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Convergence of Least-Square Learning Mechanisms in Self-Referential Linear Stochastic Models
Abstract
We study a class of models in which the law of motion perceived by agents influences the law of motion that they actually face. We assume that agents update their perceived law of motion by least squares. We show how the perceived law of motion and the actual one may converge to one another, depending on the behavior of a particular ordinary differential equation. The differential equation involves the operator that maps the perceived law of motion into the actual one.
... That is, instead of imposing -as the rational expectations assumption does -that agents know the correct (and extremely complex) transition probabilities of equilibrium prices, we instead impose that agents learn about these transition probabilities over time. This approach has a long tradition in the economics literature, typically in the form of "least-squares learning" (Bray, 1982;Marcet and Sargent, 1989;Evans and Honkapohja, 2001), and has recently been applied in the MFG literature to the case without common noise (Laurière et al., 2022(Laurière et al., , 2024Xu et al., 2023;Bertucci, 2023), mostly in the form of reinforcement learning. ...
... More specifically, this is a special form of the future prices predictor Θ(s, t; p ≤t ) that appears in the system (6.2). One simple version of adaptive learning is least-squares learning (Bray, 1982;Marcet and Sargent, 1989;Evans and Honkapohja, 2001). Jacobson (2025) implements such an approach in a heterogeneous-agent model. ...
... Finally, we show how to write the adaptive learning model of Section 7 in discrete time. The economics literature typically formulates such models in discrete time (Bray, 1982;Marcet and Sargent, 1989;Evans and Honkapohja, 2001;Jacobson, 2025). The key assumption is that, to forecast prices, agents use a perceived law of motion in the form of a Markov process: p s+1,t ∼ q p (·| p s,t , Z s , θ), s ≥ t, p t,t = p t , (8.18) where θ ∈ R d is a parameter vector. ...
Mean Field Game (MFG) models implicitly assume "rational expectations", meaning that the heterogeneous agents being modeled correctly know all relevant transition probabilities for the complex system they inhabit. When there is common noise, this assumption results in the "Master equation" (a.k.a. "Monster equation"), a Hamilton-Jacobi-Bellman equation in which the infinite-dimensional density of agents is a state variable. The rational expectations assumption and the implication that agents solve Master equations is unrealistic in many applications. We show how to instead formulate MFGs with non-rational expectations. Departing from rational expectations is particularly relevant in "MFGs with a low-dimensional coupling", i.e. MFGs in which agents' running reward function depends on the density only through low-dimensional functionals of this density. This happens, for example, in most macroeconomics MFGs in which these low-dimensional functionals have the interpretation of "equilibrium prices." In MFGs with a low-dimensional coupling, departing from rational expectations allows for completely sidestepping the Master equation and for instead solving much simpler finite-dimensional HJB equations. We introduce an adaptive learning model as a particular example of non-rational expectations and discuss its properties.
... This inconsistency was relatively innocuous in the context of the early learning literature, which focused on equilibrium selection and asymptotic behaviors of learning economies. 4 However, it could represent a more serious hurdle in recent works aiming to explain the business cycle with perpetual learning dynamics. ...
... (24) solves for the SL equilibrium: α = a, (Bcρ + χ) = c and Bd + B = d. The solutions a =ā = 0 and c =c are the same as under RE given in Eq. (4). SL introduces an additional term pertaining to the effect of the average deviation from RE d = B(I − B) −1 . ...
... See, in particular, the adaptive learning literature(Evans & Honkapohja 2001). Heuristic-switching models are another example of this temporary equilibrium approach(Hommes 2021).4 See, inter alia,Evans (1985),Marcet & Sargent (1989),Evans & Honkapohja (1995),Arifovic et al. (2013). ...
Social learning (SL) is a behavioral model in which expectations and the resulting aggregate dynamics stem from the interactions of a large number of heterogeneous agents. Nonetheless, this framework has so far lacked a parsimonious development with a general-solution method. This paper bridges this gap and introduces a Dynare toolbox to solve any linear state-space model with SL expectations, opening up a wide range of potential applications. As an illustration, optimal monetary policy rules are studied in a microfounded New Keynesian (NK) model under SL and rational expectations (RE).
... According to such explanations, conditional on the same information set, economic agents make homogeneous predictions about the reaction of the economy to shocks. Alternatively, disagreement could be due to heterogeneity in subjective models, that is, the way agents think about the functioning of the economy (Bray and Savin, 1986;Marcet and Sargent, 1989;Molavi, 2019;Angeletos, Huo and Sastry, 2020). Such heterogeneity generates disagreement in expectations even when all agents observe the same shock and have the same information about previous realizations of macroeconomic variables. ...
... In contrast to this view, we document strong heterogeneity in unemployment and inflation forecasts even in a setting where all individuals observe the same shock and hold similar information about current realizations of macroeconomic variables. This finding is more in line with the alternative view that dispersion in expectations is (partially) due to individuals relying on different subjective models of the economy (Bray and Savin, 1986;Marcet and Sargent, 1989;Andrade et al., 2016;Molavi, 2019;Angeletos et al., 2020). Accordingly, economic agents evaluate the same news about the economy through the lens of their own model. ...
... Can existing theories featuring disagreement about the model of the economy explain our findings? For instance, in theories of learning and model misspecification, agents may disagree about structural parameters of the economy, such as the persistence of inflation (Bray and Savin, 1986;Marcet and Sargent, 1989;Orphanides and Williams, 2005;Milani, 2007;Evans and Honkapohja, 2012;Bhandari, Borovicka and Ho, 2019;Molavi, 2019;Angeletos et al., 2020). In models of learning from experience (Malmendier and Nagel, 2016), individuals only use realizations of macroeconomic variables observed during their lifetimes to estimate the data-generating process, leading to disagreement in inflation expectations across cohorts even if everyone observes the same current realization. ...
We study people’s subjective models of the macroeconomy and shed light on their attentional foundations. To do so, we measure beliefs about the effects of macroeconomic shocks on unemployment and inflation, providing respondents with identical information about the parameters of the shocks and previous realizations of macroeconomic variables. Within samples of 6,500 US households and 1,500 experts, beliefs are widely dispersed, even about the directional effects of shocks, and there are large differences in average beliefs between households and experts. Part of this disagreement seems to arise because respondents think of different propagation channels of the shocks, in particular demand- versus supply-side mechanisms. We provide evidence on the role of associative memory in driving heterogeneity in thoughts and forecasts: Contextual cues and prior experiences shape which propagation channels individuals retrieve and thereby which forecasts they make. Our findings offer a new perspective on the widely documented disagreement in macroeconomic expectations.
... First, our use of micro-data to study demographic differences in inflation expectation is similar to Madeira and Zafar (2015) who used the University of Michigan Survey of Consumers and find demographic differences in inflation expectations, namely larger heterogeneity of expectations and slower updating of expectations for those with less education, as well as for women and ethnic minorities. Our work is also related to a large literature that investigates subjective inflation expectations formed on the basis of agents' life-time experiences (Malmendier and Nagel, 2016). 1 Second, our empirical model of adaptive learning follows the methodologies of Branch and Evans (2006) and Weber (2010), and is in the spirit of the seminal contributions of Evans and Honkapohja (2001) and Marcet and Sargent (1989). Finally, the DSGE model with adaptive learning updates the work of Slobodyan and Wouters (2012a), whose model is based on the DSGE model of the U.S. in Smets and Wouters (2007). ...
... As shown by Marcet and Sargent (1989) the learning process converges to equilibrium only when the law of motion of the parameters is time invariant. 4 In other words, convergence requires Q t = 0. Within the Kalman filter framework it is hence possible to test whether learning is perpetual or whether it converges to equilibrium by examining whether the variance of the state variables is significantly different from zero. ...
... It has been shown by Marcet and Sargent (1989) that convergence of RLS learning can be achieved with probability one if the algorithm is augmented by a projection facility. This variation of the algorithm can be generally written aŝ ...
... It can also be shown that, if the parameters of the model are such that ja 1 + a 2 j < 1, the unique stationary REE solution is always E-stable, and the RLS algorithm always converges (see Evans and Honkapohja, 2001). 13 For the PFL model, the E-stability condition is a 1 < 1. Note that for many well known models, parameter restrictions imply that this is always true (e.g. for the Lucas tree model, a 1 = 2 (0; 1) and thus a 1 is always less than one). ...
We analyze practical aspects of implementing adaptive learning in the context of forward looking linear models. We focus on how to set initial conditions for three popular algorithms, namely recursive least squares, stochastic gradient and constant gain learning. We propose three ways of initializing, one that uses randomly generated data, one that is ad hoc and one that uses an appropriate distribution. We illustrate via standard examples, that the behavior of macroeconomic variables not only depends on the learning algorithm, but on the initial conditions as well. Furthermore, we provide a computing toolbox for analyzing the quantitative properties of dynamic stochastic macroeconomic models under adaptive learning. r
... Our level-0 learning is based on the AL literature developed in Bray and Savin (1986), Marcet and Sargent (1989), Evans (1989), and Evans and Honkapohja (2001). AL is a versatile technique that has been applied in both nonexperimental and experimental settings. ...
We propose a model of boundedly rational and heterogeneous expectations that unifies adaptive learning, k-level reasoning, and replicator dynamics. Level-0 forecasts evolve over time via adaptive learning. Agents revise over time their depth of reasoning in response to forecast errors, observed and counterfactual. The unified model makes sharp predictions for when and how quickly markets converge in Learning-to-Forecast Experiments, including novel predictions for individual and market behavior in response to announced events. We present experimental results that support these predictions. We apply our unified approach in the New Keynesian model to study forward guidance policy. (JEL D83, D84, E12, E31, E32, E71)
... (2021), Gagnon-Bartsch and Bushong (2022) and Danenberg and Fudenberg (2024) for some other recent applications of stochastic approximation in economics and Marcet and Sargent (1989) for an earlier application. Frick et al. (2023) propose a general model of Bayesian learning, where the true and perceived data generating processes p * q ∈ ∆(X) and p θ,q ∈ ∆(X), θ ∈ Θ, may depend on the current belief q ∈ ∆(Θ). ...
Deviations from Bayesian updating are traditionally categorized as biases, errors, or fallacies, thus implying their inherent ``sub-optimality.'' We offer a more nuanced view. We demonstrate that, in learning problems with misspecified models, non-Bayesian updating can outperform Bayesian updating.
... (3) extends the classical least-squares recursion of, e.g., Marcet and Sargent (1989) by making the updating weight a function of age t − s rather than merely time t. ...
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 esti-mator and corresponding test statistics in finite samples, while an application to a panel of survey expectations demonstrates the usefulness of the theory developed.
... Both aspects of the filtering method are mainly treated by Sargent (1987) and Hamilton (1994), respectively. Some other, similar types of perceptions have been discussed in Marcet and Sargent (1989a), Marcet and Sargent (1989b), Hansen and Sargent (2007), and Hansen, Polson, and Sargent (2010). Rather than considering a specific economic or econometric model, this paper characterizes general perceptions that are concealed in abstract models where both the active and passive arguments can be integrated. ...
Filtering has had a profound impact as a device of perceiving information and deriving agent expectations in dynamic economic models. For an abstract economic system, this paper shows that the foundation of applying the filtering method corresponds to the existence of a conditional expectation as an equilibrium process. Agent-based rational behavior of looking backward and looking forward is generalized to a conditional expectation process where the economic system is approximated by a class of models, which can be represented and estimated without information loss. The proposed framework elucidates the range of applications of a general filtering device and is not limited to a particular model class such as rational expectations.
... Regarding model misspecification, our approach is thus akin to the literature on learning, Bray (1982), Marcet and Sargent (1989), and Sargent (1993), where agents use a misspecified least squares approach to infer unknown model parameters. Rothschild (1974) and ...
We study a large economy in which firms cannot compute exact solutions to the non-linear equations that characterize the equilibrium price at which they can sell future output. Instead, firms use polynomial expansions to approximate prices. The precision with which they can compute prices is endogenous and depends on the overall level of supply. At the same time, firms' individual supplies, and thus aggregate supply, depend on the precision with which they approximate prices. This interrelation between supply and price forecast induces multiple equilibria, with inefficiently low output, in economies that otherwise have a unique, efficient equilibrium. Moreover, exogenous parameter changes, which would increase output were there no computational frictions, can diminish agents' ability to approximate future prices, and reduce output. Our model therefore accommodates the intuition that interventions, such as unprecedented quantitative easing, can put agents into "uncharted territory".
... Least Squares Learning. Under least squares learning (Marcet and Sargent (1989), Evans and Honkapohja (2001)), agents update their beliefs about the laws of motion of endogenous variables via recursive least squares. For simplicity, we assume agents can directly observe the monetary shock ζ t . ...
Central banks wish to avoid self‐fulfilling fluctuations. Interest rate rules with a unit response to real rates achieve this under the weakest possible assumptions about the behavior of households and firms. They are robust to household heterogeneity, hand‐to‐mouth consumers, non‐rational household or firm expectations, active fiscal policy, and to any form of intertemporal or nominal‐real links. They are easy to employ in practice, using inflation‐protected bonds to infer real rates. With a time‐varying short‐term inflation target, they can implement an arbitrary inflation path, including optimal policy. This provides a way to translate policy makers' desired path for inflation into one for nominal rates. U.S. Federal Reserve behavior is remarkably close to that predicted by a real rate rule, given the desired inflation path of U.S. monetary policy makers. Real rate rules work thanks to the key role played by the Fisher equation in monetary transmission.
... A common feature in these studies is that they document the shortcomings of REE models along the expectations dimension and argue for the usefulness of incorporating data from survey expectations into these models. Much of the literature on adaptive learning focuses on dynamics under MSVlearning of a correctly specified model (see, e.g., Marcet and Sargent (1989), Evans andHonkapohja (2001), Milani (2007)) and studies conditions under which the learning process converges on the underlying REE. Orphanides and Williams (2004) study monetary policy under MSV-learning and find that optimal policy is typically more aggressive to inflation under learning. ...
We introduce Behavioral Learning Equilibria (BLE) into a multivariate linear framework and apply it to New Keynesian DSGE models. In a BLE, boundedly rational agents use simple, but optimal AR(1) forecasting rules whose parameters are consistent with the observed sample mean and autocorrelation of past data. We study the BLE concept in a standard 3‐equation New Keynesian model and develop an estimation methodology for the canonical Smets and Wouters (2007) model. A horse race between Rational Expectations (REE), BLE, and constant gain learning models shows that the BLE model outperforms the REE benchmark and is competitive with constant gain learning models in terms of in‐sample and out‐of‐sample fitness. Sample‐autocorrelation learning of optimal AR(1) beliefs provides the best fit when short‐term survey data on inflation expectations are taken into account in the estimation. As a policy application, we show that optimal Taylor rules under AR(1) expectations inherit history dependence and require a lower degrees of interest rate smoothing than REE.
... This mechanism has been previously explored in Sethi (1996), and is formally equivalent to models of learning in which individuals adopt forecasting rules that have proved to be lucrative in the past (Brock and Hommes, 1997). This may be contrasted with learning from data, by comparing expectations to realizations, as in the literature descended from Blume and Easley (1982), Marcet and Sargent (1989) and Evans and Honkapohja (2001). 1 In our model such learning from data would lead agents to adopt beliefs that are more accurate, but that erode their wealth in the long run. ...
The scale and terms of aggregate borrowing in an economy depend on the manner in which wealth is distributed across potential creditors with heterogeneous beliefs about the future. This distribution evolves over time as uncertainty is resolved, in favour of optimists if loans are repaid in full, and in favour of pessimists if there is widespread default. We model this process in an economy with two assets - risky bonds and risk-free cash. Within periods, given the inherited distribution of wealth across belief types, the scale and terms of borrowing are endogenously determined. Following good states, aggregate borrowing and the face value of debt both rise, and the interest rate falls. In the absence of noise, wealth converges to beliefs that differ systematically from the objective probability governing state realisations, with greater risk-aversion associated with greater optimism. In the presence of noise, the economy exhibits periods of high performance, punctuated by periods of crisis and stagnation.
... How does experience-based learning work? The authors builds on adaptive learning by assuming that individuals form expectations based on historical data (Marcet and Sargent, 1989;Evans and Honkapohja, 2001). Individuals try to estimate an AR(1) process as the perceived law of motion (e.g., Orphanides and Williams (2004)) recursively from past data. ...
As a schematic model of the complexity economic agents are confronted with, we introduce the “Sherrington-Kirkpatrick game,” a discrete time binary choice model inspired from mean-field spin glasses. We show that, even in a completely static environment, agents are unable to learn collectively optimal strategies. This is either because the learning process gets trapped in a suboptimal fixed point or because learning never converges and leads to a never-ending evolution of agent intentions. Contrarily to the hope that learning might save the standard “rational expectation” framework in economics, we argue that complex situations are generically and agents must do with solutions, as argued long ago by Simon []. Only a centralized, omniscient agent endowed with enormous computing power could qualify to determine the optimal strategy of all agents. Using a mix of analytical arguments and numerical simulations, we find that (i) long memory of past rewards is beneficial to learning, whereas overreaction to recent past is detrimental and leads to cycles or chaos; (ii) increased competition (nonreciprocity) destabilizes fixed points and leads first to chaos and, in the high competition limit, to quasicycles; (iii) some amount of randomness in the learning process, perhaps paradoxically, allows the system to reach better collective decisions; (iv) nonstationary, “aging” behavior spontaneously emerges in a large swath of parameter space of our complex but static world. On the positive side, we find that the learning process allows cooperative systems to coordinate around satisficing solutions with rather high (but markedly suboptimal) average reward. However, hypersensitivity to the game parameters makes it impossible to predict who will be better or worse off in our stylized economy. The statistical description of the space of satisficing solutions is an open problem.
Published by the American Physical Society 2024
This paper evaluates how initial beliefs uncertainty can affect data weighting and the estimation of models with adaptive learning. One key finding is that misspecification of initial beliefs uncertainty, particularly with the common approach of artificially inflating initials uncertainty to accelerate convergence of estimates, generates time-varying profiles of weights given to past observations in what should otherwise follow a fixed profile of decaying weights. The effect of this misspecification, denoted as diffuse initials, is shown to distort the estimation and interpretation of learning in finite samples. Simulations of a forward-looking Phillips curve model indicate that (i) diffuse initials lead to downward biased estimates of expectations relevance in the determination of actual inflation, and (ii) these biases spill over to estimates of inflation responsiveness to output gaps. An empirical application with U.S. data shows the relevance of these effects for the determination of expectational stability over decadal subsamples of data. The use of diffuse initials is also found to lead to downward biased estimates of learning gains, both estimated from an aggregate representative model and estimated to match individual expectations from survey expectations data.
In this article, we describe the research area of evolving systems starting with a brief history, the basic concepts, and definitions and moving to classes of evolving systems. We further provide illustrative examples of evolving systems to various problems. The research area of evolving systems is closely related to adaptive systems, machine learning, mathematical modeling, data science, and computer systems. Some systems with such properties emerged at the end of last century, but the area was formed and recognized in the current century. The importance of this research – which is now a recognized niche topic with its own scientific journal ( Evolving Systems ), annual IEEE conferences, technical committees, etc. – is even clearer now when the era of Big Data, data streams, machine learning, and intelligence is a topic not only of research but also of publicity. The importance of evolving systems was stressed at the very beginning of its formation as a subdiscipline because a true intelligence can only be evolving. Indeed, the majority of the real processes and phenomena that people are interested in, such as modeling, predicting, classifying, controlling, or simply monitoring, are nonstationary; they are complex, nonlinear, and dynamically evolving. This includes climate, human behavior, social and biomedical systems, and even contemporary technical systems. The reality in which we are living now, couple of decades after the appearance of evolving systems, is characterized not only by a huge volume of exponentially growing data but also by their dynamically evolving nature, heterogeneous form, uncertainty, and often lack of structure. The classical theories and practical toolset (techniques, algorithms, and methods) have their own limitations and often rely on unrealistic assumptions. The concept of evolving systems was revolutionary and broke a number of these assumptions bringing the results closer to the reality of the problems, processes, and phenomena we study. One such key restrictive assumption usually made is about the fixed structure of the model of the system that describes processes or phenomena we are interested in. This is closely linked with the millennia old principle of approaching complex issues – “divide et Impera.” Adding to it “evolve” is vital.
This paper studies how adaptive learning affects the interactions between monetary policy, stock prices and the optimal degree of Rogoff conservatism in a New Keynesian DSGE model with non-Ricardian agents whose heterogeneous portfolios generate a financial wealth channel of monetary transmission. A positive intergenerational portfolio turnover in the stock market could improve the dual-mandate central bank’s intertemporal trade-off allowed by learning and implies a positive weight on financial stability in the social welfare criterion. For plausible learning gains and turnover rates, a rise in the turnover rate could lead the central bank to reduce the aggressiveness in its policy needed to manage private beliefs if the learning gain is high enough. Both the distortion due to learning and the central bank’s lack of concern for financial stability could lead the government to appoint a liberal central banker, i.e. less conservative than society. A rise in the turnover rate implies a higher (lower) degree of liberalism for low (high) learning gains.
Inflation expectations are central to economics because they affect the effectiveness of fiscal and monetary policy as well as realized inflation. We survey the recent literature with a focus on the inflation expectations of households. We first review standard data sources and discuss their advantages and disadvantages. We then document that household inflation expectations are biased upwards, dispersed across individuals, and volatile in the time series. We also provide evidence of systematic differences by gender, income, education, and race. Turning to the underlying expectations formation process, we highlight the role of individuals' exposure to price signals in their daily lives, such as price changes in groceries, the role of lifetime experiences, and the role of cognition. We then discuss the literature that links inflation expectations to economic decisions at the individual level, including consumption-savings and financial decisions. We conclude with an outlook for future research.
This paper studies the effects of a temporary increase in the inflation target in a model with adaptive learning and imperfect credibility of the central bank. In such a setup the private sector has to learn the model governing the economy and assess if the change in the interest rate is due to a standard interest rate shock or to a persistent shock to the inflation target. We find that the analytical condition that guarantees asymptotical E–stability under full credibility is the same that we obtain for the calibrated version of our model under imperfect credibility. Under full credibility the transition to the long-run equilibrium is faster and the variance of inflation is lower. Our results are consistent with the view that increasing the inflation target can be used as a policy instrument without unanchoring expectations.
This article proposes an agent-based policy simulation framework that can be applied to the cases satisfying: 1) the agents try to maximize some intertemporal preference and 2) the impacts of different factors on agents’ behavioral tendency are monotonic. By combining the simulation and optimization methods, this framework balances the flexibility and validity of agent-based models (ABMs): the sigmoid function is modified and used to model agents’ decision-making rules, and the evolutionary training method is used to calibrate agents’ behavioral parameters. Based on an example for the emission trading scheme, the application of the framework is presented and evaluated in detail.
In a broad class of non-linear representative agent models, represented by a system of difference equations, we replace rational expectations with linear forecast models conditioning on a predetermined set of regressors. Within this framework, a restricted perceptions equilibrium (RPE) corresponds to a forecast rule that is optimal within that class of models. Local uniqueness of a stationary rational expectations equilibrium (REE) near the non-stochastic steady state is shown to guarantee the existence, uniqueness and E-stability of an RPE local to that steady state. A benchmark RBC model with government spending shocks illustrates the theoretical results.
We draw from research that finds local weather conditions affect individuals’ belief-formation process and asset prices to investigate whether “extreme” local temperatures impact individuals’ beliefs about U.S. economic conditions (i.e., economic sentiment) and the stock returns of local firms. We combine Gallup’s U.S. Daily Poll, which provides the daily-level economic sentiment of a population-representative random sample of 1.5 million individuals, with daily weather conditions based on survey respondent location. We document that extreme local temperatures decrease individuals’ sentiment about the U.S. economy and that this decrease relates to declines in the stock returns of local firms. Further tests distinguish this extreme local temperature-sentiment effect from the effect of perceived life satisfaction on individuals’ economic sentiment, suggesting that the potential effect of extreme temperatures on individuals’ moods is not driving our results. We conclude that extreme local temperatures affect individuals’ sentiment about economic activity beyond the potential effect of temperature on firm- or local-level economic variables, with implications for stock returns.
This paper introduces new economic dynamics induced by learning. Under monetary policy with regime switches, private sector agents learn about output and inflation, which are functions of their beliefs about those variables. The learners’ models are misspecified because they don’t observe the regimes. The novel dynamics occur if the regime switches are slow relative to the learning speed: Learners’ beliefs do not converge to the single process that usually describes their asymptotic beliefs, but to a Markov switching process. Checking for this possibility is crucial: When it occurs, the standard analysis gives incorrect results about the equilibria to which beliefs and other endogenous variables will converge. In an extension in which the learners choose their learning speed optimally, the result is strengthened.
We discuss the impact of shocks from a sudden pandemic in a recessionary economy. The framework is one of imperfect knowledge with a standard New Keynesian model. Global aspects of dynamics with possible interest-rate lower bound are discussed.KeywordsStagnationExpectationsProductivity shocksAdaptive learningNew Keynesian modelJEL classification:E62E63E52D84E71
"Nowhere does history indulge in repetitions so often or so uniformly as in Wall Street," observed legendary speculator Jesse Livermore. History tells us that periods of major technological innovation are typically accompanied by speculative bubbles as economic agents overreact to genuine advancements in productivity. Excessive run-ups in asset prices can have important consequences for the economy as firms and investors respond to the price signals, resulting in capital misallocation. On the one hand, speculation can magnify the volatility of economic and financial variables, thus harming the welfare of those who are averse to uncertainty and fluctuations. But on the other hand, speculation can increase investment in risky ventures, thus yielding benefits to a society that suffers from an underinvestment problem.
Recursive algorithms where random observations enter are studied in a fairly general framework. An important feature is that the observations my depend on previous "outputs" of the algorithm. The considered class of algorithms contains, e.g., stochastic approximation algorithm, recursive identification algorithm, and algorithms for adaptive control of linear systems. It is shown how a deterministic differential equation can be associated with the algorithm. Problems like convergence with probability one, possible convergence points and asymptotic behavior of the algorithm can all be studied in terms of this differential equation. Theorems stating the precise relationships between the differential equation and the algorithm are given as well as examples of applications of the results to problems in identification and adaptive control.
Growing out of a conference on Expectations Formation and Economic Disequilibrium held in New York City in 1981, the papers in this volume provide a complex view of market processes in which individual rationality is no guarantee of convergence to the 'correct' model and the equilibrium coordination of agents' plans. They reject the 'optimality' argument for the rational expectations hypothesis, opening the door to other hypotheses of optimal expectations of agents in the decentralized market economy.
Much recent work in the economics of information has stemmed from the observation that demand functions, and therefore prices, reflect agents’ private information. Given an adequate understanding of the relationship between private information and equilibrium prices, it is possible to infer some or all of the private information. It seems natural to suppose, therefore, that agents will endeavour to use the information contained in equilibrium prices.
Let M(x) denote the expected value at level x of the response to a certain experiment. M(x) is assumed to be a monotone function of x but is unknown to the experimenter, and it is desired to find the solution of the equation , where is a given constant. We give a method for making successive experiments at levels in such a way that will tend to in probability.
The stability of the rational expectations equilibrium of a simple asset market model is studied in a situation where a group of traders learn about the relationship between the price and return on the asset using ordinary least squares estimation, and then use their estimates in predicting the return from the price. The model which they estimate is a well-specified model of the rational expectations equilibrium, but a misspecified model of the situation in which the traders are learning. It is shown that for appropriate values of a stability parameter the situation converges almost surely to the rational expectations equilibrium.
The authors study the convergence of recursive least-squares learning schemes in economic environments in which there is private information. The presence of private information leads to the presence of hidden state variables from the viewpoint of particular agents. By applying theorems of Lennart Ljung, the authors extend some of their earlier results to characterize conditions under which a system governed by least-squares learning will eventually converge to a rational expectations equilibrium. They apply insights from the learning results to formulate and compute the equilibrium of a version of Robert Townsend's model. Copyright 1989 by University of Chicago Press.
Parallel computation has a long history in econometric computing, but is not at all wide spread. We believe that a major impediment is the labour cost of coding for parallel architectures. Moreover, programs for specific hardware often become obsolete quite quickly. Our approach is to take a popular matrix programming language (Ox), and implement a message-passing interface using MPI. Next, object-oriented programming allows us to hide the specific parallelization code, so that a program does not need to be rewritten when it is ported from the desktop to a distributed network of computers. Our focus is on so-called embarrassingly parallel computations, and we address the issue of parallel random number generation.
Linear models involving expectations of future endogenous variables generally have multiple rational expectations equilibria.
This paper investigates the stability of solutions in the disequilibrium sense of whether, given a small deviation of expectations
functions from some rational expectations equilibrium, the system returns to that solution under a natural revision rule.
Weak and strong local stability are distinguished. Stability conditions are calculated for a simple general linear model and
applied to two macroeconomic examples. In some cases there is a unique stable equilibrium. In other cases a continuum of equilibria
forms a weakly but not strongly stable class.
An adaptive learning rule is exhibited for the Azariadis (1981) overlapping generations model of a monetary economy with multiple equilibria, under which the economy may converge to a stationary sunspot equilibrium, even if agents do not initially believe that outcomes are significantly different in different "sunspot" states. The learning rule studied is of the "stochastic approximation" form studied by H. Robbins and S. Monro (1951); methods for analyzing the convergence of this form of algorithm are presented that may be of use in many other contexts as well. Conditions are given under which convergence to a sunspot equilibrium occurs with probability one. Copyright 1990 by The Econometric Society.
This paper investigates whether agents can learn how to form rational expectations using standard econometric techniques in the case of a linear stochastic supply and demand model with a production lag. This model has a unique rational expectations equilibrium in which the expected price is a linear function of an observable exogenous random variable. Outside of rational expectations equilibrium agents predict the price by using a regression of past prices on the exogenous random variable where the regression is estimated by either ordinary least squares or Bayesian methods. If the agents are Bayesians, they may have diverse prior beliefs on the mean of the estimated parameter, but all have the same precision. This estimation procedure would be appropriate for an outside observer estimating the parameters of the model in rational expectations equilibrium the coefficient of the equation relating the mathematical conditional expectation of the price to the exogenous variable is constant through time. Outside rational expectations equilibrium this coefficient, which changes each time new data change the regression coefficient. The data are generated by a time-varying parameter model where the varying parameter is determined by past data and the estimation procedure. Agents fail to take this feedback into account and so are estimating a misspecific model.
This paper studies estimators that make sample analogues of population orthogonality conditions close to zero. Strong consistency and asymptotic normality of such estimators is established under the assumption that the observable variables are stationary and ergodic. Since many linear and nonlinear econometric estimators reside within the class of estimators studied in this paper, a convenient summary of the large sample properties of these estimators, including some whose large sample properties have not heretofore been discussed, is provided.
Macroeconomic models with rational expectations find a new justification if these models appear as limits of some learning procedures. In this paper we consider the case in which, during the learning period, the predictions are obtained by regression. We exhibit the necessary and sufficient condition on the parameter of the model ensuring the convergence of the learning process. The limit is the solution of a rational expectations model in which the information set only includes the exogenous variables used in the auxiliary regression.
"Nowhere does history indulge in repetitions so often or so uniformly as in Wall Street," observed legendary speculator Jesse Livermore. History tells us that periods of major technological innovation are typically accompanied by speculative bubbles as economic agents overreact to genuine advancements in productivity. Excessive run-ups in asset prices can have important consequences for the economy as firms and investors respond to the price signals, resulting in capital misallocation. On the one hand, speculation can magnify the volatility of economic and financial variables, thus harming the welfare of those who are averse to uncertainty and fluctuations. But on the other hand, speculation can increase investment in risky ventures, thus yielding benefits to a society that suffers from an underinvestment problem.
The statistical properties of the certainty equivalence control rule and of the least squares estimates generated by this rule are examined experimentally in a linear model with two unknown parameters. It is found that the least squares certainty equivalence rule converges to its true value with probability one and is asymptotically efficient, having an asymptotic distribution with a variance as small as any other strongly consistent rule. However, while a linear combination of the parameter estimates is consistent, the evidence does not confirm that the individual estimates themselves are consistent. If these converge to their true values at all, they do so very slowly (on the order of (log t)^-^1).
This paper determines the time series behavior of investment, output, and prices in a competitive industry with a stochastic demand. It is shown, first, that the equilibrium development for the industry solves a particular dynamic programming problem (maximization of "consumer surplus"). This problem is then studied to determine the characteristics of the equilibrium paths.
We will first check the set of conditions B on page 554 of Ljung c191
- D Satisfied
- D Co
- Proof
is satisfied and D, c D,, then p, --f ar almost surely as t + CO. Proof. We will first check the set of conditions B on page 554 of Ljung c191. Assumptions B.l and B.2 in Ljung are implied by our (A.5). B.3, B.4, and B.5 in Ljung are implied by the smoothness assumptions on T, A, B, V in our (A.2).
6 in Ljung, we see that the following limits exist and are given by Finally
- B For
For B.6 in Ljung, we see that the following limits exist and are given by Finally, assumptions B.8 to B.ll are implied by our (A.4).
In this section, we briefly describe how the apparatus of this paper would apply in such setups. To accommodate contemporary information, we replace (4a) by 8:=P:~1+(cc
- Fourgeaud
- Gourieroux
and Fourgeaud, Gourieroux, and Pradel [ 111). In this section, we briefly describe how the apparatus of this paper would apply in such setups. To accommodate contemporary information, we replace (4a) by 8:=P:~1+(cc,/t)R~1,z,,-,Cz;,-z;
Convergence to Rational Expectations Equilibrium, in " Individual Forecasts and Aggregate Outcomes
- M M Bray
M. M. BRAY, Convergence to Rational Expectations Equilibrium, in " Individual Forecasts and Aggregate Outcomes " (R. Frydman and E. S. Phelps. Eds.), Cambridge Univ. Press. Cambridge, 1983.
at t to depend on data (z i, zzl-r) This amendment makes p, and z, simultaneously determined, and complicates convergence proofs based on previous methods
- However
However, in many other applications it is desirable to alter a model to permit 8, at t to depend on data (z i,, zzl-r). This amendment makes p, and z, simultaneously determined, and complicates convergence proofs based on previous methods (e.g., see Bray and Savin
+ bf almost surely as t -+ co. and let u(t, A, c), where 0 < A < 1, and c is a positive constant, be given by o(t, 1, c)=h(t-1, A, c)
- D Satisfied
is satisfied and D, c D,, then /3, + bf almost surely as t -+ co. and let u(t, A, c), where 0 < A < 1, and c is a positive constant, be given by o(t, 1, c)=h(t-1, A, c)+c le(t)l, ~(0, A, c)=O. (I-16)
The monetary dynamics of hyperinflation, in " Studies in the Quantity Theory of Money
- P Cagan
P. CAGAN, The monetary dynamics of hyperinflation, in " Studies in the Quantity Theory of Money " (M. Friedman, Ed.), Univ. of Chicago Press, Chicago, 1956.
The model (3), (4) shares with many examples in the literature the feature that the estimate /II at t is a function of zls, zzs-i for
- Learning From
- Data
LEARNING FROM CONTEMPORARY DATA The model (3), (4) shares with many examples in the literature the feature that the estimate /II at t is a function of zls, zzs-i for sd t -1.
The stability of rational expectations in macroeconomic models, in " Individual Forecasting and Aggregate Outcomes: Rational 'Expectations' " Examined
- George Evans
GEORGE EVANS, The stability of rational expectations in macroeconomic models, in " Individual Forecasting and Aggregate Outcomes: Rational 'Expectations' " Examined. " (Roman Frydman and Edmund S. Phelps, Eds.). Cambridge Univ. Press, Cambridge, 1983.
Toward an understanding of market processes
- Frydman