Partially linear functional-coefficient dynamic panel data models: sieve estimation and specification testing

Abstract

We study the nonparametric estimation and specification testing for partially linear functional-coefficient dynamic panel data models, where the effects of some covariates on the dependent variable vary nonparametrically according to a set of low-dimensional variables. Based on the sieve approximation of unknown slope functions, we propose a sieve 2SLS procedure to estimate the model. The asymptotic properties of the estimators of both parametric and nonparametric components are established when sample size N and T tend to infinity jointly. A nonparametric specification test for the constancy of slopes is also proposed. We show that after being appropriately standardized, the test is asymptotically normally distributed under the null hypothesis. The asymptotic properties of the test is also studied under a sequence of local Pitman alternatives and global alternatives. A set of Monte Carlo simulations show that our sieve 2SLS estimators and specification test perform remarkably well in finite samples. We apply our method to study the impact of income on democracy, and find strong evidence of nonlinear/nonconstant effect of income on democracy.

Keywords:

• Dynamic panel data models
• Sieve approximation
• functional-coefficient
• 2SLS estimation
• specification testing

JEL Classification:

• C12
• C23
• C26
• C33
• C38

Notes

1 Another more flexible specification is $y_{it}={\sum }_{k=1}^{p_z}{d}_{k,it}{\theta }_l({\mathbf{z}}_{k,it})+{\mathbf{x}}_{it}^{\prime }\mathit{\beta }+{\eta }_i+u_{it},$ where conditional variable ${\mathbf{z}}_{k,it}$ are different in different coefficient function ${\theta }_k(·).$ In addition, the functional-coefficient specification can cover the time-varying coefficient as special case by letting ${\mathbf{z}}_{it}=t/T.$

2 The lagged dependent variables may enter the vectors of ${\mathbf{d}}_{it},{\mathbf{x}}_{it}$ or ${\mathbf{z}}_{it}.$ In particular, when ${\mathbf{d}}_{it}={(y_{i,t-1},\dots y_{i,t-p^{\ast }})}^{\prime }$ and ${\mathbf{z}}_{it}=y_{i,t-q}$ where $1\le q\le p^{\ast }$ for some $p^{\ast }\ge 1,$ we obtain the panel version of functional-coefficient autoregressive model in Chen and Tsay (1993).

3 Different number of basis functions or different basis functions for different unknown function are also allowed. To simplify the illustration, the same number and the same sieve basis functions in the approximation of different coefficient functions are used in the paper.

4 An illustrating example is if $\Delta {\mathbf{x}}_{it}=\Delta y_{it-1},$ then we have $E(\Delta {\mathbf{x}}_{it}\Delta {\epsilon }_{it})\ne 0.$

5 Generally speaking, we can consider the sieve GMM estimate defined by: ${({\widehat{\Gamma }}^{\prime },{\widehat{\mathit{\beta }}}^{\prime })}^{\prime }={[\Delta {\tilde{\mathbf{X}}}^{\prime }\overline{\mathbf{W}}{\overline{\mathbf{A}}}_{NT}{\overline{\mathbf{W}}}^{\prime }\Delta \tilde{\mathbf{X}}]}^{-} \Delta {\tilde{\mathbf{X}}}^{\prime }\overline{\mathbf{W}}{\overline{\mathbf{A}}}_{NT}{\overline{\mathbf{W}}}^{\prime }\Delta \mathbf{Y}, {\overline{\mathbf{A}}}_{NT}$ is a $p_w\times p_w$ weighting matrix that is symmetric and asymptotically positive definite. The asymptotic properties are similar to the 2SLS estimator but the notation is slightly more complicated.

6 One can also construct a test statistic by comparing the two estimates for the whole conditional mean function under the null and the alternative:

$D_{NT}^{\diamond }=\frac{1}{N(T-1)}{\sum }_{i=1}^N{\sum }_{t=2}^T{\Vert {(\widehat{\mathit{\theta }}({\mathbf{z}}_{it})-\stackrel{⌣}{\mathit{\gamma }})}^{\prime }{\mathbf{d}}_{it}+{\mathbf{x}}_{it}(\widehat{\mathit{\beta }}-\stackrel{⌣}{\mathit{\beta }})\Vert }^{2}a({\mathbf{z}}_{it}).$

7 For a simple dynamic model such as (5.6)(2.3) $$\Delta y_{it}=\Delta {\mathbf{H}}_{it}^{\prime }\mathbf{\Gamma }+\Delta {\mathbf{x}}_{it}^{\prime }\mathit{\beta }+\Delta {\epsilon }_{it}, t=2,\dots ,T,i=1,\dots ,N.$$(2.3) with functional coefficient on lagged variable, the within group estimator is given by (Lee, 2014)

$${\widehat{\theta }}_{WG}(z)=\mathbf{h}{(z)}^{\prime }{({\sum }_{i=1}^N{\mathbf{H}}_i^{\prime }{\mathbf{M}}_{T_1}{\mathbf{H}}_i)}^{-1}{\sum }_{i=1}^N{\mathbf{H}}_i^{\prime }{\mathbf{M}}_{T_1}{\mathbf{y}}_i,$$

where $\mathbf{h}(z) \text{and} {\mathbf{H}}_i$ are defined in (2.2)(2.2) $$y_{it}={\mathbf{H}}_{it}^{\prime }\mathbf{\Gamma }+{\mathbf{x}}_{it}^{\prime }\mathit{\beta }+{\eta }_i+{\epsilon }_{it},$$(2.2) , ${\mathbf{M}}_{T_1}={\mathbf{I}}_{T_1}-{\iota }_{T_1}{\iota }_{T_1}^{\prime }/T_1,T_1=T-1,{\iota }_T$ is the $T\times 1$ vector of ones. The bias-corrected estimator is given by (e.g., Lee, 2014)

$${\widehat{\theta }}_{\mathit{BCWG}}(z)={\widehat{\theta }}_{FE}(z)+{\widehat{b}}_k(z)/T_1,$$

where

$${\widehat{b}}_k(z)=\mathbf{h}{(z)}^{\prime }{(\frac{1}{N{T}_1}{\sum }_{i=1}^N{\mathbf{H}}_i^{\prime }{\mathbf{M}}_{T_1}{\mathbf{H}}_i)}^{-1}{\sum }_{j=0}^J\frac{1}{N(T-j)}\sum _{i=1}^N\sum _{t=1}^{T-j-1}\left(1-\frac{j}{J+1}\right)y_{i,t+j}\mathbf{h}(z_{i,t+j+1}){\widehat{\epsilon }}_{it},$$

with ${\widehat{\epsilon }}_{it}=y_{it}-\widehat{\theta }(z_{it})y_{i,t-1}-T_1^{-1}{\sum }_{s=2}^T(y_{is}-\widehat{\theta }(z_{is})y_{i,s-1}).$ For simplicity, we just follow Lee (2014) and set $J=\lfloor T^{1/3}\rfloor$ as the truncation parameter.

8 How to exactly derive the bias term and propose an iterative bias correction procedure for the sieve fixed effects estimator of general functional coefficient dynamic panel models deserves a separate work.

9 Figures of the approximation using L₁, L₂ and L₃ are available upon request.

10 After deletion of missing values, the total observation used in this section is 682, with 92 countries in total, while the five-year varies from 4 to 8, i.e., the data is unbalanced.

Additional information

Funding

Zhang gratefully acknowledges the financial support from National Natural Science Foundation of China10.13039/501100001809 (Grant No.71401166, No.71973141, and No.71873033).