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 I.3 Trend Inflation Volatility This part presents evidence on the volatility of trend inflation. The focus here is on time series ideas. It matters how you de-trend the data. There is significant low frequency volatility in inflation, over typical sample periods. Contrast the Great Depression with the Great Inflation and think of Japan’s lost decades. Under inflation targeting it is natural to think of the headline target as the trend inflation rate. This makes trend inflation much less volatile. Nevertheless, there has been discussion about changing the inflation target which should be factored in by rational agents, (see for example Blanchard et al. [2010], Ball [2013] and Cecchetti and Schoenholtz [2017]). The European Central Bank’s recent move to a symmetric inflation target might work out like this (Ignazio and Gros [2021]). The asymptotic justification is a little delicate though. As shocks shrink, are trend inflation shocks sufficiently large to be considered first order? Typically, trend inflation is not observed but estimated via a statistical decomposition. It is difficult to get away from the influence of tuning parameters set a priori. In the popular Hodrick and Prescott [1997] framework, the quadratic loss function weights deviations in trend λ = 1600 times more than variation in the cycle for quarterly data (Ravn and Uhlig [2002]). This corresponds to a standard deviation ratio of 40, which can be viewed as a Bayesian prior. Take the standard deviation of quarterly output to be 0.5%. This is plausible for advanced economies in normal times. It is roughly consistent with a variety of evidence presented by Cogley and Sargent [2002], Stock and Watson [2002], Stock and Watson [2005], Keating and Valcarcel [2012] and Hulseman and Detmeister [29 July, 2017]. Throughout the subsequent discussion I use asymptotic notation x ≍ y to mean x and y are of the same order of magnitude. I take this to mean that one variable is on average no more than a tenth of the other. [147] Hamilton’s technique is controversial. The literature has long been aware of problems with the standard filter. It generates spurious cycles, uses future information to construct present trend components and does not account for events that might be best considered structural breaks; for more on this view consult Harvey and Jaeger [1993], King and Rebelo [1993], Cogley and Nason [1995], Canova [1998], Baxter and King [1999] and Christiano and Fitzgerald [2003]. Nevertheless, recent work by Hodrick [2020] shows that his filter outperforms others, when there are slow-moving changes in persistent components which might be a plausible consequence of, for example, long-run structural change. [148] This calibration exercise and surrounding discussion have justified the claim that there is no decisive empirical answer to whet her shocks to trend inflation are first order or not. This should be an important field for future work. We are in a far better place than before as a discipline, if we are discussing the econometric implementation of our main model, rather than debating its underlying contents. This paper is available on arxiv under CC 4.0 license. 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 I.3 Trend Inflation Volatility This part presents evidence on the volatility of trend inflation. The focus here is on time series ideas. It matters how you de-trend the data. There is significant low frequency volatility in inflation, over typical sample periods. Contrast the Great Depression with the Great Inflation and think of Japan’s lost decades. Under inflation targeting it is natural to think of the headline target as the trend inflation rate. This makes trend inflation much less volatile. Nevertheless, there has been discussion about changing the inflation target which should be factored in by rational agents, (see for example Blanchard et al. [2010], Ball [2013] and Cecchetti and Schoenholtz [2017]). The European Central Bank’s recent move to a symmetric inflation target might work out like this (Ignazio and Gros [2021]). The asymptotic justification is a little delicate though. As shocks shrink, are trend inflation shocks sufficiently large to be considered first order? Typically, trend inflation is not observed but estimated via a statistical decomposition. It is difficult to get away from the influence of tuning parameters set a priori. In the popular Hodrick and Prescott [1997] framework, the quadratic loss function weights deviations in trend λ = 1600 times more than variation in the cycle for quarterly data (Ravn and Uhlig [2002]). This corresponds to a standard deviation ratio of 40, which can be viewed as a Bayesian prior. Take the standard deviation of quarterly output to be 0.5%. This is plausible for advanced economies in normal times. It is roughly consistent with a variety of evidence presented by Cogley and Sargent [2002], Stock and Watson [2002], Stock and Watson [2005], Keating and Valcarcel [2012] and Hulseman and Detmeister [29 July, 2017]. Throughout the subsequent discussion I use asymptotic notation x ≍ y to mean x and y are of the same order of magnitude. I take this to mean that one variable is on average no more than a tenth of the other. [147] Hamilton’s technique is controversial. The literature has long been aware of problems with the standard filter. It generates spurious cycles, uses future information to construct present trend components and does not account for events that might be best considered structural breaks; for more on this view consult Harvey and Jaeger [1993], King and Rebelo [1993], Cogley and Nason [1995], Canova [1998], Baxter and King [1999] and Christiano and Fitzgerald [2003]. Nevertheless, recent work by Hodrick [2020] shows that his filter outperforms others, when there are slow-moving changes in persistent components which might be a plausible consequence of, for example, long-run structural change. [148] This calibration exercise and surrounding discussion have justified the claim that there is no decisive empirical answer to whet her shocks to trend inflation are first order or not. This should be an important field for future work. We are in a far better place than before as a discipline, if we are discussing the econometric implementation of our main model, rather than debating its underlying contents. 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

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

How the Lucas Critique Impacts the Rotemberg Economic Model

Keynesian.TECH

Too Long; Didn't Read

Boost your HackerNoon story @ $159.99! 🚀

Why Trend Inflation Volatility is Key for Future Economic Forecasts

About Author

Comments

TOPICS

THIS ARTICLE WAS FEATURED IN

Related Stories

Untitled Story

A Beginner's Guide to Homology

Bifurcation Analysis of the Keynesian Cross Model: Conclusion and References

Bifurcation Analysis of the Keynesian Cross Model: Method and G is constant

Bifurcation Analysis of the Keynesian Cross Model: Abstract and Introduction

Bifurcation Analysis of the Keynesian Cross Model: Results

A Beginner's Guide to Homology

Bifurcation Analysis of the Keynesian Cross Model: Conclusion and References

Bifurcation Analysis of the Keynesian Cross Model: Method and G is constant

Bifurcation Analysis of the Keynesian Cross Model: Abstract and Introduction

Bifurcation Analysis of the Keynesian Cross Model: Results

Light-Mode

Classic

Newspaper

Dark-Mode

Neon Noir

Minty

HN StartUps