P1: GOPAL JOSHI November 3, 2010 17:3 C7035 C7035˙C015 446 Handbook of Empirical Economics and Finance wrong restrictions on the ␳ parameters, which in turn, introduce bias and lead to bad MSE performance of the resulting MLEs. Fortunately, this does not translate fully into bad MSE performance for the regression coefficients. The pretest estimator of the regression coefficients always performs better than the misspecified MLE and is recommended in practice. 15.4 Forecasts Using Panel Data with Spatial Error Correlation The literature on forecasting is rich with time series applications, but this is not the case for spatial panel data applications. Exceptions are Baltagi and Li (2004, 2006) with applications to forecasting sales of cigarette and liquor per capita for U.S. states over time. In order to explain how spatial autocorrela- tion may arise in the demand for cigarettes, we note that cigarette prices vary among states primarily due to variation in state taxes on cigarettes. Border effect purchases not included in the cigarette demand equation can cause spa- tial autocorrelation among the disturbances. In forecasting sales of cigarettes, the spatial autocorrelation due to neighboring states and the individual het- erogeneity across states is taken explicitly into account. Baltagi and Li (2004) derive the best linear unbiased predictor for the random error component model with spatial correlation using a simple demand equation for cigarettes based on a panel of 46 states over the period 1963–1992. They compare the performance of several predictors of the states demand for cigarettes for 1 year and 5 years ahead. The estimators whose predictions are compared in- clude OLS, fixed effects ignoring spatial correlation, fixed effects with spatial correlation, random effects GLS estimator ignoring spatial correlation and random effects estimator accounting for the spatial correlation. Based on the RMSE criteria, the fixed effects and the random effects spatial estimators gave the best out of sample forecast performance. Best linear unbiased prediction (BLUP) in panel data using an error com- ponent model have been surveyed in Baltagi (2008b). However, these panel forecasting applications do not deal with spatial dependence across the panel units. Following Baltagi and Li (2004), Baltagi, Bresson, and Pirotte (2010) compare various forecasts using panel data with spatial error correlation. This is done using a Monte Carlo setup rather than empirical applications. The true data generating process is assumed to be a simple error component regression model with spatial remainder disturbances of the autoregressive or moving average type. The best linear unbiased predictor is compared with other forecasts ignoring spatial correlation, or ignoring heterogeneity due to the individual effects. The paper checks the performance of these forecasts under misspecification of the spatial error process, different spatial weight matrices, and various sample sizes. Goldberger (1962) has shown that, for a given , the best linear unbiased predictor (BLUP) for the ith individual at a future period T + ␶ is given by:  y i,T+␶ = X i,T+␶  ␤ GLS + ␻   −1  u GLS (15.28) P1: GOPAL JOSHI November 3, 2010 17:3 C7035 C7035˙C015 Spatial Panels 447 where ␻ = E[u i,T+␶ u] is the covariance between the future disturbance u i,T+␶ and the sample disturbances u.  ␤ GLS is the GLS estimator of ␤ based on  and  u GLS denotes the corresponding GLS residual vector. For the error component without spatial autocorrelation (␳ = 0), this BLUP reduces to  y i,T+␶ = X i,T+␶  ␤ GLS + ␴ 2 ␮ ␴ 2 1  ␫  T ⊗l  i   u GLS (15.29) where ␴ 2 1 = T␴ 2 ␮ +␴ 2 v and l i is the ith column of I N . The typical element of the last term of Equation 15.29 is (T␪) u i.,GLS , where u i.,GLS =  T t=1  u ti,GLS /T and ␪ = ␴ 2 ␮ /␴ 2 v ; see Baltagi (2008b). Therefore, the BLUP of y i,T+␶ for the RE model modifies the usual GLS forecasts by adding a fraction of the mean of the GLS residuals corresponding to the ith individual. In order to make this forecast operational,  ␤ GLS is replaced by its feasible GLS estimate and the variance components are replaced by their feasible estimates. Baltagi and Li (2004, 2006) derived the BLUP correction term when both errorcomponentsandspatialautocorrelation arepresent and ⑀ t follows a SAR process. So, the predictor for the SAR is given by:  y i,T+␶ = X i,T+␶  ␤ MLE + ␪  ␫  T ⊗l  i C −1 1   u MLE = X i,T+␶  ␤ MLE + T␪ N  j=1 c 1,j u j.,MLE (15.30) where c 1 j is the jth element of the ith row of C −1 1 with C 1 = [T␪I N +(B  B) −1 ] and u j.,MLE =  T t=1  u tj,MLE /T. In other words, the BLUP of y i,T+␶ adds to X i,T+␶  ␤ MLE a weighted average of the MLE residuals for the N individuals averaged over time. The weights depend upon the spatial matrix W N and the spatial autoregressive coefficient ␳. To make these predictors operational, we replace ␪ and ␳ by their estimates from the RE-spatial MLE with SAR. When there are no random individual effects, so that ␴ 2 ␮ = 0, then ␪ = 0 and the BLUP prediction terms drop out completely from Equation 15.30. In these cases,  reduces to ␴ 2 v [I T ⊗(B  B) −1 ] for SAR, and the corresponding MLE for these models yield the pooled spatial MLE with SAR remainder disturbances. This result can be extended to the spatial moving average model (SMA); see Baltagi, Bresson, and Pirotte (2010). For the Kapoor, Kelejian, and Prucha (2007) model, the BLUP of y i,T+␶ for the SAR-RE also modifies the usual GLS forecasts by adding a fraction of the mean of the GLS residuals corresponding to the ith individual. More specifically, the predictor is given by  y i,T+␶ = X i,T+␶  ␤ FGLS +  ␴ 2 ␮ ␴ 2 1  b i  ␫  T ⊗ B N   u FGLS = X i,T+␶  ␤ FGLS +  ␴ 2 ␮ ␴ 2 1 )(␫  T ⊗l  i   u FGLS (15.31) P1: GOPAL JOSHI November 3, 2010 17:3 C7035 C7035˙C015 448 Handbook of Empirical Economics and Finance whereb i is the ith row of the matrix B −1 N . This holds because b i (␫  T ⊗B N ) = (1⊗ b i )(␫  T ⊗B N ) = (␫  T ⊗l  i ),wherel  i is the ith row of I N as defined above. B −1 N B N = I N and therefore b i B N = l  i . This proof applies to both the Kapoor, Kelejian, and Prucha (2007) SAR-RE specification and the Fingleton (2008) SMA-RE specification. Therefore, the BLUP of y i,T+␶ for the SAR-RE and the SMA- RE, like the usual RE model with no spatial effects, modifies the usual GLS forecasts by adding a fraction of the mean of the GLS residuals corresponding to the ith individual. While the predictor formula is the same, the MLEs for these specifications yield different estimates which in turn yield different residuals and hence different forecasts. The results of the Monte Carlo study by Baltagi, Bresson, and Pirotte (2010) findthatwhenthe trueDGPis RE withaSAR orSMAremainderdisturbances, estimators that ignore heterogeneity/spatial correlation perform badly in RMSE forecasts. Accounting for heterogeneity improves the forecast perfor- mance by a big margin and accounting for spatial correlation improves the forecast but by a smaller margin. Ignoring both leads to the worst forecasting performance. Heterogeneous estimators based on averaging perform worse than homogeneous estimators in forecasting performance. This performance improves with a larger sample size and seems robust to the type of spatial error structure imposed on the remainder disturbances. These Monte Carlo experiments confirm earlier empirical studies that report similar findings. 15.5 Panel Unit Root Tests and Spatial Dependence Baltagi, Bresson, and Pirotte (2007) studied the performance of panel unit root tests when spatial effects are present that account for cross-section cor- relation. Monte Carlo simulations show that there can be considerable size distortions in panel unit root tests when the true specification exhibits spatial error correlation. Panel data unit root tests have been proposed as alternative more powerful tests than those based on individual time series unit roots tests; see Baltagi (2008a) and Breitung and Pesaran (2008) for some recent reviews of this liter- ature. One of the advantages of panel unit root tests is that their asymptotic distribution is standard normal. This is in contrast to individual time series unit roots which have nonstandard asymptotic distributions. But these tests are not without their critics. The first generation panel unit root tests assumed cross-section independence. These tests include the one proposed by Levin, Lin, and Chu (2002), hereafter denoted by LLC, where the null hypothesis is that each individual time seriescontainsaunitrootagainstthealternativethat each time series is stationary. As Maddala (1999) pointed out, the null may be fine for testing convergence in growth among countries, but the alternative restricts every country to converge at the same rate. Im, Pesaran, and Shin (2003), hereafter denoted by IPS, allow for heterogeneous panels and propose P1: GOPAL JOSHI November 3, 2010 17:3 C7035 C7035˙C015 Spatial Panels 449 panel unit root tests which are based on the average of the individual ADF unit root tests computed from each time series. The null hypothesis is that each individual time series contains a unit root while the alternative allows for some but not all of the individual series to have unit roots. One major criticism of both the LLC and IPS tests is that they require cross-sectional independence. This is a restrictive assumption given the cross-section corre- lation and spillovers across countries, states, and regions. Maddala and Wu (1999) and Choi (2001) proposed combining the p-values from the individual unit root ADF tests applied to each time series. Once again, these tests follow a standard normal limiting distribution. They have the advantage that N, thenumber of cross sections, can be finite or infinite; the time series can be of different length; and the alternative allows some groups to have unit roots while others may not. Recent studies that try to account for cross-sectional dependence in panel unit root testing include the following: Chang (2002) who explored the non- linear IV methodology to solve the inferential difficulties in the panel unit root testing which arise from the intrinsic heterogeneities and dependencies of panel models. Chang (2002) suggests an average of individual nonlinear IV t-ratio statistics of the autoregressive coefficient obtained from using an integrable transformation of the lagged level as instrument. These methods assume cross-sectional correlation in the innovation terms driving the autore- gressive processes. Choi (2002), on the other hand, generalizes the three unit root tests (inverse chi-square, inverse normal and logit) to the case where the cross-sectional correlation is modeled by error component models. The tests are formulated by combining p-values from the ADF test applied to each in- dividual time series whose stochastic trend components and cross-sectional correlations are eliminated using GLS-demeaning and GLS-detrending. Choi (2002) shows that the combination tests have a standard normal limiting dis- tributions under the sequential asymptotics T →∞and N →∞. To avoid the restrictive nature of cross-section demeaning procedure, Bai and Ng (2004), and Phillips and Sul (2003), among others, propose dynamic factor models by allowing the common factors to have differential effects on cross-section units. Phillips and Sul’s model is a one-factor model where the factor is independently distributed across time. They propose a moment- based method to eliminate the common factor which is different from prin- cipal components. More specifically, in the context of a residual one-factor model, Phillips and Sul (2003) provide an orthogonalization procedure which in effect asymptotically eliminates the common factors before preceding to the application of standard unit root tests. Pesaran (2007) suggests a simple way of getting rid of cross-sectional dependence that does not require the estimation of factor loading. His method is based on augmenting the usual ADF regression with the lagged cross-sectional mean and its first-difference to capture the cross-sectional dependence that arises through a single factor model. Baltagi, Bresson, and Pirotte (2007) run Monte Carlo simulations to com- pare the empirical size of panel unit root tests with and without spatial error P1: GOPAL JOSHI November 3, 2010 17:3 C7035 C7035˙C015 450 Handbook of Empirical Economics and Finance dependence. The structure of the dependence is based on some commonly used spatial error processes: the spatial autoregressive (SAR) and the spatial moving average (SMA) error process and the spatial error components model (SEC). For each experiment, they perform nine panel unit root test statistics: the Levin, Lin, and Chu test (2002), the Breitung (2000) test, the Im, Pesaran, and Shin test (2003), the Maddala and Wu test (1999), the Choi tests (2001, 2002) with and without cross-sectional correlation, the Chang IV test (2002), the Phillips and Sul test (2003), and the Pesaran test (2007). The experiments include a case of no spatial correlation as well as four types of spatial corre- lation (SAR, SMA, SEC1, and SEC3), with two values of the parameters indi- cating weak versus strong spatial dependence. They also consider 10 weight matrices, differing in their degree of sparseness, four pairs of (N, T) and two models includingindividualeffectsandindividual deterministic trends.Even with this modest design, the total number of experiments considered is 1600. They find that ignoring spatial dependence when present can seriously bias the size of panel unit root tests. 15.6 Extensions Elhorst (2003) considers theMLestimationofafixed and random effects panel data model extended either to include spatial error autocorrelation or a spa- tially lagged dependent variable. This is also extended to the case of random coefficients model. In another paper, Elhorst (2005) considers the estimation of a fixed effects dynamic panel data model extended either to include spa- tial error autocorrelation or a spatially lagged dependent variable. The latter models arefirstdifferencedtoeliminate the fixedeffectsandthen the uncondi- tional likelihood function is derived taking into account the density function of the first-differenced observations on each spatial unit. Lee and Yu (2010) consider the estimation of a SAR panel model with fixed effects and SAR dis- turbances. If T is finite but N is large, they show that direct ML estimation of all the parameters including the fixed effects will yield consistent estimators except for the variance of disturbances. Using a transformation that elimi- nates the individual fixed effects, they provide consistent estimates for all the parameters including the variance of disturbances. The transformation ap- proach is shown to be a conditional likelihood approach if the disturbances are normally distributed. Next, they extend their results to the SAR model with both individual and time-fixed effects. In this case, the transformation approach yields consistent estimators of all the parameters when either N or T are large. For the direct approach, consistency of the variance parameter requires both N and T to be large and consistency of other parameters re- quires N to be large. Monte Carlo results are provided illustrating the finite sample properties of the various estimators with N and/or T being small or moderately large. P1: GOPAL JOSHI November 3, 2010 17:3 C7035 C7035˙C015 Spatial Panels 451 Yu, de Jong, and Lee (2007, 2008) study the asymptotic properties of quasi- maximum likelihood estimators for spatial dynamic panel data with fixed effects when both the number of individuals N and the number of time periods T are large. They cover both the stationary and nonstationary cases. When the roots in the DGP are not all unitary, the estimators’ rates of con- vergence will be the same as the stationary case, and the estimators can be asymptotically normal.Infact,for the distribution of thecommonparameters, when T is asymptotically large relative to N, the estimators are √ NT con- sistent and asymptotically normal, with the limiting distribution centered around 0. When N is asymptotically proportional to T, the estimators are √ NT consistent and asymptotically normal, but the limiting distribution is not centered around 0. When N is large relative to T, the estimators are con- sistent with rate T, and have a degenerate limiting distribution. Compared to the stationary case, the estimators’ rate of convergence will be the same, but the asymptotic variance matrix will be driven by the nonstationary com- ponent and it is singular. Consequently, a linear combination of the spatial and dynamic effects can converge at a higher rate. They also propose a bias correction which performs well when T grows faster than N 1/3 . Pesaran and Tosetti (2008) study large panel data sets where even after con- ditioning on common observed effects the cross-section units might remain dependently distributed. This could be due to unobserved common factors and/or spatial effects. They introduce the concepts of time-specific weak and strong cross-section dependence and show that the commonly used spatial modelsareexamples ofweakcross-sectiondependence.Pesaran’s (2006)com- mon correlated effects (CCE) estimator of paneldata model with a multifactor error structure continues to provide consistent estimates of the slope coeffi- cient, even in the presence of spatial error processes. This chapter highlights some of the recent research in spatial panels. Due to space limitations, several applications and related extensions have not been discussed. Hopefully, this will entice the reader to read more papers on this subject and spur some needed research in this area. 15.7 Acknowledgment A preliminary version of this chapter was presented as a keynote speech at the 13th African Econometric Society meeting held at the University of Pre- toria, South Africa, July 9–11, 2008. Also as the keynote address for the 10th Econometrics and Statistics Symposium held at Ataturk University, Turkey, May 27–29, 2009, and in a session in honor of Cheng Hsiao at the 15th Interna- tional Conference on Panel Data at the University of Bonn, Germany, July 3–5, 2009. Iwouldliketo thank my coauthors Georges Bresson, AlainPirotte,Dong Li, Seuck Heun Song, Peter Egger, Michael Pfaffermayer, Byoung Cheol Jung, Jae Hyeok Kwon, and Won Koh for allowing me to draw freely on our work. P1: GOPAL JOSHI November 3, 2010 17:3 C7035 C7035˙C015 452 Handbook of Empirical Economics and Finance References Anselin, L. 1988. Spatial Econometrics: Methods and Models. Dordrecht: Kluwer Aca- demic Publishers. Anselin, L. 2001. Spatial econometrics. In B. Baltagi, (ed.). A Companion to Theoretical Econometrics. pp. 310–330. Oxford, U.K.: Blackwell. Anselin, L., and A. K. Bera. 1998. Spatial dependence in linear regression models with an introduction tospatial econometrics. In: A.Ullah, D.E.A. 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P1: BINAYA KUMAR DASH November 1, 2010 17:9 C7035 C7035˙C016 16 Nonparametric and Semiparametric Panel Econometric Models: Estimation and Testing Liangjun Su and Aman Ullah CONTENTS 16.1 Introduction 456 16.2 Nonparametric Panel Data Models with Random Effects 458 16.2.1 Local Linear Least Squares Estimator 458 16.2.2 More Efficient Estimation 459 16.3 Nonparametric Panel Data Model with Fixed Effects 460 16.3.1 Profile Least Squares Estimators 461 16.3.2 Measure of Goodness-of-Fit 463 16.3.3 Differencing Method 464 16.3.4 Series Estimation 468 16.3.5 A Nonparametric Hausman Test 468 16.4 Partially Linear Panel Data Models 469 16.4.1 Partially Linear Panel Data Models with Random Effects 469 16.4.2 Partially Linear Panel Data Models with Fixed Effects 471 16.4.3 Extensions 474 16.4.4 Specification Tests 474 16.5 Varying Coefficient Panel Data Models 476 16.5.1 Profile Least Squares Method 476 16.5.2 Differencing Method 477 16.5.3 Nonparametric GMM Estimation 478 16.5.4 Testing Random Effects versus Fixed Effects 482 16.6 Nonparametric Panel Data Models with Cross-Section Dependence 482 16.6.1 Common Correlated Effect (CCE) Estimator 483 16.6.2 Estimating the Homogenous Relationship 484 16.6.3 Specification Tests 485 16.7 Nonseparable Nonparametric Panel Data Models 486 16.7.1 Partially Separable Nonparametric Panel Data Models 486 16.7.2 Fully Nonseparable Nonparametric Panel Data Models 487 455 [...]... 17:9 C7035 C7035˙C016 468 Handbook of Empirical Economics and Finance See Pagan and Ullah (1999) and Li and Racine (2007) However, the magnitude of bias differs with −h 2 m(2) (x) f (1) (x)␬2 /(2 f (x)) which arises due to the local weighted average differencing, but the magnitude of variance remains the same A similar idea can be applied to the case of time differenced model Lee and Mukherjee (2008) suggest...P1: BINAYA KUMAR DASH November 1, 2010 17:9 C7035 C7035˙C016 456 Handbook of Empirical Economics and Finance 16.7.2.1 16.7.2.2 Local Average Response (LAR) Estimator 488 Structural Function and Distribution (SFD) Estimator 490 16.7.2.3 Nonparametric Identification and Estimation without Monotonicity 491 16.7.3 Testing of Monotonicity in Nonseparable Nonparametric Panel Data Models ... T, (16.46) where xit and zit are of dimensions p × 1 and q × 1, respectively, ␤0 is a p × 1 vector of unknown parameters, m(·) is an unknown smooth function, ␣i is random or fixed effects, and uit is the idiosyncratic disturbance We will first discuss the estimation of Equation 16.46 when ␣i represents the random effects and then the fixed effects model We also comment on extensions and specification tests... Models with Random Effects Let εit = ␣i + uit We can rewrite Equation 16.46 as yit = xit ␤0 + m(zit ) + εit (16.47) In the literature, it is frequently assumed that E(εit |zit ) = 0 (16.48) P1: BINAYA KUMAR DASH November 1, 2010 17:9 C7035 C7035˙C016 470 Handbook of Empirical Economics and Finance Note that this assumption does not rule out the dependence between xit and εit As a matter of fact, some... suggested by Li and Stengos (1996) and Li and Ullah (1998) 16.4.2 Partially Linear Panel Data Models with Fixed Effects We now discuss the estimation of Equation 16.46 when ␣i represents the n fixed effect For the identification purpose, we can impose i=1 ␣i = 0 For P1: BINAYA KUMAR DASH November 1, 2010 17:9 472 C7035 C7035˙C016 Handbook of Empirical Economics and Finance simplicity, we assume that... element yi,t−1 Assume the existence of an IV wit ∈ Rl with l ≥ p such that E(Uit |wit , zit ) = 0, and Cov(wit , Xit ) = 0 (16.66) We can estimate ␤0 = (␤0,1 , ␤0,2 ) by the IV method for the case l = p: ˆ ˜ ˜ ˜ ˜ ␤ I V = [(W − W) ( X − X)]− (W − W) (Y − Y), (16.67) P1: BINAYA KUMAR DASH November 1, 2010 17:9 C7035 C7035˙C016 474 Handbook of Empirical Economics and Finance and estimate m(z) by ˆ m I V (z)... null and alternative and compared the squared disa tance between the estimated models For example, to test H0 : Equation 16.71 a versus H1 : Equation 16.72, they estimate both Equations 16.71 and 16.72 and base their test statistic on a Jn = 1 nT n T ˜ ˆ [xit ␤ + zit ␥ − xit ␤ − m(zit )]2 ˜ ˆ i=1 t=1 a ˆ ˜ where ␤ and ␥ are estimates of ␤0 and ␥0 under H0 , ␤ and m(zit ) are estimates ˜ ˆ a of ␤0 and. .. 16.40, Li and Stengos (1996) suggest estimation of m(xit , xi,t−1 ) = m(xit ) − m(xi,t−1 ) by doing a local linear regression of yit on xit and xi,t−1 Then we can obtain estimates of m(x) by the method of estimating nonparametric additive models, e.g., by the marginal integration method of Linton and Nielson (1995) or by the backfitting method For example, after we obtain estimates m(x, xi,t−1 ) of ˆ m(x,... 1, 2010 17:9 C7035 C7035˙C016 466 Handbook of Empirical Economics and Finance where xi = T −1 follows as T t=1 xit The local linear within-group estimator of ␤(x) then −1 T n ˆ ␤W (x) = (xit − xi )(xit − xi ) K h (xit − x) i=1 t=1 n T (xit − xi )( yit − yi ) K h (xit − x) ¯ i=1 t=1 Similarly, if we use the first differencing method, then the local linear estimator of ␤(x) for some fixed element x in... i.i.d (0, ␴2 ), and ␣i and u jt are ␣ u uncorrelated for all i, j = 1, , n and t = 1, , T We remark that some of these assumptions can be relaxed and specification testing is also possible Let εit = ␣i + uit , εi = (εi1 , , εi T ) and εi = (ε1 , , εn ) Then ≡ E(εi εi ) = ␴2 IT + ␴2 l T l T and ≡ E(εε ) = In ⊗ We first discuss local u ␣ linear least squares (LLLS) estimator of m and its first-order . C7035˙C015 454 Handbook of Empirical Economics and Finance Maddala, G. S., and S. Wu. 1999. A comparative study of unit root tests with panel data and a new simple test. Oxford Bulletin of Economics and. com- pare the empirical size of panel unit root tests with and without spatial error P1: GOPAL JOSHI November 3, 2010 17:3 C7035 C7035˙C015 450 Handbook of Empirical Economics and Finance dependence Models 486 16. 7.2 Fully Nonseparable Nonparametric Panel Data Models 487 455 P1: BINAYA KUMAR DASH November 1, 2010 17:9 C7035 C7035˙C 016 456 Handbook of Empirical Economics and Finance 16. 7.2.1