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First Draft of the paper - University of Toronto

Pro**of** **of** model identification The following pro**of** is little more than anexercise, but it illustrates how model identification is established for structuralequation models in general, and it also leads to several points we wantto make. Collecting X 1 , X 2 and Y into a single long data vector, we writeits variance-covariance matrix as a partitioned matrix:⎡⎤Σ 11 Σ 12 Σ 13Σ = ⎣ Σ ′ 12 Σ 22 Σ 23⎦ , (8)Σ ′ 13 Σ ′ 23 Σ 33where **the** covariance matrix **of** X 1 is Σ 11 , **the** covariance matrix **of** X 2 is Σ 22 ,**the** matrix **of** covariances between X 1 and Y is Σ 13 , and so on.The parameters **of** **the** model consist **of** **the** non-redundant elements **of** **the**matrices Γ, Φ, Ψ, Θ 1 and Θ 2 . Assuming multivariate normality, **the** probabilitydistribution **of** **the** observable random variables corresponds uniquelyto Σ. Thus, to prove model identification, we need to show we can express**the** model parameters in terms **of** **the** Σ ij quantities. **First**, we use Model (7)to write **the** Σ ij matrices in terms **of** **the** parameter matrices.Σ 11 = Φ + Θ 1 (9)Σ 12 = ΦΣ 13 = ΦΓ ′Σ 22 = Φ + Θ 2Σ 23 = ΦΓ ′Σ 33 = ΓΦΓ ′ + ΨThis system **of** matrix equations is readily solved for **the** parameter matricesto yieldΦ = Σ 12 (10)Θ 1 = Σ 11 − Σ 12Θ 2 = Σ 22 − Σ 12Γ = Σ ′ 13Σ −112 = Σ ′ 23Σ −112Ψ = Σ 33 − Σ ′ 13Σ −112 Σ 13 .This shows that Model (7) is identified, so that if data are collectedfollowing **the** test-retest recipe, **the**n **the** data analyst may proceed without24

giving fur**the**r thought to model identification. Again, **the** test-retest recipeis to measure **the** independent variables on more than one occasion, in sucha way that errors **of** measurement may be assumed independent betweenoccasions. We emphasize that most data sets do not look like this, at present.An exception are data collected according to Campbell and Fiske’s (reference)“multi-trait multi-method matrix” scheme for ascertaining convergent anddivergent validity **of** psychological measurements.We are suggesting that **the** independent variables be measured twice.Measuring **the** dependent variable(s) twice has no effect on **the** issue **of** modelidentification. We do not know if it affects precision **of** estimation or **the**quality **of** inference.Multivariate Normality Our discussion **of** model identification mentionedmultivariate normality, but this is not necessary. Suppose that Model (7)holds, and that **the** distributions **of** **of** **the** latent independent variables anderror terms are unknown, except for possessing covariance matrices. In thiscase **the** parameter **of** **the** model could be expressed as (Γ, Φ, Ψ, Θ 1 , Θ 2 ,F Φ , F ζ , F δ1 , F δ2 ), where F Φ , F ζ , F δ1 and F δ2 are **the** cumulative distributionfunctions **of** Φ, ζ, δ 1 and δ 2 respectively.Note that **the** parameter in this “non-parametric” problem is **of** infinitedimension, but this presents no conceptual difficulty. The probability distribution**of** **the** observed data is still a function **of** **the** parameter, and to showmodel identification, we would have to be able to recover **the** parameter from**the** probability distribution **of** **the** data. While in general we cannot recover**the** entire parameter vector, we certainly can recover a useful function **of** it,namely Γ. In fact, Γ is **the** only quantity **of** interest; **the** remainder **of** **the**parameter vector consists only **of** nuisance parameters, whe**the**r **the** model isnormal or not.Again using Σ to denote **the** covariance matrix **of** **the** observed data, wesee that Σ is a function **of** **the** probability distribution **of** **the** observed data.The calculations leading to (10) still hold, showing that Γ is a function **of**Σ, and hence **of** **the** probability distribution **of** **the** data. This means thatΓ is identified, and consistent estimation **of** it is possible; for example, areasonable though non-standard estimator iŝΓ = 1 2 ( ̂Σ ′ −113 ̂Σ 12 + ̂Σ ′ −123 ̂Σ 12 ), (11)where ̂Σ is **the** sample variance-covariance matrix. Consistency follows from25

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