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Functional Network Autoregressive Models for Panel Data

Tomohiro Ando, Tadao Hoshino

Abstract

This study proposes a novel functional vector autoregressive framework for analyzing network interactions of functional outcomes in panel data settings. In this framework, an individual's outcome function is influenced by the outcomes of others through a simultaneous equation system. To estimate the functional parameters of interest, we need to address the endogeneity issue arising from these simultaneous interactions among outcome functions. This issue is carefully handled by developing a novel functional moment-based estimator. We establish the consistency, convergence rate, and pointwise asymptotic normality of the proposed estimator. Additionally, we discuss the estimation of marginal effects and impulse response analysis. As an empirical illustration, we analyze the demand for a bike-sharing service in the U.S. The results reveal statistically significant spatial interactions in bike availability across stations, with interaction patterns varying over the time of day.

Functional Network Autoregressive Models for Panel Data

Abstract

This study proposes a novel functional vector autoregressive framework for analyzing network interactions of functional outcomes in panel data settings. In this framework, an individual's outcome function is influenced by the outcomes of others through a simultaneous equation system. To estimate the functional parameters of interest, we need to address the endogeneity issue arising from these simultaneous interactions among outcome functions. This issue is carefully handled by developing a novel functional moment-based estimator. We establish the consistency, convergence rate, and pointwise asymptotic normality of the proposed estimator. Additionally, we discuss the estimation of marginal effects and impulse response analysis. As an empirical illustration, we analyze the demand for a bike-sharing service in the U.S. The results reveal statistically significant spatial interactions in bike availability across stations, with interaction patterns varying over the time of day.

Paper Structure

This paper contains 43 sections, 16 theorems, 198 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Paper organization
  3. Notation
  4. Functional Network Autoregressive Model
  5. The model
  6. Completeness
  7. Estimation and Asymptotic Theory
  8. Integrated-GMM estimation
  9. Asymptotic theory
  10. Network Multiplier Effects: Marginal Effects and Impulse Responses
  11. Monte Carlo Simulation
  12. Extensions
  13. Functional NAR models with two-way fixed effects
  14. More general interaction structures
  15. Incompletely observed response function
  16. ...and 28 more sections

Key Result

Proposition 2.1

Suppose that Assumption as:inverse holds. Then, $(\text{Id} - \mathcal{A})^{-1}$ exists, and for each $t \in [T]$, $Y_t$ is the only solution of eq:model in the Banach space $(\mathcal{H}_{n, 2}, ||\cdot||_{\infty, 2})$, where $||H||_{\infty, p} \coloneqq \max_{1 \le i \le n} ||h_i||_{L^p}$.

Figures (7)

  • Figure 7.1: Availability of bikes at each station (averaged over 5-9 May, 2014)
  • Figure 7.2: Estimated $\alpha_0(s)$
  • Figure 7.3: Impulse responses
  • Figure E.1: Locations of bike stations
  • Figure E.2: Distribution of potential bike relocation events
  • ...and 2 more figures

Theorems & Definitions (34)

  • Example 1.1: Health data analysis
  • Example 1.2: Demographic data analysis
  • Example 1.3: Transportation data analysis
  • Proposition 2.1
  • Remark 3.1: Choice of the locations and the number of grid points
  • Theorem 3.1: Rates of convergence
  • Theorem 3.2: Asymptotic normality
  • Corollary 3.1
  • Remark 3.2: Choice of $K$
  • Proposition 4.1
  • ...and 24 more