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24-08-2020

An iterative method for the computation of the correlation matrix implied by a recursive path model

Authors: M’barek Iaousse, Zouhair El Hadri, Amal Hmimou, Yousfi El Kettani

Published in: Quality & Quantity | Issue 3/2021

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Abstract

In Path Analysis, especially in social sciences studies, many researchers usually assume that errors in the model are uncorrelated with all exogenous variables as well as with each other. These assumptions, in most cases, are not valid in reality and were introduced to facilitate the model estimation. This article establishes a new algorithm for the computation of the correlation matrix implied by a recursive path model that overcomes these drawbacks. We compare our algorithm to two other methods used in the literature. The comparison was made mathematically through an illustrated example and numerically with a simulation study. The findings show that, unlike the classical methods, the proposed method gives more accurate results.

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Bollen (1989) and others choose to define a recursive model by the fact that \({\varvec{B}}\) is lower triangular and \({\varvec{\varPsi }}\) is a diagonal matrix.

more details about the limits of assuming uncorrelated errors are presented in Sect. 2.

Many other, econometric, methods are used in dealing with the estimation step such as Two-Stage Least Squares, Three-Stage Least Squares. See Greene (2018) and Wooldridge (2010) for more details.

Bollen (1989) added the recursiveness of a model is sufficient condition for its identification. However this is based on his definition for recursiveness, which includes a diagonal matrix of \({\varvec{\varPsi }}\).

It should be noted that \({\varvec{0}}\) is used to define the null matrix whose order depends on where it is used. For example: \({\mathbb {E}}[{\varvec{\xi }}{\varvec{\zeta }}^{t}]={\varvec{0}}={\varvec{0}}_{1:q,1:p}\) since \({\varvec{\xi }}\) is vector of order q and \({\varvec{\zeta }}\) is vector of order p. Therefore, The reader shall know from the expression the order of the matrix (eventually vector) \({\varvec{0}}\).

The term augmented set of exogenous variables is used in Kang and Seneta (1980) and it refers to all model variables whose causes are not explicit in the model (i.e. observed exogenous variables and the unmeasured errors.

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Title: An iterative method for the computation of the correlation matrix implied by a recursive path model
Authors: M’barek Iaousse
Zouhair El Hadri
Amal Hmimou
Yousfi El Kettani
Publication date: 24-08-2020
Publisher: Springer Netherlands
Published in: Quality & Quantity / Issue 3/2021
Print ISSN: 0033-5177
Electronic ISSN: 1573-7845
DOI: https://doi.org/10.1007/s11135-020-01034-1

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