Identification, Equivalent Models, and Computer Algebra

1 Introduction


1.1 Themes of this Book


1.2 An Overview of the Book


1.3 Limitations


1.4 On Notation


1.5 Tie Commutation Matrix


1.6 Computer Algebra


2 Identification, Equivalence and the Computerized Evaluation of Rank Conditions


2.1 Introduction


2.2 Basic Concepts


2.3 Identification and the Rank of the Information Matrix


2.4 The Jacobian Matrix Criterium


2.5 Prior Information


2.6 Handling the Rank in Practice


2.7 Partial Identification


2.8 Identification and Equivalence


2.9 The Exact Rank Analyzer (ERA)


3 Simultaneous Equations Models


3.1 Introduction


3.2 The Conventional Simultaneous Equations Model


3.3 Simultaneous Equations Systems with Covariance Restrictions


3.4 An Example


3.5 A General Model


3.6 Simultaneous Equations Models with Measurement Error


3.7 An Example


3.8 Elaboration of the Example


3.9 Another Jacobian Matrix


4 Identification in Factor Analysis Models


4.1 Introduction


4.2 The Jacobian Matrix in Confirmatory Factor Analysis


4.3 An Example


4.4 Exploratory Factor Analysis


4.5 The Indeterminacy Problem


5 Restrictions on the Reduced Form and its Computerized Parametrization


5.1 Introduction


5.2 The Number of Restrictions on the Reduced Form


5.3 Exclusion Restrictions


5.4 Transformable Restrictions


5.5 A Minimal Parametrization


5.6 Examples of a Minimal Parametrization


5.7 An Algorithm for the Computation of a Minimal Parametrization of ∏


5.8 MINPARAM


5.9 Examples of MINPARAM


6 Identification in the Lisrel Model


6.1 Introduction


6.2 The Lisrel Model


6.3 Identification Using a Minimal Parametrization


6.4 Identification Using an Alternative Parametrization


6.5 An Example


6.6 Identification Using the Augmented Jacobian Matrix Method


6.7 Order Conditions for Lisrel

Appendix: Derivation of the Jacobian Matrix for the Original Parameters


7 Equivalence of Non-Nested Models


7.1 Introduction


7.2 Equivalence and Underidentification


7.3 Equivalence and Linear Manifolds


7.4 A General Condition for Equivalence


7.5 Equivalence of Systems of Simultaneous Equations


8 Program Description


8.1 General Description


8.2 Main Programs


8.3 Subprograms


8.4 ERA


8.5 IDSIMCO and SIMCOJAC


8.6 IDSIMEV and SIMEVJAC


8.7 IDFAC and FACJAC


8.8 IDLISEX and LISEXJAC


8.9 IDLIS and LISJAC


8.10 MINPARAM


8.11 LISORDER


8.12 EQUISIM

Bibliography

Subject Index

Author Index