Chapter 8
EXACT SMALL SAMPLE THEORY
IN THE SIMULTANEOUS EQUATIONS MODEL
P. C. B. PHILLIPS*
Yale University
Contents
1. Introduction 451
2. Simple mechanics of distribution theory 454
2. I. Primitive exact relations and useful inversion formulae 454
2.2. Approach via sample moments of the data 455
2.3. Asymptotic expansions and approximations 457
2.4. The Wishart distribution and related issues 459
3. Exact theory in the simultaneous equations model 463
The model and notation
Generic statistical forms of common single equation estimators
The standardizing transformations
The analysis of leading cases
The exact distribution of the IV estimator in the general single equation case
The case of two endogenous variables
Structural variance estimators
Test statistics
Systems estimators and reduced-form coefficients
Improved estimation of structural coefficients
Supplementary results on moments
Misspecification
4. A new approach to small sample theory
4.1 Intuitive ideas
4.2. Rational approximation
4.3. Curve fitting or constructive functional approximation?
5. Concluding remarks
References

Chapter 8
EXACT SMALL SAMPLE THEORY
IN THE SIMULTANEOUS EQUATIONS MODEL
P. C. B. PHILLIPS*
Yale University
Contents
1. Introduction 451
2. Simple mechanics of distribution theory 454
2. I. Primitive exact relations and useful inversion formulae 454
2.2. Approach via sample moments of the data 455
2.3. Asymptotic expansions and approximations 457
2.4. The Wishart distribution and related issues 459
3. Exact theory in the simultaneous equations model 463
The model and notation
Generic statistical forms of common single equation estimators
The standardizing transformations
The analysis of leading cases
The exact distribution of the IV estimator in the general single equation case
The case of two endogenous variables
Structural variance estimators
Test statistics
Systems estimators and reduced-form coefficients
Improved estimation of structural coefficients
Supplementary results on moments
Misspecification
4. A new approach to small sample theory
4.1 Intuitive ideas
4.2. Rational approximation
4.3. Curve fitting or constructive functional approximation?
5. Concluding remarks
References

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