Luca

luca

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.2 Lucas Critique Proposition 11 and the parametizations in Appendix H.1.2 suggest that the observational equivalence part of the Lucas critique applies to the Rotemberg model (at ZINSS) only if the microeconomic evidence supports ˜ω > 1. At the headline parametization Lastly, the parametization could be effected by the addition of real frictions. These suppress the relationship between real output and marginal costs (see Gopinath and Itskhoki [2011]), which would show up as where R > 0 measures the size of the real friction. These have been popular because the slope of the Rotemberg Phillips curve appears too high, from a macroeconometric standpoint. The difficulty is that microeconometric evidence favours smaller values, which are too small to overturn these conclusions (see Beck and Lein [2020]). This is probably as far as one should take this exercise. 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.2 Lucas Critique Proposition 11 and the parametizations in Appendix H.1.2 suggest that the observational equivalence part of the Lucas critique applies to the Rotemberg model (at ZINSS) only if the microeconomic evidence supports ˜ω > 1. At the headline parametization Lastly, the parametization could be effected by the addition of real frictions. These suppress the relationship between real output and marginal costs (see Gopinath and Itskhoki [2011]), which would show up as where R > 0 measures the size of the real friction. These have been popular because the slope of the Rotemberg Phillips curve appears too high, from a macroeconometric standpoint. The difficulty is that microeconometric evidence favours smaller values, which are too small to overturn these conclusions (see Beck and Lein [2020]). This is probably as far as one should take this exercise. 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

How the Lucas Critique Impacts the Rotemberg Economic Model

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 6.2 Equilibrium Construction Here I characterize the stochastic equilibrium of the Calvo New Keynesian economy. The main tool is Birkhoff’s ergodic theorem. The stochastic steady state resembles its non-stochastic counterpart, except there are expectation terms attached to future realizations of non-linear functions of the variables (π, y, ∆, ψ, A) underpinning the recursive equilibrium, described back in Section 4.8 and Proposition 4. Current values of this set can be interpreted as their long run mathematical expectation. Existence results will be established in Section 8. Starting with the Phillips curve, in particular the numerator of the reset price relationship from Proposition 1 given by combining infinite horizon expression (28) and recursive form (30) combining gives the basis for the Stochastic Equilibrium Phillips curve. Price dispersion evolves as follows It is clear that the asymptote in all these equations corresponds to the case where ∆ grows unbounded. The Euler equation takes the form Hence the equilibrium real interest rate, known as the natural rate of interest rate, is Similar expressions could be written for the quantities U and Π but are suppressed in the interest of space. Author:
(1) David Staines. This paper is available on arxiv under CC 4.0 license. [44] This integral is technically a Bochner or strong integral, the natural extension of Lebesgue integral to infinite dimensional space, whereas that of (28) is a more general Pettis integral. 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 6.2 Equilibrium Construction Here I characterize the stochastic equilibrium of the Calvo New Keynesian economy. The main tool is Birkhoff’s ergodic theorem. The stochastic steady state resembles its non-stochastic counterpart, except there are expectation terms attached to future realizations of non-linear functions of the variables (π, y, ∆, ψ, A) underpinning the recursive equilibrium, described back in Section 4.8 and Proposition 4. Current values of this set can be interpreted as their long run mathematical expectation. Existence results will be established in Section 8. Starting with the Phillips curve, in particular the numerator of the reset price relationship from Proposition 1 given by combining infinite horizon expression (28) and recursive form (30) combining gives the basis for the Stochastic Equilibrium Phillips curve. Price dispersion evolves as follows It is clear that the asymptote in all these equations corresponds to the case where ∆ grows unbounded. The Euler equation takes the form Hence the equilibrium real interest rate, known as the natural rate of interest rate, is Similar expressions could be written for the quantities U and Π but are suppressed in the interest of space. Author: (1) David Staines. Author: Author: (1) David Staines. 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 [44] This integral is technically a Bochner or strong integral, the natural extension of Lebesgue integral to infinite dimensional space, whereas that of (28) is a more general Pettis integral.

Stochastic Equilibrium the Lucas Critique and Keynesian Economics: Equilibrium Construction

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 5.4 Lucas Critique This part begins by explaining the Lucas critique and its various interpretations. The second subsection sets out observational equivalences and relates them to the Lucas critique framework. This should be viewed as a failure of existing New Keynesian economics. I will go onto demonstrate that the approximation I construct "passes" this crucial aspect of the Lucas critique by breaking this equivalence. In Appendix C.2.3, I confirm that there are (correctly solved) multiple models of recent vintage that are still ensnared. 5.4.1 Interpretations 1. Mapping Micro to Macro Lucas Jr [1976] heralded a new approach to macroeconomics that emphasised that models should be based on aggregation of the optimizing behaviour of agents. This methodology has become ubiquitous in modern research, providing an underlying motivation for all the New Keynesian models in this text. Researchers however have neglected to check solutions to DSGE models preserve the mapping from micro to macro. This paper analyses and exemplifies two threats to this connection. There are a priori reduced form approximations used in econometric work that do not represent neighboring stochastic systems because of failure to account for singularities. Principle 1 links this phenomenon back to the breakdown of constraints in individual optimization problems, akin to the common understanding of micro foundations. Secondly, there arise instances where the local approximation would be correct but ex post the dynamics of the reduced form are not consistent with the existence of nearby non-stochastic systems. In fact, these two forces combine to obscure any connection between the behavior of approximation solutions and the developments and mechanisms of an underlying non-linear New Keynesian economy. 2. Observational Equivalence The sole formal contribution of Lucas Jr [1976] was to demonstrate the equivalence between a new classical Phillips curve and a then standard Keynesian formulation of the relationship. It is a seminal result of macroeconometrics. I am able to show that it still applies to popular New Keynesian formulations but breaks down when Calvo is solved correctly. It demonstrates a fundamental advantage of Keynesian over Classical monetary models. 3. Breakdown of Statistical Relationships Lucas argued that statistical relationships like the Phillips curve[38] would break down when exploited for policy purposes.[39] This argument gained credence in the 1970s when there was stagflation (simultaneous increases in unemployment and inflation). The idea was central to the rise of New Classical economics, indeed it spurred the creation of DSGE. The result is only partially correct. He was right that the Phillips curve is not policy invariant, as the policy rule parameters affected the optimal supply decisions with drawbacks to monetary activism. He was also correct that there need not be an upward sloping relationship between inflation and output in equilibrium. On the other hand, this does not rule out systematic countercylical policy consistent with the Keynesian consensus (see Snowdon and Vane [2005]). Lastly, the comparative statics, part of Theorem 2, can be viewed as an analysis of policy regime changes of the kind considered by Lucas. 5.4.2 Observational Equivalence This part formalizes the traditional interpretation of the Lucas critique and proves that, subject to parametric conditions, it extends to the current benchmark solution of Calvo and the Rotemberg Phillips curves. The equivalence between these approximations is not novel, although the link back to Lucas is. Proposition 11. (Lucas Critique) There can arise an observational equivalence between the Singular Calvo, Rotemberg and Lucas Phillips curves provided that ω > η. Proof. Proposition 8 ensures the expectation terms vanish from the two forward-looking models. The rest of the proof is constructive. Equating (54) and (91) yields which sets Rotemberg equal to Calvo. Deploying (81) and equating shows equivalence to the Lucas model when the bound follows because these variances have to be strictly positive. The interpretation is that provided the Phillips curve is sufficiently steep, then the methods of macro econometrics cannot be used to distinguish between these three underlying designs for the Phillips curve. As an empirical matter, this is doubtful. Empirical specifications of the Phillips curve strongly favor a flat slope (ω < 1), whilst micro econometric evidence strongly supports an inelastic labor supply (η > 1).[40] Furthermore, there appears to be greater volatility of monetary shocks relative to technology shocks than the model would support. These issues are taken up in detail, in Appendices H and I. The broader interpretation is that these solutions are just barely "passing" the observational equivalence aspect of the Lucas critique. There are many dynamic variants of the Lucas model, featuring more sophisticated information frictions (see Angeletos et al. [2021] and Maćkowiak et al. [2023]). They typically generating flatter inflation-output trade-offs (Afrouzi and Yang [2021]). Thus with respect to these versions, forward-looking Phillips curves may still be ensnared by the critique. Conversely, by overturning the singular Calvo solution, I can rule out all these models too. We can do better than this. The Divine Coincidence and lack of dynamics have helped motivate a move away from DSGE back to ad hoc models. With new methods and better results, I anticipate a reversal. Moreover, armed with corrected approximations to Calvo around ZINSS, it should be possible to search for similar observational equivalences, in a wider class of DSGE. The link with Taylor pricing, established in Section 10, is a starting point. Author:
(1) David Staines. This paper is available on arxiv under CC 4.0 license. [38] He had in mind the original Phillips curve, a negative relationship between unemployment and inflation estimated by Phillips [1958]. [39] Similar ideas were advanced in the context of monetary targeting by Goodhart [1975] without formal mathematical underpinning. The principles are of wide application throughout economics. 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 5.4 Lucas Critique This part begins by explaining the Lucas critique and its various interpretations. The second subsection sets out observational equivalences and relates them to the Lucas critique framework. This should be viewed as a failure of existing New Keynesian economics. I will go onto demonstrate that the approximation I construct "passes" this crucial aspect of the Lucas critique by breaking this equivalence. In Appendix C.2.3, I confirm that there are (correctly solved) multiple models of recent vintage that are still ensnared. 5.4.1 Interpretations 1. Mapping Micro to Macro 1. Mapping Micro to Macro Lucas Jr [1976] heralded a new approach to macroeconomics that emphasised that models should be based on aggregation of the optimizing behaviour of agents. This methodology has become ubiquitous in modern research, providing an underlying motivation for all the New Keynesian models in this text. Researchers however have neglected to check solutions to DSGE models preserve the mapping from micro to macro. This paper analyses and exemplifies two threats to this connection. There are a priori reduced form approximations used in econometric work that do not represent neighboring stochastic systems because of failure to account for singularities. Principle 1 links this phenomenon back to the breakdown of constraints in individual optimization problems, akin to the common understanding of micro foundations. Secondly, there arise instances where the local approximation would be correct but ex post the dynamics of the reduced form are not consistent with the existence of nearby non-stochastic systems. In fact, these two forces combine to obscure any connection between the behavior of approximation solutions and the developments and mechanisms of an underlying non-linear New Keynesian economy. 2. Observational Equivalence 2. Observational Equivalence The sole formal contribution of Lucas Jr [1976] was to demonstrate the equivalence between a new classical Phillips curve and a then standard Keynesian formulation of the relationship. It is a seminal result of macroeconometrics. I am able to show that it still applies to popular New Keynesian formulations but breaks down when Calvo is solved correctly. It demonstrates a fundamental advantage of Keynesian over Classical monetary models. 3. Breakdown of Statistical Relationships 3. Breakdown of Statistical Relationships Lucas argued that statistical relationships like the Phillips curve[38] would break down when exploited for policy purposes.[39] This argument gained credence in the 1970s when there was stagflation (simultaneous increases in unemployment and inflation). The idea was central to the rise of New Classical economics, indeed it spurred the creation of DSGE. The result is only partially correct. He was right that the Phillips curve is not policy invariant, as the policy rule parameters affected the optimal supply decisions with drawbacks to monetary activism. He was also correct that there need not be an upward sloping relationship between inflation and output in equilibrium. On the other hand, this does not rule out systematic countercylical policy consistent with the Keynesian consensus (see Snowdon and Vane [2005]). Lastly, the comparative statics, part of Theorem 2, can be viewed as an analysis of policy regime changes of the kind considered by Lucas. 5.4.2 Observational Equivalence This part formalizes the traditional interpretation of the Lucas critique and proves that, subject to parametric conditions, it extends to the current benchmark solution of Calvo and the Rotemberg Phillips curves. The equivalence between these approximations is not novel, although the link back to Lucas is. Proposition 11. (Lucas Critique) There can arise an observational equivalence between the Singular Calvo, Rotemberg and Lucas Phillips curves provided that ω > η. Proposition 11. (Lucas Critique) There can arise an observational equivalence between the Singular Calvo, Rotemberg and Lucas Phillips curves provided that ω > η. Proof . Proposition 8 ensures the expectation terms vanish from the two forward-looking models. The rest of the proof is constructive. Equating (54) and (91) yields Proof which sets Rotemberg equal to Calvo. Deploying (81) and equating shows equivalence to the Lucas model when the bound follows because these variances have to be strictly positive. The interpretation is that provided the Phillips curve is sufficiently steep, then the methods of macro econometrics cannot be used to distinguish between these three underlying designs for the Phillips curve. As an empirical matter, this is doubtful. Empirical specifications of the Phillips curve strongly favor a flat slope (ω < 1), whilst micro econometric evidence strongly supports an inelastic labor supply (η > 1).[40] Furthermore, there appears to be greater volatility of monetary shocks relative to technology shocks than the model would support. These issues are taken up in detail, in Appendices H and I. The broader interpretation is that these solutions are just barely "passing" the observational equivalence aspect of the Lucas critique. There are many dynamic variants of the Lucas model, featuring more sophisticated information frictions (see Angeletos et al. [2021] and Maćkowiak et al. [2023]). They typically generating flatter inflation-output trade-offs (Afrouzi and Yang [2021]). Thus with respect to these versions, forward-looking Phillips curves may still be ensnared by the critique. Conversely, by overturning the singular Calvo solution, I can rule out all these models too. We can do better than this. The Divine Coincidence and lack of dynamics have helped motivate a move away from DSGE back to ad hoc models. With new methods and better results, I anticipate a reversal. Moreover, armed with corrected approximations to Calvo around ZINSS, it should be possible to search for similar observational equivalences, in a wider class of DSGE. The link with Taylor pricing, established in Section 10, is a starting point. Author: (1) David Staines. Author: Author: (1) David Staines. 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 [38] He had in mind the original Phillips curve, a negative relationship between unemployment and inflation estimated by Phillips [1958]. [39] Similar ideas were advanced in the context of monetary targeting by Goodhart [1975] without formal mathematical underpinning. The principles are of wide application throughout economics.

How Correcting Calvo Solutions Reshapes the Lucas Critique

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 5 Existing Solutions This section sets out the existing perturbation solutions, taken from ZINSS. There is a natural focus on the first order approximation, although I am able to generalize salient results to non-linear solutions. I set out familiar problems with their predictions concerning persistence and optimal policy. Next, I set out two alternative models of nominal rigidity the Lucas and Rotemberg Phillips curves. Up to parametric conditions, I connect these three models in a pseudo-observational equivalence, which I later disprove. This allows me to link this wrong claim back to a particular argument in Lucas Jr [1976]. Finally, I review all aspects of this famous paper. Author:
(1) David Staines. This paper is available on arxiv under CC 4.0 license. 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 5 Existing Solutions This section sets out the existing perturbation solutions, taken from ZINSS. There is a natural focus on the first order approximation, although I am able to generalize salient results to non-linear solutions. I set out familiar problems with their predictions concerning persistence and optimal policy. Next, I set out two alternative models of nominal rigidity the Lucas and Rotemberg Phillips curves. Up to parametric conditions, I connect these three models in a pseudo-observational equivalence, which I later disprove. This allows me to link this wrong claim back to a particular argument in Lucas Jr [1976]. Finally, I review all aspects of this famous paper. Author: (1) David Staines. Author: Author: (1) David Staines. 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

A Critical Review of Lucas Jr's Famous Paper on Nominal Rigidity

So, let’s start! Tell us a bit about yourself. For example, name, profession, and personal interests. Hey Everyone, my name is Lucas Matheus. I’m from Brazil, and I currently work in the technology area at a supercomputing and applied sciences company here in my country. I have interests in a range of topics in human knowledge; I really like culture, music and I consider programming an art. Interesting! What was your latest Hackernoon Top story about? It's called “The Surprising Lessons I've Learned as a Tech Lead,” and it's about my first adventures and learnings leading a project. Do you usually write on similar topics? If not, what do you usually write about? To be honest, it was the first time I chose to write about a topic that deals with soft skills and behaviors that are necessary for a successful project. These were things that I learned the importance of over time and that I ended up enjoying the experience of writing about. I typically focused on writing on more technical topics in business, physics, and programming. Great! What is your usual writing routine like (if you have one?) Good question! I always set aside time on the weekend to think about a topic that I'm really excited to share. I filter what is important to talk about this topic and select the most valuable learning to be exposed to. It's not a very methodical process, but it's very content-based. Depending on the subject, I like to put a context beforehand and explain the reason for writing to make sense to the audience and so as not to appear to be writing just for the sake of writing. Being a writer in tech can be a challenge. It’s not often our main role, but it's an addition to another one. What is the biggest challenge you have when it comes to writing? Beat the blank paper or, in this case, the blank computer screen. Just kidding! Although there is some truth to it and I believe it is a very complicated problem for any writer. My biggest challenge is to end up maintaining the coherence of what is written and ensuring that what was written maintains my identity. Nowadays, with artificial intelligence, which has the ability to produce text faster and maintain authenticity, the essence of writing has become increasingly complicated! In more technical texts, I always try to maintain objectivity and not leave room for other interpretations. A challenge of mine is to always keep the writing as impersonal and accessible in all cases. What is the next thing you hope to achieve in your career? I'm currently focused on being able to share as much knowledge as I've acquired in my 10-year programming career. I feel like it's time to be able to share and learn more from people. I think about evolving in technical leadership, perhaps reaching a C-level in some companies and contributing even more to the technology community around the world. I'm also thinking about improving a project for a social recommendation network that I created with some friends here in Brazil. I hope to end up evolving a lot over the years and helping many people on my journey in the technology area. Wow, that’s admirable. Now, something more casual: What is your guilty pleasure of choice? Ok, ok, I admit that from time to time, I end up reviewing films considered very bad by the general public. I have a list of these movies, and for some reason, I like the 2005 Fantastic Four and that hacker movie with Jonny Lee Miller. Do you have a non-tech-related hobby? If yes, what is it? Well, I really enjoy working out and playing soccer sometimes. Lately, I've been focusing on relaxing a bit with short walks around where I live. I also really like watching movies and playing on the PS4. Every now and then, I end up listening to an album on Spotify from some obscure 2000s rock band. What can the Hacker Noon community expect to read from you next? More stories about my experiences as a tech lead in different projects, more technical articles from the hacker world and perhaps some content more focused on the startup area. I'm already working on some related to meetings and their importance in development (I think some myths should be broken about this practice). What’s your opinion on HackerNoon as a platform for writers? One of the best platforms for sharing knowledge. It's incredible how it was thought of in every detail with a very motivating environment. I felt welcomed by the community and the fact that the article was analyzed. Thank you very much for the opportunity, and I hope to contribute more to this incredible platform Thanks for taking time to join our “Meet the writer” series. It was a pleasure. Do you have any closing words? Thanks a lot! There's a funny story from Brazil: a TV network once ran a show where they had a random guy dressed as an alien. They hyped it up, saying the alien had an important message for everyone. And when the big moment came, the alien simply said: 'Just seek it out. Knowledge.' So, I'll leave you with those words! So, let’s start! Tell us a bit about yourself. For example, name, profession, and personal interests. Hey Everyone, my name is Lucas Matheus. I’m from Brazil, and I currently work in the technology area at a supercomputing and applied sciences company here in my country. I have interests in a range of topics in human knowledge; I really like culture, music and I consider programming an art. Interesting! What was your latest Hackernoon Top story about? It's called “The Surprising Lessons I've Learned as a Tech Lead,” and it's about my first adventures and learnings leading a project. Do you usually write on similar topics? If not, what do you usually write about? To be honest, it was the first time I chose to write about a topic that deals with soft skills and behaviors that are necessary for a successful project. These were things that I learned the importance of over time and that I ended up enjoying the experience of writing about. I typically focused on writing on more technical topics in business, physics, and programming. Great! What is your usual writing routine like (if you have one?) Good question! I always set aside time on the weekend to think about a topic that I'm really excited to share. I filter what is important to talk about this topic and select the most valuable learning to be exposed to. It's not a very methodical process, but it's very content-based. Depending on the subject, I like to put a context beforehand and explain the reason for writing to make sense to the audience and so as not to appear to be writing just for the sake of writing. Being a writer in tech can be a challenge. It’s not often our main role, but it's an addition to another one. What is the biggest challenge you have when it comes to writing? Beat the blank paper or, in this case, the blank computer screen. Just kidding! Although there is some truth to it and I believe it is a very complicated problem for any writer. My biggest challenge is to end up maintaining the coherence of what is written and ensuring that what was written maintains my identity. Nowadays, with artificial intelligence, which has the ability to produce text faster and maintain authenticity, the essence of writing has become increasingly complicated! In more technical texts, I always try to maintain objectivity and not leave room for other interpretations. A challenge of mine is to always keep the writing as impersonal and accessible in all cases. What is the next thing you hope to achieve in your career? I'm currently focused on being able to share as much knowledge as I've acquired in my 10-year programming career. I feel like it's time to be able to share and learn more from people. I think about evolving in technical leadership, perhaps reaching a C-level in some companies and contributing even more to the technology community around the world. I'm also thinking about improving a project for a social recommendation network that I created with some friends here in Brazil. I hope to end up evolving a lot over the years and helping many people on my journey in the technology area. Wow, that’s admirable. Now, something more casual: What is your guilty pleasure of choice? Ok, ok, I admit that from time to time, I end up reviewing films considered very bad by the general public. I have a list of these movies, and for some reason, I like the 2005 Fantastic Four and that hacker movie with Jonny Lee Miller. Do you have a non-tech-related hobby? If yes, what is it? Well, I really enjoy working out and playing soccer sometimes. Lately, I've been focusing on relaxing a bit with short walks around where I live. I also really like watching movies and playing on the PS4. Every now and then, I end up listening to an album on Spotify from some obscure 2000s rock band. What can the Hacker Noon community expect to read from you next? More stories about my experiences as a tech lead in different projects, more technical articles from the hacker world and perhaps some content more focused on the startup area. I'm already working on some related to meetings and their importance in development (I think some myths should be broken about this practice). What’s your opinion on HackerNoon as a platform for writers? One of the best platforms for sharing knowledge. It's incredible how it was thought of in every detail with a very motivating environment. I felt welcomed by the community and the fact that the article was analyzed. Thank you very much for the opportunity, and I hope to contribute more to this incredible platform Thanks for taking time to join our “ Meet the writer ” series. It was a pleasure. Do you have any closing words? Meet the writer Thanks a lot! There's a funny story from Brazil: a TV network once ran a show where they had a random guy dressed as an alien. They hyped it up, saying the alien had an important message for everyone. And when the big moment came, the alien simply said: 'Just seek it out. Knowledge.' So, I'll leave you with those words!

Meet the Writer: Hacker Noon's Contributor Lucas Matheus, Tech Entrepreneur, Tech Lead and Hacker

Discussing the future of finance and technology with Lucas Vogelsang, CEO and Co-Founder at Centrifuge Ishan Pandey: Hi Lucas, welcome to our series “Behind the Startup.” Please tell us about yourself and the story behind ? Centrifuge Lucas Vogelsang: I’ve spent my entire career working in startups, first in web2.0, where I quickly realized the inefficiencies with our current financial system. Finance gives people the power to transform ideas into reality, yet the current system is inefficient, not transparent, and serves the largest corporations at the expense of the small businesses that account for over 90% of companies globally. I was always excited about building technology to solve exciting and hard challenges. When I started diving deeper into Ethereum in 2017, I realized the potential of crypto to build a more open, transparent, and affordable financial system. Centrifuge started in 2018 with a vision to change global trade rules. We were the first protocol to bring real-world assets on-chain with MakerDAO and NewSilver and continue to remain focused on getting the entire world of structured credit on-chain. : Vested Interest Disclosure The author is an independent contributor publishing via our . Be it through direct compensation, media partnerships, or networking, the author has a vested interest in the company/ies mentioned in this story. HackerNoon has reviewed the story for quality, but the claims hereon belong to the author. #DYOR brand-as-author program Ishan Pandey: Please tell us a little bit about Centrifuge and Real-World DeFi. Lucas Vogelsang: Centrifuge connects real-world assets (RWAs) to DeFi - hence Real-World DeFi - creating a fully transparent market which allows borrowers and lenders to transact without unnecessary intermediaries. Ultimately, the protocol aims to lower the cost of borrowing for businesses worldwide, while providing DeFi users with a stable source of collateralized yield that is uncorrelated to the volatile crypto markets. Centrifuge was the first DeFi protocol to bridge RWAs on-chain and has integrated with some of the leading DeFi protocols, including MakerDAO and Aave. Underlying RWAs and liquidity providers fully collateralize centrifuge asset pools and have legal rights to the collateral in the case of a default. This makes Centrifuge unique versus many of our RWA peers that largely focus on undercollateralized lending. NFTs are used as digital representations of the real-world collateral on-chain, allowing liquidity providers to have visibility on the exact assets they finance. The protocol is asset class agnostic and has created liquidity pools for assets such as mortgages, invoices, microlending and consumer finance. Ishan Pandey: Ethereum will switch the cryptocurrency consensus method from Proof-of-Work (PoW) to Proof-of-Stake as it approaches the Merge. What will the post-merge scenario look like for Ethereum at large? Lucas Vogelsang: The switch to proof-of-stake is a great foundational change at the heart of how Ethereum works. A successful merge should be good for the price of ETH in the short term, but the real impact will take place over a much longer timeline. The new block production mechanism will directly impact Ethereum’s underlying infrastructure, especially when considering how the new MEV implications will affect the behaviour of Ethereum’s fee market. The competition for low fees should eventually translate to the benefit of users, but there will be friction as the market takes time to adjust and react to the realities of proof-of-stake. The long-term positive impact on L2s, the inclusion of sharding, and the continued diversity and innovation that the merge will encourage will be exciting. Ishan Pandey: The hoopla surrounding the new technology has been matched by growing worries that NFTs may become the focal point of traditional financial crimes like money laundering and wire fraud given the ongoing popularity of some NFTs and the possibility of their use in the Metaverse and beyond. According to you, will the industry face a regulatory crackdown in light of such issues? Lucas Vogelsang: While we will continue to see enforcement actions for fraud and sanctions violations, we don't expect a general crackdown. Key regulators in the US and EU seem to be bringing parts of the crypto space into a regulated environment. It would not make sense to launch a major crackdown while the regulatory perimeter is being discussed and defined at the legislative level. Ishan Pandey: According to you, what are the best practices that should be followed in relation to cybersecurity in the DeFi industry? Lucas Vogelsang: When designing DeFi protocols, emphasizing simplicity is key. Complex protocols are hard to develop and even harder to audit. For example, upgradeability in Solidity smart contracts is a pattern that adds a layer of indirection and complexity to a protocol, which is difficult to check and thus has led to many exploits. Beyond the design of protocols itself, teams should spend more time ensuring the security of their front end. We have seen many exploits recently, e.g. the Curve Finance exploit hack, that was caused by issues in the security of the hosting or domain management of the frontend. These kinds of supply chain issues are hard to prevent due to the reliance on centralized services such as DNS, but are more regularly becoming targets for blackhats and more time should be spent on preventing these issues. Ishan Pandey: According to you, what are the certain limitations surrounding Dapps and Defi applications running on the ethereum blockchain? Lucas Vogelsang: Any Dapp or DeFi developer knows that gas optimization is the key metric to focus on. This means reducing the amount of state stored on-chain and the number of transactions required for interacting with their dapp or protocol. Unfortunately, this can also limit the possibilities of an app. E.g. this makes many gaming use cases almost impossible on Ethereum, which has led these projects to find other ecosystems to build on. Another limitation is the ability to schedule code automatically. Everything needs to be user-triggered. Being able to run code automatically is a powerful feature that can unlock many new use cases. Ishan Pandey: Though both DeFi and CeFi provide a variety of financial services relating to cryptocurrencies, they are vastly different from one another. Kindly elaborate on some of the major differences for our readers. Lucas Vogelsang: CeFi is opaque as users have no idea how their funds are deployed and need to “trust” a third party. In the last few months, we have seen many CeFi platforms struggle following the Three Arrows Capital (3AC) collapse and reckless lending. DeFi, on the other hand, gives you visibility on the all process without introducing any “trust” assumptions or need to KYC - not your keys not your coins. Ishan Pandey: According to you, what next trends are we going to see in the blockchain and DeFi space? Lucas Vogelsang: 3 trends I am excited about: Structured Products: these are pre-packaged strategies that allow anyone without prior knowledge of finance or derivatives to access new yield opportunities. DeFi has the potential to bring these products to mainstream usage. Cross-chain dapp: It is now possible to create composable cross-chain protocols that are no longer restricted to an ecosystem, thanks to the development of new general messaging passing bridges. Privacy: Tornado crackdown highlighted how vital privacy is - I expect a new surge of privacy projects, particularly those utilizing ZK technology. And, of course, keep an eye on Real-World Assets (RWAs)! Don’t forget to like and share the story!

Understanding the Impact of DeFi on Real Word Assets with Lucas Vogelsang

Learn how Lucas Liew scaled his online course for algorithmic trading from 0 to 30k students. Davis Baer : What’s your background, and what are you working on? Hey guys. I’m Lucas Liew, founder of , an online course for algorithmic trading (i.e. trading using code). Lucas Liew: AlgoTrading101 Essentially, we teach our students how to generate, code and execute their own trading ideas. We have almost 30k students from 160 countries at the moment. What motivated you to get started with your company? I was trading at a proprietary trading firm when I started AlgoTrading101 as a side project. The first version of the idea was to sell code templates for trading robots. Customers can then build their own trading robots by editing these templates. Bad idea. Turns out no one wants to pay for templated code (especially when there is plenty of free code floating around). We then pivoted to an online course for algorithmic trading. There were (and still are) many “get rich quick” trading courses around. Our angle was to appeal to those who were not looking for a quick buck, or who had tried those “get rich quick” schemes and failed. What went into building the initial product? We tried to validate the idea by talking to “users”, aka my friends who were in the finance industry. Their feedback was honest but bad. My friends were genuinely concerned for me pursuing this idea. In hindsight, my friends weren’t my target audience. We should have spoken to those seeking a career change, or college undergraduates looking at trading as a possible career path. We went ahead with the idea anyway. The plan was to launch fast and get feedback from the market directly. Our initial investment was $100; it was a Blue Snowball microphone. The launch was supposed to happen within 3 months, but self-doubt and a massive underestimation of the work required dragged that out to 7 months. It would have been much longer had we decided to perfect the course. We launched (with much hesitation) a scrappy version of our course. We should have launched earlier. Coding and testing a trading robot How have you attracted users and grown your company? We got lucky. We decided to launch on Udemy (an educational course marketplace) instead of our own hosted site. Moreover, we launched right before their massive Black Friday sale. We didn’t even know that they had a Black Friday sale. By leeching off Udemy’s user base, we had sales on our first day of launch. By the end of the first month we had over a thousand students. Thousands of free coupons were given away in the following months for publicity, reviews, and feedback. This helped us become one of the top courses on Udemy for algorithmic trading. What’s your business model, and how have you grown your revenue? We managed to grow our revenue by 10% month-on-month for the first 24 months. We got traction from content marketing and press exposure on other platforms (such as Quora and Investopedia) and SEO. For SEO, it took us about 2 years to rank first for “Learn algorithmic trading”. Unfortunately, revenue plateaued as we let our foot off the gas for SEO and content marketing efforts to look at new products, which didn’t contribute to growth. In those first 24 months, our pricing strategy contributed to our growth. We were raising prices regularly to find our optimal price point. For customers who were on the fence, this tipped them over. Our prices started from $49 (for lifetime access) in 2015 and now it is $570, but we have also added significant amount of paid content during this period. Prices were raised yearly. We usually see double the revenue in the month before a price increase as compared to an average month. If your product doesn’t solve a “hair-on-fire” problem, you will get plenty of resistance at the last mile. Having some sort of urgency to purchase your product will help greatly. What are your goals for the future? We joined Y Combinator’s Start Up School 4 weeks ago. With the guidance of our YC mentor , revenue has been growing at 20% week-on-week. Time to keep the momentum going! Igor The current focus is on SEO and content marketing. After which, we will be looking to create more courses and online tools in nearby verticals, like machine learning for finance. What are the biggest challenges you’ve faced and obstacles you’ve overcome? If you had to start over, what would you do differently? Changing my mentality made a big difference. AlgoTrading101 started out as a side project. I never thought of it as a “real” business until many months after its launch. Once we sat down and decided to focus on this business, we pushed it to grow more than we had ever thought possible. Some things I would do differently: Launch (much) earlier Presale my product Focus on building an email list of happy users Be consistent about content marketing Launch more courses in verticals close to my niche To those of you running a side project, treat it like a real startup! Have you found anything particularly helpful or advantageous? Former VP of Growth at Facebook, Chamath Palihapitiya, once . talked about how they thought of growth at Facebook The framework: Get users through the door Hit them with the “Aha” moment (within seconds) Deliver core product value as often as possible To stand out from other courses, our course had to deliver value as often as possible. Most other coding courses tend to teach serially, starting from tons of nitty-gritty basics, followed by mundane coding exercises. Our aim was for our students to feel good about their achievements within the first day. Thus, we arranged the course in a way so that most of our students could code their first robot in one day. This made them engaged, happy, and motivated enough to push through the harder parts that would come later. What’s your advice for entrepreneurs who are just starting out? Minimise the duration of your feedback loop. You need a fast way to know if what you’re doing works. Put something out there ASAP to gauge interest. Spending 7 months building an MVP without feedback is dangerous (I got lucky). Money is the best validation. Not sign-ups, free trials, or press coverage. When in doubt, charge now and charge more. Where can we go to learn more? To learn more about AlgoTrading101, check out . AlgoTrading101.com This interview is brought to you by , a tool to schedule and automatically repeat your posts on Facebook, Instagram, Twitter, Pinterest, LinkedIn, and Google My Business OneUp

Founder Interviews: Lucas Liew of AlgoTrading101

Mindeye2: References You Should Check Out

Benchmarking Long-Form Factuality in Large Language Models

Table of Links Introduction
Related Work
Methodology
Experiments
Conclusion and References Introduction Introduction Related Work Related Work Methodology Methodology Experiments Experiments Conclusion and References Conclusion and References 4. Experiments 4.1. Experimental Setup To investigate the early-bird ticket hypothesis in Transformer models, we conducted experiments on four different architectures: ViT, Swin-T, GPT-2, and RoBERTa. The experiments were performed using the following setup: Hardware The experiments were conducted on the Georgia Tech PACE ICE clusters, utilizing A100, A40, or V100 GPUs with a minimum of 32GB memory, depending on availability. Hardware Software The experiments were implemented using PyTorch and Hugging Face libraries for language models. Software Datasets For the vision transformers, we used the CIFAR10 dataset. For fine-tuning the language models, we utilized the IMDB dataset. Datasets 4.2. Results and Analysis 4.2.1 ViT 4.2.1 ViT For the ViT model, the early-bird ticket was found around epoch 20 [3a]. When retrained, the pruned model with a Figure 3. Heatmaps and mask distance plots at p = 0.1 (to the left) and 0.3 (to the right) for all models pruning ratio of 0.1 (p=0.1) fully recovered the baseline performance [2a], achieving an accuracy of 84.3% compared to the unpruned baseline of 85.11%. The model with a higher pruning ratio of 0.3 (p=0.3) also came close to the baseline, with an accuracy of 82.05%. These results demonstrate the trade-off between model sparsity and performance, indicating that a moderate pruning ratio can lead to significant resource savings while maintaining comparable accuracy. 4.2.2 Swin-T 4.2.2 Swin-T Similar to ViT, the early-bird ticket for the Swin-T model was found around epoch 20 [3b]. When retrained, the p=0.1 model fully recovered the baseline performance, achieving an accuracy of 89.54% compared to the unpruned baseline of 89.5%. Interestingly, the p=0.3 model also managed to recover the baseline, with an accuracy of 88.95%. These results suggest that the Swin-T architecture is particularly well-suited for the early-bird ticket hypothesis, as it can maintain high performance even with significant pruning. 4.2.3 GPT-2 4.2.3 GPT-2 For GPT-2, we focused on identifying early-bird tickets during the fine-tuning stage. Remarkably, the early-bird ticket was discovered as early as epoch 2 of fine-tuning [3c]. When fine-tuned with pruning, both the p=0.1 and p=0.3 pruned models achieved a validation accuracy of 83.4%, slightly surpassing the unpruned baseline accuracy of 83.3% [2c]. This finding highlights the potential for early-bird tickets to emerge rapidly during the fine-tuning process, enabling efficient adaptation of pre-trained language models to downstream tasks. 4.2.4 RoBERTa 4.2.4 RoBERTa Similar to GPT-2, we identified the early-bird ticket for RoBERTa at epoch 2 [3d] of the fine-tuning stage. When fine-tuned with pruning, the p=0.1 and p=0.3 pruned models achieved validation accuracies of 86.0% and 86.02%, respectively [2d]. Although these accuracies are slightly lower than the unpruned baseline accuracy of 93.9%, the pruned models still maintain a high level of performance while significantly reducing the computational requirements. This phenomnenon could be associated with the architecture of the model and why we see GPT-2 perform in terms of recovered performance. The architectural differences between RoBERTa and GPT-2, such as the use of dynamic masking and a different pre-training objective in RoBERTa, may contribute to the variations in their ability to recover performance after pruning [7]. 4.2.5 Memory Usage 4.2.5 Memory Usage In addition to the performance evaluation, we also analyzed the memory usage of the pruned models compared to their unpruned counterparts. Table [1] presents the memory usage comparison for each model. The percent change in memory based on the pruned amounts stayed relatively the same for both 0.1 and 0.3 levels. Moreover, upon testing other metrics such as FLOPs, parameters, and inference time, we noticed no change except for the memory utilization. This is because the models inherently are the same, but their pruned nature reduces the amount of memory that is taken up storing the full precision weights. The unpruned ViT model consumed 157.26 MB of memory, while the pruned models with p=0.1 and p=0.3 required only 83.61 MB, resulting in a significant reduction of 46.8%. Similarly, the Swin-T model achieved a memory usage reduction of 49.0%, with the unpruned model consuming 423.43 MB and the pruned models using 216.03 MB. For the language models, GPT-2 exhibited a memory usage reduction of 20.6%, and RoBERTa demonstrated a notable memory usage reduction of 26.9% comparing the unpruned and p=0.1 pruned models. These results highlight the substantial memory savings achieved through pruning, making the models more efficient in terms of resource utilization while maintaining comparable performance to the unpruned models. A change in pruning method based on specific architecture could yield different results from just pruning the linear layers. 4.2.6 Discussion 4.2.6 Discussion The experimental results provide strong evidence for the existence of early-bird tickets in Transformer models across both vision and language domains. The early-bird tickets were consistently found within the first few epochs of training or fine-tuning, indicating the potential for significant resource optimization and cost reduction. The performance of the pruned models obtained from early-bird tickets was comparable to or even surpassed the unpruned baselines in some cases. This suggests that the early-bird ticket hypothesis holds true for Transformer architectures, and that pruning can be effectively applied to reduce the computational requirements without compromising performance. Furthermore, the comparative analysis across different Transformer models highlights the generalizability of the early-bird ticket phenomenon. The successful identification of early-bird tickets in ViT, Swin-T, GPT-2, and RoBERTa demonstrates the applicability of this approach to a wide range of Transformer architectures. However, it is important to note that the optimal pruning ratio may vary depending on the specific model and task. While higher pruning ratios can lead to greater resource savings, they may also result in a slight degradation in performance. 5. Conclusion In this research, we investigated the early-bird ticket hypothesis in Transformer models across vision and language domains. By employing a methodology based on iterative pruning, masked distance calculation, and selective retraining, we successfully identified early-bird tickets in various Transformer architectures. Our experimental results demonstrate that these early-bird tickets can achieve comparable or even superior performance to the unpruned models while significantly reducing the computational re- quirements. The consistent emergence of early-bird tickets within the first few epochs of training or fine-tuning highlights the potential for substantial resource optimization and cost reduction in Transformer model development. This study contributes to the growing body of research on efficient training strategies for Transformer models and paves the way for further exploration of the early-bird ticket hypothesis across a wider range of architectures and tasks. By leveraging the insights gained from this research, practitioners can develop more efficient and accessible Transformer models, enabling their deployment in resource-constrained environments and accelerating the progress of natural language processing and computer vision applications. References [1] Tianlong Chen, Jonathan Frankle, Shiyu Chang, Sijia Liu, Yang Zhang, Zhangyang Wang, and Michael Carbin. The lottery ticket hypothesis for pre-trained bert networks. Advances in neural information processing systems, 33:15834– 15846, 2020. 1 [2] Xiaohan Chen, Yu Cheng, Shuohang Wang, Zhe Gan, Zhangyang Wang, and Jingjing Liu. Earlybert: Efficient bert training via early-bird lottery tickets. arXiv preprint arXiv:2101.00063, 2020. 1 [3] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018. 1 [4] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020. 1 [5] Jonathan Frankle and Michael Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks. arXiv preprint arXiv:1803.03635, 2018. 1 [6] Song Han, Jeff Pool, John Tran, and William Dally. Learning both weights and connections for efficient neural network. Advances in neural information processing systems, 28, 2015. 1 [7] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692, 2019. 1, 2, 4 [8] Paul Michel, Omer Levy, and Graham Neubig. Are sixteen heads really better than one? Advances in neural information processing systems, 32, 2019. 1 [9] Victor Sanh, Lysandre Debut, Julien Chaumond, and Thomas Wolf. Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter. arXiv preprint arXiv:1910.01108, 2019. 1 [10] Emma Strubell, Ananya Ganesh, and Andrew McCallum. Energy and policy considerations for deep learning in nlp. arXiv preprint arXiv:1906.02243, 2019. 1 [11] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017. 1 [12] Ziheng Wang, Jeremy Wohlwend, and Tao Lei. Structured pruning of large language models. arXiv preprint arXiv:1910.04732, 2019. 1 [13] Haoran You, Chaojian Li, Pengfei Xu, Yonggan Fu, Yue Wang, Xiaohan Chen, Richard G Baraniuk, Zhangyang Wang, and Yingyan Lin. Drawing early-bird tickets: Towards more efficient training of deep networks. arXiv preprint arXiv:1909.11957, 2019. 1, 2 Author:
(1) Shravan Cheekati, Georgia Institute of Technology (shravan.cheekati@gatech.edu). Author: Author: (1) Shravan Cheekati, Georgia Institute of Technology (shravan.cheekati@gatech.edu). This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv available on arxiv

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