Semiparametric spatial autoregressive models with nonlinear endogeneity

Abstract.

This article constructs nonparametric two-step least squares (2SLS) and generalized method of moments (GMM) sieve estimators to estimate a functional-coefficient spatial autoregressive model with an endogenous environment variable. We derive the consistency and asymptotic normality results for our proposed sieve estimators. A small Monte Carlo study shows that our proposed estimators exhibit good finite-sample performance. An empirical application is used to illustrate the usefulness of our methods.

Keywords:

• Nonlinear endogeneity
• sieve estimator
• spatial autoregressive
• varying coefficients

JEL Classification::

• C13
• C14
• C21
• R32

Acknowledgments

This paper uses the data set of Baltagi et al. (2016) which is published at the Journal of Applied Econometrics Archive.

Disclosure statement

The author declares no conflict of interest that influences the author’s objectivity of the research.

Notes

1. Different orthonormal basis functions are allowed to approximate the unknown curves.

2. The “elect80” dataset comes with spData package in the R software.

3. Differing from the kernel estimation method, the optimal smoothing parameter, κ_n, is unknown for the sieve estimation method because the exact order of the sieve approximation error is unknown. In practice, one may always begin with a large κ_n before applying any basis function selection method.

4. As a common measure of market concentration in the literature, the higher the HHI index, the higher the market’s concentration, the closer a market is to a monopoly, and the lower the market’s competition.

5. Inverse power functions are not used because firms located in the same city share the same latitude and longitude data.

6. We have set $d^{\ast }=200$, 300, and 400 for robustness check. To save space, we only report the results for $d^{\ast }=300$, as the qualitative patterns of our empirical results do not change across different choices of d^∗.

7. The control function approach can be easily adapted to handle the potential endogeneity of some of the input variables if first-stage IV regression is available. Specifically, we can: (i) construct the first-stage IV regression model of the endogenous variables in x_i to obtain an error term μ_i, which is assumed to be correlated with the error term (u_i) in model (6.3); (ii) define the control function as $E(u_i|X_i,v_i,{\mu }_i)$$=E(u_i|v_i,{\mu }_i)\equiv \omega (v_i,{\mu }_i)$. By following these steps, our theory remains valid with the proper adjustment of assumptions. However, due to data limitations, further exploration of the possibility that some of the x_i variables are endogenous in model (6.3) falls outside the scope of this article. We intend to investigate this aspect in our future research.

8. For instance, we sort the coefficient estimates ${\tilde{\rho }}_{\mathrm{opt}}\left(z_1\right)$,…, ${\tilde{\rho }}_{\mathrm{opt}}\left(z_n\right)$ in ascending order. The (100α)th percentile represents the coefficient estimate below which a percentage of 100α of the estimates falls, where $\alpha \in (0,1)$.

9. Our computer does not have the capacity to meet the memory requirements for the calculation of the GMM sieve estimator for the POFs data with n = 9,117. Out of fair comparison, the optimal 2SLS estimator is used in the empirical application. Additionally, we report the results with κ_n = 4 and 8 for the SOFs and POFs data, respectively. The estimated patterns remain unchanged with ${\kappa }_n\in \{4,\dots ,9\}$ for the SOFs data and ${\kappa }_n\in \{8,9,\dots ,15\}$ for the POFs data.

Additional information

Funding

The author would like to thank the Social Science and Humanities Research Council of Canada for financial support from Insight Grant 435-2016-0340