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Estimation and inference in models with multiple behavioural equilibria

Alexander Mayer, Davide Raggi

Abstract

We develop estimation and inference methods for a stylized macroeconomic model with potentially multiple behavioural equilibria, where agents form expectations using a constant-gain learning rule. We first show geometric ergodicity of the underlying process to study in a second step (strong) consistency and asymptotic normality of the nonlinear least squares estimator for the structural parameters. We propose inference procedures for the structural parameters and uniform confidence bands for the equilibria. When equilibrium solutions are repeated, mixed convergence rates and non-standard limit distributions emerge. Monte Carlo simulations and an empirical application illustrate the finite-sample performance of our methods.

Estimation and inference in models with multiple behavioural equilibria

Abstract

We develop estimation and inference methods for a stylized macroeconomic model with potentially multiple behavioural equilibria, where agents form expectations using a constant-gain learning rule. We first show geometric ergodicity of the underlying process to study in a second step (strong) consistency and asymptotic normality of the nonlinear least squares estimator for the structural parameters. We propose inference procedures for the structural parameters and uniform confidence bands for the equilibria. When equilibrium solutions are repeated, mixed convergence rates and non-standard limit distributions emerge. Monte Carlo simulations and an empirical application illustrate the finite-sample performance of our methods.

Paper Structure

This paper contains 18 sections, 15 theorems, 172 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Probabilistic properties of the dgp
  4. Estimation and inference
  5. Consistency of the NLS estimator
  6. Asymptotic normality of the NLS estimator
  7. Asymptotic distribution of the roots and the number of roots
  8. Inference
  9. Monte Carlo simulation
  10. Empirical application
  11. Conclusion
  12. Proofs
  13. Results of Section \ref{['sec:prop']}
  14. Results of Section \ref{['sec:cons']}
  15. Results of Section \ref{['sec:dis']}
  16. ...and 3 more sections

Key Result

Proposition 1

If Assumptions ass:density and ass:para hold, then the process $s_t$ is geometrically ergodic, with $\textnormal{E}[|r_t|]<\infty$ and $\textnormal{E}[\|z_t\|^2]<\infty$, where $z_t \coloneqq (\pi_t,\alpha_t,y_t)^{\textnormal{T}}$. Moreover, if, in addition, $\textnormal{E}[\|v_t\|^k] < \infty$, $v_

Figures (4)

  • Figure 1: The solid black line is the true $G(\beta)$ at $r_0 = 2$ with two roots indicated by the two grey vertical lines. The dashed and the dotted lines indicate a realized $G_n(\beta)$ conditioning on $r_n = 1$ and $r_n = 3$, respectively.
  • Figure 2: The solid and dashed lines indicate the true $G(\beta)$ for scenario (A) and (B), respectively. The $r_0 = 3$ and $r_0=2$ roots are indicate with grey vertical lines.
  • Figure 3: Percentage change of the CPI (upper panel) and, in the lower panel, the output gap and the percentage change of unit labour costs .
  • Figure 4: Plot of $\beta \mapsto G_n(\beta)$ together with 90% confidence bands based on $B=$ 4,999 bootstrap replications.

Theorems & Definitions (18)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Proposition 2
  • Lemma 2
  • Corollary 1
  • Lemma 3
  • Proposition 3
  • Corollary 2
  • ...and 8 more