Estimation in Semiparametric Single Index Panel Data Models with Fixed Effects

Transcription

1 Estimation in Semiparametric Single Index Panel Data Models with Fixed Effects By Degui Li, Jiti Gao and Jia Chen School of Economics, The University of Adelaide, Adelaide, Australia Abstract In this paper, we consider semiparametric estimation in a partially linear single index panel data model with fixed effects. Without taking the difference explicitly, we propose using a semiparametric minimum average variance estimation SMAVE based on a dummy variable method to remove the fixed effects and obtain consistent estimators for both the parameters and the unknown link function. As both the cross section size and the time series length tend to infinity, we not only establish an asymptotically normal distribution for the estimators of the parameters in the single index and the linear component of the model, but also obtain an asymptotically normal distribution for the nonparametric local linear estimator of the unknown link function. The asymptotically normal distributions of the proposed estimators are similar to those obtained in the random effects case. Furthermore, we study several partially linear single index dynamic panel data models. Both simulated examples and an application to a real data set are provided to illustrate the finite sample behavior of both the theory and estimation method proposed in this paper. Keywords: Fixed effects, local linear smoothing, panel data, semiparametric estimation, single index models. Abbreviated Title: Partially Linear Single Index.

2 . Introduction Panel data analysis has become increasingly popular in many fields, such as, climatology, economics and finance. The double index models enable researchers to estimate complex models and extract information that may be difficult to obtain by applying purely cross section or time series models. There exists rich literature on parametric linear and nonlinear panel data models. For an overview of statistical inference and econometric analysis of parametric panel data models, we refer to the books by Baltagi 995, Arellano 2003 and Hsiao As in both the cross section and time series cases, parametric panel data models may be misspecified, and estimators obtained from such misspecified models are often inconsistent. To address such issues, some nonparametric methods have been used in both panel data model estimation and specification testing. Recent studies include Ullah & Roy 998, Hjellvik et al 2004, Cai & Li 2008, Henderson et al 2008, and Mammen et al In the multivariate setting with more than three covariates, the underlying regression function cannot be estimated with reasonable accuracy due to the so called curse of dimensionality. How to circumvent the curse of dimensionality is an important issue in both nonlinear time series and panel data analysis. Many useful approaches are developed to address this issue see, recent books by Fan & Yao 2003; Gao 2007; Li & Racine 2007 for example. One commonly used way is the so called semiparametric partially linear modeling. An advantage of the semiparametric partially linear modeling is that any existing information concerning possible linearity of some of the components can be taken into account in such models. This has been studied extensively in both the time series and panel data cases see, for example, Gao 2007; Li & Racine As is well known, however, the nonparametric components in the partially linear models may only accommodate covariates X with low dimension and they are also subject to the curse of dimensionality when the dimension of X is larger than three. To address such an issue, we propose using a dimension reduction model in this paper by taking into account the single index form X θ. Specifically, we consider a partially 2

3 linear single index panel data model of the form Y it = Z itβ 0 + ηx itθ 0 + α i + v it, i n, t T,. where {Z it = Z it,,, Z it,d } is of dimension d, {X it = X it,,, X it,p } is of dimension p, β 0 and θ 0 are unknown parameters with dimension d and p, respectively, η is an unknown link function, {α i } is unobserved i.i.d. random sequence with zero mean and a finite variance, and {v it } is the random error. Model. is called a fixed effects model if {α i } is allowed to be correlated with {Z it } and or {X it } with an unknown correlation structure. Model. is called a random effects model if {α i } is uncorrelated with both {Z it } and {X it }. In this paper, we are concerned with the fixed effects case. For the purpose of identification, we assume that n α i = 0 and θ 0 =,.2 i= where := 2 is the L 2 distance. The first term in.2 is a commonly used identification condition on the fixed effects see, for example, Su & Ullah 2006, Sun et al The second one in.2 is an identification condition for the single index structure in our model see, for example, Carroll et al 997, Xia et al Without loss of generality, we assume that the first entry of θ 0 is positive in this paper. Model. covers many interesting panel data models. When β 0 0, model. reduces to a single index panel data model Bai et al When {X it } is scalar, model. becomes to a partially linear panel data model with fixed effects Su & Ullah When β 0 0 and η is known, model. is a generalized linear panel data model with fixed effects Hsiao Existing literature mainly focuses on both nonparametric and semiparametric estimation of random effects panel data models see Li & Stengos 996; Ullah & Roy 998; Lin & Ying 200; Lin & Carroll 2000, 200, 2006; Henderson & Ullah 2005 for example. Note that the random effects estimators are inconsistent if the true model is one with fixed effects. In this paper, we will develop a semiparametric estimation method associated with a local linear dummy variable approach for model.. The estimation method is consistent under either the random effects setting or the fixed effects setting. 3

4 In this paper, we also allow that either {Z it } or {X it } contains the lagged values of {Y it } for each i. In this case, model. covers some useful partially linear single index dynamic panel data model. In Section 4, we show that under some mild conditions, {Y it, t } is geometrically ergodic when it is generated by a type of partially linear autoregression models. This implies that stationarity and mixing conditions on the underlying process are satisfied for each i. Furthermore, we apply the partially linear single index panel data model to analyze the dynamic demand of cigarettes based on a panel data set from 46 states in the US. The data set contains the consumption of cigarettes, the lagged consumption of cigarettes, the average retail price, disposable income and the minimum price of cigarettes in any neighboring state. Baltagi et al 2000 and Mammen et al 2009 respectively used a parametric linear model and a nonparametric additive model to analyze the relationship among the variables. From the study by Mammen et al 2009, we can see that there is some linear relationship between the consumption of cigarettes and its lagged consumption. This suggests that model. might be a better option for such data set see Section 5 for detail. The main contribution of this paper can be summarized as follows. We first propose using a semiparametric minimum average variance estimation SMAVE approach associated with a dummy variable method to estimate the parameters β 0 and θ 0 as well as the unknown link function η. Under certain regularity conditions, we are able to establish asymptotically normal distributions for the proposed parametric estimators and nonparametric estimator of the unknown link function when both n and T tend to infinity. Furthermore, we find that the dummy variable approach proposed for the fixed effects case helps us derive the same asymptotically normal distributions as in the case where random effects are involved. The rest of the paper is organized as follows. In Section 2, we introduce the so called SMAVE method to estimate β 0, θ 0 and η. Section 3 establishes an asymptotic theory for the proposed estimators. Section 4 gives some autoregression extensions of the proposed model. Section 5 illustrates the performance of the proposed models and estimation methods using both simulated and real data examples. Technical assumptions and proofs of the main results are provided in Appendices A C. An additional appendix as Appendix D is given in a supplemental document. 4

5 2. SMAVE procedure There are several estimation methods introduced for model. for the time series case of n = and α i 0 see, for example, Carroll et al 997; Liang et al 200; Wang et al 200 for the profile likelihood method; Yu & Ruppert 2002 for the penalized spline method; Xia and Härdle 2006 for the SMAVE method. To construct an estimation method for our panel data model., as one might expect, we face several challenges. The first one is that there are fixed effects involved in our model, which need to be removed in the estimation procedure. The conventional method is to use the difference approach by deducting each equation from the cross time average of the system Henderson et al However, it is difficult to directly apply the difference technique in our case as there is a single index structure. Instead, our estimation will be constructed with the help of a local linear dummy variable method, which is motivated by the least squares dummy variable procedure proposed for parametric panel data analysis Hsiao The second challenge is that there are two indices involved in our model: the time series dimension T and the cross section size n. We will consider the case where both n and T tend to infinity simultaneously and establish the asymptotic theory by using the joint limit approach introduced by Phillips & Moon 999. The detailed proof for such joint limiting distribution results is more complicated than that for asymptotic distribution theory in the time series case. We next introduce the so called SMAVE method to estimate both the parameters and the unknown link function by minimizing a single common loss function, and we will show that it can produce root nt consistent parametric estimators. The minimum average variance estimation MAVE method was first introduced by Xia et al 2002 for single index time series models, Recently, Xia 2006 established an asymptotic theory for the MAVE approach in single index models and Xia & Härdle 2006 extended the MAVE method and its asymptotic theory to partially linear single index models. In the above literature, the authors only considered such an estimation method for the time series case. As mentioned before, there are various challenges for us to develop a substantially extended method for the panel data case. We first introduce some notations for brevity of the presentation of our estimation 5

6 method. Let Y = Y,, Y T, Y 2,, Y nt, Z = Z,, Z T, Z 2,, Z nt, V = v,, v T, v 2,, v nt, ηx, θ = ηx θ,, ηx T θ, ηx 2θ,, ηx nt θ, D 0 = I n e T, α 0 = α,, α n, where I n is an n n identity matrix, e T is a T dimensional vector with all elements being, and denotes the Kronecker product. We can now rewrite model. as Y = Zβ 0 + ηx, θ 0 + D 0 α 0 + V. 2. Furthermore, by the identification assumption n α i = 0, we have α = n α i. Letting D = [ e n, I n ] e T written as i= i=2 and α = α 2,, α n, model 2. can then be Y = Zβ 0 + ηx, θ 0 + Dα + V. 2.2 For X it close to x R p, we have the following local linear approximation: Y it Z itβ 0 ηx itθ 0 α i Y it Z itβ 0 α i ηx θ 0 η x θ 0 X it x θ 0, where η u is the derivative of ηu at point u. The basic idea for MAVE method is to minimize n T Y Zβ Dα a it e nt b it X it θ W it Y Zβ Dα a it e nt b it X it θ 2.3 i= t= with respect to β, θ, a it and b it, where X it θ = X X it θ,, X T X it θ, X 2 X it θ,, X nt X it θ and W it = diagw,it,, w T,it, w 2,it,, w nt,it is a diagonal matrix with the elements satisfying n T w js,it = for all i, t. j= s= In view of 2.3, the detailed iteration procedure can be described as follows. 6

7 Step i: Taking derivative of Y Zβ Dα a it e nt b it X it θ W it Y Zβ Dα a it e nt b it X it θ 2.4 with respect to α, we get α it = D W it D D W it Y Zβ a it e nt b it X it θ. Then, letting α in 2.4 replaced by α it, we obtain the local linear estimator see Fan & Gijbels 996 for detail for given β, θ given by where a it, b it = X it, θw it X it, θ X it, θw it Y it, Z it, β, 2.5 X it, θ = e it,, X it, θ, e it, = e nt D D W it D D W it e nt, X it, θ = X it θ D D W it D D W it X it θ, Y it, = Y D D W it D D W it Y, Z it, = Z D D W it D D W it Z. Step ii: Given a it, b it for i n and t T, by 2.3 and 2.5 we construct a pair of parametric estimators of the form β, θ = Z WZ X WZ Z WX X WX Z where W = diag W,, W T, W 2,, W nt, Y = Y,,, Y T,, Y 2,,, Y nt,, Z = Z,,, Z T,, Z 2,,, Z nt,, X = b X,,, b T X T,, b 2 X 2,,, b nt X nt,, X it, = X it D D W it D D W it X it, X W Y A, 2.6 X it = X X it,, X T X it, X 2 X it,, X nt X it, A = a e,,, a T e T,, a 2 e 2,,, a nt e nt,. 7

8 Step iii: Repeat the above two steps until the successive values of β and θ differ insignificantly. As discussed in Xia et al 2002 for the time series case, we use two sets of weights in the above estimation procedure. The first set of weights is w js,it = H X js X it /h, 2.7 n T H X js X it /h j= s= where H is a p variate symmetric kernel function and h is a bandwidth. Following steps i iii, we obtain consistent initial estimators of β 0 and θ 0 defined by β and θ, respectively. However, the estimation based on the p variate kernel H is not efficient due to the curse of dimensionality. To circumvent this problem, we take the second set of single index weights wjs,it θ = K X js X it θ/h 2. n T K X js X it θ/h j= s= where K is a univariate symmetric kernel function and h 2 is a bandwidth. Letting β and θ be the initial values and following steps i iii, we then obtain the final estimators β and θ. 3. Asymptotic theory In this section, we establish the weak consistency of β and θ and then state the asymptotically normal distributions of β, θ and the nonparametric local linear estimator of the link function. Theorem 3.. Let Assumptions A A7 in Appendix A hold. Then, we have β β 0 = o P and θ θ 0 = o P. 3. The proof of Theorem 3. is given in Appendix B below. Theorem 3. establishes the weak consistency of β and θ. Note that it follows from the detailed proof of Theorem 3. and some related technical lemmas in Appendix D of the supplemental document that one may also strengthen the weak consistency to the strong consistency. The consistency of the initial value for the iteration procedure, based on the 8

9 second set of weights defined in 2.8, will help us to establish the final parameter estimators of β 0 and θ 0 with root nt rate of convergence. Before we establish an asymptotic distribution for β and θ, we introduce some notations. Let Z it,θ = Z it v θ X it and X it,θ = X it µ θ X it, where v θ x = E Z X θ = x θ and µ θ x = E X X θ = x θ. Define where Σ 0 = Σ 0 Σ 0 2 Σ 0 2 Σ 0 3 Σ 0 = E Z Z, and Σ = Σ 0 2 = E [ Z η X θ 0 X ], [ η Σ 0 3 = E X θ 0 ] 2 X X, Σ = Σ 2 = Σ 3 = t= t= t= E Z i Z itv i v it, E [ Z i η X itθ 0 X itv i v it ] Σ Σ 2 Σ 2 Σ 3 and E [ η X iθ 0 η X itθ 0 X i X itv i v it ]., 3.2 Letting β and θ be the initial values in the iteration procedure, we then establish the asymptotically normal distributions of β and θ in the following theorem. Theorem 3.2. Assume that Assumptions A A7 and B B4 in Appendix A are satisfied. Then we have β β nt 0 d N 0, Σ 0 Σ Σ 0, 3.3 θ θ 0 where 0 is a null vector with dimension d + p. In the above theorem, we have shown that the resulting estimators in the iterative procedure associated with the second set of weights can achieve the root nt rate of convergence. The asymptotic distribution in 3.3 can be regarded as a non trivial extension of some existing results for the time series case, such as Theorems 2 and 3 in Carroll et al 997, Theorem in Xia & Härdle 2006 and Theorem in Liang 9

10 et al 200. Furthermore, if we assume that the error process {v it } is independent of {Z it } and {X it }, and {v it, t } is independent and identically distributed i.i.d., the asymptotic variance in 3.3 is σ 2 Σ 0, where σ 2 = E[v 2 it]. This implies that the SMAVE method can achieve an semiparametrically efficient bound following the same argument as in Carroll et al 997. Under some mild conditions, we can find that the joint limit as both n and T tend to infinity is identical to the sequential limit as T first and then n or the sequential limit as n first and then T see, for example, Phillips & Moon 999. Additionally, we also find that, as T, the dummy variable approach proposed for the fixed effects case helps us derive the same asymptotically normal distributions as in the case where random effects are involved. To the best of our knowledge, this is a set of new findings for this type of nonlinear panel data models. Let us turn to the establishment of the asymptotic distribution of the nonparametric local linear estimator defined in Section 2. Let µ k = u k Kudu, ν k = u k K 2 udu, b η u = µ 2 2η uh 2 2 and σ 2 ηu = ν 0 σ 2 θ 0 ufθ 0 u, where σ 2 θ 0 u = Ev 2 it X itθ 0 = u and f θ0 is the density function X itθ 0. We establish an asymptotically normal distribution for the nonparametric local linear estimator ηu, which is defined by 2.5 with β and θ being replaced by β and θ, respectively. Theorem 3.3. Assume that the conditions of Theorem 3.2 are satisfied. As n, T simultaneously, nt h 2 ηx θ ηx θ 0 b η x θ 0 d N 0, σ 2 ηx θ From the above theorem, the forms of the bias term b η and the asymptotic variance σ 2 η are similar to those of the local linear estimator for panel data models with random effects see, for example, Theorem 3 in Cai & Li This implies that the dummy variable approach proposed for the fixed effects case is also applicable to derive the same asymptotically normal distributions as in the case where random effects are involved. 4. Dynamic partially linear single index panel data models This section introduces several dynamic models of. for the case where the regressors {Z it } and {X it } may contain lagged values of {Y it }. In this section, we consider three types of partially linear single index dynamic panel data models. 0

11 Case i Letting Z it = Y i,t,, Y i,t d, model. then becomes d Y it = Y i,t j β 0,j + ηx itθ + α i + v it, i n, t T. 4. j= To deduce the geometrical ergodicity of {Y it }, w assume that both {X it, t } and {v it, t } are i.i.d. for each i, and X it, v it is independent of Y i,t j, j. It is well known that a sufficient condition for the stationarity of {Y it, t } defined by 4. is that β 0,,, β 0,d satisfy y d β 0, y d β 0,d y β 0,d 0, for any y. 4.2 Under condition 4.2, {Y it, t } is geometrically ergodic for each i. Case ii Consider the case where {X it } contains lagged values of {Y it }, X it = Y i,t,, Y i,t p. Model. then reduces to p Y it = Z itβ 0 + η Y i,t j θ 0,j + α i + v it, i n, t T. 4.3 j= For model 4.3, we suppose that both {Z it, t } and {v it, t } are i.i.d. for each i, and Z it, v it is independent of Y i,t j, j. Furthermore, assume that for any u R, ηu λ u / p + c, 4.4 where 0 < λ < and 0 < c <. Following the same argument as in Example 3.5 of An and Huang 996, we can show that {Y it, t } is geometrically ergodic for each i. Case iii Consider the case that both {Z it } and {X it } contain lagged values of {Y it }. In this case, we have d p Y it = Y i,t j β 0,j + η Y i,t j θ 0,j + α i + v it, i n, t T. 4.5 j= j= Xia et al 999 considered the time series case of 4.5 with α i = 0 and gave some conditions on β and θ to make sure that the model is identifiable. We now deduce the geometrical ergodicity of {Y it, t } in the panel data model 4.5 with α i 0 generally. Let η i u = ηu + α i. Then 4.5 can be written as d p Y it = Y i,t j β 0,j + η i Y i,t j θ 0,j + v it. 4.6 j= j=

12 η i u u u ηu u u Suppose that β 0,j for j d satisfy 4.2, lim = lim = 0, max α i < i, and the probability density function of {v it } is positive everywhere. Then following the proof of Theroem 3 in Xia et al 999, we can show that {Y it, t } is geometrically ergodic for each i. 5. Numerical Examples To examine the finite sample performance of the proposed estimation method, we carry out a Monte Carlo simulation experiment and then use the method to analyze a set of US cigarette demand data. Throughout this section, we use a Gaussian kernel in the local linear smoothing. For the multivariate smoothing at the first stage to obtain initial estimates of β 0 and θ 0, we use a product Gaussian kernel and equal bandwidths for each variate. As the the choice of bandwidth at the first stage has little effect on the performance of the estimators at the second stage, we choose the initial bandwidth at the first stage as h = σ X nt /4+p, where σ X is the sample standard deviation of X it for i n and t T. At the second stage, we use a leave one unit out cross validation method as proposed in Sun et al The leave one unit out cross validation method is an extension of the conventional leave one out cross validation method. The idea is to remove {Z it, X it, Y it, t T } from the data and use the rest of the n T observations as the training data to obtain estimates of β 0, θ 0 and η, which are denoted as β i, θ i and η i. We thus choose an optimal bandwidth that minimizes a weighted squared prediction error of the form Y BZ, β ηx, θ M M Y BZ, β ηx, θ, 5. where M = I n T T I n e T e T, and BZ, β = Z β,, Z β T, Z 2 β 2,, Z β 2T 2,, Z n β n, Z β nt n ηx, θ = η X θ,, η X T θ, η 2 X 2 θ 2,, η n X nt θ n. 2

13 The weight matrix M is constructed to satisfy MD = 0 so that the fixed effects α i are eliminated from 5.. As a consequence, M removes a cross-time average from each variable. For example, MY = Y Y A,, Y T Y A, Y 2 Y 2A,, Y nt Y na, where Y ia = T T Y it for i =,, n. t= 5.. Simulated Example We use the following data generating process Y it = Z itβ 0 + ηx itθ 0 + α i + v it, i n, t T, 5.2 Z it = 0.8Z i,t + z it, X it = 0.5X i,t + x it, v it = 0.4v i,t + e it where β 0 = 2,, θ 0 = 2,,, 2 {zit } is a two dimensional i.i.d. random vector of independent uniform U, components, {x it } is a three dimensional i.i.d. vector of independent normal N0, components, {e it } is an i.i.d. normal N0, scalar variable, α i = 0.5ZiA + u i for i =,, n, and α n = n α i, in which ZiA = i= T Z 2T it, + Z it,2 and {u i } is an i.i.d. N0, random error. The link function t= η is chosen as ηu = 2. u 3 Finally, {zit }, {x it }, {u i } and {e it } are mutually independent. To measure the performance of the proposed estimators, we define the mean absolute errors of β and θ as MAE β = R R r= d respectively, where β r = d β r l l= β 0,l β r,, and MAE θ = R r β d and θ r = R r= p θ r,, p θ r l l= θ 0,l, r θ p are the respective estimates of β 0 = β 0,,, β 0,d and θ0 = θ 0,l,, θ 0,p from r-th replication, and R is the number of replications with R = 200 being chosen here. The simulation results for model 5.2 are summarized in Table 5.. Table 5. 3

14 MAE s and SD s in parentheses of the estimated parameters N\T β θ β θ β θ Table 5. shows that the proposed estimation method works quite well numerically A Real Data Example The real data example is about the cigarette demand in 46 states of the USA over the period The data set is from Baltagi et al 2000, who used a linear dynamic panel data model of the form ln Y it = α + β 0 ln Y i,t + β ln X it, + β 2 ln X it,2 + β 3 ln X it,3 + u it 5.3 to analyze the demand for cigarettes, where i =,, 46, denotes the i th state, t =,, 29 denotes the t th year, {Y it } is the real per capita sale of cigarettes, {X it, } is the average retail price per pack of cigarettes, {X it,2 } is the real per capita disposable income, {X it,3 } is the minimum real price of cigarettes in any neighboring state, and the disturbance term u it in 5.3 is specified as u it = µ i + λ t + v it, 5.4 where µ i denoted a state-specific effect, and λ t denoted a year-specific effect, which could also be interpreted as a trend in t. Due to the presence of the time specific effect or trend λ t in ln Y it and the explanatory variables ln X it,j, j =, 2, 3, we first remove the trend from the log-transformed 4

15 !4!3!2! !2!2!3!2! !2!4!3!2! N=0, T=0 N=0, T=20 N=0, T= !2!4!3!2! !2!3!2! !2!4!3!2! N=20, T=0 N=20, T=20 N=20, T= !2!4!3!2! $! $# $% ' &! # %!#!!!"!#!$ % $ # "! $ $' $! $# $% '! # %!#!!!"!#!$ % $ # "! & N=30, T=0 N=30, T=20 N=30, T=30 Figure 5.. Simulation result from typical data sets in Example 5.. The solid line is the true function of ηx it θ 0, the dashed line is the estimated function of ηx it θ, the dots denote Y it Z it β α i plotted against X it θ. observations as used in Mammen et al 2009, Y it = ln Y it s Y t, X it,l = ln X it,l s X,l t, l =, 2, 3, where s Y and s X,l are the nonparametric estimates of the trends in ln Y it and ln X it,l, i =,, 46, t =,, 29. Define Z it = Y i,t. We plot the de-trended observations Y it against the de-trended explanatory variables Z it, X it,, X it,2, X it,3 in Figure 5.2. It is clear from Figure 5.2 that Y it exhibits strong linearity with Z it i.e. the lagged variable Y i,t. For the other three covariates, their linearities are not as strong as that for the lagged-variable. Hence, we choose Z it as the covariate in the parametric part of our semiparametric single index model Y it = Z itβ + gx it θ + α i + v it, where X it = X it,, X it,2, X it,3, αi is a state specific effect which corresponds to µ i 5

16 '!!'!!"#!!"$!!"%!!"&!!"&!"%!"$!"# ' '!!'!!"%!!"*!!"&!!"'!!"'!"&!"*!"%!"+,' '!!'!!"#!!"$!!"%!!"&!!"&!"%!"$,& '!!'!!"*!!"&!!"'!!"'!"&!"*!"%,* Figure 5.2. From top to bottom: the detrended observations of Y against that of Z, X, X 2 and X 3. in as in Baltagi et al Note that {α i } may include effects resulting from religion, race, tourism, tax, and education. After implementing the proposed algorithm in Section 2 to the de trended observations Yit, Zit, X it, we obtain the estimates for the coefficients: β = and θ = , , The estimated link function η is given in Figure 5.3. Comparison of these results with that in Baltagi et al 2000 indicates that our estimate of β is smaller than that in Baltagi et al 2000, who got an estimate of 0.90 in the OLS method and 0.9 in the GLS method. In addition, compared to θ = , 0.22, in the OLS and θ = , 0.602, in the GLS, our estimate of θ is smaller, while that of θ 2 and θ 3 are larger Note that as we have the identification condition θ =, one has to normalize the estimates of θ in Baltagi et al 2000 before making comparison.. The plot of the estimate of the link function shows a possible linear relation between X it θ and Y it Z it β α i. 6

17 !"'!"&!"#!"%!!!"%!!"#!!"&!!"#$!!"#!!"%$!!"%!!"!$!!"!$!"%!"%$!"#!"#$ Figure 5.3. Estimation results for the real data example. Dots denote Y it Z it β α i plotted against X it θ. The solid line denotes the estimated link function ηx it θ. 6. Conclusion This paper has proposed a semiparametric partially linear single index panel data model with fixed effects. A semiparametric minimum average variance estimation method associated with dummy variable approach has been proposed to deal with the estimation of both the parametric and nonparametric components of the model. We have established new asymptotic distributions for the proposed estimators and shown that the proposed estimators all have asymptotically normal distributions regardless of whether the effects involved are random or fixed. We have then assessed the finite sample performance of the proposed estimation method through using both simulated and real data examples. The real data example has shown that both the model and the estimation method proposed in this paper are more suitable than existing ones for the particular data set. The paper certainly has some limitations. One question is whether the established theory may be extended to the case where both {X it } amd {Z it } are nonstationary in t and cross sectional dependent in i. How to answer such a question should be left in future research. 7. Acknowledgments The authors would like to acknowledge the financial support from the Australian Research Council Discovery Grants Program under Grant Number: DP The 7

18 authors also acknowledge the useful comments by the seminar participants at the University of Adelaide and Monash University. Appendix A: Assumptions Let Z i = Z it : t, X i = X it : t and V i = v it : t. consistency of β and θ, we need the following set of regularity conditions. To derive the A Z i, X i, V i, i =,, n, are i.i.d. and {Z it, X it, v it, t } is a stationary α mixing sequence with mixing coefficient α i t for each i. Furthermore, there exists a positive coefficient function αt such that sup α i t αt with αt C α t γ 0, i where C α > 0 and γ 0 > 2+δ 2+δ 2δ δ, in which δ > δ > 0 are involved in A3 and A4 below. A2 The kernel function H : R d R + is a bounded and Lipschitz continuous probability density function with a compact support. Furthermore, x Hxdx = 0 and xx Hxdx is positive definite. A3 The density function f X of X it is second order continuous and has gradient f X }. Let f X be positive and bounded in X := {x : x MnT 2+δ for any M > 0 and E X it 2+δ <, where is the L 2 distance. [ ] A4 Let g x := E [Z it X it = x] and g 2 x = E Z it Z it X it = x. Suppose that g x and g 2 x have bounded and continuous derivatives. Let E Z it 2+δ < and E {Z it EZ it X it Z it EZ it X it } be a positive definite matrix, where δ > 0 is some small constant. A5 Suppose that {v it } is independent of {Z it, X it } with E[v it ] = 0, 0 < σ 2 := E[v 2 it ] < and E[ v it 2+δ ] < for some δ > 0. A6 The link function η has continuous derivatives of up to the second order. A7 The bandwidth h involved in the first set of the weights satisfies h 0, T h p+2 log T, nt 2γ0 4p+ 2+δ 3 h 2pγ 0 +4p2 +9p+2 log 2γ 0 4p+ nt. 8

19 To establish asymptotic distributions for the final parametric estimators β and θ with β and θ being the initial values in the iteration procedure, we further need to introduce the following set of regularity conditions. B The kernel function K : R R + is a bounded and symmetric probability density function. Furthermore, K is Lipschitz continuous and has a compact support. B2 The density function f θ of X itθ is positive and second order continuous for θ. } Let f θ0 be positive and bounded in U := {u = x θ 0 : x MNT 2+δ for any M > 0 and E X it 2+δ <, where δ is defined as in A. B3 The conditional expectation g 3 u := E[Z it X itθ = u] has a bounded and continuous derivative for θ in a neighborhood of θ 0, and E Z it 2+δ <. B4 The bandwidth h 2 involved in the second set of the weights satisfies lim h 2 = 0, n,t lim nt h 2 = and 0 < lim nt n,t n,t h5 2 <. Furthermore, T = o and γ 0 > 5p3+δ 22+δ + 9+6p 4. n 5δ 2+δ 2 δ2+δ 2 In A, we assume that Z i, X i, V i, i n, are cross sectional independent see Su & Ullah 2006, Sun et al 2009 for example and each component time series is α mixing dependent, which can be satisfied by many linear and nonlinear time series see, for example, the discussion in Section 4. Assumption A2 is a set of some mild conditions on the multi variate kernel function H. A3 and A4 are similar to the corresponding conditions in Xia & Härdle Since {α i } is allowed to correlate with X it, Z it, u it = α i + v it is thus correlated with X it, Z it even though {v it } is assumed to be independent of X it, Z it. Meanwhile, the independence between {Z it, X it } and {v it } in A5 is imposed to simplify our proofs and it can be removed if we involve more tedious proofs. A6 is a usual condition for local linear estimators see, for example, Fan & Gijbels 996; Fan & Yao We next show that the bandwidth conditions in A7 are satisfied under mild conditions if we take h NT θ, 0 < θ < /p + 2. It is easy to check that h NT θ = o and the second bandwidth condition in A7 is also satisfied when n = o T θ p+2 / log θ p+2 T. Letting p = 2γ 0 4p3+δ 2+δ 3, p 2 = 2pγ 0 + 4p 2 + 9p + 2 and p 3 = 2γ 0 4p +, the last part in A7 becomes nt p h p 2 log p 3 nt = nt p θ p2 log p 3 nt, which implies that p > p 2 θ. By some elementary calculation, we can show that when γ 0 > 4p2 + 9p + 2θ 4p 22 + δ + 8p + 2 2pθ 2 + δ2 2pθ, 9

20 p > p 2 θ and thus the third bandwidth condition in A7 is satisfied. Assumptions B B3 are similar to such conditions in C2, C4 and C5 in Xia & Härdle 2006 for the time series case. The rate of the bandwidth h 2 in B4 is optimal for pooled local linear estimators. In particular, if we take δ = δ/2 and δ = 2, Then, the condition T = o n 5δ 2 + δ 2 δ2 + δ 2 = δ 2+δ 2 δ2+δ 2 in B4 becomes T = on 3/9. Appendix B: Proof of Theorem 3. Define a x = ηx θ 0, a it = ηx it θ 0, b x = η x θ 0 and b it = η X it θ 0. Let ã x, ã it, b x, and b it be the local linear estimators obtained from 2.5 using the first set of weights defined in 2.7. Let e x,, X x,, X x,, W x and Z x, be defined as e it,, X it,, X it,, W it and Z it, with X it being replaced by x. Furthermore, define For simplicity, define τ T = and ζ θ = θ θ 0. D x, = V D D W x D D W x D, V x, = V D D W x D D W x V, V it, = V D D W it D D W it V. log T T h p, τ nt = log nt nt h p, τ nt 2 = log nt nt h 2, ζ β = β β 0 To prove the weak consistency of β and θ established in Theorem 3., we need asymptotic uniform expansions of ã x and b x in {x : x M nt } for M nt = M 0 nt /2+δ and 0 < M 0 <. Lemma B.. Let Assumptions A A7 in Appendix A hold. Then, ã x = a x + g xβ 0 β + O P h 2 + τ T, B. and b x = θ θ 0 b x + O P ζ β + h + h τ T B.2 uniformly in {x : x M nt }, where g x is defined in A4. 20

21 Proof. By the definition of ã x and b x, we have ã x, b x = X x, θw x X x, θ X x, θw x Z x, β 0 β + X x, θw x X x, θ X x, θw x D x, α + X x, θw x X x, θ X x, θw x η x, X, θ 0 + X x, θw x X x, θ X x, θw x V x, = X x, θw x X x, θ X x, θw x Z x, β 0 β + X x, θw x X x, θ X x, θw x η x, X, θ 0 + X x, θw x X x, θ X x, θw x V x,, where η x, X, θ 0 = ηx, θ 0 D D W x D D W x ηx, θ 0. Then, by Taylor expansion for ηx it θ 0 we have 2 3 ηx itθ 0 = ηx θ 0 + η x θ 0 d itxθ 0 + η x θ 0 d itxθ 0 + O d itxθ 0, B.3 where d it x = X it x. have Following the proof of Lemma D.2 in Appendix D of the supplemental document, we X x, θw x X x, θ X x, θw x η x, X, θ 0 = a x + O P h 2, θ θ 0 b x + O P h. By Lemmas D.4 and D.5 in Appendix D of the supplemental document, we also have B.4, 0 X x, θw x X x, θ X x, θw x Z x, = g x + O P h 2 + τ T B.5 and X x, θw x X x, θ X x, θw x V x, = O P τ nt = o P τ T B.6 uniformly in x M nt. By B.3 B.6, we have proved that B. holds. uniformly in x M nt, Again by Lemma D.4, we have, 0, X x, θw x X x, θ X x, θw x Z x, = O P h + h τ T. B.7 By B.3, B.4, B.6 and B.7, we have shown that B.2 holds. We next give the proof of Theorem 3. with the help of Lemma B.. 2

22 Proof of Theorem 3.. Note that for any small ε > 0, P max max X it > MnT /2+δ i n t T n T E X it 2+δ /M 2+δ nt = M 2+δ < ε i= t= n T i= t= P X it > MnT /2+δ if M > ε 2+δ. Hence, we need only to consider the case of max max X it MnT i n t T By 2.6 and B., we have β β 0 = 2+δ. [ ] [ ] E Z it Z it E g X it g X it β β 0 + o P. B.8 Since we use the multivariate kernel H in the first set of weights, B.8 does not involve θ. In the iterative process, we have from B.8, β k+ β 0 = [ ] [ ] E Z it Z it E g X it g X it β k β 0 + o P. B.9 [ ] [ ] By A4 in Appendix A, we can show that the matrix E Z it Z it E g X it g X it is positive definite. Similarly to the proofs of Lemma and Theorem in Xia & Härdle [ ] [ ] 2006, the eigenvalues of the matrix E Z it Z it E g X it g X it are all less than. Hence, after a sufficiently large number of iterations, β k β = o P, which implies that the first result in 3. holds. By 2.6 and B.2, we have θ θ 0 = θ θ 0 θ θ 0 θ 0 + O ζ β + o P, B.0 which implies that θ = θ θ 0 θ0 + O ζ β + o P. B. Following the proof of Lemma in Xia & Härdle 2006, we have shown that the second result in 3. holds. Appendix C: Proofs of Theorems 3.2 and 3.3 Let W it θ be defined as W it with the weights in 2.6 being replaced by those in 2.7. For simplicity, e it,, X it,, X it,, V it and Z it, are defined in the same way as above, but W it involved is being replaced by W it θ. Throughout this appendix, â x, â it, b x, and b it are the local linear estimators obtained from 2.5 using the second set of weights defined in

23 As in Appendix B, e x,, X x,, X x,, W x, V x, and Z x, are defined similarly to e it,, X it,, X it,, W it, V it, and Z it, with X it being replaced by x. Furthermore, define d x θ = d xθ,, d T xθ, d 2xθ,, d nt xθ, d x, θ = d x θ D D WθD D Wθd x θ, where d it x is defined as in the proof of Lemma B.. To establish the asymptotic distributions of the proposed estimators, we need the following asymptotic uniform expansions of â x and b x in {x : x M nt }. Lemma C.. Let Assumptions A A7 and B B4 in Appendix A hold. Then, uniformly in {x : x M nt }, â x = a x + b x U x ζ θ + U x 2ζ β + R x + h 2 2η x θ 0 U x 3 + O P h 3 2 C. and b x = b x + R x 2 + O P h ζ β + ζ θ, C.2 where Ux =, 0 X x, θw x θx x, θ X x, θw x θx x,, Ux 2 =, 0 X x, θw x θx x, θ X x, θw x θz x,, U x 3 =, 0 X x, θw x θx x, θ X x, θw x θd x, θ 0, R x, R x 2 = X x, θw x θx x, θ X x, θw x θv x,. Proof. By the definition of â x and b x, we have â x, b x = X x, θw x θx x, θ X x, θw x θz x, β 0 β + X x, θw x θx x, θ X x, θw x θd x, α + X x, θw x θx x, θ X x, θw x θη x, X, θ 0 + X x, θw x θx x, θ X x, θw x θv x, = X x, θw x θx x, θ X x, θw x θz x, β 0 β + X x, θw x θx x, θ X x, θw x θη x, X, θ 0 + X x, θw x θx x, θ X x, θw x θv x,, C.3 23

24 where η x, X, θ 0 is defined in the same way as in Appendix B with W it being replaced by W it θ. By C.3, Lemma D.3 and the same Taylor expansion for ηx it θ 0 as in the proof of Lemma B., we complete the proofs of C. and C.2. Before we prove Theorem 3.2, we introduce the following notations. Let U j = U, j,, U T, j, U 2, j,, U nt, j, U it, j = e nt Uit j DD W it θd D W it θe nt Uit j, j =, 2, Ṽ = Ṽ,,, Ṽ T,, Ṽ 2,,, Ṽ nt,, where U it j is defined in the same way as U x j with x replaced by X it and Ṽit, is defined as V it, with V replaced by V R it e nt, and R it is defined as R x with x being replaced by X it. Proof of Theorem 3.2. Since the main idea of the proof is similar to the proof of Theorem in Xia & Härdle 2006, we only provide an outline here. By Lemma C. and following the proof of Lemma 6.3 in Xia & Härdle 2006, we have where ζ r = β β 0 θ θ 0 = JnT M nt + JnT U nt β β 0 θ θ 0 + O P ζ r 2 + h 2 + h 2 τ nt 2 ζ r + h 3 2, C.4, ζ β, ζ θ ζβ and ζ θ are defined as in Appendix B, J nt = U nt = diag nt Z WθZ nt Z WθX, nt X WθZ nt X WθX nt Z WθU 2,, M nt = nt nt X WθU. Z X WθṼ, have Following the proof of Lemma D.4 in Appendix D of the supplemental document, we J nt P J J 2 J 2 J 22 =: J C.5 and U nt [ ] P diag E v θ0 X vθ 0 X, 2 J 22 =: U, C.6 24

25 where J = E Z Z [ J 22 = 2E η X µ θ x = E, J 2 = E [Z η X θ 0 ] µ θ0 X X, 2 X θ 0 µ θ0 X X µ θ0 X ],, v θ x = E Z X θ = x θ. X X θ = x θ Following the proof of Theorem in Xia & Härdle 2006, we can show that Ñ := J /2 U J /2 is a semi positive definite matrix with rank d+p and all eigenvalues being less than. Let > λ λ 2 λ d+p > 0 be the eigenvalues of Ñ. Let J nt k and U nt k be versions of J nt and U nt at k th iteration. Furthermore, by C.5 and C.6, the eigenvalues of Ñk := JnT /2 k UnT k JnT /2 k satisfy > λ k λ 2 k λ d+p k > 0, λ k = λ + o P for all k. Define r k = J /2 nt k β k β 0, θ k θ 0. By C.4, we have r k+ = JnT /2 k MnT k+ñkr k+o P ζ rk + h 2 + h 2 τ nt 2 ζ rk + h 3 2, C.7 which, together with the proof of Lemma D.5, implies ζ rk+ τ nt 2 + λ k ζ rk + c 0 ζ rk 2 + c 0 h 2 + h 2 τ nt 2 ζ rk + c 0 h 3 2, C.8 where c 0 > is a constant. By Theorem 3., we have ζ r λ /3c 0. C.9 By the definitions of τ nt 2 and B4, we have τ nt 2 + c 0 h 3 2 λ 2 /9c 0, c 0 h 2 + h 2 τ nt 2 λ /3. By C.8 C.0 and the fact that λ k λ, we have ζ rk λ /3c 0. C.0 C. for all k. Then, following the proof of Theorem in Xia & Härdle 2006, we have for sufficiently large k, ζ rk+ = OP τ nt 2 + h 3 2. C.2 Note that nt h 6 2 0, nt τ 2 nt 2 + h 6 2 0, nt h 2 + h 2 τ nt 2 h τ nt 2 0. C.3 25

26 By C.5 and C.6, we have J nt U nt P Σ 0. C.4 By standard argument, the leading term of M nt is MnT = n T Z it nt i= t= η X it θ 0 X v it it as nt. Applying the central limit theorem for α mixing processes see, for example, Theorem 2.2 of Fan and Yao 2003, we have nt M nt d N 0, Σ. C.5 By C.4 and C.3 C.5, we have shown that Theorem 3.2 holds. Proof of Theorem 3.3. By the definition of local linear estimators, it is easy to check that ηx θ ηx θ 0 =, 0 X x, θw x θx x, θ X x, θw x θ Y x, Z x, β ηx θ 0 = Sx θη x, X, θ 0 ηx θ 0 + Sx θv x, + Sx θz x, β 0 β =: Π nt + Π nt 2 + Π nt 3, where Sx θ =, 0 X x, θw x θx x, θ X x, θw x θ. By Theorem 3.2, we then have Let us now consider Π nt and Π nt 2. Note that Π nt = = Π nt 3 = O P nt /2. C.6 Sx θη x, X, θ 0 ηx θ 0 Sx θ 0 η x, X, θ 0 ηx θ 0 =: Π nt, + Π nt, 2. + Sx θ Sx θ 0 η x, X, θ 0 Noticing that θ θ 0 = O P nt /2 by Theorem 3.2, we have Π nt, 2 = O P nt /2. Meanwhile, by the property of local linear smoothing, we have Sx θ 0 η x, X, θ 0 ηx θ 0 = 2 h2 2η x θ 0 u 2 Kudu + o P h 2 2 = b η x θ 0 + o P h

27 Hence, we have Π nt = b η x θ 0 + o P h 2 2. C.7 We next turn to the proof of the central limit theorem for Π nt 2. By B, we have Xit x K θ Xit x θ 0 = K h 2 h 2 + K Xit x θ Xit x θ θ 0, h 2 h 2 where K is the first order derivative of K and θ = θ 0 +λ θ θ 0 for some 0 < λ <. Hence, n T Xit x K θ n T Xit x θ v it = 0 K v it nt h2 h i= t= 2 nt h2 h i= t= 2 n T + K Xit x θ Xit x θ θ 0 v it nt h2 h i= t= 2 h 2 =: Π nt 2, + Π nt 2, 2. By Theorem 3.2 and following the same argument as in the proof of Lemma D.5 of the supplemental document, we have Π nt 2, 2 = o P, C.8 n T which implies that the leading term of X nt h2 K it x θ h 2 v it is Π nt 2,. i= t= In a similar way to the proof of Theorem 2.2 of Fan and Yao 2003, applying Doob s large block and small block argument in the proof of asymptotic distribution for nonparametric kernel estimator under α mixing dependence, we can show that Π nt 2, d N 0, σ 2, C.9 where σ 2 = σ 2 θ 0 x θ 0 f θ0 x θ 0 ν 0. By C.8, C.9 and the uniform convergence results in Appendix D of the supplemental document, we have By C.6, C.7 and C.20, Theorem 3.3 holds. Π nt 2 d N 0, σ 2 ηx θ 0, C.20 References An, H. Z. & Huang, F. C The geometrical ergodicity of nonlinear autoregressive models. Statistica Sinica 6, Arellano, M Panel Data Econometrics. Oxford University Press: Oxford. 27

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