Why Trend Inflation Volatility is Key for Future Economic Forecasts

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.