BOOTSTRAP FOR PANEL DATA MODELS

Transcription

1 BOOSRAP FOR PAEL DAA MODELS Bertrand HOUKAOUO Université de Montréal, CIREQ July 3, 2008 Preliminary, do not quote without ermission. Astract his aer considers ootstra methods for anel data models with xed regressors. It is shown that simle resamling methods (i.i.d., individual only or temoral only) are not always valid in simle cases of interest, while a doule resamling that comines resamling in oth individual and temoral dimensions is valid. his aroach also ermits to avoid multiles asymtotic theories that may occur in large anel models. In articular, it is shown that this resamling method rovides valid inference in the one-way and two-way error comonent models and in the factor models. Simulations con rm these theoretical results. JEL Classi cation: C5, C23. Keywords: Bootstra, Panel Data Models. I am grateful to Benoit Perron and Silvia Gonçalves for helful comments. All errors are mine. Financial suort from the Deartment of Economics, Université de Montréal and CIREQ, is gratefully acknowledged. Address for corresondance : Université de Montréal, Déartement de Sciences Economiques, C.P. 628, succ. Centre-Ville, Montréal, Qc H3C 3J7, Canada. .hounkannounon@umontreal.ca. 3J7, Canada. .hounkannounon@umontreal.ca.

2 Introduction he true roaility distriution of a test statistic is rarely known. Generally, its asymtotic law is used as aroximation of the true law. If the samle size is not large enough, the asymtotic ehavior of the statistics could lead to a oor aroximation of the true one. Using ootstra methods, under some regularity conditions, it is ossile to otain a more accurate aroximation of the distriution of the test statistic. Original ootstra rocedure has een roosed y Efron (979) for statistical analysis of indeendent and identically distriuted (i.i.d.) oservations. It is a owerful tool for aroximating the distriution of comlicated statistics ased on i.i.d. data. here is an extensive literature for the case of i.i.d. oservations. Bickel & Freedman (98) estalished some asymtotic roerties for ootstra. Freedman (98) analyzed the use of ootstra for least squares estimator in linear regression models. In ractice, oservations are not i.i.d. Since Efron (979) there is an extensive research to extend ootstra to statistical analysis of non i.i.d. data. Wild ootstra is develoed in Liu (988) following suggestions in Wu (986) and Beran (986) for indeendent ut not identically distriuted data. Several ootstra rocedures have een roosed for time series. he two most oular aroaches are sieve ootstra and lock ootstra. Sieve ootstra attemts to model the deendence using a arametric model. he idea ehind it is to ostulate a arametric form for the data generating rocess, to estimate the arameters and to transform the model in order to have i.i.d. elements to resamle. he weakness of this aroach is that results are sensitive to model misseci cation and the attractive nonarametric feature of ootstra is lost. On the other hand, lock ootstra resamles locks of consecutive oservations. In this case, the user is not oliged to secify a articular arametric model. For an overview of ootstra methods for deendent data, see Lahiri (2003). Alication of ootstra methods to several indices data is an emryonic research eld. he exression "several indices data" regrous : clustered data, multilevel data, and anel data. he term "anel data" refers to the ooling of oservations on a cross-section of statistical units over several eriods. Because of their two dimensions (individual -or cross-sectional- and temoral), anel data have the imortant advantage to allow to control for unoservale heterogeneity, which is a systematic di erence across individuals or eriods. For an overview aout anel data models, see for examle Baltagi (995) or Hsiao (2003). here is an aounding literature aout asymtotic theory for anel data models. Some recent develoments treat of large anels, when temoral and cross-sectional dimensions are oth imortant. Paradoxically, literature aout ootstra for anel data is rather restricted. In general, simulation results suggest that some resamling methods work well in ractice ut theoretical results are rather limited or exosed with strong assumtions. As revious references aout ootstra methods for anel models, it can e quoted Bellman et al. (989), Andersson & Karlsson (200), Carvajal (2000), Kaetanios (2004), Focarelli (2005), Everaert & Pozzi (2007) and Herwartz (2006; 2007). In error comonent models, Bellman et al. (989) uses ootstra to correct ias after feasile generalized least squares. Andersson & Karlsson (200) resents ootstra resamling methods for one-way error comonent model. For two-way error comonent models, Carvajal (2000) evaluates y simulations, di erent ootstra resamling methods. Kaetanios (2004) resents theoretical results when cross-sectional dimension goes to in nity, under the assumtion that cross-sectional vectors of regressors and errors terms are i.i.d.. his assumtion does not ermit time varying regressors or temoral aggregate shocks in errors terms. Focarelli (2005) and Everaert & Pozzi (2007) uses ootstra to reduce ias in dynamic anel models with xed e ects when is xed and goes to in nity, ias quoted y ickell (98). Herwartz (2006 & 2007) deliver a ootstra version to Breusch-Pagan test in anel data models under cross-sectional deendence. Recently, Hounkannounon (2008) gives some theoretical results 2

3 aout ootstra methods used with anel data models. Its theoretical results are aout a model without regressor and concern the samle mean. his aer aims to extend these results to linear regression model. Various ootstra resamling methods will e confronted with anel models commonly used to evaluate their validity. he aer is organized as follows. In the second section, di erent anel data models are resented. Section 3 resents ve ootstra resamling methods for anel data. he fourth section resents theoretical results, analyzing validity of each resamling method. In section 5, simulation results are resented and con rm theoretical results. he sixth section concludes. 2 Panel Data Models It is ractical to reresent anel data as a rectangle. By convention, in this document, rows corresond to the individuals and columns reresent time eriods. A anel dataset with individuals and time eriods is reresented y a matrix Y of rows and columns. Y contains thus elements. y it is i 0 s oservation at eriod t: Y = (; ) Consider the following linear model 0 y y 2 ::: ::: y y 2 y 22 ::: ::: y 2 ::: ::: ::: ::: ::: ::: ::: ::: :: ::: y y 2 :: ::: y y it = + V i + W t + X it + it = Z it + it (2. ) = (K;) 0 C A C A (2.2 ) hree kinds of variales are considered : cross-section varying variales V i, time varying variale W t and doule dimensions varying variales X it. is an unknown vector of arameters. Inference will e aout these arameters and consists in uilding con dence intervals for each comonent k of. Assumtions aout it de ne di erent anel data models. he seci cations of error terms commonly used can e summarized y (2.3) under assumtions elow. it = i + f t + i F t + " it (2.3 ) Assumtions A A : ( ; 2 ; ::::::; ) i:i:d: 0; 2, 2 2 (0; ) A2 : ff t g is a stationary and strong -mixing rocess with E (f t ) = 0, 9 2 (0; ) : E jf t j 2+ <, X j= (j) =(2+) <, and V f = X h= A3 : ( ; 2 ; ::::::; ) i:i:d: 0; 2, 2 2 (0; ) A4 : f" it g i=:::; t=:: i:i:d: 0; 2 ", 2 " 2 (0; ) See Aendix for de nition of an -mixing rocess. Cov (f t ; f t+h ) 2 (0; ) 3

4 A5 : ff t g is a stationary and strong -mixing rocess with E (F t ) = 0, 9 2 (0; ) : X E jf t j 2+ <, (j) =(2+) <, and VF = X Cov (F t ; F t+h ) 2 (0; ) j= A6 : he ve series are indeendent. h= hese assumtions can seem strong. hey are made in order to have strong convergence and to simlify demonstrations. Considering secial cases with di erent cominations of rocesses in (2.3.), gives the following anel data models : one-way error comonent model, two-way error comonent model and factor model. One-way it = i + " it (2.4) it = f t + " it (2.5) wo seci cations are considered. he term, one-way error comonent model (), comes from the structure of error terms : only one kind of heterogeneity, that is systematic di erences across cross-sectional units or time eriods, is taken into account. he seci cation 2.4 (res. 2.5) allows to control unoservale individual (res. temoral) heterogeneity. he seci cation (2.4) is called individual one-way, (2.5) is temoral one-way. It is imortant to emhasize that here, unoservale heterogeneity is a random variale, not a arameter to e estimated. he alternative is to use xed e ects model in which heterogeneity must e estimated. wo-way it = i + f t + " it (2.6 ) wo-way error comonent model allows to control for individual and temoral heterogeneity, hence the term two-way. Like in one-way, individual and temoral heterogeneities are random variales. Classical aers on error comonent models include Balestra & erlove (966), Fuller & Battese (974) and Mundlak (978). Factor Model it = i + i F t + " it (2.7 ) In (2.7), the di erence with one-way, is the term i F t. he roduct allows the common factor F t to have di erential e ects on cross-section units. his seci cation is used y Bai & g (2004), Moon & Perron (2004) and Phillis & Sul (2003). It is a way to introduce deendence among cross-sectional units. An other way is to use satial model in which, the structure of the deendence can e related to geograhic, economic or social distance (see Anselin (988)) 2. 2 Bootstra methods studied in this aer do not take into account satial deendence. Reader insterested y resamling methods for satial data, can see for examle Lahiri (2003), cha. 2. 4

5 3 Bootstra Methods his section resents the ootstra methodology and ve ways to resamle anel data. Bootstra Methodology From initial data (Y; X), create seudo data (Y ; X ) y resamling with relacement elements of (Y; X) : his oeration must e reeated B times in order to have B + seudo-samles : fy ; X g =::B+. Statistics are comuted with these seudo-samles in order to make inference. In this aer, inference is aout and consists in uilding con dence intervals and testing hyothesis for each arameter of the element of the vector. here are two main ootstra aroaches with regression models :the residual-ased ootstra an the aired ootstra. he aired ootstra resamles deendent variale and regressors whereas residuals resamles rst ste residuals to comute seudo values of the deendent variale. his aer analyzes only the residual-ased ootstra which stes are the followings : Ste : Run ooling regression to otain OLS estimator and the residuals it = ez = e Z ez = e Y it = y it Z it Ste 2 : Rescale the residuals in order to have etter roerties in small samles. u it = K it By OLS roerties, the residuals have mean equal to zero : centering is not necessary. he matrix of rescaled residuals is noted U. Ste 3 : Use a resamling method to create seudo-samle of residuals U : U = fu it g resamling U = fu itg Use the seudo- residuals to seudo-values of the deendent variale. Run ooling regression with (Y ; X) y it = Z it + u it = ez = e Z ez = e Y Ste 4 : Reeat ste 3 B times in order to have B + realizations of fy ; Xg and n o : hese realizations are quoted fy ; Xg =::B+ and =::B+ : he roaility measure induced y the resamling method conditionally on U is noted P. E () and V ar () are resectively exectation and variance associated to P. In this aer, he resamling methods used to comute seudo-samles are exosed elow. 5

6 Iid Bootstra In this document i.i.d ootstra refers to original ootstra as de ned y Efron (979). It was designed for one dimensional data, ut it s easy to adat it to anel data. For matrix U, i.i.d. resamling is the oeration of constructing a matrix U where each element u it is selected with relacement from Y: Conditionally on U, all the elements of U are indeendent and identically distriuted. here is a roaility = that each u it is one of the elements u it of U: Individual Bootstra For a matrix U, individual resamling is the oeration of constructing a matrix U with rows otained y resamling with relacement rows of U: Conditionally on U, the rows of U are indeendent and identically distriuted. Contrary to i.i.d. ootstra case, u it cannot take any value. u it can just take one of the values fu itg i=;:::. emoral Bootstra For matrix U, temoral resamling is the oeration of constructing a matrix Y with columns otained y resamling with relacement columns of Y: Conditionally on U, the columns of U are indeendent and identically distriuted. u it can just take one of the values fu it g t=;:::. Block Bootstra Block ootstra for anel data is a direct accommodation of non-overlaing lock ootstra for time series, due to Carlstein (986). he idea is to resamle in temoral dimension, ut not single eriod like in temoral ootstra case, ut locks of consecutive eriods in order to cature temoral deendence. Assume that = Kl, with l the length of a lock, then there are K non-overlaing locks. For matrix Y, lock ootstra resamling is the oeration of constructing a matrix U with columns otained y resamling with relacement the K locks of columns of U: ote that temoral ootstra is a secial case of lock ootstra, when l = : Moving lock ootstra (Kunsch (989), Liu & Singh (992)), circular lock ootstra (Politis & Romano (992)) and stationary lock ootstra (Politis & Romano (994)) can also e accommodated to anel data. Doule Resamling Bootstra For a matrix Y, doule resamling is the oeration of construction a matrix U with columns and rows otained y resamling columns and rows of U. wo schemes are exlored. he rst scheme is a comination of individual and temoral ootstra. he second scheme is a comination of individual and lock ootstra. Carvajal (2000) and Kaetanios (2004) imrove this resamling method y Monte Carlo simulations, ut give no theoretical suort. Doule stars are used to distinguish estimator, roaility measure, exectation and variance induced y doule resamling. 6

7 4 heoretical Results his section resents theoretical results aout resamling methods exosed in section 3, using models seci ed in section 2. Multile Asymtotics In the study of asymtotic distriutions for anel data, there are many ossiilities. One index can e xed and the other goes to in nity. In the second case, how and go to in nity, is not always without consequence. Hsiao ( ) distinguishes three aroaches : sequential limit, diagonal ath limit and joint limit. A sequential limit is otained when an index is xed and the other asses to in nity, to have intermediate result. he nal result is otained y assing the xed index to in nity. In case of diagonal ath limit, and ass to in nity along a seci c ath, for examle = () and : With joint limit, and ass to in nity simultaneously without a seci c restrictions. In some times it can e necessary to control relative exansion rate of and. It is ovious that joint limit imlies diagonal ath limit. For equivalence conditions etween sequential and joint limits, see Phillis & Moon (999). In ractice, it is not always clear how to choose among these multile asymtotic distriutions which may e di erent. Assumtions B (Regressors) B : he regressors are xed. B2 : (Rank condition) : he K K matrix e Z = e Z is not singular. B3 : B4 : ez = e Z Q > 0 (4. ) (K;K) B5 : B6 : Z = Z Z = Z Q (4.2 ) Q (4.3 ) > 0 (4.4 ) In addition to these assumtions, some Lindeerg conditions will e aroriately assumed, deending on the case of analysis. Assumtion A4 imlies that time varying regressors are excluded. Assumtions A5 excludes cross-section varying regressors. ale summarizes asymtotic distriutions for the di erent anel models. For i.i.d. anel model, summarizes three cases of asymtotic : is xed and goes to in nity, is xed and goes to in nity, and nally and ass to in nity simultaneously. wo asymtotic theories are availale for one-way. In the case of two-way, and must go to in nity. he relative convergence rate etween the two indexes, de nes a continuum of asymtotic distriutions. Finally, factor has a unique asymtotic distriution, when the two dimensions go to in nity. Details aout these convergences are exosed in aendix. 7

8 M odel Asymtotic distriution V ariance () Individual One way =) 2 Q QQ + 2 " Q =) (0; ) ; 2 Q QQ emoral One way =) (0; ) 2 f Q QQ 2 " Q =) (0; ) ; 2 f Q QQ wo way =) (0; ) ; 2 Q QQ + : f 2[0;) =) (0; ) ; f F actor model =) (0; ) ; 2 Q QQ ale : Asymtotic distriutions 8

9 Bootstra Con dence Intervals In the literature, there are several ootstra con dence interval. he methods commonly used are ercentile interval and ercentile-t interval. Bootstra Percentile Interval With each seudo-samle Y, comute and the K statistics r k = k he emirical distriution of these (B + ) realizations is : R k (x) = B+ X I B + = k (4.5 ) rk x (4.6 ) he ercentile-t con dence interval of level ( ) for the arameter k is: h CI ;k = k + rk; ; i 2 k + rk; (4.7 ) 2 where rk;=2 and r k; =2 are resectively is the lower emirical =2-ercentage oint and ( =2)-ercentage oint of R k : B must e chosen so that (B + ) =2 is an integer. Using equality 4.5, 4.7 ecomes : h CI ;k = k; ; i k; (4.9 ) 2 2 where k;=2 and k; =2 are resectively is the lower emirical =2-ercentage oint and ( =2)- n o ercentage oint of the emirical distriution of k; =::B+ : Bootstra Percentile-t Interval With each seudo-samle Y, comute and the K statistics k = k r k (4.0 ) V ar k t In the denominator of (4.0), there is an estimator of the ootstra-variance that must e comuted for every ootstra resamle. he emirical distriution of the (B + ) realizations of t k is : F k (x) = B+ X I B + = t k x he ercentile-t con dence interval of level ( ) for the arameter k is: " r r # CI ;k = k V ar k :t k; ; 2 k V ar k :t k; 2 (4. ) (4.2 ) he strength of ercentile-t is that it ermits theoretical demonstrations aout asymtotic re nements. 9

10 su x2r K Bootstra Consistency here are several ways to rove consistency of a resamling method. For an overview, see Shao & u (995, cha. 3). he method commonly used is to show that the distance etween the cumulative distriution function on the classical estimator and the ootstra estimator goes to zero when the samle grows-u. Di erent notions of distance can e used : su-norm, Mallow s distance... Su-norm is the commonly used. he notations used for one dimension data must e to anel data, in order to e more formal. Because of multile asymtotic distriutions, there are several consistency de nitions. he resentation takes into account ercentile con dence interval. Similar de nitions can e formulated for ercentile-t con dence interval. A ootstra method is said consistent for if : P x P or or su x2r K su x2r K P P x x P P x P 0 (4.3 ) x P 0 (4.4 ) x P 0 (4.5 ) De nitions 4.2, 4.3 and 4.4 are given with convergence in roaility ( P ). his case imlies a weak consistency. he case of almost surely (a:s.) convergence rovides a strong consistency. hese de nitions of consistency does not require that the ootstra estimator or the classical estimator has asymtotic distriution. he idea ehind it, is the mimic analysis : when the samle grows, the ootstra estimator mimics very well the ehavior of the classical estimator. In the secial when the samle mean asymtotic distriution is availale, consistency can e estalished y showing that ootstra-samle mean has the same distriution. he next roosition exresses this idea. Proosition Assume continuous, then su x2r K =) L and P x =) L. If L and L are identical and P x P 0 Proof. he fact that and have the same asymtotic distriution, imlies that jp (::) P (::)j converges to zero. Under continuity assumtion, the uniform convergence is given y Pólya theorem (Pólya (920) or Ser ing (980),. 8) Similar roositions similar can e formulated for de nitions 4.3 and 4.4. Using Proosition, the methodology adoted in this document is, for each resamling method, to nd the asymtotic distriution of the ootstra-estimator. Comaring theses distriutions with those in ale, ermits to nd consistent and inconsistent ootstra resamling methods for each anel model. Consistent resamling methods can e used to uild con dence intervals. 0

11 Remarks - he de nitions of consistency are given for the vector of arameters, ut con dence intervals are given for each comonent of this vector. 2 - he given de nitions of consistency are aroriate to uild ercentile con dence interval. Similar de nitions can e given using t k and its classical counterart. In this case, for each arameter k, the ercentile-t con dence interval is valid if on the de nition given for consistency,.we must have : V ar k P V ar k (4.6 ) 3 - Consistency in an asymtotic roerty. It must e taken in mind that ootstra rocedure has een originally designed for small samles and its validity deend of the fact that the ootstra data mimic as well as ossile the ehavior of the original data. In this aer, this aroach will e called mimic analysis. Residual-ased ootstra estimator mimics very well the ehavior the classical asymtotic estimator if the ootstraed residuals mimic very well the ehavior of the original error terms. In the following for each ootstra resamling method, the mimic analysis is resented followed y the consistency analysis. I.i.d. Bootstra In a mimic analysis, it can e said that iid resamling method does not take into account the structure of deendence of the error terms. hat leads to inconsistent estimator for all the anel model seci cations. M odel Asymtotic distriution V ariance () Consistency Individual =) (0; ) " Q One way o =) (0; ) ; " Q emoral One wo F actor model way way =) (0; ) =) (0; ) ; =) (0; ) ; 2 f + 2 " Q 2 f + 2 " Q f + 2 " Q =) (0; ) ; F + 2 " Q ale 2 : Asymtotic distriutions with i.i.d. ootstra o o o

12 Individual Bootstra U = + f + = F + " ; U ind = + f + = F + ["col ] Uind E (U ) = ( ) + = = F + (["row ] ") (4.7 ) M odel Asymtotic distriution V ariance Consistency () Individual =) (0; ) 2 Q QQ + 2 " Q One way =) (0; ) ; 2 Q QQ Y es emoral One way =) (0; ) 2 f Q QQ + 2 " Q =) (0; ) ; 2 f Q QQ o wo way =) (0; ) 2 Q QQ + 2 " Q =) (0; ) ; 2 Q QQ o F actor model =) (0; ) ; 2 Q QQ Y es ale 3 : Asymtotic distriution with individual ootstra 2

13 U tem E (U ) = emoral Bootstra f f + = F F + ([" col ] ") (4.8 ) M odel Asymtotic distriution V ariance Consistency () Individual =) (0; ) 2 "Q One way o =) (0; ) ; 2 "Q emoral One way =) (0; ) 2 f Q QQ o if + 2 " Q f t is =) (0; ) ; 2 f Q QQ correlated wo way =) (0; ) 2 f Q QQ + 2 " Q =) (0; ) ; 2 f Q QQ o F actor model m:s: ; 0 ale 4 : Asymtotic distriutions with temoral ootstra o 3

14 Block Bootstra U l E (U l ) = f l f + = F l F + ([" col ] l ") M odel Asymtotic distriution V ariance Consistency () Individual =) (0; ) 2 "Q One way o =) (0; ) ; 2 "Q emoral One way =) (0; ) f + 2 " Q =) (0; ) ; f Y es wo way =) (0; ) f + 2 " Q =) (0; ) ; f o F actor model m:s: ; 0 ale 4 : Asymtotic distriutions with lock ootstra o 4

15 Doule Resamling Bootstra U = + f + = F + " U E (U ) = ( ) + f f + = = F F + (" ") M odel Asymtotic distriution V ariance Consistency () Individual One way emoral One wo way way F actor model =) ; (0; ) 2 Q QQ Y es =) (0; ) ; 2 f Q QQ Y es =) ; (0; ) 2 Q QQ o if 2[0;) + 2 f Q QQ f t is =) (0; ) ; 2 f Q QQ correlated =) (0; ) ; 2 Q QQ Y es ale 6 : Asymtotic distriutions with doule resamling ootstra : scheme 5

16 U l E (Ul ) = ( ) + f l f + = = F l F + (" l ") (4.2 ) M odel Asymtotic distriution V ariance Consistency () Individual One way emoral One wo way way =) ; (0; ) 2 Q QQ Y es =) (0; ) ; f Y es =) (0; ) ; 2 Q QQ + f 2[0;) =) (0; ) ; f Y es F actor model =) (0; ) ; 2 Q QQ Y es ale 7 : Asymtotic distriutions with doule resamling ootstra : scheme 2 5 Simulations Data Generating Process for errors is the following : i i:i:d: (0; ), i i:i:d: (0; ) ; f t i:i:d: (0; ), " it i:i:d: (0; ) ; F t = F t + t, t i:i:d: 0; 2 = 0 and = 0:5: Data Generating Process for data for regressors is the following : = ; U i i:i:d: (; ), W t i:i:d: (; ) ; X it i:i:d: (; ) : For each ootstra resamling method, 999 Relications and 000 Simulations are used. Six samle sizes are considered : (; ) = (30; 30) ; (50; 50) ; (00; 00), (50; 0) and (0; 50) : ales 7, 8, 9, 0 and give rejection rates, for theoretical level = 5%: 6

17 I.i.d. Ind. em. Block 2Res- 2Res Individual One-way emoral One-way = 0: wo-way = wo-way = 0: Factor model ale 7 : Simulations with (; )=(30;30) 7

18 I.i.d. Ind. em. Block 2Res- 2Res Individual One-way emoral One-way wo-way = wo-way = 0: Factor model ale 8 : Simulations with (; )=(50;50) 8

19 I.i.d. Ind. em. Block 2Res- 2Res-2 Individual One-way emoral One-way = 0: wo-way = wo-way = 0: Factor model ale 9 : Simulations with (; )=(00;00) 9

20 I.i.d. Ind. em. Block 2Res- 2Res-2 Individual One-way emoral One-way wo-way = wo-way = 0: Factor model ale 0 : Simulations with (; )=(50;0) 20

21 I.i.d. Ind. em. Block 2Res- 2Res-2 Individual One-way emoral One-way wo-way = wo-way = 0: Factor model ale : Simulations with (; )=(0;50) 2

22 6 Conclusion his aer considers the issue of ootstra methods for anel data models. Four seci cations of anel data have een considered, namely, individual one-way error comonent model, temoral one-way error comonent model, two-way error comonent model and lastly factor model. Five ootstra methods are exlored in order to make inference aout vector of arameters.it is demonstrated that simle resamling methods (i.i.d., individual only or temoral only) are not valid in simle cases of interest, while doule resamling that comines resamling in oth individual and temoral dimensions is valid in these situations. Simulations con rm these results. A logical follow-u of this aer is to extend the results to dymamic anel models. 22

23 References [] Andersson M. K. & Karlsson S. (200) Bootstraing Error Comonent Models, Comutational Statistics, Vol. 6, o. 2, [2] Anselin L. (988) Satial Econometrics: Methods and Models, Kluwer Academic Pulishers, Dordrecht. [3] Bai J. & g S. (2004) A Panic attack on unit roots and cointegration, Econometrica, Vol. 72, o 4, [4] Balestra P. & erlove M. (966) Pooling Cross Section and ime Series data in the estimation of a Dynamic Model : he Demand for atural Gas, Econometrica, Vol. 34, o 3, [5] Baltagi B. (995) Econometrics Analysis of Panel Data, John Wiley & Sons. [6] Bellman L., Breitung J. & Wagner J. (989) Bias Correction and Bootstraing of Error Comonent Models for Panel Data : heory and Alications, Emirical Economics, Vol. 4, [7] Beran R. (988) Preivoting test statistics: a ootstra view of asymtotic re nements, Journal of the American Statistical Association, Vol. 83, o 403, [8] Bickel P. J. & Freedman D. A. (98) Some Asymtotic heory for the Bootstra, he Annals of Statistics, Vol. 9, o 6, [9] Carvajal A. (2000) BC Bootstra Con dence Intervals for Random E ects Panel Data Models, Working Paer Brown University, Deartment of Economics. [0] Davidson R. (2007) Bootstraing Econometric Models, his aer has aeared, translated into Russian, in Quantile, Vol. 3, [] Driscoll J. C. &. Kraay A. C. (998) Consistent Covariance Matrix Estimation with Satially Deendent Panel Data, he Review of Economics and Statistics, Vol. 80, o 4, [2] Efron B. (979) Bootstra Methods : Another Look at the Jackknife, he Annals of Statistics, Vol. 7, o, -26. [3] Everaert G. & Pozzi L. (2007) Bootstra-Based Bias Correction for Dynamic Panels, Journal of Economic Dynamics & Control, Vol. 3, [4] Focarelli D. (2005) Bootstra ias-correction rocedure in estimating long-run relationshis from dynamic anels, with an alication to money demand in the euro area, Economic Modelling, Vol. 22, [5] Freedman D.A. (98) Bootstraing Regression Models, he Annals of Statistics, Vol. 9, o 6, [6] Fuller W. A. & Battese G. E. (974) Estimation of linear models with crossed-error structure, Journal of Econometrics, Vol. 2, o, [7] Herwartz H. (2007) esting for random e ects in anel models with satially correlated disturances, Statistica eerlandica, Vol. 6, o 4,

24 [8] Herwartz H. (2006) esting for random e ects in anel data under cross-sectional error correlation a ootstra aroach to the Breusch Pagan est, Comutational Statistics and Data Analysis, Vol. 50, [9] Hounkannounon B. (2008) Bootstra for Panel Data, Working Paer, Deartment of Economics, Université de Montréal. [20] Hsiao C. (2003) Analysis of Panel Data, 2nd Edition. Camridge University Press. [2] Iragimov I.A. (962) Some limit theorems for stationary rocesses, heory of Proaility and Its Alications Vol. 7, [22] Kaetanios G. (2004) A ootstra rocedure for anel datasets with many cross-sectional units, Working Paer o. 523, Deartment of Economics, Queen Mary, London. Forthcoming in Econometrics Journal. [23] Künsch H. R. (989) he jackknife and the ootstra for general stationary oservations, Annals of Statistics, Vol. 7, [24] Lahiri S.. (2003) Resamling Methods for Deendent Data, Sringer Series in Statistics, Sinring-Verlag, ew York. [25] Liu R. Y. (988) Bootstra Procedure Under ome on-i.i.d. Models, Annals of Statistics, Vol. 6, [26] Liu R. Y. & Singh K. (992) Moving Block Jackknife and ootstra cature weak deendence, in Exloring the Limits of Bootstra, eds, y Leage and Billard, John Wiley, ewyork. [27] Moon H. R. & Perron B. (2004) esting for unit root in anels with dynamic factors, Journal of Econometrics, Vol. 22, [28] Mundlak Y. (978) On the Pooling of ime Series and Cross Section Data, Econometrica, Vol. 46,, [29] ewey W. K. & West K. D. (987) A Simle, Positive Semi-De nite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica, Vol. 55, o 3, [30] Phillis P.C.B. & Sul D. (2003) Dynamic anel estimation and homogeneity testing under cross-section deendence, Econometrics Journal, Vol. 6, [3] Phillis P.C.B. & Moon H.R. (999) Linear Regression Limit heory for onstationary Panel Data, Econometrica, Vol. 67, o 5, [32] Politis D. & Romano J. (994) he stationary ootstra, Journal of the American Statistical Association, Vol. 89, [33] Politis D. & Romano J. (992) A circular lock resamling rocedure for stationary data, in Exloring the Limits of Bootstra eds. y Leage and Billard, John Wiley, ewyork. [34] Pólya G. (920) Uer den zentralen Grenzwertsatz der Wahrscheinlinchkeitsrechnug und Momenten rolem, Math. Zeitschrift, 8. [35] Ser ing R. J. (980) Aroximation heorems of Mathematical Statistics, ew York: Wiley. 24

25 [36] Shao J. & u D. (995) he Jackknife and ootstra, Sinring-Verlag, ew York. [37] Singh K. (98) On the Asymtotic Accuracy of Efron s Bootstra, he Annals of Statistics, Vol. 9, o. 6, [38] Wu C. F. J. (986) Jackknife, ootstra and other resamling methods in regression analysis, Annals of Statistics, Vol. 4, [39] Yang S.S. (988) A Central Limit heorem for the Bootstra Mean, he American Statistician, Vol. 42, APPEDIX Error terms : Regressors : U = (; ) 0 Aendix : Matrix otations U U 2 ::: ::: U U 2 U 22 ::: ::: U 2 ::: ::: ::: ::: ::: ::: ::: ::: :: ::: U U 2 :: ::: U eu = (;) Z (i;t) = (;K) 0 C A = U = () U = (2) ::: ::: U = () ez (;K) = 0 Z () Z (2) ::: ::: Z ( ) C A Z () (i;t) Z (2) (i;t) :::: Z (K) (i;t) U () U (2) ::: ::: U () Z (i) = Z (i;) Z (i;2) ::: = Z (i; ) ; Z(t) = Z (;t) Z (2;t) ::: = Z (;t) (;K) (;K) Suar is ut for aggeegation in individual dimension. Uar refers to aggregation in temoral dimension. = C A Z (i) = (;K) X t= Z (it) ; Z (t) = (;K) X i= Z (it) Z = Z = = () Z (2) ::: Z () ; Z = Z () Z (2) ::: Z ( ) ; (;K) (;K) 25

26 Mixing Process De nition Let ff t g t2z e a sequence of random variales. he strong mixing or -mixing coe cient of ff t g t2z is de ned as : (j) = su fjp (A \ B) P (A) P (B)jg ; j 2 with A 2 hff t : t kgi ; B 2 hff t : t k + j + gi ; k 2 Z ff t g t2z is called strongly mixing (or -mixing ) if (j) 0 as j : Aendix 2 : Classical Asymtotic heory Individual One-way i = X Z = (it) E it = Z= (i) E (i) t= E ( i ) = 0 V ar ( i ) = h i V ar E = (i) Z (i) 2 Z= (i) h i V ar E = (i) = 2 "I + 2 J where I is ( ) identity matrix, and J is ( ) matrix with each element equal to one. i are indeendent. Lindeerg condition can e written : lim max i Z= (i) Z (i) Z Z = = 0 with X V ar ( i ) = i= X i= h i 2 Z= (i) V ar E = ez (i) Z (i) = 2 = Z e " 2 + Z = Z 2 P 2 " Q + 2 Q Aly Linderg-Feller CL to ( ; 2 ; ::::::; ) : X i= i =) 0; 2 " Q + 2 Q =) (0; ) = Q Q = 2 Q QQ + 2 " Q 26

27 emoral One-way a) 2 [0; ) Z = e " = Z e X i= wo-way Z = (i) i + X i= X i= t= X t= X Z = (it) " it Z = (t) f t + m:s: 0 Z = (i) i =) 0; 2 Q X i= t= # X Z = (it) " it ) X t= Z = (t) f t =) 0; 2 f Q =) 0; ; 2 Q QQ + 2 f Q QQ Z = e " = Z e X i= Z = (i) i X i= t= X i= + X t= X Z = (it) " it Z = (i) i Z = (t) f t + m:s: 0 m:s: ; 0 X i= t= # X Z = (it) " it he result follows. X t= Z = (t) f t =) 0; 2 f Q 27

28 Factor model i = X Z = (it) F t = Z= (i) F t= Z = e " = Z e X i= Z = (i) i + X i= i i + X i= t= # X Z = (it) " it he result follows. X i= Z = (i) i =) 0; 2 Q X i= t= X i= X Z = (it) " it m:s: i i 0 m:s: 0 Aendix 3 : Iid Bootstra Proosition 2 : CL for i.i.d. ootstra Under assumtions A and B, and aroriate Lindeerg conditions, if the ootstra-variance of error term V ar E it P 2, then iid =) 0; ; 2 Q Proof of roosition 2. variance. Aly Lindeerg-Feller CL to ( ; 2 ; ::::::; ) and correct the E ( it) = 0 V ar ( it) Z = (it) hv i ar ^E it Z (it) = Z = (it) Z (it)v ar E it ( ; 2 ; ::::::; ) are indeendent with di erent variances. X X i= t= V ar ( it) = e Z = e Z V ar E it P ; Q2 28

29 By Linderg-Feller CL, then : X X i= t= iid it =) 0; 2 Q =) 0; 2 Q Aendix 4 : Individual Bootstra Proosition 3 : CL for individual ootstra Under assumtions A and B, and aroriate Lindeerg conditions, ind =) (0; ) with = Q "P lim X i= Z = (i) h # i V ar ^E= (i) Z (i) Q ind i = Z = e= Z e X Z = ^E (it) it t= " X i i= = Z= (i) ^E (i) # Proof of Proosition 3. V ar ( t ) = 2 Z= (i) h i V ar ^E= (i) ( ; i ; ::::::; ) are indeendent with di erent variances. Under assumtions G, aly Linderg- Feller CL to ( ; i ; ::::::; ) : = Q " Lim X i= Z = (i) h Z (i) # i V ar ^E= (i) Z (i) Q Aendix 5 : emoral Bootstra tem t = Z = e= Z e X Z = ^E (it) it i= " X t t= = Z= (t) ^E (t) # 29

30 Proosition 4 : CL for temoral ootstra Under assumtions A and B, and aroriate Lindeerg conditions, tem: =) (0; ) with = Q "P lim X t= Z = (t) h # i V ar ^E= (t) Z (t) Q Demonstrations are similar to individual ootstra case. Aendix 6 : Block Bootstra Aendix 7 : Doule Resamling Bootstra 30