Unified M-estimation of matrix exponential spatial dynamic panel specification

Abstract

In this paper, a unified M-estimation method in Yang (2018) is extended to the matrix exponential spatial dynamic panel specification (MESDPS) with fixed effects in short panels. Similar to the STLE model which includes the spatial lag effect, the space-time effect and the spatial error effect in Yang (2018), the quasi-maximum likelihood (QML) estimation for MESDPS also has the initial condition specification problem. The initial-condition free M-estimator in this paper solves this problem and is proved to be consistent and asymptotically normal. An outer product of martingale difference (OPMD) estimator for the variance-covariance (VC) matrix of the M-estimator is also derived and proved to be consistent. The finite sample property of the M-estimator is studied through an extensive Monte Carlo study. The method is applied to US outward FDI data to show its validity.

Keywords:

• Matrix exponential
• dynamic panels
• martingale difference
• OPMD
• initial-condition free estimator

JEL CLASSIFICATION:

• C10
• C13
• C15
• C21
• C23

Notes

1 As kindly pointed out by a referee, since the estimation approach is based on the first difference of the model, we cannot estimate the parameters of time-invariant variables.

2 In this paper the commutability is not required since it is a dynamic panel data setting instead of panel data setting.

3 Note the differences in the definition of matrix D and ${\mathbf{D}}_{-1}$ with those in Yang (2018).

4 STLE specification is the comprehensive model which contains the spatial lag effect, dynamic effect, space-time effect and spatial error effect. It corresponds to our MESDPS(1,1,1). See section 2.2 for the detailed model specification.

5 As kindly pointed out by a referee, note here $0.7255>$ ln2, which implies the corresponding STLE coefficient is $1-e^{0.7255}=-1.0658.$ This is one of the advantages of the MESDPS compared with the STLE model: the parameter space of the MESS coefficient for the disturbance term is unrestricted.