Nonparametric Estimation in a One-Way Error Component Model: A Monte Carlo Analysis

Transcription

1 Nonparametric Estimation in a One-Way Error Component Model: A Monte Carlo Analysis Daniel J. Henderson Department of Economics State University of New York at Binghamton Aman Ullah Department of Economics University of California, Riverside August 16, 2005 Abstract This paper considers the problem of nonparametric kernel estimation in panel data models. We examine the finite sample performance of several estimators for estimating a one-way error component model. Monte Carlo experiments show that the pooled estimator that ignores the dependence structure in the modelperformswellineachtrial,butitisnotthemostefficient estimator since it is generally outperformed in the mean squared sense by both the two-step estimator and the nonparametric feasible generalized least squares estimator. Although the asymptotic bias and variance of most of the estimators converge at the same rate, the two-step estimator, which has smaller asymptotic variance and can have smaller asymptotic bias, generally outperforms most of the other estimators incorporating dependence when the technology is nonlinear and the variance of the error component is large relative to the variance of the random disturbance. Finally, we use an empirical example regarding the public capital productivity puzzle to showcase the estimators in a real data setting. Keywords: Nonparametric Kernel, Panel Data, Productivity, Public Capital JEL Classification: C1, C14, C33, H4, O4 Daniel J. Henderson, Department of Economics, State University of New York, Binghamton, NY, 13902, (607) , Fax: (607) , djhender@binghamton.edu. Aman Ullah, Department of Economics, University of California, Riverside, CA 92521, (951) , Fax (909) , aman.ullah@mail.ucr.edu.

2 1 Introduction Economic research has been enriched by the availability of panel data that measure individual cross-sectional behavior over time (for a review of the literature as well as estimation and inference in parametric panel data models see Baltagi 2001). Recently nonparametric modeling and estimation has attracted much attention among statisticians and econometricians in panel data models. One popular model is the nonparametric one-way error component model. In addition to relaxing the restrictive parametric assumptions on the functional form of the model, effort has been made to incorporate the dependence structure in the data within the model. In this setup several estimators of the nonparametric one-way error component have been developed. The recently developed nonparametric estimators for a one-way error component model consists of pooled estimators ignoring the dependence structure in the errors and the estimators which incorporate the dependence structure, see Henderson and Ullah (2005), Li and Ullah (1998), Lin and Carroll (2000), Ruckstuhl, Welsh and Carroll (2000), and Ullah and Roy (1998), among others. Ruckstuhl, Welsh and Carroll (2000) state that the simple pooled estimator which ignores the dependence structure performs well asymptotically. Intuitively this is because dependence is a global property of the error structure which is not important to methods which act locally in the covariate space. Although the asymptotic performance of the pooled estimator raises the question as to whether there exists an estimator which exploits the dependence structure sufficiently well such that it performs better than the pooled estimator asymptotically, one can always ask how current alternative models compare to the pooled estimator and against one another in a finite setting. In this paper we examine the finite sample performance of several nonparametric techniques for estimating a one-way error component model. Specifically we employ Monte Carlo simulations to compare the finite sample performance of the pooled estimator versus estimators which take the inherent dependence structure into account as well as make comparisons amongst the alternative estimators. Our results show that the pooled estimator performs very well. However, it is generally not the most efficient estimator in our trials, since a feasible generalized least squares estimator 1

3 incorporating the dependence structure is found to outperform it in a mean squared sense. Further, a two-step estimator, which has smaller asymptotic variance and can have smaller asymptotic bias, outperforms most models when the variance of the error component is large relative to the variance of the random disturbance and the technology is nonlinear. We therefore suggest that there is still good reason to examine nonparametric estimators specifically designed for the one-way error component model. The remainder of the paper is organized as follows: section 2 presents both the pooled and alternative models whereas the third section defines the Monte Carlo setup and summarizes the results of the experiment. Section 4 showcases the estimators in an empirical exercise regarding the public capital productivity puzzle and the fifth section concludes. 2 Methodology Here we consider the nonparametric panel data model y it = m(x it )+ε it, (1) where i =1, 2,...,N, t=1, 2,...,T, y it is the endogenous variable, x it is a vector of k exogenous variables and m( ) is an unknown smooth function. We consider the case that the error ε it follows a one-way error component specification ε it = u i + v it, (2) where u i is i.i.d. (0,σ 2 u), v it is i.i.d. (0,σ 2 v) and u i and v jt are uncorrelated for all i and j, j =1, 2,...,N. If ε i =[ε i1,ε i2,...,ε it ] 0 is a T 1 vector, then V E(ε i ε 0 i ), takes the form V = σ 2 vi T + σ 2 ui T i 0 T, (3) where I T is an identity matrix of dimension T and i T is a T 1 column vector of ones. Since the observations are independent over i and j, thecovariancematrixfor the full NT 1 disturbance vector ε, Ω = E(εε 0 ) is Ω = I N V. (4) 2

4 We are interested in estimating the unknown function m(x) at a point x and the slope of m(x), β(x) = m(x), where is the gradient vector of m(x). The parameter β(x) is interpreted as a varying coefficient. We consider the usual panel data situation of large N and finite T. Nonparametric kernel estimation of m(x) and β(x) can be obtained by using local linear least squares (LLLS) estimation. This is obtained by minimizing the local least squares of errors X X µ (y it X it δ(x)) 2 xit x K =(y Xδ(x)) 0 K(x)(y Xδ(x)) (5) h i t with respect to m(x) and β(x), wherey is a NT 1 vector, X is a NT (k+1) matrix generated by X it =(1, (x it x)), δ(x) =(m(x),β 0 (x)) 0 is a (k +1) 1 vector, K(x) is an NT NT diagonal matrix of kernel functions K( x it x h ) and h is the bandwidth (smoothing) parameter. The estimator obtained is b δ(x) =(X 0 K(x)X) 1 X 0 K(x)y (6) and is called the LLLS estimator (see Fan and Gijbels 1992 or Pagan and Ullah 1999 for details on this estimator and the choice of h). The LLLS estimator ignores the dependence structure in the model and it is a pooled estimator that simply fits a single nonparametric regression model through all the data. An alternative approach, which Ruckstuhl, Welsh and Carroll (2000, pp. 54) define as the component estimator, involves fitting separate nonparametric models relating the tth component (time period) of y to the tth component (time period) of x and then combines these estimators to produce an overall estimator of the common regression function. The component estimator of δ(x) is defined as the weighted average of the component estimators given by X e T TX δ(x) = c t b δt (x), c t =1 (7) t=1 t=1 where ( T 1 X c t = f t (x) f k (x)) for t =1, 2,...,T, k=1 b δt (x) is the local linear kernel regression estimator of the y it on the x it with bandwidth h t and f l is the marginal density of x il. Further, Ruckstuhl, Welsh and Carroll (2000, 3

5 pp. 63) suggest using the common optimal bandwidth for the component estimator, which minimizes its mean squared error, defined as h RW C = γ(0) σ 2 u + σ 2 " n o T # 1 2 X ε m (2) (x) N f t (x) t=1 1 5, (8) where γ(0) = R K 2 (z)dz and m (2) (x) is the second derivative of m(x). Here we note that Ruckstuhl, Welsh and Carroll (2000) only define the result for m(x) and not β(x). Although the LLLS estimator ignores the dependence, Ruckstuhl, Welsh and Carroll (2000) note that the LLLS and component estimators are asymptotically equivalent. Further, LLLS estimation has the advantage of simplicity. Both of these estimators, however, ignore the information contained in the disturbance vector covariance matrix Ω. In view of this, Ruckstuhl, Welsh and Carroll (2000) develop the two-step estimator. This estimator attempts to make use of the known variance structure to achieve asymptotic improvement over the pooled LLLS estimator. Specifically, the estimator is obtained by first defining z = τω 1 2 y + ³I NT τω 1 2 bm(x), (9) where τ is a constant and bm(x) is the first stage estimator of b δ(x) in (6). Estimation of δ TS (x) = m TS (x),β 0 TS (x) 0 can now be obtained by running LLLS on the following equation z it = m TS (x it )+v it. (10) Further, the value for τ can be obtained from τ 2 = σ 2 v h n o i 1 1 (1 d T ) /T, (11) where d T = Tσ 2 u/ σ 2 v + Tσ 2 u. Ruckstuhl, Welsh and Carroll (2000) results show that the order of magnitudes of the bias and variance of the two-step estimator are the same as with the pooled LLLS estimator. They further show that the asymptotic bias can also be smaller than that of the pooled LLLS estimator. For example, when m is a quadratic function, the bias decreases monotonically to zero as the ratio σ 2 u/σ 2 v increases. 1 1 Ruckstuhl, Welsh and Carroll (2000) alternatively suggest defining τ to be equal to σ 1 v,which when employed in the estimation of (10) has smaller asymptotic variance than estimation performed 4

6 Henderson and Ullah (2005) also attempt to model the information contained in the distrubance vector covariance matrix. They introduce an estimator, Local Linear Weighted Least Squares (LLWLS), by minimizing (y Xδ(x)) 0 W (x)(y Xδ(x)) (12) with respect to δ(x), where W (x) is a kernel based weight matrix. This provides the kernel estimating equations for δ(x) as X 0 W (x)(y Xδ(x)) = 0, whichgives They consider the following cases of (13), d(x) = X 0 W (x)x 1 X 0 W (x)y. (13) d r (x) = X 0 W r (x)x 1 X 0 W r (x)y, (14) where d r (x) = m r (x),β 0 r(x) 0, and for r =1, 2, 3, W1 (x) = p K(x)Ω 1p K(x), W 2 (x) =Ω 1 K(x) and W 3 (x) =Ω 1 2 K(x)Ω 1 2. The estimators d 1 (x) and d 2 (x) are as given in Lin and Carroll (2000), and d 3 (x) is as given in Ullah and Roy (1998). When the matrix V, and hence Ω, is a diagonal matrix, then W 1 (x) =W 2 (x) =W 3 (x), and hence d 1 (x) = d 2 (x) = d 3 (x). Further, in the special case when Ω = I NT, d 1 (x) =d 2 (x) =d 3 (x) = b δ(x) in (6). In general, however, d 1 (x), d 2 (x), and d 3 (x) are often different. Ruckstuhl, Welsh and Carroll (2000) and Lin and Carroll (2000) provide the asymptotic bias and variance of d 1 (x) and d 2 (x). These results provide the consistency of these estimators. However, Henderson and Ullah (2005) note that the rates of convergence of d 1 (x), d 2 (x) and d 3 (x) are the same as with the pooled LLLS estimator. Two-step estimation in (10) and the LLWLS estimator in (13), however, depend upon the unknown parameters σ 2 u and σ 2 v. Henderson and Ullah (2005) use the spectral decomposition of Ω to obtain consistent estimators of the variance components as bσ 2 1 = T X bε 2 i /N, bσ 2 1 X X ³ v = bε it bε i 2, (15) N(T 1) i i t with (11). Although defining τ in this manner works well in Monte Carlo, it does not perform as well as with (11) in small samples. Further, in our empirical example we found it to give weaker estimates then when using (11). The results for both the Monte Carlo and the empirical exercise using the alternative τ can be obtained from the authors upon request. 5

7 where σ 2 1 Tσ2 u + σ 2 v, bε i = T 1 P t bε it and bε it = y it bm(x it ) is the LLLS residual based on the first stage estimator of b δ(x) in (6). Further, the estimate of σ 2 u is obtained as bσ 2 u = bσ 2 1 bσ 2 v /T. Ruckstuhl, Welsh and Carroll (2000, pp. 59) provide an alternative approach to estimating the variance components. They estimate the variance-covariance matrix by pretending that the residuals have mean zero and that the covariance matrix is the same as if m( ) were known. Specifically, the consistent estimators of the elements of eω are obtained as and eσ 2 1 = T N NX i=1 eσ 2 v = when eσ 2 1 > eσ 2 v and ³ y i bm i 2, where yi = 1 T 1 N(T 1) NX TX i=1 t=1 eσ 2 1 = eσ 2 v = 1 NT TX y it, bm i = 1 T t=1 TX bm(x it ), (16) t=1 ³ ³ y it bm(x it ) y i bm i 2, (17) NX i=1 t=1 TX (y it bm(x it )) 2 (18) otherwise. Similarly, the estimate of σ 2 u is obtained as eσ 2 u = eσ 2 1 eσ 2 v /T. Substituting estimates of σ 2 u and σ 2 v from (15), (16) and (17), or (18) into (10) and (14) give the feasible two-step and the Local Linear Feasible Weighted Least Squares (LLFWLS) or Nonparametric Feasible Weighted Least Squares (NPFWLS) estimators as b δts (x) =(X 0 K(x)X) 1 X 0 K(x)bz, (19) and e δts (x) =(X 0 K(x)X) 1 X 0 K(x)ez, (20) b δr (x) =(X 0 c Wr (x)x) 1 X c W r (x)y, (21) e δr (x) =(X 0 fw r (x)x) 1 XfW r (x)y, (22) where bz, ez, W c r (x) and W f r (x) are the same as above, with Ω replaced by the consistent estimators Ω b and Ω e respectively. Further, following Li and Ullah (1998), we can show that the consistency of b δ TS (x), b δ TS (x), b δ r (x) and e δ r (x) follow from the consistency of δ TS (x) and d r (x) for known Ω. Finally, for the remainder of the paper we will refer to δ 1 (x) as the NPGLS estimator, δ 2 (x) as the Lin and Carroll estimator and δ 3 (x) as the Ullah and Roy estimator. 6

8 3 Monte Carlo This section uses Monte Carlo simulations to examine the finite sample performance of the panel data estimators. Following the methodology of Baltagi, Chang and Li (1992), the following data generating process is used: y it = α + x it β + x 2 itγ + u i + v it, where x it is generated by the method of Nerlove (1971). 2 The value of α is chosen to be 5, β is chosen to be 0.5 and γ takes the values of 0 (linear technology) and 2 (quadratic technology). The distribution of u i and v it are generated separately as i.i.d. Normal. The total variance σ 2 v + σ 2 u =20and ρ = σ 2 u/(σ 2 u + σ 2 v) is varied to be 0.1, 0.4, and0.8. For comparison, we compute the following estimators of δ: (I) Parametric (linear) feasible GLS (FGLS) estimator b δ =(X 0 Ω b 1 X) 1 X 0 Ω b 1 y. (II) LLLS estimator b δ(x) =(X 0 K(x)X) 1 X 0 K(x)y. (III) Component estimator X e T δ(x) = c t b δt (x). t=1 (IV) Feasible Two-Step estimator b δts (x) =(X 0 K(x)X) 1 X 0 K(x)bz (V) Feasible NPGLS (NPFGLS) estimator b δ1 (x) =(X 0p K(x) b Ω 1p K(x)X) 1 X 0p K(x) b Ω 1p K(x)y. (VI) Feasible Lin and Carroll estimator b δ2 (x) =(X 0 Ω b 1 K(x)X) 1 X 0 Ω b 1 K(x)y. 2 The x it were generated as follows: x it =0.1t +0.5x it 1 + w it, where x i0 =10+5w i0 and w it U[ 1, 1 ]

9 (VII) Feasible Ullah and Roy 3 estimator b δ3 (x) =(X 0 b Ω 1 2 K(x) b Ω 1 2 X) 1 X 0 b Ω 1 2 K(x) b Ω 1 2 y. In addition, we also estimate each of the four feasible nonparametric estimators using the estimated omega matrix described in Ruckstuhl, Welsh and Carroll (2000). (VIII) Feasible Two-Step estimator e δts (x) =(X 0 K(x)X) 1 X 0 K(x)ez. (IX) NPFGLS estimator e δ1 (x) =(X 0p K(x) e Ω 1p K(x)X) 1 X 0p K(x) e Ω 1p K(x)y. (X) Feasible Lin and Carroll estimator e δ2 (x) =(X 0 e Ω 1 K(x)X) 1 X 0 e Ω 1 K(x)y. (XI) Feasible Ullah and Roy estimator e δ3 (x) =(X 0 e Ω 1 2 K(x) e Ω 1 2 X) 1 X 0 e Ω 1 2 K(x) e Ω 1 2 y. The parametric estimator is expected to perform best when the parametric model is correctly specified. However, when the parametric model is incorrectly specified, it is expected to lead to inconsistent estimation of m and β. Further, although the asymptotic bias and variance of most of the nonparametric estimators converge at the same rate, the finite sample performance of the estimators against one another is unknown. Reported is the average estimated mean squared error (MSE) for each estimator. These are computed via MSE( bm) =M 1 P j MSE j( bm), wheremse j ( bm) = 1 P NT ( bm(x) m (x)) 2 is the MSE of bm at the jth replication. Further, M (j = 1, 2,...,M) is the number of replications, and bm(x) is the estimated value of m (x) = α + xβ + x 2 γ, evaluated at each x. Similarly, for the varying coefficient parameter, MSE( β)=m b 1 P j MSE j( β), b wheremse j ( β)= b 1 P NT ( β(x) b β (x)) 2 is 3 This feasible estimator considers a consistent estimator of the omega matrix which is different from that given in Ullah and Roy (1998) and also in Li and Ullah (1998), see Henderson and Ullah (2005). 8

10 the MSE of β b at the jth replication. Again, M (j =1, 2,...,M)isthenumberof replications, and β(x) b is the estimated value of β (x) =β +2xγ, evaluated at each x. M =500is used in all simulations. T is varied to be 5 and 10, whilen takes the values 25 and 50. The simulation results are given in Tables 1 and 2. The smallest MSE for each case (for a given N, T, ρ and γ) is shown as a boldface number. Table 1 reports the results for γ =0(linear technology). As expected, the correctly specified linear parametric model outperforms each of the nonparametric estimators in terms of MSE. However, the results for the nonparametric estimators are not as straightforward. 4 First we note that the component estimator performs best among the nonparametric estimators in terms of estimating m when ρ =0.1, and better than all the nonparametric estimators except the two-step estimator for all ρ. That is, the Ruckstuhl, Welsh and Carroll (2000) asymptotic results go through for small samples. However, our attempts at estimating β with the component estimator (which was not defined in Ruckstuhl, Welsh and Carroll 2000) proved to be poor. One possible explanation for this is that the optimal bandwidth for the component estimator is designed to minimize the MSE of m and not necessarily β (see Ruckstuhl, Welsh and Carroll 2000, pp. 63). However, when ρ =0.4 or 0.8, the two-step estimator using the Henderson and Ullah (2005) omega matrix performs best with the estimation of m. Regarding the estimation of β, the Ullah and Roy estimator performs best on average amongst the nonparametric estimators. Unfortunately, it performs poorly in terms of estimating m in our exercise when using Ω. b A similar result holds for the Lin and Carroll estimator. It performs poorly in terms of estimating β when using bω. Estimation of both m and β improves for the Lin and Carroll estimator and for m (but not for β) with the Ullah and Roy estimator when employing Ω. e However they are still less efficient than several of the other estimation techniques. Two-step, NPFGLS (whose MSE are smaller with Ω) b and LLLS are more consistent in terms of their performance relative to the other nonparametric estimators. Although they are not consistently the top nonparametric performers in terms of m or β, the two-step and NPFGLS estimators perform well and better than LLLS in most situations in 4 Except for the component estimator whose bandwidth is shown in (8), we use the Silverman (1986) rule of thumb bandwidth in our Monte Carlo simulations. 9

11 the table. On the other hand, the results for the quadratic technology (γ =2) show quite well for both the two-step and NPFGLS estimators and are presented in Table 2. In all but two cases (in the estimation of m and β with N =50,T=5and ρ =0.1) the two-step estimator outperforms each of the other models. In the two cases where the NPFGLS estimator outperforms the two-step estimator, ρ is relatively small (that is, the ratio σ 2 u/σ 2 v iss relatively small). As is the case with the linear technology, the performance of the two-step estimator improves greatly relative to the other estimators when ρ increases. Again, the LLLS estimates perform well, but less efficient generally than the NPFGLS estimator (except for a limited number of cases when it outperforms it in terms of estimating m). Regarding the remaining estimators, the now misspecified parametric estimator has a large MSE which does not decrease as the number of cross-sections grow. The Lin and Carroll estimator again performs poorly in terms of β and now the Ullah and Roy estimator performs poorly in terms of both m and β. Even more so than in the linear case, estimation of both estimators improves greatly when employing Ω e instead of Ω. b Again, the component estimator performs well in terms of the estimation of m, but not as well in the estimation of β. Finally, we estimated each of the feasible estimators assuming that the omega matrix was known. The conclusions of this experiment are not significantly different from the estimated omega matrix calculations for the feasible estimators and are not reported here for sake of brevity but are available from the authors upon request. Based on the above results, we note that although the results differed between the two data generating processes, we were able to learn a great deal about the finite sample performance of the estimators. In summary, our principal conclusions are as follows: (1) When correctly specified, the parametric estimator outperforms each of the nonparametric estimators. (2) With the linear data generating process, there is no clear cut winner amongst the nonparametric estimators. Although the component estimator performed best in terms of estimating m when ρ was small, it performed poorly in estimating β and vice versa for the Ullah and Roy estimator. Although less efficient, the two-step estimator, the NPFGLS and to a lesser extent the LLLS estimator consistently performed well in terms of the estimation of m and β. (3) 10

12 When the technology became nonlinear, the two-step estimator performed best in all but two cases, in which it was outperformed by the NPFGLS (when ρ =0.1). LLLS also performed well and occasionally gave a lower MSE than the NPFGLS estimator. (4) Finally, the Henderson and Ullah (2005) omega matrix performed well in both the estimation of the two-step and the NPFGLS estimators, but caused both the Lin and Carroll, and Ullah and Roy estimators to give less efficient results in our exercise. The performance of the latter two estimators improves when employing the Ruckstuhl, Welsh and Carroll (2000) estimator of omega. 4 Empirical Example The Monte Carlo results in the previous section compared the finite sample performance of several nonparametric panel data estimators. In this section we apply the aforementioned estimation procedures to the well known public capital productivity puzzle. Although numerous authors have examined this puzzle, we will compare our results to the more recent study by Baltagi and Pinnoi (1995). In their paper they consider the following production function: y it = α + β 1 KG it + β 2 KP it + β 3 L it + β 4 unem it + ε it, (23) where y it denotes the gross state product of state i (i =1,...,48) inperiodt (t = 1970,...,1986), public capital (KG) aggregates highways and streets (KH), water and sewer facilities (KW), and other public buildings and structures (KO), KP is the Bureau of Economic Analysis private capital stock estimates, and labor (L) is employment in non-agricultural payrolls. Details on these variables can be found in Munnell (1990) as well as Baltagi and Pinnoi (1995). Following Baltagi and Pinnoi (1995) we use the unemployment rate (unem) to control for business cycle effects. The results based on the Cobb-Douglas production function (linear in logs) are the same as in Baltagi (2001, pp. 25) and are reported in Table 3. The coefficients on both labor and private capital are found to be positive and statistically significant. On the other hand, the coefficient on public capital is quite small and statistically insignificant. These results have caused some to suggest that public capital is unproductive. However, assuming a Cobb-Douglas production function assumes a particular form for the underlying production function, which may or may not be correct. 11

13 Further, by construction, the elasticities of the model are exactly the same across all states and over all years. Thus, it seems natural to ask whether the results from the Cobb-Douglas model can be trusted. In fact, if the true model is nonlinear and one ignores it, the resulting estimates of returns to inputs are likely to be inconsistent. The results for the nonparametric models are also reported in Table 3. It should be noted that on average we find similar results as in the Cobb-Douglas case (by using nonparametric regression that captures nonlinearity in the functional form) in terms of private capital, labor and unemployment. However, our results are significantly different in terms of the returns to public capital. Specifically, in a majority of the cases, we find evidence of a significant positive return to public capital. The elasticities here are similar to those found in Henderson and Kumbhakar (2004) and Henderson and Millimet (2005) who first showed the return to public capital to be positive and significant when employing nonparametric techniques. Somewhat striking are the results for the Lin and Carroll ( b δ 2 (x)) and the Ullah and Roy ( b δ 3 (x)) estimators which use the Henderson and Ullah (2005) omega matrix ( Ω). b In each case the median estimated elasticity for public capital is small and not significantly different from zero. However, when employing the Ruckstuhl, Welsh and Carroll (2000) estimator of Ω, each of these estimates becomes positive and significant (here they actually give results nearly identical to one another and that of the LLLS estimator). These results should not be surprising. The Monte Carlo results in Table 2 show that the performance of the Lin and Carroll, and Ullah and Roy estimators when using the Henderson and Ullah (2005) omega matrix are less efficient in small samples and that they can be improved by employing the Ruckstuhl, Welsh and Carroll (2000) omega matrix. The results using the two-step estimator are equally interesting. When employing the Henderson and Ullah (2005) omega matrix, the median estimate corresponding to public capital (0.136) isslightlylessthanthatofthenpfglsandlllsestimates. However, when employing the Ruckstuhl, Welsh and Carroll (2000) omega matrix, the return to public capital (0.153) is closer to that of the other nonparmetric estimators. As stated by Ruckstuhl, Welsh and Carroll (2000) and as shown in the Monte Carlo section of this paper, the estimates of the two-step estimator are greatly improved when the ratio σ 2 u/σ 2 v increases. The Henderson and Ullah (2005) estimated 12

14 ρ = σ 2 u/(σ 2 u + σ 2 v) for this particular data set is 0.621, which is between the 0.4 and 0.8 values in the Monte Carlo experiment which show strong performance of the two-step estimator. Thus, there is some reason to believe that the other estimators may be slightly overstating the elasticity of public capital in this data set. When examining the point estimates, we find that for most states the estimated return to public capital is positive a majority of the time when looking at the LLLS, two-step or NPFGLS estimates. However, several states still give negative values for a majority of the time periods. For example, when employing the NPFGLS estimator we find that some of these states (New Mexico, North Dakota and South Dakota) give relatively small negative returns ( 0.05) while others (Wyoming) give larger negative returns ( 0.15) on average (we should further note that although Idaho, Iowa and Montana have positive returns on average, they also posses several negative values). It is interesting to note that this group of states are all plains states. One possible explanation for these negative returns to (or over-investment in) public capital is that they each have large relative investments in highways. These states have major highways running through them (designed to transport goods through their respective states) while at the same time their gross state products are relatively small. In summary, these results suggest that the Cobb-Douglas model is too simple and fails to capture the non-linearity inherent in the functional relationship underlying the technology. Former research which employed the Cobb-Douglas model caused many to believe that public capital was unproductive. Although this example does not necessarily prove the opposite, it does show that the previous research is not sufficient to condemn the idea that public capital is productive. 5 Concluding Remarks In this paper we considered the problem of estimating a nonparametric panel data model with errors that exhibit a one-way error component structure. Specifically, we examined the finite sample performance of several nonparametric kernel estimators for estimating a panel data model. When the data generating process used in the exercise was linear, (of the nonparametric estimators) the component estimator performed best in terms of the estimation of the conditional mean when ρ was small whereas 13

15 the two-step estimator performed best when ρ was larger. At the same time, the Ullah and Roy estimator performed best in terms of the estimation of the varying coefficient parameter. Although less efficient, the NPFGLS and to a lesser extent the LLLS estimator consistently performed well in terms of the estimation of both m and β. However, when the technology used became nonlinear, the two-step estimator performed best in nearly each trial (occasionally being outperformed by the NPFGLS estimator). Interestingly, the LLLS estimator which ignores the dependence structure in the model and the information contained in the disturbance vector covariance matrix also performed well and occasionaly gave a lower MSE than the NPFGLS estimator. Asalastcommentwewouldliketonotethatthroughoutthepaperweassume the existence of random individual effects. In practice one way want to test for the existence of random individual effects. 14

16 References [1] Baltagi, B. H. (2001). Econometric Analysis of Panel Data, Wiley and Sons, New York. [2] Baltagi, B. H., Y.-J. Chang, and Q. Li (1992). Monte Carlo Results on Several New and Existing Tests for the Error Component Model, Journal of Econometrics, 54, [3] Baltagi, B. H., and N. Pinnoi (1995). Public Capital Stock and State Productivity Growth: Further Evidence from an Error Components Model, Empirical Economics, 20, [4] Fan, J., and I. Gijbels (1992). Variable Bandwidth and Local Linear Regression Smoothers, Annals of Statistics, 20, [5] Henderson, D. J., and S. C. Kumbhakar (2004). Public and Private Capital Productivity Puzzle: A Nonparametric Approach, manuscript, State University of New York at Binghamton. [6] Henderson, D. J., and D. L. Millimet (2005). Environmental Regulation and U.S. State-Level Production, Economics Letters, 87, [7] Henderson, D. J., and A. Ullah (2005). A Nonparametric Random Effects Estimator, Economics Letters, 88, [8] Li, Q., and A. Ullah (1998). Estimating Partially Linear Panel Data Models with One-Way Error Components, Econometric Reviews, 17, [9] Lin, X., and R. J. Carroll (2000). Nonparametric Function Estimation for Clustered Data when the Predictor is Measured Without/With Error, Journal of the American Statistical Association, 95, [10] Munnell, A. H. (1990). How Does Public Infrastructure Affect Regional Economic Performance? New England Economic Review, [11] Pagan, A., and A. Ullah (1999). Nonparametric Econometrics, Cambridge University Press, Cambridge. [12] Ruckstuhl, A. F., A. H. Welsh, and R. J. Carroll (2000). Nonparametric Function Estimation of the Relationship Between Two Repeatedly Measured Variables, Statistica Sinica, 10, [13] Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, Chapman and Hall, London. [14] Ullah, A. and N. Roy (1998). Nonparametric and Semiparametric Econometrics of Panel Data, in A. Ullah and D. E. A. Giles, eds., Handbook of Applied Economics Statistics, 1, Marcel Dekker, New York,

17 Table 1: Monte Carlo Results (Average MSE) -- Linear Technology (γ = 0) N = 25, T = 5 m β ρ = 0.1 ρ = 0.4 ρ = 0.8 ρ = 0.1 ρ = 0.4 ρ = 0.8 FGLS LLLS Components Two-Step NPFGLS Lin and Carroll Ullah and Roy Two-Step RWC NPFGLS RWC Lin and Carroll RWC Ullah and Roy RWC N = 50, T = 5 FGLS LLLS Components Two-Step NPFGLS Lin and Carroll Ullah and Roy Two-Step RWC NPFGLS RWC Lin and Carroll RWC Ullah and Roy RWC N = 25, T = 10 FGLS LLLS Components Two-Step NPFGLS Lin and Carroll Ullah and Roy Two-Step RWC NPFGLS RWC Lin and Carroll RWC Ullah and Roy RWC N = 50, T = 10 FGLS LLLS Components Two-Step NPFGLS Lin and Carroll Ullah and Roy Two-Step RWC NPFGLS RWC Lin and Carroll RWC Ullah and Roy RWC

18 Table 2: Monte Carlo Results (Average MSE) -- Quadratic Technology (γ = 2) N = 25, T = 5 m β ρ = 0.1 ρ = 0.4 ρ = 0.8 ρ = 0.1 ρ = 0.4 ρ = 0.8 FGLS LLLS Components Two-Step NPFGLS Lin and Carroll Ullah and Roy Two-Step RWC NPFGLS RWC Lin and Carroll RWC Ullah and Roy RWC N = 50, T = 5 FGLS LLLS Components Two-Step NPFGLS Lin and Carroll Ullah and Roy Two-Step RWC NPFGLS RWC Lin and Carroll RWC Ullah and Roy RWC N = 25, T = 10 FGLS LLLS Components Two-Step NPFGLS Lin and Carroll Ullah and Roy Two-Step RWC NPFGLS RWC Lin and Carroll RWC Ullah and Roy RWC N = 50, T = 10 FGLS LLLS Components Two-Step NPFGLS Lin and Carroll Ullah and Roy Two-Step RWC NPFGLS RWC Lin and Carroll RWC Ullah and Roy RWC

19 Table 3: Empirical Example -- Median Estimated Elasticities β(kg) β(kp) β(l) β(unem) FGLS LLLS Components Two-Step NPFGLS Lin and Carroll Ullah and Roy Two-Step RWC NPFGLS RWC Lin and Carroll RWC Ullah and Roy RWC The endogenous variable in each regression is gross state product. Each variable (besides the unemployment rate) is measured in logarithims. Except for the component estimator whose bandwidth is shown in (8), we use the Silverman (1986) rule of thumb bandwidth for each estimator.