Annals of Statistics

Global identifiability of linear structural equation models

Mathias Drton, Rina Foygel, and Seth Sullivant

Full-text: Open access


Structural equation models are multivariate statistical models that are defined by specifying noisy functional relationships among random variables. We consider the classical case of linear relationships and additive Gaussian noise terms. We give a necessary and sufficient condition for global identifiability of the model in terms of a mixed graph encoding the linear structural equations and the correlation structure of the error terms. Global identifiability is understood to mean injectivity of the parametrization of the model and is fundamental in particular for applicability of standard statistical methodology.

Article information

Ann. Statist., Volume 39, Number 2 (2011), 865-886.

First available in Project Euclid: 9 March 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H05: Characterization and structure theory 62J05: Linear regression

Covariance matrix Gaussian distribution graphical model multivariate normal distribution parameter identification structural equation model


Drton, Mathias; Foygel, Rina; Sullivant, Seth. Global identifiability of linear structural equation models. Ann. Statist. 39 (2011), no. 2, 865--886. doi:10.1214/10-AOS859.

Export citation


  • [1] Andrews, D. W. K. and Guggenberger, P. (2010). Asymptotic size and a problem with subsampling and with the m out of n bootstrap. Econometric Theory 26 426–468.
  • [2] Bollen, K. A. (1989). Structural Equations With Latent Variables. Wiley, New York.
  • [3] Brito, C. and Pearl, J. (2002). A new identification condition for recursive models with correlated errors. Struct. Equ. Model. 9 459–474.
  • [4] Chung, F. R. K. (1997). Spectral Graph Theory. CBMS Regional Conference Series in Mathematics 92. Amer. Math. Soc., Providence, RI.
  • [5] Cox, D., Little, J. and O’Shea, D. (2007). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd ed. Springer, New York.
  • [6] Drton, M., Eichler, M. and Richardson, T. S. (2009). Computing maximum likelihood estimates in recursive linear models with correlated errors. J. Mach. Learn. Res. 10 2329–2348.
  • [7] Drton, M. (2009). Likelihood ratio tests and singularities. Ann. Statist. 37 979–1012.
  • [8] Drton, M. and Yu, J. (2010). On a parametrization of positive semidefinite matrices with zeros. SIAM J. Matrix Anal. Appl. 31 2665–2680.
  • [9] McDonald, R. P. (2002). What can we learn from the path equations?: Identifiability, constraints, equivalence. Psychometrika 67 225–249.
  • [10] Okamoto, M. (1973). Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 763–765.
  • [11] Pearl, J. (2009). Causality: Models, Reasoning, and Inference, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [12] Richardson, T. and Spirtes, P. (2002). Ancestral graph Markov models. Ann. Statist. 30 962–1030.
  • [13] Spirtes, P., Glymour, C. and Scheines, R. (2000). Causation, Prediction, and Search, 2nd ed. MIT Press, Cambridge, MA.
  • [14] Shpitser, I. and Pearl, J. (2006). Identification of joint interventional distributions in recursive semi-Markovian causal models. In Proceedings of the 21st National Conference on Artificial Intelligence 1219–1226. AAAI Press, Menlo Park, CA.
  • [15] Tian, J. (2002). Studies in causal reasoning and learning. Ph.D. thesis, Computer Science Dept., Univ. California, Los Angeles.
  • [16] Tian, J. (2009). Parameter identification in a class of linear structural equation models. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), Pasadena, California 1970–1975. Morgan Kaufmann, San Francisco, CA.
  • [17] Wermuth, N. (2010). Probability distributions with summary graph structure. Bernoulli. To appear. Available at arXiv:1003.3259.