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First Draft of the paper - University of Toronto

**the** Strong Law **of** Large Numbers and a continuity argument. All this assumes**the** existence only **of** second moments and cross-moments. With **the** assumption**of** fourth moments, **the** multivariate Central Limit Theorem wouldprovide a routine basis for large-sample interval estimation and testing.Intercepts We now expand Model (7) to include intercepts and non-zeroexpected values. However, we will see that this leads to complications thatare seldom worth **the** trouble, and **the** classical models with zero expectedvalue and no intercepts are usually preferable. LetY = α + Γξ + ζ (12)X 1 = ν 1 + ξ + δ 1X 2 = ν 2 + ξ + δ 2 ,where α, ν 1 and ν 2 are vectors **of** constants, and E(ξ) = κ. Everything elseis as in Model (7).Under a convenient initial assumption **of** multivariate normality, **the** probabilitydistribution **of** **the** observable data corresponds uniquely to **the** pair(µ, Σ). Since **the** addition **of** constants has no effect on variances or covariances,**the** contents **of** Σ are given by (8) and (9), as before. The expectedvalue µ is **the** partitioned vector⎡µ = ⎣⎤µ 1µ 2⎦ =µ 3⎡⎣E(X 1 )E(X 2 )E(Y)⎤⎡⎦ = ⎣ν 1 + κν 2 + κα + Γκ⎤⎦ . (13)To demonstrate **the** identification **of** **the** model with intercepts, that is,Model (12), one would need to solve **the** equations in (13) uniquely for ν 1 ,ν 2 , κ and α. Even with Γ considered known and fixed because it is identifiedin (10), this is impossible, because **the**re are still more unknowns thanequations.If ei**the**r ν 1 or ν 2 can be assumed zero (or if κ = 0) **the**n **the** system canbe solved uniquely and **the** model is identified, but we doubt that such anassumption could be justified very **of**ten in practice. Most **of** **the** time, allwe can do is identify **the** parameter matrices that appear in **the** covariancematrix, and also **the** functions µ 1 , µ 2 and µ 3 **of** **the** parameter vector. Thiscan be viewed as a re-parameterization **of** **the** model.26

It is instructive to see how this works in **the** multivariate normal case,where **the** parameters would be estimated by maximum likelihood. For i =1, . . . , n, we collect **the** observed data x i,1 , x i,2 and y i into a vector d i , **of**length m + 2p. We **the**n write -2 times **the** log likelihood as a function **of** µand Σ. Simplifying, we obtain−2 log L(µ, Σ) = n[ (m + 2p)(log |Σ| + log 2π) + tr(Σ −1 ̂Σ) (14)+ (d − µ) ′ Σ −1 (d − µ) ].The goal, **of** course, is to minimize (14) over all **the** parameters making upµ and Σ. Now for any value **of** Σ (so long as it is non-singular), **the** quadraticform in **the** second line **of** (14) is zero and **the** entire function is minimizedwhen µ equals d. This means that “centering **the** data” by subtracting **of**fsample means and **the**n pretending that all variables have expected valuezero is equivalent to starting with a model like (12) that contains intercepts,re-parameterizing **the** components **of** µ in (13) as µ 1 , µ 2 and µ 3 , and **the**nestimating those functions by **the** corresponding sample means (yielding **the**MLE **of** µ).Notice that this minimization works for any value **of** **the** matrix **of** regressionslopes Γ, so that **the** MLE **of** Γ is determined entirely by **the** firstline **of** (14). In this sense, **the** mean vector contains no information about**the** relationships between independent and dependent variables. We believethat except in special circumstances, this makes it reasonable to employ **the**classical no-intercept structural equation models to do latent variable regression.And again, while **the** foregoing discussion assumes multivariate normality,it is not really necessary. One can easily assume a model like (12) without anydistributional assumptions beyond **the** existence **of** moments, re-parameterizein terms **of** µ 1 , µ 2 and µ 3 to purchase identification **of** Γ and **the** variancecovarianceparameters, estimate **the** functions µ 1 , µ 2 and µ 3 consistentlywith **the** corresponding sample means, and use an expression like (11) asa consistent estimator **of** Γ. In fact, though, **the** normal-**the**ory maximumlikelihood estimators have very nice robustness properties, and are muchmore convenient.27

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