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First Draft of the paper - University of Toronto

sometimes this requires a bit **of** work), **the** standard advice is re-examine **the**model, seeking re-parameterizations or simplifications that will make it identifiedwithout doing too much violence to scientific plausibility. Sometimes,additional variables can be included in **the** analysis, and **the** model that includes**the**se new variables may be identified even when **the** original modelis not. Such variables, included in **the** analysis primarily to obtain modelidentification, are called “instrumental variables” (Fuller, 1987).This is great when it works, but in our view it requires too much expertise.Fixing up a non-identified model requires a combination **of** quantitativesophistication and subject-matter sophistication that is not always easy t**of**ind in **the** same person, unless that person is an econometrician or a psychometrician.And even when an individual or research team can muster **the**right combination **of** expertise, **the** results can be disappointing. Our recommendationis to plan **the** statistical analysis in advance, and to ensure modelidentification by collecting **the** right kind **of** data. The key to **the** method wepropose is to measure **the** independent variables on more than one occasion,preferably using different methods or measuring instruments.3.2 The test-retest designThe model identification problem is solved if we measure all **the** independentvariables twice, in such a way that errors **of** measurement on **the** two occasionsare independent. We begin with a classical structural equation model inwhich all random variables have expected value zero and **the**re no intercepts,and **the**n later extend it to a model with intercepts and non-zero expectedvalues.For each **of** n independent observations, we assume **the** following simultaneousequation model. Implicitly, all **the** random quantities involved havea subscript i, i = 1, . . . , n.whereX 1 = ξ + δ 1 (7)X 2 = ξ + δ 2 ,Y = Γξ + ζY is an m × 1 random vector **of** observable dependent variables, so **the**regression can be multivariate.22

Γ is an m × p matrix **of** unknown constants. These are **the** regressioncoefficients, with one row for each dependent variable and one columnfor each independent variable.ξ is a p×1 random vector **of** latent independent variables, with expectedvalue zero and variance-covariance matrix Φ, an m × m symmetric andpositive definite matrix **of** unknown constants.ζ is **the** error term **of** **the** latent regression. It is an m × 1 randomvector with expected value zero and variance-covariance matrix Ψ, anm × m symmetric and positive definite matrix **of** unknown constants.X 1 and X 2 are p × 1 observable random vectors, each representing ξplus a different piece **of** random error.δ 1 is **the** measurement error in X 1 . It is a p × 1 random vector **of** errorterms, with expected value zero and variance-covariance matrix Θ 1 , ap × p symmetric and positive definite matrix **of** unknown constants.δ 2 is **the** measurement error in X 2 . It is a p × 1 random vector **of** errorterms, with expected value zero and variance-covariance matrix Θ 2 , ap × p symmetric and positive definite matrix **of** unknown constants.ξ, ζ, δ 1 and δ 2 are all uncorrelated.Notice that in this model, measurement errors in **the** independent variablescan be correlated in one sense, but not in ano**the**r. Because **the** variancecovariancematrices **of** **the** error terms (Θ 1 and Θ 2 ) need not be diagonal, **the**model allows, for example, farmers who overestimate **the**ir number **of** pigs toalso overestimate **the**ir number **of** cows. On **the** o**the**r hand, if one thinks **of**X 1 and X 2 as measurements **of** **the** independent variables by two differentmethods, **the**n **the** errors **of** measurement by different methods must not becorrelated. For example, if **the** number **of** pigs were counted once by **the** farmmanager at feeding time (an element **of** X 1 ) and on ano**the**r occasion by aresearch assistant from an areal photograph (**the** corresponding element **of**X 2 ), **the**n **the** requirement **of** uncorrelated measurement errors would surelybe satisfied.To emphasize an important practical point, **the** matrices Θ 1 and Θ 2 mustbe **of** **the** same size, but none **of** **the**ir corresponding elements need be equal.This means that if measurements **of** **the** independent variables are obtainedby two different methods, **the** methods need not be equally precise.23

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