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First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

Proof

Proof of model identification The following proof 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 thematrices 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 expressthe 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, then the data analyst may proceed without24

giving further 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 thequality 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 distributionof the observed data is still a function of the parameter, and to showmodel identification, we would have to be able to recover the parameter fromthe probability distribution of the data. While in general we cannot recoverthe 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 theparameter vector consists only of nuisance parameters, whether 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 ofΣ, 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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