, divided by ˆδ λ which is the denominator o ˆϕ λ. V λ KT = 2σ4 tra λ2 KT, with Aλ KT = /2Λ Q λ + Q λ Λ Ψ λ Ψ λ, is the variance o the limiting distribution o ˆϕ ˆδ λ λ ˆb λ /ˆδ λ and Ψ λ is a T T dimension matrix having in its main diagonal the corresponding elements 0

11 o matrix Λ Q λ and zeros elsewhere. In implementing the KT test statistic, note that a consistent estimator o σ 2 under H 0 : c = 0 is given by ˆσ 2 = [ / trψ λ ] i= trψλ y i y i, where y i = y i y i,. Matrix Ψ λ implies that trψ λ = trλ Q λ. It is designed so as, in adjusting the numerator o the estimator ˆϕ λ or its inconsistency, some sample inormation is let to test the null hypothesis. In particular, Ψ λ has two properties. First, it restricts the estimator / i= y i y i, which constitutes a consistent estimator o σ 2 I based on all the available sample inormation, to its main diagonal. 2 This restriction uses inormation coming only rom contemporaneous observations and, because there is no inormation about σ 2 in the o-diagonal elements o σ 2 I, it preserves then consistency o the estimator. Second, matrix Ψ λ weights the diagonal elements o / i= y i y i in the same way that are weighted by trλ Q λ, mimicking the part o the bias which is due to the within transormation matrix. In the next theorem, we give the limiting distribution o the HT and KT test statistics or model M, under the sequence o local alternatives ϕ = c/. Theorem For model M, let Assumptions A and D hold and u i IID0, σ 2 I. Then, under ϕ = c/, we have and V λ /2 HT ˆϕ λ Bλ V ˆδλ λ /2 KT ˆϕ λ d ck HT, ˆb λ ˆδλ d ck KT,, 2 otice that, under H 0 : c = 0, we have y i = u i and, thus, p lim ˆσ 2 = p lim[/trψ λ ] i= trψλ y i y i = σ2 trλ Q λ /trψ λ = σ 2, since trψ λ = trλ Q λ.

12 as, where k HT = T T 2 [ T 2 3λ 2 3λ + ] T 4 Φ + T 2 Φ T 2 2λ 2 2λ + 8 T 6 R + T 5 R 2 + T 4 R 3 + T 2 R T 36 and k KT = 3T 2, T 2 2λ 2 2λ + + 6T T +2λ λt λ λ where R, R 2, R 3, R 4 and Φ, Φ 2 are polynomials o λ deined in the appendix see proo o the theorem. The limiting distributions given by Theorem imply that the asymptotic local power unction o test statistics HT and KT depend on the values o slope parameters k HT and k KT, respectively. In Table, we present values o these parameters, or dierent values o T and λ. The results o this table indicate that the asymptotic local power behaviour o the two test statistics is dierent. The HT statistic has much higher power than the KT. The power o this statistic is much bigger when the break is in the beginning, or towards the end o the sample, i.e., or λ = {0.25, 0.75}. 3 On the other hand, the power o test statistic KT reaches its maximum point when the break is in the middle o the sample, λ = {0.50}. The power o statistic HT increases with T, i.e., k HT = OT. The power o the KT test increases with T, but or relatively small T. As T grows large, the test has no power gains. This can be seen rom lim T k KT = 3/ 2λ 2 2λ +, which is independent o T. These results can be more clearly seen by the three-dimension Figures and 2, presenting values o k HT and k KT, or dierent values o λ and T. The above dierences between test statistics HT and KT can be attributed to the way 3 Analogous evidence is provided or single time series unit root tests allowing or breaks, based on a model selection Bayesian approach see Meligkotsidou et al

13 that each o them corrects or the inconsistency o the LS estimator ˆϕ λ. As mentioned beore, HT is based on a correction o LS estimator ˆϕ λ or the inconsistency o both its numerator and denominator. On the other hand, the KT test statistic is based on an adjustment o estimator ˆϕ λ only or the inconsistency o its numerator, which additionally requires a consistent estimator o the variance o error term u i,t, σ 2. The later reduces the local power o the test. Finally, another result o Theorem is that, under the sequence o local alternatives considered, the break unction parameters do not enter the asymptotic distribution o both test statistics HT and KT. Thus, the magnitude o the break does not aect local power o the tests. Furthermore, local power is also robust, asymptotically, to the initial conditions o the panel y i,0, which means that their magnitudes also do not aect the power o the test see also Harvey and Leybourne 2005 and Harris et al Scaling appropriately test statistics HT and KT by T and assuming that T,, with /T 0, it can be shown see appendix that, under ϕ,t = c/ T, the limiting distributions o the large-t versions o these statistics are given as ollows: Corollary For model M, let Assumptions A and D hold and u i IID0, σ 2 I. Then, under ϕ,t = c/ T, we have and V λ /2 HT T ˆϕ λ Bλ V λ /2 ˆδλ KT T ˆϕ λ L ckht,, ˆb λ ˆδλ d ck KT,, as T,, with /T 0, where kht = 3λ2 3λ + Φ 42λ 2 and kkt = 0, 2λ + R 3

14 and V λ HT = 36R λ Φ 2λ 2 and V 2λ + 2 KT p / lim ˆδλ = 362λ4 4λ 3 + 3λ 2 λ,t 2λ λ2λ 2 2λ +, 2 respectively, denote the local power slope coei cients and the variances o the limiting distributions o the large-t versions o the HT and KT test statistics. Values o power slope coei cients k HT and k KT, or dierent values o λ, are reported in Table 2. These indicate that, in contrast to the HT test, the large-t extension o test statistic KT does not have asymptotic local power. 4 Thus, the KT test can be thought o as more appropriate or short panels. The results o the table also indicate that the large-t extension o the HT test has less power than its ixed-t version. We have also ound that power takes its highest values in the beginning and towards the end o the sample, i.e., or λ = {0.0, 0.90}, as with its ixed-t version. The smaller power o the large-t versions o test statistics HT and KT, compared to their ixed-t ones, can be attributed to the aster rate o convergence o the alternative hypotheses to the null, i.e. ϕ,t = c/ T compared to ϕ = c/ see also Harris et al The test statistics given by Theorem and Corollary can be readily applied in practice. To this end, or test statistic HT, irst the annihilator matrix Q λ must be built, where Q λ is a deterministic matrix based on vectors e and e 2. Then, given Λ which is a ixed matrix and Ψ λ which is a restricted orm o Λ Q λ, the bias Bλ = trλ Q λ /trλ Q λ Λ and matrix A λ HT = /2Λ Q λ + Q λ Λ BλΛ Q λ Λ can be easily calculated. With these quantities at hand, the last step is to calculate the LS estimator and the Z λ HT test 4 ote that an analogous result has been derived by Moon and Perron 2008 or this test in the case o no break. 4

15 statistic. Similar steps to those above can be ollowed or the application o test statistic KT. ˆσ 2 = [ / trψ λ ] i= trψλ y i y i must also be calculated as part o ˆb λ λ. ˆδ is the denominator o the LS estimator. As a inal note, i error terms u i,t are non-normal as is the most probable case, only the ormula o the variance o statistic HT changes in the [ above theorems. This becomes V λ HT = ] k 4 j= aλ2 HT,jj + 2σ4 tra λ2 HT / [ σ 2 trλ Q λ Λ ] 2, where A λ HT = [aλ HT,ij ] and parameter k 4 can be estimated as in Harris and Tzavalis I the date o the break is unknown, as oten assumed in practice, then to test null hypothesis H 0 : c = 0 we may either estimate it as in Bai 200 or rely on the minimum values o the known break test statistics HT and KT over all possible break points o the sample, denoted respectively as min λ Iλ Z λ HT and min λ I λ Z λ KT. As shown in Karavias and Tzavalis 204a, the limiting distributions o these test statistics behave like the minimum o a ixed number o correlated normal variables. The pd o this minimum is given by Karavias and Tzavalis 204b. Because inverting this pd is a numerical problem and because the results are qualitatively similar to those o the known date break tests presented above, we do not pursue this issue any urther. 3.2 Model M2 For model M 2, which additionally considers incidental trends in the deterministic components o panel data series y i,t, the HT and KT test statistics are deined analogously to those or model M. Test statistic HT admits the same ormulas, but now matrix Q λ is based on X λ = e, e 2, τ, τ 2. Bλ = p limˆϕ λ = trλ Q λ /trλ Q λ Λ denotes the inconsistency o LS estimator ˆϕ λ, or model M2 and V λ HT = 2trAλ2 HT /trλ Q λ Λ 2, with A λ HT = 2 Λ Q λ + Q λ Λ BλΛ Q λ Λ, is the variance o the limiting distribution o 5

16 ˆϕ λ Bλ. However, or test statistic KT, ˆσ 2 = [ / trψ λ ] i= tr Ψ λ y i y i is no longer a consistent estimator o σ 2 in the case o model M2, due to the presence o individual coei cients eects β i under null hypothesis H 0 : c = 0. These imply that y i = β i e + β 2 i e 2 + u i and it can be easily seen that p lim i= y i y i = Eβ 2 i e e + Eβ 2 i e 2 e 2 + σ 2 I. 2 To render test statistic KT invariant to nuisance parameters β i, Karavias and Tzavalis 204b suggested the ollowing estimator o σ 2 : ˆσ 2 = trθ λ tr Θ λ y i y i, i= with Θ λ = Ψ λ trψλ e e trm e e M trψλ e 2 e 2 trm 2 e 2 e 2 M 2, 3 where Ψ λ is deined as beore i.e., it is a T T diagonal matrix having in its main diagonal the elements o the main diagonal o the matrix Λ Q λ, M = e e diag{e e, 0} and M 2 = e 2 e 2 diag{e 2 e 2, 0}, where diag{e r e r, p}, r = {, 2}, denotes two selection matrices which have zeros everywhere except rom their main and p upper and p lower diagonals in which they have the elements o the matrices e r e r. Matrices M r, or r = {, 2}, select the elements o p lim / i= y i y i containing individual eects Eβ r2 i. In particular, matrices M and M 2 respectively select the o-diagonal elements o the right hand side o 2 where nuisance parameters Eβ 2 i and Eβ 22 i reside. This can be seen by noticing that p lim trm i= y i y i/trm e e = Eβ 2 i and 6

17 p lim trm 2 i= y i y i/trm 2 e 2 e 2 = Eβ 22 i. Thus, these matices are used in the adjustment o the LS estimator ˆϕ λ or its inconsistency to render the limiting distribution o this estimator net o the individual eects β i. Having deined matrices M r, one can see that Θ λ, given by 3, plays the same role that Ψ λ does or test statistic KT in the case o model M. However, in addition to rendering the limiting distribution o ˆϕ λ ˆb λ /ˆδ λ net o the diagonal elements o σ 2 I which is done through matrix Ψ λ, matrix Θ λ also makes this limiting distribution net o individual eects Eβ r2 i, or r = {, 2}. The latter is done through matrices [ trψ λ e e /trm e e ] M and [ trψ λ e 2 e 2 /trm 2 e 2 e 2 ] M 2. Given the deinition o Θ λ, the bias adjustment unction and the variance o the limiting distribution o ˆϕ λ ˆb λ /ˆδ λ will be respectively given as ollows: ˆb λ /ˆδ λ = trθ λ / i= y i y i/ i= y i, Q λ y i, and V λ KT = vecqλ Λ Θ λ ΠvecQ λ Λ Θ λ, with ˆΠ = / i= vec y i y ivec y i y i. The next theorem derives the limiting distribution o test statistics HT and KT or model M2 under the sequence o local alternatives ϕ = c. Theorem 2 For model M2, let Assumptions A, C and D hold with u i IID0, σ 2 I and β j i be IID, or j =, 2. Then, under ϕ = c, we have V λ /2 HT ˆϕ λ L Bλ ck HT, and V ˆδλ λ /2 KT ˆϕ λ ˆb λ ˆδλ d ck KT,, as, where k HT = 0 and k KT = 0. 7

18 The results o the theorem indicate that the well-known incidental trends problem o panel data unit root tests see e.g. Moon et al also exists even i the tests allow or a break and T is ixed. Both, the HT and KT test statistics have trivial power, or model M2. This result also holds or the case that T grows large and is established in the next corollary. Corollary 2 For model M2, let Assumptions A, C and D hold with u i IID0, σ 2 I and β r i be IID, or r =, 2. Then, under ϕ,t = c/ T, we have and V λ /2 HT T ˆϕ λ Bλ V λ /2 ˆδλ KT T ˆϕ λ λ ˆb ˆδλ L ckht,, d ck KT,, as T,, with /T 0, with k HT = 0 and k KT = 0, 4 where k HT and k KT denote the local power slope coei cients o the large-t versions o the HT and KT test statistics. The implementation o the test statistics given by Theorem 2 or Corollary 2, or model M2, ollows the same steps to those or the test statistics or model M, given by Theorem or Corollary. More speciically, or test statistic HT matrix Q λ, with X λ = e, e 2, τ, τ 2, must be employed. For test statistic KT, the ixed matrices M r, or r = {, 2}, and Θ λ must be built. Then, the variance unction V λ /2 KT must be calculated, by plugging in the estimator ˆΠ = / i= vec y i y ivec y i y i. 8

19 4 Power o the KT tests i error terms u i are serially correlated In this section, we consider the case that the variance-covariance matrix o the vector o error terms u i has a more general orm than Γ = σ 2 I, assumed in the previous section. That is, we assume that Γ i = [γ i,ts ], where γ i,ts = Eu i,t u i,s = 0 or s = t + p max +,..., T and t < s. This means that errors u i,t allow or heteroscedasticity and serial correlation o maximum lag order p = max p i, or i =,...,. We urther allow or cross-sectional heterogeneity, by allowing the type o heteroscedasticity and serial correlation to change with i. The order o serial correlation p, considered may dier across the units o the panel, but we still impose the ollowing bound or it: p p max, which is smaller than T and is determined in Assumption B. This assumption provides some common nuisance parameter ree moments which are exploited in the cross section dimension o the panel. We no longer impose the IID assumption on incidental trends slope coei cients β r i, used or simplicity in the previous section. These less restrictive assumptions enable us to investigate the combined eects o a structural break and serial correlation in u i,t on the asymptotic local power o panel unit root tests. As only the KT test is extended to allow or serially correlated errors u i,t see, e.g., Karavias and Tzavalis 204b, our analysis will be ocused on this test. For both models M and M2, the KT test statistic under the above assumptions about u i has analogous orms to those presented in the previous section. What changes is that, in order to take into account the p-th order serial correlation in u i,t which appears in the p-upper and p-lower secondary diagonals o matrix Γ, the T T selection matrix Ψ λ now is deined as having in its main diagonal and its p-lower and p-upper diagonals the corresponding elements o matrix Λ Q λ, and zeroes elsewhere. Similarly, M and M 2, deined beore, will 9

20 have zeroes in their p-lower and p-upper diagonals as opposed to zeroes only in their main diagonal, assumed or the simple case o Γ = σ 2 I, i.e. M r = e r e r diag{e r e r, p}. This is needed, because in these diagonals the trend nuisance parameters appear together with the higher order serial correlation nuisance parameters. For the M model, which assumes that X λ = e, e 2, the inconsistency o LS estimator ˆϕ λ under null hypothesis H 0 : c = 0 and Assumptions A, B and D is given by trλ Q λ Γ/trΛ Q λ ΛΓ, since p lim ˆϕ λ trλ Q λ Γ/trΛ Q λ Λ Γ = 0. This ormula o the inconsistency o ˆϕ λ indicates that in order to correct ˆϕ λ we need an estimator o matrix Γ. To this end, deine the ollowing estimator: ˆΓ = / i= y i y i. ] Then, by Chebyshev s Weak Law o Large umbers, we have p lim [ˆΓ Γ = 0 and, [ thus, p lim trψ λˆγ trλ Q λ Γ ] = 0. The last result implies that ˆϕ λ can be adjusted or its inconsistency, by deining ˆb λ as ˆbλ = trψ λˆγ. Then, test statistic ˆϕ λ ˆb λ /ˆδ λ λ will be centred around zero, where ˆδ is the denominator o ˆϕ λ. The variance unction o this statistic is given by V λ KT = 2tr A λ KT, Γ2 where A λ KT = /2Λ Q λ + Q λ Λ Ψ λ Ψ λ. The above ormula o the inconsistency o LS estimator ˆϕ λ, i.e., trλ Q λ Γ/trΛ Q λ ΛΓ also holds or model M2, which assumes that X λ = e, e 2, τ, τ 2. Under the above assumptions and H 0 : c = 0, it can be shown p lim ˆϕ λ trλ Q λ Γ/trΛ Q λ Λ Γ = 0. However, ˆΓ is an inconsistent estimator o Γ due to the presence o β r i under the null hypothesis. It can be easily shown that p lim [ˆΓ Γ β2 e e β 22 e 2 e 2 ] = 0. In this case, to adjust LS estimator ˆϕ λ or its inconsistency, due to nuisance parameters β r i and the ] presence o serial correlation in u i,t both implying p lim [ˆΓ Γ 0, we will employ matrix Θ λ. This matrix now is based on the modiied or p-order serial correlation matrices Ψ λ and M r, or r = {, 2}, deined above. Then, p lim [trm rˆγ/trm r e r e r β r2] = 20

21 0, or r = {, 2} and, hence, p lim [ trθ λˆγ trλ Q λ Γ ] = 0. The last result establishes that ˆb λ = trθ λˆγ constitutes a consistent estimator o the bias o the numerator o ˆϕ λ, and thus statistic ˆϕ λ ˆb λ /ˆδ λ is centred around 0. Its variance has the same ormula as beore, i.e., V λ KT = vecqλ Λ Θ λ ΠvecQ λ Λ Θ λ, with ˆΠ = / i= vec y i y ivec y i y i. In the next theorem, we provide the limiting distribution o test statistic KT under the sequence o local alternatives ϕ = c/, or model M allowing or serial correlation in u i,t. Theorem 3 For model M, let Assumptions A, B, and D hold. Then, under ϕ = c/, we have V ˆδλ ˆb λ /2 KT ˆϕ λ λ d ck KT,, or model M, ˆδλ as, with k KT = trf Q λ Γ + trλ Q λ ΛΓ trψ λ ΛΓ trλ Ψ λ Γ. 2trA λ KT Γ2 where F is a T T deterministic matrix independent o the order o serial correlation, deined in the Appendix. The results o the theorem indicate that the asymptotic local power o the KT test now depends also on the values o the variance-covariance parameters γ i,ts, aecting the power slope parameter k KT. This can increase, or reduce, the local power o the test depending on the sign and orm o γ i,ts. To see this more clearly, in Panel A o Table 3 we present estimates o the power slope parameter k KT assuming that error terms u i,t ollow a MA 2

22 process: u i,t = ε i,t + θε i,t, where ε i,t IID0, σ 2 ε. ote that the table also considers the case that θ = 0 i.e., there is no serial correlation in u i,t, but test statistic KT allows or serial correlation o order p =. This case can show i this test loses signiicant power i a higher order o serial correlation p is assumed than the correct one. The results o the table also show that the KT test has always power i θ 0. The inding that the test has power even i θ = 0, or all cases o T 0 considered, indicates that it may be applied to test or unit roots even i higher than the correct order o serial correlation is assumed. 5 As was expected, the power o the test in this case is always less, compared to that when the correct lag order p = 0 is considered. This happens because in this case the test exploits less moment conditions in drawing inerence about unit roots, by assuming p = when θ = 0. Another conclusion that can be drawn rom the results o the table is that, when θ > 0, the power o test statistic KT becomes bigger than that o its version which does not allow or serial correlation u i,t, presented in the previous section see Table 2. We have ound that this result can be mainly attributed to the presence o selection matrix Ψ λ in terms trψ λ ΛΓ and trλ Ψ λ Γ o the unction o the slope coei cient k KT, given by Theorem 3. These terms have a positive eect on k KT i.e., trψ λ ΛΓ + trλ Ψ λ Γ < 0 when θ > 0 and a negative eect when θ < 0 i.e., trψ λ ΛΓ + trλ Ψ λ Γ > 0. 6 As T increases, the above sign eects o θ on the KT test are ampliied. These power gains o the KT test or model M, when θ > 0, may be also attributed to the act that a positive value o θ adds to 5 We have ound that this is true even or p >. 6 The sum o traces trf Q λ Γ + trλ Q λ ΛΓ aects the power o the KT test, too. However, because this constitutes a parabola unction which opens upwards, its eect on k KT is almost symmetrical with respect to the sign o θ. Thus, the relationship between k KT and θ is mainly determined by trψ λ ΛΓ+trΛ Ψ λ Γ. 22

23 the variability o individual panel series y i,t driving urther away the limiting distributions o the test under the null and alternative hypotheses. For model M2, the limiting distributions o test statistic KT under ϕ = c/ and serially correlated error terms u i,t are given in the next theorem. Theorem 4 For model M2, let Assumptions A, B, C and D hold. Then, under ϕ = c/, we have V ˆδλ λ /2 KT ˆϕ λ ˆb λ ˆδλ d ck KT,, as, where k KT = trf Q λ Γ + trλ Q λ ΛΓ trθ λ ΛΓ trλ Θ λ Γ. V λ KT The above theorem shows that, i we allow or serial correlation in u i,t, the KT test can have non-trivial power even in the case o incidental trends. Panel B o Table 3 presents values o k KT or the case that u i,t = ε i,t + θε i,t. This is done or dierent values o θ and T. As in Panel A, we also consider the case that θ = 0. The results o Panel B o Table 3 indicate that, or model M2, test statistic KT has non-trivial power only i θ < 0. I θ = 0, the test has trivial power, while, or θ > 0 the test is biased. For θ < 0, the power o the test increases with T. For a given T, it becomes bigger i the break point T 0 is located towards the end o the sample, i.e. λ = These results are in contrast to those or model M, presented in Panel A, where the KT test is ound to have more power i θ > 0. This can be attributed to the trace terms on the power slope parameter k KT and, in particular, trθ λ ΛΓ and trλ Θ λ Γ. Evaluations o these terms show that 23

24 negative values o θ mitigate the power reduction eects coming rom the detrending o the individual panel series. In contrast to model M, this now happens only when θ < 0. To see this more clearly, notice the identity trf Q λ +trλ Q λ Λ+trΛ Q λ = 0 which holds or all models. I p = 0 we have trθ λ Λ+trΛ Θ λ = trλ Q λ and thus, the numerator o k KT becomes 0. However, i p > 0, then trθ λ ΛΓ + trλ Θ λ Γ trλ Q λ Γ and thus the numerator is non-zero see also the proo o Theorems 2 and 4. 7 The above analysis indicates that the power o the KT test can be attributed to the properties o selection matrix Θ λ, when p 0. To implement Theorems 3 and 4 one must irst speciy the deterministic matrices Ψ λ and Θ λ which depend on the appropriate order o serial correlation. Ψ λ is a restricted version o Λ Q λ while Θ λ requires the deterministic selection matrices M r, or r = {, 2}, where all o these quantities are deined above. Then one must calculate estimators ˆΓ = / i= y i y i and ˆΠ = / i= vec y i y ivec y i y i. With these at hand, next we can calculate ˆb λ = trψ λˆγ or Theorem 3 and ˆbλ = trθ λˆγ or Theorem λ 4. The variances can be calculated accordingly, i.e. ˆV KT = 2tr A λ ˆΓ KT 2, where A λ KT = /2Λ Q λ +Q λ Λ Ψ λ Ψ λ or Theorem 3 and ˆV λ KT = vecqλ Λ Θ λ ˆΠvecQ λ Λ Θ λ or Theorem 4. It is straightorward to show that k KT = O in the ixed-t case and, because o the 7 I p = 0, trθ λ Λ = trψ λ Λ trψ λ e e trm Λ trm e e trψλ e 2 e 2 trm 2 Λ trm 2 e 2 e 2 = = trψλ e e 2 = trλ Q λ. 2 trψλ e 2 e 2 2 because trm j e j e j = 2trM j Λ and trψ λ Λ = 0. 24

25 required scaling o test statistic KT when T is asymptotic, the test always has trivial power. This is true or both models M and M2, and under serially correlated errors. In act, the test has zero local power in the /2 T neighbourhood o unity. This result corresponds to that o Moon and Perron 2004, denoted as MP, considering the case o test statistic KT without breaks and large T. Although the two tests have the same asymptotic local power or large T, they may have dierent properties in small samples. This can be attributed to the act that they rely on a dierent type o bias correction o LS estimator ˆϕ λ. The MP test statistic assumes that error terms u i,t are given as u i,t = j=0 d i,jε i,t j, subject to usual restrictions. This structure o u i,t results in an asymptotic bias o ϕ λ which equals a unction o the one sided long run variance o u i,t, given as λ e,i = l= j=0 d i,jd i,j+l. Estimating this long run variance requires imposing urther assumptions that ensure the consistency o the kernel estimators employed. For test statistic KT, the bias adjustment o ˆϕ λ relies on selection matrices Ψ λ, M and M 2, which have zero and non-zero diagonals based on the order o serial correlation p. This test does not require an estimate o the long-run variance o u i,t, which may be proved problematic in small samples see, e.g., Moon and Perron To see this more clearly, consider the case o model M, where Ψ λ has its main, its p upper and its p lower diagonals non-zero, catching all the non-zero elements o variance-covariance matrix Γ. Then, note that the not-yet standardized test statistic KT can be written as ollows: ˆb ˆδλ ˆϕ λ λ ˆδλ = = ˆδ λ y i, Q λ y i i= y i, Qλ y i, i= i= y iψ λ y i i= y i, Qλ y i, u iλ Q λ Ψ λ u i, 5 i= 25

26 where Λ Q λ Ψ λ is a matrix whose main diagonal, its p upper and its p lower diagonals are zero. This means that the quadratic orm u iλ Q λ Ψ λ u i is the sum o products o the ollowing orm: u i,t u i,t+j, where j > p max. These products have means and variances which are ree rom the serial correlation nuisance parameters. Denoting Λ Q λ Ψ λ = C λ, with elements C λ = [c λ k,j ] or k, j =,..., T, the test statistic given by 5 can be written as u iλ Q λ Ψ λ u i = i= i= T u i,t u i,j c λ j,t, t= j / ĉ where ĉ = [t p max, t + p max ] and c λ j,t are known deterministic quantities, by construction. The limiting distribution o this statistic can be ound by applying the central limit theorem to a scaled version o i= T t= j / ĉ u i,tu i,j c λ j,t. The ollowing corollary gives the limiting distribution o the non-standardised version o KT test statistic or the cases that: i and ii,t, respectively, based on the above representation o i= u iλ Q λ Ψ λ u i. Corollary 3 For model M, let Assumptions A, B and D hold. Then, under null hypothesis H 0 : c = 0, we have: i as, ˆb ˆδλ ˆϕ λ λ T d 0, 4σ 4 ˆδλ t= j>t c λ2 j,t and ii as,t jointly, T ˆδ λ ˆb λ ˆϕ λ ˆδλ d 0, p lim,t [ 4σ 4 ] T c λ2 j,t /T 2. t= j>t The results o this corollary show that the unknown nuisance long run variances, which 26

27 appear in the Moon and Perron 2004 version o the test statistic, are replaced by the known weights c λ j,t in test statistic KT. These weights are unctions o known quantities, which depend on the individual deterministic components o panel series y i,t. This eature o the tests skips the problem o estimating the long run variance which is a dii cult econometric task in small samples, as noted above. 5 Common actors In this section, we extend the HT and KT test statistics to allow or cross sectional dependence in error terms u i,t taking the orm o common actors. The assumption o cross section independence may be restrictive in panel data macroeconomic studies see, e.g., Saraidis and Wansbeek 202. There are only a ew studies examining the eect o common actors on the local power o unit root tests. Hansen 995, in the single time series literature, considers additional exogenous covariates which lead to more powerul unit root tests. On the other hand, Moon and Perron 2004, who examine the local power o a large-t version o the KT test, ind that power is unaected by the presence o common actors in u i,t. In our analysis, we consider the common actors to be known observed. This is without loss o generality, as our results would be qualitatively the same even i the common actors had to be estimated in a irst step as in Moon and Perron Our aim is to explore the impact o cross section dependence on the power o test statistics HT and KT. Consider the ollowing speciications o models M and M2 including a single common actor : M: y i = a i e + a 2 i e 2 + ζ i, i =, 2,..,, and 27

28 M2: y i = a i e + a 2 i e 2 + β i τ + β 2 i τ 2 + ζ i, with ζ i = ϕζ i, + ε i, and ε i = ξ i + ξ 2 i 2 + u i, where = t i t T 0, and zero otherwise, and 2 = 2 t i t > T 0, and zero otherwise. 8 That is, we assume that there is a common actor in errors u i,t which also undergoes a structural break at the same time. For ξ i = ξ 2 i, both models M and M2 can consider the case that there is no break in the common actor process. Also note that the assumption that there is only one common actor is not restrictive, and it is made only or ease o exposition. A more general speciication would be F ξ i +F 2 ξ 2 i, where ξ j i = ξ j,i,..., ξj K,i is a K vector o actor loadings and F j = j,..., j is a T K matrix o K observed actors, or j =, 2. The reduced orm o the above models can be written as ollows: K M : y i = ϕy i, + ϕa i e + a 2 i e 2 + ξ i + ξ 2 i 2 + u i, and 2 [ M2 : y i = ϕy i, + j= ϕa j i e j + ϕβ j i ] e j + ϕβ j i τ j + ξ j i j + u i. Deine Q λ = I X λ X λ X λ X λ, where X λ = e, e 2,, 2, Λ, Λ 2 or model M and X λ = e, e 2, τ, τ 2,, 2, Λ, Λ 2 or model M2. For our 8 In the presentation o our results we assume that there is a break in the actor. We do so to maintain the ocus o the structural break in this section as well. Equally, without loss o generality and with the same results, we could have considered two actors instead o a broken one. Breitung and Eickmeier 20 provide the intuition that structural breaks severely inlate the number o actors. 28

29 asymptotic results, next we make the ollowing assumption. Assumption E: e ξ i and ξ 2 i are sequences o independent random variables with inite 4+δ moments. They are also independent rom u i. e2 lim maxeξ j2 i / ξ j2 = 0, where ξ j2 = / i= Eξj2 i, or j =, 2. Also, ξ j2 = lim ξj2 is inite. e3 and 2 are T inite, non-random vectors. Assumption F: T > colx λ, where col denotes the column dimension o a matrix. 2 Product matrix X λ X λ which appears in Q λ is invertible. 3 Variance unction V is non-zero. Conditions e and e2 guarantee that ξ j i obey the Lindeberg-Feller CLT and condition e3 states that the common actor can be seen as another type o deterministic component, which is a common approach in the large- - ixed-t panel data literature see Saraidis and Wansbeek 202. ote that condition e3 is weaker than that made by Moon and Perron 2004, or their large T test. This is because in the panel data actor models, the values o a common actor variable are treated like a set o parameters which are removed rom the model, just like the individual intercepts and the individual linear trends. We thereore need not assume them linear or restrict their order o integration. Assumption F determines the maximum number o common actors and the position o the breaks. Condition puts a limit to the number o actors that can appear in models M and M2. This is common in the literature see, e.g., Saraidis and Wansbeek 202. Violation o this assumption may lead to an invertible X λ which will result in Q λ = 0. Condition 2 also guarantees that Q λ exists. I there is no serial correlation, the above conditions are sui cient or the application o the HT and KT test statistics. In the presence o serial correlation in u i,t, a case relevant only or the KT test, condition 3 must be satisied as well. Since serial correlation in u i,t limits the available moments or 29

30 estimation, condition 3 guarantees that there are sui cient observations beore and ater the break or the identiication o nuisance parameters β j2 and ξ j2, or j =, 2. We do not need to assume that the variance unction o the limiting distribution o the adjusted or its inconsistency LS estimator ˆϕ λ is known. An easy way to check this condition will be presented below. Overall, Assumption F is more general than Assumptions B and D. Assumptions E and F accommodate both models M and M2. The ollowing theorem derives the limiting distribution o statistic HT under the sequence o local alternatives ϕ = c/ or model M, as diverges to ininity. Theorem 5 Let Assumptions A, E and F hold or model M and ui IID0, σ 2 I. Then, under ϕ = c/, we have V λ /2 HT ˆϕ λ d Bλ ck HT, as, with k HT = trf Q λ + trλ Q λ Λ 2BλtrF Q λ Λ 2trA λ2 HT where Bλ = p lim ˆϕ λ = trλ Q λ /trλ Q λ Λ, V λ HT = 2trAλ2 HT /trλ Q λ Λ 2, with A λ HT = 2 Λ Q λ + Q λ Λ BλΛ Q λ Λ, and F is deined in the Appendix. To calculate the value o k HT, given by the above theorem, we have considered various types o processes or and 2. Panel A o Table 4 contains the average k HT, denoted k HT or 5000 realizations, i t is generated by process t = ρ t, + η t, where ρ = 0.8 and η t IID0,. We see that the HT test statistic has reasonable power. This power is however lower than that or model M, without a common actor. An explanation o this 30

31 result could be that the existence o common actor reduces variation o the individual series o the panel, and thus the inormation available in the sample. Mathematically, or the HT test, less power comes rom the large dimension o X λ ; every actor increases it by two and a broken actor, like in our case, increases it by our. Summing up, our indings indicate that the existence o a actor in the error terms u i,t is what determines the magnitude o power o statistic HT, or model M. This is veriied by using as common actors various processes, even non-stationary ones. The limiting distribution o statistic HT under ϕ = c/ or model M2, where X λ = e, e 2, τ, τ 2,, 2, Λ, Λ 2, is given next. This theorem shows that the power o the HT test in the case o incidental trends remains trivial. The presence o common actor does not change this result. Theorem 6 Let Assumptions A, E and F hold or model M2 and ui IID0, σ 2 I. Then, under ϕ = c/, we have V λ /2 HT ˆϕ λ d Bλ ck HT, as, with k HT = 0, where Bλ = p limˆϕ λ = trλ Q λ /trλ Q λ Λ, V λ HT = 2trAλ2 HT /trλ Q λ Λ 2, with A λ HT = 2 Λ Q λ + Q λ Λ BλΛ Q λ Λ, and F is deined in the Appendix. To study the eects o cross-section dependence presence o common actor on the power o test statistic KT, we consider the more general version o it allowing or serial correlation in error terms u i,t. Beore presenting our main results, next we make all necessary 3

32 deinitions to derive the liming distribution o the KT test statistic or models M and M2. For model M, the inconsistency o ˆϕ λ is given as trλ Q λ Γ/trΛ Q λ ΛΓ, where Q λ is based on X λ = e, e 2,, 2, Λ, Λ 2. The estimator ˆΓ is inconsistent, with p lim [ˆΓ Γ ξ2 ξ ] = 0, since under null hypothesis H 0 : c = 0 we have y i = ξ i + ξ 2 i 2 + ε i. To adjust ˆΓ and, hence, ˆϕ λ or their inconsistency, we will rely on selection matrices M j = j j diag j j, p, or j =, 2. These enable [ us to identiy nuisance parameters ξ j, as p lim trm j ˆΓ/trM j j j ξ j2] = 0, or j =, 2. Given matrix M j, selection matrix Θ λ now becomes: Θ λ = Ψ λ trψλ trm M trψλ 2 2 trm M 2 [ with p lim trθ λˆγ trλ Q λ Γ ] = 0. Statistic ˆϕ λ ˆb λ /ˆδ λ, where ˆb λ = trθ λˆγ, is centred around 0 and its variance is given as V λ KT = vecqλ Λ Θ λ ΠvecQ λ Λ Θ λ, with ˆΠ = / i= vec y i y ivec y i y i. For model M2, matrix Q λ is based on X λ = e, e 2, τ, τ 2,, 2, Λ, Λ 2., ˆΓ is an inconsistent estimator o Γ due to the presence o nuisance parameters β j i and ξ j i, or j =, 2 with p lim [ˆΓ Γ β2 e e β 22 e 2 e 2 ξ 2 ξ ] = 0, since under H 0 : c = 0 we have y i = β i e + β 2 i e 2 + ξ i + ξ ε i. To adjust ˆΓ or its inconsistency due to nuisance parameters β j i and ξ j i, we will rely on selection matrices M j = e j e j diag{e j e j, p} and M j [ p lim trm jˆγ/trm j e j e j β j2] [ = 0 and p lim 0. Given these deinitions, selection matrix Θ λ now becomes i = j j diag j j, p, which imply trm j ˆΓ/trM j j j ξ j2] = Θ λ = Ψ λ 2 j= trψ λ e j e j trm j e j e j M j 32 2 trψ λ j j trm j j j M j, j=

33 with p lim [ trθ λˆγ trλ Q λ Γ ] = 0, and then the appropriate deinitions o ˆb λ and V λ KT will apply as above.9 The next theorem gives the limiting distributions o tests statistic KT or models M and M2 assuming serially correlated u i,t. Theorem 7 Let Assumptions A, E, and F hold or models M and M2. Then, under ϕ = c/ we have V ˆδλ ˆb λ /2 KT ˆϕ λ λ d ck KT,, ˆδλ as, with k KT = trf Q λ Γ + trλ Q λ ΛΓ trθ λ ΛΓ trλ Θ λ Γ, V λ KT where matrices Q λ, Θ λ are appropriately speciied or each model. Panels A and B o Table 4 present average values o k KT, denoted k KT, or models M and M2, respectively, when errors terms u i,t are IID, over 5000 repetitions. As beore, t is generated as t = ρ t, + η t, where ρ = 0.8 and η t IID0,. The results o Panel A indicate that the presence o common actor leads to power reduction or model M. The values o k KT are all positive, but smaller than the case without common actor. In contrast, Panel B indicates that the inclusion o actor in model M2 leads to non-trivial power o the KT test. This result was rather expected ater the indings o Section 4, which considers the case o the KT test allowing or serial correlation in u i,t. The power o the test 9 To check condition 3 it is sui cient to check that the denominators in Θ λ, which are based on quantities known to the researcher, are dierent than 0, i.e. trm j e j e j 0 and trm j j j 0. These denominators represent the number o elements the selection matrices M j and M j choose so that they estimate β j2 and ξ j2. I they are equal to zero, this means that there are zero elements available or M j and M j and thereore the corresponding β j2 and ξ j2 cannot be identiied. 33

34 can be attributed to the interaction between individual trends and common actors in the bias adjustment o the LS estimator ˆϕ λ. This result is notably dierent than that in the large-t case, where the test is robust to the eects o common actors see, e.g., Moon and Perron To implement Theorems 5, 6 and 7 one must irst speciy the annihilator matrix Q λ which contains actors, 2, i.e. or Theorem 7, Q λ = I X λ X λ X λ X λ where X λ = e, e 2, τ, τ 2,, 2, Λ, Λ 2. Then, the HT test can be applied as beore. For the KT test, the steps are also similar to those o the previous section, but care must be taken to appropriately speciy Θ λ because it is inluenced by both the number o actors and by the order o serial correlation. It must be made sure that Θ λ exists as described in ootnote 9. ˆΓ and ˆΠ can be estimated as in the previous sections. Summing up, the results o this section indicate that cross section correlation in error terms u i,t aects the power perormance o the HT and KT test statistics, i T is ixed. The tests are not robust to the presence o a common actor in u i,t, as in the large-t case. For the large-t case, it can be easily seen that test statistics HT and KT have zero local power, or both models M and M2. 6 Monte Carlo results In this section, we conduct a Monte Carlo study to examine i the asymptotic local power unctions o the HT and KT test statistics, implied by the results o the previous section, provide good approximations o their small sample ones. This is done based on 5000 repetitions and or dierent values o and T, oten met in microeconomic and macroeconomic studies, i.e., = {00, 300, 000} and T = {8, 0, 2, 5, 20, 50, 00, 200, 300}. For each iter- 34

35 ation, we calculate the size o the tests at 5% level i.e., or c = 0 and their power i.e., or c =. This is done separately or the cases that u i,t IID0, and u i,t = ε i,t + θε i,t, with θ { 0.8, 0.5, 0, 0.5, 0.8}. The nuisance parameters o the models are set to the ollowing values: y i,0 = 0, a j i = 0 and β j i = 0, or all i. Table 5 presents the results o our simulation study or the case that u i,t IID0,. The last column o the tables gives the theoretical values T V o the power unction and the nominal size o the tests, at a = 5%. For model M, the results indicate that both the HT and KT tests have size and power values which are very close to their theoretical ones. Furthermore, the results conirm that the HT test has more power towards the beginning and the end o the sample, while the KT test has more power in the middle. As was also predicted by the theory, the HT test has higher power than the KT test. The small sample power o this test is very close to that predicted by its asymptotic local power unction see column T V even or small e.g., = 00. However, this is not always true or the KT test, which needs very high in order its power to converge to its theoretical value. For model M2, the results o Table 5 indicate that, or large, both HT and KT tests have trivial power, as it was expected. However, in small samples e.g., = 00, both tests have some non-trivial power. This can be obviously attributed to second, or higher, order eects o the true power unction, which cannot be approximated by the irst-order approximation considered in our analysis. ote that, or model M2, the KT test has slightly higher small sample power than the HT. Tables 6 and 7 present the results o our simulation study or the KT test statistic allowing or serial correlation in error terms u i,t, assuming u i,t = ε i,t + θε i,t. This is done or models M and M2, and T {8, 0}. The maximum order o serial correlation allowed by the KT test is set to p =, which matches that o the MA process o u i,t. The results 35

36 o these tables are also consistent with theory. For model M, the KT test has signiicant power when θ > 0. As increases, this power converges quite ast to its theoretical value, reported in the last column o the table. ote that both the theoretical and small sample values o the power unction o the KT test statistic are higher than their corresponding values in the absence o serial correlation see Table 5. This is also consistent with the theory. It can be attributed to the serial correlation eects o u i,t on the power unction o the test, discussed in the previous section. For negative values o θ, the test has also signiicant power. This happens or λ = {0.75}, as was predicted by the theory. For model M2, the results o Table 7 indicate that the KT test statistic has smaller power than or model M. As was expected by the theory, the power o the test is non-trivial i θ < 0. The KT test has also some small sample power i θ > 0, which qualiies its use in practice. As was argued beore, this power can be attributed to second, or higher, order eects o the true power unction, which are not captured by our asymptotic approximations. Finally, another conclusion which can be drawn rom the results o our simulation study reported in Tables 6 and 7 is that, when θ < 0, a break towards the end o the sample increases the power o the KT test. When θ > 0, the power o the test is maximized at the middle o the sample. These results apply to both models M and M2. They are also consistent with the theoretical results reported in Table 3. Table 8 presents the results o the tests or models M and M2 including a common actor in error terms. This actor t is generated as beore see Section 5, i.e., t = ρ t, + η t, with ρ = 0.8 and η t IID0,. The slope coei cients o this actor ξ j i are set to zero, i.e., ξ j i = 0, or all i and j. The results o the table clearly indicate that both the size and power o the HT and KT test statistics are close to their theoretical values, or all cases o, T and λ considered. For model M both tests have non-trivial power, while or model 36

37 M2 only the KT test has non-trivial power, which is in accordance to our theoretical rsults o Section 5. Finally, Table 9 shows how good the large-t approximations the HT and KT tests statistics, given by Corollaries and 2, are or a variety o dierent values o and T. In particular, we consider the ollowing cases o and T : = {0, 20, 50} and T = {50, 00, 200, 300}, oten used in macroeconomic studies. The results o the table indicate that, or model M, the HT test statistic is a bit oversized when T is much larger than. For instance, the size o the test is equal to 0.07, or = 0, T = 300 and λ = 0.5. As increases, the size o the test tends to its nominal value o The power o the test is also close to its theoretical values, with the approximation becoming better as increases. The KT test statistic displays similar behaviour to that o HT, but it is a bit undersized or large T and very small. Consistently with the theory, the KT test does not have power anywhere. For model M2, this result holds or both HT and KT test statistics. When T becomes large, both the HT and KT tests have trivial power. The quality o the approximations seems to be unaected by the relative position o the break in the sample. 7 Conclusions This paper analyses the asymptotic local power properties o least-squares based ixed-t panel unit root tests allowing or a structural break in the deterministic components o the AR panel data model, namely its individual eects and/or slope coei cients o its individual linear incidental trends. This is done by assuming that the cross-section dimension o the panel data models grows large. 37

38 The paper derives the limiting distributions o the panel unit root test proposed in Karavias and Tzavalis 204a denoted HT under local alternatives. This is studied together with the test o Karavias and Tzavalis 204b, denoted as KT, which allows or a structural break and serial correlation in the error terms o the AR panel data model. In this paper, we have extended the above tests to allow or cross section dependence. Both o these tests are based on the least squares dummy variables estimator o the autoregressive coei cient o the AR panel data model, which is corrected or its inconsistency due to the deterministic components o the panel, the cross section dependence and/or serial correlation eects o the error term. The results o the paper lead to a number o conclusions. First, they show that, or the standard AR panel data model with white noise error terms and individual eects, both the HT and KT tests have signiicant asymptotic local power. The HT test has much higher power than the KT test. This can be attributed to the act that, in order to adjust or the inconsistency o the least squares estimator, the KT test requires consistent estimation o the variance o the error term. The HT test does not depend on this nuisance parameter, as it adjusts the least squares estimator or both the inconsistency o its numerator and denominator, and thus the variance o the error terms is cancelled out. The HT test is ound to have more power when the break is towards the beginning or the end o the sample, while the KT test has more power when the break is towards the middle o the sample. Second, both the HT and KT tests have asymptotically trivial power in the case that the AR allows also or incidental trends. The allowance or a common break in the slope coei cients o the incidental trends does not change the behaviour o the tests. This problem does not always exist or the KT test extended or serial correlation o the error terms. In this case, the paper presents circumstances that the KT test has non-trivial 38

39 power. In particular, this happens when the error terms ollow a MA procedure with negative serial correlation. The power o the KT in this case can be attributed to the eects o the serial correlation o error term on the adjustment o the least squares estimator o the autoregressive coei cient or its inconsistency, upon which the KT test is based on. In contrast to large-t panel data unit root tests, the power unction o ixed-t tests depends on the values o nuisance parameters capturing serial correlation eects which can aect the asymptotic over power o the tests. Third, we ind that the existence o a common actor in the error terms changes the behaviour o the tests. For the model with intercepts, the presence o the common actor reduces the power o both the HT and KT tests. For the model with trends, the HT has trivial power, while the KT has positive. Fourth, we compare our results to those o the large-t literature and we show that the desirable properties o the KT test presented in this paper, i.e. non-trivial power or the case with incidental trends, are derived rom the ixed-t estimator o the bias o the within groups estimator. The above results are conirmed through a Monte Carlo simulation exercise. This exercise has shown that the empirical probabilities o rejection are very close to their theoretical values, which means that the asymptotic theory provides a good approximation o small sample results o ixed-t panel data unit roots. The above indings suggest that there are several theoretical arguments in avour o the use o ixed T tests in practice, especially in the case where incidental trends are in the model. 39

40 Reerences [] Abowd, J.M., Card, D., 989. On the Covariance Structure o Earnings and Hours Changes. Econometrica. Econometric Society, vol. 572, pages [2] Andrews, D. W. K., 993. Tests or Parameter Instability and Structural Change with Unknown Change Point. Econometrica, Econometric Society, Econometric Society, vol. 64, p [3] Arellano, M., 990.Testing or Autocorrelation in Dynamic Random Eects Models. Review o Economic Studies, Wiley Blackwell, vol. 57, pages [4] Arellano, M., Panel data econometrics. Oxord University Press. [5] Bai, J., 200. Common breaks in means and variances or panel data. Journal o Econometrics, Elsevier, vol. 57, pages [6] Bai J., Carrion-I-Silvestre, J.L., Structural Changes, Common Stochastic Trends, and Unit Roots in Panel Data. Review o Economic Studies, vol. 762, [7] Baltagi, B., 2008, Econometric analysis o panel data. John Wiley & Sons. [8] Breitung, J., Eickmeier, S., 20. Testing or structural breaks in dynamic actor models. Journal o Econometrics, Elsevier, vol. 63, pages [9] Carrion-i-Silvestre, J.L., Del Barrio-Castro, T., Lopez-Bazo, E., Breaking the panels: An application to real per capita GDP. Econometrics Journal, 8, [0] Chan, F., Pauwels, L.L., 20. Model speciication in panel data unit root tests with an unknown break. Mathematics and Computers in Simulation. 8,

41 [] Hahn, J., Kuersteiner, G., Asymptotically unbiased inerence or a dynamic panel model with ixed eects when both n and T are large. Econometrica. 70, [2] Hadri K., Larsson, R., Rao, Y., 202. Testing or stationarity with a break in panels where the time dimension is inite. Bulletin o Economic Research, 64, s23-s48. [3] Hansen, B., E., 995. Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase Power. Econometric Theory, Cambridge University Press, vol. 05, pages [4] Harris D., Harvey D., Leybourne S., and Sakkas., 200. Local asymptotic power o the Im-Pesaran-Shin panel unit root test and the impact o initial observations. Econometric Theory 26, [5] Harris, D., Leybourne, S., and McCabe, B., Panel Stationarity Tests or Purchasing Power Parity with Cross-Sectional Dependence. Journal o Business & Economic Statistics, vol. 23, [6] Harris, R., Tzavalis, E., 999. Inerence or unit roots in dynamic panels where the time dimension is ixed. Journal o Econometrics, 9, [7] Harris, R., Tzavalis, E., Inerence or unit roots or dynamic panels in the presence o deterministic trends: Do stock prices and dividends ollow a random walk? Econometric Reviews 23, [8] Harvey, D.I., Leybourne S.J., On testing or unit roots and the initial observation. Econometrics Journal 8, 97. 4

42 [9] Karavias, Y., Tzavalis, E., 204a. Testing or unit roots in short panels allowing or a structural break. Computational Statistics & Data Analysis, Elsevier, vol. 76C, pages [20] Karavias, Y., and Tzavalis, E., 204b. Testing or unit roots in panels with structural changes, spatial and temporal dependence when the time dimension is inite. Granger Centre Discussion Paper Series, o 4/03. [2] Karavias, Y., Tzavalis, E., 204c. A ixed-t version o Breitung s panel data unit root test. Economics Letters, Elsevier, vol. 24, pages [22] Kruiniger, H., Maximum Likelihood Estimation and Inerence Methods or the Covariance Stationary Panel AR/Unit Root Model. Journal o Econometrics 44, [23] Kruiniger, H., and E., Tzavalis, Testing or unit roots in short dynamic panels with serially correlated and heteroscedastic disturbance terms. Working Papers 459, Department o Economics, Queen Mary, University o London, London. [24] Madsen E., 200. Unit root inerence in panel data models where the time-series dimension is ixed: a comparison o dierent tests. Econometrics Journal 3, [25] Meligkotsidou, L., Tzavalis, E., Vrontos I.D., 20. A Bayesian analysis o unit roots and structural breaks in the level, the trend and the error variance o autoregressive models o economic series. Econometric Reviews, 30 2, [26] Moon, H.R., Perron, B., Asymptotic local power o pooled t-ratio tests or unit roots in panels with ixed eects. Econometrics Journal,

43 [27] Moon, H.R., Perron, B., Phillips, P.C.B., Incidental trends and the power o panel unit root tests. Journal o Econometrics, 42, [28] Perron, P., 989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica, 57, [29] Saraidis, V., Wansbeek, T., 202. Cross-Sectional Dependence in Panel Data Analysis. Econometric Reviews, Taylor & Francis Journals, Taylor & Francis Journals, vol. 35, pages Appendix In this appendix, we provide proos o the theorems and the corollary presented in the main text o the paper. Proo o Theorem : First, we derive the limiting distribution o the HT test statistic, under the sequence o local alternatives ϕ = c/. Deine vector w =, ϕ, ϕ 2,..., ϕt and matrix Ω = ϕ.. ϕ 2 ϕ ϕ T 2 ϕ T 3.. ϕ 0 43

44 Under null hypothesis H 0 : c = 0, we have Ω = Λ. The irst order Taylor expansions o Ω and w yields Ω = Λ + F ϕ + o p and 6 w = e + ϕ + o P, 7 respectively, where F = dω/dϕ c=0 and = dw/dϕ c=0. Based on the above deinitions o w and Ω, vector y i, can be written as y i, = wy i,0 + ΩX λ γ λ i + Ωu i, 8 where γ λ i = a i ϕ, a 2 i ϕ = ϕ a i, a 2 i. Using last relationship o y i,, the HT test statistic or model M can be written under ϕ = c/ as ollows: ˆϕ λ ϕ Bλ 9 = y i, Q λ ϕ y i, + X λ γ λ i + u i i= ϕ Bλ, y i, Qλ y i, = = i= i= y i, Q λ u i i= Bλ y i, Qλ y i, i= y i, Q λ u i Bλ y i, Q λ y i,, y i, Qλ y i, i= i= y i, Q λ y i, i= = y i, Qλ y i, i= A B. 0 C ext, we derive asymptotic results o each o quantities A, B and C, deined by 0. 44

45 Substituting 8 in A, we have A = i= i= y i, Q λ u i = i= y i,0 w + γ λ i X λ Ω + u iω Q λ u i y i,0 w Q λ u i + γ λ i X λ Ω Q λ u i + u iω Q λ u i Using relationships 6-7, we can ind the ollowing limits o the summands entering into the last relationship o A. First, it can be shown that i= y i,0 w Q λ u i = = i= i= y i,0 e + ϕ Q λ u i + o P, y i,0 e Q λ u i + c y i,0 Q λ u i + o P, i= = o P, since e Q λ = 0 and Ey i,0 u i = 0 by assumption a4, and = = = c i= i= i= = o p, i= a γ λ i X λ Ω Q λ u i γ λ i X λ Λ + F ϕ + o p Q λ u i, γ λ i X λ Λ Q λ u i + c i, a 2 i i= X λ Λ Q λ u i + c2 3/2 γ λ i X λ F Q λ u i + o p, i= a i, a 2 i X λ F Q λ u i + o p, 2 45

46 since Ea λ i u i = 0 by assumption a4. Finally, we have i= u iω Q λ u i = = i= i= u i Λ + F ϕ + o p Q λ u i, u iλ Q λ u i c u if Q λ u i + o p, i= where c i= u if Q λ u i p cσ 2 trf Q λ and 3 u iλ Q λ u i σ 2 trλ Q λ d 0, V HT,A, 4 i= where V HT,A is the variance o the last limiting distribution. Based on the asymptotic results given by equations -4, we can show that A i= y i, Q λ u i d cσ 2 trf Q λ, V HT,A. 5 To derive asymptotic results or summand B, write it as ollows: B Bλ = Bλ y i, Q λ y i, i= i= y i,0 w + γ λ i X λ Ω + u iω Q λ y i,0 w + ΩX λ γ λ i + Ωu i. By similar arguments to those applied to derive results -4, we can prove the ollowing 46

47 asymptotic results: i= γ λ i i= y 2 i,0w Q λ w + y i,0 w Q λ ΩX λ γ λ i + y i,0 w Q λ Ωu i = o p,6 X λ Ω Q λ wy i,0 +γ λ i X λ Ω Q λ ΩX λ γ λ i +ΩX λ γ λ i Ωu i = o p,7 i= u iω Q λ wy i,0 + u iω Q λ ΩX λ γ λ i = o p,8 where i= u iω Q λ Ωu i = i= u iλ + F ϕ Q λ Λ + F ϕ u i + o p, 9 c c u iλ Q λ Λu i σ 2 trλ Q λ d Λ 0, V HT,B, 20 i= u if Q λ p Λu i σ 2 trf Q λ Λ, 2 i= u iλ Q λ p F u i σ 2 trλ Q λ F and 22 i= c 2 3/2 u if Q λ F u i = o p. 23 i= Based on the above results, given by equations 6-23, it can be shown that B Bλ y i, Q λ y d i, 24 i= cσ 2 Bλ[trF Q λ Λ + trλ Q λ F ], B 2 λv HT,B. Finally, ollowing similar arguments to the above, we can easily show that, or quantity C, 47

48 the ollowing asymptotic result holds: C y i, Q λ p y i, σ 2 trλ Q λ Λ. 25 i= Using asymptotic results 5, 24 and 25, equation 0 implies that ˆϕ λ ϕ Bλ d c trf Q λ 2BλtrF Q λ Λ, V λ trλ Q λ HT Λ, 26 or ˆϕ λ Bλ d c trf Q λ + trλ Q λ Λ 2BλtrF Q λ Λ, V λ trλ Q λ HT, Λ since trλ Q λ BλtrΛ Q λ Λ = 0. ote that the analytic ormula o variance V λ HT o the last limiting distribution is the same with that o the HT test under null hypothesis H 0 : c = 0, given by V λ HT = 2trAλ2 HT /trλ Q λ Λ 2. This does not depend on local parameter c. It remains the same under the null and sequence o local alternative hypotheses see, e.g., Madsen 200 and Karavias and Tzavalis 204c, given as V λ HT = 2trAλ2 HT /trλ Q λ Λ 2. Scaling by V λ /2 HT the above limiting distribution yields V λ /2 HT ˆϕ λ Bλ d ck HT,, with 27 k HT = trf Q λ + trλ Q λ Λ 2BλtrF Q λ Λ. 2trA λ2 HT 48

49 Substituting into the above ormula o k HT the ollowing identities: trf Q λ Λ = trλ Q λ F = 28 = λ2 3λ + T 3 2 2λ2 2λ + T 2 24 T + 6 trλ Q λ Λ + trf Q λ + trλ Q λ = 0, 29 trf Q λ = T 2 6 2λ2 2λ + + T 2 4 6, 30 trλ Q λ = T 2, 2 3 trλ Q λ Λ = T 2 6 2λ2 2λ + 2 6, 32 [ ] 2 tra λ2 HT = tr 2 Λ Q λ + Q λ Λ BλΛ Q λ Λ,33 Λ tr Q λ + Q λ Λ 2 = T 2 6 2λ2 2λ + + T 7 3, 34 tr Λ Q λ Λ 2 = 90 2λ4 4λ 3 + 6λ 2 4λ + T λ2 2λ + T , tr Λ Q λ + Q λ Λ Λ Q λ Λ = T 2, 36 2 yields the results o Theorem, or the HT test statistic. ote that 2trA λ2 HT can be analytically written as 2trA λ2 HT = D S, where 49

50 D = T 6 R + T 5 R 2 + T 4 R 3 + T 2 R T 36, S = T 4 Φ + T 2 Φ , R = 40λ 6 20λ λ 4 208λ λ 2 78λ + 7, R 2 = 26λ λ 3 528λ λ 78, R 3 = 26λ 4 432λ λ 2 372λ + 08, R 4 = 20λ λ 44, Φ = 240λ 4 480λ λ 2 240λ + 60 and Φ 2 = 480λ λ 240. To derive the limiting distribution o the KT test under the sequence o local alternatives ϕ = c/, write ˆδλ ˆb ˆϕ λ λ ϕ ˆδλ = ˆδ λ ϕ + = = y i, Q λ u i λ i= ˆb ϕ ˆδλ, y i, Qλ y i, i= y i, Q λ u i ˆσ 2 trλ Q λ, i= y i, Q λ u i i= y iψ λ y i, i= = i= y i, Q λ u i i= y iψ λ y i, 37 where y i can be written as y i = u i + ϕ y i, + X λ γ λ i

51 The limiting distribution o the KT test under ϕ = c/ can be proved by obtaining asymptotic results or the two summands entering into equation 37, i.e., / and / i= y iψ λ y i, ollowing analogous to the proo o 27 steps. The ormula o slope power parameter k KT is given as i= y i, Q λ u i k KT = trf Q λ + trλ Q λ Λ. 39 2trA λ2 KT Substituting the ollowing identities into the above ormula o k KT : 2 tra λ2 KT = tr 2 Λ Q λ + Q λ Λ Ψ λ Ψ λ, 40 trψ λ Λ = trλ Ψ λ = 0, 4 2trA λ2 KT = 2trP λ 2trZ λ2, with Z λ = 2 Ψλ + Ψ λ 42 and P λ = 2 Λ Q λ Λ Q λ Λ, 43 tr Λ Q λ 2 = T 2 2 2λ2 2λ + T 2 5 and 6 44 trz λ2 = + 2λ λt T 6λ λ 45 yields the results o Theorem, or the KT test statistic. Proo o Corollary : The results o the corollary and, in particular, those o equation can be derived based on analogous arguments to those applied or the proo o Theorem. To obtain the analytic ormula o k HT, given by equation, scale 9 by T, replace ϕ 5

52 with ϕ,t, and apply asymptotic theory or, as in Theorem. Then, we will have T ˆϕ λ ϕ,t Bλ d Multiplying with /2 T 2 V λ HT and using ϕ,t = c/ T, the last limiting distribution can be written as c trf Q λ 2BλtrF Q λ Λ, T 2 V λ trλ Q λ HT Λ. T /2 T 2 V λ HT ˆϕ λ Bλ d c T k HT, 46 where k HT = [ trf Q λ + trλ Q λ Λ 2BλtrF Q λ Λ ] / 2trA λ2 HT see proo o Theorem. By taking the limit or T o k HT and T 2 V λ HT, 46 can be written as T V λ HT /2 ˆϕ λ Bλ d ck HT,, where kht lim T T k HT = 3λ2 3λ + Φ 42λ 2 2λ + R V λ HT lim T T 2 V λ HT = 36R Φ 2λ 2 2λ 2. and The analytic ormulas o the last two limits are derived based on the results o identities The above results have been derived by taking limits sequentially, irst or and then or T. Joint convergence in, T requires the extra assumption that /T 0, see also Moon and Perron However, or c = 0 there is no need to speciy the relative rate o convergence between and T see Hahn and Kuersteiner 2002 and Karavias and Tzavalis 204a. Considering now the KT test, by the deinition o V λ KT, we have that lim T V λ KT = +. 52

53 To clariy how this is not a problem or the implementation o the statistic, notice that the deinition o the variance V λ KT = 2trAλ2 KT is made or convenience our notation gives weight to ˆδ λ and thus to the bias correction aspect o the test and does not correspond exactly to the variance o ˆϕ λ ˆb λ /ˆδ λ. oticing that p lim ˆδλ = trλ Q λ Λ and that in the HT test, V λ HT = 2trAλ2 HT /trλ Q λ Λ 2, the KT test maybe written as ˆδλ2 / 2trA λ2 KT ˆϕ λ ˆb λ /ˆδ λ. The ormulas o k KT λ and V, given by the corollary or the large-t version o the KT KT test, can be derived by ollowing similar steps to the above. Then, using the results o identities 40-45, we can obtain k KT lim T T k KT = 0 and V λ KT lim T T 2 V λ KT / p lim ˆδλ = 362λ4 4λ 3 + 3λ 2 λ 2λ λ2λ 2 2λ + 2. Proo o Theorem 2: To prove the theorem, we will ollow analogous steps to those or the proo o Theorem. We now will rely on relationships 8 and 38, where now vector γ λ i is deined as γ λ i = ϕ a i + ϕ β i ϕ a 2 i + ϕ β 2 i ϕ β i ϕ β 2 i = e ν i + ϕ µ i, due to the presence o individual trends under ϕ,t = c/, where µ i = α β 2 i, β i, β 2 i, e = i β i, α and ν i = β i, β 2 i. The non-standardized HT test i 53

54 statistic or model M2 can be written as ollows: where A ˆϕ λ ϕ Bλ = = i= i= y i, Q λ ϕ y i, + X λ γ λ i + u i y i, Qλ y i, i= y i, Q λ u i Bλ i= ϕ Bλ y i, Q λ y i, i= A B =, C y i, Qλ y i, / i= y i, Q λ u i, B Bλ / i= y i, Q λ y i, and C / i= y i, Q λ y i,. As in the proo o Theorem, next we derive asymptotic results o A, B and C, using γ λ i = e ν i + ϕ µ i. The most important ones are the ollowing: c c c c ν ie X λ Λ Q λ ΛX λ e ν i tre X λ Λ Q λ ΛX λ e Eν i ν d i 0, V HT,4 i= ν ie X λ F Q λ Λe ν i i= ν ie X λ Λ Q λ F X λ e ν i i= p ctre X λ F Q λ Λe Eν i ν i p ctre X λ Λ Q λ F e Eν i ν i µ ix λ Ω Q λ ΩX λ p e ν i ctrx λ Λ Q λ ΛX λ e Eν i µ i i= ν ie X λ Λ Q λ ΛX λ µ i i= p ctre X λ Λ Q λ ΛX λ Eµ i ν i Given these results, the proo o Theorem 2 or the test statistic HT ollows immediately, 54

55 ater using the ollowing identities: tre X λ Λ Q λ ΛX λ e Eν i ν i = 0 tre X λ F Q λ Λe Eν i ν i tre X λ Λ Q λ F e Eν i ν i = 0 and trx λ Λ Q λ ΛX λ e Eν i µ i tre X λ Λ Q λ ΛX λ Eµ i ν i = 0. The proo o the second result o the theorem, i.e., k KT = 0, can be proved by ollowing analogous steps to the above and using the ollowing identities: tre X λ Θ λ ΛX λ e Eν i ν i tre X λ Λ Θ λ X λ e Eν i ν i = 0 and trx λ Θ λ X λ e Eν i µ i tre X λ Θ λ X λ Eµ i ν i = 0. Proo o Corollary 2: The proo comes directly rom Theorem 2 by scaling the results with T. Proo o Theorem 3: This can be proved by ollowing analogous steps to the proo o Theorem, or the KT test statistic, by inserting Γ instead o σ 2 I and by using the corresponding asymptotic theorems see Karavias and Tzavalis 204b or proo o Theorem 7 or an example. Proo o Theorem 4: This can be proved by ollowing analogous steps to the proo o Theorems 2 and 3, or the KT test statistic. Proo o Theorem 5: Under the null hypothesis, model M becomes: y i = y i, + ξ i + ξ 2 i 2 + u i, 47 55

56 or i =,...,. Solving backwards last rlationship yields y i, = y i,0 e + ξ i Λ + ξ 2 i Λ 2 + Λu i. 48 The ollowing proo is as in Karavias and Tzavalis 204a. Equation 48 corresponds to equation 8, under the null hypothesis and the presence o common actors. otice that multiplying 47 and 48 with Q λ based on the augmented X λ = e, e 2,, 2, Λ, Λ 2 removes the nuisance parameters such that: Q λ y i = Q λ y i, + Q λ u i Q λ y i, = Q λ Λu i. 49 Substituting 49 in the inconsistency o ˆϕ λ : ˆϕ λ = y i, Q λ y i i= = y i, Q λ y i, i= u iλ Q λ u i i=. u i Λ Q λ u i i= By applying standard properties o the quadratic orms: Eu iλ Q λ u i = σ 2 trλ Q λ, Eu iλ Q λ Λu i = σ 2 trλ Q λ Λ and thus Bλ = p lim ˆϕ λ = trλ Q λ trλ Q λ Λ. 56

57 To derive the limiting distribution under the null ˆϕ λ Bλ = i= [ u i Λ Q λ u i Bλu iλ Q λ u i ]. u i Λ Q λ u i i= Then, p lim / i= u iλ Q λ u i = σ 2 trλ Q λ Λ. Also, /2 [ ] i= u i Λ Q λ u i Bλu iλ Q λ u i = / i= u ia λ HT u i where Eu ia λ HT u i = 0 and V aru ia λ HT u i = 2σ 4 tra λ2 HT. The result ollows rom the Lindeberg-Levy CLT and the CMT. The proo o the distribution under the local alternatives is the same with the proo o Theorem. Proo o Theorem 6: Under the null hypothesis, model M2 becomes: y i = y i, + β i e + β 2 i e 2 + ξ i + ξ 2 i 2 + u i, 50 or i =,...,. Solving backwards last relationship gives y i, = y i,0 e + β i Λe + β 2 i Λe 2 + ξ i Λ + ξ 2 i Λ 2 + Λu i. 5 Then, by multiplying with Q λ based on X λ = e, e 2, τ, τ 2,, 2, Λ, Λ 2 we remove the nuisance parameters such that: Q λ y i = Q λ y i, + Q λ u i Q λ y i, = Q λ Λu i. 52 The proo then ollows the steps o Theorem 5. 57

58 Proo o Theorem 7: We irst begin by proving some claims in the text beore Theorem 7. Under null hypothesis H 0 :c = 0, we have y i = ξ i + ξ 2 i 2 + u i. Then, ˆΓ can be written as ˆΓ = = = y i y i i= i= i= ξ i + ξ 2 i 2 + u i ξ i + ξ 2 i 2 + u i ξ 2 i + ξ 22 i u i u i + ξ i ξ 2 i 2 + ξ i u i +ξ 2 i ξ i 2 + ξ 2 i 2 u i + ξ i u i + ξ 2 i u i 2 o ˆΓ as To make the exposition simpler and without loss o generality, write the top let element where E i= ξ 2 i 2 + u 2 i, + 2ξ i u i, = E ξ 2 i 2 + u 2 i, + 2ξ i u i,, ξ 2 i 2 + Eu 2 i, by condition e. Also, by Condition e2 it is straightorward that V arξ 2 i 2 +u 2 i,+2ξ i u i, is inite. Then, by Chebyshev s Weak Law o Large umbers, we obtain the ollowing result: p lim i= [ ξ 2 i 2 + u 2 i, + 2ξ i u i, E ξ 2 i ] 2 Eu 2 i, = 0. The arguments used here apply to all elements o ˆΓ and thus we have that p lim [ˆΓ Γ ξ 2 + ξ ] = 0. ext, we will show that ξ 2 and ξ 22 can be consistently estimated. First, write trm ˆΓ = 58

59 / i= y im y i. Then, we have [ ] E y im y i = tr M E y i y i and E y i y i = Eξ 2 i + Eξ 22 i Γ i [ by Assumption E. The above imply that tr Eξ 22 i tr [ M 2 2 ] + tr [ M Γ i ] wherever Γ i is non-zero, we have tr ] E y i y i M. But since M j [ M j Γ i ] [ ] = Eξ 2 i tr M + has zeroes in its central diagonals [ ] = 0 and also tr M 2 2 = 0, because the block orms that the matrices have due to the structural break. Thus, or all i, it holds [ ] that tr M E y i y i = Eξ 2 i tr M. Since variances o y i y i are inite, or all i, by Assumptions A and E, the ollowing results hold [ p lim trm ˆΓ ξ 2 tr p lim tr M ] trm ˆΓ M ξ 2 = 0, or = 0, by Chebyshev s Weak Law o Large. A similar result to the above hods or ξ j2, i j = 2. Finally, we will show that p lim [ trθ λˆγ trλ Q λ Γ ] = 0. To this end, irst note that trλ Q λ Γ = / i= trλ Q λ Γ i. Also, note that trθ λˆγ = / i= y iθ λ y i with E y iθ λ y i = trθ λ E y i y i = Eξ 2 i tr Θ λ +Eξ 22 i tr Θ λ tr Θ λ Γ i. ext note the ollowing result: tr Θ λ = tr Ψ λ trψλ tr trψλ tr trm 2 trm M 2 = tr Ψ λ trψ λ = 0, 59 M

60 which holds because tr M 2 = 0. Similarly, we can prove that tr Θ λ 2 2 = 0 and, hence, trθ λ E y i y i = tr Θ λ Γ i. Also, note that tr Θ λ Γ i = tr Ψ λ Γ i because tr M j Γ i = 0, by the construction o M j, or j =, 2. By the construction o Ψ λ, we have tr Ψ λ Γ i = trλ Q λ Γ i. Taking together the above results, we have that p lim [ trθ λˆγ trλ Q λ Γ ] = 0. The rest o the proo ollows similar steps to those o the proo o Theorem. Following analogous steps to the above, we can prove the results o Theorem 7 or model M2. 60

61 8 Tables Table : Values o k HT and k KT or model M k HT k KT λ\t

62 Table 2: Values o slope parameters kht and k KT or model M λ kht kkt

63 Table 3: Values o k KT with u i,t = ε i,t + θε i,t Panel A: Model M T T 0 θ = 0.8 θ = 0.5 θ = 0.0 θ = 0.5 θ = Panel B: Model M

64 Table 4: Values o k HT and k KT Panel A: Model M λ\t k HT k KT Panel B: Model M

65 Table 5: Simulated size and power o the HT and KT tests or u i,t IID0, σ 2 Model M Model M T V T V T = 8 λ = 0.25 c = 0 HT KT c = HT KT λ = 0.5 c = 0 HT KT c = HT KT λ = 0.75 c = 0 HT KT c = HT KT T = 0 λ = 0.25 c = 0 HT KT c = HT KT λ = 0.5 c = 0 HT KT c = HT KT λ = 0.75 c = 0 HT KT c = HT KT

66 Table 6: Simulated size and power o the KT test or model M with u i,t = ε i,t + θε i,t. T T V T V θ = 0.8 λ = 0.25 c = c = λ = 0.50 c = c = λ = 0.75 c = c = θ = 0.5 λ = 0.25 c = c = λ = 0.50 c = c = λ = 0.75 c = c = θ = 0.5 λ = 0.25 c = c = λ = 0.50 c = c = λ = 0.75 c = c = θ = 0.8 λ = 0.25 c = c = λ = 0.50 c = c = λ = 0.75 c = c =

67 Table 7: Simulated size and power o the KT test or model M2 with u i,t = ε i,t + θε i,t T T V T V θ = 0.8 λ = 0.50 c = c = λ = 0.75 c = c = θ = 0.5 λ = 0.50 c = c = λ = 0.75 c = c = θ = 0.5 λ = 0.50 c = c = λ = 0.75 c = c = θ = 0.8 λ = 0.50 c = c = λ = 0.75 c = c =

68 Table 8: Simulated size and power o the HT and KT tests or M and M2, with u i,t IID0, σ 2 Model M Model M T V T V T = 2 λ = 0.25 c = 0 HT KT c = HT KT λ = 0.5 c = 0 HT KT c = HT KT λ = 0.75 c = 0 HT KT c = HT KT T = 5 λ = 0.25 c = 0 HT KT c = HT KT λ = 0.5 c = 0 HT KT c = HT KT λ = 0.75 c = 0 HT KT c = HT KT

69 Table 9: Simulated size and power o the HT and KT tests when T is large and u i,t IID0, σ 2 /T 0/00 0/200 0/300 0/50 20/50 50/50 T V Model M λ = 0.25 c = 0 HT KT c = HT KT λ = 0.5 c = 0 HT KT c = HT KT λ = 0.75 c = 0 HT KT c = HT KT Model M2 λ = 0.25 c = 0 HT KT c = HT KT λ = 0.5 c = 0 HT KT c = HT KT λ = 0.75 c = 0 HT KT c = HT KT

70 70

71 7