Network vector autoregression with individual effects

Abstract

In recent years, there has been great interest in using network structure to improve classic statistical models in cases where individuals are dependent. The network vector autoregressive (NAR) model assumes that each node’s response can be affected by the average of its connected neighbors. This article focuses on the problem of individual effects in NAR models, as different nodes have different effects on others. We propose a penalty method to estimate the NAR model with different individual effects and investigate some theoretical properties. Two simulation experiments are performed to verify effectiveness and tolerance compared with the original NAR model. The proposed model is also applied to an international trade data set.

Appendix

1. Proof of Theorem 1. Denote by $\lambda _i(M)$ the ith eigenvalue of any arbitrary matrix $M\in \mathbb {R}^{N\times N}$. To ensure the existence of a strict stationary solution, it is only required that $\max _i |\lambda _i(G)|<1$. Note that $\max _i|\lambda _i(W)|\le 1$ by Zhu et al. (2017) and Carlin et al. (2014); thus, we can obtain the following proof using (284) in Petersen and Pedersen (2008) and the Gelfand corollaries:

$$\begin{aligned} \begin{aligned} \max _{1\le i\le N}|\lambda _i(G)|&=\max _{1\le i\le N}|\lambda _i(W\varvec{\eta }+\beta _1I)|\\&=\max _{1\le i\le N}|\beta _1+\lambda _i(W\varvec{\eta })|\\&\le |\beta _1|+\max _{1\le i\le N}|\lambda _i(W\varvec{\eta })|\\&\le |\beta _1|+\max _{1\le i\le N}|\lambda _i(W)|\cdot \max _{1\le i\le N}|\lambda _i(\varvec{\eta })|\\&\le |\beta _1|+\max _{1\le i\le N}|\eta _i|<1. \end{aligned} \end{aligned}$$

After proving the existence of the strict stationary solution, it is easy to obtain the form and the distribution of the solution given $\mathbb {Z}$, using Theorem 1 and Proposition 1 in Zhu et al. (2017).

2. Proof of Theorem 2.

Denote

$$\begin{aligned} {\hat{\varSigma }}=\frac{1}{T}\begin{pmatrix} \sum \limits _{t=1}^T\mathbb {V}_{t-1}^\top \mathbb {V}_{t-1} &{}\ \sum \limits _{t=1}^T\mathbb {V}_{t-1}^\top \mathbb {X}_{t-1}\\ \sum \limits _{t=1}^T\mathbb {X}_{t-1}^\top \mathbb {V}_{t-1} &{}\ \sum \limits _{t=1}^T\mathbb {X}_{t-1}^\top \mathbb {X}_{t-1}+\lambda L\\ \end{pmatrix}\quad \text {and}\quad {\hat{\varSigma }}_{xe}=\frac{1}{T}\begin{pmatrix} \sum \limits _{t=1}^T\mathbb {V}_{t-1}^\top {\mathcal {E}}_t\\ \sum \limits _{t=1}^T\mathbb {X}_{t-1}^\top {\mathcal {E}}_t\\ \end{pmatrix}. \end{aligned}$$

According to (6) and $\mathbb {Y}_t=\mathbb {V}_{t-1}\theta +\mathbb {X}_{t-1}+{\mathcal {E}}_t$, we have

$$\begin{aligned} \begin{aligned} \begin{pmatrix} {\hat{\theta }}\\ {\hat{\eta }}\\ \end{pmatrix}&=\begin{pmatrix} \theta \\ \eta \end{pmatrix}-{\hat{\varSigma }}^{-1} \begin{pmatrix} \mathbf {0} &{}\mathbf {0} \\ \mathbf {0} &{}\lambda L/T \end{pmatrix}\begin{pmatrix} \theta \\ \eta \\ \end{pmatrix} +{\hat{\varSigma }}^{-1} {\hat{\varSigma }}_{xe}. \end{aligned} \end{aligned}$$

Then Theorem 2 holds if

$$\begin{aligned}&{\hat{\varSigma }}{\mathop {\longrightarrow }\limits ^{P}}\varSigma , \end{aligned}$$

(A.1)

$$\begin{aligned}&\sqrt{T}{\hat{\varSigma }}_{xe}{\mathop {\longrightarrow }\limits ^{D}}N(0,\sigma ^2\varSigma ), \end{aligned}$$

(A.2)

as $T\rightarrow \infty$. We will prove (A.1) and (A.2) respectively.

Step 1 In this step, we attempt to prove that

$$\begin{aligned} {\hat{\varSigma }}=\begin{pmatrix} N &{} S_{12} &{} S_{13} &{} S_{14}\\ &{} S_{22} &{} S_{23} &{} S_{24}\\ &{} &{} S_{33} &{} S_{34}\\ &{} &{} &{} S_{44}\\ \end{pmatrix}{\mathop {\longrightarrow }\limits ^{P}}\begin{pmatrix} N, &{} {\mathbf {1}}^\top \mu , &{} {\mathbf {1}}^\top \mathbb {Z}, &{} ({\mathbf {1}}^\top W)\circ \mu ^\top \\ &{} \mu ^\top \mu +\text {tr}\{\varGamma (0)\}, &{} \mu ^\top \mathbb {Z}, &{} \text {diag}[W^\top (\varGamma (0)+\mu \mu ^\top )]\\ &{} &{} \mathbb {Z}^\top \mathbb {Z}, &{} (\mathbb {Z}^\top W)\circ \mu ^\top \\ &{} &{} &{} \mu \mu ^\top +\varGamma (0)\\ \end{pmatrix} \end{aligned}$$

where

$$\begin{aligned} S_{12}= & {} \frac{1}{T}\sum _{t=1}^T {\mathbf {1}}^\top \mathbb {Y}_{t-1},\quad S_{13}=\frac{1}{T}\sum _{t=1}^T {\mathbf {1}}^\top \mathbb {Z}, \quad S_{14}=\frac{1}{T}\sum _{t=1}^T ({\mathbf {1}}^\top W)\circ \mathbb {Y}_{t-1}^\top , \\ S_{22}= & {} \frac{1}{T}\sum _{t=1}^T \mathbb {Y}_{t-1}^\top \mathbb {Y}_{t-1}, \quad S_{23}=\frac{1}{T}\sum _{t=1}^T \mathbb {Y}_{t-1}^\top \mathbb {Z}, \quad S_{24}=\frac{1}{T}\sum _{t=1}^T (\mathbb {Y}_{t-1}^\top W)\circ \mathbb {Y}_{t-1}^\top , \\ S_{33}= & {} \frac{1}{T}\sum _{t=1}^T \mathbb {Z}^\top \mathbb {Z}, \quad S_{34}=\frac{1}{T}\sum _{t=1}^T (\mathbb {Z}^\top W)\circ \mathbb {Y}_{t-1}^\top , \quad \\ S_{44}= & {} \frac{1}{T}\sum _{t=1}^T (W^\top W)\circ (\mathbb {Y}_{t-1}\mathbb {Y}_{t-1}^\top )+\frac{\lambda }{T}L. \end{aligned}$$

Due to the fact that N is fixed, $S_{13}={\mathbf {1}}^\top \mathbb {Z}$ and $S_{33}=\mathbb {Z}^\top \mathbb {Z}$. According to (7), $\mathbb {Y}_t=(I-G)^{-1}{\mathcal {B}}_0+\sum _{j=0}^{\infty } G^j{\mathcal {E}}_{t-j},$ and $\mu =(I-G)^{-1}{\mathcal {B}}_0$, we show the convergence of the other entries in ${\hat{\varSigma }}$. By the ergodic theorem as described in Walters (2000), we have $S_{12}{\mathop {\longrightarrow }\limits ^{P}}{\mathbf {1}}^\top \mu$, $S_{14}{\mathop {\longrightarrow }\limits ^{P}}({\mathbf {1}}^\top W)\circ \mu ^\top$, $S_{22}{\mathop {\longrightarrow }\limits ^{P}}\mu ^\top \mu +\text {tr}\{\varGamma (0)\}$, $S_{23}{\mathop {\longrightarrow }\limits ^{P}}\mu ^\top \mathbb {Z}$, $S_{34}{\mathop {\longrightarrow }\limits ^{P}}(\mathbb {Z}^\top W)\circ \mu ^\top$, $S_{44}{\mathop {\longrightarrow }\limits ^{P}}\mu \mu ^\top +\varGamma (0)$, where $\varGamma (0)$ is defined in Proposition 1. As for the convergence of $S_{24}$, we use the basic property of the covariance of any n-dimensional random variable $\mathbf {y}$, $\mathbb {E}(\mathbf {y}\mathbf {y}^\top )=\text {cov}(\mathbf {y})+\mu _{\mathbf {y}}\mu _{\mathbf {y}}^\top$, where $\mu _{\mathbf {y}}=\mathbb {E}(\mathbf {y})$. $(\mathbb {Y}_t^\top W)\circ \mathbb {Y}_t^\top =(w_{11}Y_{1t}+\ldots +w_{N1}Y_{1t},\cdots ,w_{1N}Y_{1t}+\ldots +w_{NN}Y_{Nt})= \text {diag}(W^\top \mathbb {Y}_t\mathbb {Y}_t^\top )$. So $S_{24}{\mathop {\longrightarrow }\limits ^{P}}\text {diag}[W^\top (\varGamma (0)+\mu \mu ^\top )]$. This completes the proof of (A.1).

Step 2 Proof of (A.2). It suffices to show that $T^{1/2}\delta ^\top {\hat{\varSigma }}_{xe}=T^{-1/2}\sum _t\delta ^\top (\mathbb {V}_{t-1}, \mathbb {X}_{t-1})^\top {\mathcal {E}}_t{\mathop {\longrightarrow }\limits ^{D}}N(0,\delta ^\top \varSigma \delta )$ for any $\delta \in \mathbb {R}^{2+p+N}$, where $\sigma ^2$ is set to be 1 in this step for simplicity. To this end, denote $\xi _t=T^{-1/2}\delta ^\top (\mathbb {V}_{t-1}, \mathbb {X}_{t-1})^\top {\mathcal {E}}_t$, $\mathbb {S}_t=\sum _{s=1}^t \xi _s$ and ${\mathscr {F}}_t=\sigma \{{\mathcal {E}}_s,-\infty<s<t\}$, then $\{\mathbb {S}_t,{\mathscr {F}}_t,t>-\infty \}$ is a zero-mean martingale and it is equivalent to prove $\mathbb {S}_t{\mathop {\longrightarrow }\limits ^{P}}N(0,\delta ^\top \varSigma \delta )$. According to (A.1), we have

$$\begin{aligned} (\delta ^\top \varSigma \delta )^{-1} \sum _{t=1}^T\mathbb {E}(\xi _t^2|{\mathscr {F}}_{t-1}){\mathop {\longrightarrow }\limits ^{P}}1, \end{aligned}$$

(A.3)

and as $T\rightarrow \infty$,

$$\begin{aligned}&(\delta ^\top \varSigma \delta )^{-1}\sum _{t=1}^T\mathbb {E}[\xi _t^2 I(|\xi _t|\ge \epsilon (\delta ^\top \varSigma \delta )^{1/2})] \nonumber \\&\quad \le \frac{\epsilon ^{-2}}{(\delta ^\top \varSigma \delta )^2}\sum _{t=1}^T\mathbb {E}(\xi _t^4)\nonumber \\&\quad \le \frac{\epsilon ^{-2}}{(\delta ^\top \varSigma \delta )^2T^2}\sum _{t=1}^T\mathbb {E}\Big (\delta ^\top (\mathbb {V}_{t-1}, \mathbb {X}_{t-1})^\top (\mathbb {V}_{t-1}, \mathbb {X}_{t-1})\delta \Big )^2 \rightarrow 0. \end{aligned}$$

(A.4)

The limit is satisfied because $\Big (\delta ^\top (\mathbb {V}_{t-1}, \mathbb {X}_{t-1})^\top (\mathbb {V}_{t-1}, \mathbb {X}_{t-1})\delta \Big )^2$ does not tend to infinity. Therefore, by (A.3), (A.4) and the central limit theorem for martingale sequences (Hall and Heyde 1980, pp. 9–10), we have $\mathbb {S}_t=T^{1/2}\delta ^\top {\hat{\varSigma }}_{xe}{\mathop {\longrightarrow }\limits ^{D}}N(0,\delta ^\top \varSigma \delta )$. This completes the proof.