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l71 - Structural Coefficients in Recursive Models Evils of...

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Structural Coefficients in Recursive Models/Evils of Standardization Page 1 Structural Coefficients in Recursive Models/ Evils of Standardization R ECURSIVE DEFINED . A model is said to be recursive if all the causal linkages run “one way”, that is, no two variables are reciprocally related in such a way that each affects and depends on the other, and no variable feeds back upon itself through any indirect concatenation of causal linkages. The model we have been looking at is recursive: X1 X2 X3 X4 u v w The model would be non-recursive if, for example, X4 also affected X3, i.e. the causation ran in both directions. Non- recursive models are much more difficult to work with, and we’ll discuss them later in the course. R ECURSIVE MODELS WITH STANDARDIZED VARIABLES . We have been examining a 4-variable recursive model in which variables were standardized. The b’s in such models are referred to as the path coefficients. The advantages of standardized variables are: 1. Certain algebraic steps are simplified 2. Sewell Wright’s rule for expressing correlations in terms of path coefficients can be applied without modification 3. Continuity is maintained with the earlier literature on path analysis and causal models in Sociology 4. It shows how an investigator whose data are only available in the form of a correlation matrix can, nevertheless, make use of a clearly specified model in interpreting those correlations. Nevertheless, standardization should generally be avoided. Standardization tends to obscure the distinction between the structural coefficients of the model and the several variances and covariances that describe the joint distribution of the variables in a certain population. To illustrate this, we will first see what happens when variables are not standardized.
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Structural Coefficients in Recursive Models/Evils of Standardization Page 2 R ECURSIVE MODELS WITH UNSTANDARDIZED VARIABLES . We will continue to assume that al l X’s have a mean of 0. (The only thing this affects is the intercepts.) We will not assume that variances all = 1. Under these conditions, E(X i 2 ) = E(X i X i ) = V(X i ) = σ i 2 E(X i X j ) = Cov(X i X j ) = σ ij To get the normal equations, we proceed as before: Multiply each structural equation by the predetermined variables and then take expectations. In addition, to get the variances, we multiply the structural equation by the DV of the equation and take expectations. Hence, (1) For X2, the structural equation is X X u 2 21 1 The only predetermined variable is X1. Hence, if we multiply both sides of the above equation by X1 and then take expectations, we get the normal equation 2 1 21 21 1 2 1 21 2 1 ) ( ) ( ) ( u X E X E X X E
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