A fixed-t version of Breitung s panel data unit root test and its asymptotic local power

Transcription

1 A fixed-t version of Breitung s panel data unit root test and its asymptotic local power by Yiannis Karavias and Elias Tzavalis Granger Centre Discussion Paper o. 4/02

2 A fixed-t Version of Breitung s Panel Data Unit Root Test and its Asymptotic Local Power Yiannis Karavias a, and Elias Tzavalis b a: School of Economics, University of ottingham b: Athens University of Economics & Business Abstract We extend Breitung s 2000 large-t panel data unit root test to the case of fixed time dimension while still allowing for heteroscedastic and serially correlated error terms. The analytic local power function of the new test is derived assuming that only the cross section dimension of the panel grows large. It is found that if the errors are serially uncorrelated the test also has trivial power, but, if not, this is no longer the case. Monte Carlo experiments show that the suggested test is more powerful than its large-t, original version when the number of cross section units is moderate or large, regardless of the number of time series observations. JEL classification: C22, C23 Keywords: Panel unit root; local power function; serial correlation; incidental trends *Corresponding author. University of ottingham, ottingham G7 2RD, UK. Tel: , addresses: ioannis.karavias@nottingham.ac.uk Y. Karavias, etzavalis@aueb.gr E. Tzavalis. The authors would like to thank Jörg Breitung and Steve Leybourne for their helpful comments.

3 Introduction To improve the power performance of panel data unit root tests in the presence of heterogeneous individual trends, Breitung 2000 proposed a statistic for large-t panels based on an orthogonal transformation of the individual series. The test does not require an inconsistency adjustment of the estimator of the autoregressive parameter ϕ as opposed to other tests in the literature, see e.g., Baltagi 203 for a survey. Although it was found to be consistent and have superior power in small samples for values of ϕ not far from unity e.g., ϕ = 0.95, its asymptotic local power in a T /2 neighbourhood of unity is trivial and equivalent to that of the asymptotically bias corrected tests see, Moon et al In this paper, we extend Breitung s 2000 test in two directions. First, we allow the time dimension T of the panel to be finite fixed while allowing for heterogeneity, heteroscedasticity, and serial correlation in the error terms. Second, we derive the fixed-t asymptotic local power function of the new test. These extensions make the application of the test valid in cases of short-t panels, often met in practice, and under higher than first order serial correlation. The paper provides a number of interesting results. First, it shows that the fixed-t version of the test can further improve its small sample size and power performance in short panels, compared to its large-t version. Second, the new test also has trivial asymptotic local power in a /2 neighbourhood of unity when the error terms are independently distributed over time, which explains analytically Breitung s 2000 findings in his Monte Carlo experiment. Third, when the error terms are serially correlated, the estimator of ϕ becomes inconsistent and thus, the test needs an inconsistency correction. Fourth, there are forms of serial correlation for which the test has non-trivial local power. The paper is organized as follows. Section 2 introduces the new test and provides its asymptotic local power function. Section 3 presents the results of our Monte Carlo exercise, while Section 4 concludes the paper. All proofs are relegated to the Appendix. 2 The test statistic and its asymptotic local power Consider the following first order autoregressive panel data model with individual effects: y i = ϕy i + ϕa i e + ϕβ i e + ϕβ i τ + u i, where y i = y i,..., y it and y i = y i0,..., y it are T X vectors, u i is the T X vector of error terms u it, and a i and β i are the individual coeffi cients of the deterministic components of the model. The coeffi cients a i reflect the individual effects of the panel, while β i capture the slopes of individual linear trends, referred to as incidental trends. The T X vector e has 2

4 elements e t =, for t =...T, and τ t = t is the time trend. ext, define the autoregressive coeffi cient ϕ as ϕ = c. Then, the null hypothesis of a unit root in ϕ against its alternative of stationarity i.e., ϕ < can be respectively written as H 0 : c = 0 and H : c > 0, where c is the local to unity parameter. The asymptotic distribution of the extension of Breitung s 2000 test statistic denoted as U B is derived by making the following assumption. Assumption A i {u i }, i {, 2,..., }, are independent random vectors with means Eu i = 0, heterogeneous variance-covariance matrices Γ i Eu i u i [γ i,ts ], where γ i,ts = Eu it u is = 0 for t < s and s = t + p +,..., T. The maximum order of serial correlation in u i is p = T 2. All 4 + ɛ mixed moments are finite. ii Γ = lim Γ i is a finite, positive definite matrix and limγ Γ i =lim i Γ Γi = 0, for all i. iii The random variables y io, a i and β i have finite 4 + ɛ moments and are independent from u it for i =,..., and t =,..., T. Assumption A allows us to derive the distribution of the fixed-t version of Breitung s 2000 large-t panel unit root test statistic under the null hypothesis. Condition i determines the order of serial correlation in error terms u it and together with condition ii provide the necessary assumptions for the application of the Lindeberg-Feller multivariate CLT. Condition iii is needed for the derivation of the asymptotic local power function. Breitung s 2000 test is based on the forward orthogonal deviations transformation of the individual series y it of model. In a first step, the initial observations y i0 are subtracted from y it, i.e. z it = y it y i0. Then, define the following T XT matrices: G = A = 0 XT GH and B = T 2 0 T T 3 T XT and H = I T 2 0 T 2X T τ T 2, where T... T T 2 T with dimensions T 2XT 2 and T 2XT respectively, and vector τ T 2 =, 2,..., T 2. In the case that u it IIID0, σ 2, multiplying z i with matrix A and z i with matrix., 3

5 B implies the following orthogonal moment conditions under null hypothesis H 0 : c = 0: Ez ib A z i = 0. 2 These can be tested based on the following least squares estimator of ϕ: ˆϕ F OD = + z ib Bz i z ib A z i, which is equal to that of Breitung 2000 plus. This estimator is consistent under H 0 : c = 0, i.e., p lim ˆϕ F OD =. In the more general case where Γ σ 2 I T, estimator ˆϕ F OD becomes inconsistent and its asymptotic bias is equal to p lim ˆϕ F OD = trλ+i B AΓ, where trλ+i B BΛ+IΓ Λ is a T XT matrix which has unities at its lower than its main diagonals, and zero elsewhere, and I is a T XT identity matrix. Thus, to test moment conditions 2, ˆϕ F OD needs to be corrected for its inconsistency see, e.g., Harris and Tzavalis 999. Theorem Let conditions i and ii of Assumption A hold and +, then under H 0 : c = 0 we have UB T = V /2ˆδ ˆϕ F OD ˆb d 0,, ˆδ where ˆbˆδ trφ = p ˆΓ, ˆΓ = z i B Bz i z i z i, Φ p = Ψ p e Ψ pe M with Ψ e Me p a T XT matrix having in its diagonals { p,.., 0,...p} the corresponding elements of matrix Ξ = Λ+I B A, and zero elsewhere, M is a T XT selection matrix with elements m ts = 0, if γ ts 0, and m ts =, if γ ts = 0, and V = vecξ Φ p ΘvecΞ Φ p where Θ = V arvec z i z i. 2 Implementing test statistic UB T requires a consistent estimator of variance V, given under H 0 as ˆV = vecξ Φ p ˆΘvecΞ Φ p where ˆΘ = vec z i z ivec z i z i. The main difference between UB T and Breitung s statistic UB is the replacement of a T - consistent variance estimator of u i with a -consistent one. To study the asymptotic local power of UB T under H : c > 0, we will rely on a "slope" parameter, denoted as k, which is defined in local power functions of form Φz a + ck, where Φ is the standard normal cumulative distribution function and z a denotes the α-level percentile. Since Φ is strictly monotonic, a larger k means greater power for the same value This happens because trλ + I B A = 0 and trλ + I B AΓ 0. 2 An alternative specification of UB T for u it IID0, σ 2 is UB T,2 = V /2 d 2 ˆϕ F OD 0,, where V 2 = vecξ ΘvecΞ and Ξ = Λ + I B A. If u it are also normally distributed, V 2 becomes 2trA V 2 = 2 Ξ trλ+i B BΛ+I, with A 2 Ξ = 2 Ξ + Ξ. 4

6 of c. If k > 0, then test statistic UB T will have non-trivial power. If k = 0, it will have trivial power, which is equal to a. Finally, if k < 0, it will be biased. In the next theorem, we derive the limiting distribution of UB T under H : c > 0. Theorem 2 Under Assumption A and H : c > 0, we have as, where UB T = V /2ˆδ ˆϕ F OD ˆb d ck,, 3 ˆδ k = trλ B AΛΓ + trb AΛΓ + trλ B AΓ + trf B AΓ trλ Φ p Γ trφ p ΛΓ V, 4 where F is defined in the Appendix. The result of Theorem 2 implies that UB T can have non-trivial power, as k can be positive. Power becomes trivial if u it are serially uncorrelated. Then, UB T will suffer from the problem of zero asymptotic local power due to incidental trends, noted by Moon et al for large- T panel unit root tests. 3 This explains Breitung s 2000 Monte Carlo findings. ote that this power also depends on the moments of nuisance parameters β i, entered in the denominator of k through the variance function V. For instance, if u it and β i are zero-mean normally distributed random variables, then V is given as V = 2trA F OD Γ+Eβ 2 i A F OD ee 2, where A F OD = Ξ + 2 Ξ Φ p Φ p see proof of Theorem. 3 Simulation Results The aim of our simulation study is twofold: first, to examine if the size and power performance of the fixed-t test statistic UB T in small samples is satisfactory compared to its large-t version and, second, to investigate how well the asymptotic local power function can approximate the actual power of the test. In our analysis, we assume that error terms u it are generated from the MA process u it = ε it + θε it, with innovations ε it IID0, and values of θ { 0.8, 0.4, 0, 0.4, 0.8}. We set y i0 = 0 and a i = 0, without loss of generality as these parameters do not appear in the local power function. For β i, we consider β i = 0 or β i IID0,. Finally, ϕ {, 0.95}, {20, 50, 00} and T {7, 0, 5, 20, 50}. 3 The limiting distribution of UB T,2 under H :c > 0 becomes UB T,2 = V /2 2 ˆϕ F OD ck 2,, where k 2 = 0, which means that the test has trivial power. d 5

7 Rejection frequencies are computed from 0000 replications at the 5% significance level ϕ/t UB T U B Table : Size and size-adjusted power of test statistics UB T and UB, for θ = 0. Table presents the size and the size-adjusted power of UB T and Breitung s statistic UB. This is done for θ = 0 and β i = 0, for all i see also Breuitung The results of the table clearly indicate that both the size and power of UB T are satisfactory see De Blander and Dhaene 202. The size of the test is very close to its nominal 5% level. Its power increases with or T, but faster with T than. For small i.e., = 20 and large T, statistic UB has better size and more power than UB T. However, as increases UB T improves its size and is more powerful than the UB test irrespective of T. This qualifies application of UB T also in cases where both dimensions and T of the panel are large θ ϕ/t Table 2: Size and power of the fixed-t panel root test statistic UB T when θ 0. Regarding the effects of serial correlation on the test, the results of Table 2, which presents size and power of statistic UB T for non-zero θ, indicate that positive serial correlation θ > 0 in the errors u it increases considerably the power of UB T, even for very small values of T and. Also, the size performance of UB T is unaffected when error terms u it are negatively correlated θ < 0. This result is in contrast to that of single time series unit root tests 6

8 which are critically oversized for θ < 0 see, e.g., Schwert 989. To see how well the asymptotic theory approximates the local power of UB T in the neighbourhood of unity, Table 3 presents power values when ϕ = c/, for c =, {50, 00, 300, 000}, T = 0 and two cases of β i : β i = 0 and β i IID0,. The results of Table 3 indicate that the estimates of the power obtained by our Monte Carlo experiment tend to approximate their theoretical values T V. For θ < 0, the test has nontrivial local power while for θ > 0, it is biased. The non-trivial local power of the test for θ < 0 can be attributed to the fact that the individual series of the panel y it become close to those of a panel data autoregressive model with a common trend, for all i. In this case, the incidental trends problem does not apply see Moon et al Finally, the power losses for β i IID0, are not very large. They become minimal for θ = 0, where β i does not affect the local power function. β i = 0, i =,..., β i 0,, i =,..., θ\ TV θ\ TV Table 3: Local power values of statistic UB T for T = 0, when u it = ε it +θε it. 4 Conclusions This paper extends Breitung s 2000 panel unit root test to the case of fixed-t time dimension and derives its asymptotic local power. It shows that the new test can further improve its small sample size and power performance in short panels, compared to its large-t version. In addition to this, allowing for serial correlation in error terms leads to a test which can have non-trivial local asymptotic power in the presence of incidental trends. Monte Carlo analysis confirms the asymptotic results provided by the paper. 7

9 References [] Baltagi, B.H., Eonometric analysis of panel data, 5th Edition. Wiley and Sons, Y. [2] Breitung J., The local power of some unit root tests for panel data. In Badi H. Baltagi, Thomas B. Fomby, R. Carter Hill ed. onstationary Panels, Panel Cointegration, and Dynamic Panels Advances in Econometrics, Volume 5, Emerald Group Publishing Limited, pp [3] De Blander, R., Dhaene, G., 202. Unit root tests for panel data with AR errors and small T. The Econometrics Journal, 5, [4] Harris R.D.F. and E. Tzavalis 999. Inference for unit roots in dynamic panels where the time dimension is fixed. Journal of Econometrics, 9, [5] Madsen E., 200. Unit root inference in panel data models where the time-series dimension is fixed: a comparison of different tests. Econometrics Journal 3, [6] Moon H.R., Perron B. & Phillips P.C.B., Incidental trends and the power of panel unit root tests. Journal of Econometrics, 42, [7] Schott J.R Matrix Analysis for Statistics, Wiley-Interscience. [8] Schwert, G.W., 989. Tests for unit roots:a Monte Carlo investigation. Journal of Business and Economic Statistics 7, Appendix Theorem : Under H 0 : c = 0, we have z i = z i + β i e + u i and z i = Λeβ i + Λu i. Then, the denominator of ˆϕ F OD, denoted as ˆδ, is z ib A z i = z i + β i e + u i B Aβ i e + u i = u iλ + I T + β i τ B Aβ i e + u i = u iλ + I T B Au i, since Λ + I T e = τ and τ B = 0 XT, B Ae = 0 T X by construction. By Khinchine s Weak Law of Large umbers: u iλ + I T B Au i = u p iξu i trξγ. 8

10 Similarly, it can be shown that the denominator of ˆϕ F OD has the following limit: z ib Bz i = u iλ + I T B p BΛ + I T u i trλ + I T B BΛ + I T Γ. The last two relationships imply that the inconsistency of ˆϕ F OD is given as p lim ˆϕ F OD trξγ =. Thus ˆϕ trλ +I T B BΛ+I T Γ F OD becomes unbiased, if trξγ = 0 i.e. Γ = σ 2 I T. Combining the above, the limiting distribution of UB T can be derived as follows: ˆδ ˆϕ F OD ˆb/ˆδ = u iλ + I T B Au i z iφ p z i = z iξ Φ p z i since z iξ z i = u iξu i, where E z iξ Φ p z i = 0 by construction of Φ p and V ar z iξ Φ p z i = vecξ Φ p V arvec z i z ivecξ Φ p. The result follows by applying the Lindeberg-Feller CLT. If u i and β i are zero-mean normally distributed random variables, then z i is also normal and V ar z iξ Φ p z i = 2tr A F OD Γ + Eβ 2 i ee 2. Theorem 2: To prove the theorem, we will employ following relationships: ϕ T 2 z i = ϕ z i + Xζ i + u i, i =, 2,..., 5 z i = ΩXζ i + Ωu i + w ey i0, 6 and z i = ϕ z i + Xζ i + u i, 7 ϕ where ζ i = a i y i0 + ϕβ i, X = e, τ, w =, ϕ, ϕ 2,..., ϕt ϕ β and i ϕ.. Ω = ϕ 2 ϕ.... ote that, for ϕ =, we have Ω Λ. The first ϕ T 3.. ϕ 0 order Taylor expansions of Ω and w yield Ω = Λ + F ϕ + o and w = e + fϕ + o, 8 respectively, where F = dω dϕ ϕ = and f = dw dϕ ϕ =see also Madsen 200. ζ i can be 9

11 written in more compact form as ζ i = c µ i + β i e 2, 9 where hold: c = ϕ, µ i = a i y i0 β i, β i and e 2 =, 0. The following equalities also trξ = 0 and trλ B A = trb A, e Ξ = 0 XT and Ξe = 0 T X, B AXe 2 = 0 T X, 0 e 2X Λ B AΛXe 2 = e 2X Λ B AXẽ, e 2X B AΛXe 2 = e 2X B AXẽ, e 2X Φ p ΛXe 2 = e 2X Φ p Xẽ, 2 e 2X Λ Φ p Xe 2 = ẽ X Φ p Xe 2, 3 where ẽ =,. Consider the following formula of test statistic UB T : ˆδ ˆϕ F OD ϕ ˆb = 4 ˆδ = ˆδ + z ib A z i ϕ z iφ p z i z i B Bz 5 i z i B Bz i = c z ib Bz i + z ib A z i z iφ p z i = I + II + III. The limiting distribution of the above statistic is derived by taking limits of I, II and III, for. To derive the limit of I, we will employ 5. Then, I can be written as c z ib Bz i = c φ 2 z i B Bz i +φ z i B BXζ i +φ z i B Bu i +φ ζ ix B Bz i + ζ ix B BXζ i + ζ ix B Bu i + φ u ib Bz i + u ibb Xζ i + u ib Bu i. Using 6 and 8 and 9, the first term of the last relationship can be written as c φ2 z i B Bz i = c z i B Bz i + o p = c β ie 2X Λ + u iλ B BΛXe 2 β i + Λu i + o p. Since the sum is multiplied by, any summand coming from the expansion of it which is also multiplied by, or, will be asymptotically negligible, o p. By KWLL and stan- dard results on quadratic forms see Schott 996, we can show that c β ie 2X Λ + u iλ B p BΛXe 2 β i + Λu i c [ Eβ 2 i e 2X Λ B BΛXe 2 + trλ B BΛΓ ]. Following analo- 0

12 gous arguments to the above, it can be shown that I : c trλ B BΛΓ + trλ B BΓ + trb BΛΓ + trb BΓ z ib p Bz i c +Eβ 2 i e 2X Λ B BΛXe 2 + Eβ 2 i e 2X Λ B BXe 2 6 Similarly, we can show II : +Eβ 2 i e 2X B BΛXe 2 + Eβ 2 i e 2X B BXe 2 z ib A z i Eβ 2 i e 2X B AΛXe 2 + Eβ 2 i e 2X B AXẽ p cµ, V II 7 trλ B AΛΓ trλ B AΓ trb AΛΓ trf B AΓ where µ = c Eβ 2 i e 2X Λ B AΛXe 2 + Eβ 2 i ẽ X Λ B AXẽ+ + trλ B AΓ + trb AΓ and III : z iφ p z i p cµ 2, V III 8 trλ Φ p Γ + trφ p ΛΓ where µ 2 = c +Eβ 2 i e 2X Λ Φ p Xe 2 + Eβ 2 i e 2X Φ p ΛXe 2 trφ p Γ. Summing up the Eβ 2 i e 2X Φ p Xẽ Eβ 2 i ẽ X Φ p Xe 2 results in 6, 7 and 8 and using the results of equations 3, we can prove the result of Theorem 2. ote that the variance functions of the limiting distributions of quantities I and II: V II and V III, as well as their covariance do not need to be calculated, given that they are equal to variance V of the test statistic UB T, under H 0 : c = 0. This happens because these functions are independent of c see also Breitung 2000.