In statistics, causal inference describes the process of determining causal effects within a system. In order to accurately model observed variables, statisticians often derive estimators from graphical models such as the linear structural equation model (LSEM). These models are constructed under the assumption that it is possible to model unobserved (latent) variables by accounting for their effects through the usage of correlated error terms of the observed variables. The main objective of this thesis is to investigate, given an observation, how the uniqueness of structural parameters in such a model is affected by external equality constraints. In this case, the constraints of interest are caused by equiconfounded random variables, i.e. multiple observed variables that are influenced by a common unobserved confounder. The main tool used to quantify such homogeneities are edge-colorings that are applied to the underlying causal graph. Using this terminology, one can establish results that are similar in nature to those found when studying uncolored graphs, like an upper bound for the number of edges in an identifiable graph. However, it quickly becomes clear that the usage of edge-colorings severely influences the identifiability of certain graphs. For instance, it is possible to greatly improve the number of identifiable graphs with a fixed bidirected structure by using a certain coloring, as discussed in section 3.2. Section 3.4 explores how colorings can also cause adverse effects by limiting identifiability for some graphs, which has significant effects on the applicability of graphical criteria for identifiability to colored graphs. All computational results in this thesis have been produced using concepts from algebraic geometry, as described in sections 3.1 and 3.2. Specifically, section 3.1 contains multiple results that that are similar to well-known methods from the analysis of uncolored LSEMs, which have been applied to the colored case. Finally, every section is accompanied by multiple examples and analyses of special graph layouts, the most notable being several graphs in the appendix that highlight the undesired properties of graph colorings as discussed in section 3.4.      «

In statistics, causal inference describes the process of determining causal effects within a system. In order to accurately model observed variables, statisticians often derive estimators from graphical models such as the linear structural equation model (LSEM). These models are constructed under the assumption that it is possible to model unobserved (latent) variables by accounting for their effects through the usage of correlated error terms of the observed variables. The main objective of th...     »