A New Diagnostic Test for Cross Section Independence in Nonparametric Panel Data Model

Transcription

1 e University of Adelaide Scool of Economics Researc Paper No October 2009 A New Diagnostic est for Cross Section Independence in Nonparametric Panel Data Model Jia Cen, Jiti Gao and Degui Li

2 e University of Adelaide, Scool of Economics Working Paper Series No: ) A New Diagnostic est for Cross Section Independence in Nonparametric Panel Data Models Jia Cen, Jiti Gao and Degui Li e University of Adelaide, SA 5005, Australia Abstract In tis paper, we propose a new diagnostic test for residual cross section independence in a nonparametric panel data model. e proposed nonparametric cross section dependence CD) test is a nonparametric counterpart of an existing parametric CD test proposed in Pesaren 2004) for te parametric case. We establis an asymptotic distribution of te proposed test statistic under te null ypotesis. As in te parametric case, te proposed test as an asymptotically normal distribution. We ten analyze te power function of te proposed test under an alternative ypotesis tat involves a nonlinear multi factor model. We also provide several numerical examples. e small sample studies sow tat te nonparametric CD test associated wit an asymptotic critical value works well numerically in eac individual case. An empirical analysis of a set of CPI data in Australian capital cities is given to examine te applicability of te proposed nonparametric CD test. Keywords: Cross section independence; local linear smooter; nonlinear panel data model; nonparametric diagnostic test, size and power function Jiti Gao is from te Scool of Economics, e University of Adelaide. Adelaide SA 5005, Australia. jiti.gao@adelaide.edu.au.

3 2 J. Cen, J. Gao and D. Li. Introduction Panel data analysis as become increasingly popular in many fields, suc as economics, finance and biology, since it provides te researcer wit a wide variety of double index models rater tan just purely cross section or time series data models. ere exists a ric literature on parametric linear and nonlinear panel data models. For an overview of statistical inference and econometric analysis of te parametric panel data models, we refer to te books by Baltagi 995), Arellano 2003) and Hsiao 2003). As in bot te cross sectional and time series cases, parametric models may be too restrictive in some cases. As a consequence, existing parametric tests may not be applicable in suc cases. o address suc issues, nonparametric and semiparametric metods ave been used in bot model estimation and specification testing. Recent studies include Li and Hsiao 998), Ulla and Roy 998), Hjellvik, Cen and jøsteim 2004), Li and Racine 2007), Cai and Li 2008), and Henderson, Carroll and Li 2008). Existing studies in nonparametric and semiparametric estimation and model specification testing mainly assume cross section independence. Suc an assumption is far from realistic, since cross section dependence may arise in practice due to te presence of common socks, unobserved components tat become part of te error term ultimately, economic distance and spatial correlations. If observations are cross section dependent, parametric and nonparametric estimators based on te assumption of cross section independence may be inconsistent. As pointed out by Hsiao 2003), meanwile, tere is no natural ordering for cross section indices, and appropriate modelling and estimation of cross section dependence is difficult particularly wen te dimension of cross section observations N is large. Hence, it is appealing to test for cross section independence before one attempts to make some statistical inference for a panel data model. ere is a substantial literature on diagnostic tests for cross section independence in parametric panel data models. Breusc and Pagan 980) proposed an Lagrange multiplier LM) test statistic, wic is based on te average of te squared pair wise correlation coefficients of te residuals. e LM test requires tat is muc larger tan N, were and N are te time dimension and te cross section dimension, respectively. Note tat te mean of te squared correlation coefficients is, owever, not correctly centered wen is small. Frees 995) tus proposed a test statistic tat is based on te squared Spearman rank correlation coefficients and allows N to be larger tan. Recently, Pesaran 2004) introduced te so called cross section dependence CD) test. e main idea of proposing te CD test is to use te simple average of all pair wise correlation coefficients of te residuals from te individual parametric linear regressions in te panel. e advantage of te CD test is tat it is correctly centered wen bot N and are fixed. Ng 2006) employed spacing variance ratio statistics to test te severity of cross section correlation in panels by partitioning te pair wise cross correlations into groups from ig to low. Ng 2006) s test statistics are proposed as agnostic tools for

4 Nonparametric est for Cross Section Independence 3 identifying and caracterizing correlations across groups. More recently, Hsiao, Pesaran and Pick 2007) extended te LM and CD tests from parametric linear panel data models to parametric nonlinear models. For oter recent contributions to diagnostic tests of cross section independence, we refer to Huang, Kab and Urga 2008), Pesaran, Ulla and Yamagata 2008), and Sarafidis, Yamagata and Robertson 2009). By contrast, tere is little study on diagnostic testing of te null ypotesis tat te residuals are cross section independence in a nonparametric nonlinear panel data model. We terefore propose a new diagnostic test for cross section independence in a nonparametric nonlinear panel data model. e main contributions of tis paper can be summarized as follows. i) We construct a local linear estimator of an individual regression function in te case were and ten propose a nonparametric CD test statistic in a similar fasion to tat proposed in Pesaran 2004) for te parametric case. As a sequence of using te local linear estimation metod, te first order biases involved are all eliminated in te construction of te proposed test. As sown in Sections 3 and 5, respectively, te proposed test as bot sound large and small sample properties. ii) We ten establis an asymptotically normal distribution under te null ypotesis, and also an asymptotically normal distribution under a sequence of local alternatives in Section 3 below. In te small sample studies in Section 5 below, we examine te performance of bot te size and te power functions under various cases were te conditional mean function and te residual may take te form of eiter linear, nonlinear or a mixture of bot. iii) We conclude from te small sample studies in Section 5 tat te proposed nonparametric CD test performs well wen te data satisfy a nonparametric panel data model. By comparison, existing tests for te parametric case are not applicable. In addition, te proposed nonparametric CD test also performs well in bot te size and power even wen te conditional mean function is of a parametric form. In tis case, te nonparametric CD test is just sligtly less powerful tat te parametric CD test. iv) In summary, te construction in Section 2 and te small sample analysis in Section 5 bot sow tat te proposed nonparametric CD test is easily computable and implementable. e simulation study in Section 5 sows tat te proposed nonparametric CD test is generally more applicable tan te corresponding parametric CD test. As an empirical application, we apply te proposed test for testing te cross section independence of a set of CPI data in Australian capital cities.

5 4 J. Cen, J. Gao and D. Li e rest of tis paper is organized as follows. A nonparametric test for cross section independence in a nonlinear panel data model is proposed in Section 2. An asymptotic distribution of te proposed nonparametric CD test statistic is establised in Section 3. Section 3 also establises an asymptotic normality under an alternative ypotesis. Section 4 discusses possible extensions. Several simulated examples are given in Section 5. An empirical analysis of a set of CPI data in Australian capital cities is given in Section 6. All te matematical proofs of te asymptotic results are given in Appendix A. 2. Nonparametric panel data model and CD test statistic Consider a nonparametric nonlinear panel data model of te form Y it = g i X it ) + u it, i =,, N; t =,,, 2.) were g i ) is te individual regression function, {X it } is random and satisfies some mild conditions see A2 below), and {u it } is independent of {X it } wit E[u it ] = 0. e aim of tis paper is to test te null ypotesis H 0 : {u it } is independent of {u jt } for all i j. 2.2) e above testing problem as been studied by many autors in te context of parametric panel data models. In te parametric case, te so called CD test statistic was introduced by Pesaran 2004) in te parametric linear panel data case. e main idea is to use te simple average of all pair wise correlation coefficients of te residuals from te individual nonparametric nonlinear regression in te panel. Before proposing a nonparametric CD test statistic, we need to decide wic kernel metod sould be used in te construction of our nonparametric CD test. Existing studies see, for example, Capter 3 of Gao 2007) already sow tat te use of te Nadaraya Watson kernel estimation metod in te construction of a nonparametric kernel test may ave severe size distortion due to te first order bias issue inerited from te Nadaraya Watson kernel estimation metod. In tis paper, we tus coose to use a local linear estimation metod in te construction of our nonparametric CD test. As sown in Section 3, te proposed nonparametric CD test as sound large sample teory under some mild conditions. Section 5 sows tat te proposed nonparametric CD test also as good small sample properties witout using a bootstrap metod. We now introduce te local linear estimator of te individual regression function g i ). Assume tat g i ) as derivatives up to te second order at te point x 0. By aylor s expansion, for x in a neigborood of x 0, we ave g i x) = g i x 0 ) + g ix 0 )x x 0 ) + O x x 0 ) 2). 2.3)

6 en, we find α 0, α ) to minimize Nonparametric est for Cross Section Independence 5 ) Y it α 0 α X it x 0 )) 2 Xit x 0 K, 2.4) were K ) is some kernel function and := is te bandwidt. e local linear estimator for g i x 0 ) is defined as ĝ i x 0 ) = α 0i, were α 0i, α i ) is te unique pair tat minimizes 2.4). For more details about te local linear estimators, we refer to Fan and Gijbels 996). In general, one probably sould use a kernel function and a bandwidt indexed by i for eac cross section. For notational simplicity, tis paper uses te same kernel and bandwidt for bot te large and small sample discussion. In practice, te bandwidt can be cosen using te conventional leave one out cross validation metod. By an elementary calculation, te local linear estimator of g i x 0 ) can be expressed as were w it x 0 ) = K x0,x it ) K x0,x it ) ĝ i x 0 ) = w it x 0 )Y it, i N, 2.5), in wic K x0,x it ) = K Xit x 0 ) S i2 x 0 ) ) Xit x 0 ) S i x 0 ) wit S ij x 0 ) = ) j ) Xit x 0 K Xit x 0 for j = 0,, 2. Wit te elp of te local linear smooter defined above, we estimate u it by ũ it = Y it ĝ i X it ). were We are now ready to propose a nonparametric CD test statistic of te form N N NCD = ρ ij, 2.6) NN ) u it u jt ρ ij = ρ ji = = u 2 it u 2 jt u it u jt, u 2 it u 2 jt in wic u it = ũ f it i X it ) and f i x 0 ) = K x0,x it ). Note tat te test statistic NCD is invariant to σui 2 = E[u2 i ]. e aim of using u it instead of ũ it in te test statistic 2.6) is to eliminate te random denominator problem involved in te nonparametric estimator ĝ i. e construction of te nonparametric CD test in 2.6) is motivated by a similar form proposed in Pesaran 2004) for

7 6 J. Cen, J. Gao and D. Li te parametric case. e main step is te involvement of a nonparametric estimate ũ it, wic is equivalent to te OLS estimate in te parametric case. As sown in Sections 3 and 5 below, te nonparametric CD test as bot good large and small sample properties. In Section 3 below, we sow tat te nonparametric CD test statistic 2.6) is asymptotically centered wen first and ten N. Furtermore, asymptotic distributions of te test statistic are establised under eiter te null ypotesis or a sequence of local alternatives. 3. Large sample teory 3. Asymptotic teory under te null ypotesis o study te asymptotic teory of te test statistic, we need te following conditions. A i). e probability kernel function K ) is a symmetric and continuous function wit some compact support. ii). e individual regression function g i ), i N, as derivatives up to te second order and te derivatives are continuous. Furtermore, max i E [ g i X i) 2] <, were g i ) is te second order derivative of g i ). A2 i). For eac individual series for eac fixed i N), {X it } is a sequence of stationary α mixing random regressors wit max i E [ X i 2] < and te mixing coefficient {α xi )} satisfying α xi k) C 0 k β uniformly in i and for some 0 < C 0 < and β > 3. ii). Let f i ) be te density function of {X it }. Suppose tat f i x) is continuous and bounded in x R. ere exists a joint density function f is,is 2,,is l,jt,jt 2,,jt k,, ) of X is, X is2,, X isl, X jt, X jt2,, X jtk ), i, j N, l, k 4, suc tat f is,is 2,,is l,jt,jt 2,,jt k,, ) is continuous and bounded. iii). Let {u is, i N, s } and {X jt, j N, t } be independent for all i, j) and s, t). For eac individual series for eac fixed i N), {u it } is a sequence of stationary α mixing random errors wit te mixing coefficient {α ui )} satisfying δ 0 2+δ α 0 ui k) < for some δ 0 > 0. In addition, E [ u 2 i] = σ 2 ui > 0 and [ k= ] max i E u i 2+δ 0 <. iv). Let τi,j 2 = µ4 2 µ4 0 κ i,j σui 2 σ2 uj + 2 ) E[u i u it ]E[u j u jt ]κ i,j t), were µ k = u k Ku)du, t=2 κ i,j = f 4 i x)f 4 j y)f i,jx, y)dxdy and κ i,j t) = fi 2 x )fi 2 x 2 )fj 2 y )fj 2 y 2 )f i,it,j,jt x, x 2, y, y 2 )dx dx 2 dy dy 2,

8 Nonparametric est for Cross Section Independence 7 and σi 2 = µ2 2 µ2 0 σ2 ui f 5 i x)dx. Let 0 < τi,j 2 < and σ2 i tat tere exists some τ 0 > 0 suc tat as N, > 0 for all i, j N. Suppose NN ) N N τ 2 i,j σ 2 i σ2 j τ 0. 3.) A3. e bandwidt satisfies were θ = β 3 β+2. θ log as and N as and N, 3.2) Remark 3.. e above assumptions are mild and can be satisfied in many cases. For example, Ai) is a mild condition on te kernel function and is assumed by many autors in nonparametric inference of bot stationary time series and panel data see, for example, Fan and Yao 2003; Gao 2007; Cai and Li 2008). Aii) and A2ii) are some mild conditions on te individual regression functions and density functions. e α mixing condition assumed in A2i) and A2iii) is a commonly used condition in te time series case see, for example, Auestad and jøsteim 990, Cen and say 993, Fan and Yao 2003; Gao 2007; Li and Racine 2007). It is introduced in tis paper for te nonparametric panel data case. Note tat wen {u is } and {u it } are mutually independent for all s t and eac fixed i, and {X it } and {X jt } are independent for all i j and eac given t, we ave κ i,j = f 5 i x)f 5 j y)dxdy, κ i,jt) 0 and τ 2 ij = µ4 2 µ4 0 κ i,jσ 2 ui σ2 uj = σ2 i σ2 j. us, τ 0. Condition A3 is a set of conditions on te bandwidt as well as on te restriction on and N. e first bandwidt condition in A3 is proposed in order to apply te uniform consistency of te nonparametric kernel estimator in te proofs of eorems 3. and 3.2 below. e second bandwidt condition in A3 is also needed in te proofs of eorems 3. and 3.2. In addition, te second part of A3 allows for te case were rate of is slower tan tat of N. is basically implies tat condition A3 allows for bot medium and small integers for in practice wile te asymptotic teory requires bot N and in teory. e simulation studies in Section 5 support tat te nonparametric CD test works well even wen as small as = 0, altoug it cannot be sown at tis stage tat te conclusions of eorems 3. and 3.2 remain true wen is fixed. In te following teorem, we sow tat te nonparametric CD test statistic, defined by 2.6), as an asymptotically normal distribution as tat obtained by Pesaran 2004) and Hsiao, Pesaran and Pick 2007), wo considered similar testing problems in te context of parametric linear and nonlinear panel data models.

9 8 J. Cen, J. Gao and D. Li eorem 3.. Assume tat 2.) and te conditions A A3 are satisfied. en under H 0, N N NCD = ρ ij d N0, τ 0 ) 3.3) NN ) as first and ten N. e proof of eorem 3. is given in Appendix A below. Remark 3.2. i) Note tat τ 0 = wen {u is } and {u it } are mutually independent for all s t and eac fixed i, and {X it } and {X jt } are independent for all i j and eac given t. ii) In general, τ 0 is an unknown parameter to be estimated. Define ρ ij = ρ ij σ i σ j τ i,j, were τ i,j and σ i are consistent estimators of τ i,j and σ i, respectively. In tis case, it can be sown tat under H 0 N N NCD =: ρ NN ) ij d N0, ) as first and ten N. Remark 3.3. e asymptotic distribution in eorem 3. is obtained by letting first and ten N. A natural question is wat will appen if eiter N first and ten or and N simultaneously. i) o see tis, we define Z it = u itfi 2X it)µ 0 µ 2 σ i. Following te proof of eorem 3. in Appendix A, we can find tat te leading term of NCD is N NN ) N were ω N t) = NN ) Z it Z jt = N N N N Z it Z jt =: NN ) { Ni= ) } 2 Z it Z jt = Z it Ni= Z 2 NN ) it. ω N t), It is obvious tat, for eac fixed t, {ω N t)} is a sequence of U statistics. By eorem in Serfling 980), under H 0, ω N t) d χ 2 t, as N, were χ 2 t, is te ci square distribution wit one degree of freedom. If bot {u it} and {X it } are i.i.d. for all i, t), ten it can be seen tat {χ 2 t,, t } is a sequence of i.i.d. ci square random variables. By te conventional central limit teorem for te i.i.d. case see, for example, Cow and eicer 988), te conclusion of Corollary 3. remains true wen N first and ten.

10 Nonparametric est for Cross Section Independence 9 ii) If bot {u it } and {X it } are i.i.d. for all i, t), moreover, {ω N t} is also a sequence of i.i.d. errors. us, it follows from te conventional central limit teorem for te i.i.d. case tat as bot N and simultaneously ω N t) d N0, ), wic implies tat te conclusion of Corollary 3. remains true. In summary, eorem 3. and te discussion given in Remarks 3.2i) and 3.3 imply te following corollary; its implementation is given troug te simulation studies in Section 5 and te empirical application in Section 6. Corollary 3.. Assume tat 2.) and te conditions A and A3 are satisfied. If, in addition, {u is } and {u it } are mutually independent for all s t and eac fixed i, and {X it } and {X jt } are independent for all i j and eac given t, ten under H 0, N N NCD = ρ ij d N0, ) 3.4) NN ) as eiter first and ten N, or N first and ten, or bot N and simultaneously. In summary, te limiting distribution of te suitably normalized test statistic depends on te independence or dependence assumption on {X it } and {u it } as well as te treatment of te two indices N and. Pillips and Moon 999) introduced tree limit approaces: sequential limit teory, diagonal pat limit teory and joint limit teory. ey also discuss some relations between sequential and joint limits. e asymptotic distribution given in eorem 3. is obtained by a sequential limit approac first and ten N ). It is not clear weter te conclusion of eorem 3. remains true wen eiter N first and ten or bot N and simultaneously. Suc issues are tus left to future researc Asymptotic teory under an alternative ypotesis In tis section, we analyze te power of te proposed test under a sequence of local alternatives. Naturally, te power of te proposed test for te cross section dependence relies on te form of an alternative ypotesis. We now consider a sequence of cross sectional dependence alternatives via a nonlinear multi factor model of te form H : u it = F N z t, β i ) + ε it wit F N z t, β i ) = N /2 /4) k Gz t, β i ) 3.5) for k = 0,, were {Gz t, β i )} is a sequence of known parametric linear or nonlinear functions indexed by {β i }, {z t, t } is a sequence of stationary α mixing random variables,

11 0 J. Cen, J. Gao and D. Li {β i, i N} is a sequence of common factors, {ε it } is a sequence of stationary α mixing random variables for fixed i and is independent of {z t }, and {ε it } is independent of {ε jt } for all t and i j. Note tat form 3.5) defines a global alternative wen k = 0, wile it gives a sequence of local alternatives wen k =. Before establising an asymptotic distribution of te nonparametric CD test statistic under te alternative ypotesis H, we need te following set of conditions. A4 i) {z t } is a sequence of stationary α mixing random variables wit te mixing coefficient {α z )} satisfying α δ /2+δ ) z t) < for some δ > 0. ii) e nonlinear function G, ) satisfies te following conditions, E[Gz t, β i )] = 0, 3.6) E[Gz t, β i )] 2+δ <. 3.7) In addition, tere exists an array of constants {ψ ij ; i N, j N} wit ψ ij = ψ ji suc tat were ψ is a constant. E[Gz t, β i )Gz t, β j )] = ψ ij 3.8) N N ψ ij ψ NN ) as N, 3.9) iii) A2iii) and A.2iv) are bot satisfied wen {u it } is replaced by {ε it }. Moreover, {ε it } is independent of {z t }. Let τ be defined in te same way as for τ 0 wit {u it } being replaced by {ɛ it }. Condition A4 allows for a general class of forms for Gz t, β i ). It obviously covers te linear multi factor case: Gz t, β i ) = z t β i, wic was studied by Pesaran 2004). Wen te alternative ypotesis H olds, we ave te following asymptotic distribution for te nonparametric CD test statistic NCD. e proof of eorem 3.2 below is given in Appendix A below. eorem 3.2. Assume tat 2.) and te conditions A, A2 i), A3 and A4 are satisfied. i) Under H wit k = 0, we ave as first and ten N NCD P. 3.0)

12 Nonparametric est for Cross Section Independence ii) Under H wit k =, we ave as first and ten N NCD d Nψ, τ ). 3.) e divergence result in 3.0) is quite common in te case were we assume tis kind of global alternative. e asymptotic distribution in 3.) is similar to te result obtained by Pesaran 2004). F N z t, β i ) can be viewed as te measure of te dependence between individual time series. By an elementary calculation and 3.8), we can sow tat E[u it u jt ] = ψ ij N. Furtermore, 3.9) implies tat te nonparametric CD test statistic allows te detection of te alternatives wen te nonlinear multi factor function as a decreasing rate of O /4 N /2), wic is te same as tat in Pesaran 2004). e simulated examples in Section 5 sow tat te power of te proposed test is satisfactory wen ψ > 0 or ψ < 0). However, wen ψ = 0, te asymptotic distribution in 3.) is te same as tat in eorem 3., wic implies tat te test will not ave a satisfactory power. In te context of parametric panel data models, Pesaran, Ulla and Yamagata 2008) proposed a bias adjusted LM test to avoid te problem of poor power for te case of ψ = 0. It is interesting to consider a nonparametric type of bias adjusted LM test statistic. Suc an issue is left for our future study. 4. Some extensions In bot teory and practice, tere are cases were we need to consider a nonlinear autoregressive panel data model of te form Y it = g i Y i,t ) + u it, i =,, N; t =,,. 4.) Pesaran 2004) and Sarafidis, Yamagata and Robertson 2008) studied te test of error cross section dependence wen all g i ), i N, are of some linear form. As far as we are aware, owever, tere is little study on diagnostic testing of cross section independence for model 4.). It seems tat we may apply te nonparametric CD test statistic NCD to test weter H 0 olds. In teory, establising an asymptotic distribution for te nonparametric CD test NCD in tis case is not straigtforward. Furter discussion is left for our future study. Meanwile, nonparametric approaces are useful for exploring idden structures. Wen tere are multiple regressor variables, owever, te nonparametric approaces face a serious problem of te so called curse of dimensionality. o address tis issue, some dimensional reduction metods ave been discussed in bot te cross section data and te time series data cases see, for example, Härdle, Liang and Gao 2000; Gao 2007; Li and Racine 2007). In te panel data case, we may consider a partially linear model of te form Y it = X τ itα + mz it ) + ε it, i =,, N; t =,,, 4.2)

13 2 J. Cen, J. Gao and D. Li were {X it } is a vector of regressors, α is a vector of unknown parameters and te coefficient functions m ) are all unknown. Recently, tere ave been some attempts on bot teoretical studies and empirical applications of tis type of partially linear models in te panel data case see, for example, Li and Hsiao 998; Henderson, Carroll and Li 2008). o te best of our knowledge, tere is little study on te testing of cross section independence for a partially linear panel data model of te form 4.2). We will extend te proposed test statistic to te partially linear model case and establis an asymptotic distribution of te proposed test statistic. Since different metods and more tecnicalities are likely to get involved, suc an issue is terefore left for future researc. 5. Small sample simulation studies In tis section, we give some simulated examples to sow te finite sample performance of te nonparametric CD test. In addition, we also compare its performance wit tat of a parametric CD test. Since bot te sizes and power values of te proposed nonparametric CD test associated wit an asymptotic critical value in eac case are already comparable wit tose of te parametric CD test based on an asymptotic critical value, our experience suggests tat tere is no need to introduce a bootstrap simulation procedure to improve te finite sample performance of te proposed nonparametric CD test. In te following experiments, te uniform kernel Ku) = 2I{ u } is used in te implementation of te proposed nonparametric CD test. e bandwidt is cosen using te conventional leave one out cross validation metod. We first examine te finite sample performance of te proposed nonparametric test wen te data set is simulated from a parametric linear panel data model of te form Y it = a i + b i X it + u it, i =, 2,, N; t =, 2,,, 5.) i.i.d. i.i.d. i.i.d. were a i U0, ), b i N, 0.04), X it N0, ), u it = fr i, β t ) + e it, β t is te time i.i.d. i.i.d. specific common effect and β t N0, ), e it N0, ), and {r i } is a sequence of non random numbers indicating te degree of cross section error correlations. Note tat {u it } and {X it } are generated independently. Under te null ypotesis of cross section independence, we ave r i = 0, and under te i.i.d. alternative ypotesis, we experiment wit r i U0., 0.3). e parameters a i, b i, and r i are drawn once for eac i =, 2,, N, and ten fixed trougout te replications. X it, β t, and e it are newly drawn for eac replication, independently of eac oter. We experiment wit bot linear and nonlinear forms for te function f, ). For te linear case, we set fr i, β t ) = r i β t, and for te nonlinear case, we set fr i, β t ) = r iβ t. It can be +ri 2β2 t

14 Nonparametric est for Cross Section Independence 3 easily seen tat te pair wise correlation coefficients are given by corru i,t, u jt ) = r i r j + r 2 i ) + r2 j ), for te parametric linear case, and )) E r i r j βt 2 / + ri 2 corru it, u jt ) = β2 t ) + rj 2β2 t ) )) + E r 2 i β2 t / + r2 i β2 t )2)) + E rj 2β2 t / + r2 j β2 t )2 for te parametric nonlinear case. Using an asymptotic critical value, we computed te two sided simulated sizes and power values of te proposed nonparametric CD test and te parametric counterpart in eac case. e experiments are carried out for N, = 0, 20, 30, 50, 00. e number of replications is 000, and te significance level is p = %, 5%, and 0%, respectively. e simulated sizes of te parametric and te nonparametric CD tests for te linear model 5.) are reported in able 5. below. able 5.a) Size of te tests for linear model 5.) at te % level parametric test nonparametric test \N able 5.b) Size of te tests for linear model 5.) at te 5% level parametric test nonparametric test \N able 5.c) Size of te tests for linear model 5.) at te 0% level parametric test nonparametric test \N

15 4 J. Cen, J. Gao and D. Li ables 5.a) 5.c) sow tat te simulated sizes look quite reasonable in eac case regardless of weter using te nonparametric CD test or using te parametric CD test. is implies tat te nonparametric CD test is still applicable even wen te data follow a parametric linear model. In addition, te results in ables 5.a) 5.c) sow tat te nonparametric CD test associated wit an asymptotic critical value works well numerically even wen and N are as small as = N = 20. In addition, te tables also sow tat te sizes of te parametric CD test are sligtly more stable tan tose of te nonparametric CD test, mainly because te true model is just parametric and te parametric CD test is supposed to perform better. e power values of te tests for model 5.) wit linear fr i, β t ) = r i β t ) and nonlinear fr i, β t ) = r i β t / + ri 2β2 t )) forms of f, ) are given in able 5.2 below. able 5.2a) Power of te tests for linear model 5.) at te % level parametric test nonparametric test \N fr i, β t) = r iβ t fr i, β t) = r iβ t/ + r 2 i β 2 t ) able 5.2b) Power of te tests for linear model 5.) at te 5% level parametric test nonparametric test \N fr i, β t) = r iβ t fr i, β t) = r iβ t/ + r 2 i β 2 t )

16 Nonparametric est for Cross Section Independence 5 able 5.2c) Power of te tests for linear model 5.) at te 0% level parametric test nonparametric test \N fr i, β t) = r iβ t fr i, β t) = r iβ t/ + r 2 i β 2 t ) ables 5.2a) 5.2c) sow tat te simulated power values are quite satisfactory in eac of te cases concerned. Meanwile, te simulated power values of te nonparametric CD test associated wit an asymptotic critical value are quite comparable wit tose of te parametric CD test based on te use of an asymptotic critical value. is may be due to te fact tat te asymptotic normality can be used as a good approximation to te sample distribution of te proposed nonparametric test in eac of te cases considered. In addition, ables 5.2a) 5.2c) sow tat te parametric CD test is more powerful tan te nonparametric CD test. is is not surprising, since te true model is just parametric and te parametric CD test is supposed to be more powerful. In te following simulation studies, we examine te finite sample performance of te proposed nonparametric test wen te data set is simulated from a parametric nonlinear panel data model of te form i.i.d. were θ i Y it = + θi 2 + u it, i =, 2,, N; t =, 2,,, 5.2) X2 it i.i.d. N, 0.04), X it U0., 0.7), and {u it } is te same as in model 5.). Wen r i = 0, te simulated sizes of te parametric and nonparametric CD test for tis model are i.i.d. reported in able 5.3 below at different significance levels, and wen r i U0., 0.3), te power values of te test are reported in able 5.4 below.

17 6 J. Cen, J. Gao and D. Li able 5.3a) Size of te tests for nonlinear model 5.2) at te % level parametric test nonparametric test \N able 5.3b) Size of te tests for nonlinear model 5.2) at te 5% level parametric test nonparametric test \N able 5.3c) Size of te tests for nonlinear model 5.2) at te 0% level parametric test nonparametric test \N ables 5.3a) 5.3c) sow tat bot te parametric CD test and te nonparametric CD test already ave reasonable simulated sizes wen using an asymptotic critical value in eac case. As in ables 5.a) 5.c), te simulated sizes of te nonparametric CD test are very comparable wit tose of te parametric CD test. able 5.4 gives te corresponding power values for bot te parametric and nonparametric CD tests.

18 Nonparametric est for Cross Section Independence 7 able 5.4a) Power of te tests for nonlinear model 5.2) at te % level parametric test nonparametric test \N fr i, β t ) = r i β t fr i, β t ) = r i β t / + r 2 i β2 t ) able 5.4b) Power of te tests for nonlinear model 5.2) at te 5% level parametric test nonparametric test \N fr i, β t ) = r i β t fr i, β t ) = r i β t / + r 2 i β2 t )

19 8 J. Cen, J. Gao and D. Li able 5.4c) Power of te tests for nonlinear model 5.2) at te 0% level parametric test nonparametric test \N fr i, β t ) = r i β t fr i, β t ) = r i β t / + r 2 i β2 t ) ables 5.4a) 5.4c) sow tat te simulated power values are quite satisfactory in eac of te cases concerned. Meanwile, te simulated power values of te nonparametric test sow tat te nonparametric CD test is only sligtly less powerful tan te parametric CD test. In summary, we can conclude tat in bot te parametric linear and nonlinear models, te nonparametric CD test as te correct size even for small N and. Wile te power of te proposed nonparametric CD test increases as N or increases. it increases faster wit N tan wit. Similar findings ave been drawn from Hsiao, Pesaran and Pick 2007) for te parametric CD test. is sows tat te proposed nonparametric CD test is a generally applicable test in tis kind of testing for cross section independence, as te applicability does not require a model to be parametrically specified. In oter words, it still works well witout necessarily pre specifying te conditional mean function. In te following example, we sow tat te proposed nonparametric CD test is needed wen te data follow a nonparametric panel data model, since existing tests for te parametric case are not applicable. Consider a nonparametric panel data model of te form X it Y it = + Xit 2 + u it, i =, 2,, N; t =, 2,,, 5.3) were X it i.i.d. N0, ), and {u it } is te same as used in model 5.). For r i = 0, te sizes of te proposed nonparametric CD test are reported in able 5.5, and for r i U0., 0.3), te power values are given in able 5.6.

20 Nonparametric est for Cross Section Independence 9 able 5.5a) Size of te nonparametric test for model 5.3) at te % level \N able 5.5b) Size of te nonparametric test for model 5.3) at te 5% level \N able 5.5c) Size of te nonparametric test for model 5.3) at te 0% level \N able 5.6a) Power of te nonparametric test for model 5.3) at te % level fr i, β t ) = r i β t fr i, β t ) = r i β t / + ri 2β2 t ) \N

21 20 J. Cen, J. Gao and D. Li able 5.6b) Power of te nonparametric test for model 5.3) at te 5% level fr i, β t ) = r i β t fr i, β t ) = r i β t / + ri 2β2 t ) \N able 5.6c) Power of te nonparametric test for model 5.3) at te 0% level fr i, β t ) = r i β t fr i, β t ) = r i β t / + ri 2β2 t ) \N ables 5.5a) 5.5c) sow tat te nonparametric CD test as te correct sizes for te simulated nonparametric panel data model 5.3). Meanwile, ables 5.6a) 5.6c) sow tat te simulated power values of te nonparametric CD test are also satisfactory. 6. Empirical application: An analysis of CPI in Australian capital cities As an application of our testing metod, we test for te cross sectional independence of CPI consumer price index) between eigt Australian capital cities during te period e data set, wic is obtained from te website of te Australian Bureau of Statistics, is recorded quarterly eac year. Hence, it consists of te CPI numbers for eigt cities N = 8) at 80 different times = 80). We cose Y it as te log of te food CPI for city i at time t and X it as te log of all group CPI for city i at time t. For eac city i, we computed te nonparametric regression function of Y it on X it t =, 2,, ) using te nonparametric local linear estimation metod. en, we used te estimation residuals u it to compute te nonparametric CD test statistic. In a similar way, we also computed te regression of log of te transportation CPI on log of all group CPI for eac city. e results are summarized in able 6..

22 Nonparametric est for Cross Section Independence 2 able 6. Cross section dependence of CPI in Australian capital cities food transportation nonparametric CD test bootstrap % critical values [ 2.330, 2.600] [ , ] bootstrap 5% critical values [.8796,.857] [.8786,.8899] bootstrap 0% critical values [.6086,.5584] [.6532,.6203] Note tat te two-sided bootstrap critical values were calculated using 000 iterations. It follows from able 6. tat tere is some evidence to suggest rejecting te null ypotesis tat te cross section independence is true for bot te food and transportation indexes. Meanwile, based on te bootstrap simulated critical value in eac case, te cross section independence sould be rejected at all te levels of %, 5% and 0%. is suggests tat te assumption of cross section independence in suc empirical studies may not be appropriate. Furter studies are needed to find ways of defining a suitable cross section dependence structure in order to deal wit panel data analysis wen tere is some cross section dependence. 7. Conclusions and discussion We ave proposed a new diagnostic test for residual cross section independence in a nonparametric panel data model. e proposed test is a nonparametric counterpart of an existing test proposed in Pesaren 2004) for te parametric case. e asymptotic distribution under eiter te null or a sequence of local alternatives as been establised. e small sample performance of te proposed test as been examined in Section 5. Section 6 as given an example of empirical application. Future researc in tis field includes discussion about ow to coose a data driven bandwidt suc tat bot te resulting size and power functions are appropriately assessed. As pointed out in Section 4, certain extensions of te model may also be considered. Since study of suc topics is not trivial, tey are left for future researc. 8. Acknowledgments is work was motivated by a keynote presentation by Professor Ceng Hsiao at an International Conference on ime Series Econometrics at WISE in Xiamen, Cina in May 2008 wen te second autor was a participant at te conference. e second autor would like to tank Professor Yongmiao Hong for is invitation to participate in te conference. e autors would all like to acknowledge te Australian Researc Council Discovery Grants Program for its financial support under Grant Numbers: DP and DP Appendix A: Proofs of te main results

23 22 J. Cen, J. Gao and D. Li Before proving te main results, we need te following lemma on te uniform consistency of nonparametric estimators. Since te proposed test statistic is invariant to σui 2 = E[u2 i ], we assume witout loss of generality tat σ ui trougout tis appendix. In addition, we use a double sum of te form s t to replace eiter s=, t or s t for notational simplicity trougout tis appendix. Lemma A.. Assume tat Ai) and A2i) are satisfied. If, in addition, θ log, were θ = β 3 β+2, ten we ave for k = 0,, 2, as uniformly in i, were µ k = u k Ku)du. Proof. Observe tat sup S ik x) f i x)µ k = o P ), x R sup S ik x) f i x)µ k sup S ik x) f i x)µ k + sup S ik x) f i x)µ k x R x c x c uniformly in i, were c = /2 log. For any ɛ > 0, by eorem 6 in Hansen 2008), we know tat tere exists an integer 0 suc tat wen > 0, we ave P max i sup S ik x) f i x)µ k ɛ/2 x c Since f i ) is continuous and integrable, max i sup f i x) 0 as. Hence, wen x c µ k 0, ) P P = P + P 0, max i max i max i max i sup S ik x) f i x)µ k > ɛ/2 x c ) ). sup S ik x) > ɛ/4 + P max i sup f i x) > ɛ/4 µ k ) x c x c ) k ) ) sup Xit x x c K Xit x > ɛ/4 ) sup f i x) > ɛ/4 µ k ) x c P max i X it c C) + P C c ) 2 max i E X i 2 + P max i max i sup f i x) > ɛ/4 µ k ) x c ) sup f i x) > ɛ/4 µ k ) x c were C is some positive constant. Wen µ k = 0, from te above argument we can see tat ) )

24 P max i sup S ik x) > ɛ/2 x c Nonparametric est for Cross Section Independence 23 P ) max 0. erefore, we ave sup i x R wic completes te proof of Lemma A.. S ik x) f i x)µ k > ɛ ) 0, We ten give te well known Davydovs inequality for α mixing sequence, wic follows from Corollary A2 of Hall and Heyde 980). An updated version is given in Lemma A. of Gao 2007). Lemma A.2. Suppose tat E X p < and E Y q <, were p, q >, p + q <. en were α = EXY ) EX)EY ) 8E X p ) /p E Y q ) /q α p q, sup P AB) P A)P B). A σx),b σy ) Proof of eorem 3.. Note tat u it = Y it ĝ i X it )) f i X it ) = u it f i X it ) + g i X it ) ĝ i X it )) f i X it ). Hence, by a standard decomposition, we ave ) u it u jt = N u f it i X it )u f jt j X jt ) N u f it i X it ) u js K j st ) s= + N u f it i X it ) g j X jt ) g j X js )) K j st s= ) N u f jt j X jt ) u is K st i s= ) + N u f jt j X jt ) g i X it ) g i X is )) K st i s= + N g i X it ) ĝ i X it )) g j X jt ) ĝ j X jt )) f i X it ) f j X jt ) 6 =: ρ i, j, k), k= A.) were K i st = K Xit,X is ). below. In te following, we complete te proof of eorem 3. troug using Lemmas A.3 A.7 Lemma A.3. Assume tat te conditions of eorem 3. are satisfied. en under H 0, we ave ρ i, j, 2) = o P N ), ρ i, j, 4) = o P N ). A.2)

25 24 J. Cen, J. Gao and D. Li Proof. We only give te detailed proof for te case of ρ i, j, 2) since te proof for ρ i, j, 4) is similar. Observe tat under H 0, =: x R 2 E ρ i, j, 2) = N 4 E ) lt) 2 u it K st i u jl K j s= l= = N 4 E u it u it2 u jl u K i K jl2 s t i K j K j s 2 t 2 l t l 2 t 2 t,t 2 = l,l 2 = s,s 2 = C N 4 E u 2 itu 2 jl K st) i 2 K j lt )2 l= s= + C N 4 E u 2 itu 2 jl K s i t K s i 2 t K j lt )2 l= s s 2 + C N 4 E u it u it2 u 2 jl K i K st st i 2 K j lt )2 t t 2 l= s= + C N 4 E u 2 itu jl u jl2 K st) i 2 K j l t K j l 2 t l l 2 s= + C N 4 E u it u it2 u 2 jl K i K s t s i 2 t 2 K j lt )2 t t 2 l= s s 2 + C N 4 E u 2 itu jl u K jl2 s i t K s i 2 t K j l t K j l 2 t l l 2 s s 2 + C N 4 E u it u it2 u jl u K i K jl2 st i K j K j st 2 l t l 2 t 2 t t 2 l l 2 s= + C N 4 E u it u it2 u jl u K i K jl2 s t i K j K j s 2 t 2 l t l 2 t 2 t t 2 l l 2 s s 2 8 Π k. k= By Lemma A. and µ = 0, we ave ) )) sup K Xit x Xit x x, X it ) K f i x)µ 2 o P ) = o P ),

26 wic, by A iii), implies tat Nonparametric est for Cross Section Independence 25 K x, X it ) C K Xit x were C is independent of x and X it. Let Kst i = K Xit X is For Π, by A.3), we ave Π = O = O = O erefore, ), A.3) C N 4 4 E u 2 itu 2 jt + C N 4 4 E u 2 itu 2 jt Kst) i 2 + K j st )2) s t + N 4 4 E u 2 itu 2 jlkst) i 2 K j lt )2 s t l t N 2 3 4) + O 4 4 E [Kst) i 2 + K j st )2] s t + O 4 4 E [Kst) i 2 K j lt )2] s t l t N 2 3 4) + O ) w v 4 4 K 2 f is,it v, w)dvdw s t + O ) w v 4 4 K 2 f js,jt v, w)dvdw s t + O ) 4 4 K 2 v u s t l t ) ) K 2 u2 v 2 f is,it,jl,jt u, v, u 2, v 2 )du du 2 dv dv 2 N N N 2 ) 2. ). ) N 2 Π = O 2. A.4) On te oter and, for Π 2, we ave C N Π E u 2 itu 2 jlkstk i j lt )2 s t l= + C N 4 4 E u 2 itu 2 jlks i tks i 2 tk j lt )2 s t s 2 t,s l= = O N 4 4 E Kst i + O N 4 4 E KstK i j lt )2 s t s t l t

27 26 J. Cen, J. Gao and D. Li + O N 4 4 E Ks i tk i s 2 t s t s 2 t,s + O N 4 4 E Ks i tks i 2 tk j lt )2 s t s 2 t,s l t O N N ) w v 4 4 K f is,it v, w)dvdw i= s t + O ) v u 4 4 K s t l t ) ) K 2 v2 u 2 f is,it,jl,jt u, v, u 2, v 2 )du du 2 dv dv 2 + O ) v u 4 4 K s t s 2 t,s ) ) w u K f is,is 2,itu, v, w)dudvdw + O ) ) v u w u 4 4 K K s t s 2 t,s l t ) ) K 2 v2 u 2 f is,is2,it,jl,jtu, v, w, u 2, v 2 )du du 2 dv dv 2 dw N 2 = O N N 2 ). erefore, we ave By A2 ii) and Lemma A.2, we ave ) N 2 Π 2 = O. A.5) 2+δ E [u it u it2 ] E [u it ] E [u it2 ] C 0 α 0 u t t 2 ), A.6) were C 0 is some positive constant. Hence, by te α mixing coefficient condition in A2 ii), A.6) and following te calculation of Π 2, we ave ) N 2 Π k = O, k = 3,, 8. A.7) In view of A.4), A.5) and A.7), we ave δ 0 ρ i, j, 2) = O P N /2 ) = o P N ) A.8) since by A3.