Nonparametric Time-Varying Coefficient Panel Data Models with Fixed Effects

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1 The University of Adelaide School of Economics Research Paper No May 2010 Nonparametric Time-Varying Coefficient Panel Data Models with Fixed Effects Degui Li, Jia Chen, and Jiti Gao

2 Nonparametric Time Varying Coefficient Panel Data Models with Fixed Effects By Degui Li, Jia Chen and Jiti Gao School of Economics, The University of Adelaide, Adelaide, Australia Abstract This paper is concerned with developing a nonparametric time varying coefficient model with fixed effects to characterize nonstationarity and trending phenomenon in nonlinear panel data analysis. We develop two methods to estimate the trend function and the coefficient function without taking the first difference to eliminate the fixed effects. The first one eliminates the fixed effects by taking cross sectional averages, and then uses a nonparametric local linear approach to estimate the trend function and the coefficient function. The asymptotic theory for this approach reveals that although the estimates of both the trend function and the coefficient function are consistent, the estimate of the coefficient function has a rate of convergence of 1/2 that is slower than that of the trend function, which has a rate of N 1/2. To estimate the coefficient function more efficiently, we propose a pooled local linear dummy variable approach. This is motivated by a least squares dummy variable method proposed in parametric panel data analysis. This method removes the fixed effects by deducting a smoothed version of cross time average from each individual. It estimates the trend function and the coefficient function with a rate of convergence of N 1/2. The asymptotic distributions of both of the estimates are established when T tends to infinity and N is fixed or both T and N tend to infinity. Simulation results are provided to illustrate the finite sample behavior of the proposed estimation methods. JEL Classifications: C13, C14, C23. Abbreviated Title: Time Varying Coefficient Models Keywords: Fixed effects, local linear estimation, nonstationarity, panel data, specification testing, time varying coefficient function. 2

3 1. Introduction Panel data analysis has received a lot of attention during the last two decades due to applications in many disciplines, such as economics, finance and biology. The double index panel data models enable researchers to estimate complex models and extract information which may be difficult to obtain by applying purely cross section or time series models. There exists a rich literature on parametric linear and nonlinear panel data models, see the books by Baltagi 1995, Arellano 2003 and Hsiao However, it is known that the parametric panel data models may be misspecified, and estimators obtained from misspecified models are often inconsistent. To address such issues, some nonparametric methods have been used in both panel data model estimation and specification testing, see Ullah and Roy 1998, Lin and Ying 2001, Hjellvik, Chen and Tjøstheim 2004, Cai and Li 2008, Henderson, Carroll and Li 2008, and Mammen, Støve and Tjøstheim Meanwhile, the trending econometric modeling of nonstationary processes has also gained a great deal of attention in recent years. For example, it is generally believed that the increase in carbon dioxide emissions through the 20th century has caused global warming problem and it is important to model the trend of the global temperature. Some existing literature, such as Gao and Hawthorne 2006, revealed that the parametric linear trend does not approximate well the behavior of global temperature data. Hence, nonparametric modelling of the trending phenomenon has since attracted interest. One of the key features of the nonparametric trending model is that it allows for the data to speak for themselves with regard to choosing the form of the trend. For the recent development in nonparametric and semiparametric trending modelling of time series or panel data, see Gao and Hawthorne 2006, Cai 2007, Robinson 2008, Atak, Linton and Xiao 2009 and the references therein. While there is a rich literature on parametric and nonparametric time varying coefficient time series models Robinson 1989; Phillips 2001; Cai 2007, as far as we know, few work has been done in the panel data case. The recent work by Robinson 2008 may be among the first to introduce a trending time varying model for the panel data case where cross section dependence is incorporated. In both theory and applications, various explanatory variables are of significant interest when modeling the trend of a panel data. Thus, it may be more informative and useful to add such explanatory variables into a time varying panel data model when modeling the trend of a panel data. This paper thus proposes using a nonparametric trending time varying coefficient panel 3

4 data model of the form Y it = f t + d β t,j X it,j + α i + e it j=1 = f t + X it β t + α i + e it, i = 1,, N, t = 1,, T, 1.1 where X it = X it,1,, X it,d, β t = β t,1,, β t,d, all β t and f t are unknown functions, {α i } reflects unobserved individual effect, and {e it } is stationary and weakly dependent for each i and independent of {X it } and {α i }, T is the time series length and N is the cross section size. Model 1.1 is called a fixed effects model as {α i } is allowed to be correlated with {X it } with an unknown correlation structure. Model 1.1 is called a random effects model if {α i } is uncorrelated with {X it }. For the purpose of identification, we assume that throughout the paper. α i = 0, 1.2 i=1 Model 1.1 includes many interesting nonparametric panel data models. For example, when {β t } does not vary over time and reduces to a vector of constants, model 1.1 becomes a semiparametric trending panel data model with fixed effects. When β t 0 d 0 d is a d dimensional null vector, model 1.1 reduces to a nonparametric trending panel data model as discussed in Robinson 2008, which allows for cross sectional dependence for {e it }. The aim of this paper is to construct consistent estimates for the time trend f t and time varying coefficient vector β t before we establish asymptotic properties for the estimates. As in Robinson 1989, 2008 and Cai 2007, we suppose that the time trend function f t and the coefficient vector β t satisfy t f t = f T and β t,j = β j t T where f and β j are unknown smooth functions., t = 1,, T, 1.3 In this paper, we consider two classes of local linear estimates. As N α i i=1 = 0, the first method eliminates the fixed effects by taking cross sectional averages, and we call it the averaged local linear method. We establish asymptotic distributions for the resulting estimates of f and β under mild conditions. The asymptotic results reveals that as both T and N tend to infinity, the rate of convergence for the estimate of the coefficient function β is O P 1/2 while the rate of convergence for the estimate of the trend function 4

5 f is O P N 1/2. To improve the rate of convergence for the coefficient function, a local linear dummy variable approach is proposed. This is motivated by a least squares dummy variable method proposed in parametric panel data analysis see Chapter 3 of Hsiao 2003 for example. This method removes the fixed effects by deducting a smoothed version of cross time average from each individual. As a consequence, it is shown that the rate of convergence of O P N 1/2 for both of the estimates is achievable. The simulation study in Section 3 confirms that the local linear dummy variable estimate of β outperforms the averaged local linear estimate. The rest of this paper is organized as follows. The two classes of local linear estimates as well as their asymptotic distributions are given in Section 2. The simulated example is provided in Section 3. Section 4 summarizes some conclusions and discusses future research. All the mathematical proofs of the asymptotic results are relegated to Appendix A. 2. Nonparametric estimation method and asymptotic theory In this section, we introduce two classes of local linear estimates and establish the asymptotic distributions of the proposed estimates. In Section 2.1, we consider the averaged local linear estimation method. Section 2.2 discusses the local linear dummy variable approach Averaged local linear estimation To introduce the estimation method, we introduce some notation. Define Y t = 1 N Y it, i=1 X t = 1 N X it i=1 and e t = 1 N e it. i=1 By taking averages over i and using N α i = 0, we have i=1 Y t = f t + X t β t + e t, t = 1,, T, 2.1 where the individual effects α i s are eliminated. The averaged model 2.1 can be treated as a nonparametric time varying coefficient time series model, see Robinson 1989 and Cai 2007 for example. The formulation in model 2.1 makes it possible to directly estimate f and β j nonparametrically. Letting Y = Y 1,, Y T, f = f 1,, f T, BX, β = X 1β 1,, X T β T and e = e 1,, e T, model 2.1 can then be rewritten as the following vector form: Y = f + BX, β + e

6 We then adopt the conventional local linear approach see, for example, Fan and Gijbels 1996 to estimate β = f, β 1,, β d. For given 0 < τ < 1, define Dτ = 1 X X T 1 τt. T τt 1 τt X 1. T τt X T and 1 τt T τt W τ = diag K,, K. Assuming that β has continuous derivatives of up to the second order, by Taylor expansion we have β t = β τ + β τt τ + O t τ 2, 2.3 where 0 < τ < 1 and β is the derivative of β. Based on the local linear approximation in 2.3, β τ, β τ can be estimated by solving the optimization problem: arg min Y Dτa, b W τ Y Dτa, b. 2.4 a R d+1,b R d+1 Following the standard argument, the local linear estimator of β τ is [ 1 β τ = [I d+1, 0 d+1 ] D τw τdτ] D τw τy, 2.5 where I d+1 is a d + 1 d + 1 identity matrix and 0 d+1 is a d + 1 d + 1 null matrix. To establish asymptotic results, we need to introduce the following regularity conditions. Here and in the sequel, define µ j = u j Kudu and ν j = u j K 2 udu. A1 The probability kernel function K is symmetric and Lipschitz continuous with a compact support [ 1, 1]. A2 i {X i, e i, i 1} is a sequence of independent and identically distributed i.i.d. variables, where X i = X it, t 1 and e i = e it, t 1. Furthermore, for each i 1, {X it, e it, t 1} is stationary and α mixing with mixing coefficient α k satisfying α k = O k τ, where τ > δ + 2/δ for some δ > 0 involved in ii below. X it Xit. Fur- ii EX it = 0 d and there exists a positive definite matrix Σ X := E thermore, E X it 22+δ <, where is the L 2 distance. 6

7 iii The error process {e it } is independent of {X it } with E [e it ] = 0 and σe 2 = E [ e 2 it] <. Furthermore, E [ e it 2+δ] < and σeσ 2 X + 2 c X tc e t is positive definite, where c X t = E X is Xi,s+t and c e t = E e is e i,s+t. A3 The trend function f and the coefficient function β have continuous derivatives up to the second order. A4 The bandwidth h satisfies that h 0 and as T. Remark 2.1. The conditions on the kernel function K in A1 are imposed for brevity of our proofs and they can be weakened. For example, the compact support assumption can be removed if we impose certain restriction on the tail of the kernel function. In A2 i, we assume that {X i, e i, i 1} is cross sectional independent and each time series component is α mixing. Such assumptions are reasonable and verifiable and cover many linear and nonlinear time series models see, for example, Fan and Yao 2003; Gao 2007; Li and Racine Condition A2iii imposes the homoscedasticity assumption on {e it }. Since model 1.1 allows for the inclusion of the fixed effects term {α i } and therefore model 1.1 allows for endogeneity, the homoscedasticity assumption is less restrictive than in the time series case. Having mentioned this, the independence between {e it } and {X it } may be weakened through allowing e it = σx it, tɛ it, where σx, t is a positive function and Lipschitz continuous in t, and {ɛ it } satisfies A2iii with E[ɛ it ] = 0 and E[ɛ 2 it ] = 1. Both A3 and A4 are mild common conditions on the smoothness of the functions and the bandwidth involved in the local linear fitting. For the brevity of the proofs in Appendix A, we use Assumptions A.1 A.4 throughout this paper. Define X = 1 0 d 0 d Σ X, Λ X = ν 0 σe c e t 0 d 0 d σeσ 2 X + 2 c X tc e t. We state an asymptotic distribution for β τ in the following theorem. Theorem 2.1. Consider models 1.1 and 1.2. Suppose that A1 A4 are satisfied. Then, as T, D NT β τ β τ bτh 2 + o P h 2 d N 7 0 d+1, 1 X Λ X 1 X 2.6

8 for given 0 < τ < 1, where D NT = diag N, Id, bτ = 1 2 µ 2β τ and β = f, β 1,, β d. In particular, and N fτ fτ b f τh 2 + o P h 2 d N βτ βτ b β τh 2 + o P h 2 d N 0, ν 0 σe c e t d, Σ 1 X ν 0 σeσ 2 X + 2 c X tc e t Σ 1 X, 2.8 where b f τ = 1 2 µ 2f τ, b β τ = 1 2 µ 2β τ and β = β 1,, β d. Remark 2.2. The above theorem complements some existing results see Robinson 1989, 2008; Cai 2007 for example. Theorem 2.1 considers only the interior point in the interval 0, 1 and the case of the boundary points can be dealt with analogously see Theorem 4 in Cai 2007 for example. Remark 2.3. The asymptotic distribution for the case of EX it 0 d can be obtained as a corollary of Theorem 2.1. Note that 1.1 can be rewritten as Y it = f t + X it β t + α i + e it = f t + X it β t + α i + e it, where ft = f t + EXit β t and X it = X it EX it. By Theorem 2.1, we can establish the asymptotic distribution for the averaged local linear estimator of f τ, β τ. Then by the Cramér Wold device, we can further obtain an asymptotic distribution for the averaged local linear estimate of fτ, β τ by noting that f τ, β τ = 1 EX it 0 d I d fτ, β τ. Remark 2.4. The asymptotic distribution 2.6 is established by letting T. This implies that two cases are included: i T tends to infinity and N is fixed and ii both T and N tend to infinity. It can be seen from 2.7 and 2.8 that the rate of convergence of βτ is slower than that of fτ, which is confirmed by the simulation study in Section 3. To get an asymptotically more efficient estimate of β, we will introduce a local linear dummy variable approach in Section 2.2 below Local linear dummy variable approach 8

9 As shown in Theorem 2.1, the rate of convergence of the averaged local linear estimate of β is 1/2. To get an estimate that has faster rate of convergence, we propose a local linear dummy variable approach. Recently, Su and Ullah 2006 considered a profile likelihood dummy variable approach in partially linear models with fixed effects, and Sun, Carroll and Li 2009 discussed a local linear dummy variable method in varying coefficient models with fixed effects. We now propose using a nonparametric dummy variable technique for the trending time varying coefficient panel data model 1.1. Note that 1.1 can be rewritten as Ỹ = f + BX, β + D 0 α 0 + ẽ, 2.9 where Ỹ = Y1,, YN, ẽ = e 1,, e N, Y i = Y i1,, Y it, e i = e i1,, e it, f = I N f 1,, f T = I N f, BX, β = X11β 1,, X1T β T, X21β 1,, XNT β T, α 0 = α 1,, α N, D 0 = I N I T, in which I k is a k dimensional vector of ones and f is as defined in Section 2.1. As N α i = 0, 2.9 can be further rewritten as i=1 where α = α 2,, α N and D = Ỹ = f + BX, β + Dα + ẽ, 2.10 I N 1, I N 1 IT. The two step algorithm for the local linear dummy variable method is described as follows. Step i: For given β and 0 < τ < 1, we first solve the optimization problem: where Kτ = I N W τ. min Ỹ f BX, β Dα Kτ Ỹ f BX, Dα β, 2.11 α Taking derivative of 2.11 with respect to α and setting the result to zero, we obtain α := ατ = D Kτ D 1 D Kτ Ỹ f BX, β. 9

10 Step ii: Letting α in 2.11 replaced by α and based on the Taylor expansion 2.3, we obtain the local linear estimators of β τ and β τ through minimizing the weighted least squares: min a,b Ỹ Dτa, b W τ Ỹ Dτa, b, 2.12 where D τ = D 1 τ,, D N τ with 1 Xi1 D i τ =.. 1 XiT 1 τt. T τt 1 τt X i1. T τt X it W τ = W τkτwτ and Wτ = I NT D D Kτ D 1 D Kτ. Observe that for any τ, Wτ Dα = 0. Hence, the fixed effects term Dα is eliminated in By simple calculation, we obtain the solution to the minimization problem 2.12 as β τ = [I d+1, 0 d+1 ] [ D τw τ Dτ] 1 D τw τỹ. 2.13, β τ is the local linear dummy variable estimator of β τ and its asymptotic distribution is given in the following theorem. Theorem 2.2. Consider models 1.1 and 1.2. Suppose that A1 A4 are satisfied. For given 0 < τ < 1, as T, N β τ β τ bτh 2 + o P h 2 d N In particular, and N fτ fτ b f τh 2 + o P h 2 d N N βτ βτ b β τh 2 + o P h 2 d N 0 d+1, 1 X Λ X 1 X , ν 0 σe c e t 0 d, Σ 1 X ν σeσ 2 X + 2 c X tc e t Σ 1 X Remark 2.5. When E [X it ] 0 d, define X = [ ] 1 E Xit E [X it ] Σ X, 10

11 Λ X = ν 0 σe c e t σe c e t EX it σ 2 e + 2 c e t EXit σeσ 2 X + 2 c X tc e t. Then, the asymptotic distribution in 2.14 still holds if X and Λ X are positive definite and X and Λ X are replaced by X and Λ X, respectively. On the other hand, if either X or Λ X is non-positive definite, we need to use the transformation in Remark 2.3 to obtain the estimates of f and β. Remark 2.6. As in Theorem 2.1, both N being fixed or N tending to infinity are allowed in Theorem 2.2. Moreover, the local linear dummy variable estimates of both f and β have a rate of convergence of N 1/2. This implies that the local linear dummy variable estimate of β is asymptotically more efficient than the averaged local linear estimate of β. In addition, as the averaged local linear method, the individual fixed effects are eliminated in the estimation procedure without taking the first difference, and the fixed effects do not affect the asymptotic distributions of the two estimates. 3. Simulations In this section, we provide a simulated example to compare the small sample behavior of the proposed local linear estimation methods with that of the ordinary local linear approach ignoring the fixed effects in nonparametric random effects panel data models. Throughout this section, we use a uniform kernel function of the form Ku = 1 2 I [ 1,1]x and apply the leave-one-out cross validation method to choose the bandwidth involved in each of the local linear estimates. Example 3.1. Consider a trending time varying coefficient model of the form Y it = ft/t + βt/t X it + α i + e it, i = 1,, N, t = 1,, T, 3.1 where fu = u 2 + u + 1, βu = sinπu, {X it } is generated by the AR 1 process X it = 1 2 X i,t 1 + x it, t 1, X i,0 = 0, {x it, t 1} is a sequence of independent and identically distributed N0, 1 random variables, {x it, t 1} is independent of {x jt, t 1} for i j, N 1 α i = θ 0 X i + u i, i = 1,, N 1, α N = α i, i=1

12 X i = 1 T X it, θ 0 = 0, 1, 2, {u i, i 1} is a sequence of independent and identically distributed N0, 1 random variables, {e it } is an AR1 process generated by e it = 1 2 e i,t 1 + η it, {η it, t 1} is a sequence of independent and identically distributed N0, 1 random variables, {η it, t 1} is independent of {η jt, t 1} for i j, and {η it, 1 i N, 1 t T } is independent of {X it, 1 i N, 1 t T }. Model 3.1 is similar to the simulated example in Sun, Carroll and Li 2009 and they studied the case of random design time varying coefficient panel data models. It is easy to check that 3.1 becomes the random effects case when θ 0 = 0 in 3.2. Otherwise θ 0 = 1, 2, 3.1 is the fixed effects panel data model. We next compare three classes of local linear estimation methods: the averaged local linear estimates ALLE, local linear dummy variable estimates LLDVE and the ordinary local linear estimates OLLE ignoring the fixed effects. We compare the average mean squared errors AMSE of the three estimators. The AMSE of an estimator f of f is defined as AMSE f = 1 R R r=1 1 T 2 f r t/t ft/t, where f r denotes the estimate of f in the rth replication and R is the number of replications which is chosen as R = 500 in our simulation. estimator of β is AMSE β = 1 R R r=1 1 T 2 β r t/t βt/t. Similarly, the AMSE of an The simulation results for estimates of f and β with different values of N and T N, T = 5, 10, 20 and with both random effects θ 0 = 0 and fixed effects θ 0 = 1, 2 are summarized in Tables 3.1a c and Tables 3.2a c. Tables 3.1a c contain the AMSEs and their standard deviations in parentheses of the three estimates of f for the three cases of θ 0 = 0, 1 and 2, and those of β are listed in Tables 3.2a c. From Tables , we can see the performances of all the three estimates of the trend function f are satisfactory even when both N and T are small. And all of the three estimates of f improve as either T or N increases. However, when we observe the simulation results for estimates of β, we observe that when keep T fixed, the performance of the average local linear estimate of the coefficient function β does not improve as N 12

13 increases. But as T increases, its performance improves significantly. In contrast, the local linear dummy variable estimate and the ordinary local linear estimate perform better and better as either N or T increases. This confirms the asymptotic theory given in Section 2: the rate of convergence of the averaged local linear estimate of β, which is 1/2, is slower than that of the local linear dummy variable estimate, which is N 1/2. However, the averaged local linear estimate and the local linear dummy variable estimate of f have the same rate of convergence of N 1/2. The simulation results also reveal that in the random effects case θ 0 = 0, the performances of the local linear dummy variable estimate and the ordinary local linear estimator are comparable with the local linear dummy variable estimate performing slightly better than the ordinary local linear estimate. As θ 0 increases, the performance of the ordinary local linear estimate becomes worse, while the performance of the dummy variable local linear dummy variable estimate isn t influenced by the increase of θ 0 and remains quite satisfactory. 13

14 Table 3.1a AMSE for estimators of f with random effects θ 0 = 0 AMSE for estimators of f N\T ALLE LLDVE OLLE ALLE LLDVE OLLE ALLE LLDVE OLLE Table 3.1b AMSE for estimators of f with fixed effects θ 0 = 1 AMSE for estimators of f N\T ALLE LLDVE OLLE ALLE LLDVE OLLE ALLE LLDVE OLLE

15 Table 3.1c AMSE for estimators of f with fixed effects θ 0 = 2 AMSE for estimators of f N\T ALLE LLDVE OLLE ALLE LLDVE OLLE ALLE LLDVE OLLE Table 3.2a AMSE for estimators of β with random effects θ 0 = 0 AMSE for estimators of β N\T ALLE LLDVE OLLE ALLE LLDVE OLLE ALLE LLDVE OLLE

16 Table 3.2b AMSE for estimators of β with fixed effects θ 0 = 1 AMSE for estimators of β N\T ALLE LLDVE OLLE ALLE LLDVE OLLE ALLE LLDVE OLLE Table 3.2c AMSE for estimators of β with fixed effects θ 0 = 2 AMSE for estimators of β N\T ALLE LLDVE OLLE ALLE LLDVE OLLE ALLE LLDVE OLLE

17 4. Conclusions and Discussion We have considered a nonparametric time varying coefficient panel data model with fixed effects. Two classes of nonparametric estimates have been proposed and studied. The first one is based on an averaged local linear estimation method while the second one relies on a local linear dummy variable approach. Asymptotic distributions of the proposed estimates have been established with the second estimation method providing a faster rate of convergence than the first one. Some detailed simulation results have been provided to illustrate the asymptotic theory and support the finite sample performance of the proposed estimates. There are some limitations in this paper. The first one is the assumption on cross sectional independence. One future topic is to extend the discussion of Robinson 2008 to model 1.1 under cross sectional dependence. The second one is that there is no endogeneity between {e it } and {X it }. Another future topic is to accommodate such endogeneity in a general model. 5. Acknowledgments The authors would like to thank Professor Peter Phillips for his constructive comments when the authors discussed the paper with him during his recent visit to Adelaide in February Thanks also go to the Australian Research Council Discovery Grant Program under Grant Number: DP for its financial support. Appendix: Proofs of the main results In this section, we provide the proofs for the asymptotic results in Section 2. We first give a central limit theorem on weighted sums of α mixing processes, which can be derived from Theorem 2.2 in Peligrad and Utev Lemma A.1. Suppose that {a nk, k = 1,, n, n 1} is a triangular array of real numbers satisfying sup n n a 2 nk < and k=1 and {X k } is a stationary α mixing sequence satisfying max a nk 0 1 k n [ E[X k ] = 0 and E X k 2+δ] < for some δ > 0. 17

18 n 2 Furthermore, E a nk X k 1 and k 2/δ α k <, where α k is the α mixing coefficient. Then, we have k=1 k=1 n d a nk X k N0, 1. k=1 We now provide the detailed proof of Theorem 2.1. Proof of Theorem 2.1. We only consider the case when T and N tend to infinity simultaneously. The proof for the case when T tends to infinity and N is fixed is similar and we therefore omit the details here. Note that β τ = [ 1 [I d+1, 0 d+1 ] D τw τdτ] D τw τy = [ 1 [I d+1, 0 d+1 ] D τw τdτ] D τw τ BX, β + e A.1 and β τ β τ = { [ } 1 [I d+1, 0 d+1 ] D τw τdτ] D τw τbx, β β τ [ 1 + [I d+1, 0 d+1 ] D τw τdτ] D τw τe where BX, β = =: I NT 1 + I NT 2, f 1 + X 1 β 1,, f T + X T β T. A.2 Note that 2.7 and 2.8 can be derived from 2.6. Hence, we only provide the detailed proof of 2.6, which follows from the following three propositions. Proposition A.1. Suppose that A1, A2 i, ii and A4 are satisfied. Then, as T, N simultaneously, PNT D τw τdτp NT Λ µ X = o P 1, A.3 1 where P NT = diag, N I 1 d,, N I d, Λ µ = diag µ 0, µ 2 and X is as defined in Section 2. Proof. Observe that P NT D τw τdτp NT = X t X t K t τt X t X t τt t K t τt X t X t X t X t t τt t τt K t τt 2 K t τt, 18

19 where X t = PNT 1, X t and P 1 NT = diag, N I d. We need only to prove that X t X t K t τt = µ 0 X + o P 1, A.4 since the proofs for the other components in PNT D τw τdτp NT are analogous. For simplicity, define Q t = X t X t, K t = K t τt and X it = PNT 1, X it. By A2iii, we have N N 2 E N X it i=1 i=1 X it = 1 0 d 0 d Σ X = X. By A1 and the above equation, it is easy to check that as T, N, E [Q t K t ] = 1 N 2 = 1 = µ 0 X K t E X it X it i=1 i=1 K t E X N 2 it X it i=1 i=1 1 + O 1. A.5 We then consider the variance of Q t K t. Note that T Var Q t K t = Var Q t K t + Cov Q t K t, Q s K s s t =: Π NT 1 + Π NT 2. It follows from A2 iii that [ sup E X t 22+γ ] C 2+γ, A.6 1 t T for some 0 γ δ and some constant C > 0, where δ is as defined in A2. Furthermore, by A1 and A.6 with γ = 0, we have Π NT 1 = T Kt 2 VarQ t = T Kt 2 Var X t X t C T 2 h 2 Kt 2 = O 1. A.7 19

20 Meanwhile, for Π NT 2, notice that Π NT 2 = T 0< s t q T Cov Q t K t, Q s K s + T = Π NT 3 + Π NT 4, s t >q T Cov Q t K t, Q s K s A.8 where q T and q T = o. As {X i, i 1} is cross sectional independent and for each i, {X it, t 1} is stationary and α mixing, both {X t } and {Q t } are still stationary and α mixing with the same mixing coefficient α t. As a consequence, we are able to employ some existing results for α mixing processes. By A1, A2 iii, A.6 and Corollary A.2 in Hall and Heyde 1980, we have Π NT 4 C T K t C K T 2 h 2 t α δ/2+δ k k>q T as q T and α k = Ok τ for τ > δ + 2/δ. s t >q T Cov Q t, Q s Since K is Lipschitz continuous by A1, we have Standard calculation gives = o 1, A.9 K t K s C q T when t s q T. A.10 Π NT 3 = T Kt 2 T K t Cov Q t, Q s 0< s t q T K t K s Cov Q t, Q s 0< s t q T =: Π NT 5 + Π NT 6. By A.10 and the fact that CovQ 1, Q t = O 2, we have t=2 Π NT 6 Cq T T 3 h 3 Furthermore, similarly to the proof of A.7, we can show that Π NT 5 C T 2 h 2 In view of A.7 A.9 and A.11 A.13, T Var Q t K t = O 20 A.11 1 K t = o. A.12 1 Kt 2 = O. A A.14

21 By A.5 and A.14, we have shown that A.4 holds. The proof of Proposition A.1 is completed. Proposition A.2. Suppose that A1, A2 iii, A3 and A4 are satisfied. Then, as T, N, I NT 1 = 1 2 µ 2β τh 2 + o P h 2. Proof. By A3 and Taylor expansion, we have β t/t β τ + β τt/t τ β τt/t τ 2. A.15 Equation A.15 then follows from the definition of I NT 1, the above equation and Proposition A.1. Proposition A.3. Suppose that A1, A2 and A4 are satisfied. Then, as T, N, D NT I NT 2 d N 0 d+1, 1 X where D NT and Λ X are defined as in Theorem 2.1. Proof. By A2 iiii, it is easy to check that Λ X 1 X, A.16 E I NT 2 = 0 d+1. A.17 We then calculate the variance of D NT I NT 2. Noticing that N 2 E X t X t e 2 t = N 2 N 4 E N X it Xjt N e kt = 1 N 2 = Σ X σ 2 e i=1 i=1 k=1 j=1 k=1 EX it X it Ee 2 kt l=1 e lt and N 2 Cov X t e t, X s e s = E X it Xis E e it e is = c X t s c e t s, we have as T, N, N 2 T T Var X t e t σeσ 2 X + 2 c X tc e t. A.18 1 By A.18, the fact that Kt 2 = ν 0 + o P 1 and following the proof of A.14, we have N 2 T Var K t X t e t = ν 0 σeσ 2 X + 2 c X tc e t + o1. A.19 21

22 and Analogously, we have N T Var K t e t = ν 0 σe c e t + o1 A.20 N 3/2 T E Kt 2 X t e 2 t = 0 d. A.21 Furthermore, by A.19 A.21 and Proposition A.1 and noting that D NT I NT 2 = D NT [I d+1, 0 d+1 ] P NT [P NT D τw τdτp NT ] 1 P NT D τw τe, we have Var D NT I NT 2 = 1 X Λ X 1 X + o1. A.22 By A2, A.17, A.22 and Lemma A.1, we have shown that A.16 holds. Proof of Theorem 2.2. Note that [ β Dτ] 1 τ β τ = [I d+1, 0 d+1 ] D τw τ D τw τỹ β τ { [ Dτ] 1 = [I d+1, 0 d+1 ] D τw τ D τw τ f + BX, } β β τ + [I d+1, 0 d+1 ] [ D τw τ Dτ] 1 D τw τ Dα + [I d+1, 0 d+1 ] [ D τw τ Dτ] 1 D τw τẽ =: Ξ NT 1 + Ξ NT 2 + Ξ NT 3. By the definition of W τ, we have A.23 W τ = W τkτwτ = Kτ Kτ D D Kτ D 1 D Kτ, which implies that Ξ NT 2 [ Dτ] 1 = [I d+1, 0 d+1 ] D τw τ D τkτ Dα [ Dτ] 1 [I d+1, 0 d+1 ] D τw τ D τkτ D D 1 D Kτ D Kτ Dα [ Dτ] 1 = [I d+1, 0 d+1 ] D τw τ D τkτ Dα [ Dτ] 1 [I d+1, 0 d+1 ] D τw τ D τkτ Dα 0 d+1. 22

23 Therefore, in order to establish the asymptotic distribution 2.14 in Theorem 2.2, we need only to consider Ξ NT 1 and Ξ NT 3. The following propositions establish the asymptotic properties of Ξ NT 1 and Ξ NT 3, which lead to And 2.14 implies 2.15 and Theorem 2.2 holds for both N being fixed and N tending to infinity, but we only give the proof for the case of N tending to infinity simultaneously with T. The proof for fixed N is similar. Proposition A.4. Suppose that A1, A2 i, ii and A4 are satisfied. Then, as T, N simultaneously, where Λ µ is defined as in Proposition A.1. Proof. By the definition of W τ, we have 1 N D τw τ Dτ Λ µ X = o P 1, A.24 D τw τ Dτ = D τkτ Dτ D τkτ D D Kτ D 1 D Kτ Dτ. Following the proof of Proposition A.1, we have 1 N D τkτ Dτ = Λ µ X + o P 1. A.25 A.26 In view of A.25, we need only to prove D τkτ D D Kτ D 1 D Kτ Dτ = o P N. A.27 To do so, we first consider the term D Kτ D. Define Z T = T definition of D and standard calculation, we have 2Z T Z T Z T D Kτ D =... Z T Z T 2Z T Z T Z T Z T =... Z T Z T Z T K t T τ. By the + diag Z T,, Z T. Letting A = diag Z T,, Z T, B = Z T,, Z T, C = 1 and D = ZT,, Z T, then D Kτ D = A + BCD. Noticing that A + BCD 1 = A 1 A 1 BDA 1 B + C 1 1 DA 1, A.28 23

24 which can be found in Poirier 1995, we have 1 Z T 1 NZ T 1 NZ T 1 NZ T D Kτ D NZ = T Z T 1 NZ T 1 NZ T. A NZ T 1 1 NZ T Z T 1 NZ T Define Z T i = T K t T τ X it and Z T i = T t T τ K t T τ X it. By standard arguments, we have D τkτ D Z T 2 Z T 1 Z T 3 Z T 1 Z T N Z T 1 = Z T 2 Z T 1 Z T 3 Z T 1 Z T N Z T 1 which together with A.29 implies that D τkτ D D 1 D Kτ D Kτ Dτ = A NT 0 C NT D NT 0 B NT, A.30 where and A NT = B NT = C NT = D NT = A NT kz T k Z T 1, k=2 k=2 k=2 k=2 A NT k = Z T k N 1 NZ T 1 NZ T B NT k = Z T k N 1 NZ T 1 NZ T B NT k Z T k Z T 1, A NT k Z T k Z T 1, B NT kz T k Z T 1, j=1, k j=1, k Z T j = Z T k Z T 1 NZ T Z T j = Z T k Z T 1 NZ T Z T i i=1 Z T i. i=1 24

25 By the consistency of the nonparametric estimator in the fixed design case, we have for each i 1, 1 Z T = µ 0 + o P 1, 1 Z T i = o P 1, 1 Z T i = o P 1. A.31 By A.30 and A.31, we have shown that A.27 holds. The proof of Proposition A.4 is completed. Proposition A.5. Suppose that A1, A2 iii, A3 and A4 are satisfied. Then, as T, N simultaneously, Ξ NT 1 = 1 2 µ 2β τh 2 + o P h 2 = bτh 2 + o P 1. A.32 Proof. By A3, Taylor expansion of β at τ and Proposition A.4, we can show that A.32 holds. Proposition A.6. Suppose that A1, A2 and A4 are satisfied. Then, we have NΞNT 3 d N 0 d+1, 1 X Λ X 1 X A.33 as T, N simultaneously. Proof. By Propositions A.4 and A.5, to prove A.33, it suffices to prove 1 D τw τẽ d N 0 2d+2, Λ ν Λ X, N A.34 where Λ ν = diag ν 0, ν 2. By the definition of W τ, we have D τw τẽ = D τkτẽ D τkτ D D Kτ D 1 D Kτẽ. A.35 By A.35, it is enough to show that 1 D τkτẽ d N 0 2d+2, Λ ν Λ X. N A.36 and Define D τkτ D D Kτ D 1 D Kτẽ = o P N. A.37 t T τ L j = t T τ j t T τ K 25

26 for j = 0, 1. To prove A.36, we need only to prove By A2, we have 1 N N i=1 t T τ 1 L j X it e it 1 N Var t T τ 1 L j i=1 X it d N 0 d+1, ν 2j Λ X. e it = ν 2j Λ X + o1. A.38 A.39 Denote {ξ j, j = 1,, NT } = {X 11, e 11,, X 1T, e 1T, X 21, e 21,, X NT, e NT }, then {ξ} is stationary and mixng with α mixing coefficient α k or 0, k < T, α k = 0, k T. By A.39 and Lemma A.1, A.38 holds. We then turn to the proof of A.37. Let e T i = T of A.30, we have K t T τ e it. Following the proof D Kτẽ = e T 2 e T 1,, e T N e T 1. A.40 By A.29, A.30 and A.40, we have D τkτ D D Kτ D 1 D Kτẽ = 0, U NT, 0, V NT, A.41 where U NT = A NT ke T k e T 1 = A NT ke T k k=2 k=1 and V NT = B NT ke T k e T 1 = B NT ke T k, k=2 k=1 in which A NT k and B NT k are defined in the proof of Proposition A.4. By A2 i and iii, we can show E UNT 2 = on and E VNT 2 = on, which implies U NT = o P N and V NT = o P N. A.42 26

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