Toward a formal science of economics : the axiomatic method in economics and econometrics

Toward a Formal Science of Economics provides a unifying way to look at the concept of economic science. Toward a Formal Science of Economics provides a unifying way to look at the concept of economic science. It lays a foundation for the axiomatic method, focusing on applications in economics and econometrics, and including discussions in logic, epistemology, and probability theory. Each chapter deals with a topic of fundamental importance to a rigorous science of economics while illustrating an aspect of the axiomatic method. Stigum describes an introductory course in mathematical logic, developing a symbolic language for mathematics and discussing the strengths and weaknesses of the axiomatic method. He presents the standard theory of consumer choice, illustrating different aspects of the use of the axiomatic method and evaluating economic theories of individual behavior. He takes up problems in the foundations of econometrics and choice under uncertainty and offers an introduction to nonstandard analysis that leads to discussion of exchange and probability in hyperspace. A section on epistemology completes Stigum's construction of a formal unitary methodological basis for theoretical and empirical science. The last three parts of the book apply these methodological tools to various topics in economics and econometrics including empirical analyses of the permanent income hypothesis and consumer choice among risky and nonrisky assets; discussion of determinism, uncertainty, and the utility hypothesis; and study of topics of importance to the analysis of economic time series. 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