Estimation and inference in models with multiple behavioural equilibria

This paper develops estimation and inference methods for a macroeconomic model with multiple behavioural equilibria and constant-gain learning, establishing the consistency and asymptotic normality of the nonlinear least squares estimator while addressing non-standard limit distributions that arise from repeated equilibrium solutions.

The Gist

Imagine the economy as a giant, chaotic dance floor. In the center, there are thousands of dancers (the "agents" or people making economic decisions). Traditionally, economists assumed these dancers were all geniuses who knew the exact choreography, the music, and the moves of everyone else perfectly. This is called Rational Expectations.

But in real life, people aren't supercomputers. They get tired, they forget, and they often guess based on what happened yesterday. This paper is about a new way to model that dance floor, where the dancers are learning as they go, and sometimes, the dance gets stuck in a weird loop.

Here is the breakdown of the paper in simple terms:

1. The Problem: The "Broken" Dance Floor

The authors are studying a specific type of economic model (the New Keynesian Phillips Curve) that tries to explain how inflation (rising prices) and the economy's "slack" (how busy or lazy the economy is) interact.

• The Old Way: Everyone knows the rules perfectly. There is only one "correct" way the dance can go.
• The New Way (This Paper): The dancers don't know the full rules. They only see the last few steps and try to guess the next one. They use a simple rule of thumb: "If prices went up last time, they'll probably go up a bit more this time."
• The Twist: Because everyone is guessing based on limited info, the dance floor can get stuck in multiple different patterns (equilibria).
  • Pattern A: Everyone expects low inflation, so prices stay low.
  • Pattern B: Everyone expects high inflation, so prices spiral up.
  • Both patterns can be "stable" even though they are very different. It's like a ball rolling on a landscape with two valleys; it can get stuck in either one depending on where you started.

2. The Challenge: Finding the Rules in the Chaos

The authors want to figure out the "structural parameters"—the hidden rules of the dance floor (like how much people react to news, or how fast they learn).

The problem is that because the dancers are learning and the system has multiple possible patterns, the math gets incredibly messy. It's like trying to figure out the rules of a game while the players are changing the rules as they play, and the game can end in three different ways.

Standard statistical tools (the usual math used by economists) often break down here because they assume there is only one "true" answer and that the system is calm. This system is neither.

3. The Solution: A New Toolkit

The authors developed a new set of mathematical tools to handle this mess. Think of it as a new pair of glasses that lets you see clearly through the fog of the learning process.

• Step 1: Proving the Dance Floor is Stable.
  First, they proved that even though the dancers are guessing, the system doesn't go crazy forever. It settles down into a predictable rhythm (mathematically called "geometric ergodicity"). It's like proving that no matter how wild the dance starts, the music eventually forces everyone into a steady beat.

• Step 2: The "Best Guess" Estimator.
  They created a method called Nonlinear Least Squares (NLS). Imagine you are trying to tune a radio to find the clearest station. You turn the dial (adjusting your guess of the rules) until the static (the error between your guess and reality) is as low as possible. They proved that if you keep turning the dial long enough, you will find the true rules of the dance.

• Step 3: Handling the "Double Roots" (The Tricky Part).
  Sometimes, the math hits a "bump" where two possible patterns merge into one. This is like a ball sitting exactly on the peak of a hill between two valleys.

  • In normal math, you can easily tell which way the ball will roll.
  • Here, the ball is so balanced that it might roll left, right, or stay put, and the math behaves strangely (it converges slower).
  • The authors figured out exactly how to measure the uncertainty in these tricky spots. They showed that while you can't be 100% sure how many patterns exist at a specific moment, you can draw a "confidence band" (a safety zone) that covers all the likely patterns.

4. The Real-World Test: US Inflation

To prove their tools work, they applied them to real US data from 1960 to 2019.

• The Result: When they used their new method on the "Output Gap" (how much the economy is underperforming), they found three possible stable patterns for inflation expectations.
  • One where expectations are very low (anchored).
  • One where they are moderate.
  • One where they are very high.
• The Insight: This explains why inflation sometimes feels stuck. The economy isn't just moving randomly; it might be jumping between these different "belief regimes." If people start believing inflation will be high, the economy actually becomes high-inflation, even if the underlying economic fundamentals haven't changed much.

The Big Picture Analogy

Imagine you are trying to predict the weather.

• Rational Expectations: You have a perfect satellite and know every physics equation. You know exactly what the weather will be.
• This Paper's Model: You are a farmer looking at the sky. You look at yesterday's clouds to guess today's rain.
  • Sometimes, if everyone looks at the clouds and thinks "It's going to rain," they all carry umbrellas, and the economy changes because of that belief.
  • The authors built a new calculator that can tell you: "Based on how farmers are looking at the sky, here are the three different weather patterns that could happen, and here is how likely each one is."

Why This Matters

This paper gives economists a way to stop pretending people are perfect geniuses. It acknowledges that we are all just learning as we go, and that our collective guesses can create different "realities" for the economy. By understanding these multiple realities, policymakers can better understand why inflation spikes or stalls and how to steer the economy out of a bad "belief loop."

Technical Summary

Here is a detailed technical summary of the paper "Estimation and inference in models with multiple behavioural equilibria" by Alexander Mayer and Davide Raggi.

1. Problem Statement

The paper addresses the econometric challenges associated with estimating and conducting inference in macroeconomic models featuring adaptive learning and multiple behavioural equilibria.

• Context: Traditional Rational Expectations (RE) models assume agents know the true model structure. However, empirical evidence suggests agents often form expectations using bounded rationality, updating beliefs via learning rules (e.g., constant-gain learning).
• The Specific Model: The authors focus on a New Keynesian Phillips Curve (NKPC) where agents incorrectly perceive inflation dynamics as a first-order autoregression (AR(1)) rather than the true structural model.
• The Core Challenge:
  1. Nonlinearity: The data generating process (DGP) is highly nonlinear because agents recursively estimate parameters that feed back into the actual law of motion (ALM).
  2. Multiple Equilibria: The interaction between learning and the structural model can yield multiple "behavioural equilibria" (fixed points of the perceived law of motion).
  3. Identification Issues: Standard asymptotic theory often breaks down when equilibria are repeated (non-unique roots) or when parameters (like the learning gain) are not jointly identified under the null hypothesis.
  4. Lack of Theory: Prior to this work, there was a lack of rigorous frequentist properties (consistency, asymptotic normality) for estimators in such nonlinear learning models with constant gains.

2. Methodology

The authors develop a comprehensive framework for estimation and inference using Nonlinear Least Squares (NLS).

A. Model Specification

• Structural Equation: $\pi_t = \delta \pi^e_{t+1} + \psi y_t + u_t$ (NKPC).
• Perceived Law of Motion (PLM): Agents believe $\pi_t = \alpha + \beta(\pi_{t-1} - \alpha) + e_t$.
• Learning Rule: Agents use a Constant-Gain Sample Autocorrelation (SAC) algorithm to update $(\alpha_t, \beta_t)$. The gain parameter $\gamma > 0$ ensures recent data is weighted more heavily, preventing convergence to a single point and inducing a stationary distribution for the recursive estimators.
• Actual Law of Motion (ALM): Derived by substituting the agents' forecasts into the NKPC, resulting in a nonlinear state-space system involving the state vector $s_t = (\pi_t, \alpha_t, y_t, \beta_t, r_t)$.

B. Probabilistic Properties (Ergodicity)

Before estimation, the authors establish the time-series properties of the DGP:

• Geometric Ergodicity: They prove the underlying Markov chain is geometrically ergodic. This is non-trivial because standard Lipschitz conditions (e.g., Tong, 1990) fail due to the high nonlinearity of the recursive autocorrelation estimator $\beta_t$.
• Method: They utilize a drift condition (Lyapunov-Foster) and show the system is a $\phi$-irreducible, aperiodic T-chain. This guarantees the existence of a unique stationary distribution and validates the use of Law of Large Numbers (LLN) and Central Limit Theorem (CLT) for sample averages.

C. Estimation Strategy

• Estimator: Nonlinear Least Squares (NLS) minimizes the sum of squared residuals between observed inflation and the model-implied inflation.
• Identification: They prove strong consistency of the NLS estimator $\hat{\theta}_n$ for the structural parameters $\theta = (\gamma, \delta, \psi)$.
  • Challenge: The population objective function is analytically intractable.
  • Solution: They adapt arguments from Francq et al. (2024), showing that the regression function is separated for different parameter values on tail events, ensuring identification without explicit evaluation of the population objective.

D. Asymptotic Theory

• Asymptotic Normality: Under standard regularity conditions (finite moments of errors), the NLS estimator is $\sqrt{n}$-consistent and asymptotically normal: $\sqrt{n}(\hat{\theta}_n - \theta_0) \xrightarrow{d} N(0, \Sigma)$.
• Boundary Cases: They address the case where $\delta = 0$ (learning gain $\gamma$ is not identified). Following Saikkonen (1995) and Seo (2011), they show the remaining structural parameters remain $\sqrt{n}$-consistent.
• Inference on Equilibria: The paper derives the asymptotic distribution of the estimated equilibria (roots of a polynomial function).
  • Simple Roots: Converge at rate $\sqrt{n}$.
  • Repeated Roots: When the equilibrium is a double root (tangency), the convergence rate slows to $n^{1/4}$, and the number of roots becomes a random variable (Bernoulli) asymptotically. This mirrors second-order identification mechanisms.

E. Inference Procedures

• Wald Tests: Standard tests for structural parameters using numerical derivatives to estimate the Hessian.
• Supremum Tests: For testing $\delta = 0$ (where $\gamma$ is a nuisance parameter), they propose a Sup-F test based on the profiled objective function, with critical values obtained via a Gaussian multiplier bootstrap.
• Uniform Confidence Bands: To handle the uncertainty around the location of equilibria (especially when roots are repeated), they construct uniform confidence bands for the function $G(\beta) = \beta - F(\beta; \lambda)$ using a multiplier bootstrap approach.

3. Key Contributions

1. Rigorous Ergodicity Proof: Provides the first proof of geometric ergodicity for a constant-gain learning NKPC model with recursive sample autocorrelation, overcoming the failure of standard Lipschitz conditions.
2. NLS Consistency & Normality: Establishes strong consistency and asymptotic normality for NLS estimators in this specific class of nonlinear learning models, filling a gap in the econometric literature.
3. Analysis of Multiple Equilibria:
   • Derives the non-standard asymptotic distribution for repeated roots (slower $n^{1/4}$ convergence).
   • Shows that the number of estimated equilibria does not necessarily converge to the true number when roots are repeated (it oscillates between 1 and 3).
   • Proposes a method to average split roots to recover consistency.
4. Practical Inference Tools: Develops feasible procedures (bootstrap-based) for hypothesis testing and constructing uniform confidence bands for the equilibrium function, which are robust to the non-standard behavior of repeated roots.

4. Results

Monte Carlo Simulations

• Scenario A (3 Equilibria): The NLS estimator is consistent, and the normal approximation for t-tests is accurate. The estimated number of roots converges to the true value (3) as sample size increases.
• Scenario B (2 Equilibria, one repeated):
  • The estimated number of roots oscillates between 1 and 3 with roughly equal probability (as predicted by theory).
  • The convergence of the repeated root estimate is significantly slower ($n^{1/4}$) compared to simple roots.
  • Averaging the two split estimates corresponding to the repeated root significantly improves performance.
  • Studentized confidence bands have coverage slightly below the nominal 95% level in small samples but improve with $n$.

Empirical Application (US Data, 1960–2019)

• Data: Quarterly US inflation, output gap, and unit labor costs.
• Findings:
  • Specification A (Output Gap): The model estimates three behavioural equilibria. Two are stable (low and high persistence), while the intermediate one is unstable. This suggests the US economy can switch between different inflation persistence regimes.
  • Specification B (Unit Labor Costs): The model estimates a unique equilibrium with low persistence.
  • Implication: The existence of multiple equilibria depends on the persistence of the driving variable ($\rho$). High persistence in the output gap supports multiple equilibria, whereas low persistence in labor costs supports a single equilibrium.
  • Learning Gain: Estimated gains imply agents weigh roughly the last 3–5 years of data, confirming the relevance of adaptive learning.

5. Significance

• Theoretical Advancement: The paper bridges the gap between the macroeconomic literature on adaptive learning and rigorous econometric theory. It provides the necessary mathematical justification for using frequentist methods in models previously dominated by Bayesian approaches or heuristic simulations.
• Policy Relevance: By characterizing the uncertainty around equilibrium locations and the possibility of multiple regimes, the methods allow policymakers to assess the risk of the economy shifting from a "well-anchored" low-inflation equilibrium to a high-persistence, unstable regime.
• Methodological Innovation: The treatment of repeated roots and the development of uniform confidence bands for equilibrium functions offer a template for analyzing other nonlinear economic models where identification is weak or second-order.

In summary, Mayer and Raggi provide a robust toolkit for estimating and testing New Keynesian models with adaptive learning, demonstrating that multiple behavioural equilibria are not just theoretical curiosities but empirically relevant features that can be rigorously identified and quantified.