Nonparametric Time-Varying Coefficient Models for Panel Data

Abstract

The collection rate of contributions to public pension (CRCP), expressed as the ratio of the actual contributions to the expected contributions from insurers, is a key component of the public pension system in China. Recent years have seen various patterns of change in CRCPs at the provincial level. In order to study the drastic changes in a short time and understand their underlying implications, we propose a nonparametric time-varying coefficients model for longitudinal data with pre-specified finite time points, also known as panel data. By utilizing a penalized least squares method, the proposed method enables estimation of a large number of parameters, which can exceed the sample size. The resulting estimator is shown to be efficient, robust, and computationally feasible. Furthermore, it possesses desirable theoretical properties such as $n^{1/2}$-consistency, asymptotic normality, and the oracle property.

Appendices

Appendix

Proof of Theorem 1

Let $\alpha _n=n^{-1/2}+a_n$. Denote by $\theta _0$ the true value of $\theta$. We want to show that for any given $\varepsilon >0$, there exists a large constant C such that

$$\begin{aligned} \text{ Pr }\left\{ \inf \limits _{\Vert u\Vert =C} L_n(\theta _0+ \alpha _n \cdot u) > L_n(\theta _0) \right\} \ge 1-\varepsilon . \end{aligned}$$

(5.1)

This implies with a probability larger than $1-\varepsilon$ that there exists a local minimum in the ball $\{\theta _0+ \alpha _n \cdot u: \Vert u\Vert \le C\}.$ Hence, there exists a local minimizer such that $\Vert {\widehat{\theta }}-\theta _0\Vert =O_p(\alpha _n)$.

Define $\theta ^{*}=\theta _0 + \alpha _n \cdot u=(\theta _1^{*},\ldots ,\theta _{Tp+T-1}^{*})^{\prime }$, using $p_{\lambda }(0)=0$, we have

$$\begin{aligned} D_n(\theta ^{*})= L_n(\theta ^{*} ) - L_n(\theta _0) \ge S (\theta ^{*}) -S(\theta _0)+ \sum \limits _{j=1}^{m} \{p_{\lambda }(|\theta ^{*}_{j}|)-p_{\lambda }(|\theta _{j0}|)\}, \end{aligned}$$

where $S(\theta )=\frac{1}{n}\sum _{i=1}^{n}\sum _{t>s}\Big \{Y_{it} -Y_{is}-\sum _{d=s+1}^t g_d-\sum _{j=1}^p\Big (\sum _{d=1}^t \gamma _{dj}X_{it,j}-\sum _{d=1}^s \gamma _{dj}X_{is,j}\Big ) \Big \}^2$, m is the number of components of $\theta ^{(1)}_0$. Let ${\dot{S}}$ be the gradient vector of S; by the standard argument of the Taylor expansion, we have

$$\begin{aligned} D_n(\theta ^{*})\ge & {} {\dot{S}}(\theta _0)^{\prime }(\theta ^{*}-\theta _0) +(\theta ^{*}-\theta _0)^{\prime } \ddot{S}(\theta _0)(\theta ^{*} -\theta _0)\{1+o_p(1)\}\nonumber \\&+\sum \limits _{j=1}^m [{\dot{p}}_{\lambda }(|\theta _{j0}|) \text{ sgn }(\theta _{j0})(\theta _{j}^{*}-\theta _{j0}) +\ddot{p}_{\lambda }(|\theta _{j0}|)\left( \theta _{j}^{*} -\theta _{j0}\right) ^2\{1+o(1)\}]\nonumber \\&\widehat{=}&I_1+I_2+I_3. \end{aligned}$$

(5.2)

Noting that $E\{{\dot{S}}(\theta _0)\}=0$ and $Var\{{\dot{S}}(\theta _0)\}=O(n^{-1})$, by the central limit theory we have

$$\begin{aligned} I_1 = \left\{ E\{{\dot{S}}(\theta _0)\} +O_p\left( \sqrt{\text{ Var }({\dot{S}}(\theta _0)}\right) \right\} (\theta ^{*}-\theta _0) =O_p \left( \frac{\alpha _n}{\sqrt{n}}\right) . \end{aligned}$$

(5.3)

Similarly, we get

$$\begin{aligned} I_2 =O(\alpha _n^2C^2). \end{aligned}$$

(5.4)

For $I_3$, it is easy to see that it is bounded by

$$\begin{aligned} m\alpha _n a_n C+\alpha _n^2 \max \{|\ddot{p}_{\lambda } (|\theta _{j0}|)|:|\theta _{j0}| \ne 0\}C^2. \end{aligned}$$

(5.5)

From (5.3), (5.4), and (5.5), $I_1$ and $I_3$ are dominated by $I_2$. Hence, by choosing a sufficiently large C, (5.1) holds. $\square$

Proof of Theorem 2

We first show that with a probability tending to 1, for any given $\theta ^{(1)}$ satisfying $\Vert \theta ^{(1)}-\theta ^{(1)}_0\Vert =O_p(n^{-1/2})$ and any constant C,

$$\begin{aligned} L_n(({\theta ^{(1)}}',{0'})')=\min \limits _{\Vert \theta ^{(2)}\Vert \le Cn^{-1/2}} L_n(({\theta ^{(1)}}',{\theta ^{(2)}}')'). \end{aligned}$$

(5.6)

To show (5.6), by Taylor’s expansion, we have

$$\begin{aligned} \frac{\partial L_n(\theta )}{\partial \theta _r}= & {} -\frac{2}{n}\sum _{i=1}^{n}\sum _{t>s}\left\{ Y_{it} -Y_{is}-\sum _{d=s+1}^t g_d-\sum _{j=1}^p\left( \sum _{d=1}^t\gamma _{dj}X_{it,j} -\sum _{d=1}^s \gamma _{dj}X_{is,j}\right) \right\} \nonumber \\&\times \frac{\partial }{\partial \theta _r}\left\{ \sum _{d=s+1}^t g_d+\sum _{j=1}^p\left( \sum _{d=1}^t \gamma _{dj}X_{it,j}-\sum _{d=1}^s \gamma _{dj}X_{is,j}\right) \right\} \\&+{\dot{p}}_{\lambda }(|\theta _r|) \text{ sgn }(\theta _r). \end{aligned}$$

By the central limit theorem, we have

$$\begin{aligned} \frac{\partial L_n(\theta )}{\partial \theta _r} =\lambda \{-\lambda ^{-1}{\dot{p}}_{\lambda }(|\theta _r|) \text{ sgn }(\theta _r)+O_p(n^{-1/2}/\lambda )\}, \end{aligned}$$

where $\liminf \limits _{n\rightarrow \infty }\liminf \limits _{\theta \rightarrow 0^{+}} \lambda ^{-1}{\dot{p}}_{\lambda }(\theta )>0$ and $n^{-1/2}/\lambda \rightarrow 0$. The sign of the derivative is completely determined by that of $\theta _r$. Hence (5.6) follows.

By (5.6), Part (a) follows. Now we prove Part (b). It can be shown that there exists ${\widehat{\theta }}^{(1)}$ in Theorem 1 that is a $n^{1/2}$- consistent local maximizer of $L_n(({\theta ^{(1)}}', 0')')$, which is regarded as a function of $\theta ^{(1)}$, and that satisfies the following equation:

$$\begin{aligned} \frac{\partial L_n(\theta )}{\partial \theta _r}\Bigg |_{\theta =({\theta }^{(1)},0)^{\prime }} =0, \qquad \text{ for } \quad r=1,\ldots ,m. \end{aligned}$$

Note that ${\widehat{\theta }}^{(1)}$ is a constant estimator. Thus, we have

$$\begin{aligned} 0= & {} \frac{\partial L_n(\theta )}{\partial \theta _r}\Bigg |_{\theta =({\theta }^{(1)},0)^{\prime }}= \frac{S(\theta )}{\partial \theta _r} \Bigg |_{\theta =({\theta }^{(1)},0)^{\prime }}+{\dot{p}}_{\lambda } (|{\widehat{\theta }}_r|)\text{ sgn }({\widehat{\theta }}_r) \\= & {} \frac{\partial S(\theta _0)}{\partial \theta _r}+\sum \limits _{l=1}^m \left\{ \frac{\partial ^2 S(\theta _0)}{\partial \theta _r \theta _l}+o(1) \right\} ({\widehat{\theta }}_l-\theta _{l0})\\&+{\dot{p}}_{\lambda }(|\theta _{r0}|)\text{ sgn }(\theta _{r0}) +\{\ddot{p}_{\lambda }(|\theta _{r0}|) +o_p(1)\}({\widehat{\theta }}_r-\theta _{r0}). \end{aligned}$$

Furthermore, we have

$$\begin{aligned} \frac{\partial ^2 S(\theta _0)}{\partial \theta ^{(1)}\partial {\theta ^{(1)}}^{\prime }}= 2\varLambda (1+o_p(1)), \end{aligned}$$

where

$$\begin{aligned} \varLambda =\lim _{n\rightarrow \infty } \frac{1}{n}\sum _{i=1}^{n} \sum _{t>s} \left[ \frac{\partial }{\partial \theta ^{(1)}} \left\{ \sum _{d=s+1}^t g_d+\sum _{j=1}^p \left( \sum _{d=1}^t \gamma _{dj}X_{it,j}-\sum _{d=1}^s \gamma _{dj}X_{is,j}\right) \right\} \right] ^{\otimes 2}\big |_{\theta =\theta _0}. \end{aligned}$$

Hence following by Slutsky’s theorem we have

$$\begin{aligned} \sqrt{n}(2\varLambda +\varSigma )\left\{ {\widehat{\theta }}^{(1)} -\theta ^{(1)}_0+(2\varLambda +\varSigma )^{-1}{b}\right\} = \sqrt{n}\frac{\partial S(\theta _0)}{\partial \theta ^{(1)}}+o_p(1). \end{aligned}$$

This completes the proof of Part (b). $\square$