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Least squares estimation in nonstationary nonlinear cohort panels with learning from experience

Alexander Mayer, Michael Massmann

Abstract

We discuss techniques of estimation and inference for nonstationary nonlinear cohort panels with learning from experience, showing, inter alia, the consistency and asymptotic normality of the nonlinear least squares estimator used in empirical practice. Potential pitfalls for hypothesis testing are identified and solutions proposed. Monte Carlo simulations verify the properties of the estimator and corresponding test statistics in finite samples, while an application to a panel of survey expectations demonstrates the usefulness of the theory developed.

Least squares estimation in nonstationary nonlinear cohort panels with learning from experience

Abstract

We discuss techniques of estimation and inference for nonstationary nonlinear cohort panels with learning from experience, showing, inter alia, the consistency and asymptotic normality of the nonlinear least squares estimator used in empirical practice. Potential pitfalls for hypothesis testing are identified and solutions proposed. Monte Carlo simulations verify the properties of the estimator and corresponding test statistics in finite samples, while an application to a panel of survey expectations demonstrates the usefulness of the theory developed.
Paper Structure (18 sections, 12 theorems, 177 equations, 2 figures, 4 tables)

This paper contains 18 sections, 12 theorems, 177 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Estimation
  4. Assumptions, consistency, and asymptotic normality
  5. Assumptions
  6. Consistency
  7. Asymptotic normality
  8. Standard errors and inference
  9. Monte Carlo simulation
  10. Empirical application
  11. Concluding remarks
  12. Proofs
  13. Auxiliary lemmata
  14. Proof of the main results
  15. Proofs of auxiliary results
  16. ...and 3 more sections

Key Result

Proposition 1

. If, in addition, $\beta_0 \neq 0$, then $\normalfont\theta \mapsto \textsf{D}_{m}(\theta)$ and $\normalfont\theta \mapsto \textsf{D}(\theta)$ are uniquely minimised at $\theta = \theta_0$. Consequently, $\theta_n \rightarrow_p \theta_0$.

Figures (2)

  • Figure 1: Theoretical asymptotic local power of the $t$-statistic for $H_0$: $\gamma = \gamma_0$ (solid line) and for $H_0$: $\beta = \beta_0$ (dashed line) as a function of $\gamma \in (1/2,3]$ for $\beta = \sigma = \omega = \lambda = \Delta_\beta = \Delta_\gamma =1$. The dotted horizontal line indicates the five per cent significance level.
  • Figure 2: Four-quarter moving-averages of inflation expectations by age group relative to the cross-sectional mean using quarterly MSC data from 1978Q1 to 2023Q3.

Theorems & Definitions (14)

  • Proposition 1
  • Example 1
  • Corollary 1
  • Proposition 2
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Lemma A.0
  • Remark A.0
  • ...and 4 more