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First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

sometimes this requires

sometimes this requires a bit of work), the standard advice is re-examine themodel, 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 includesthese 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 tofind in the same person, unless that person is an econometrician or a psychometrician.And even when an individual or research team can muster theright 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 there no intercepts,and then 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 theregression 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 another. Because the variancecovariancematrices of the error terms (Θ 1 and Θ 2 ) need not be diagonal, themodel allows, for example, farmers who overestimate their number of pigs toalso overestimate their number of cows. On the other hand, if one thinks ofX 1 and X 2 as measurements of the independent variables by two differentmethods, then 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 another occasion by aresearch assistant from an areal photograph (the corresponding element ofX 2 ), then 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 their 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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