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EXACT LIKELIHOOD INFERENCE IN GROUP INTERACTION NETWORK MODELS

Published online by Cambridge University Press:  19 December 2016

Grant Hillier  and
Federico Martellosio
Grant Hillier*
Affiliation:
University of Southampton
Federico Martellosio
Affiliation:
University of Surrey
*
*Address correspondence to Grant Hillier, CeMMAP and Department of Economics, University of Southampton, Highfield, Southampton SO17 1BJ, UK; e-mail: ghh@soton.ac.uk.
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Abstract

The paper studies spatial autoregressive models with group interaction structure, focussing on estimation and inference for the spatial autoregressive parameter λ. The quasi-maximum likelihood estimator for λ usually cannot be written in closed form, but using an exact result obtained earlier by the authors for its distribution function, we are able to provide a complete analysis of the properties of the estimator, and exact inference that can be based on it, in models that are balanced. This is presented first for the so-called pure model, with no regression component, but is also extended to some special cases of the more general model. We then study the much more difficult case of unbalanced models, giving analogues of some, but by no means all, of the results obtained for the balanced case earlier. In both balanced and unbalanced models, results obtained for the pure model generalize immediately to the model with group-specific regression components.

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 2 , April 2018 , pp. 383 - 415
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

We thank the co-editor, Rob Taylor, the three referees, and Peter Phillips for comments on the first version of the paper that improved its presentation considerably.

References

REFERENCES

Andrews, D.W.K. (1993) Exactly median-unbiased estimation of first-order autoregressive/unit root models. Econometrica 61, 139–165.CrossRefGoogle Scholar
Arnold, S.F. (1979) Linear models with exchangeably distributed errors. Journal of the American Statistical Association 74, 194–199.Google Scholar
Baltagi, B.H. (2006) Random effects and spatial autocorrelation with equal weights. Econometric Theory 22, 973–984.Google Scholar
Bao, Y. (2013) Finite-sample bias of the QMLE in spatial autoregressive models. Econometric Theory 29, 68–88.Google Scholar
Barndorff-Nielsen, O. (1980) Conditionality resolutions. Biometrika 67, 293–310.Google Scholar
Buckley, M.J. & Eagleson, G.K. (1988) An approximation to the distribution of quadratic forms in normal random variables. Australian Journal of Statistics 30A, 150–159.Google Scholar
Bramoullé, Y., Djebbari, H., & Fortin, B. (2009) Identification of peer effects through social networks. Journal of Econometrics 150, 41–55.Google Scholar
Boucher, V., Bramoullé, Y., Djebbari, H., & Fortin, B. (2014) Do peers affect student achievement? Evidence from Canada using group size variation. Journal of Applied Econometrics 29, 91–109.Google Scholar
Case, A. (1992) Neighborhood influence and technological change. Regional Science and Urban Economics 22, 491–508.Google Scholar
Daniels, H.E. (1987) Tail probability approximations. International Statistical Review 55, 37–48.Google Scholar
Davezies, L., D’Haultfoeuille, X., & Fougère, D. (2009) Identification of peer effects using group size variation. Econometrics Journal 12, 397–413.Google Scholar
de Paula, A. (2016) Econometrics of Network Models, Advances in Economics and Econometrics: Theory and Applications, Eleventh World Congress, forthcoming.CrossRefGoogle Scholar
Hall, P. (1983) Chi squared approximations to the distribution of a sum of independent random variables. The Annals of Probability 11, 1028–1036.CrossRefGoogle Scholar
Hillier, G.H. & Martellosio, F. (2016) Exact and higher-order properties of the MLE in spatial autoregressive models, with applications to inference. Manuscript (This is a substantially revised version of Hillier, G.H. & F. Martellosio (2013) Properties of the Maximum Likelihood Estimator in Spatial Autoregressive Models. Cemmap Working papers, CWP44/13.)CrossRefGoogle Scholar
Hsieh, C.-S. & Lee, L.F. (2016) A social interactions model with endogenous friendship formation and selectivity. Journal of Applied Econometrics 31, 301–319.CrossRefGoogle Scholar
James, A.T. (1964) Distributions of matrix variates and latent roots derived from normal samples. Annals of Mathematical Statistics 35, 475–501.CrossRefGoogle Scholar
Johnson, N.L., Kotz, S., & Balakrishnan, N. (1994) Distributions in Statistics—Continuous Univariate Distributions, 2nd ed., vol. 1. John Wiley & Sons.Google Scholar
Kelejian, H.H., Prucha, I.R., & Yuzefovich, Y. (2006) Estimation problems in models with spatial weighting matrices which have blocks of equal elements. Journal of Regional Science 46, 507–515.Google Scholar
King, M.L. (1981) A small sample property of the Cliff-Ord test for spatial autocorrelation. Journal of the Royal Statistical Society B 43, 263–264.Google Scholar
Lee, L.F. (2004) Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica 72, 1899–1925.Google Scholar
Lee, L.F. (2007) Identification and estimation of econometric models with group interactions, contextual factors and fixed effects. Journal of Econometrics 140, 333–374.Google Scholar
Lee, L.F. & Liu, X. (2010) Efficient GMM estimation of high order spatial autoregressive models with autoregressive disturbances. Econometric Theory 26, 187–230.Google Scholar
Lee, L.F., Liu, X., & Lin, X. (2010) Specification and estimation of social interaction models with network structures. The Econometrics Journal 13, 145–176.Google Scholar
Lee, L.-F. & Yu, J. (2010) Estimation of spatial autoregressive panel data models with fixed effects. Journal of Econometrics 154, 165–185.Google Scholar
Lugannani, R. & Rice, S.O. (1980) Saddlepoint approximations for the distribution of the sum of independent random variables. Advances in Applied Probability 12, 475–490.Google Scholar
Manski, C.F. (1993) Identification of endogenous social effects: The reflection problem. The Review of Economic Studies 60, 531–542.Google Scholar
Moffitt, R. (2001) Policy interventions, low-level equilibria and social interactions. In Durlauf, S. & Young, P. (eds.), Social Dynamics, pp. 45–82. MIT Press.Google Scholar
Robinson, P.M. & Rossi, F. (2015a) Refinements in maximum likelihood inference on spatial autocorrelation in panel data. Journal of Econometrics 189, 447–456.Google Scholar
Robinson, P.M. & Rossi, F. (2015b) Refined tests for spatial correlation. Econometric Theory 31, 1249–1280.Google Scholar
Ruben, H. (1962) Probability content of regions under spherical normal distributions, IV: The distribution of homogeneous and non-homogeneous quadratic functions of normal variables. The Annals of Mathematical Statistics 33, 542–570.Google Scholar
Smith, R.J. (1997) Alternative semi-parametric likelihood approaches to generalised method of moments estimation. The Economic Journal 107, 503–519.Google Scholar
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