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Partially Linear Functional-Coeffi Cient Dynamic Panel Data Models: Sieve

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Partially Linear Functional-Coeffi Cient Dynamic Panel Data Models: Sieve Partially Linear Functional-Coeffi cient Dynamic Panel Data Models: Sieve Estimation and Specification Testing Yonghui Zhangy Qiankun Zhouz Job Market Paper This version: November 10, 2016 Abstract In this paper, we study the nonparametric estimation and testing for the partially linear functional-coeffi cient dynamic panel data models where the effects of some covariates on the dependent variable vary according to a set of low-dimensional variables nanparametrically. Based on the sieve approximation of unknown functions, we propose a sieve 2SLS procedure to estimate the model. The asymptotic properties for both parametric and nonparametric components are established when sample size N and T tend to infinity jointly or only N goes to infinity. We also propose a specification test for the constancy of slopes, and we show that after being appropriately standardized, our test is asymptotically normally distributed under the null hypothesis. Monte Carlo simulations show that our sieve 2SLS estimators and test perform remarkably well in finite samples. We apply our method to study the effect of income on democracy and find strong evidence of nonconstant effect of income on democracy. Key words: Dynamic panel models, Sieve approximation, Functional-coeffi cient, 2SLS estimation, Specification testing JEL Classification: C12, C23, C26, C33, C38. Address correspondence to: Qiankun Zhou, Department of Economics, State University of New York at Binghamton, Binghamton, NY 13902, USA. Email: [email protected]. Zhang gratefully acknowledges the financial support from the National Science Foundation of China under Grant 71401166. All errors are the authors’sole responsibilities. ySchool of Economics, Renmin University of China, Beijing, China. zDepartment of Economics, State University of New York at Binghamton, Binghamton, NY, 13902, USA. 1 1 Introduction ≥ Since the seminal work of Balestra and Nerlove (1966), there is rich literature on the research of dynamic panel data models among both theoretical and empirical economists. Based on the influential work of Anderson and Hsiao (1981, 1982), using the two stage least squares (2SLS) or generalized method of moments (GMM) to estimate the dynamic panel data model has received lots of attention in the literature. To name a few, see Arellano and Bond (1991) and Alvarez and Arellano (2003), among others. However, it should be pointed out that linear parametric form is generally assumed in the aforementioned researches of dynamic panel models, and it is well-known that parametric dynamic panel data models might not be flexible enough to capture nonlinear structure in practice, such a failure may result in model misspecification issue. To deal with this issue, various nonparametric or semiparametric dynamic panel data models have been proposed. For example, in earlier work, Li and Ullah (1998) and Baltagi and Li (2002) consider semiparametric estimation of partially linear dynamic panel data models using instrumental variable methods. More recently, in order to allow coeffi cients to depend on some informative variables, research of varying-coeffi cient models has received lots of attention. For the varying coeffi cient models, it has wide application in the economics literature. As for the first example, in the traditional labor economics literature of return to schooling, researchers usually apply linear IV regression model. However, Card (2001) finds that the returns to education tend to be underestimated by using the 2SLS method when one ignores the nonlinearity and the interaction between schooling and working experience, and Schultz (2003) argues that the marginal returns to education may vary with different levels of working experience and schooling. This motivates Cai et al. (2010) and Su et al. (2014) to consider the partially linear functional coeffi cient model. Apparently, both models allow the impact of education on the log-wage to vary with working experience. Another application of varying coeffi cient model is the heterogenous effects of FDI on economic growth. Based on the finding of Kottaridi and Stengos (2010), Cai et al. (2010) find that the effect of FDI on economic growth varies across initial income levels, and thus varying coeffi cient model is adapted for such purpose. For other applications of varying- coeffi cient models in economics and finance, refer to Baglan (2010), Cai (2010), Cai et al (2000, 2010) and Cai and Hong (2009), among others. In this paper, we consider a new class of partially linear varying-coeffi cient additive dynamic models, which allows for linearity in some regressors and nonlinearity in other regressors. In 2 other words, some coeffi cients are constant and others are varying over some variables. This new class model is flexible enough to include many existing models as special cases. By extending the model in Cai and Li (2008) to a partially varying-coeffi cient model with fixed effects, we reduce the model dimension without influencing the degree of the model flexibility, and the pNT consistent estimation of parametric coeffi cients can be achieved. We also extend the work of Cai et al. (2015) to sieve estimation instead of kernel estimation. The choice of sieve estimation over kernel estimation is simply because series estimation methods are more convenient than kernel methods under certain type of restrictions (such as additivity or shape- preserving estimation, see Dechevsky and Penez (1997)). It is also computationally convenient because the results can be summarized by a relatively small number of coeffi cients. Based on the sieve approximation of unknown varying-coeffi cient functions, we use the standard approach of taking the first difference to eliminate the fixed effects and use the lagged variables as instruments. This results in a sieve two stage least squares (2SLS) estimation for partially linear functional-coeffi cient dynamic panel models. The asymptotic properties for both parametric and nonparametric components are established when sample size N and T tend to infinity jointly or only N goes to infinity. We also discuss the plausibility of extending the proposed sieve 2SLS estimation procedure to unbalanced dynamic panels. We also propose a nonparametric test for the linearity of the nonparametric component, i.e., slopes of the nonparametric part is constant. This specification test for the constancy of slopes is based on a weighted empirical L2-norm distance between the two estimates under the null and the alternative, respectively. We show that after being appropriately standardized, our test is asymptotically normally distributed under the null hypothesis. Compared with the existing literature of estimation of varying-coeffi cient additive dynamic models, our paper has the following merits. On the first hand, in the existing literature, it is common to use within-group transformation to eliminate the fixed effects, however, such a transformation for dynamic panels will in general lead to biased estimation and bias correction is needed (e.g., Cai and Li (2008), Tran (2014), Rodriguez-Poo and Soberon (2015) and reference therein). However, our paper considers the first difference transformation to remove the fixed effects, and we use the lagged variables as instruments and propose the 2SLS estimation of the constant and varying coeffi cients. It is shown in this paper that the 2SLS estimators are free of asymptotical bias. On the other, instead of assuming the time series dimension is short for dynamic panels (e.g., An et al (2016) and Cai et al (2015)), we establish the asymptotics of the 3 2SLS estimation when both N and T are large, thus our asymptotic results cover the foregoing results as special cases. We also discuss the applicability of the 2SLS estimation when the panel is unbalanced, and it is shown in the simulation that the proposed sieve 2SLS estimation works remarkably well even if the panel is unbalanced. The small sample properties of the sieve 2SLS estimation and specification testing for partial linear varying-coeffi cient additive dynamic models are investigated through Monte Carlo simulation, using six different data generating processes (DGPs). The first four DGPs are designed to check the performance of the sieve 2SLS estimation for the balanced panels, when T is large or fixed, and the fifth DGP is to verify the applicability of the sieve 2SLS estimation for unbalanced panels. From the simulation results, we can observe that the proposed sieve 2SLS works remarkably well for the estimation of both parametric and nonparametric part in the model. Namely, for the estimation of parametric part of the model, the constant coeffi cient can always be consistently estimated, and the shape of estimated functional-coeffi cients is close enough to the true pre-specified functions. Similar findings can be applied to the case when the panel is unbalanced. The last DGP is to investigate the performance of specification test. From the simulation results, we can notice that the empirical size of the specification test is very close to the nominal value and the empirical power increases steadily with the increase of either N or T. Finally, We apply the proposed method to study the relationship of income and democracy as in Acemoglu et al (2008) and Cervellati et al (2014). Through the sieve 2SLS estimation, we find substantial nonlinearity in the relationship between a country’s degree of democracy and its lagged value and a nonlinear relationship between income and democracy. The rest of the paper is organized as follows. We introduce the model and sieve 2SLS estimation in Section 2. Asymptotics for the sieve 2SLS estimation is established in Section 3. We propose the specification test in Section 4, and in Section 5 we conduct a small set of Monte Carlo simulations to evaluate the finite sample performance of the sieve 2SLS estimation and specification testing. We apply our method to study to study the relationship of income per capita and democracy in Section 6. Conclusion are made in Section 7. All technical details are relegated to the Appendix. Notations: For a real matrix A, let A = [tr (A A)]1/2 denotes its Frobenius norm and k k 0 1/2 A = [max (A A)] denotes its spectral norm where max ( ) is the largest eigenvalue of “ ”.
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