A feasible Spline-kernel estimate for short cross-sectional dependence panel data models

Abstract In this paper, we take two different perspectives to cross-sectional dependence panel data model. We introduce the Spline-kernel method and the Quasi-Difference method to estimate the cross-sectional dependence panel data models in large n small T framework. Asymptotic distributions for the Spline-kernel estimators are derived. Monte Carlo simulations show good performances in finite samples. Empirical analysis of the impact of China’s infrastructure investment on economic growth suggests that infrastructure investment and urbanization rate showed very significant positive effects on current China’s economic growth.


Introduction
Recently large panels allowing for spatial correlation have aroused great enthusiasm among econometricians. To account for the spatial correlation, a typical panel model often incorporates a factor structure into its errors. For example, Pesaran (2006) considered a parametric regression model in which the error terms have a factor structure and suggested the Common Correlated Effects (CCE) estimation procedure. Su and Jin (2012) extended this parametric model into a nonparametric one which can be written as y it ¼ a i c t þ gðz it Þ þ it , i ¼ 1, :::, n, t ¼ 1, :::, T, where z it is a q Â 1 regressors, gðÁÞ is an unknown smooth function, a i and c t are unobservable loadings and factors, it is the idiosyncratic error term. Bai (2009) calls the term a i c t as the Interactive Fixed Effects. Since the common factors influence all y it simultaneously, the interactive fixed effects is considered as a means to model the spatial correlation. However, in this paper, we take another perspective with respect to model (1). We think of a i as the heterogeneous characters that is unique to the i-th unit, c t as the time effects which make the heterogeneous characters vary when time elapses. In fact, a i c t is more often interpreted as heterogenous impacts of common fators/shocks when T is fixed (Holtz-Eakin, Newey, and Rosen 1988;Bai 2003;Ahn, Lee, and Schmidt 2013). This kind of interpretation is quite relevant in microeconomics. For example, when we study the relationship between salaries and years of educations, we could think of a i as the individual ability that cannot be observed directly and c t as a time effect to measure the change of ability with time.
Since commonly obtained panel data have observations on a large number of individuals and a small number of periods, another obstacle in estimating such a model involves dealing with the interactive fixed effects in the case of large n and small T. This feature is contrast to Pesaran (2006), Bai (2009) and Su and Jin (2012) who consider large n and large T.
To solve these difficulties in a framework of nonparametric panel data model with interactive fixed effects, we extend two main approaches to our nonparametric models. Our approach is to propose a relationship (Mundlak 1978) between a i and z i which is the averaging over t for a given i of z it : Another approach is to eliminate the interaction fixed effect in the model by quasi difference method, which has been used by Nauges and Thomas (2003), Hayakawa, Pesaran, and Smith (2018).
The paper is structured as follows. In section 2 we introduce the model and describes the estimation procedures. Furthermore, we study the asymptotic properties in Section 3. In section 4 we examines the finite sample performances of our estimators via Monte Carlo simulations. Empirical analysis of the impact of China's infrastructure investment on economic growth can be found in Section 5. All the technical proofs are listed in the appendix.

Estimation procedures
In this section, we consider two different methods to estimate the model (1). In the first method, the unobservable effects a i considered to be unknown coefficients. Then the model (1) can be estimated by Spline-kernel estimators. The alternative method is about Quasi-Difference method to sweep out the fixed effects a i . The detailed estimation procedures are illustrated as follows.

Asymptotic properties
In this section, we will present the asymptotic properties of the Spline-kernel estimatorŝ GðÁÞ: The Quasi-Difference estimators can be obtained by similar arguments. First, we introduce some notation. Let jjPjj represent Euclidean norm when P is a vector, and Frobenius norm when P is a matrix. Moreover, let € g ðÁÞ be the second derivative of gðÁÞ, I q be an identity matrix of dimension q, j ¼ maxfj 1 , j 2 g, f ðzÞ ¼ P T t¼1 f t ðzÞ, r 2 ðzÞ ¼ Eð 2 it jzÞ: Before giving the large sample distributions of the estimators, we first put forward some conditions, which may not be the possible weakest ones but convenient for technical proofs.

A1
(1) ðY i , Z i Þ, i ¼ 1, :::, n are i.i.d; (2) The errors term it are uncorrelated across i and t; A2 (1) The density functions f t ðÁÞ of z it with compact support Z q , have continuous second partial derivatives, and is bounded away from zero and infinite.
(2) The kernel function KðÁÞ on R q is a continuous symmetric density function with a compact support; A3 (1) The functions /ðÁÞ and gðÁÞ are d-smooth functions where d ! 2: (2) The basis functions, fp i ðÁÞi ¼ 1, :::, jg are twice continuously differentiable on the compact support, and the eigenvalue of Q has the smallest eigenvalues bounded away from zero uniformly in j.

Simulation study
In this section we evaluate the finite sample performances of Spline-kernel estimators and Quasi-Difference estimators by Monte Carlo simulations. We consider the following data generating process (DGP): where gðz it Þ ¼ z it À z 2 it and a i ¼ ð 1 T P T t¼1 z it Þ a þ t i : Let t i , e it and c t are i.i.d.N(0, 1), z it is i.i.d.U½À1, 1: To implement our estimation procedure, the number of interior knots j is chosen by rule of thumb (Wang and Yang 2007), that is, j ¼ 0:5ðnTÞ 2=5 log ðnTÞ: We use Epanechnikovs kernel and the bandwidth h is chosen by the leave-one-unit-out cross validation method. In order to evaluate the estimation accuracy of the estimators, we report the average mean squared errors (AMSE) for the nonparametric function where M is the number of replications. In Table 1, we report the average mean square errors (AMSE) and average fixed effect noise to signal ratiosr ac =r y of Spline-kernel estimators and Quasi-Difference estimators, wherer ac andr y are the sample standard deviations of a i c t and y it . we firstly observe the AMSE of the both estimators reduce fast as n and T increase. It is obvious that the both estimates behave better and better when the sample size increases. By comparing the both estimators at a ¼ 0.5 and a ¼ 1, It is obvious that the the Quasi-Difference estimators perform less accurate than Spline-kernel estimators, where a indicates the correlations between covariates. Moreover, we found that the average fixed-effect-noise-to-signal ratios increase evidently, but the both estimators still work robustly.
Meanwhile, Figures 1 and 2 show the performances of curve fit where the mean, the 10 percentile and 90 percentile of the 1000 estimates ofĝ ðÁÞ are given. Figure 1 displays the performances of curve fit under the case n ¼ 300, T ¼ 4 and a ¼ 0.5. Figure 2 displays the performances of curve fit under the case n ¼ 300, T ¼ 4 and a ¼ 1. In summary, the both estimators all perform well. However, the Spline-kernel estimators perform more robustly, When z it and a i are strongly correlated.

Application: the impact of China's infrastructure investment on economic growth
It is generally accepted in the literatures that strengthening infrastructure investment is conducived to economic growth. Expanding aggregate demand by increasing investment in infrastructure is one of the important measures of counter-cyclical macroeconomic regulations in China. In order to cushion the impact on China's economy during the global financial crisis in 2008, the Chinese government launched a 4 trillion yuan stimulus plan, most of which was spent on infrastructure construction. After 2012, with the economic growth slowing down, the Chinese government has once again increased its investment in infrastructure such as railways, water conservancy and urban rail transit. Thus, researchers focus on how China's infrastructure investment affects economic growth. Economic theory has always had different views on whether infrastructure investment can promote economic growth. At the beginning of the 20th century, Keynes (1936) believed that expanding fiscal and monetary policies to increase infrastructure investment could promote economic growth. However, the neoclassical economists believe that the expansion of infrastructure will ultimately be subject to the law of diminishing marginal returns. Thus, there is a reverse U relationship with economic growth. China's long-term and vigorous infrastructure investment will provide the most favorable practical evidence for our research. The data comes from the CEIC database and our analysis uses a panel of roughly 31 provinces of china between 2004 to 2019. We deflate investments using the annual CPI-U so that they are in year 2004 RMB.
We first find that there are significant individual effects and time effects through Wald-Test. It is obvious that our proposed model (1) is well suited to to study the impact of the infrastructure investment on China's economic growth. Let i denote the provinces in China, t denote years, a i denote the fixed effects which can be used to control possible invariant unobserved characteristics, c t denote the common factor, growth it denote the real GDP growth rate, Inv it denote the logarithm of real per capita infrastructure investment. The linear part vectors x it Contains six control variables: Ins it be the industrial ratio which represents the industrial structure of different provinces, Exp it be the export dependence which represented by the proportion of exports to GDP. Fir it be the fixed Investment Ratio which represented by the proportion of fixed capital to GDP. Lgr it be the labor force growth rate, Dou it be the degrees of Urbanization, Lpg itÀ1 be the the lagged GDP per capita. We use the following model to estimate the effect of infrastructure investment on China's economic growth: where e it is any time-varying unobserved shocks to the GDP growth rate. Next, we use the Hausman test to check if only the time mean is important in the regression. Let the null hypothesis H 0 : Since the Quasi-Difference estimation is consistent when null hypothesis does not hold, but  the Spline-kernel estimation is in consistent. Therefore, we can compute the Huasman statistic as: Which has a v 2 distribution under the null hypothesis. The results showed that the null hypothesis could not be rejected. Therefore, we use two methods to estimate this model. Tables 2 and 3 lists the parametric estimators and Figure 3 shows the curve of our nonparametric estimators. By comparing the conclusion of the two methods, we can get the following conclusions directly: (1) the fixed investment Ratio showed negatively effects on Economic growth. (2) The proportion of urbanization rate helped to speed up economic growth.
(3) The current infrastructure investment showed significant positive effect on economic growth.