Partially Linear Functional-Coeffi cient Dynamic Panel Data Models: Sieve Estimation and Specification Testing

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1 Partially Linear Functional-Coeffi cient Dynamic Panel Data Models: Sieve Estimation and Specification Testing Yonghui Zhang Qiankun Zhou Job Market Paper This version: November, 26 Abstract In this paper, we study the nonparametric estimation and testing for the partially linear functional-coeffi cient dynamic panel data models where the effects of some covariates on the dependent variable vary according to a set of low-dimensional variables nanparametrically. Based on the sieve approximation of unknown functions, we propose a sieve 2SLS procedure to estimate the model. The asymptotic properties for both parametric and nonparametric components are established when sample size N and T tend to infinity jointly or only N goes to infinity. We also propose a specification test for the constancy of slopes, and we show that after being appropriately standardized, our test is asymptotically normally distributed under the null hypothesis. Monte Carlo simulations show that our sieve 2SLS estimators and test perform remarkably well in finite samples. We apply our method to study the effect of income on democracy and find strong evidence of nonconstant effect of income on democracy. Key words: Dynamic panel models, Sieve approximation, Functional-coeffi cient, 2SLS estimation, Specification testing JEL Classification: C2, C23, C26, C33, C38. Address correspondence to: Qiankun Zhou, Department of Economics, State University of New York at Binghamton, Binghamton, NY 392, USA. qzhou@binghamton.edu. Zhang gratefully acknowledges the financial support from the National Science Foundation of China under Grant All errors are the authors sole responsibilities. School of Economics, Renmin University of China, Beijing, China. Department of Economics, State University of New York at Binghamton, Binghamton, NY, 392, USA.

2 Introduction Since the seminal work of Balestra and Nerlove 966), there is rich literature on the research of dynamic panel data models among both theoretical and empirical economists. Based on the influential work of Anderson and Hsiao 98, 982), using the two stage least squares 2SLS) or generalized method of moments GMM) to estimate the dynamic panel data model has received lots of attention in the literature. To name a few, see Arellano and Bond 99) and Alvarez and Arellano 23), among others. However, it should be pointed out that linear parametric form is generally assumed in the aforementioned researches of dynamic panel models, and it is well-known that parametric dynamic panel data models might not be flexible enough to capture nonlinear structure in practice, such a failure may result in model misspecification issue. To deal with this issue, various nonparametric or semiparametric dynamic panel data models have been proposed. For example, in earlier work, Li and Ullah 998) and Baltagi and Li 22) consider semiparametric estimation of partially linear dynamic panel data models using instrumental variable methods. More recently, in order to allow coeffi cients to depend on some informative variables, research of varying-coeffi cient models has received lots of attention. For the varying coeffi cient models, it has wide application in the economics literature. As for the first example, in the traditional labor economics literature of return to schooling, researchers usually apply linear IV regression model. However, Card 2) finds that the returns to education tend to be underestimated by using the 2SLS method when one ignores the nonlinearity and the interaction between schooling and working experience, and Schultz 23) argues that the marginal returns to education may vary with different levels of working experience and schooling. This motivates Cai et al. 2) and Su et al. 24) to consider the partially linear functional coeffi cient model. Apparently, both models allow the impact of education on the log-wage to vary with working experience. Another application of varying coeffi cient model is the heterogenous effects of FDI on economic growth. Based on the finding of Kottaridi and Stengos 2), Cai et al. 2) find that the effect of FDI on economic growth varies across initial income levels, and thus varying coeffi cient model is adapted for such purpose. For other applications of varyingcoeffi cient models in economics and finance, refer to Baglan 2), Cai 2), Cai et al 2, 2) and Cai and Hong 29), among others. In this paper, we consider a new class of partially linear varying-coeffi cient additive dynamic models, which allows for linearity in some regressors and nonlinearity in other regressors. In 2

3 other words, some coeffi cients are constant and others are varying over some variables. This new class model is flexible enough to include many existing models as special cases. By extending the model in Cai and Li 28) to a partially varying-coeffi cient model with fixed effects, we reduce the model dimension without influencing the degree of the model flexibility, and the consistent estimation of parametric coeffi cients can be achieved. We also extend the work of Cai et al. 25) to sieve estimation instead of kernel estimation. The choice of sieve estimation over kernel estimation is simply because series estimation methods are more convenient than kernel methods under certain type of restrictions such as additivity or shapepreserving estimation, see Dechevsky and Penez 997)). It is also computationally convenient because the results can be summarized by a relatively small number of coeffi cients. Based on the sieve approximation of unknown varying-coeffi cient functions, we use the standard approach of taking the first difference to eliminate the fixed effects and use the lagged variables as instruments. This results in a sieve two stage least squares 2SLS) estimation for partially linear functional-coeffi cient dynamic panel models. The asymptotic properties for both parametric and nonparametric components are established when sample size N and T tend to infinity jointly or only N goes to infinity. We also discuss the plausibility of extending the proposed sieve 2SLS estimation procedure to unbalanced dynamic panels. We also propose a nonparametric test for the linearity of the nonparametric component, i.e., slopes of the nonparametric part is constant. This specification test for the constancy of slopes is based on a weighted empirical L 2 -norm distance between the two estimates under the null and the alternative, respectively. We show that after being appropriately standardized, our test is asymptotically normally distributed under the null hypothesis. Compared with the existing literature of estimation of varying-coeffi cient additive dynamic models, our paper has the following merits. On the first hand, in the existing literature, it is common to use within-group transformation to eliminate the fixed effects, however, such a transformation for dynamic panels will in general lead to biased estimation and bias correction is needed e.g., Cai and Li 28), Tran 24), Rodriguez-Poo and Soberon 25) and reference therein). However, our paper considers the first difference transformation to remove the fixed effects, and we use the lagged variables as instruments and propose the 2SLS estimation of the constant and varying coeffi cients. It is shown in this paper that the 2SLS estimators are free of asymptotical bias. On the other, instead of assuming the time series dimension is short for dynamic panels e.g., An et al 26) and Cai et al 25)), we establish the asymptotics of the 3

4 2SLS estimation when both N and T are large, thus our asymptotic results cover the foregoing results as special cases. We also discuss the applicability of the 2SLS estimation when the panel is unbalanced, and it is shown in the simulation that the proposed sieve 2SLS estimation works remarkably well even if the panel is unbalanced. The small sample properties of the sieve 2SLS estimation and specification testing for partial linear varying-coeffi cient additive dynamic models are investigated through Monte Carlo simulation, using six different data generating processes DGPs). The first four DGPs are designed to check the performance of the sieve 2SLS estimation for the balanced panels, when T is large or fixed, and the fifth DGP is to verify the applicability of the sieve 2SLS estimation for unbalanced panels. From the simulation results, we can observe that the proposed sieve 2SLS works remarkably well for the estimation of both parametric and nonparametric part in the model. Namely, for the estimation of parametric part of the model, the constant coeffi cient can always be consistently estimated, and the shape of estimated functional-coeffi cients is close enough to the true pre-specified functions. Similar findings can be applied to the case when the panel is unbalanced. The last DGP is to investigate the performance of specification test. From the simulation results, we can notice that the empirical size of the specification test is very close to the nominal value and the empirical power increases steadily with the increase of either N or T. Finally, We apply the proposed method to study the relationship of income and democracy as in Acemoglu et al 28) and Cervellati et al 24). Through the sieve 2SLS estimation, we find substantial nonlinearity in the relationship between a country s degree of democracy and its lagged value and a nonlinear relationship between income and democracy. The rest of the paper is organized as follows. We introduce the model and sieve 2SLS estimation in Section 2. Asymptotics for the sieve 2SLS estimation is established in Section 3. We propose the specification test in Section 4, and in Section 5 we conduct a small set of Monte Carlo simulations to evaluate the finite sample performance of the sieve 2SLS estimation and specification testing. We apply our method to study to study the relationship of income per capita and democracy in Section 6. Conclusion are made in Section 7. All technical details are relegated to the Appendix. Notations: For a real matrix A, let A = [tr A A)] /2 denotes its Frobenius norm and A 2 = [λ max A A)] /2 denotes its spectral norm where λ max ) is the largest eigenvalue of. Define P A A A A) A, and M A = I cola) P A where col A) denotes the column number 4

5 of A. Throughout, C denotes a generic non-zero positive constant that does not depend on N or T and may vary case by case. The symbols p and d denote convergence in probability and in distribution, respectively. 2 Model and Sieve Estimation In this section, we first introduce the model and then propose a nonparametric estimation procedure based on sieve approximation. 2. The models We consider the following partially linear functional-coeffi cient dynamic panel data models with fixed effects y it = d itθ z it ) + x itβ + η i + u it, i =,..., N, t =,..., T, 2.) where d it and x it are p d and p x vectors of covariates, respectively, and z it is a p z 3 p z 3) vector of covariates which enter the unknown functions θ ) = θ ),..., θ pd )) nonparametrically, η i represents the unobserved heterogeneity of the i-th individual, and u it is the idiosyncratic error. For identification, we follow Chen and Liu 2) to assume that there is no common variables between d it and x it. As classical fixed effects, η i can be correlated with d it, x it, z it ). In this paper, we are interested in the estimation of θ ) and β when N and T go to infinity jointly or only N goes to infinity while T is fixed. The specification in 2.) is natural extensions of classical parametric models with good interpretability and are becoming more and more popular in data analysis. Thanks to the flexibility and interpretability, the panel version of varying coeffi cient models have many potential applications. For instance, in the study of return to schooling, the impact of education on the wage may vary with working experience Cai et al. 2) and Su et al. 24)) while the impact of other variables are constant. Moreover, model 2.) is a generalization of Cai and Li 28), Sun et al 29) and Cai et al 25) by including fixed effects in the model. However, if α i is assumed to be random, then model 2.) is similar to the one considered by Zhou et al 2). Furthermore, model 2.) is also an extension of Feng et al 27) and Cai and Li 28) by allowing partially linear in the model. Finally, model 2.) extends the nonparametric Another more flexible specification is y it = p k= d k,itθ l z k,it ) + x itβ + η i + u it, where different coeffi cient function θ k ) may have different conditional variables 5

6 dynamic model of Lee 24) and Su and Zhang 26) to partially linear functional-coeffi cient dynamic models which can further alleviate the problem of "curse of dimensionality". Before we move to the estimation, two important features for the model 2.) need to be emphasized. First, model 2.) allows for general dynamic pattern. The lagged dependent variables may enter the vectors of d it, x it or z it. In particular, when d it = y i,t,...y i,t p ) and z it = y i,t q where q p for some p, we obtain the panel version of functionalcoeffi cient autoregressive model in Chen and Tsay 993). Second, either d it or x it may include endogenous covariates Sieve estimation In principle, one can choose either kernel method or sieve method to estimate the unknown nonparametric component in the model 2.). But in this paper, we focus on the sieve method. There are mainly two reasons. First, the conditional variables for different coeffi cient functions θ l ) may be different, which is complicated to use kernel method. Second, as argued by Su and Hoshino 26), even if kernel method has the advantage of capturing the local properties of the coeffi cient functionals and its asymptotic properties are also well documented in the literature, the kernel method for functional-coeffi cient models usually require iterative methods which are particular computationally demanding. However, it is convenient to use sieve approximation to handle different conditional variables; in our case, there exists explicit expression for our estimates and the computation is fast. See Chen 27) and Li and Racine 27) for an overview on sieve methods. Let h L ) = h ),..., h L )) be an L sequence of basis functions where the number of sieve basis functions L L increases as either N or T increases. Then for k =,..., p d, we have θ k ) h L ) γ k, where γ k = γ k,..., γ kl ) is an L vector of corresponding coeffi cients of the basis functions. 3 For notational simplicity, we suppress the dependence of h L ) on L and let h ) = h L ) and h it = h z it ). Define a p d L vector H d, z) = d h z) = d h z),..., d pd h z) ), 2 However, we do not consider the case with endogeneous z it. Othewise, we face the problem of nonparametric instrumental variables NPIV) regression which raise different issues in identifcations and estimation. This is beyond the scope of this paper. 3 For simplicity, we use the same sieve basis functions in the approximation of different coeffi cient functions. We also allow for different number of basis functions for different unknown function. 6

7 where is the Kronecker product and H it H d it, z it ). Then we can rewrite the model 2.) as follows y it = H itγ + x itβ + η i + ε it, 2.2) where Γ γ,..., γ p d ) denotes the vector of sieve approximation coeffi cients and ε it = u it +r it, r it = p d k= d k,itr L k,it with rl k,it θ k z it ) γ k h z it) signifying the sieve approximation error. Now we have a linear dynamic panel data models with two new features: one is the increasing dimension of H it and the other is the composite structure of the new error ε it. When establish the asymptotic properties for our estimates, we have to take these feature into consideration. Since the fixed effects α i enters model 2.2) linearly, one can apply some linear transformation to eliminate α i. When there is neither lagged dependent variables nor endogenous variables, and all the variables are strictly exogenous, we can use the within-group transformation to remove α i. However, for dynamic panel data models, the most commonly used transformation is the so-called first time difference Anderson and Hsiao 98, 982) and Arellano and Bond 99)). Let A it = A it A it be the first time difference FD) of sequence {A it } T t= for A = y, H, x, ε, u and r, then the first differenced model of 2.2) is given by y it = H itγ + x itβ + ε it, t = 2,..., T ; i =,..., N. 2.3) As L when N, T ), the sieve approximation errors become asymptotic negligible and u it dominates in ε it. For the FD model 2.3), there is endogeneity problem E x it ε it ) = E u it x it ) + o ) or E H it ε it ) = E u it H it ) + o ) which may be caused by the lagged dependent variables in either H it or x it, or the endogenous covariates in d it or x it. 4 To handle the problem, we suppose there exists a p w vector w it such that E u it w it ) =, which can be used as instruments for H it and x it, then we can apply the IV or 2SLS estimation to obtain consistent estimators of Γ and β for model 2.3). Remark 2. How to choose IVs or construct moment conditions depends on the model specification case by case. When only one lagged dependent variable on the right-hand side of 2.3), we should include all the lagged levels y i,t 2,..., y i or lagged differences y i,t 2,..., y i2 as IVs when is T small; when T is large, we focus on the consistency instead of effi ciency for the estimator because the latter is still an open question in the literature of nonparametric/semiparametric dynamic panel data models. For simplicity, we assume there exists only one lagged dependent variable y i,t in d it, x it,or z it : 4 An illustrating example is if x it = y it, then we have E x it ε it). 7

8 i) Linear dynamic panel: x it = y i,t, x,it ) where x,it = x 2,it,..., x px,it) are sequentially exogenous. Choose w it = H it, y i,t 2, x,it ) or w it = H it, y i,t 2, x,it ). 5 ii) Lagged dependent variable with functional coeffi cient: d,it = y i,t. Let w it = y i,t 2 h i,t 2, H,it, x it ) or w it = H,i,t,..., H L,i,t, H,it, x it ), where H,it = H L+,it,..., H pd L,it) with H k,it being the kth element of H it. iii) Nonparametric dynamic coeffi cient functions: z it = y i,t. Set w it = w it = H i,t 2 it), x. H i,t 2, x it) or iv) When endogenous covariates are included in x it or d it, we can use additional variables as IVs or construct IVs from the lags of x it, d it, z it or y it according to the specific dependent structure of the model. Remark 2.2 How to choose optimal instruments for the nonparametric/semiparametric dynamic panel model when T is large, it is still an open question due to the curse of dimensionality and the possible problems caused by many weak IVs e.g., Newey and Windmeijer 29), Okui 29) and reference therein). Let W i w i2,..., w it ), W W, W 2,..., W N ), y i y i3,..., y it ), and Y y, y 2,..., y N ). Similarly define H i, H, x i,and X. Then the sieve IV/2SLS estimates of Γ and β based on the model 2.3) are given by 6 ˆΓ, ˆβ ) = [ X P W X] X P W Y, where X = H, X), and P W = WW W) W is a projection matrix with A denoting the Moore-Penrose generalized inverse of square matrix A e.g., Horn and Johnson 22)). Let Y W P W Y, H W P W H, X W P W X and M XW = I ) X W X W X W) X W. By the formula for partitioned regressions, we can write the estimators for Γ and β separately by ˆΓ = H WM XW H W ) H W M XW Y W, 2.4) ˆβ = X WM HW X W ) X W M HW Y W. 2.5) 5 In practice, one usually adds more several lags but not all lags in IVs to improve finite sample performance in dynamic panel models. The use of all lags raise new issues such as weak and many IVs. The choice of optimal IVs is still an open question. See Okui 29) for an overview. 6 More generally, we can consider the sieve GMM estimate defined by: ˆΓ, ˆβ ) = [ X WĀ W X] X WĀ W Y, Ā is a p w p w weighting matrix that is symmetric and asymptotically positive definite. The asymptotic properties are similar to the 2SLS estimator but the notation is slightly more complicated. So we decide to focus on the sieve 2SLS estimation here. 8

9 Given the estimator of ˆΓ = ˆγ,..., ˆγ p d ), we can estimate the varying coeffi cient θ u) by ˆθ u) = H S u) ˆΓ, where H S u) = h u) S. h u) S pd 2.6) where S k = i k I L and i k is the p d unity vector with the only nonzero element being at its k-th place. The L p d L matrix S k selects the estimator of γ l. That is ˆγ k = S kˆγ and ˆθ k u) = h u) ˆγ k for k =,..., p d. 3 Asymptotic Properties of Sieve Estimators In this section, we study the asymptotic properties of the sieve estimators ˆβ and ˆθ ). In order to derive the asymptotics, we focus on the case of large T and large N and discuss the extension to the case of large N and small T briefly. The latter case is much simpler because we do not need to impose the stationarity and mixing conditions on the system 2.), whilst we need to impose extra conditions e.g., mixing conditions) to ensure the system 2.) is ergodic and stationary when T is large. 3. Assumptions To apply the method of sieves, we assume that θ l ) s satisfy some smoothness conditions. Let U R pz be the support of z it. To allow for the possible unboundedness of U, we follow Chen et al. 25), Su and Jin 22), and Lee 24) to use a weighted sup-norm: m,ϖ sup u U m u) [ + u 2 ] ϖ/2 for some ϖ. When ϖ =, the norm is the usual sup-norm which is suitable for the case with compact support Newey 997) and Andrews 25)). Let α α,..., α pz ) be a p z -vector of non-negative integers and α p z k= α k. For any u = u,..., u pz ) U, the α-th derivative of m is denoted as α mu) α mu)/ u α... uαpz p z ) and the l-th derivatives of m include all α mu) s with α = l. The Hölder space Λ γ U) with order γ > is the set of functions m : U R such that the first γ derivatives are bounded and the γ th derivatives are Hölder continuous with the exponent γ γ, ]. The Hölder norm is defined by m Λ γ sup m u) + max sup α mu) α mu ) u U α = γ u u u u γ γ. 9

10 The following definition is adopted from Chen et al. 25). Definition. Let Λ γ U, ϖ) {m : U R such that m )[ + 2 ] ϖ/2 Λ γ U)} denote a weighted Hölder space of functions. A weighted Hölder ball with radius c is { } Λ γ c U, ϖ) m Λ γ U, ϖ) : m )[ + 2 ] ϖ/2 Λ c <. γ Function m ) is said to be Hγ, ϖ)-smooth on U if it belongs to a weighted Hölder ball Λ γ c U, ϖ) for some γ >, c > and ϖ. Let y i y i,..., y it ) and define d i, z i, x i and u i analogously. Let y i,t y i,t, y i,t 2,..., y i ) and define d it, x it and z it in the same way. Denote Q wx, N T i= t=2 w it x it, Q wx EQ wx, ), Q ww, N T i= t=2 w itw it, Q ww EQ ww, ), Q wh, N T i= t=2 w it H it, Q wh EQ wh, ), Q hh, N T i= t=2 H it H it and Q hh = E Q hh, ). Then define Q Q wxq wwq wx Q wxq w Q wh Q wh Q wwq wh ) Q wh Q wwq wx, 3.) Q 2 Q wxq ww Q wxq w Q wh Q wh Q wwq wh ) Q wh Q ww, 3.2) Q 3 Q wh Q wwq wh Q wh Q wwq wx Q wx Q ) wwq wx Q wx Q wwq wh, 3.3) Q 4 Q wh Q ww Q wh Q w Q wx Q wx Q wwq wx ) Q wx Q ww, and 3.4) Q 5 Q hh Q wh Q wwq hw. 3.5) For model 2.), we make the following assumptions. Assumption A. i) y i, d i, x i, z i, η i, u i ) are independently across i and Eu it y i,t,d it, x it,z it ) =. ii) There exists a p w vector w it such that p w p x + p d L and E u it w it ) =. iii) For each i, {y it, d it, x it, z it, u it ) : t =, 2,...} is strong-mixing with mixing coeffi cient α i ) given the fixed effects. α ) = max i N α i ) satisfies s= s2 α δ 4+δρ s) < C < for some δ >. Assumption A.2 i) θ l ) s l =,..., p d ) are all Hγ, ϖ)-smooth on U for some γ > p z +)/2 and ϖ. ii) For any Hγ, ϖ)-smooth function θ u), there exists a linear combination of basis functions Π,L θ γ θ h ) in the linear sieve space G L = {θ ) = γ h )} such that θ Π,L θ, ϖ = O L γ/pz) for some ϖ > ϖ + γ. iii) plim N,T ) N i= T t=2 + z it 2 ) ϖ αz it ) <. iv) There are a sequence of constants ζ L) and a sequence of increasing compact set U satisfying that sup u U u = Oζ L) / ϖ ), sup u U h u) ζ L), and ζ L) 2 L/ ) as N, T ).

11 Assumption A.3 i) sup i N sup 2 t T E χ it 4+ɛ C < for some ɛ > and χ it = u it, z it, d it, w it, and H it. ii) Q ww is invertible, Q wxq wwq wx is invertible, Q and Q 3 are invertible, and Q wh, Q wx ) has full rank p x + L. C >. iii) The eigenvalues of Q and Q 3 are all bounded and bounded away from, and λ min Q 4 Q 4 ) > iv) λ max Q hhω ) < where Q hhω u U H S u) H S u) ω u) du and ω ) is a nonnegative weight function. v) Ω = lim N,T ) N i= E W i u i u i W i) > and λ max Ω) < C <. Assumption A.4 As N, T ), L 3 / ), L γ/pz. Most of the above assumptions are very similar to those that are used in Lee 24), and Su and Zhang 26), we modify a few of them for the purpose of our analysis. Assumption A.i) is standard for dynamic panel data models e.g., Alvarez and Arellano, 23); A.ii) requires the existence of a vector of IVs; and Aiii) imposes the strong mixing condition on the data generating process, which can be easily satisfied for a wide class of nonlinear autoregressive functions in time series context e.g., Chen and Shen, 998). Assumptions A.2 is widely used in the literature on sieve estimation with infinite support. A.2i) imposes a smoothness conditions on the unknown functions; A.2ii) states the uniform sieve approximation errors; A.2iii) is used to obtain the convergence of our sieve estimator in the empirical L 2 -norm; and A.2iv) is used to derive the uniform convergence rate on an increasing compact support U. Assumption A3i) gives some moment conditions on the variables and basis functions; A.3ii)-iii) are used in identification condition for sieve 2SLS estimation. In the sieve literature, it is known that many series function satisfy assumptions A.2 and A.3, for example, power series, orthogonal polynomial, trigonometric series and splines. Assumption A.4 impose some rate conditions on L to control the sieve approximation error and variance. 3.2 Asymptotic properties of sieve estimators We establish the asymptotic normality of ˆβ in the following theorem. Theorem 3. Suppose Assumptions A.-A.4 hold. Then ˆβ β) d N, Q Q 2ΩQ 2Q ). 3.6)

12 The above theorem gives the asymptotic distribution of ˆβ, which is shown to be - consistent, asymptotically unbiased and asymptotically normally distributed. Now let s turn to the asymptotic properties of sieve estimator 2.6). The following theorem reports the convergence rates and asymptotic normality of ˆθ u). Theorem 3.2 Suppose Assumptions A.-A.4 hold and inf u U h u) C >. Then i) ˆθ u) θ u) 2 ω u) du = O L p + L 2γ/pz) ; ii) sup u U ˆθ u) θ u) L = O p ζ L) + L γ/pz )); iii) [ˆθ ] Ξ u) /2 u) θ u) d N, ), where Ξ u) = H S u) Q 3 Q 4ΩQ 4Q 3 H S u). Several remarks can be made for the above asymptotic results. Remark 3.3 The most important feature about the above result is that with oversmoothing the sieve estimation of β and θ ) is unbiased in the sense that the limiting distributions of ˆβ and ˆθ ) are centered at zero, unlike the sieve estimation based on within group transformation, which is shown by Lee 24) and Tran 24) to be asymptotically biased of order O T ) and bias correction method is needed for statistical inference. It will be interesting to compare the bias-corrected estimators with our sieve 2SLS estimators. Remark 3.4 For the above asymptotic results, it is assumed that both N and T go to infinity. However, similar asymptotic results for both ˆβ and ˆθ u) still hold if T is fixed and N goes to infinity. When T is fixed, the sieve 2SLS estimation procedure for β and θ u) remain the same as the case when T is large. However, the assumption for asymptotics for large N and fixed T can be relaxed, namely, we don t need to impose weak dependence condition strong mixing) on the variables. Also, all assumptions regarding the limit of T can be relaxed. For instance, Assumption A3iii) can be relaxed as Ω = lim N N i= E W i u i u i W i) >. Finally, Assumption A.4 can be relaxed to L/N, NL γ/pz,and L 3 /N as N. Under these assumptions, the above asymptotics for both ˆβ and ˆθ ) can be easily established by following the derivation in this paper. Remark 3.5 In the above sieve 2SLS estimation process, it is assumed that the panel structure is balanced, i.e., time dimension T is the same for diff erent cross-sectional units. However, the proposed sieve 2SLS estimation procedure can be easily modified to suit the unbalanced panels. In such a case, the sieve 2SLS estimation still works, namely, the only thing needs 2

13 to be changed in both 2.4) and 2.5) is the dimension of the data matrix. For example, for 2.4), the dimension of data H and Y W changes to T + T T N 2N) p d L and T + T T N 2N), respectively, where T i is the number of observation of i-th crosssectional unit. As shown in the simulation below, the sieve 2SLS estimation works remarkably well for unbalanced panel regardless min i N T i ) is large large T case) or max i N T i ) is fixed fixed T case). 4 Specification Test for the Constant Slopes In this section we maintain the correct specification of the partially linear panel data model and consider testing for the constancy of the nonparametric component θ ) in the partially linear model. The null hypothesis is H : Pr θ z it ) = γ ) = for some γ Θ R p d, 4.) where i =,..., N, t =,..., T. The alternative hypothesis is given by H : Pr [θ z it ) = γ] < for all γ Θ R p d. 4.2) To facilitate the asymptotic local power analysis, we consider the following sequence of Pitman local alternatives: H δ ) : Pr θ z it ) = γ + δ Ψ z it ) ) = for some γ Θ R p d where Ψ ) = Ψ ) is a measurable nonlinear function and δ as N, T ). Here we follow Su and Zhang 26) and propose a test for H versus H by comparing the weighted empirical L 2 -norm distance between two estimators of the slopes of d it, i.e., the nonparametric estimator ˆθ ) and parametric estimator γ. Intuitively, both estimators are consistent under the null hypothesis while only the sieve estimator is consistent under the alternative. So if there is any deviation from the null, the distance between two estimators will 3

14 signal it out asymptotically. This motivates us to consider the following test statistic 7 D = N i= t=2 T ˆθ z it ) γ 2 a z it ), 4.3) where a ) is a user-specified nonnegative weighting function, and γ is the usual IV/2SLS estimate for γ in the linear panel data model under H. Similar test statistics have been proposed in various other contexts in the literature; see, e.g., Härdle and Mammen 993), Hong and White 995), and Su and Zhang 26). We will show that after being appropriately centered and scaled, D is asymptotically normally distributed under the null hypothesis of constant slopes. Under H, taking the first difference on the linear panel data model leads to y it = d itγ + x itβ + u it, for i =,..., N and t = 2,..., T. Due to the possible endogeneity of d it or x it, let v it be a p v vector of instrumental variables. Let D = d 2,..., d T,..., d N2,..., d ) and V = v 2,..., v T,..., v N2,..., v ). Denote P V = V V V) V. Define the linear projection matrix of A on V by A V = P V A where A = D, Y, or X. Then the 2SLS estimator for the slope of γ can be written as γ = D VM XV D V ) D V M XV Y V, where M XV = I ) P XV. Denote Q vx, N T i= t=2 v it x it, Q vx EQ vx, ), Q vv, N T i= t=2 v itv it, Q vv EQ vv, ), Q vd, N T i= t=2 v it d it, Q vd EQ vd, ), q vψ, = N T i= t=2 v it Ψ it and q vψ = E Q vψ, ), where Ψ it = Ψ z it ) Ψ z i,t ). Define Q 6 = Q vxq vv Q vx Q vxq vv Q vd Q vd Q vv Q vd ) Q vd Q vv Q vx, q Ψ = Q vxq vv q vψ Q vxq vv Q vd Q vd Q vv Q vd ) Q vd Q vv q vψ, 7 One can also construct a test statistic by comparing the two estimates for the whole conditional mean function under the null and the alternative:. D = N T ) N i= t=2 T ˆθ z it) γ) dit + x it ˆβ β) 2 a zit) 4

15 and γ Ψ = Q 6 q Ψ. Let Q a) h = N T s, i= t=2 H S z it ) H S z it ) a u it ) and Q a) h s = EQ a) h ). s, Then define ) Q 7, = Q wh, Q wx, Q wx, Q ww, Q wx, Q wx, Q ww, Q wh,, Q 7 = Q wh Q wx Q wx Q wwq wx ) Q wx Q wwq wh, Q = Q ww, Q 7, Q 3, Qa) h s, Q 3, Q 7, Q ww,, Q = Q w Q 7 Q 3 Qa) h s Q 3 Q 7Q w. Now we give some additional assumptions which is used in deriving the asymptotic properties for our test statistic. Assumption A.5. i) plim N,T ) D V M X V D V exists and is invertible. ii) plim N,T ) N i= T t=2 v it Ψ it exists. iii) D V M X V P V u = O p ) /2 ). Assumption A.6. i) sup i N sup 2 t T E χ it 8+8ɛ C < for some ɛ > and χ it = u it, z it, d it, w it, and h it. ii) λ max Ω i ) < C < where Ω i = E W i u i u i W i) for i =,..., N. iii) 4δ d= dα 4δ+ d) and d= d2 α δ δ+ d) < for some δ > /4. Assumption A.7. As N, T ), L 4 / ), L γ/pz, and L 3 /N. Remark 4. Assumption A.5 is used to study the asymptotic behavior of parametric estimator γ under the local alternatives. Assumption A.6i) imposes more higher order moment on the variables in testing; A.6ii) require the variance-covariance matrix of T /2 u i W i has bounded eigenvalues. Assumption A.7 gives some more strict rate conditions on L diverging to in testing. We define the test statistic J = D B ) / V where B =trqω ) and V = 2trQΩ QΩ ) are asymptotical bias and variance terms, respectively. Noting that J is infeasible due to the unknown B and V, then we define a feasible test statistic Ĵ = D ˆB ) / ˆV 5

16 where B and V are respectively estimated by ˆB = trq Ω, ) and ˆV = 2trQ Ω, Q Ω, ) with Ω, = N T 2) i= t=2 w itw it û it) 2 and û it = y it H itˆγ x itˆβ. Let µ Ψ = plim N T N,T ) i= t=2 Ψ z it) γ Ψ 2 a z it ) where γ Ψ = Q 6 q Ψ. The following theorem establishes the asymptotic distribution of Ĵ under H δ ). Theorem 4.2 Suppose that Assumptions A.-A.3, and A.5-A.6 hold. Under H δ ) with δ ) /2 V /4, as N, T ),Ĵ d N µψ, ). Remark 4.3 The proof for the above theorem is tedious and is relegated to the appendix. We d complete the proof by showing that i) J N µψ, ), and ii) Ĵ J = o p ). The idea to prove i) is to write J as a degenerated second order U-statistic plus some smaller order terms and then apply dejong s 987) CLT for independent but non-identically distributed inid) observations. Remark 4.4 In view of the fact that V = O L), we have δ ) /2 L /4 which indicates that our test has power to detect the local alternatives that converge to the null hypothesis at the rate ) /2 L /4. The asymptotic local power function is given by ) Pr Ĵ z α H δ ) Φ z α µ Ψ ) as N, T ), where z α is the upper αth percentile from the standard normal distribution, and Φ ) is the standard normal cumulative distribution function CDF). Remark 4.5 To study the asymptotic behavior of Ĵ study the asymptotic properties of γ under H. under global alternatives, we need to We can define pseudo-true parameter γ as the probability limit of γ. Then Pr Π z it ) θ z it ) γ ) > C for some C >. Let Π = Π z ),..., Π z T ),..., Π z N ),..., Π z ) ). With the additional assumption that Π = O p ) /2 ), we can show that D = N T i= t=2 ˆθ z it ) γ 2 a z it ) = N T i= t=2 Π z it ) 2 a z it ) + o p ) = O p ). Then, together with the fact that ˆB = O p L) and ˆV = O p L), we can demonstrate that Ĵ diverges to infinity at rate O p / L) under H as L/ ). That is, PrĴ > b H ) for any nonstochastic sequence b = O/ L). So our test achieves the consistency against the global alternatives. 6

17 Remark 4.6 With a slightly modification, our test can be applied to testing for the parametric specification of θ ). One can still construct test statistics based on the empirical L 2 -norm distance between the sieve estimate and parametric estimate for the slope vector of d it. The null hypothesis H can be seen a special case of H δ ) when µ Ψ =. Clearly, Ĵ is asymptotic distributed N, ). This result is stated as a corollary. Corollary 4.7 Suppose Assumptions A.-A.3, A.6-A.7. hold. Under H, Ĵ d N, ) as N, T ). Remark 4.8 In principle, we can compare Ĵ with the one-sided critical value z α from the standard normal distribution, and reject the null when Ĵ z α. In finite sample, tests based on standard normal critical values tend to suffer from severe size distortion due to the nonparametric nature of our test. Therefore, we propose to implement the test based on bootstrap p-value. To improve the finite sample performance of our test, we propose a fixed-regressor bootstrap Hansen, 2) procedure as follows:. Estimate the restricted model under H and obtain the residuals ŭ it = y it d it γ x it β, where γ and β are the IV or GMM estimates of γ and β under the null; under H, obtain the sieve estimator ˆθ z it ). Calculate the test statistic Ĵ based on the original sample {y it, d it, x it, z it }. Let η i T T t= ŭit. 2. Obtain the bootstrap error u it = ŭ it η i ) ɛ it for i =, 2,..., N and t = 2,..., T, and ɛ it s are IID across both i and t and follow a two-point distribution: ɛ it = 5 2 with probability and with probability 5 2. We generate the bootstrap analogue 5 yit of y it as y it = d itˇγ + x it β + η i + u it for i =, 2,..., N and t = 2,..., T, where yi = y i. 3. Given the bootstrap resample {yit, d it, x it, z it }, estimate both the restricted linear) and unrestricted semi-parametric) panel data model and calculate the bootstrap test statistic Ĵ. 7

18 4. Repeat steps 2 and 3 for B times and index the bootstrap test statistics as {Ĵ,b }B b=. The bootstrap p-value is calculated by p = B B b= Ĵ,b > Ĵ ). It is straightforward to implement the above bootstrap procedure. Clearly, we impose the null hypothesis of constant slopes for d it in step 2. Noting that there is no dynamic and endogeneity in the bootstrap world, we can estimate the model with/without using the IV approach. Conditional on the data, yit, u it ) are independently but not identically distributed INID) across i, and u it are also independently distributed across t. So we need to resort to the CLT for second order U-statistics with INID data e.g., de Jong 987)) to justify the asymptotic validity of the above bootstrap procedure. Following Su and Lu 23) and Su and Zhang 26), we can easily justify the validity of our bootstrap procedure. 5 Simulations In this section, we conduct a small set of Monte Carlo simulations to examine the finite sample performance of our proposed sieve 2SLS estimation for partially linear functional-coeffi cient dynamic panel models. We consider the following five data generating processes DGPs) with functional coeffi cient by allowing the panel structure could be either balanced or unbalanced. DGP. functional coeffi cient on lagged variable): y it =.5 e 2z2 it )yi,t +.5x it + η i + ε it, 5.) so the functional coeffi cient θ z) =.5 e 2z2, similar setting can be found in Cai et al 25). We also assume ε it are IID N, ) across both i and t, η i are IID N, ), and x it = ρ x,i x it +.5η i + ε x,it, z it = ρ z,i z it +.5η i + ε z,it, with ρ x,i and ρ z,i are independent draws from U.2,.8) for i =, 2,..., N. DGP 2. functional coeffi cient on lagged variable): y it = e z2 it z 2 it + z it )y i,t +.5x it + η i + ε it, 5.2) so the functional coeffi cient θ z) = e z2 z 2 + z), 8 and the generation of x it, z it, η i and ε it are the same as in 5.). 8 For this function θ z), it can be verified that min z R θ z) =.4 and max z R θ z) =.85. 8

19 DGP 3. functional coeffi cient on lagged variable): y it =.8 sinz it )y i,t +.5x it + η i + ε it, 5.3) so the functional coeffi cient θ z) =.8 sinz) with Φ ) being the CDF of standard normal distribution, and the generation of x it, z it, η i and ε it are the same as in 5.). DGP 4. functional coeffi cients on both lagged and exogenous variables): y it =.5 e 2z2,it)y i,t φ z 2,it ))d it +.5x it + η i + ε it. 5.4) so the functional coeffi cient θ z) =.5 e z2 and θ 2 z) =.5 + φ z) where φ ) is the standard normal PDF. We assume the generation of ε it, α i and x it are the same as in DGP, and z,it = ρ z,iz j,it +.5η i + ε z,it, z 2,it = η i + ε z2,it, d it = ρ d,i d it +.5η i + ε d,it, with {ε z,it, ε d,it } being IID N, ) and ε z2,it being IID χ 2 ) across both i and t and independent of {η i } and {ε it }. Also, ρ z,i and ρ d,i are independent draws from U.2,.8) for i =, 2,..., N. DGP 5. unbalanced model with functional coeffi cient on lagged variable): For this DGP, we assume the generation of θ z), y it, z it and x it are the same as of DGP 5.), but we assume the panel is unbalanced in the sense that T i T j for some i j. To this end, we consider two cases of generation of T i i =,..., N), where we assume T i are integers randomly drawn from [5, ], i.e, we assume the time period of the panel is fixed. While in the second case, we assume T i are integers randomly drawn from [4, 5], i.e, we consider a relatively large panel. DGP 6. linear dynamic panels with constant lag coeffi cient) For this DGP, we consider a linear dynamic panels with constant lag coeffi cient, and we perform the hypothesis testing of whether the lag coeffi cient is indeed constant. The prototype model under the null hypothesis that the lag coeffi cient is constant) is given by y it =.5y i,t +.5x it + η i + ε it, 5.5) and we assume the model under alternative hypothesis is given by y it = θ z it ) y i,t +.5x it + η i + ε it, 5.6) 9

20 where θ z) =.5 e 2z2 as in 5.) or θ z) = e z2 z 2 + z) as in 5.2). The purpose of DGP 5.6) is to verify the power of the proposed test statistics. The generation of x it, η i and z it are the same as in DGP 5.). For the N, T ) pair, we consider N =, 2 and T = 5,, 5 for DGP 5.)-DGP 5.4), and we set the number of replications as for the estimation. In the estimation, we consider the following sieve estimation for θ ) and β, ˆθ sieve ) and ˆβ sieve, respectively. Since the coeffi cient of y i,t is assumed to be functional, we follow the suggestion i) of Remark 2.) for choice of IVs. For the sieve estimates, we choose the cubic B-spline as the sieve basis and include the tensor product terms to approximate the function θ ). Along each dimension of the covariate in θ ), we let L c = c ) /5 + and choose L c sieve approximating terms, where a denotes the integer part of a and c =, 2, 3. For the sieve estimates of ˆθ sieve ) for DGPs 5.)-5.3) and θ,sieve ) and θ 2,sieve ) for DGP 5.4), we calculate the median bias which is computed as the difference of the medians of θ ) and θ sieve )) and RMSE which is computed as square root of the pointwise difference of θ ) and θ sieve )). For estimation of ˆβ sieve for DGPs 5.)-5.4), we calculate the bias and RMSE around the true value for comparison. Finally, for the specification testing of constant lag coeffi cient of DGP 5.5), we use 5 replications and 3 bootstrap resamples for the empirical size and power study. The simulation results are summarized in Table -5 and Fig -6. Several interesting findings can be observed from the simulation results. On the first hand, when the panel is balanced DGP-4), Table -2 reports the median bias m.b) and root mean squared error RMSE) of various estimates of θ ) or θ ) and θ 2 )), and Table 4 reports the median bias m.b) and root mean squared error RMSE) of various estimates of β. For all DGPs under investigation, the RMSEs of both nonparametric part and parametric part decrease as either N or T increases and are roughly halved as N is quadrupled regardless of the sieve choice of approximation terms L, L 2 and L 3. The observation is valid when N is large or T is large. However, it should be noticed that the choice of approximation terms L, L 2 and L 3 do have impact on the estimation for the sieve estimation of nonparametric part, namely, compared to using a small number of sieve approximation terms, estimation based larger number of sieve approximation terms will have relatively smaller RMSE, but the impact is almost negligible for the sieve estimation of parametric part of the model. These suggest that the sieve 2SLS estimation for balanced dynamic panels is indeed consistent. On the other, when the panel is unbalanced DGP5), from the simulation results, we can 2

21 still observe that the RMSE of the sieve estimation of both θ ) and β decreases as either N or T increases regardless of whether N is large or T is large, which shows the effectiveness of applying the sieve 2SLS estimation to unbalanced panels. When coming the specification testing, Table 5 gives the empirical rejection frequency for our proposed test. From this table, we can see that the empirical size behave reasonably well for DGP 5.5), and they are very close to the nominal values %, 5% and % regardless the choice of number of sieve approximation numbers. The powers are reasonably good, and increase quite fast with the increase of either N or T. These empirical size and power suggest that the proposed test is applicable to testing whether the lag coeffi cient is constant for linear dynamic panels. Finally, we compare the approximation of the estimated functional coeffi cient with the true functional coeffi cient for DGP -5 in Fig -6. It is obvious that the estimated functional coeffi - cient using sieve 2SLS estimation fits the true functional coeffi cient quite well in all simulation designs regardless of whether the panel is balanced or not and whether T is large or fixed. In most cases, the shape of estimated functional coeffi cient are almost coincide with the true function coeffi cients, which illustrates the validity of using sieve approximation for the unknown functional coeffi cient. In all, we can conclude that the simulation results confirm our theoretical findings in the paper and the sieve 2SLS estimation perform reasonably good. 2

22 Table : Simulation results for sieve estimates θ ) for DGPs 5.)-5.3) L L 2 L 3 DGP T N m.b RMSE m.b RMSE m.b RMSE Notes:. "m.b" refers to the median bias. 2. "L", "L2" and "L3" refers to sieve estimation using L c sieve terms in the approximation, which is determined by L c = c ) /5 for c =, 2, 3. 22

23 Table 2: Simulation results for sieve estimates θ ) and θ 2 ) for DGP 5.4) L L 2 L 3 θ ) T N m.b RMSE m.b RMSE m.b RMSE θ 2 ) See notes of table. Table 3: Simulation results for sieve estimates θ ) and β for DGP 5 with unbalanced panel L L 2 L 3 θ ) T i N m.b RMSE m.b RMSE m.b RMSE [5, ] [4, 5] β [5, ] [4, 5] See notes of table. 23

24 Table 4: Simulation results for β for DGPs 5.)-5.4) L L 2 L 3 DGP T N Bias RMSE Bias RMSE Bias RMSE Notes: The true value of β is β =.5. See also Note 2 of Table. 24

25 Table 5: Empirical size and power for specification test of DGP 5.5) size study for H : θ z) =.5 L L 2 L 3 size size size T N % 5% % % 5% % % 5% % 5.4% 2% 4.4%.4% 2.4% 5.4%.4%.2% 4% 2.4% 4.4% 7.8% % 2.4% 7.2%.% 2.8% 7%.4% 3.6% % % 4.4% 9.2%.8% 5.4% 9.2% 2.8% 4.6% 8.8%.4% 3.4% 8.8%.8% 3.6% 9.6% 5.4% 6.6% 2.2%.4% 6.6%.2%.4% 7%.2% 2 % 5% 9.8%.8% 4.2% 9%.8% 4.6% 9% Power study for H : θ z) =.5 e 2z2 L L 2 L 3 Power Power Power T N % 5% % % 5% % % 5% % 5 7% 32.6% 53.6% 5.8% 24.4% 46.4% 3% 3.8% 3% % 65.4% 84.6% 2.6% 63.8% 85.4% 2.2% 46.2% 75.2% 97.8% 99.6% % 97.4% 99.8% % 9.8% 99.2% 99.8% 2 % % % % % % 99.8% % % 5 % % % % % % % % % 2 % % % % % % % % % Power study for H : θ z) = e z2 z 2 + z) L L 2 L 3 Power Power Power T N % 5% % % 5% % % 5% % 5 7.4% 27.6% 54.8% 3.8% 9.6% 4.6%.4% % 25.4% 2 32% 75.2% 89.8% 7.2% 59.8% 82.8%.2% 4.4% 7.8% 97.6% % % 98% % % 9% 99.6% % 2 % % % % % % % % % 5 % % % % % % % % % 2 % % % % % % % % % See notes of Table. 25

26 Fig. Sieve approximation of θ z) =.5 e z2 for DGP 5.) N =, T = 5 N = 2, T = N =, T = N = 2, T = L L2 L3 True N =, T = 5 N = 2, T = 5 Note: "L" refers to sieve approximation using L terms, "L2" refers to sieve approximation using L 2 terms, "L3" refers to sieve approximation using L 3 terms, and "True" refers to the true function. 26

27 Fig 2. Sieve approximation of θ z) = e z2 z 2 + z) for DGP 5.2) N =, T = 5 N = 2, T = N =, T = N = 2, T = L L2 L3 True N =, T = 5 N = 2, T = 5 See note of Fig. 27

28 Fig 3. Sieve approximation of θ z) =.8 sinz) for DGP 5.3) N =, T = 5 N = 2, T = N =, T = N = 2, T = L L2 L3 True N =, T = 5 N = 2, T = 5 See note of Fig. 28

29 Fig 4. Sieve approximation of θ z) =.5 e z2 for DGP 5.4) N =, T = 5 N = 2, T = N =, T = N = 2, T = L L2 L3 True N =, T = 5 N = 2, T = 5 See note of Fig. 29

30 Fig 5. Sieve approximation of θ 2 z) =.5 + φ z) for DGP 5.4) N =, T = 5 N = 2, T = N =, T = N = 2, T = L L2 L3 True N =, T = 5 N = 2, T = 5 See note of Fig. 3