Nonparametric Estimation of the Marginal Effect in Fixed-Effect Panel Data Models

Transcription

1 Nonparametric Estimation of the Marginal Effect in Fixed-Effect Panel Data Models Yoonseok Lee Debasri Mukherjee Aman Ullah October 207 Abstract This paper considers local linear least squares estimation of panel data models when fixed effects present. The local marginal effect is of the main interest. A within-group type nonparametric estimator is developed, where the fixed effects are eliminated by subtracting individual-specific locally weighted time average (i.e., using the local within transformation). It is shown that the local-within-transformation-based estimator satisfies the standard properties of the local linear estimator. In comparison, the nonparametric estimators based on the conventional (i.e., global) within transformation or first difference result in biased estimators, where the bias does not degenerate even with large samples. The new estimation is applied to examine the nonlinear relationship between the U.S. income and nitrogen-oxide level (i.e., the environmental Kuznets curve) using U.S. state-level panel data. Keywords: Nonparametric estimation, panel data, fixed effects, local linear least squares, local within transformation, environmental Kuznets curve. JEL Classifications: C4, C23 First Draft: May Corresponding author. Department of Economics and Center for Policy Research, Syracuse University, Syracuse, NY 3244; ylee4@maxwell.syr.edu. Economics Department, Western Michigan University, Kalamazoo, MI 49008; debasri.mukherjee@wmich.edu. Department of Economics, University of California, Riverside, CA 9252; aman.ullah@ucr.edu.

2 Introduction Recently much attention has been given to nonparametric fixed effect panel data models. The main interest is in identifying and estimating the nonparametric component when the unobserved individual effects can be correlated with the observed covariates. Like the nonlinear parametric case (e.g., Hahn and Newey, 2004; Arellano and Hahn, 2007; Lee and Phillips, 205), the nonparametric estimator suffers incidental parameters problem particularly with large cross section but small time series observations, unless fixed effects are not properly treated. Even when both and are large, it is required that is finite in the limit in order to obtain asymptotic normality. A limited list of recent studies on nonparametric conditional mean estimation of fixed effect regression models includes Ullah and Roy (998), Mukherjee (2002), Henderson and Ullah (2005), Su and Ullah (2006), Henderson et al. (2008), Evdokimov (200), Qian and Wang (202), Su and Jin (202), Lee (204), and recent survey articles by Ai and Li (2008), Su and Ullah (20), and Li, Sun and Zhang (203). When the fixed effect is separable from the nonparametric component, an easy recipe is to eliminate it using first difference or within transformation. Such transformations are most likely to alter the structure of the nonparametric part, so some additional normalization assumptions (e.g., Su and Ullah, 2006) or some additional steps (e.g., Qian and Wang, 202) are required to obtain the desired estimate, unless estimation is conducted using the series estimation (e.g., Su and Jin, 202; Lee, 204). In this paper, we focus on kernel-based estimation using the local linear least squares, but we assume a weaker normalization condition than that of Su and Ullah (2006) or Henderson et al. (2008). We investigate if the standard (global) within-transformation or the first-difference approach can be used in the panel nonparametric estimation when kernel-based (local) estimation is considered. In particular, we analytically compare the conventional within transformation approach with the local within transformation (i.e., the fixed effects are eliminated by subtracting individual-specific locally weighted time average; e.g., Mukherjee, 2002). Showing some pitfalls of the conventional within transformation approach in this context, we bring attention from empirical researchers using panel semiparametric estimation. More precisely, we consider local linear least squares estimation, where the fixed effects are eliminated by the local within transformation, which can be also obtained from local concentration of the fixed effects. The panel observations can have unknown serial correlation as long as it satisfies

3 some mixing conditions. The local marginal effect is of the main interest, which corresponds to the first derivative of the nonparametric function. It is shown that the local-within-transformationbased estimator satisfies the standard asymptotic properties of the local linear estimator. On the other hand, the nonparametric estimators based on the conventional within transformation or the first difference result in biased estimators, where the bias does not degenerate even with large samples. These results are obtained under ( )-asymptotics, but they do not require that converges to some nonzero and finite constant as in the standard large panel studies. Interestingly, the results hold as long as converges to some finite value that includes zero, and hence can be quite large comparing with. As an empirical illustration, the new estimation is applied to analyze the nonlinear relationship between pollution and income (i.e., the environmental Kuznets curve) using U.S. state-level panel data on nitrogen oxide levels. The estimation results demonstrate inverted -shaped relationship between pollution and income in general, which affirms the environmental Kuznets curve hypothesis: At some initial phases of income, pollution level grows with income; but beyond some threshold level of income, pollution level goes down with income. The turning point is estimated as the income level of $2,60 per capita. Several studies are closely related to the kernel-based nonparametric estimation developed in this paper. Su and Ullah (2006) considers local linear version least squares dummy variables (LSDV) estimator, which treats the fixed effects using individual specific dummy variables. Henderson et al. (2008) considers first-difference approach to eliminate the fixed effects and applies iterative backfitting approach in the framework of generalized estimating equations (GEE), though the initial consistent estimator for the iteration is obtained using series estimator. Qian and Wang (202) also considers first-difference approach but retrieves the original nonparametric function through marginal integration. Comparing with these approaches, the computation cost of the new estimation method developed in this paper is much lower especially when either or is large. However, our estimation produces the marginal effect (i.e., derivatives of the unknown function), and hence unlike those studies, we need to conduct one more step to obtain the nonparametric function estimator itself though it is still computationally straightforward. The remainder of the paper is organized as follows. Section 2 introduces the locally weighted within transformation and develops new local linear estimation. Section 3 derives the asymptotic distribution of the new estimator as well as its approximate bias and variance; a discussion on bootstrap confidence interval is also included. Section 4 compares this new estimator with the local 2

4 linear estimator using the conventional (global) within transformation and shows the conventional approach results in non-degenerating bias even in the limit. Section 5 applies the new estimator to examine the nonlinear relationship between the U.S. income and nitrogen-oxide level (i.e., the environmental Kuznets curve) as an illustration. Section 6 concludes this paper with some remarks. All the mathematical proofs are collected in Appendix. 2 Fixed-Effect Local Linear Estimation and Local Within Transformation We suppose that we observe a stationary process ( ) ( 2 2 ) ( ) for each satisfying E[ ]=( )+ () for =2 and =2 almost surely. : X R is an unknown Borel measurable function normalized as (0) = 0, wherex R is the support of,and is unobserved heterogeneity that can be arbitrarily correlated with (i.e., a fixed effect). The effects from, which is possibly nonlinear, is described by ( ) whereas some unobserved (and possibly omitted) individual specific characteristics on the level of is controlled by. In this partially linear specification, the nonlinear functional form ( ) is assumed to be common over and, andthe individual specific effects are additively separable from this unknown function similarly as Su and Ullah (2006) and Henderson et al. (2008). The main interest is in estimating local marginal effects 0 () = () for a given X. Assuming ( ) is smooth enough, we can Taylor expand ( ) around X so that one can carry a local linear regression by minimizing the following objective function (e.g., Fan and Gijbels, 992; Ruppert and Wand, 994): ( 0 ; )= ( 0 ( ) ) 2 ( ), (2) = = where ( ) = (( )), ( ) is a non-negative kernel function and is a bandwidth parameter. When the model includes fixed effects as (2), the incidental parameter problem prevails as increases (e.g., Neyman and Scott, 948) and the conventional approach is to eliminate We can extend the model to the case of multi-dimensional with a properly defined kernel function and a bandwidth parameter vector, but we focus on the one-dimensional case for notational simplicity. 3

5 fixed effects before estimation. By concentrating out in (2) with 0 and being fixed, we obtain the concentrated objective function given as ( )= = = () () 2 ( ), (3) where, for the weight () = ( ) P = ( ), wedefine () = () (4) and similarly for (). Notethat () is nonnegative for all and P = () =for any. The local linear estimator b () is then obtained as P = = b () =P () () ( ) P, (5) = ( ())2 ( ) P = = which estimates 0 (). 2 The local linear estimator b () in (5) can be alternatively understood as the within-group local linear estimator of () on (). The transformed variables () and () are deviations from the individual specific local means, or locally within-transformed variables, if we understand P = () in (4) as the locally weighted average of over. This local within transformation is different from the conventional within transformation (i.e., P = ), which subtracts the individual global means. More precisely, if we let = E[ ],wehaveapanel partially linear regression model given by = ( )+ +. (6) In this case, if we linearize ( ) around X, the regression model (6) becomes = ()+( ) 0 ()+ + () (7) with () = + (), where the remainder term is given as () = 00 () 2 ( ) 2 (8) 2 In practice, we could consider the leave-one-out local-within-transformation (i.e., () = 6= () with () = ( ) 6= ( )) inordertohavesmallerfinite sample bias. Furthermore, in order to prevent the denominator from being close to zero, we could consider = = ( ()) 2 ( )+( ) 2 for the denominator in (5) as Fan (992), but such modification does not change the main results. 4

6 with 00 () = 2 () 2 and for some X between and. The local within transformation on (7) then eliminates the fixed effects and yields () = 0 () ()+ () for P = () = by construction, where we let () = ()+ () and () = 00 () 2 ( ( ) 2 ) ( ) 2 (). (9) Apparently, b () in (5) corresponds to the local linear least squares estimator b 0 () of this transformed regression model. Though we focus on estimation of the marginal effect 0 () in this paper, one could be interested in the estimator of (). However, the local constant term 0 in the approximation (2) is deleted from the concentration step (3); or equivalently, the term () in (7) is deleted from local within-transformation. Therefore, in this additively separable fixed-effect framework, estimates of the original function ( ) cannot be obtained directly from the local linear or local polynomial estimation, which is distinct from the standard local linear least squares estimation. Under the normalization condition (0) = 0, however, we can obtain an estimate for () as = b () = (b () b (0)), where b () = = = ³ b ()( ) () since from the expression (7) = = = ()+ 0 ()( ) () 0 (0) (0) ª = = { () () (0) (0)} and the second term can be verified as () from Lemma A.2 in the Appendix. Alternatively, one could recover () using the Riemann sum approximation as b () = Z b 0 () 2 () X =2 [] [ ] ³ b [] + b [ ] and redefine b () as b () b (0), where [()] = and [] [() ] [()] [()+] [ ] are sorted observations from the pooled data { } = ;=. 5

7 3 Statistical Properties In order to derive the asymptotic properties of b () in (5), we firstassumethefollowingconditions similarly as Masry and Fan (997), where we only consider the second order kernel. For notational simplicity, we denote as in what follows. For each, we define the -algebra F 2 = ({ } 2 = ) with 2 for the stationary process { }.Wealsodefine () =sup sup F 0 F () () () as the strong mixing (Rosenblatt, 956) coefficient of the panel process { }, which goes to zero as. Assumption A (A) ( ) is Borel measurable and twice continuously differentiable at in the interior of X with bounded derivatives, where X is the support of that is bounded. (A2) The fixed effect is i.i.d. with mean zero and finite variance. (A3) The process { } is independent across but strong mixing over with mixing parameter () satisfying P = () 2 for some 2. (A4) The density of satisfies 0 () and twice continuously differentiable with bounded second order derivatives for in the interior of X. (A5) is independent across andseriallyuncorrelatedover conditional on and = ( ) 0. It also satisfies E[ ] = 0, E[ 2 ] = 2 ( ) (0 ) and E[ 4+ ] for some 0, where 2 ( ) is continuous in the interior of X. (A6) The kernel function is compactly supported, bounded, and symmetric; it also satisfies R () =and R 2 () 6= 0. 3 (A7) The bandwidth parameter satisfies 0, and = () as. (A8) There exists a sequence of positive integers satisfying and = (( ) 2 ) such that () 2 ( ) 0 as. 3 It follows that () for =2 4 and = 2 3, and () for = 2 and for some 2, which are frequently used in the main proof. 6

8 Note that we assume the panel process { } as a stationary mixing process so that we allow for moderate level of serial correlations. On the other hand, they are cross sectionally independent as in the standard panel literature. For, it is assumed to be serially uncorrelated, which can be relaxed provided that its (conditional) serial correlations are uniformly bounded. The condition (A7) implies =( ) 2 and 3 =( )() 2,which resembles conditions for the standard local linear estimator. Though we assume large panel so that both and go to infinity here, we do not impose an explicit condition on the limit ratio between and as in the standard large penal data literature, particularly whether or not lim [0 ). In fact, we can allow for to be quite small relative to sinceweonlyneed = () and hence it is possible that = () 0. Therefore, the large condition here is much weaker than the standard large panel data regression literature (cf. Hahn and Kuersteiner, 2002; Lee, 202), in the sense that can be quite large relative to. The condition (A8) is required to obtain the CLT as Masry and Fan (997). The rest of the conditions are quite standard for the local polynomial estimator in the time series setup. The first result shows that the local linear least squares estimator () b = b 0 () in (5) follows asymptotic normal distribution. Theorem. Under Assumption A, we have 3 ( () b 0 () ()) N (0()) for in the interior of X,where () = 2 00 () 0 () 2 () 4 2 and () = 2 () () with = R () for =2 4 and 2 = R 2 2 (). For later use, we summarize the approximate bias and variance expressions of b () in the following corollary. Corollary. Under Assumption A, we have for in the interior of X. E[ b () 0 ()] = 2 2 [ b () 0 ()] = 3 00 () 0 () () 2 () () , (0) () Note that, even after the local within transformation, the asymptotic orders of the bias and the variance are the same as those of the standard local polynomial estimator (e.g., Fan and Gijbels, 7

9 992; Ruppert and Wand, 994; Masry and Fan, 997). Since 0 and 3 as, both the bias (0) and the variance () are asymptotically negligible, which yields b () 0 () for each as. From (0) and (), the optimal bandwidth parameter can be obtained as =( ) 7 " by minimizing the approximated mean squared error h i b () = Z 00 () 0 () 3 2 R ( 2 () ()) () 2 4 R ( 00 () 0 () ()) 2 () 2 () + 3 # 7 (2) Z 2 () (), () where () is some positive weight function chosen to ensure that the integral converges. Remark (Bootstrap Confidence Interval) It can be challenging to estimate () and (), and so can be the confidence bounds of b () in practice. We can use bootstrap to construct the pointwise confidence intervals instead, where we undersmooth the bootstrap estimates to reduce the bias. One caveat is that, though we assume cross sectional independence, we still allows for serial correlations of the observations. In order to preserve the (unknown) serial dependence structure, we bootstrap the vectors of the fitted errors over, instead of bootrapping for each and. More precisely, we can summarize the bootstrap procedure as follows: Using the local linear estimate b (), weobtainb () = () b () () for each and let b () =(b () b ())0. We recenter them by subtracting () P = b () for each. For the case of residual-based bootstrap, we resample from {b ()} = with replacement to obtain the bootstrap residuals, say {b ()} =, where stands for the size of the boostrapped samples. We then let () =b () ()+b () and using the boostrapped sample { () ()} we redo local linear estimation in (5) for each with undersmoothing to obtain b (). We repeat this procedure over =2 and construct the pointwise confidence interval as [2 b () b ( 2) () 2 b () b (2) ()], where b ( 2) () denotes the ( 2)-percentile of the bootstrapped b () s. Remark 2 (Time Specific Fixed Effects) In general, a panel regression model (6) could include both the time fixed effects and the individual fixed effects as = ( )+ + +,where is useful to capture a common fluctuation across (e.g., a common macroeconomic fluctuation in the economy) and some degree of cross sectional dependence. Note that can be correlated with and, but it is assumed to be independent of.inthiscase, 0 () can be 8

10 estimated as P P = = () () ( ) P P, (3) = = ( ())2 ( ) where the local transformation () is defined as () = () = () + = = = () (4) with () = ( )/ P = ( ) () = ( )/ P = ( ) () = ( )/ P P = = ( ) and similarly for (). It can be understood as a local version of the transformation proposed by Wallas and Hussain (969): the transformation (4) successfully eliminates the fixed effects and in the linearized equation = ()+( ) 0 ()+ + + () of (3) asymptotically, provided is identically distributed over and stationary over. Using a similar argument as Theorem, we can readily conjecture that the approximate bias and the variance of (3) remain the same as (0) and (), provided is i.i.d. with zero mean and finite variance, and Assumption A-(A5) holds with conditioning on both and. 4 Comparison with the Standard Within Transformation It would be interesting to compare the local within transformation (4) with the conventional (nonlocal or global) within transformation on the linearized regression model (7) to eliminate the fixed effects: 0 = 0 () (), (5) where 0 () = () ( ) P = () and similarly for 0 and 0.Notethatwecanrewrite 0 () =0 + 0 () with 0 () = 00 () 2 ( ( ) 2 ) ( ) 2 for some between and, where 00 ( ) is assumed to be bounded and is chosen such that with probability approaching to one for given and ; becomes negligible as 0 for given and. Since the regressor is assumed to be strictly exogenous, therefore, the remainder term before within transformation (i.e., () =(2) 00 ()( ) 2 ) also becomes = (6) 9

11 negligible as 0, which is the case of the standard local polynomial estimation. However, the averaged remainder (2) 00 () ( ) P = ( ) 2 does not need to be small even with large (and small ) because we cannot assume is small for a given for every =2. Note that we need enough amount of variation in { } for the regression analysis to be valid. In comparison, the proposed local within transformation uses locally weighted average (2) 00 () P = ( ) 2 () that vanishes as 0 atthesamerateof () =(2) 00 ()( ) 2. It thus follows that, unlike () in (9), we cannot simply regard the approximation error 0 () in (6) to be small enough to be neglected. As a consequence, the local least squares estimator based on the standard within-transformed regression (5), P = = e () =P 0 0 ( ) P P = = (0 )2 ( ), (7) is supposed to have non-degenerating bias from the large approximation error, even under very strong assumptions such as is large and the regression errors are i.i.d. Intuitively, this bias is because the conventional within transformation equally weighs the entire history of each individual, which thus accumulates all the approximation biases in for 6=, where is chosen local to for given and. In other words, while the regression equation (7) is a local approximation, the subtracting equations (i.e., the standard within transformation) are not locally weighted. More precisely, we can compare the local within transformation with the standard within transformation by investigating their approximate biases, where the approximate bias of e () in (7) can be obtained as follows. Theorem 2. Suppose E[ 2 ] and Assumption A hold. Then E[e () 0 ()] = () for in the interior of X\{E( )}. Inparticular,when{ } is serially uncorrelated, we can obtain E[ e () 0 ()] = 00 () 2 () 2 () where () =E[( ) ] for = , (8) Note that 2 () =( )+(E[ ] ) 2 is nonzero in general. Therefore, the approximate bias (8) of e () is () and it does not vanish even when 0 unlike b () in Corollary. Even when is large, the leading bias of e () does not disappear since the approximation error in (7) prevails. Moreover, the approximate bias can be very large when is near E[ ] since () =0 when = E[ ]. Another interesting point is that, unlike b (), the bias of e () can be large when issmallbecauseofthe ( ) remaining term. This ( ) bias term can be understood 0

12 as a similar vein as the ( ) bias in the nonlinear fixed-effects models (e.g., Hahn and Newey, 2004), which is due to the estimation error of the fixed effect parameter estimates. As we show in the previous section, such problem can be solved if we use the appropriate weights when we define the within transformation. In particular, the weights should use only the local information around as we described in (4), which yields the local within transformation, so that the transformed remainder terms die out with the same speed as diminishes (i.e., () in (9)). The local within transformation imposes very little weights on the observations whose locations are far from (or ). Therefore, it eventually controls for () being small enough to be asymptotically ignored. One remark is that, if we use series approximation for nonparametric estimation instead of kernel estimation, we can eliminate the fixed effects by the standard within transformation (e.g., Lee, 204), where both transformations are global. Table I: IMSE and IMAE comparison b e b e Note: IMSE is the integrated mean squared errors and IMAE is the integrated mean absolute errors of the estimators of = 0. b is the new estimator proposed in (5) based on the local-within-transformation, whereas e is based on the standard (global) within-transformation in (7). Table I shows how the curve fitting improves using the local within transformation comparing with the conventional approach. We consider the panel regression model (6) with the sigmoid function () = ( + exp( )),whichsatisfies the smoothness condition in Assumption A., and are randomly drawn from U( 05 05), N (0 ), N ( 4), respectively. Six sets of samples of = and = are generated. The Gaussian kernel is used and the bandwidth parameter is chosen as ( ( d )) 2 ( ) 7 to satisfy = (( ) 7 ). The entire estimation procedure is repeated 500 times. The integrated mean squared errors (IMSE) and the integrated mean absolute errors (IMAE) are obtained by averaging pointwise MSE and MAE over

13 the different values. 4 Note that the overall curve fit of the new estimator b dominates that of e even with the moderate size of,andthefit notably improves as increases. In comparison, the improvement with is quite marginal for e. Another interesting finding is that the curve fit improves with for the case of,whereaswehardlyseeanyimprovementforthecaseof b e.in addition, the IMSE and the IMAE of b remain quite similar between the cases of ( ) = (00 50) and (200 25), which implies that the fitting depends on the entire number of observations rather than the individual or relative sizes of and. Remark 3 (First-difference) The discussion above extends to the first-difference-based approach. More precisely, for (7), the standard local linear estimator based on the first-differenced equation is given by P = = e () =P ( ) P P = = ( ) 2 ( ), (9) where = and =. However, this estimator is also well expected to yield non-degenerating bias similarly as e () for the same reason. An alternative estimator, which preserves the standard properties of the local linear estimator like b () in (5), can be defined as P P = = ( ) P P = = ( ) 2 ( ), where ( ) = 2 (( ) ( )) is a bivariate kernel. A product kernel (i.e., ( ) = 2 (( ))(( ))) is a simple candidate. Note that it uses the local information of as well as. 5 Empirical Illustration: Environmental Kuznets Curve As an empirical illustration, we investigate income-pollution relationship (i.e., the environmental Kuznets curve hypothesis; EKC hypothesis hereafter) using the U.S. state-level panel data. The basic hypothesis states that as income per capita rises, the level of pollution rises at the initial phase of economic growth but it falls beyond some level of income. So it suggests an inverted -shape for the relationship between income and emission levels. However, the empirical literature finds mixed results yielding different policy implications (e.g., Grossman and Krueger, 995; List 4 More precisely, we estimate the IMSE and the IMAE as = {(500) 500 = ( [+] [])( 0 ( []) ( [])) 2 } and = {(500) 500 = ( [+] []) 0 ( []) ( []) }, respectively, where [] [] [ ] are sorted at the th replication for =

14 and Gallet, 999): It is found to be inverted -shaped (i.e., containing one turning point and thus supporting the EKC hypothesis), -shaped (i.e., containing two turning points instead of one) or simply linear (i.e., no turning point). 5 Instead of imposing ex ante parametric restrictions, such as a quadratic or a cubic parametric form, we employ nonparametric approach to investigate the nature of the nonlinear relationship between income and pollution (e.g., Taskin and Zaim, 2000; Millimet et al., 2003; Bertinelli and Strobl, 2005; Azomahou et al., 2006). Since the main issue of the EKC can be summarized as to identify the number of turning points, we focus on nonparametric estimation of the marginal effect, which is based on the local linear estimator that we develop in the previous sections. We note that local linear (or polynomial) estimator is more suitable for the EKC analysis than the Nadaraya-Watson-type (local constant) estimator: Unlike the Nadaraya-Watson estimator, the local linear estimator is more robust to the boundary effect, where the inverted -shape versus -shape discussion is mostly focused on the right tail of the EKC. Following List and Gallet (999) and Millimet et al. (2003), we consider the U.S. state-level panel data, especially on the emission of Nitrogen Oxides (NO x ). It is obtained from U.S. Environmental Protection Agency s (EPA) National Air Pollutant Emission Trend, , which includes state-wise yearly observations from 929 to 994. We consider two balanced panel data subsets: one from 929 to 984 ( = 48; =56)thatreflects a large case; and one from 985 to 994 ( = 48; =0)thatreflects a small case. 6 Unlike cross-country data, we could suppose that the major economic characteristics of the observations are rather homogeneous in this U.S. state-level data set and hence we can examine a simple regression model without worrying much about other control variables if we include state-specific fixed effects in the regression model. More precisely, we consider the standard EKC regression model given by = ( )+ + +, where is the real per capita income and is the per capita pollutant (NO x in this example) emission level of state at year. ( ) is an unknown function, is the state-specific fixed effect, and is the time-specific fixed effect. The state-specific effects can capture institutional 5 -shaped EKC poses the concerning question of whether the decrease is only in local pollutants and pollution is simply exported to poorer developing countries that increases the global level of pollution. 6 Note that the original data set is available for the period of but the emission measurement methodologies differ between the period and the period So they can not be pulled together. A detailed description of the data can be found in List and Gallet (999) and Millimet et al. (2003). 3

15 Figure : EKC Estimation of NOx (year ) 4.00E 05 m'(x) 0.08 m(x) 2.00E E+00 $0 $5,000 $0,000 $5,000 $20, E E $0 $5,000 $0,000 $5,000 $20,000 particulars such as differences in state-level administrative rigidities in emission regulation. The time effect can capture common trends in the pollution abatement technologies and control for some degree of cross section dependence from common time factors. We also considered a regression model without time effects, but we found that the main conclusions remains the same qualitatively. Although pollution potentially affects income with a sufficient time lag through health hazards and other channels, income affects pollution more directly since it is highly correlated with economic development and industrialization. The regression model, in addition, includes fixed effects, which can control for possible endogeneity to some degree. Following the standard EKC studies on the U.S. data, therefore, we consider income as an exogenous regressor. One might be concerned about possible spatial correlation in the pollution data. But it is normally believed that NO x pollution is more soil-based than airborne (more specifically, it is known that more than 70% of NO x pollution is soil-based) and thus it is less likely to impose significant level of spatial correlation. Since spatial correlation tests under fixed-effect nonparametric regression is not available, we conduct a parametric panel spatial correlation test of Baltagi et al. (2003) to get some rough idea though. It fails to reject the null of no spatial correlation under the quadratic parametrization. 7 The estimation results are depicted in Figures and 2. The main interest is in the number of turning points that the income-pollution relationship ( ) has, which can be readily told by 7 Note that we consider quadratic parametrization based on the fact that our nonparametric estimation results in the inverted -shaped function. We consider two test statistics from Baltagi et al. (2003). The LM statistic (which is asymptotically 2 ) is about 60, and the standardized LM statistic (which is asymptotically N (0 )) is about 07. Testing for spatial dependence in nonparametric fixed-effect regression is beyond the scope of this paper though further works can be carried on in the lights of Robinson (20). 4

16 Figure 2: EKC Estimation of NOx (year ) 3.00E 05 m'(x) 0.02 m(x) 2.00E 05.00E E+00 $0,000 $3,000 $6,000 $9,000 $22,000.00E E E 05 0 $0,000 $3,000 $6,000 $9,000 $22, theestimateof 0 ( ). Note that the exact number and the location of the turning points can be visualized by the levels of income s at which b 0 () is close to zero. Figure shows b 0 () and b () for the period of (for relatively large ), where the -axis shows the nominal income per capita in U.S. dollar; Figure 2 shows the same estimation results for the period of (for relatively small ). The dashed lines with b 0 () depict the 95% pointwise confidence intervals that are obtained based on the procedure described in Remark in Section 3. b () are obtained using the empirical Riemann sum of b 0 () asdescribedinsection2. For the period of (Figure ), we can roughly find inverted -shaped relationship, where b 0 () crosses zero around the income level of $2,60 per capita. Though some fluctuations are observed at the right tail of the graph, but it is not statistically significant. For the period of (Figure 2), which is relatively more recent periods, we find almost a monotonically downward-sloping relation, indicating that the level of pollutant decreases steadily with an increase in income. Combining these results, we can conclude that pollution level rises with income at the first stage of development (with relatively lower level of income) but once a threshold level of income is reached, pollution level decreases with income. The current analysis supports for inverted -shaped EKC, similarly as Millimet et al. (2003). 6 Concluding Remarks This paper questions the common practice in kernel estimation of fixed-effects panel data regression models, where the fixed effects are eliminated using the standard within transformation. We demonstrate that such an approach produce biased and inconsistent estimators of the marginal 5

17 effects. We analytically show that the bias is because the conventional within transformation generates sum of non-negligible distances between each observation (for 6= ) andafixed location, which is introduced to approximate the unknown function locally about a given observation value. To overcome such problem, we propose local within transformation, which use locally weighted averages over time around the particular point. The local linear kernel estimation based on the local within transformation yields asymptotically unbiased and consistent estimator as for the conventional local polynomial estimation. We only consider the simplest form of the nonparametric fixed-effects model in this paper. Despite its simplicity, it is enough to draw attentions from the empirical researchers by presenting the problem of one of the commonly used estimator, and to explain the main intuition behind the newly proposed estimator as an alternative. The main idea in this paper can be readily generalized to the general partially linear frameworks (i.e., including other linearly additive control variables; e.g., Robinson,988). 6

18 A Appendix: Mathematical Proofs A. Useful Lemmas We first obtain several useful lemmas. The first lemma extends the standard results from kernel estimation (e.g., Fan and Yao, 2003, p.204), where the results can be shown using the Taylor expansion and the Lebesgue dominated convergence theorem. Lemma A.. Let () have the second bounded derivative that is continuous at an interior point of the support X. Assume that isafunctionsuchthat R () for integers 0 and = 2. Then,as 0, we have Z () ( + ) = ()+ + 0 () ()+ 2, where = R (). The following lemma gives the basic building blocks in proving the main theorem. We let () = (( )). Lemma A.2. Under Assumption A, for a non-negative integer 0, we have () =() () ; (A.) = and + () (A.2) = = ½ () + + = 2 00 () (if is odd) 0 () (if is even); 2 () () (A.3) = = = ½ ( ) () +2 + = 3 +() (if is odd) ( ) 0 () () (if is even) for large and,where are defined as in Lemma A.. Proof. Results (A.) and (A.2) are standard from Lemma A. since ( ) is even and is stationary. We denote [] () =(( )) () for an integer 0. For (A.3), by taking expectation, we note that " 2 E X = = # [] ()[] () = E[[] ()[] ()] + = n o E[ [] ()[] + ()] + E[[] ()[] ()]. (A.4) 7

19 First, from Lemma A. we have ½ E[ [] () ()[] ()] = () () (if is odd) (if is even) (A.5) since 2 ( ) is even. It follows that the first term in (A.4) is ( ) () +2 + ( 2 ) when is odd or ( ) 0 () ( 2 ) when is even. For the covariance terms in (A.4), similarly as Masry and Fan (997), we decompose = E[ [] ()[] + ()] = + + = = = E[ [] ()[] + ()] ( [] ()[] + ()) E[ [] ()]E[[] + ()] ()+ 2 ()+ 3 () for some satisfying 0 as.for (), by Cauchy-Schwarz, we have n E[ [] ()[] + ()] E( [] ())2 E( [] + ())2o 2 = () from Lemma A.. Therefore, for some positive constant 0, wehave () = = since 0. For 2 (), using the mixing inequality (e.g., Hall and Heyde, 980, Theorem A.5), for some 2, wehave n ( [] ()[] + ()) 8 () 2 E [] () E [] () o 2 () 2 2 for some positive constant 2. Note that for any integer 0 andevenfunction( ), () = () is even and thus it can be shown that E[ [] () ]= R () ( + ) = () similarlyaslemmaa.providedthat R (). It follows that for some 2, Ã! 2 X 2 () 2 () 2 2 X 2 () 2 2 = = = since P = () 2 P = () 2 and. By taking = 2,which satisfies the condition and 0, wehave 2 () =( ). Finally,for 3 (), Lemma A. yields E[ [] ½ ()] = 2 0 () () () (if is odd) (if is even) 8

20 and therefore for some positive constant 3 ( ( 3 4 ) P = 3 () ( ( )) = 3 ( 3 3 ) P = ( ( )) = 2 (if is odd) (if is even). We note that ( ) P = ( ( )) 0 as since ( ) P and as. By combining these results, we conclude that = and (A.3) follows. E[ [] ()[] ½ + ()] = 3 +() 2 +() = ( ( )) 2 (if is odd) (if is even) Lemma A.3. Under Assumption A, we have () N 0 2 () () 22 = = as,where 22 = R 2 2 () as defined in Lemma A.. Proof. First note that Assumption A gives the CLT as Masry and Fan (997, Section 3). Moreover, we have E [(( )) ()] = E [(( )) ()E [ ]] = 0 since E [ ]= 0 by construction. Therefore, we only need to verify the asymptotic variance. Note that " = = " = X = = [ ()] + = () () # ( () + ()), where () =(( )) (). SinceE [ ()] = 0, LemmaA.2gives [ ()] = E 2 () " = E E 2 # 2 = 2 () For the covariance term, we note that = 2 () () () 00 () ( () + ()) = E ( () + ()) = + E E [ + ] () + () =0 since is serially uncorrelated conditional on ( ). The desired result immediately follows. # 9

21 A.2 Proof of Theorem We write where () = 2 () = 3 (3) = b() () = 2() () + 3() (), = = = = = = ( ()) 2 (), () () () ()() (). Recall that we define () = (( )). We claim that () = () () 4 2+ ( 2 )+ ( ), 3 () = 2 00 () 0 () 4 2+ ( 3 )+ ( ) and 3 2 () N 0 2 () () 22 for each X as. The desired result then follows immediately by Slutsky from 3 ( () b () 3 () ()) = 3 2 () () and the continuity of 00 ( ). First note that () = = () = and ()+ () since () P = () () 0 as. We let () =( ) and ' denote the equality with probability approaching to one as. Then, we can write () = ' ' = = = = " () " = = 2 () ()# () = () () 2 () () () 2 () ()# () = 2 2 = = = () () () (). However, from Lemma A.2, the first term is () () 4 2+ ( 2 ) whereas the second term is simply ( ). The expression of 3 () can be obtained similarly since, for some between and, 3 () " #" # = 00 () () () () 2 2 () ()() 2 () ' 00 () 2 ( = = = = = 3 () () () = = = = ) ()() 2 () (),

22 where the first term in the curly bracket is 0 () 4 + ( 2 ) whereas the second term is ( ) from Lemma A.2. The continuity of 00 ( ) gives the desired result. For 2 (), notethatitsplimis simply zero for E [ ]=0.However, 3 2 () " #" # = () () () () () ' = = " = = = () () = #" () () = () # () () ' () () () = = () () (), = = = where the first term satisfies the CLT in Lemma A.3. For the second term, since it can be shown that " # E () () () = = = " = # E X ()E [ ] () () = () = = similarly as Lemma A.2, it is ( p ) assuming that does not diverge. A.3 Proof of Theorem 2 For notational simplicity, we let = R () and () = R ( ) () for =2. Recall that we define () = (( )). Wefirst write e () 0 () = 2 () () + 3 () (), where () = 2 () = 3 () = = = = = = = ( ) 2 (), ( )( ) () and = ( ) () () () with () =( ) P = (), =( ) P = and =( ) P =. We claim that both () and 3 () are () as,whereastheplimof 2 () is simply zero for E [ ]=0. The desired result then follows immediately. Particularly when { } is serially uncorrelated, moreover, we can show that () = 2 ()()+ 2 + and 3 () = 2 00 () () 2 ()()

23 as, so that the approximate bias form (8) is obtained. To find the expressions of () and 2 (), first note that () = = = ( ()) 2 () 2 2 = = = + 3 () () () = = = = () 2 2 ()+ 3 (), () () () where () =( ). From Lemma A.2, we obtain () 2 () since is symmetric and 2 is assumed. Moreover, from the stationarity, we have " E 2 () = # 2 E X () () () (A.6) = = = i h( E ()) 2 () + X X 2 E [ () () ()], = 6= where the first term is ( 2 ) similarly as (). However, using a similar approach as the proof of Lemma A.2, it can be verified that the rest of the term is ( 2 ). For example, even when there is no serial correlation in { }, the second term in (A.6) becomes 2 = 2 = X = E [ () ()] E [ + ()] 0 () 2 + ( 2 ) ª ½ ¾ () 2 () 0 () 2 as with () since ( ) P = ( ( )) 2. Using a similar argument, however, it can be verified that " E 3 () = # 3 E X () () () = = = = i h( 2 E ()) 2 () + X X E 3 () 2 () +2E [ () () ()] ª = 6= + X X X 3 E [ () () ()] 2 = = 6= 6= () + 2. The first term is straightforward from () above. Particularly when there is no serial correlation 22

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