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First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

the

the Strong Law of Large Numbers and a continuity argument. All this assumesthe existence only of second moments and cross-moments. With the assumptionof 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 there are still more unknowns thanequations.If either ν 1 or ν 2 can be assumed zero (or if κ = 0) then the system canbe solved uniquely and the model is identified, but we doubt that such anassumption could be justified very often 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 , oflength m + 2p. We then 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 offsample means and then 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 thenestimating those functions by the corresponding sample means (yielding theMLE 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 aboutthe relationships between independent and dependent variables. We believethat except in special circumstances, this makes it reasonable to employ theclassical 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-theory maximumlikelihood estimators have very nice robustness properties, and are muchmore convenient.27

draft - Toronto and Region Conservation Authority
draft - Toronto and Region Conservation Authority
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