Toward a Formal Science of Economics: The Axiomatic Method in Economics and Econometrics

Toward a Formal Science of Economics
CONTENTS
1— Introduction
1.1— The Need for a Formal Unitary Methodological Basis for the Science of Economics
1.2— The Axiomatic Method and the Development of a Formal Science of Economics
1.2.1— The Rise of Formal Economics
1.2.2— The Rise of Formal Logic
1.2.3— The Development of a Formal Science of Economics
1.3— Formalism and the Unity of Science
1.3.1— The Unity of Science
1.3.2— Advantages of Formalism in Science
1.3.3— Formalism, Formalization, and the Scientific Method
1.4— Noteworthy Results
1.4.1— Parts I and II: Mathematical Logic
1.4.2— Part III: Consumer Choice
1.4.3— Part IV: Chance, Ignorance, and Choice
1.4.4— Part V: Nonstandard Analysis
1.4.5— Part VI: Epistemology
1.4.6— Part VII: Empirical Analysis of Economic Theories
1.4.7— Part VIII: Determinism, Uncertainty, and the Utility Hypothesis
1.4.8— Part IX: Prediction, Distributed Lags, and Stochastic Difference Equations
1.5— Acknowledgments
2— The Axiomatic Method
2.1— Axioms and Undefined Terms
2.2— Rules of Inference and Definition
2.3— Universal Terms and Theorems
2.4— Theorizing and the Axiomatic Method
2.5— Pitfalls in the Axiomatic Method
2.6— Theories and Models
2.7— An Example
I— MATHEMATICAL LOGIC I: FIRST-ORDER LANGUAGES
3— Meaning and Truth
3.1— A Technical Vocabulary
3.1.1— Names
3.1.2— Declarative Sentences
3.1.3— Constants and Variables
3.1.4— Functions and Predicates
3.2— Logical Syntax
3.3— Semantics
3.4— The Semantic Conception of Truth
3.5— Truth and Meaning
4— Propositional Calculus
4.1— Symbols, Well-Formed Formulas, and Rules of Inference
4.1.1— Symbols
4.1.2— Well-Formed Formulas
4.1.3— Rules of Inference
4.2— Sample Theorems
4.3— The Intended Interpretation
4.3.1— Tautologies
4.3.2— Theorems and Tautologies
4.4— Interesting Tautological Structures
4.5— Disjunction, Conjunction, and Material Equivalence
4.5.1— Either-Or and Both-And Sentences
4.5.2— Material Equivalence
4.6— Syntactical Properties of the Propositional Calculus
4.7— Proof of the Tautology Theorem
5— The First-Order Predicate Calculus
5.1— Symbols, Well-Formed Formulas, and Rules of Inference
5.1.1— The Symbols
5.1.2— The Well-Formed Formulas
5.1.3— The Axioms
5.2— Sample Theorems
5.2.1— Equality
5.2.2— The Quantifiers
5.2.3— Material Equivalence
5.3— Semantic Properties
5.3.1— Structures
5.3.2— Structures and the Interpretation of a First-Order Language
5.3.3— Tautologies and Valid Well-Formed Formulas
5.3.4— Valid Well-Formed Formulas and Theorems
5.4— Philosophical Misgivings
5.4.1— For-All Sentences
5.4.2— There-Exist Sentences
5.5— Concluding Remarks
II— MATHEMATICAL LOGIC II: THEORIES AND MODELS
6— Consistent Theories and Models
6.1— First-Order Theories
6.2— Proofs and Proofs from Hypotheses
6.3— The Deduction Theorem
6.4— Consistent Theories and Their Models
6.5— The Compactness Theorem
6.6— Appendix: Proofs
6.6.1— A Proof of TM 6.10
6.6.2— A Proof of TM 6.11
6.6.3— A Proof of TM 5.19
6.6.4— A Proof of TM 6.13
7— Complete Theories and Their Models
7.1— Extension of Theories by Definitions
7.1.1— Predicates
7.1.2— Functions
7.1.3— Valid Definitional Schemes
7.2— Isomorphic Structures
7.3— Elementarily Equivalent Structures
7.4— Concluding Remarks
7.5— Appendix
8— The Axiomatic Method and Natural Numbers
8.1— Recursive Functions and Predicates
8.1.1— Recursive Functions
8.1.2— G[dieresis(o)]del's [beta] Function
8.1.3— Recursive Predicates
8.1.4— Sequence Numbers
8.2— Expression Numbers
8.3— Representable Functions and Predicates
8.4 Incompleteness of Consistent, Axiomatized Extensions of T(N)
8.5 The Consistency of T(P)
8.6— Concluding Remarks
9— Elementary Set Theory
9.1— The Axioms of KPU
9.2— The Null Set and Russell's Antinomy
9.3— Unions, Intersections, and Differences
9.3.1— Unions
9.3.2— Intersections and Differences
9.4— Product Sets
9.5— Relations and Functions
9.6— Extensions
9.7— Natural Numbers
9.8— Admissible Structures and Models of KPU
9.9— Concluding Remarks
III— ECONOMIC THEORY I: CONSUMER CHOICE
10— Consumer Choice under Certainty
10.1— Universal Terms and Theorems
10.2— A Theory of Choice, T(H 1,
10.2.1— Axioms
10.2.2 The Intended Interpretation of T(H. . . , H 6)
10.2.3— Sample Theorems
10.3— The Fundamental Theorem of Consumer Choice
10.4— The Hicks-Leontief Aggregation Theorem
11— Time Preference and Consumption Strategies
11.1— An Alternative Interpretation of T(H 1,
11.2— The Time Structure of Consumer Preferences
11.2.1— Independent Preference Structures
11.2.2— Stationary Preference Structures
11.3— The Rate of Time Preference and Consumption Strategies
11.3.1— The Induced Ordering of Consumption Strategies
11.3.2— Irving Fisher's Rate of Time Preference
11.3.3— Stationary Price Expectations and Monotonic Optimal Consumption Strategies
11.3.4— Optimal Consumption Strategies and Age
11.4— Consumption Strategies and Price Indices
12— Risk Aversion and Choice of Safe and Risky Assets
12.1— An Axiomatization of Arrow's Theory
12.1.1— The Axioms
12.1.2— The Intended Interpretation
12.1.3— Sample Theorems
12.2— Absolute and Proportional Risk Aversion
12.2.1— The Absolute Risk-Aversion Function
12.2.2— Absolute Risk Aversion and Ordering of ([mu], m) Pairs
12.2.3— Absolute Risk Aversion and Investment in Risky Assets
12.2.4— The Proportional Risk-Aversion Function
12.3— The Fundamental Theorems of Arrow
12.3.1— Risky Assets and Absolute Risk Aversion
12.3.2— Safe Assets and Proportional Risk Aversion
12.4— New Axioms
12.5— An Aggregation Problem
12.6— Resolution of the Aggregation Problem
12.6.1— Preliminary Remarks
12.6.2— The Separation Property
12.6.3— Arrow's Theorems and the Separation Property
12.7— Appendix: Proofs
12.7.1— Proof of T 12.1
12.7.2— Proof of T 12.2
12.7.3— Proof of T 12.4
12.7.4— Proof of T 12.5
12.7.5— Proof of T 12.6
12.7.6— Proof of T 12.7
12.7.7— Proof of T 12.8 and T 12.9
12.7.8— Proof of T 12.10
12.7.9— Proof of 12.11
12.7.10 Proof of T 12.13
12.7.11— Proof of T 12.14
12.7.12— Proof of T 12.15
12.7.13— Proof of T 12.16
12.7.14— Proof of T 12.17
13— Consumer Choice and Revealed Preference
13.1— An Alternative Set of Axioms, S 1 ,
13.2— The Fundamental Theorem of Revealed Preference
13.2.1— A Rough Contour of S[sup (+)](x[sup(0)])
13.2.2— Salient Characteristics of the Lower Boundary Points of S[super(+)](x[super(0)])
13.2.3— Characteristics of Vectors in S[sup(+)](x[sup(0)]) [Union] (R[sup(n)sub(+)] – [overline (S[s...
13.2.4— The Fundamental Theorem
13.3— The Equivalence of T(S 1 ,
13.3.1— A Counterexample
13.3.2— Homothetic Utility Functions and the Fundamental Theorem
13.3.3— Additively Separable Utility Functions and the Fundamental Theorem
13.4— Concluding Remarks
14— Consumer Choice and Resource Allocation
14.1— Competitive Equilibria in Exchange Economies
14.1.1— A Scenario for Commodity Exchange
14.1.2— Competitive Equilibria in E
14.2— Resource Allocation in Exchange Economies
14.2.1 Pareto-Optimal Allocations and Fair Allocations
14.2.2— Pareto-Optimal Allocations and Competitive Equilibria
14.3— The Formation of Prices in an Exchange Economy
14.3.1— On the Stability of Competitive Equilibria
14.3.2— Concluding Remarks
14.4— Temporary Equilibria in an Exchange Economy
14.4.1— Consumption-Investment Strategies
14.4.2— The Current-Period Utility Function
14.4.3— Current-Period Temporary Equilibria
14.4.4— Feasible Sequences of Temporary Equilibria
14.5— Admissible Allocations and Temporary Equilibria
14.6— On the Stability of Temporary Equilibria
IV— PROBABILITY THEORY: CHANCE, IGNORANCE, AND CHOICE
15— The Measurement of Probable Things
15.1— Experiments and Random Variables
15.1.1— Events
15.1.2— Random Variables
15.2— Belief Functions
15.2.1— Basic Probability Assignments and Belief Functions
15.2.2— Orthogonal Sums of Belief Functions
15.2.3— Support Functions
15.2.4— Additive Versus Nonadditive Belief Functions
15.2.5— Additive Belief Functions
15.3— Probability Measures
15.3.1— Finitely Additive Probability Measures
15.3.2— The Bayes Theorem
15.3.3— Posterior Probabilities and Conditional Belief Functions
15.3.4— [sigma]-Additive Probability Measures
15.4— Probability Distributions
15.4.1— The Probability Distribution of a Random Variable
15.4.2— The Joint Probability Distribution of n Random Variables
15.4.3— Integrable Random Variables
15.4.4— Probability Distributions in Econometrics
15.4.5— Convergence in Distributions
15.5— Random Processes and Kolmogorov's Consistency Theorem
15.5.1— Random Processes
15.5.2— Kolmogorov's Consistency Theorem
15.5.3— The Measurement of Random Processes
15.6— Two Useful Universal Theorems
16— Chance
16.1— Purely Random Processes
16.1.1— Independent Events and Variables
16.1.2— A Purely Random Process
16.2— Games of Chance
16.2.1— The Absence of Successful Gambling Systems
16.2.2— The Arc Sine Law
16.2.3— The Classical Ruin Problem
16.3— The Law of Large Numbers
16.3.1— Tail Events and Functions
16.3.2— Kolmogorov's Strong Law of Large Numbers
16.3.3— The Central Limit Theorem
16.4— An Empirical Characterization of Chance
16.4.1— The Collectives of Von Mises
16.4.2— Church's Concept of Chance
16.5— Chance and the Characteristics of Purely Random Processes
17— Ignorance
17.1— Epistemic Versus Aleatory Probabilities
17.1.1— Risk and Epistemic Probability
17.1.2— Uncertainty and the Principle of Insufficient Reason
17.1.3— Modeling Ignorance [grave(a)] La Laplace and Edgeworth
17.1.4— Measuring Uncertainty with Entropy
17.2— The Bayes Theorem and Epistemic Probabilities
17.2.1— Learning by Observing
17.2.2— An Example
17.2.3— A Paradox
17.3— Noninformative Priors
17.3.1— Locally Uniform Priors
17.3.2— Exact Data-Translated Likelihoods
17.3.3— Approximate Data-Translated Likelihoods
17.4— Measuring the Performance of Probability Assessors
18— Exchangeable Random Processes
18.1— Conditional Expectations and Probabilities
18.2— Exchangeable Random Variables
18.2.1— Finite Sequences of Binary Exchangeable Random Variables
18.2.2 Sequences of Infinitely Many Binary Exchangeable Variables
18.2.3— Integrable Exchangeable Random Processes
18.3— Exchangeable Processes and Econometric Practice
18.3.1— Consistent Parameter Estimates
18.3.2— Finite-Sample Interval Estimates
18.3.3— Concluding Remarks
18.4— Conditional Probability Spaces
18.4.1— Conditional Probability Spaces
18.4.2— Renyi's Fundamental Theorem
18.5— Exchangeable Processes On a Full Conditional Probability Space
18.6— Probability Versus Conditional Probability
19— Choice under Uncertainty
19.1— The Decision Maker and His Experiment
19.1.1— The Decision Maker
19.1.2— The State of the World
19.1.3— Acts and Consequences
19.2— The Decision Maker's Risk Preferences
19.2.1— Risk Preferences
19.2.2— The Sure-Thing Principle
19.2.3— Constant Acts
19.3— Risk Preferences and Subjective Probability
19.3.1— Bets and Prizes
19.3.2— Qualitative Probability
19.3.3— Subjective Probability
19.4— Expected Utility
19.4.1— Savage's Fundamental Theorem
19.4.2— Measurable Utility
19.4.3— Expected Utility with a Finite Number of States of the World
19.5— Assessing Probabilities and Measuring Utilities
19.5.1— Assessing Subjective Probabilities
19.5.2— Measuring Utility Functions
19.5.3— A Test of Savage's Theory
19.6— Belief Functions and Choice under Uncertainty
19.6.1— Belief Functions and the Axioms of Savage
19.6.2— Belief Functions, Qualitative Probability, and Expected Utility
19.6.3— Belief Functions and Uncertainty Aversion
19.6.4— Examples
19.6.5— Concluding Remarks
V— NONSTANDARD ANALYSIS
20— Nonstandard Analysis
20.1— The Set of Urelements U
20.1.1— The Axioms for U
20.1.2— Structural Characteristics of U
20.2— A Model of the Axioms for U
20.2.1— Free Ultrafilters over N
20.2.2— An Ordered Field of Hyperreal Numbers *R
20.3— Elementarily Equivalent Structures and Transfer
20.3.1— Two Elementarily Equivalent Structures
20.3.2— Transfer
20.3.3— Transfer Versus Elementary Extension of Structures
20.4— Superstructures and Superstructure Embeddings
20.4.1— Superstructures W(·) over Sets of Urelements
20.4.2— The Superstructure over R
20.4.3— Superstructure Embeddings
20.5— Transfer and Superstructure Embeddings
20.5.1— Leibniz's Principle
20.5.2— [stroke(L)]os's Theorem and the Validity of Leibniz's Principle
20.6— Internal Subsets of W(*R)
20.6.1— A Classification of the Elements of W(*R)
20.6.2— Elementary Properties of Internal Sets
20.6.3— Hyperfinite Sets in W(*R)
20.7— Admissible Structures and the Nonstandard Universe
20.7.1 Admissible Structures
20.7.2— Cardinal Numbers
20.7.3— An Admissible Model of T
20.7.4— Admissible Models of T and Superstructures
21— Exchange in Hyperspace
21.1— The Saturation Principle
21.1.1— The Saturation Principle
21.1.2— Useful Consequences
21.2— Two Nonstandard Topologies
21.2.1— The * Topology
21.2.2— The S Topology
21.3— Exchange in Hyperspace by Transfer
21.3.1— A Hyperfinite Exchange Economy
21.3.2— A Nonstandard Version of a Theorem of Debreu and Scarf
21.4— Exchange in Hyperspace Without Transfer
21.4.1— On Exchange in the S Topology
21.4.2— An Auxiliary Lemma
21.4.3— The Fundamental Equivalence
21.5— Concluding Remarks
22— Probability and Exchange in Hyperspace
22.1— Loeb Probability Spaces
22.2— Standard Versions of Loeb Probability Spaces
22.2.1— Examples
22.2.2— A Hyperfinite Alias of Lebesgue's Probability Space
22.3— Random Variables and Integration in Hyperspace
22.3.1— Random Variables in Hyperspace
22.3.2— Integration in Hyperspace
22.4— Exchange in Hyperspace Revisited
22.5— A Hyperfinite Construction of the Brownian Motion
22.5.1— Independent Random Variables in Hyperspace
22.5.2— Brownian Motion
22.5.3— The Wiener Measure
VI— EPISTEMOLOGY
23— Truth, Knowledge, and Necessity
23.1— The Semantical Concept of Truth Revisited
23.2— Truth and Knowledge
23.3— The Possibility of Knowledge
23.3.1— The Universe Is Not Empty, PE 1
23.3.2— Induction
23.3.3— The Uniformity of Nature, PE 2
23.3.4— Identity and the Closest-Continuer Schema
23.3.5— Analogy
23.3.6— The Principle of Limited Variety, PLV
23.4 Different Kinds of Knowledge
23.4.1 Knowledge of Logical Propositions
23.4.2— Knowledge of Extralogical Propositions
23.4.3— Knowledge of Variable Hypotheticals
23.4.2.1— Knowledge by Definition, Analysis, Intuition, and Enumeration
23.4.3.2— Accidental, Nomological, and Derivative Laws
23.5— Necessity and Modal Logic
23.5.1— A Modal-Logical System, ML
23.5.2— Sample Theorems in ML
23.5.3— The Intended Interpretation of ML
23.5.4— Salient Properties of the Intented Interpretation of ML
23.5.5— Universals, Nomological Laws, and Modal Logic
23.5.6— Concluding Remarks
24— The Private Epistemological Universe, Belief, and Knowledge
24.1— The Private Epistemological Universe
24.1— A Reformulation of PLV, PE 3
24.1.2— Epistemological Universes
24.1.3— A Private Universe for the Theory of Knowledge and PE 4
24.2— Logical Probabilities and Their Possible-World Interpretation
24.2.1— Additive Logical Probabilities
24.2.2— Superadditive Logical Probabilities
24.2.3— Concluding Remarks
24.3— An Axiomatization of Knowledge
24.3.1— The Symbols
24.3.2— The Logical Axioms
24.3.3— The Nonlogical Axioms
24.3.4— The Rules of Inference
24.5— The Intended Interpretation of EL
24.3.6— Salient Properties of the Interpretation of EL
24.3.7— Theorems of EL
24.3.7.1— Useful Properties of P(·|·)
24.3.7.2— Good Inductive Rules of Inference and the Properties of P(·|·)
24.3.7.3— The Existence of P(·|·)
24.3.7— Theorems Concerning Kn(·) and Bl(·x)
24.3.7.5— Substitution in Referentially Opaque Contexts
24.3.7.6— The Epistemological Concept of Truth
24.4— Other Concepts of Knowledge
24.4.1— Peirce's Concept of Knowledge
24.4.2— Hintikka's Concept of Knowledge
24.4.3— Chisholm's Concept of Knowledge
24.4.4— Sundry Comments and a Look Ahead
25— An Epistemological Language for Science
25.1— Simple, Autonomous Relations
25.2— Analogy and the Generation of Scientific Hypotheses
25.2.1— Analogy and Inductive Inference
25.2.2— Models
25.2.3— Representative Individuals and Aggregates
25.2.4— Observations, Theoretical Hypotheses, and Analogy
25.3— Induction and Meaningful Sampling Schemes
25.4— Many-Sorted Languages
25.4.1— The Symbols
25.4.2— The Terms and the Well-Formed Formulas
25.4.3— The Axioms and the Rules of Inference
25.4.4— Sample Theorems
25.4.5— Structures and the Interpretation of Many-Sorted Languages
25.5— Semantic Properties of Many-Sorted Languages
25.6— A Language for Science
25.7— A Modal-Logical Apparatus for Testing Scientific Hypotheses
25.8— Appendix: Proof of the Completeness Theorem for Many-Sorted Languages
25.8.1— Predicate-Calculus Aliases of Many-Sorted Languages
25.8.2— The Completeness Theorem
VII— ECONOMETRICS I: EMPIRICAL ANALYSIS OF ECONOMIC THEORIES
26— Empirical Analysis of Economic Theories
26.1— Four Kinds of Theorems
26.2— The Structure of an Empirical Analysis
26.2.1— The Undefined Terms: S, ([Omega], [script (F)]), and P(·)
26.2.2— The Axioms Concerning [Omega]
26.2.3— The Axioms Concerning P(·) and ([Omega], [script(F)])
26.2.4— Sample Theorems
26.2.5— Testing an Economic Theory
26.3— New Axioms and New Tests
26.3.1— New Axioms Versus New Tests
26.3.2— An Example
26.4— Superstructures, Data-Generating Mechanisms, the Encompassing Principle, and Meaningful Sampli...
27— The Permanent-Income Hypothesis
27.1— Formulation of the Hypothesis
27.2— The Axioms of a Test of the Certainty Model: F 1,
27.3— Theorems of T(F 1,
27.4— Confronting T(F 1,
27.4.1— Budget Data Versus Time-Series Data
27.4.2— A Factor-Analytic Test
27.4.3— The Rate of Time Preference and the Human-Nonhuman Wealth Ratio
27.4.4— Concluding Remarks
27.5— A Test of the Uncertainty Version of Friedman's Theory
27.5.1— New Axioms
27.5.2— New Theorems
27.5.3— The Test
27.5.4— Concluding Remarks
27.6— Appendix: Standard Errors of Factor-Analytic Estimates
27.6.1— The Asymptotic Distribution of the Sample Covariance Matrix
27.6.2— The Asymptotic Distribution of Factor-Analytic Estimates
27.6.3— Bootstrap Estimates of Factor-Analytic Parameters
28— An Empirical Analysis of Consumer Choice among Risky and Nonrisky Assets
28.1— The Axioms of the Empirical Analysis
28.1.1— Axioms Concerning the Components of [omega][sub(T)]
28.1.2— Axioms Concerning the Components of [omega][sub(p)]
28.1.3— Axioms Concerning the Images of F
28.1.4— An Example
28.1.5— Axioms Concerning P(·) and [script(F)]
28.2— Arrow's Risk-Aversion Functions and the Data
28.2.1— The Data and the Axioms
28.2.2— Sample Theorems
28.2.3— An Indirect Test of SA 7 and SA 11-SA 17
28.2.4— A Test of Arrow's Hypotheses
28.3— Comparative Risk Aversion
28.3.1— One-Way Analysis of Variance: Theory
28.3.2— One-Way Analysis of Variance of the Data
28.3.3— Two-Way Analysis of Variance: Theory
28.3.4— Multiple-Classification Analysis of the Data
28.3.5— Education and Income
28.4— Concluding Remarks
VIII— ECONOMIC THEORY II: DETERMINISM, UNCERTAINTY, AND THE UTILITY HYPOTHESIS
29— Time-Series Tests of the Utility Hypothesis
29.1— A Nonparametric Test of the Utility Hypothesis
29.2— Testing for Homotheticity of the Utility Function
29.3— Testing for Homothetic Separability of the Utility Function
29.4— Excess Demand Functions and the Utility Hypothesis
29.4.1— Testing the Utility Hypothesis with Group Data That Satisfy GARP
29.4— Testing for the Homotheticity of Individual Utility Functions with Group Data That Satisfy GAR...
29.4.3— A Characterization of Excess Demand Functions
29.4.5— Constructing a "Test" of the Utility Hypothesis When the Group Data Do Not Satisfy GARP.
29.4.6— Summing up
29.5— Nonparametric Versus Parametric Tests of the Utility Hypothesis and a Counterexample
30— Temporary Equilibria under Uncertainty
30.1— The Arrow-Debreu Consumer[sup(2)]
30.1.1— Nature
30.1.2— The Consumer
30.1.3— Markets and Expenditure Plans
30.1.4— Concluding Remarks
30.2— The Radner Consumer
30.2.1— Notational Matters
30.2.2— The Consumer
30.2.3— Markets and Expenditure Plans
30.2.4— Concluding Remarks
30.3— Consumer Choice under Uncertainty
30.3.1— Definitional Axioms
30.3.2— The Intended Interpretation
30.3.3— Axioms Concerning the Properties of V(·) and Q(·)
30.3.4— The Fundamental Theorem of Consumer Choice under Uncertainty
30.3.5— Concluding Remarks
30.4— The Arrow-Debreu Producer
30.5— Entrepreneurial Choice under Uncertainty
30.5.1— Definitional Axioms
30.5.2— The Intended Interpretation
30.5.3— Axioms Concerning the Properties of g(·), V(·), and Q(·)
30.5.4— The Fundamental Theorem of Entrepreneurial Choice under Uncertainty
30.6— Temporary Equilibria under Uncertainty
30.6.1— Notational Matters
30.6.2— Axioms for a Production Economy
30.6.3— The Existence of Temporary Equilibria
30.6.4— Concluding Remarks
30.7— Appendix: Proofs of Theorems
30.7.1— Proof of T30.1 and T 30.2
30.7.2— Proof of T 30.5
30.7.3— Proof of T 30.7
31— Balanced Growth under Uncertainty[sup(1)]
31.1— Balanced Growth under Certainty
31.1.1— The Indecomposable Case
31.1.2— The Decomposable Case
31.2— Balanced Growth under Uncertainty in an Indecomposable Economy
31.2.1— Balanced Growth When n = 1
31.2.2— Balanced Growth When n [greater/equal to] 2
31.3— Balanced Growth under Uncertainty in a Decomposable Economy
IX— ECONOMETRICS II: PREDICTION, DISTRIBUTED LAGS, AND STOCHASTIC DIFFERENCE EQUATIONS
32— Distributed Lags and Wide-Sense Stationary Processes
32.1— A Characterization of Wide-Sense Stationary Processes
32.1.1— Examples
32.1.2— Orthogonal Set Functions and Stochastic Integrals[sup(1)]
32.1.3— The Spectral Distribution Function
32.1.4— The Spectral Representation of a Wide-Sense Stationary Process
32.2— Linear Least-Squares Prediction
32.2.1— The Best Linear Least-Squares Predictor
32.2.2— Examples
32.2.3— Wold's Decomposition Theorem
32.2.4 Kolmogorov's Theorem
32.3— Distributed Lags and Optimal Stochastic Control
32.3.1— Distributed Lags
32.3.2— Examples
32.3.3— A Stochastic Control Problem
32.3.4— Rational Distributed Lags and Control
33— Trends, Cycles, and Seasonals in Economic Time Series and Stochastic Difference Equations
33.1— Modeling Trends, Cycles, and Seasonals in Economic Time Series
33.1.1— Trends
33.1.2— Cycles and Seasonals
33.1.3— Concluding Remarks
33.2— ARIMA Processes
33.2.1— The Short and Long Run Behavior of ARIMA Processes
33.2.2— An Invariance Principle and the Associated Wiener Measures
33.2.3— The Invariance Principle and the Long Run of ARIMA Processes
33.3— Dynamic Stochastic Processes
33.3.1— A Definition and Illustrative Examples
33.3.2— Fundamental Theorems
33.4— Concluding Remarks On Multivariate Dynamic Stochastic Processes
34— Least Squares and Stochastic Difference Equations
34.1— The Elimination of Trend, Cycle, and Seasonal Factors in Time Series
34.1— Basic Assumptions
34.1.2— Linear SCT-Adjustment Procedures
34.1.3— Notational Matters
34.1.4— Least-Squares Estimates of the Deterministic Components of a Time Series
34.1.5— Removal of Seasonal, Cyclical, and Trend Factors in Time Series
34.2— Estimating the Coefficients in a Stochastic Difference Equation: Consistency
34.2.1— Equations with Fixed Initial Conditions
34.2.2— Equations with Random Initial Conditions: Special Cases
34.2.3— Equations with Random Initial Conditions: The Fundamental Theorem
34.2.4— Concluding Remarks
34.3— Estimating the Coefficients in a Stochastic Difference Equation: Limiting Distributions
34.3.1— Equations with Fixed Initial Conditions
34.3.2— Equations with Random Initial Conditions
34.3.3— A Simulation Experiment
34.4— Concluding Remarks
NOTES
Chapter 1
Chapter 2
Chapter 3
Chapter 5
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31
Chapter 32
Chapter 33
Chapter 34
BIBLIOGRAPHY
INDEX
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
W
Z