Integrating Deep Learning into Semiparametric Network Vector AutoRegressive Models

Abstract

Network vector AutoRegressive models play a vital role in multivariate time series analysis. However, previous research in the classic Network vector AutoRegressive (NAR) model is limited to strict assumptions of linearity and time-invariance of node-specific covariates. In this study, we propose a Semiparametric NAR (SNAR) model to broaden existing research horizons by (1) extending node-specific covariates to a nonlinear framework, (2) incorporating high-dimensional time-varying covariates for a more comprehensive analysis, and (3) maintaining the interpretability of the autoregressive effects of the NAR. A deep learning-based method is presented to simultaneously estimate the nonparametric function and the parameters in SNAR. We also provide theoretical proof for the convergence rate of the nonparametric deep neural network estimator to support linear-to-nonlinear extension and show that the proposed method is capable of avoiding the curse of dimensionality. Furthermore, we prove the asymptotic normality of the parametric estimators for autoregressive effects to demonstrate the maintenance of interpretability. Experiments on various numerical simulated data show that the proposed method can avoid the curse of dimensionality; for instance, in nonlinear settings, the SNAR model reduces the prediction MSE by approximately 69% compared to the classic NAR model (decreasing from 3.44 to 1.06). Furthermore, in real-world stock return analysis, the SNAR model achieves an MSE of 0.9930, significantly outperforming the NAR baseline (MSE 1.6540) and other state-of-the-art methods.

1. Introduction

The Network vector AutoRegressive (NAR) model is designed to explicitly characterize node-to-node dependence structures in ultra-high-dimensional multivariate time series. In the NAR model, the value of each node is determined by (a) its own past values, (b) the past values of connected nodes, (c) a set of node-specific external covariates, and (d) an error term. Through these components, NAR models capture the temporal propagation of shocks across a network—such as how stress originating from a major bank diffuses through a financial system—thereby supporting risk assessment and policy design.

Existing studies [1,2] establish asymptotic properties for NAR parameter estimators. However, these works rely on a linear specification of external covariates, which is restrictive in modern applications where covariates are high-dimensional and often exhibit nonlinear effects. This raises a natural question:

• How can we construct a nonlinear model for high-dimensional time-varying covariates while preserving the interpretability of NAR?

A common remedy is to replace the linear component with a flexible nonparametric function, but this immediately leads to two further questions:

• How should such a model be estimated?
• What are its theoretical guarantees?

Classical nonparametric methods—such as kernel or spline estimators—enjoy well-understood statistical properties, but they suffer from the curse of dimensionality: as the covariate dimension q increases, the required sample size grows exponentially, making computation infeasible.

This study proposes a Semiparametric NAR (SNAR) model to overcome these limitations. We relax the linearity constraint on the covariate component to accommodate nonlinear, complex, and high-dimensional effects and incorporate time-varying covariates—an essential feature in real-world applications (e.g., GDP dynamics in macroeconomic analyses). Specifically, we employ a deep neural network (DNN) to estimate the nonlinear component, enabling both modeling flexibility and the avoidance of dimensionality issues. We further establish the DNN estimator’s consistency and convergence rate under Hölder smoothness and intrinsic dimension assumptions, achieving a minimax-optimal rate up to logarithmic factors.

Beyond modeling nonlinear covariate effects, we also retain the interpretability of temporal dependence structures by accurately estimating autoregressive components. Prior studies [1] show that these parameters in the NAR model are asymptotically normal and thus interpretable; for example, a region with a larger autoregressive coefficient may exhibit stronger “inertial” behavior, such as more persistent weather patterns. In the SNAR model, we jointly estimate the nonlinear function and the autoregressive effects through maximum likelihood. The results show that the autoregressive parameter estimators remain unbiased, consistent with $\sqrt{NT}$, and asymptotically normal—preserving the interpretability of the original NAR framework. As illustrated in Figure 1, applying the SNAR model to international trade data yields intuitive insights into how countries influence one another through the direction, thickness, and color of network edges.

To validate the theoretical results and demonstrate forecasting advantages, we conduct extensive simulations under various configurations. The SNAR model consistently outperforms the classical NAR model and other state-of-the-art baselines. We further evaluate the model on three real-world datasets, where the SNAR model achieves markedly better predictive accuracy and parameter estimation.

Overall, our contributions are summarized as follows:

• In this paper, we propose a Semiparametric NAR (SNAR) model that incorporates nonlinear, high-dimensional, time-varying covariates through DNN-based estimation and establish the consistency and convergence rate of the nonlinear component.
• The results show that the SNAR model preserves the interpretability of the autoregressive effects and prove that the corresponding estimators remain consistent with $\sqrt{NT}$ and asymptotically normal.

2. Related Works

2.1. From AutoRegressive (AR) to Vector AutoRegressive (VAR) Models

The autoregressive (AR) model is one of the most well-known models in time series analysis. It considers one time-varying variable and specifies that the variable depends linearly on its own previous values and on a random error term ([3]). The error term is usually assumed to be independent over time. When considering a set of time-varying variables, a vector AutoRegressive (VAR) model is used to capture the relationship between multiple quantities. Like the AR model, each variable in the VAR model has an equation modeling its evolution over time, including the variable’s past values, the lagged values of the other variables in the model, and an error term. A general VAR model containing N variables with v lags can be written as

$${\mathbb{Y}}_t=c+A_1{\mathbb{Y}}_{t-1}+A_2{\mathbb{Y}}_{t-2}+\cdots +A_v{\mathbb{Y}}_{t-v}+{\mathcal{E}}_t,$$

(1)

where ${\mathbb{Y}}_t\in {\mathbb{R}}^N$ is the variable’s value at time point t, c is a N-vector of constants serving as the intercept, $A_1,\dots ,A_v$ are time-invariant $N\times N$ matrices, and ${\mathcal{E}}_t$ is an N-vector of error terms. Every error term is assumed to have a zero mean and to be independent across time.

In an ordinary VAR model in Equation (1), N is the number of variables of one individual. If we assume N to be the number of individuals and ${\mathbb{Y}}_t={(Y_t^1,\dots ,Y_t^N)}^{\top }$ to be the N- dimensional response vector, Equation (1) can be used to model a multivariate time series.

2.2. Advancements from VAR to Network Autoregressive (NAR) Models

The Network vector AutoRegressive (NAR) model ([1]) innovatively incorporates network structures to elucidate the interdependencies among N entities. This model conceptualizes each individual’s response at a specific time as a linear combination of four principal components, as stated in the first paragraph in the Introduction. The coefficients corresponding to these components are the momentum effect, network effect, and nodal effect, respectively. The momentum and network effects collectively embody the model’s autoregressive dimension, encapsulating both temporal and spatial dependencies.

A pivotal advantage of the NAR model over the conventional vector AutoRegressive (VAR) framework lies in its parameter efficiency. While the parameter count in a traditional VAR model escalates with the number of entities N, potentially leading to over-parameterization and computational burdens, the NAR model maintains a constant number of parameters regardless of N. This feature significantly enhances the model’s scalability and feasibility for large networks.

We briefly introduce the NAR model. Consider a social network with N nodes indexed by $1\le i\le N$. The network structure can be described by its adjacency matrix $A=\left(a_{ij}\right)\in {\mathbb{R}}^{N\times N}$, where $a_{ij}=1$ indicates the existence of a link from node i to node j, and $a_{ij}=0$ otherwise, $\{i,j\}\subset \{1,\dots ,N\}$. This study allows the network to be directed, i.e., $a_{ij}\ne a_{ji}$, and it is assumed that no node is self-related, i.e., $a_{ii}=0$ for any $1\le i\le N$. With the defined adjacency matrix [1] considers a one-order NAR model as follows:

$$Y_t^i={\beta }_0+Z^{i\top }\gamma +{\beta }_1{Y}_{t-1}^i+{\beta }_2{n}_i^{-1}\sum _{j=1}^N{a}_{ij}{Y}_{t-1}^j+{\epsilon }_t^i,$$

(2)

where $Y_t^i\in {\mathbb{R}}^1$ is the univariate response obtained from node i at time point t, and $Z^i\in {\mathbb{R}}^q$ is the time-invariant covariate of node i. Here, $n_i={\sum }_{j\ne i}{a}_{ij}$ is the total number of neighbors that node i links to, so $n_i^{-1}{\sum }_{j=1}^N{a}_{ij}{Y}_{t-1}^j$ represents the average impact from ith neighbors. $({\beta }_0,{\beta }_1,{\beta }_2,\gamma )\in {\mathbb{R}}^{3+q}$ are coefficients to be estimated, and ${\beta }_1,{\beta }_2,\gamma$ correspond to the momentum, network, and nodal effects, respectively. Since the relationship between nodes and the correlation over time are characterized by the network structure and the autoregressive process, respectively, it is acceptable to allow the error term, ${\epsilon }_t^i$’s, to be i.i.d. and independent of $Z^i$’s. For convenience, ${\epsilon }_t^i$ is assumed to follow $N(0,{\sigma }^2)$ for any possible i and t. Under some stationary assumptions, it is convenient to estimate $({\beta }_0,{\beta }_1,{\beta }_2,\gamma )$ using the least square method or maximum likelihood method. More contents about the NAR model can be found in [2,4,5,6,7,8].

3. Methods

3.1. Semiparametric NAR (SNAR) Model with Time-Varying Covariates

Similar to the definition of the NAR model, let $Y_t^i$ be the response variable from the ith subject at time point t, $i=1,\dots ,N$, $t=1,\dots ,T$. A q-dimensional covariate variable ${\mathbf{Z}}_t^i={(Z_t^{i}1,\dots ,Z_t^{i}q)}^{\top }\in {\mathbb{R}}^q$ for each node i is assumed to be observable at time t. Given the adjacency matrix A, the lag-1 SNAR model can be expressed as

$$Y_t^i=g_0\left({\mathbf{Z}}_{t-1}^i\right)+{\beta }_1{Y}_{t-1}^i+{\beta }_2{n}_i^{-1}\sum _{j=1}^N{a}_{ij}{Y}_{t-1}^j+{\epsilon }_t^i,$$

(3)

where $g_0:{\mathbb{R}}^q\to \mathbb{R}$ is an unknown function. The architecture of the model is shown in Figure 2. Its vector form can be written as

$${\mathbb{Y}}_t={\mathbb{G}}_{t-1}+B{\mathbb{Y}}_{t-1}+{\mathcal{E}}_t,$$

(4)

where ${\mathbb{Y}}_t={(Y_t^1,\dots ,Y_t^N)}^{\top }\in {\mathbb{R}}^N$, ${\mathbb{G}}_t={(g_0\left({\mathbf{Z}}_t^1\right),\dots ,g_0\left({\mathbf{Z}}_t^N\right))}^{\top }\in {\mathbb{R}}^N$, $B={\beta }_{1}I+{\beta }_{2}W$, $W=\mathrm{diag}\{n_1^{-1},\dots ,n_N^{-1}\}A$ is the row-normalized adjacency matrix, I is an identity matrix with compatible dimension, and ${\mathcal{E}}_t={({\epsilon }_t^1,\dots ,{\epsilon }_t^N)}^{\top }\in {\mathbb{R}}^N$. It is obvious that a NAR model in Equation (2) is a special form of the SNAR model in Equation (3).

Remark 1. 

It is noted that the SNAR model in Equation (2) considers only one lag. One could consider a high-order lag model as a flexible extension. The model setup and assumptions employed in this study are directly based on the framework presented in [1].

Remark 2. 

In classic semiparametric models, identifiability is typically ensured by imposing a mean-zero constraint, i.e., $\mathbb{E}\left[g_0(·)\right]=0$, to separate the nonparametric function from the intercept term ${\beta }_0$. Our SNAR formulation adopts an alternative but equivalent approach for identifying the model: we omit the separate intercept parameter ${\beta }_0$ and allow $g_0(·)$ to be a general function that absorbs the intercept. This formulation avoids confounding between the constant term and the nonparametric component. Theoretically, one can always decompose the estimated function as $g_0(·)=\mu +\tilde{g}(·)$, where $\mu =\mathbb{E}\left[g_0(·)\right]$ and $\mathbb{E}\left[\tilde{g}(·)\right]=0$, ensuring that the components remain interpretable.

3.2. Estimation

To estimate the unknown function $g_0$, deep neural networks (DNNs) are used as function approximators ([9,10]). A simple DNN is defined as shown in the appendix. To obtain some theoretical properties shown in the next subsection, we define some classes of DNN. Let ${\mathbb{N}}_{+}$ be the set of all positive natural numbers. Given $K\in {\mathbb{N}}_{+}$ and $\mathit{p}\in {\mathbb{N}}_{+}^{K+2}$, consider a class of DNN:

$$\begin{array}{cc}\hfill \mathcal{G}(K,\mathit{p})=& \{g:g\mathrm{is}\mathrm{a}\mathrm{DNN}\mathrm{with}(K+1)\mathrm{layers}\mathrm{and}\mathrm{widthvector}\mathit{p}\mathrm{such}\mathrm{that}\hfill \\ \hfill & max\{\parallel W_k{\parallel }_{\infty },\parallel v_k{\parallel }_{\infty }\}\le 1,\mathrm{for}\mathrm{all}k=0,\dots ,K\},\hfill \end{array}$$

(5)

where ${\parallel ·\parallel }_{\infty }$ denotes the sup-norm of matrix or vector, the matrix entries ${\left(W_k\right)}_{i,j}$ are the weight linking the jth neuron in layer k to the ith neuron in layer $k+1$, and the vector entries ${\left(v_k\right)}_i$ represent a shift term associated with the ith neuron in layer $k+1$.

To avoid the overfitting problem, consider, for $s\in N_{+}$ and $D>0$, a class of sparse neural networks:

$$\begin{array}{c}\hfill \mathcal{G}(K,s,\mathit{p},D)=\left\{g\in \mathcal{G}(K,\mathit{p}):\sum _{k=1}^K\parallel W_k{\parallel }_0+\parallel v_k{\parallel }_0\le s,{\parallel g\parallel }_{\infty }\le D\right\},\end{array}$$

(6)

where ${\parallel ·\parallel }_0$ is the number of nonzero entries of the matrix or vector, and ${\parallel g\parallel }_{\infty }$ is the sup-norm of function g.

We approximate $g_0$ using a DNN $g\in \mathcal{G}(K,s,\mathit{p},D)$, whose input is the q-dimensional vector $\mathbf{Z}$ and the output is a scalar value. With the shorthand $\mathcal{G}=\mathcal{G}(K,s,\mathit{p},\infty )$ and denotation $n:=NT$, we estimate $({\beta }_1,{\beta }_2,g_0)$ by maximizing the log likelihood as

$$({\widehat{\beta }}_1,{\widehat{\beta }}_2,\widehat{g})=\underset{({\beta }_1,{\beta }_2,g)\in \mathbb{R}\times \mathbb{R}\times \mathcal{G}}{argmax}{L}_n({\beta }_1,{\beta }_2,g),$$

(7)

where $L_n({\beta }_1,{\beta }_2,g)={\displaystyle \frac{1}{n}}{\sum }_{i=1}^N{\sum }_{t=1}^T\left\{-log\left(\sqrt{2\pi }\right)-log\sigma -{\displaystyle \frac{1}{2{\sigma }^2}}{[Y_t^i-g\left({\mathbf{Z}}_{t-1}^i\right)-{\beta }_1{Y}_{t-1}^i-{\beta }_2{w}_i^{\top }{\mathbb{Y}}_{t-1}]}^2\right\}$. The Adam optimizer ([11]) is used to implement the maximization procedure. We summarize the entire estimation process in Algorithm 1. To prevent overfitting during the training of the nonlinear component $g(·)$, we employ standard regularization techniques within the network architecture, specifically Dropout layers, alongside an Early Stopping mechanism based on validation loss.

Algorithm 1 SNAR

Require:  {$Y_t^i$, ${\mathbf{Z}}_t^i$ for $t=0,\dots ,T$ and $i=1,\dots ,N$}, Objective Function $L_n$, Number of Layers, Neurons per Layer, Epochs, Batch Size, Early Stop, Threshold for Minimum Improvement, Learning Rate $\alpha$, Dropout (Optional) Ensure:  {$\theta$: The set of ${\widehat{\beta }}_1$, ${\widehat{\beta }}_2$ and parameters defining the neural network $\widehat{g}$} 1: Set Hyperparameters: Number of Layers, Neurons per Layer, Epochs, Batch Size, Early Stop, Threshold for Minimum Improvement, Dropout (Optional) 2: Initialize ${\beta }_1$, ${\beta }_2$ (0 as usual) and define the DNN g 3: Let U be the reduction in loss from two subsequent training iterations. Initialize as 0 4: Let $V>0$ be the threshold for minimum improvement 5: Let $k=0$ 6: while U < V do 7:    Forward Pass 8:       Pass ${\mathbf{Z}}_t^i$ to the deep neural network, output $g\left({\mathbf{Z}}_t^i\right)$ 9:       Calculate MSE loss: $L_n\left(\theta \right)$ 10:   Backward Pass 11:       ${\theta }^{\left(k\right)}={\theta }^{(k-1)}-\alpha \nabla L_n\left({\theta }^{(k-1)}\right)$ 12:   Stopping Criteria Updates 13:       If k = 0: $U=L_n\left(\theta \right)$ Else $U=U-L_n\left(\theta \right)$ 14:       $k=k+1$ 15: end while 16: return $({\widehat{\beta }}_1,{\widehat{\beta }}_2,\widehat{g})={\theta }^{\left(k\right)}$

3.3. Theoretical Properties

We then study the asymptotic properties of the estimators in Equation (7). Some restrictions on the nonparametric function $g_0$ are needed. Assume that it belongs to a Hölder class of smooth functions, which has been discussed in [9,10,12]. A Hölder class of smooth functions with parameters $\alpha$, $M>0$ and domain $\mathbb{D}\in {\mathbb{R}}^q$ is computed as follows:

$${\mathcal{H}}_q^{\alpha }(\mathbb{D},M)=\left\{g:\mathbb{D}\to \mathbb{R}:\sum _{\beta :\left|\beta \right|<\alpha }{\parallel {\partial }^{\beta }g\parallel }_{\infty }+\sum _{\beta :\left|\beta \right|=\lfloor \alpha \rfloor }\underset{x,y\in \mathbb{D},x\ne y}{sup}{\displaystyle \frac{|{\partial }^{\beta }g\left(x\right)-{\partial }^{\beta }g\left(y\right)|}{{\parallel x-y\parallel }_{\infty }^{\alpha -\lfloor \alpha \rfloor }}}\le M\right\},$$

(8)

where $\lfloor \alpha \rfloor$ is the largest integer strictly smaller than $\alpha$, ${\partial }^{\beta }:={\partial }_1^{\beta }\dots {\partial }_q^{\beta }$ with $\beta =({\beta }_1,\dots ,{\beta }_q$, and $\left|\beta \right|={\sum }_{k=1}^q{\beta }_k$. Let $r\in \mathbb{N}$, $M>0$, $\alpha =({\alpha }_0,\dots ,{\alpha }_r)\in {\mathbb{R}}_{+}^{r+1}$ and $\mathit{d}=(d_0,\dots .d_{r+1})\in {\mathbb{N}}_{+}^{r+2}$, $\tilde{\mathit{d}}=({\tilde{d}}_0,\dots ,{\tilde{d}}_r)\in {\mathbb{N}}_{+}^{r+1}$ with ${\tilde{d}}_j\le d_j$, $j=0,\dots ,r$, where ${\mathbb{N}}_{+}$ is the set of all positive real numbers. We further assume that $g_0$ belongs to a composite smoothness function class:

$$\begin{array}{cc}\hfill \mathcal{H}(r,\alpha ,\mathit{d},\tilde{\mathit{d}},M):=& \{g=g_q\circ \cdots \circ g_1:g_i=(g_{i1},\cdots ,\hfill \\ \hfill & g_{i{d}_{i+1}}{)}^{\top }\mathrm{and}{g}_{ij}\in {\mathcal{H}}_{{\tilde{d}}_i}^{{\alpha }_i}({[a_i,b_i]}^{{\tilde{d}}_i},M),\hfill \\ \hfill & \mathrm{for}\mathrm{some}|a_i|,|b_i|\le M\}.\hfill \end{array}$$

(9)

Functions in this class are characterized by two kinds of dimensions, $\mathit{d}$ and $\tilde{\mathit{d}}$, where the latter represents the intrinsic dimension of the function. For this function space, we denote ${\alpha }_i^{*}={\alpha }_i{\prod }_{k=i+1}^r({\alpha }_k\wedge 1)$, ${\varphi }_n={max}_{i=0,\dots ,r}{n}^{-{\displaystyle \frac{{\alpha }_i^{*}}{2{\alpha }_i^{*}+{\tilde{d}}_i}}}$ with notation $a\wedge b:=min\{a,b\}$. To obtain the theoretical properties, the following technical conditions are required:

(A1) The nonparametric function $g_0$ is an element of $\mathcal{H}(r,\alpha ,\mathit{d},\tilde{\mathit{d}},M)$.

(A2) The neural network family $\mathcal{G}(K,s,\mathit{p},D)$ is defined as shown in Equation (6), where $K=O(logn)$, $s=O(n{\varphi }_n^{2}logn)$, and $n{\varphi }_n^2\lesssim min{\left(p_k\right)}_{k=1,\dots ,K}\le max{\left(p_k\right)}_{k=1,\dots ,K}\lesssim n$.

Assumption (A1) assumes that the real $g_0$ is the composite of some univariate Hölder smoothness functions. Assumption (A2) determines the structure of the neural network family in Equation (6). According to [10,12,13], more flexible neural networks can achieve smaller approximation error. The approximation error is defined as the distance between true functions $g_0$ and $\tilde{g}$, and $g_0$ is projected onto the space of functions that can be implemented by a neural network from $\mathcal{G}(K,s,\mathit{p},D)$. However, a larger neural network often leads to a larger estimation error, shown as the distance between $\widehat{g}$ and $\tilde{g}$. Assumption (A2) is used to provide a trade-off between the approximation error and estimation error.

For the social network structure, assume W is a sequence of matrices indexed by N. They are assumed to be nonstochastic.

(B1) (Connectivity) Treat W as a transition probability matrix of a Markov chain, whose state space is defined as the set of all the nodes in the network. The Markov chain is irreducible and aperiodic.

(B2) (Uniformity) Define $W^{*}=W+W^{\top }$ as a symmetric matrix. Assume the largest absolute eigenvalues of $W^{*}$ is $O(logn)$.

For the covariates and error term in model Equation (3), there is a standard assumption for the NAR model.

(B3) Assume that ${\mathbf{Z}}_t^i$ forms a stationary $\alpha$-mixing process with a finite fourth-order moment. This weak dependence condition is standard in time series analysis and is required to ensure the validity of concentration inequalities and central limit theorem-type results used in deriving the consistency and asymptotic properties of the proposed estimators. Without loss of generality, we assume that the domain of ${\mathbf{Z}}_t^i$ is taken to be ${\left[\mathrm{0,\; 1}\right]}^q$. ${\epsilon }_t^i$ are assumed to follow i.i.d. normal distribution with zero mean across both $1\le i\le N$ and $1\le t\le T$. Moreover, ${\mathbf{Z}}_t^i$ and $\left\{{\epsilon }_t^i\right\}$ are assumed to be mutually independent.

Similar to a usual NAR model, we first introduce the strict stationarity of the SNAR model in Theorem 1.

Theorem 1 

(Stationarity). Suppose that N is fixed and given $\mathbf{Z}$ for all i and t. If $|{\beta }_1|+|{\beta }_2|<1$, then there exists a unique strictly stationary solution with a finite first-order moment to the model (Equation (3)). The solution has the following form:

$${\mathbb{Y}}_t=\sum _{j=0}^{\infty }{B}^j{\mathbb{G}}_{t-j}+\sum _{j=0}^{\infty }{B}^j{\mathcal{E}}_{t-j}.$$

(10)

Proposition 1. 

Assume $|{\beta }_1|+|{\beta }_2|<1$ and $G_0=\mathbb{E}\left(g_0\left(\mathbf{Z}\right)\right)$. The strictly stationary solution in (Equation (10)) follows a normal distribution with the mean and covariance given by

$$\mu ={(I-B)}^{-1}{G}_0={(I-{\beta }_{1}I-{\beta }_{2}W)}^{-1}{G}_0,$$

(11)

$$\mathrm{vec}\{\Gamma \left(0\right)\}={\sigma }^2{(I-B\otimes B)}^{-1}\mathrm{vec}\left(I\right),$$

(12)

where ⊗ is the Kronecker product and $vec(·)$ is the operator stacking the columns of a given matrix.

The proof of Theorem 1 is given in Appendix B.1. Based on the strictly stationary solution Equation (10), the conditional mean and covariance of ${\mathbb{Y}}_t$ can be obtained in Proposition 1.

Theorem 2. 

Assume the stationary condition $|{\beta }_1|+|{\beta }_2|<1$ and technical conditions (A1) (A2) (B1) (B2) (B3) hold; then, there exists an estimator $\widehat{g}$ such that

$$\parallel \widehat{g}-g_0{\parallel }_{L^2\left({[0,1]}^p\right)}=O_p\left({\varphi }_n{log}^{2}n\right),$$

(13)

where ${\parallel h\parallel }_{L^2\left({[0,1]}^p\right)}$ for any function h is the $L^2$-norm of h.

Theorem 2 reveals that the convergence rates are jointly determined by the smoothness $\alpha$ and the intrinsic dimension $\tilde{\mathit{d}}$ of the function $g_0$, rather than the dimension $\mathit{d}$, under the assumption of the representational space $\mathcal{H}(r,\alpha ,\mathit{d},\tilde{\mathit{d}},M)$ in Equation (9). Thus, under our DNN-based SNAR model, the curse of dimensionality is alleviated and the convergence rate will be faster when the intrinsic dimension $\tilde{\mathit{d}}$ is relatively low. The minimax lower bound for estimating $g_0$ is shown below.

Theorem 3. 

Assume the stationary condition $|{\beta }_1|+|{\beta }_2|<1$ and technical conditions (A1) (B1) (B2) (B3) hold; then, there exists a constant, $0<c<\infty$, such that

$$\underset{\widehat{g}}{inf}\underset{({\beta }_1,{\beta }_2,g_0)\in {\mathbb{R}}_M\times {\mathbb{R}}_M\times {\mathcal{H}}_0}{sup}\mathbb{E}{\{{\widehat{g}}_0\left(Z\right)-g_0\left(Z\right)\}}^2\ge c{\varphi }_n^2,$$

(14)

where the infimum is taken over all possible estimators $\widehat{g}$ based on the observed data.

The proofs of Theorems 2 and 3 are shown in Appendix B.2 and Appendix B.3. For more theoretical foundations, see, e.g., [10,12,14,15].

We then show the asymptotic normality of estimated parameters $({\widehat{\beta }}_1,{\widehat{\beta }}_2)$ ([16,17]). Let ${\mathcal{H}}_{g_0}$ denote the collection of all subfamilies $\{g_b\in L^2\left({[0,1]}^p\right):b\in (-1,1)\}\subset {\mathcal{H}}_0$ such that ${lim}_{b\to }{\parallel b^{-1}(g_b-g_0)-g\parallel }_{L^2\left({[0,1]}^p\right)}\to 0$, and let

$$\begin{array}{cc}\hfill {\mathbb{T}}_{g_0}=\{g\in L^2\left({[0,1]}^p\right):\underset{b\to 0}{lim}{\parallel b^{-1}(g_b-g_0)-g\parallel }_{L^2\left({[0,1]}^p\right)}=0& \\ \hfill \mathrm{for}\mathrm{some}\mathrm{subfamily}\{g_b:b\in (-1,1)\}\in {\mathcal{H}}_{g_0}\}.\end{array}$$

(15)

Set ${\overline{\mathbb{T}}}_{g_0}$ as the closed linear span of ${\mathbb{T}}_{g_0}$. The asymptotic normality and the information bound are shown in Theorem 4.

Theorem 4. 

Assume the stationary condition $|{\beta }_1|+|{\beta }_2|<1$ and technical conditions (A1) (A2) (B1) (B2) (B3) hold. If the information matrix is nonsingular and $min\{N,T\}\to \infty$, we have

$$\sqrt{NT}\left(\left[\begin{array}{c}{\widehat{\beta }}_1\\ {\widehat{\beta }}_2\end{array}\right]-\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right]\right)\stackrel{d}{\to }N(0,\Sigma ),$$

(16)

where $\Sigma =I{\left(\beta \right)}^{-1}$, $I\left(\beta \right)$ is the information bound for β.

The proof of Theorem 4 is shown in Appendix B.4, and an estimate of the asymptotic variance $\Sigma$ is also provided in Appendix C.

4. Experiments

4.1. Simulation Study

To demonstrate the finite sample performance of the proposed SNAR method, we present four cases in comparison with the traditional NAR model. The main difference is the underlying function $g_0\left(x\right)$. Other than that, they are fairly similar. In each simulation case, different network sizes and numbers of time points are considered (i.e., $N=(100,200),T=(100,200)$), and the experiment is randomly replicated $R=200$ times. The specific method used for generating simulated data, information about dataset splitting, and computation details are provided in Appendix C.

The four different cases for the underlying function $g_0\left(\mathbf{Z}\right)$ are set as follows:

• Case 1 (Linear): $g_0\left(\mathbf{Z}\right)=0.2-0.5Z_1+0.3Z_2+0.8Z_3-0.1Z_4-0.1Z_5$,
• Case 2 (Additive): $g_0\left(\mathbf{Z}\right)=5-2Z_1+0.5Z_2^2-Z_3^3-log(Z_4+3)+\sqrt{Z_5+3}$,
• Case 3 (Deep): $g_0\left(\mathbf{Z}\right)=Z_1^2-2Z_2^2+Z_2{Z}_3+Z_4{Z}_5$,
• Case 4 (High-dimensional): $g_0\left(\mathbf{Z}\right)={(-100+Z_1+Z_2+\dots +Z_{100})}^2$

Let $({\widehat{\beta }}_1^{\left(r\right)},{\widehat{\beta }}_2^{\left(r\right)})$ be the estimator obtained from the training set in the rth replication. We evaluate the bias and standard deviation of ${\widehat{\beta }}_j$ as $\mathrm{Bias}\left({\widehat{\beta }}_j\right)={\overline{\widehat{\beta }}}_j-{\beta }_j$ and $SD\left({\widehat{\beta }}_j\right)={\left[{(R-1)}^{-1}{\sum }_{r=1}^R{({\overline{\widehat{\beta }}}_j-{\widehat{\beta }}_j)}^2\right]}^{1/2}$, respectively, where ${\overline{\widehat{\beta }}}_j=R^{-1}{\sum }_{r=1}^R{\widehat{\beta }}_j^{\left(r\right)}$, $j=1,2$. The results are shown in

Table 1. The bias and standard deviation (in parentheses) of ${\widehat{\beta }}_1$ and ${\widehat{\beta }}_2$ for the SNAR and NAR models.

		SNAR				NAR
		$\mathit{T}$ = 100		$\mathit{T}$ = 200		$\mathit{T}$ = 100		$\mathit{T}$ = 200
		${\widehat{\mathit{\beta }}}_{\mathbf{1}}$	${\widehat{\mathit{\beta }}}_{\mathbf{2}}$	${\widehat{\mathit{\beta }}}_{\mathbf{1}}$	${\widehat{\mathit{\beta }}}_{\mathbf{2}}$	${\widehat{\mathit{\beta }}}_{\mathbf{1}}$	${\widehat{\mathit{\beta }}}_{\mathbf{2}}$	${\widehat{\mathit{\beta }}}_{\mathbf{1}}$	${\widehat{\mathit{\beta }}}_{\mathbf{2}}$
Case 1	N = 100	−0.0124 (0.0084)	0.0057 (0.0410)	−0.057 (0.0056)	0.0013 (0.0252)	0.0003 (0.0082)	0.0049 (0.0355)	−0.0002 (0.0052)	0.0013 (0.0251)
	N = 200	−0.0062 (0.0068)	0.0042 (0.0269)	−0.0022 (0.0041)	−0.0008 (0.0162)	−0.0006 (0.0069)	0.0035 (0.0246)	0.0003 (0.0040)	0.0000 (0.0161)
Case 2	N = 100	−0.0224 (0.0041)	0.1046 (0.0036)	−0.0177 (0.0030)	0.0152 (0.0030)	0.0244 (0.0037)	0.0075 (0.0151)	0.0368 (0.0029)	0.0095 (0.0104)
	N = 200	−0.0192 (0.0030)	0.1045 (0.0035)	−0.0139 (0.0021)	0.1026 (0.0026)	0.0370 (0.0028)	0.0354 (0.0114)	0.0361 (0.0018)	0.0104 (0.0069)
Case 3	N = 100	−0.0058 (0.0047)	0.0008 (0.0221)	−0.0028 (0.0031)	−0.0003 (0.0145)	0.1763 (0.0048)	−0.0105 (0.0250)	0.1936 (0.0034)	0.0287 (0.0157)
	N = 200	−0.0026 (0.0031)	0.0039 (0.0138)	−0.0007 (0.0019)	0.0000 (0.0087)	0.2032 (0.0033)	0.0023 (0.0165)	0.1817 (0.0023)	0.0065 (0.0107)
Case 4	N = 100	−0.0981 (0.0070)	0.0288 (0.0093)	−0.0272 (0.0035)	0.0138 (0.0070)	0.2107 (0.0013)	0.0115 (0.0053)	0.2021 (0.0010)	0.0114 (0.0046)
	N = 200	−0.0298 (0.0042)	0.0184 (0.0057)	−0.0087 (0.0016)	0.0067 (0.0034)	0.2047 (0.0010)	0.0229 (0.0039)	0.2037 (0.0007)	0.0312 (0.0034)

. Under each simulation setting, the SNAR model has a steadily unbiased estimate, and the mean square errors decrease as the network size or the number of time points increases. As expected, the NAR model has a better performance than the SNAR model in the linear setting of Case 1, where all the parameters can be estimated well by the least square method. Under the complex settings of Cases 2–4, the overly restrictive NAR method results in large biases.

The information bound $I\left(\beta \right)$ is also estimated to calculate the asymptotic variance of $\widehat{\beta }$, and we use the estimated asymptotic variance to build a $95\%$ confidence interval. The empirical coverage probability of $95\%$ confidence intervals for the SNAR and NAR methods is shown in

Table 2. Empirical coverage probability of $95\%$ confidence intervals for $({\beta }_1,{\beta }_2)$ for the SNAR and NAR methods.

		SNAR				NAR
		$\mathit{T}$ = 100		$\mathit{T}$ = 200		$\mathit{T}$ = 100		$\mathit{T}$ = 200
		${\widehat{\mathit{\beta }}}_{\mathbf{1}}$	${\widehat{\mathit{\beta }}}_{\mathbf{2}}$	${\widehat{\mathit{\beta }}}_{\mathbf{1}}$	${\widehat{\mathit{\beta }}}_{\mathbf{2}}$	${\widehat{\mathit{\beta }}}_{\mathbf{1}}$	${\widehat{\mathit{\beta }}}_{\mathbf{2}}$	${\widehat{\mathit{\beta }}}_{\mathbf{1}}$	${\widehat{\mathit{\beta }}}_{\mathbf{2}}$
Case 1	N = 100	0.890	0.925	0.900	0.920	0.930	0.960	0.975	0.950
	N = 200	0.930	0.940	0.945	0.955	0.955	0.950	0.950	0.975
Case 2	N = 100	0.910	0.965	0.905	0.975	0	1	0	0.990
	N = 200	0.935	0.965	0.925	0.960	0	0.530	0	0.980
Case 3	N = 100	0.905	0.980	0.910	0.975	0	1	0	0.980
	N = 200	0.925	0.955	0.930	0.970	0	1	0	1
Case 4	N = 100	0.905	0.980	0.910	0.975	0	1	0	1
	N = 200	0.925	0.955	0.930	0.970	0	1	0	1

. The coverage rates of SNAR estimators are near $95\%$ for each simulation run. NAR performs better in Case 1 due to the correct model setting but loses efficacy in Cases 2–4. SNAR has a relatively stable performance, and the coverage rate is close to $95\%$ when the network size N increases.

After obtaining the parameter estimation of the simulated data, we make predictions on the test set data. The prediction accuracy is evaluated by the mean squared error (MSE) of the test set data. We also train some deep-learning-based models as the benchmarks, such as RNN, LSTM ([18]), Mamba ([19]), SAN ([20]), Dish-TS ([21]), and PatchTST ([22]). We not only use past observations of the target sequence but also covariates Z for the prediction. The mean and standard deviation of the MSE over R simulation runs are reported in

Table 3. The mean and standard deviation of MSEs of test set data for the SNAR and NAR methods and other SOTA methods.

		Case 1		Case 2		Case 3		Case 4
		$\mathit{N}=\mathbf{100}$	$\mathit{N}=\mathbf{200}$	$\mathit{N}=\mathbf{100}$	$\mathit{N}=\mathbf{200}$	$\mathit{N}=\mathbf{100}$	$\mathit{N}=\mathbf{200}$	$\mathit{N}=\mathbf{100}$	$\mathit{N}=\mathbf{200}$
SNAR	$T=100$	1.2246(0.0617)	1.2120(0.0425)	1.2277(0.0508)	1.0917(0.0260)	1.2306(0.0512)	1.0926(0.0270)	15.7431(2.3526)	4.9209(0.6562)
	$T=200$	1.2167(0.0371)	1.1082(0.0220)	1.0943(0.0263)	1.0578(0.0170)	1.0960(0.0288)	1.0413(0.0172)	5.6050(0.8814)	2.1810(0.2655)
NAR	$T=100$	0.9977(0.0304)	0.9989(0.0216)	3.1851(0.0744)	2.8428(0.0517)	5.8051(0.1036)	5.8951(0.0794)	86.4123(0.4252)	88.7812(0.3238)
	$T=200$	0.9995(0.0215)	1.0005(0.0141)	3.4378(0.0450)	3.7316(0.0392)	5.8680(0.0729)	5.8454(0.0567)	91.4825(0.2802)	84.2371(0.2077)
RNN	$T=100$	1.8575(0.1513)	1.8761(0.1140)	109.8140(55.6488)	109.1282(39.5176)	28.2357(6.3268)	28.7534(4.2324)	28.2252(6.9298)	28.7712(4.2661)
	$T=200$	1.2131(0.0429)	1.2152(0.0263)	83.8075(50.1262)	91.3527(46.1620)	19.5732(5.3541)	19.5431(4.5682)	17.8723(5.2225)	18.5501(3.5581)
LSTM	$T=100$	1.7561(0.1496)	1.8342(1.1097)	108.7064(54.3160)	107.2439(44.8580)	27.1096(6.0809)	26.7078(3.5893)	29.3940(4.3958)	27.0810(4.6407)
	$T=200$	1.2006(0.4031)	1.2122(0.0249)	83.2461(49.4818)	86.1089(38.0009)	16.6341(4.3989)	16.3168(2.7757)	16.7701(3.9652)	17.4747(2.1193)
Mamba	$T=100$	1.8131(0.1483)	1.8064(0.1056)	107.7040(84.6160)	106.3311(46.8227)	26.3491(5.0596)	26.3942(3.4584)	30.3135(4.6938)	26.4837(4.0382)
	$T=200$	1.2121(0.0432)	1.2046(0.0281)	81.7960(48.5441)	83.3156(36.7941)	14.5600(5.2563)	15.1934(2.7239)	15.8423(3.5427)	16.9635(2.0712)
SAN	$T=100$	1.8354(0.1601)	1.8239(0.1324)	107.5626(53.0168)	107.5397(45.5941)	25.9918(5.1969)	26.9023(4.0909)	30.9040(5.2740)	28.1543(4.9465)
	$T=200$	1.2324(0.0433)	1.2244(0.0251)	82.6443(49.0879)	85.2087(37.1801)	16.2327(3.3458)	17.1968(3.1887)	17.8292(3.8406)	19.4162(3.6096)
Dish-TS	$T=100$	1.7984(0.1550)	1.8030(0.1122)	108.3069(54.0126)	107.5191(46.0432)	28.0037(16.1282)	27.8738(4.2564)	30.1122(4.3567)	26.4054(4.3121)
	$T=200$	1.2229(0.0421)	1.2085(0.0236)	82.2392(47.8196)	84.5083(36.9033)	15.9934(4.5022)	16.3366(2.7603)	16.5457(3.3847)	17.3776(2.2386)
PatchTST	$T=100$	1.7433(0.1465)	1.7963(0.1117)	108.0326(55.4351)	107.7972(44.8908)	26.4060(5.2275)	27.1559(3.9217)	27.7546(5.5136)	25.3066(4.4854)
	$T=200$	1.2103(0.0421)	1.2160(0.0241)	82.6860(49.7877)	83.6795(36.8187)	16.1644(5.0935)	17.4434(2.8454)	17.5520(4.0382)	16.8657(2.0614)

, and the results show that our model had relatively optimal prediction performance compared to the benchmark model. As expected, NAR performs well under the linear setting of Case 1 with close-to-one MSEs and the smallest standard deviations. The mean of MSEs of our model is near 1 under all three settings, which shows prediction stability.

Table 4. The mean and standard deviation (in parentheses) of MSEs of $\widehat{g}$ for the SNAR and NAR methods.

		SNAR		NAR
		$\mathit{T}$ = 100	$\mathit{T}$ = 200	$\mathit{T}$ = 100	$\mathit{T}$ = 200
Case 1	N = 100	0.1181 (0.0249)	0.0461 (0.0027)	0.0016 (0.0012)	0.0018 (0.0006)
	N = 200	0.0326 (0.0023)	0.0231 (0.0016)	0.0014 (0.0004)	0.0008 (0.0002)
Case 2	N = 100	0.0757 (0.0065)	0.0704 (0.0057)	0.1398 (0.0001)	0.1502 (0.0001)
	N = 200	0.0496 (0.0038)	0.0489 (0.0046)	0.1028 (0.0001)	0.1114 (0.0001)
Case 3	N = 100	0.0475 (0.0034)	0.0307 (0.0021)	0.2278 (0.0001)	0.2255 (0.0001)
	N = 200	0.0215 (0.0016)	0.0142 (0.0009)	0.1607 (0.0001)	0.1580 (0.0001)
Case 4	N = 100	0.3654 (0.0301)	0.2082 (0.0184)	0.9375 (0.0001)	0.9654 (0.0001)
	N = 200	0.1357 (0.0109)	0.0752 (0.0081)	0.6810 (0.0001)	0.6589 (0.0001)

shows the mean and standard deviation of MSEs of $\widehat{g}$, where $\mathrm{MSE}\left(\widehat{g}\right)={\displaystyle \frac{1}{NT}}{\sum }_{i=1}^N{\sum }_{t=1}^T{|\widehat{g}\left({\mathbf{Z}}_t^i\right)-g_0\left({\mathbf{Z}}_t^i\right)|}^2$. It can be seen from the table that the MSE of $\widehat{g}$ obtained by both methods decreases as N or T increases, which verifies the theorems. In addition, the NAR model has a smaller MSE of $\widehat{g}$ than the SNAR model if the model setting is correct, as shown in Case 1. The SNAR model has a much better performance than the NAR model if the real structure of g is not linear, as shown in Cases 2–4.

4.2. Real Data Analysis

To demonstrate our proposed SNAR method on real data, we consider three datasets, namely international trade data, stock return data, and weather pollution index PM2.5 data.

The international trade data shown in Figure 1 contains the yearly international trade between 58 countries from 1981 to 2000, which has been analyzed in [2,23,24]. We set the logarithm of a country’s annual total trade volume as the response variable and set the logarithm of the country’s gross domestic product (GDP) and the level of political democracy as time-varying covariates. The method used in [24] is adopted to generate the network and then establish the SNAR model.

The stock return data is similar to that introduced in [7]. They built a NAR model to fit the return series of stocks. We consider 182 constituent stocks of the CSI 300 index and collect 545 daily prices of these stocks from 1 January 2019, to 31 March 2021 and generate daily return series by calculating ${\mathrm{rate}}_t=({\mathrm{price}}_t-{\mathrm{price}}_{t-1})/{\mathrm{price}}_{t-1}$ for each stock. To generate the network between 182 stocks, we calculate the correlations of the daily return series and consider an undirected network. We collect node-specific time-varying covariates about the operation status of a company from its publicly disclosed quarterly financial statements. There are 247 covariates in total, including Total Assets, Total Liabilities, Paid-in Capital, Total Owner’s Equity, Total Revenues, Total Expenses, and Net Income. Severe multicollinearity exists between these covariates.

The hourly air pollution data from 1 March 2013, to 28 February 2017, were obtained from 12 monitoring stations in Beijing, containing the hourly concentration levels of several air pollutants such as PM2.5, NO2, and other meteorological data such as temperature and surface pressure ([25]). This paper focuses on the daily PM2.5 concentration as our response, and all the other variables are the covariates. The network structure is established according to the distance between monitoring stations.

We fit these three datasets using the SNAR model and compare it with the NAR model and other SOTA methods. The MSEs of each test set data are shown in

Table 5. The MSEs of test set data for the SNAR and NAR models and other SOTA methods.

	Trade	Stock Return	PM2.5
SNAR	0.0191	0.9930	0.0511
NAR	0.0206	1.6540	0.0516
RNN	0.0269	2.8696	0.0542
LSTM	0.0251	1.7884	0.0526
Mamba	0.0232	1.6099	0.0510
SAN	0.0265	1.6963	0.0528
Dish-TS	0.0226	1.6859	0.0521
PatchTST	0.0234	1.6398	0.0519

. For more analysis results, see Appendix C.

5. Conclusions

In this paper, we propose the Semiparametric Network vector AutoRegressive (SNAR) model to address the limitations of strict linearity and the curse of dimensionality in existing network time series analysis. By integrating deep neural networks with the statistical NAR framework, our primary objective is to accurately model the complex, nonlinear effects of high-dimensional node-specific covariates while preserving the structural interpretability of network interactions.

Our major contributions are summarized as follows: (1) This paper develops a unified estimation algorithm that simultaneously learns the nonparametric component and the autoregressive parameters. (2) We provide rigorous theoretical guarantees, establishing that the nonparametric deep neural network estimator achieves the minimax optimal convergence rate. (3) We prove the $\sqrt{NT}$ consistency and asymptotic normality of the parametric estimators for the momentum and network effects. These theoretical results ensure that the SNAR model retains the statistical interpretability of the classic NAR model while significantly enhancing predictive performance in nonlinear settings, as demonstrated by our extensive simulations and real-world experiments on trade, stock, and pollution networks.

Despite these advancements, our study has several limitations that outline promising directions for future research. Firstly, the current framework relies on a static network structure. Since social and economic relationships often evolve, future work will extend the SNAR model to accommodate dynamic networks with time-varying adjacency matrices. Secondly, our theoretical derivations assume i.i.d. Gaussian errors to facilitate the proof of convergence rates. While the proposed estimator, viewed as a Quasi-MLE, is generally robust under finite moment conditions, real-world data frequently exhibit heavy tails or dependent noise. We plan to investigate more robust loss functions to handle such non-Gaussian distributions. Finally, the autoregressive parameters are assumed homogeneous across all nodes to ensure model parsimony and statistical efficiency. However, influence structures in realistic networks may vary by cluster. Extending the framework to incorporate heterogeneous or random effects would be a valuable development to capture node-specific dynamics more precisely.