[Paper Review] Dynamic Spatial Panel Models: Networks, Common Shocks, and Sequential Exogeneity

Guido M. Kuersteiner, Ingmar R. Prucha|arXiv (Cornell University)|Feb 6, 2018

Spatial and Panel Data Analysis73 references71 citations

TL;DR

This paper develops GMM estimators for dynamic spatial panel models with endogenous networks, common shocks, and sequentially exogenous regressors. It introduces a quasi-forward differencing transformation to eliminate interactive effects while preserving orthogonality of moment conditions, enabling consistent estimation and mixed normal asymptotic distributions despite stochastic norming. The key contribution is a tractable inference framework for models with complex cross-sectional dependence and data-dependent network formation.

ABSTRACT

This paper considers a class of GMM estimators for general dynamic panel models, allowing for weakly exogenous covariates and cross sectional dependence due to spatial lags, unspecified common shocks and time-varying interactive effects. We significantly expand the scope of the existing literature by allowing for endogenous spatial weight matrices without imposing any restrictions on how the weights are generated. An important area of application is in social interaction and network models where our specification can accommodate data dependent network formation. We consider an exemplary social interaction model and show how identification of the interaction parameters is achieved through a combination of linear and quadratic moment conditions. For the general setup we develop an orthogonal forward differencing transformation to aid in the estimation of factor components while maintaining orthogonality of moment conditions. This is an important ingredient to a tractable asymptotic distribution of our estimators. In general, the asymptotic distribution of our estimators is found to be mixed normal due to random norming. However, the asymptotic distribution of our test statistics is still chi-square.

Motivation & Objective

• To develop GMM estimators for dynamic panel models with spatial lags, common shocks, and interactive fixed effects.
• To allow for endogenous spatial weight matrices without restrictions on network formation processes.
• To ensure orthogonality of moment conditions under general cross-sectional dependence and stochastic norming.
• To establish asymptotic normality of estimators and chi-square distribution of test statistics under weak moment conditions.
• To enable identification and inference in social interaction models with data-dependent network formation.

Proposed method

• Uses a general GMM framework with linear and quadratic moment conditions to identify interaction parameters.
• Introduces a quasi-forward differencing transformation to eliminate interactive effects while maintaining orthogonality of moment conditions.
• Extends classical M-estimator consistency and CLT results to stochastic objective functions arising from random norming.
• Derives sufficient conditions for diagonalization of the optimal weight matrix in the GMM framework.
• Applies Helmert transformation as a special case of the proposed differencing method.
• Employs martingale difference representation and C-mixing stability to establish asymptotic distributions.

Experimental results

Research questions

• RQ1 How can GMM estimators be constructed for dynamic spatial panels with endogenous network weights and common shocks?
• RQ2 What transformation preserves orthogonality of moment conditions while removing interactive fixed effects?
• RQ3 How can identification of interaction parameters be achieved using both linear and quadratic moment conditions?
• RQ4 What is the asymptotic distribution of estimators when the norming matrix is random and stochastic?
• RQ5 How can valid inference be conducted under general cross-sectional dependence and non-i.i.d. data?

Key findings

• The proposed quasi-forward differencing transformation ensures orthogonality of moment conditions and enables tractable asymptotic theory despite stochastic norming.
• The asymptotic distribution of the GMM estimator is mixed normal due to random norming, but test statistics converge to a chi-square distribution.
• Identification of interaction parameters is achieved through a combination of linear and quadratic moment conditions, even under data-dependent network formation.
• The optimal weight matrix can be diagonalized under sufficient conditions derived in the paper, simplifying inference.
• The estimator remains consistent and asymptotically normal under weak moment conditions, including when network density grows with n.
• The framework supports inference in models with both spatial lags and unobserved common factors, even when T is fixed.