Author:
(1) David Staines. Table of Links Abstract 1 Introduction 2 Mathematical Arguments 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 5 Existing Solutions 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.3 Literature Comparison 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.3 Discussion 9 Bifurcation Analysis 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 11 Policy Rule 12 Conclusions and References Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.3 De Rham Cohomology F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.3 Trend Inflation Volatility 10.2 Econometric Duality This crucial subsection uncovers the deep connection between the topology and goodness of fit of approximate solutions around ZINSS. I finish by discussing bias implications to stimulate future computational and econometric work. The key idea is that missing dynamics in the existing approximation can directly translate to poor econometric performance. 10.2.1 Theoretical Formulation This part sets out the mathematical foundation for the econometric analysis. Remark 34. Different bifurcation experiments correspond to different sheaves. It is convenient to work with the following parametization X = (δ, n, m) where δ is the starting rate of trend inflation, n is the order of the initial difference between stochastic and non-stochastic steady states and m is the limiting order of perturbation. Thus the |ε| limit would (0, 2, 1), √ ε would be (0, 2, 1/2). The first order non-stochastic bifurcation would be (δ, 0, 1), with (δ, 0, n) the corresponding bifurcation at a general order of approximation n. There are two informative visual representations. The first is the commuting diagram below where ∇ is the duality map and dependence on X has been suppressed. ∇ is anti-symmetric and it can be represented by the following correspondence. It represents a morphism between the sheaf of local goodness of fit metrics around ZINSS and the sheaf of approximate optimization problems, through their constraints.[93] The correspondence can be represented graphically as shown above. The hole at the origin is a consequence of Divine Coincidence. There is an unacceptable trade-off between simplifying the model by cancelling lag terms and fitting the data. 10.2.2 Discussion Duality justifies the terminology of the boundary surface. It represents constraints that are always binding on the representative firm / social planner or the econometrician. This principle demonstrates a direct connection between the microfoundations missing from the faulty approximations at ZINSS and its poor empirical performance. If we believe in our Keynesian microfoundations we should expect a substantial improvement in goodness of fit when we change to the correct approximation. This is a reassuring prospect. The general conclusion is surely that modern Keynesian economics is first and foremost about justifying the presence of lags in the Phillips Curve, operating through the constraints implied by staggered optimization. Previous approximations failed to represent the underlying microfoundations; as such they failed the Lucas critique. Econometric duality is a theoretical result that holds under the null hypothesis the model is true. If the real data generating process is sufficiently close to the theory the associated improvement in goodness of fit will still arise. This is an econometric hypothesis. However, on the basis of the exercise in Section 3.1 and Appendix H, I will proceed as though it is true that (2) fits the data better than (1). This can seriously bias inference, drawn from goodness of fit comparisons. Suppose an econometrician were trying to estimate the improvement in the goodness of fit associated with a change like modelling (non-zero) trend inflation Zπ but were unaware of the bifurcation. They would use the estimator for the appropriate (convergent) goodness of fit metric k·| k but the true improvement would be This yields an upward bias in the inferred improvement in the goodness of fit. It would distort hypothesis testing towards accepting ill-fitting alternatives to the benchmark Calvo model. This would extend to any alternative to the Calvo that did not suffer from the same singularity. This would include alternative pricing models, such as those with Taylor [1979] or menu costs. In fact any bifurcation will cause overestimation of non-linearity nearby a faulty linearized benchmark. The theory is as follows: an econometrician wishing to estimate the non-linearity compared to the singular solution would estimate the true estimator would be thus an asymptotic bias would emerge It is strictly positive if the true non-linearity is small. It remains to test whether this approximate bound holds in simulations. Nevertheless, the a priori case was typically avoided, now at the expense of irrelevance, by using Rotemberg pricing.[94] In theory for ex post bifurcation the bias would be infinite in any neighborhood of the cut-off as calculated for Calvo in the next section. An underlying theme is that complicated singularities are not amenable to inferential statistical learning. The confusion should now be over. This paper is available on arxiv under CC 4.0 license. [93] Or even a functor between the two categories of sheaves around ZINSS, as in Appendix F.2. Author: (1) David Staines. Author: Author: (1) David Staines. Table of Links Abstract Abstract 1 Introduction 1 Introduction 2 Mathematical Arguments 2 Mathematical Arguments 3 Outline and Preview 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.2 Preferences 4.3 Household Equilibrium Conditions 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 4.8 Recursive Equilibrium 5 Existing Solutions 5 Existing Solutions 5.1 Singular Phillips Curve 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.3 Two Comparison Models 5.4 Lucas Critique 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.2 Equilibrium Construction 6.3 Literature Comparison 6.3 Literature Comparison 6.4 Equilibrium Analysis 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.1 Slope Coefficients 7.2 Error Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.2 Key Proofs 8.3 Discussion 8.3 Discussion 9 Bifurcation Analysis 9 Bifurcation Analysis 9.1 Analytic Aspects 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.2 Econometric Duality 10.3 Coefficient Properties 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 10.4 Microeconomic Interpretation 11 Policy Rule 11 Policy Rule 12 Conclusions and References 12 Conclusions and References Appendices Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.2 Behaviour of ∆ A.3 Proof Part (iii) A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.2 Basic Categories F.3 De Rham Cohomology F.3 De Rham Cohomology F.4 Marginal Costs and Inflation F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.2 Lucas Critique I.3 Trend Inflation Volatility I.3 Trend Inflation Volatility 10.2 Econometric Duality This crucial subsection uncovers the deep connection between the topology and goodness of fit of approximate solutions around ZINSS. I finish by discussing bias implications to stimulate future computational and econometric work. The key idea is that missing dynamics in the existing approximation can directly translate to poor econometric performance. 10.2.1 Theoretical Formulation This part sets out the mathematical foundation for the econometric analysis. Remark 34. Different bifurcation experiments correspond to different sheaves. It is convenient to work with the following parametization X = (δ, n, m) where δ is the starting rate of trend inflation, n is the order of the initial difference between stochastic and non-stochastic steady states and m is the limiting order of perturbation. Thus the |ε| limit would (0, 2, 1), √ ε would be (0, 2, 1/2). The first order non-stochastic bifurcation would be (δ, 0, 1), with (δ, 0, n) the corresponding bifurcation at a general order of approximation n. Remark 34. Different bifurcation experiments correspond to different sheaves. It is convenient to work with the following parametization X = (δ, n, m) where δ is the starting rate of trend inflation, n is the order of the initial difference between stochastic and non-stochastic steady states and m is the limiting order of perturbation. Thus the |ε| limit would (0, 2, 1), √ ε would be (0, 2, 1/2). The first order non-stochastic bifurcation would be (δ, 0, 1), with (δ, 0, n) the corresponding bifurcation at a general order of approximation n. There are two informative visual representations. The first is the commuting diagram below where ∇ is the duality map and dependence on X has been suppressed. ∇ is anti-symmetric and it can be represented by the following correspondence. It represents a morphism between the sheaf of local goodness of fit metrics around ZINSS and the sheaf of approximate optimization problems, through their constraints.[93] The correspondence can be represented graphically as shown above. The hole at the origin is a consequence of Divine Coincidence. There is an unacceptable trade-off between simplifying the model by cancelling lag terms and fitting the data. 10.2.2 Discussion Duality justifies the terminology of the boundary surface. It represents constraints that are always binding on the representative firm / social planner or the econometrician. This principle demonstrates a direct connection between the microfoundations missing from the faulty approximations at ZINSS and its poor empirical performance. If we believe in our Keynesian microfoundations we should expect a substantial improvement in goodness of fit when we change to the correct approximation. This is a reassuring prospect. The general conclusion is surely that modern Keynesian economics is first and foremost about justifying the presence of lags in the Phillips Curve, operating through the constraints implied by staggered optimization. Previous approximations failed to represent the underlying microfoundations; as such they failed the Lucas critique. Econometric duality is a theoretical result that holds under the null hypothesis the model is true. If the real data generating process is sufficiently close to the theory the associated improvement in goodness of fit will still arise. This is an econometric hypothesis. However, on the basis of the exercise in Section 3.1 and Appendix H, I will proceed as though it is true that (2) fits the data better than (1). This can seriously bias inference, drawn from goodness of fit comparisons. Suppose an econometrician were trying to estimate the improvement in the goodness of fit associated with a change like modelling (non-zero) trend inflation Zπ but were unaware of the bifurcation. They would use the estimator for the appropriate (convergent) goodness of fit metric k·| k but the true improvement would be This yields an upward bias in the inferred improvement in the goodness of fit. It would distort hypothesis testing towards accepting ill-fitting alternatives to the benchmark Calvo model. This would extend to any alternative to the Calvo that did not suffer from the same singularity. This would include alternative pricing models, such as those with Taylor [1979] or menu costs. In fact any bifurcation will cause overestimation of non-linearity nearby a faulty linearized benchmark. The theory is as follows: an econometrician wishing to estimate the non-linearity compared to the singular solution would estimate the true estimator would be thus an asymptotic bias would emerge It is strictly positive if the true non-linearity is small. It remains to test whether this approximate bound holds in simulations. Nevertheless, the a priori case was typically avoided, now at the expense of irrelevance, by using Rotemberg pricing.[94] In theory for ex post bifurcation the bias would be infinite in any neighborhood of the cut-off as calculated for Calvo in the next section. An underlying theme is that complicated singularities are not amenable to inferential statistical learning. The confusion should now be over. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv [93] Or even a functor between the two categories of sheaves around ZINSS, as in Appendix F.2.

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