Local power of fixed-t panel unit root tests with serially correlated errors and incidental trends

Transcription

1 Local power of fixe-t panel unit root tests with serially correlate errors an inciental trens Yiannis Karavias a an Elias Tzavalis b, a :School of Economics, University of ottingham b :Department of Economics, Athens University of Economics & Business 28/0/203 Abstract The asymptotic local power properties of various fixe T panel unit root tests with serially correlate errors an inciental trens are analyse. Asymptotic (over local power functions are analytically erive an through them the effects of general forms of serial correlation are examine. These functions also show that in the presence of inciental trens an IV test base on the first ifferences of the moel has non-trivial local power in a /2 neighbourhoo of unity. Furthermore, for a test base on the within groups estimator, although it is foun that it has trivial power in the presence of inciental trens, this ceases to be the case if there is serial correlation as well. JEL classification: C22, C23 Keywors: Panel ata moels; unit roots; local power functions; serial correlation; inciental trens Acknowlegements: The authors woul like to thank Tassos Magalinos, Robert Taylor, the seminar participants of the Granger Centre for Time Series Econometrics an the participants of the 2th Conference on Research on Economic Theory an Econometrics for useful comments. *Corresponing author: e.tzavalis@aueb.gr (E. Tzavalis.

2 Introuction Panel unit root test statistics assuming fixe (finite time imension (T an large crosssectional imension ( have receive much interest in the literature over the last ecae because of their very goo small sample properties. Contributions in this area inclue the test statistics suggeste by Sargan an Bhargava (983, Breitung an Meyer (994, Harris an Tzavalis (999, 2004, Kruiniger an Tzavalis (2002, Bon et al. (2005, De Wachter et al. (2007, Kruiniger (2008, Han an Phillips (200 an De Blaner an Dhaene (20. Despite the plethora of stuies for the large-t panel unit root tests power properties, there are a few stuies in the literature investigating these for fixe-t panel unit root tests. Bon et al. (2005, Kruiniger (2008 an Masen (200 are focuse on unit root tests which assume white noise error terms an consier panel ata moels without inciental trens. De Wachter et al. (2007 erive the local power function of their IV base test for the case of MA( errors. In this paper, we exten the previous results aiming to stuy the effects of serial correlation an inciental trens. First, we erive the asymptotic local power functions of the within groups estimator base tests of Kruiniger an Tzavalis (2002 which allow for general forms of serial correlation of orer up to T 2 an for inciental trens. Secon we erive the asymptotic local power function for the IV estimator base tests of De Wachter et al. (2007. Our result is more general than their as it allows for serial correlation up to orer T 2 an also it is proven uner weaker assumptions. Thir we exten their IV test for the moel with inciental trens an we erive its local power function, also allowing for serial correlation. This comparison not only shows how serial correlation an inciental trens affect the power of the tests but also how the behaviour of tests varies with the estimator. Several contributions are mae by the paper. First, it is shown that, for the moel with iniviual effects, the instrumental variables (IV base test statistic suggeste by De Wachter et al. (2007 is a very powerful test statistic 2. To allow for serial correlation an to remove the panel ata initial conitions nuisance effects on testing for unit roots, this test statistic exploits orthogonal moments of panel ata iniviual series emeane by their initial observations uner the null hypothesis of unit roots. It is foun to have higher power than the within groups (WG base test statistic suggeste by Kruiniger an Tzavalis ( which relies on the within groups transformation matrix to become invariant to initial conitions. For large T, scale appropriately with T the IV base test statistic reaches its maximum power, which equals that of the common-point optimal test of Moon et al. (2007. See, e.g., Moon an Phillips (999, Breitung (2000, Moon an Perron (2004, Moon et al. (2006, 2007, Moon an Perron (2008, Harris et al. ( De Wachter et al. (2007 also propose a GMM counterpart of this statistic but this general set-up is not consiere as the results are less tractable. Furthermore, as argue in Han an Phillips (200, the effi ciency gains of GMM are marginal. 3 A similar statistic is also presente by Moon an Perron (2004 for large-t panels. 2

3 Secon, allowing for serial correlation has a ifferent impact on the power of each of the fixe-t panel unit root tests examine. This was expecte, since the moments use in estimation an testing proceures uner serial correlation of the error terms are exploite ifferently in every test. The power loss of the tests is more severe when the egree of serial correlation is large an negative. In this case, the within groups base test statistic becomes biase. In the case of positive serial correlation, the power reuctions of the IV base test statistic are unimportant, while the WG base test isplays power gains. Thir, the WG base test in the presence of heterogeneous iniviual trens suffers from the "inciental trens" (see e.g. Moon et al. (2007 problem, i.e. has trivial local power in a /2 neighbourhoo of unity. However, this problem appears only in the case of no time epenence in the error terms. Furthermore, contrary to the case without inciental trens, negative serial correlation increases the power of the test while positive reuces it. Fourth, we exten the IV test for the case of inciental trens. We fin that the test oes have power in all cases an that it behaves like the IV test in the iniviual intercepts only moel. As T goes to infinity it loses this property in a T /2 neighbourhoo of unity agreeing with the results of Moon et al. (2007 who also erive the power envelope for this case. Monte Carlo experiments show that in the first three cases the small sample results agree with the first orer asymptotic theory an thus valiate its usefulness. The empirical size an power are close to their theoretical counterparts for as small as 50 an as increases they converge even more. The IV test in the case of iniviual trens oes not behave well as its power is always close to its size. This may be explaine by the loss of effi ciency ue to the first ifferencing an the presence of complex eterministic components. The paper is organize as follows. Section 2 introuces the fixe-t test statistics an presents the require assumptions for the erivation of the asymptotic results. Section 3 erives the asymptotic local power functions an provies results on the behaviour of the tests. Section 4 conucts a small Monte Carlo exercise to examine the small sample performance of the asymptotic results an Section 5 conclues the paper. All proofs are relegate to the Appenix. In the following, we name the main iagonal of a matrix as iagonal 0, the first upper iagonal as iagonal +, the first lower iagonal as iagonal etc. 2 Moels an Assumptions Consier the following first orer autoregressive panel ata moels with iniviual effects: M : y i = ϕy i + ( ϕa i e + u i, i =,...,. ( M2 : y i = ϕy i + ( ϕa i e + ϕβ i e + ( ϕβ i τ + u i. (2 3

4 where y i = (y i,..., y it an y i = (y i0,..., y it are (T X vectors, u i is the (T X vector of error terms u it, an a i an β i are the iniviual coeffi cients of the eterministic components of the moels. a i coeffi cients reflect iniviual effects of the panel, while β i capture the slopes of iniviual linear trens, referre to as inciental trens. The (T X vectors e an τ have elements e t = an τ t = t for t =...T, i.e. a constant an a linear tren. To stuy the asymptotic local power of fixe-t unit root tests, efine the autoregressive coeffi cient ϕ as ϕ = c. Then, the hypothesis of interest becomes H 0 : c = 0 (3 H : c > 0, (4 where c is the local to unity parameter. The asymptotic istributions of fixe-t panel unit root test statistics allowing for serial correlation or heterosceasticity in error terms u it uner the sequence of local alternatives ϕ can be erive by making the following assumptions. Assumption : (.a{u i } constitutes a sequence of inepenent normal ranom vectors of imension (T X with mean E(u i = 0 an variance-autocovariance matrix E(u i u i = Γ [γ ts ], where γ ts = E(u it u is = 0 for s = t+p max +,..., T, an p max T 2. (.b γ tt > 0 for at least one t =,..., T. (.c The 4 + δ th population moments of y i, i =,..., are uniformly boune. That is, for every l R T such that l l =, E( l y i 4+δ < B < + for some B, where is the ifference operator. (. l V ar(vec( y i y il > 0 for every l R 0.5T (T + such that l l =. Assumption 2: The iniviual coeffi cients a i an β i, an the initial observations of moels M an M2, y i0, satisfy the following conitions: E(u it a i = 0, E(u it β i = 0 an E(u it y i0 = 0, for t =,..., T an i =,...,, an V ar(y i0 < +. Assumption (.a implies that the orer of serial correlation of error term u it can be at most T 2. It requires the existence of at least one moment conition, in conucting inference about the true value of ϕ, which is free of correlation nuisance parameters. That is, it implies that at least, γ T = γ T = 0. Assuming normality in the error terms allows for close form representations of the variances of the limiting istributions of the tests 4. Assumption (.b imposes finite fourth moments on initial conitions y i0, error terms u it an iniviual coeffi cients a i an β i of moels M an M2. Along with assumptions (.c an (., they allow application of the Markov LL an the Lineberg -Levy CLT, an ensure that all quantities in the enominators of the estimators of ϕ are non-zero. Assumption 2 is require only when c > 0. Uner null hypothesis H 0 : c = 0, all test 4 General representations of the asymptotic local power functions can be straightforwarly erive uner non-normality or cross section heterogeneity of the error term. The results will be the same but with ifferent variances. The assumptions on the errors will also be strengthene to those neee by the Lineberg-Feller CLT an to ensure that a non-parametric variance estimator is consistent. For easier exposition of results we o not assume these cases here. 4

5 statistics consiere in the paper are invariant to y i0 an/or coeffi cients α i an β i. This is achieve either by subtracting y i0 from the levels of all iniviual series y it of moels M an M2 (see IV an F DIV statistics, 5 or by the within groups transformation of y it (see WG an WGT statistics. 6 Uner the local alternative hypothesis H : c > 0, the assumption that V ar(y i0 < + allows for constant, ranom an mean stationary initial conitions. Covariance stationarity of y i0, implying V ar(y i0 = σ2 ϕ 2 (see Kruiniger (2008 an Masen (200, although it oes not affect the limiting istributions uner the null, is not allowe uner the local alternatives. This is because, as also note by Moon et al. (2007, it implies that V ar(y i0 when ϕ, which means that the variance of the initial conition increases with the number of cross-section units, which is not meaningful for cross-section ata sets. To stuy the asymptotic local power of the tests, we employ a "slope" parameter, enote as k, which is foun in local power functions of the form Φ(z a + ck, where Φ is the stanar normal cumulative istribution function an z a enotes the α-level percentile. Since Φ is strictly monotonic, a larger k means greater power, for the same value of c. If k is positive, then the tests will have non-trivial power. If it is zero, they will have trivial power, which is equal to a, an if it is negative they will be biase. 3 Asymptotic local power functions This section presents the fixe-t panel unit root test statistics consiere an it erives their limiting istributions uner the sequence of local alternatives. The first part of the section presents results for moel M, while the secon for moel M2. 3. Iniviual intercepts The IV panel unit root test statistic (see De Wachter et al. (2007: This test statistic assumes an orer of serial correlation p T 2 an it is base on transformation of the iniviual series of the panel in eviations from their initial conitions, given as z it = y it y i0. The statistic becomes invariant to the serial correlation effects by exploiting the following 5 This approach is suggeste by Schmit an Phillips (992, for single time series, an Breitung an Meyer (994 for the iniviual series of panel ata moels with iniviual effects. 6 This transformation means that one subtracts the means of the iniviual series of the panel from their levels, across all units. This transformation is also employe by the panel unit root tests of Harris an Tzavalis (999, an Levin et al. (

6 moment conitions: E [ T p t= an it is base on the IV estimator ˆϕ IV = ( i= z it u i,t+p+ (ϕ T p t= ] z it z it+p ( = 0, i =,...,, (5 i= T p t= z it z it+p+ The moments given by (5 can be rewritten in matrix notation as follows:. (6 E(z i Π p u i = 0, (7 where Π p is a (T XT matrix selecting zero-mean moments, accoring to (5, an z i = y i y i0 e. In particular, Π p has ones in the pth iagonal an zeros everywhere else. Given the efinition of Π, the above IV estimator can be rewritten as ( ( ˆϕ IV = z i Π p z i z i Π p z i i= The asymptotic istribution of the IV base unit root test statistic uner the sequence of local alternatives ϕ = i= c is erive in the next theorem. Theorem Uner Assumptions, 2, we have /2 V IV (ˆϕ IV ( ck IV,, (9 (8 as, where k IV = VIV (0 an V IV = 2tr((A IV Γ 2 tr(λ Π pλγ 2, with A IV = 2 (Λ Π p +Π pλ, is the variance of the limiting istribution of ˆϕ IV. The efinition of matrix Λ is given in the appenix (see proof of the theorem. The limiting istribution of the IV test statistic given by Theorem nests the istributions uner the null an alternative hypotheses H 0 : c = 0 an H : c > 0, respectively. For c = 0, (9 gives the istribution of the test statistic uner H 0, erive by De Wachter et al. (2007. The test statistic of Breitung an Meyer (994 can be seen as a special case of the IV test, for p = 0. 7 The only unknown quantity in the variance V IV is Γ. If Γ = σ 2 ui T, where I T is the (T XT ientity matrix, then no estimation is neee since σ 2 u cancels out from both the nominator an enominator i.e. 2tr((A IV Γ 2 tr(λ Π pλγ = 2tr((A IV 2 2 tr(λ Π pλ. In the more general 2 7 As Bon et al. (2005 show, in this case ˆϕ IV can be also seen as a maximum likelihoo estimator of ϕ. 6

7 case that Γ σ 2 ui T, an estimator of Γ can be obtaine uner null hypothesis H 0 : c = 0 as ˆΓ = y i y i, ( i= since y i = u i uner this hypothesis. The results of Theorem show that the IV test statistic has always non-trivial power, since the slope parameter of the local power function k IV is always positive. This parameter epens on the time imension of the panel T, the orer of serial correlation p an the form of serial correlation consiere by variance-covariance matrix Γ. In the case where error terms u it follow MA( process u it = v it + θv it, for all i, (2 with v it IID(0, σ 2 u, then a close form of k IV, efine as k IV (p, θ for ifferent values of p an θ, is given in the next corollary. Corollary If error terms u it follow MA( process (2, an Assumptions an 2 hol, then the slope parameter k IV (p, θ is given as k IV (0, 0 = 2 (T 2 T (3 an k IV (, θ = D,IV θ 2 + D 2,IV θ + D,IV R,IV θ 4 + R 2,IV θ 3 + R 3,IV θ 2 + R 2,IV θ + R,IV (4 where D i,iv an R j,iv, for i =, 2 an j =, 2, 3, are functions of T given in the appenix. Close form solutions of k IV (2, 0 an k IV (3, 0 are also given in the appenix. The results of Corollary can be employe to examine how the values of nuisance parameter θ affect the local power of the IV base panel unit root test statistic. To this en, Figure presents values of k IV (p, θ across T, for p {0, } an θ { 0.9, 0.5, 0, 0.5, 0.9}. Inspection of Figure clearly inicates that the IV test statistic has its maximum asymptotic local power, when p = 0 an θ = 0. This can be attribute to the fact that, in this case, the test exploits the maximum number of possible moment conitions in (5. If p = (implying that one moment conition is lost, then the power of the test ecreases. Finally, the test has much higher power if θ > 0 than θ < 0. This can be attribute to the fact that θ > 0 increases the variability of y it, thus making it easier for the test to istinguish between hypotheses H 0 : c = 0 an H : c > 0. In this case, the variance of estimator ˆϕ IV ecreases. On the other han, θ < 0 reuces the variability of y it an thus, the IV test statistic is harer to istinguish H 0 : c = 0 from H : c > 0. Inepenently of the sign of θ, the plotte 7

8 values of k IV (p, θ, given by Figure, clearly inicate that the power of the IV test increases with T. The WG panel unit root test statistic (see Kruiniger an Tzavalis (2002: This test statistic becomes invariant to initial conitions y i0 of the panel by taking the within groups transformation of the iniviual series y it, using the annihilator matrix Q = I T e(e e e, where I T is the (T XT ientity matrix. Then, the least squares estimator of the transforme series is given as ( ( ˆϕ W G = y i Qy i y i Qy i. (5 i= Since ˆϕ W G is not a consistent estimator of ϕ, ue to the above transformation of y it an the presence of serial correlation in error terms u it, Kruiniger an Tzavalis (2002 suggeste the following fixe-t WG test statistic: or i= ˆδW G (ˆϕ W G ˆb W G ˆδW G ( /2 V ˆδ W G W G ˆϕ W G ˆb W G ˆδW G (0, V W G, (6 (0,, which corrects estimator ˆϕ W G for the above two sources of its inconsistency, where ˆδ W G = y i Qy i is the enominator of estimator ˆϕ W G scale by, ˆb W G = tr(ψ p,w G ˆΓ is a ˆδW G ˆδW G i= 8

9 consistent estimator of the inconsistency of ˆϕ W G, given as tr(λ QΓ, an Ψ tr(λ QΛΓ p,w G is a (T XT - imension selection matrix having in its p,.., 0,...p iagonals the corresponing elements of matrix Λ Q, an zero everywhere else. ˆΓ = y i y i an V W G = 2tr((A W G Γ 2 is the variance of the limiting istribution of the correcte for its inconsistency WG estimator ˆϕ W G, where A W G = 2 (Λ Q + QΛ Ψ p,w G Ψ p,w G.8 i= This variance can be consistently estimate provie consistent estimates of Γ. As for the IV test statistic, this can be one base on (. The WG unit root test statistic is base on the same testing principle with the IV test statistic, escribe above. It exploits moments of the numerator of ˆϕ W G which have zero mean uner H 0 : c = 0. But, this now is one for the correcte for its inconsistency estimator ˆϕ W G ˆb W G ˆδW G through the selection matrix Ψ p,w G. 9 Moon an Perron (2004 have suggeste a version of the WG test statistic for the case that both an T go infinity. The next theorem gives the limiting istribution of the WG statistic uner the sequence of local alternatives ϕ = c. Theorem 2 Uner Assumptions, 2, we have as, where ( /2 V ˆδ W G W G ˆϕ W G ˆb W G ( ck W G,, (7 ˆδW G k W G = tr(λ QΛΓ + tr(f QΓ tr(ψ p,w G ΛΓ tr(λ Ψ p,w G Γ VW G (8 an F = Ω ϕ ϕ=, where Ω is given in the appenix. 8 ote that the WG test statistic, given by 6, has been reformulate to avoi computing selection matrix S of Kruiniger an Tzavalis (2002, which is very emaning. The relationship between the two alternative formulations of the test statistics can be seen by noticing that ( tr(ψ p,w GˆΓ = vec(qλs vec( y i y i an 2tr((A W G Γ 2 = vec(qλ (I T 2 SV ar(vec( y i y i(i T 2 Svec(QΛ, where I T 2 is the (T 2 XT 2 ientity matrix an S is a (T 2 XT 2 iagonal selection matrix, with elements s st efine as s (s T +t,(s T +t = (γ ts = 0 with s, t =, 2,..., T an (. is the Dirac function. 9 To unerstan more clearly the role of selection matrix Ψ p,w G, assume T = 3 an consier that error terms u it follow MA( process (2. Then, matrix Γ becomes Γ = σ 2 u( + θ 2 σ 2 uθ 0 2 σ 2 uθ σ 2 u( + θ 2 σ uθ an Ψ,W G is given as Ψ,W G = 0 σ 2 uθ σ 2 u( + θ i= 9

10 The results of Theorem 2 inicate that annihilator matrix Q an the inconsistency correction of estimator ˆϕ W G, ˆb W G ˆδW G, base on Ψ p,w G, makes more complex the local power function. As equation (8 shows, the slope parameter of this function k W G epens on the following quantities: tr(λ QΛΓ, tr(f QΓ, tr(ψ p,w G ΛΓ an tr(λ Ψ p,w G Γ. The first two quantities come from the annihilator matrix Q an the last two from selection matrix Ψ p,w G. For p = 0, the effects of matrix Ψ p,w G isappear, since tr(ψ p,w G ΛΓ = tr(λ Ψ p,w G Γ = 0. To stuy the effects of the serial correlation nuisance parameters an lag-orer p on k W G, next corollary gives analytic formulas of k W G, for p {0, } an θ { 0.9, 0.5, 0, 0.5, 0.9}, while Figure 2 plots values of these formulas across T. Corollary 2 If error terms u it follow the MA( process in (2, an Assumptions an 2 hol, then slope parameter k IV (p, θ is given as k W G (0, 0 = 3(T T 2 2T 4T + 5, for p = 0 an θ = 0, (9 an k W G (, θ = (T 2(T θ 2 θ 2 + 3T θ 7θ + T, (20 2T R,W G θ 4 + R 2,W G θ 3 + R 3,W G θ 2 + R 2,W G θ + R,W G where R,W G, R 2,W G an R 3,W G are functions of T efine in the appenix. The appenix also gives analytic formulas of k W G (p, θ, for p =, 2, 3 an θ = 0. As can be seen from Figure 2, the effects of θ an p on the power of the WG test iffer from those on the power of the IV test. This can be attribute to the within groups transformation of iniviual series y it an the correction of estimator ˆϕ W G for its inconsistency. For positive values of θ, the WG test statistic has more power than for θ = 0. For θ > 0, the power also increases with T. These results are in contrast to those for the IV test statistic. For θ negative, the WG test statistic becomes biase, something that never happens for the IV test statistic. This happens because the inconsistency correction affects slope parameter k W G (p, θ through quantity tr(ψ p,w G ΛΓ + tr(λ Ψ p,w G Γ. For θ < 0, this quantity takes positive values an, thus, reuces the power of the WG test statistic. For θ > 0, it becomes negative an thus, it moves the limiting istribution towars the critical region, increasing the power of the test. As T increases, the above sign effects of θ on the W G test statistic are amplifie. That is, they lea to a test with greater power an bias, if θ > 0 an θ < 0, respectively. Finally, comparison between k W G (p, θ an k IV (p, θ reveals that the IV test is more powerful than the WG test statistic. This is true for all values of θ an p consiere, an across T. It can be also seen by the results of Table, which presents values of slope parameter k for the IV an WG test statistics for T {7, 0}, p {0, } an θ { 0.9, 0.5, 0, 0.5, 0.9}. Extensions to higher orer of serial correlation are conceptually similar but less tractable. 0

11 The limiting istributions of the IV an WG test statistics given by Theorems an 2, respectively, scale appropriately by T become invariant to the serial correlation nuisance parameters, if T, jointly. This result is establishe in the next proposition, which erives the limiting istributions of the scale by T versions of the IV an WG test statistics uner the following sequence of local alternatives: ϕ T = c T, consiere in the large-t panel ata literature (see, e.g., Moon et al. (2007. Proposition Let Assumptions an 2 hol. Then, uner ϕ T = T ( 2 (ˆϕ IV T ( 3 ˆδW G ( ˆϕ W G ˆb W G ˆδW G c T, we have ( c,, an (2 2 if T, jointly an the following conition hols: /T 0. ( c0,, (22 Conition /T 0 is require only uner alternative hypothesis H : c > 0. Uner null hypothesis H 0 : c = 0, it is not neee (see, e.g., Harris an Tzavalis (999, 2004, an Hahn an Kuersteiner (2002. The results of the proposition apply for every fixe orer of

12 serial correlation p an any form of short term serial correlation. For c = 0, the limiting istribution of estimator ˆϕ IV, given by (2, coincies with that erive by De Wachter et al. (2007, while the limiting istribution of estimator ˆϕ W G ajuste for its inconsistency correspons to that erive by Moon an Perron (2008. Table : Values of slope parameter k T=7 θ k IV k W G T=0 p\θ k IV k W G For c > 0, the IV test reaches its maximum local power, which is equal to that of the common-point optimal test of Moon et al. (2007, enote as MPP. However, the WG test has trivial power, since k W G = 0. This happens because the last test ajusts only the numerator of ˆϕ W G for its inconsistency, in contrast to Harris an Tzavalis (999 (enote HT an Levin s et al. (2002 (enote LLC tests. The latter tests ajust both the numerator an enominator of ˆϕ W G for its inconsistency. Moon an Perron (2008 show that the WG test has non-trivial power in a n /4 T neighbourhoo of the null hypothesis. Values of the slope parameter of the power function of the above tests, for large T, are reporte in Table 2. For comparisons, Table 2 also reports values of k for the large-t panel unit root tests of Im et al. (2003 (enote IPS, an Sargan s (SGLS test statistic (see Moon an Perron (2008. Values of k for these tests are obtaine in Moon et al. (2007, Moon an Perron (2008 an Harris et al. (200. Table 2: Slopes of large-t tests. IV MPP LLC/HT SGLS IPS WG / 2 / 2 (3/2 (5/5 / Inciental trens To stuy the power of fixe-t panel ata unit root tests allowing for serial correlation in the case of inciental trens, this section extens the IV test presente in the previous section. This extension requires that the IV test is base on a first ifference of panel ata series y it, an it will be enote as F DIV. 2

13 FDIV panel unit root test: Taking first ifferences of moel M2 yiels y i = ϕ y i + ( ϕβ i e + u i, i =,...,, (23 where y i = (y i2,..., y it, y i = (y i,..., y it, y i 2 = (y i0,..., y it 2, u i = (u i2,..., u it, u i = (u i,..., u it an e = (,,..., are (T X vectors. Subtracting from both sies of moel (23, the vector of the first ifference of initial observation y i e gives the following first ifferences transformation of the moel: yi = ϕyi + ( ϕa i e + u i, i =,...,, (24 where y i = y i y i e, y i = y i y i e, a i = (β i y i an u i = u i. Moel (24 clearly shows that, if error terms u it are serially correlate, moments similar to (7 can be exploite to test the null hypothesis of a unit root, i.e. E(y i Π pu i = 0, (25 where Π p is a (T X(T matrix with unities in its p+ iagonal, an zeros everywhere else. If we efine E(u i u i = Θ, then. a consistent estimator of Θ uner H 0 : c = 0 is given as ˆΘ = i= y i y i, (26 which correspons to (, for y i = u i. It can be easily seen that Θ = 2Γ Γ Γ, where Γ = E(u i u i an Γ = E(u i u i. But, as will be thoroughly explaine later on, Γ an Γ cannot be consistently estimate uner H 0 : c = 0 base on y i ue to the presence of inciental trens. Theorem 3 erives the limiting istribution of the IV estimator uner the sequence of local alternatives ϕ = Theorem 3 Uner Assumptions, 2 an as, c, exploiting the above moment conitions. /2 V F DIV (ˆϕ F DIV ( ck F DIV,, (27 where k F DIV = tr(λ Π pλ Θ 2tr((AF DIV Θ 2 (28 an ˆϕ F DIV = Π ( i= y i Π py i ( i= y i Π py i p Λ. Λ is a (T X(T version of Λ., V F DIV = 2tr((A F DIV Θ 2 tr(λ Π pλ Θ 2, A F DIV = 2 (Λ Π p+ The results of Theorem 3 inicate that, as with the IV test, the power of the F DIV test 3

14 statistic epens on the serial correlation nuisance parameters an lag-orer p, as well as the time imension of the panel. Corollary 3 erives the value of the slope parameter k F DIV, if error terms u it follow MA( process. Corollary 3 If error terms u it follow MA( process (2, an Assumptions an 2 hol, then slope parameter k F DIV (p, θ is given as k F DIV (p, 0 = T p 3 2(T p 2 (29 an k F DIV (, θ = (T 4θ 2 θ + T 4, (30 2(P θ 4 + P 2 θ 3 + P 3 θ 2 + P 2 θ + P where polynomials P, P 2, an P 3 are efine in the appenix. Table 3 presents values of k F DIV (p, θ, obtaine through relationship (30, for p = {0, }, T {7, 0} an θ { 0.9, 0.5, 0, 0.5, 0.9}. The results of the table inicate that the FDIV test has non-trivial power for all values of p an θ consiere. The power of the test increases slowly with T, as with the WG test. However, if T, it can be shown that k F DIV = T p 3 T 2(T p 2 0, which means that the inciental parameter problem remains. This is ue to the normalization of the statistic with T. These results mean that the asymptotic power of the FDIV test comes from the assumption that T is fixe an the presence of serial correlation. A positive value of θ tens to increase the power of the test, as it happens with the IV test for moel M. The WG unit root statistic: The version of the WG test statistic in the case of inciental trens (enote as WGT consiers an augmente annihilator matrix, given as Q = I T X(X X X, where X = [e, τ]. Uner null hypothesis H 0 : c = 0, multiplying moel M2 with Q leas to a transforme moel without iniviual effects an inciental trens. The WGT test statistic is base on the least squares estimator of the autoregressive coeffi cient ϕ of the transforme moel, enote as ˆϕ W GT. As with ˆϕ W GT, this estimator is ajuste for its inconsistency. The latter is ue to the above transformation of iniviual series y it an the presence of serial correlation in error terms u it. To correct ˆϕ W GT for its inconsistency coming from the serial correlation in u it, we can no longer rely on the previous estimator of variance-covariance matrix Γ, ˆΓ, given as ˆΓ = y i y i (see ( This happens because y i epens on the nuisance parameters of the inciental trens β i, for moel M2, i.e. y i = β i e + u i, i= 4

15 which implies p lim ˆΓ = p lim y i y i = Γ + E(β 2 i ee. (3 i= To remove the effects of β i from the estimator of matrix Γ, the following selection matrix will be efine. 0 Let matrix M have elements m ts = 0 if γ ts 0 an m ts = if γ ts = 0. Then, tr(mγ = 0 an, thus, we have p lim tr(mee y im y i = E(β 2 i. (32 i= The last relationship can be employe to substitute out iniviual effects E(β 2 i from (3, an thus to provie a consistent estimator of Γ an tr(λ Q Γ uner null hypothesis H 0 : c = 0 which is net of β i. Base on relationships (3 an (32, we can efine selection matrix Φ p,w GT = Ψ p,w GT tr(λ Q M e Me M, where Ψ p,w GT is a (T XT matrix having in its iagonals { p,.., 0,...p} the corresponing elements of matrix Λ Q, an zero everywhere else. This matrix has the property tr(φ p,w GT ee = 0, which leas to the following consistent estimator of tr(λ Q Γ: p lim tr(φ p,w GT ˆΓ = tr(λ Q Γ. (33 The limiting istribution of ˆϕ W GT correcte for its inconsistency uner ϕ = c is given in the next theorem. Theorem 4 Uner Assumptions, 2 an as +, where ˆϕ W GT = ( ˆV 2 ˆδ W GT W GT ˆϕ W GT ˆb W GT ( ck W GT,, (34 ˆδW GT k W GT = tr(λ Q Γ + tr(f Q Γ tr(φ p,w GT ΛΓ tr(λ Φ p,w GT Γ, (35 2tr((A W GT Γ 2 ( y i Q y i y i Q tr(φ y i, p,w GT ˆΓ, an V i= y i, Q y W GT = i, ( i= i= ˆbW GT ˆδW GT = 2tr((A W GT Γ 2, with A W GT = 2 (Λ Q + Q Λ Φ p,w GT Φ p,w GT, is the variance of the limiting istribution of the W GT test. The implementation of the W G test statistic is base on the estimator of Γ given by ˆΓ. As was mae clear by our analysis above, premultiplying ˆΓ by selection matrix Φ p,w GT reners this estimator net of the inciental trens nuisance parameters effects. The results 0 ote that, as in case of moel M (see fn 8, this selection matrix simplifies consierably the computation of the WGT test statistic, compare with the selection matrix S use by Kruiniger s an Tzavalis (

16 of Theorem 4 imply that, if there is no serial correlation, test statistic WGT has trivial power. This is true for any orer of serial correlation p. These results are establishe in next corollary, which erives values of the power slope parameter k W GT (p, θ uner MA process (2 of u it, for ifferent values of p an θ. Corollary 4 If error terms u it follow MA( process (2, an Assumptions an 2 hol, then, the values of slope parameter k W GT (p, θ are given as k W GT (p, 0 = 0, for p = 0,, 2,..., T 2, (36 an k W GT (, θ 0, for θ 0. (37 Values of k W GT (p, θ, for p = {0, }, T {7, 0} an θ { 0.9, 0.5, 0, 0.5, 0.9}, are given in Table 3. These inicate that test statistic WGT has asymptotic local power, if θ < 0. This power is less than that of the FDIV for θ < 0, an it increases slowly with T. This power can be attribute to the effects of quantities tr(φ p,w GT ΛΓ an tr(λ Φ p,w GT Γ on slope parameter k W GT (p, θ. As for the FDIV test, it can be shown that the large-t version of the WGT test has trivial power when T. Table 3: Values of slope parameter p. T=7 θ k F DIV k W GT T=0 θ k F DIV k W GT Simulation Results To see how well the asymptotic local power functions of the tests erive in the previous section approximate their small sample ones, this section presents the results of a Monte Carlo stuy base on 5000 iterations. For each iteration, we calculate the size of the tests at 5% level (i.e., for c = 0 an their power (i.e., for c =, assuming that error terms u it follow MA process (2. This is one for {50, 00, 200, 300, 000}, T {7, 0}, θ { 0.9, 0.5, 0.0, 0.5, 0.9} an p {0, }. The orer of serial correlation p is zero in the case of θ = 0.0, otherwise it is set to p =. The nuisance parameters of moels M an M2 that o not appear in the above local power functions are set to zero, i.e., a i = 0, β i = 0, y i0 = 0, for all i without loss of generality. Aitional simulations (not reporte here have 6

17 shown that the results remain qualitatively the same for non-zero values of the parameters even for as small as 50. Table 4: Size an power of the IV an WG tests. T=7 T= TV TV θ = 0.9 c=0 IV WG c= IV WG θ = 0.5 c=0 IV WG c= IV WG θ = 0 c=0 IV WG c= IV WG θ = 0.5 c=0 IV WG c= IV WG θ = 0.9 c=0 IV WG c= IV WG Tables 4 an 5 present the results of our simulation stuy. Table 4 presents the results for the test statistics base on moel M, while Table 5 presents those for the test statistics base on moel M2. In the tables, "TV" enotes the theoretical values of the power of the tests obtaine from their asymptotic power functions erive in the previous section. The results of Table 4 clearly inicate that, for moel M, the IV test has higher power than that 7

18 of the WG test inepenently of T, as is preicte by the theory. For θ 0, the asymptotic power function of the test approximates suffi ciently its small sample value even for small, i.e., = {50}. Table 5: Size an local power of FDIV an WGT tests. T=7 T= TV TV θ = 0.9 c=0 FDIV WGT c= FDIV WGT θ = 0.5 c=0 FDIV WGT c= FDIV WGT θ = 0 c=0 FDIV WGT c= FDIV WGT θ = 0.5 c=0 FDIV WGT c= FDIV WGT θ = 0.9 c=0 FDIV WGT c= FDIV WGT However, for θ < 0, the power of the test consierably reuces, an its small sample estimate eviates from its theoretical value, TV. This can be obviously attribute to secon, or higher orer effects, which are not capture by the first orer approximation of the local power function. As is preicte by the theory (see Table, the WG test tens to have power only 8

19 for θ 0. ote that, for θ { 0.9, 0.5}, this test loses its power an becomes biase. Finally, note that both the IV an WG test statistics have size which is close the nominal level value 5%. The size performance of both tests improves, as an T increases. Regaring the test statistics for moel M2, the results of Table 5 inicate that the IV base test statistic, enote as FDIV, no longer performs satisfactorily. It is biase in small samples, an its power eviates substantially from that preicte by its asymptotic local power function. This is true inepenently of the values of θ, T an consiere in our simulation analysis. This result can be attribute to the poor approximation of the asymptotic local power function in small samples, ue to first ifferencing an the presence of more complicate eterministic terms (see also Moon et al. (2007 an Han an Phillips (200. In contrast to the FDIV test, the WGT test is foun to have some power in small samples. As is preicte by the theory, the test has power if θ < 0. As increases, the power of the WGT test converges to its asymptotic local power value from below. The table also inicates that the WGT test can have power in samples of small even if θ 0, where their asymptotic local power inicates that shoul be biase, or have trivial power. 5 Conclusions This paper examine the power properties of fixe-t panel ata unit root tests uner serial correlation an inciental trens, assuming that only the cross-section imension of the panel ( grows large. Two types of tests have been propose in the literature; the WG tests of Kruiniger an Tzavalis (2002 an the IV tests of De Wachter et al. (2007. The latter were extene to accommoate inciental trens. Analytic forms of the power functions were also erive for the case that the error terms of the panel follow a moving average proceure of lag-orer one, often assume in practice for many economic series. The results given by the paper lea to the following main conclusions. First, for the panel ata moel without inciental trens, the IV base test clearly outperforms the WG base test. This can be attribute to the fact that the last test requires an ajustment of the WG estimator for its inconsistency, ue to the iniviual effects an the presence of serial correlation in the error terms. The power of the IV base test is bigger uner positive correlation of the error terms than uner negative, an it is ecreasing as the orer of serial correlation increases. Secon, for the moel with inciental trens, the FDIV moel is foun to have nontrivial power in a /2 neighbourhoo of unity while the WG base test is foun to have non-trivial power only in the presence of serial correlation. The latter has always power when the serial correlation in the error term is negative. This non-trivial power can be attribute to the impact of the inconsistency correction, require by the WG estimator, for 9

20 the serial correlation nuisance parameters. For panel ata moels with inciental trens, the IV base test is foun to be biase in small samples, espite its very goo asymptotic properties. This is true inepenently of the sign of serial correlation of the error terms. The asymptotic local power of this test is foun to be a very ba approximation of its true power. These results suggest employing the above WG base fixe-t panel unit root tests in mitigating the inciental trens problem in short panels with serially correlate error terms. References [] Bon, S., auges, C., an Winmeijer, F., Unit roots: Ientification an testing in micropanels. Cemmap Working Paper CWP07/05, The Institute for Fiscal Stuies, UCL. [2] Breitung J., The local power of some unit root tests for panel ata. In Bai H. Baltagi, Thomas B. Fomby, R. Carter Hill (e. onstationary Panels, Panel Cointegration, an Dynamic Panels (Avances in Econometrics, Volume 5, Emeral Group Publishing Limite, pp [3] Breitung J., Meyer W., 994. Testing for unit roots using panel ata: are wages on ifferent bargaining levels cointegrate? Applie Economics 26, [4] De Blaner, R., Dhaene, G., 202. Unit root tests for panel ata with AR( errors an small T. The Econometrics Journal, 5, [5] De Wachter, S., Harris, R.D.F., Tzavalis, E., Panel unit root tests: the role of time imension an serial correlation. Journal of Statistical Inference an Planning, 37, [6] Dickey D.A., an Fuller W.A., 979. Distribution of estimators of autoregressive time series with a unit root. Journal of the American Statistical Association 74, [7] Hahn, J., Kuersteiner, G., Asymptotically unbiase inference for a ynamic panel moel with fixe effects when both n an T are large. Econometrica. 70, [8] Han C., & Phillips, Peter C. B., 200. Gmm Estimation For Dynamic Panels With Fixe Effects An Strong Instruments At Unity. Econometric Theory, Cambrige University Press, vol. 26(0, pages 9-5. [9] Harris R.D.F. an E. Tzavalis 999. Inference for unit roots in ynamic panels where the time imension is fixe. Journal of Econometrics, 9,

21 [0] Harris R.D.F an E.Tzavalis Inference for unit roots for ynamic panels in the presence of eterministic trens: Do stock prices an iviens follow a ranom walk? Econometric Reviews 23, [] Harris D., Harvey D., Leybourne S., an Sakkas., 200. Local asymptotic power of the Im-Pesaran-Shin panel unit root test an the impact of initial observations. Econometric Theory 26, [2] Hayakawa K., 200. A unit root test for micro panels with serially correlate errors. Working Paper, Hiroshima University. [3] Im, K.S., M.H. Pesaran, an Y. Shin, Testing for unit roots in heterogeneous panels. Journal of Econometrics 5, [4] Karavias, Y., an Tzavalis, E., 202. Testing for unit roots in short panels allowing for structural breaks. Computational Statistics an Data Analysis. (In press. [5] Kruiniger, H., Maximum likelihoo estimation an inference methos for the covariance stationary panel AR(/unit root moel. Journal of Econometrics 44, [6] Kruiniger, H., an E., Tzavalis, Testing for unit roots in short ynamic panels with serially correlate an heterosceastic isturbance terms. Working Papers 459, Department of Economics, Queen Mary, University of Lonon, Lonon. [7] Levin, A., Lin, F., an Chu, C., Unit root tests in panel ata: asymptotic an finite-sample properties. Journal of Econometrics 22, [8] Masen E., 200. Unit root inference in panel ata moels where the time-series imension is fixe: a comparison of ifferent tests. Econometrics Journal 3, [9] Moon, H.R., Perron, B., Testing for a unit root in panels with ynamic factors. Journal of Econometrics 22, [20] Moon, H.R., Perron, B., Asymptotic local power of poole t-ratio tests for unit roots in panels with fixe effects. Econometrics Journal, [2] Moon, H.R., Perron, B., Phillips, P.C.B., A note on The local power of some unit root tests for panel ata by J. Breitung. Econometric Theory 22, [22] Moon H.R., Perron B. & Phillips P.C.B., Inciental trens an the power of panel unit root tests. Journal of Econometrics, 4(2,

22 [23] Sargan, J. D. an A. Bhargava (983. Testing resiuals from least squares regression for being generate by the Gaussian ranom walk. Econometrica 5, [24] Schott J.R Matrix Analysis for Statistics, Wiley-Interscience. [25] Schmit P., an Phillips P.C.B., 992. Testing for a unit root in the presence of eterministic trens. Oxfor Bulletin for Economics an Statistics 54(3,

23 6 Appenix Proof of Theorem Uner the sequence of local alternatives ϕ = c, the IV test statistic can be written as follows: (ˆϕIV ϕ = = i= i= z i Π p u i + z i Π p u i + ( ϕ (a i y i0 z i Π p e i= z i Π pz i i= i= ( ϕ (a i y i0 z i Π p e = z i Π pz i i= + ϕ ϕ (α + (b, (38 (g where (α Uner H : i= z i Π p u i, (b i= c > 0, vector y i can be expane as ( ϕ (a i y i0 z i Π p e an(g z i Π p z i. i= y i = wy i0 + Ωe( ϕ a i + Ωu i, i =, 2,...,, (39 where Ω = ϕ.. ϕ 2 ϕ ϕ T 2 ϕ T 3.. ϕ 0 an w = (, ϕ, ϕ 2,..., ϕt. ote that, for ϕ =, we have Ω Λ. The first orer Taylor expansions of Ω an w yiels (40 respectively, where F = Ω ϕ Ω = Λ + F (ϕ + o( an (4 w = e + f(ϕ + o(, (42 ϕ = an f = w ϕ ϕ =(see also Masen (200. Using these 23

24 expansions, vector z i = y i y i0 e can be written as z i = y i ey i0 = (w ey i0 + Ωe( ϕ a i + Ωu i. (43 Substituting (43 into quantity (a, efine by (38, yiels (a = i= i= z i Π p u i = i= ((w ey i0 + Ωe( ϕ a i + Ωu i Π p u i y i0 (w e Π p u i + ( ϕ a i e Ω Π p u i + u i Ω Π p u i. This quantity has a limiting istribution (0, 2tr((A IV Γ 2, since the following results hol: i= i= i= y i0 (w e Π p u i p 0 (44 ( ϕ a i e Ω p Π p u i 0 (45 u i Ω Π p u i (0, 2tr((A IV Γ 2 (46 The results given by (44 an (45 can be erive by using (4-(42 an stanar results on quaratic forms (see Schott (997, while (46 can be prove using tr(λ Π p Γ = tr(f Π p Γ = 0 an Lineberg-Levy s CLT. For quantities (b an (g, the following results can be easily erive: (b an (g i= i= ( ϕ (a i y i0 z i Π p e p 0 (47 z i Π p p z i tr(λ Π p ΛΓ. (48 otice in (47 how the initial conition isappears uner the local alternatives as well. This 24

25 is because i= ( ϕ y i0 z i Π p e = c = c = c = c = c2 3/2 = o p ( y i0 z i Π p e i= y i0 ((w ey i0 + Ωe( ϕ a i + Ωu i Π p e i= (w e yi0π 2 p e + o p ( i= (ϕ f yi0π 2 p e + o p ( i= f yi0π 2 p e + o p ( i= Using (44-(48 yiels ( (ˆϕIV ϕ 0, 2tr((A IV Γ 2 tr(λ Π p ΛΓ 2 (ˆϕIV c ( 0, 2tr((A IV Γ 2 tr(λ Π p ΛΓ 2 (ˆϕIV ( c, V IV ( /2 V IV (ˆϕ IV cv /2 IV,, (49 where V IV = 2tr((A IV Γ 2 tr(λ Π pλγ 2, which proves Theorem. Proof of Corollary The results of the corollary can be prove base on the formula of the variance of the limiting istribution of the IV test V IV = 2tr((A IV Γ 2 tr(λ Π pλγ, A 2 IV = 2 (Λ Π p + Π pλ, given by Theorem. Uner no serial correlation (i.e., θ = 0 an Γ = σ 2 ui T, the following relationship hols: 2tr(A IV 2 = tr(λ Π p Λ, for all p. (50 25

26 This yiels tr(λ Π p Λ = 2 (T 2 T, for p = 0 tr(λ Π p Λ = [ (T 2(T ]2 2, for p = T (T tr(λ Π p Λ = T 2 2 5T + 3, for p = 2 2 an tr(λ Π p Λ = T 2 2 7T + 6, for p = 3 2 Using the last relationships, we can erive the following values of the slope parameter of the power function k IV (p, θ, for θ = 0: k IV (0, 0 = k IV (, 0 = k IV (2, 0 = an k IV (3, 0 = 2 (T 2 T, T 2 2 3T 2 +, T 2 2 5T 2 + 3, T 2 2 7T For the case of θ 0 an p =, the formula of k IV (p, θ is erive by De Wachter et al. (2007. The coeffi cients of this formula are analytically given as D,IV = T 2 2 3T 2 + D 2,IV = T 2 4T + 4 R,IV = T (T R 2,IV = 2T (T R 3,IV = 3T (T Proof of Theorem 2 For c = 0 a irect comparison with Kruiniger an Tzavalis (2002 can be mae since the formulation of the test here employs ifferent selection matrices. To erive the limiting istribution of the WG test uner local alternatives ϕ = subtract y i from both sies of moel M: c, y i = u i + (ϕ y i + ( ϕ a i e. (5 26

27 The limiting istribution of the yet unstanarize WG test statistic aroun ϕ obtaine by writing ˆδW G (ˆϕ W G ˆb W G ˆδW G ϕ where ( i= = = ˆδ W G ϕ + = ( = ( i= y i Qu i an (h can be y i Qu i i= ˆb W G ϕ ˆδW G y i Qy i i= y i Qu i tr(ψ p,w GˆΓ i= y i Qu i i= y i Qu i i= i= y iψ p,w G y i i= y iψ p,w G y i = ( (h, (52 y iψ p,w G y i. By applying Lineberg-Levy s CLT, we can fin the limiting istributions of quantities ( an (h. To this en, write ( as where ( ( = i= i= y i Qu i = = ( + ( 2 + ( 3, i= i= (wy i0 + Ωe( ϕ a i + Ωu i Qu i (53 (y i0 w Qu i + a i ( ϕ e Ω Qu i + u iω Qu i y i0 w Qu i, ( 2 = i= a i ( ϕ e Ω Qu i an ( 3 i= u iω Qu i. The limits of (, ( 2 an ( 3 can be obtaine after substituting Taylor s series expansions for w an Ω given in (4 an (42, respectively, into them an ϕ = have c. For (, we ( = i= i= y i0 w Qu i = y i0 e Qu i + i= i= y i0 (e + f(ϕ + o P ( Qu i (54 f p Qu i y i0 + o p ( 0, 27

28 since i= f Qu i y i0 f QE(u i y i0 = 0 by Assumption 2 an e Q = 0. For ( 2, we have i= y i0 e Qu i = 0, as since c ( 2 a i ( ϕ e Ω Qu i = c i= i= = c a i e Λ Qu i c2 a 3/2 i e F Qu i + o p ( i= a i e Λ Qu i i= 0, as 3/2 i= a i e (Λ + F ( c + o p (Qu i (55 p 0, p c i e Λ QE(a i u i = 0 by Assumption 2 an goes to zero. For ( 3, we have c2 3/2 a i e F p Qu i i= ( 3 = i= i= u iω Qu i = u iλ Qu i c i= u i(λ + F ( c + o p (Qu i (56 u if Qu i + o p (, i= where c i= ( u if Qu i i= p ctr(f QΓ, (57 u iλ Qu i tr(λ QΓ (0, V W G,. (58 p V W G, is the variance of the limiting istribution of ( 3 an (, since ( 0 an p ( 0. Term tr(λ QΓ, which is ae in (58, oes not have to be subtracte from the WG test statistic, as it cancels out with a similarly ae term, given as tr(ψ p,w G Γ, in (60 below, since by construction tr(λ QΓ = tr(ψ p,w G Γ. Using the results of equations (54, (55 an (56, we can obtain the limiting istribution of ( as ( y i Qu i i= ( ctr(f QΓ, V W G,. (59 The proof for the limiting istribution of quantity (h follows analogous steps to those of 28