Universit`
a di Pisa
Scuola Superiore Sant’Anna
Master of Science in Economics
The Role of Expectations in
Agent-Based Models: Pseudo
Rational ABM Expectations
Supervisors:
Candidate:
Andrea Roventini
Francesco Toni
Tania Treibich
Andrea Vandin
Acknowledgements
This thesis would not have been possible without the help of my supervisors.
To begin with, I am grateful to Andrea Vandin for having introduced me to pro-gramming in a structured way, he showed me that developing computational models cannot be a mere transposition of a theoretical idea, but rather the result of a clear designing process and a rigorous method.
Meeting Tania Treibich, even in remote mode, contributed to me shaping my re-search interests. I was fascinated by analysing the impact of firms’ behavior on the macroeconomic dynamics and I strongly intend to continue working in this field, also thanks to her.
Last but not least, Andrea Roventini, who confirmed my heartfelt desire to study Economics in each discussion we had. I admire his curiosity, his ability to challenge entire theoretical apparati by suggesting compelling alternatives at the same time. I look up to him and I wish to develop such abilities myself.
Most of my thesis I’ve written during my quarantine. Covid-19 has drained my energy, but our minds unlike our bodies cannot be contained. Unfortunately, this powerful virus took my grandmother away from me.
I want to dedicate this work entirely to her, who supported my studies since I was a child.
As she used to say referring to me “Alla mia adorata” nonna Rosanna.
CONTENTS
1 Empirical evidence on expectations 10
1.1 Rational and extrapolative expectations . . . 10
1.1.1 Rational Expectations Hypothesis . . . 10
1.1.2 Extrapolative expectations . . . 12
1.2 Survey data . . . 13
1.3 Laboratory experimental data . . . 17
2 Adaptive Learning in Keynes meeting Schumpeter model 20 2.1 Introduction . . . 20
2.2 Keynes meeting Schumpeter Model . . . 22
2.2.1 Structure of the K+S model . . . 22
2.2.2 The model . . . 24
2.3 Bounded rational expectations and heuristics . . . 28
2.3.1 RLS learning and adaptive expectations scenario . . . 29
2.3.2 Simulation results . . . 30
2.4 Heterogeneous expectations Switching model . . . 35
2.4.1 The model . . . 36
2.4.2 Simulation results . . . 36
2.A Appendix: Parametrization . . . 41
3 Pseudo rational ABM expectations 42 3.1 Introduction . . . 42
3.2 ABM Rational Expectations: The general procedure . . . 44
3.3 Pseudo Rational ABM Expectations in the K+S . . . 45
3.3.1 Expectation Revision process: an Expectation Tatonnement . . 49
3.4 Simulation results . . . 52
3.5 Conclusions . . . 70 3.A Appendix: Parametrization . . . 72 3.B Additional Simulation Results, Stability . . . 73 3.C Additional Simulation Results, performance without Tatonnement . . . 74 4 Summary and Concluding Remarks 77 4.1 Summary . . . 77 4.2 Concluding Remarks . . . 78
LIST OF FIGURES
2.1 K+S model structure . . . 23
2.2 Agents’ Performance using different number of time period observations required to perform RLS . . . 31
2.3 Illustration of the overfitting problem . . . 32
2.4 Evolution of the Logarithm of sales of RLS agents (left panel) and Heuristic-guided agents (right panel) . . . 34
2.5 Effect of changing the minimum number of observations to perform RLS 35 2.6 Share of Expectation rules over time . . . 39
3.1 Illustration of the experiment with 1 ABM firm out of N . . . 48
3.2 Illustration of the Expectations Tatonnement . . . 50
3.3 Cross-Sectional Squared Forecast Error across Agents over Time, base-line case . . . 54
3.4 Average error over Time . . . 56
3.5 Death Reason of ABM firms . . . 58
3.6 Cross-Sectional Squared Forecast Error across Agents over Time, Edu-cated Guess initialization . . . 61
3.7 Global Stability under Tatonnement . . . 63
3.8 Market Shares Evolution . . . 65
LIST OF TABLES
2.1 Set of Expectation Rules . . . 29
2.2 Mean Squared Forecast Error under different Expectation Scenarios . . 31
2.3 Macroeconomic Performance under RLS-learning ADA specification, Tmin rls = 5 . . . 34
2.4 Expectation Rules and Macroeconomic Performance, Baseline Regime . 37 2.5 Expectation Rules and Macroeconomic Performance, Low Innovation Regime . . . 38
2.6 Parameters, baseline setting . . . 41
3.1 Cases under analysis . . . 53
3.2 Error matrix under Expectations Tatonnement, na¨ıve initialization . . . 55
3.3 ABM Vs. Na¨ıve, Firm Level Performance, Baseline case . . . 56
3.4 Survival under ABM expectations and Na¨ıve expectations, Big firm Case 57 3.5 ABM Vs. Na¨ıve, Macroeconomic Performance, Baseline case . . . 60
3.6 Error matrix under Expectations Tatonnement, Educated Guess initial-ization . . . 62
3.7 Survival under ABM expectations and Na¨ıve Expectations, Small firm Case . . . 66
3.8 ABM Vs. Na¨ıve, Firm Level Performance (Level) . . . 67
3.9 ABM Vs. Na¨ıve, Firm Level Performance (Growth) . . . 68
3.10 ABM Vs. Na¨ıve expectations, Macroeconomic Performance . . . 69
3.11 ABM Vs. N¨aive, Firm Level Performance (Level), ABM without forward revision . . . 74
3.12 ABM Vs. Na¨ıve, Firm Level Performance (Growth), ABM without for-ward revision . . . 75
3.13 ABM Vs. Na¨ıve expectations, Macroeconomic Performance, ABM with-out forward revision . . . 76
Abstract
This thesis consists of 3 chapters.
Chapter 1 briefly introduces rational and extrapolative expectation models and the empirical evidence on the testable implications of such models is reviewed. Rational ex-pectations are robustly rejected by using both survey data and controlled experiments in the analysis of expectations and forecasting.
Chapter 2 compares adaptive learning “rational” expectations with the “Wilderness of bounded-rationality” by means of an agent-based model. Specifically, heuristics and recursive least squares learning are introduced in the K+S model. Agents are allowed to switch across expectation rules, according to their past performance. We find that: (i) agents rationally do not systematically choose to adopt “rational expectations”; (ii) the macroeconomic and individual level performance worsens, when sophisticated ex-pectations are introduced.
Chapter 3 provides the main contribution of this work: a new form of expecta-tions is advanced for agent-based models. Analogously to Muth’s hypothesis, agents perform forecasting by using “the true model” of the Economy. The pseudo rational ABM expectations are implemented in the K+S model and compared with static na¨ıve expectations. However, expectations are subject to the “infinite regress in expectations problem”, since the model is populated by heterogeneous interacting agents. In order to break the circularity above we propose the following exercise: only one agent is rational whereas all the other ones have adaptive expectations and the rational one knows it. We find that agents, who use the true data-generating-process to form expectations, perform perfect forecasts. Nevertheless, the sophisticated expectations have mixed ef-fects on the performance of the system, as forward-looking firms’ survival proves to be worse than in heuristic-driven ones, and they destabilize the system dynamics, which is more volatile and unstable.
Keywords— Forecasting models, Recursive learning, Rational expectations, Extrapolative expectations, Heuristics and bounded-rationality, Agent-based computational economics, Het-erogeneity, Complexity
INTRODUCTION
Expectations have been a central topic in modern macroeconomics starting with knightian distinction between probabilizable risk and radical uncertainty [1921] and Keynes’ great em-phasis on the role of expectations [1936].
Expectations, largely neglected in the Neoclassical Synthesis, have been revived by Fried-man’s debate on the Phillips Curve and his call for endogenous expectations, basically in terms of adaptive ones.
Friedman’s discussion paved the way for the introduction of the Rational Expectations Hypothesis in macroeconomics [1961] leading to the so called Rational Expectations Revo-lution (’70). On the grounds of the Lucas critique large macroeconometric models started to fall into abeyance1 , since they do not account for agents’ reactions to policy changes, so that micro-founded models with optimizing agents, who make the best use of all available information, nearly became the only game in the town. Rational expectations have become a building block of mainstream macroeconomics, from the New Classicals all the way to New Keynesian DSGE.
Even though rational expectations are still the benchmark for economic modelling, in general, we have witnessed a paradigm shift towards the behavioral view, according to which economic agents have informational and cognitive limitations; and, consequently, modelers try to unbundle the inner process of expectation formation, rather than artificially assuming stochastic properties or stability conditions.
The first behavioral approach dates back to Simon’s view [1957] that lies at the interplay between economics and psychology, see also Tversky and Kaheman [1974]. In this view be-havioral patterns and expectation formation are adequately accounted by heuristics, which according to Gigerenzer and Haissmaier [2011] are “strategies that ignore part of the informa-tion, with the goal of making predictions more quickly, frugally, and/or accurately than more complex methods”. Given this definition, heuristics are not necessarily sub-optimal behaviors, but simply guidelines and behavioral approaches to problem solving in the presence of radical and procedural uncertainty.
Concerning the expectations of bounded rational agents, another alternative to genuine rational expectations, which in a way differently declines the paradigm shift, is adaptive learning. See Sargent [1993] for an early overview and Evans and Honkapojha [1999] for a
more recent one. In this version of bounded rationality agents do not know the law of motion of the Economy (differently from canonic rational expectations), but as econometricians they use time-series data to form expectations and adapt their behavior by tuning the parameters (or the specification tout court ) of their perceived law of motions, accordingly to the new data they collect. Adaptive learning may or may not lead to REE, depending on the structure of the model and on stability conditions (such as the E-learning stability in Evans and Honkapojha). In some cases adaptive learning largely deviates from the REH (Grandmont, [1998]), in some others it is a cautious modification of rational expectations (Hansen and Sargent, [2008]). Indeed, in mainstream models under suitable conditions rational expectations and adaptive learning can converge to the same asymptotic results.
In evolutionary theory the focus is not on expectations directly (outcome), but mainly on learning (process). Agents in uncertain conditions learn about the structure of the world or the “environment” as their representation of it may be partially or totally wrong (substantive uncertainty), but also they do not know the full set of actions and they can discover them by enlarging their repertoire (procedural uncertainty). In this respect, evolutionary theory is close to Simon’s view of uncertainty and agents follow heuristics. In general, economic problems can be subject to radical uncertainty, thus the conditional objective expectations are not applicable (see Dosi et al. [2005a] for an extensive overview)
Also post Keynesians (such as Michal Kalecki, Piero Sraffa, and others) have opposed to the mainstream approach to uncertainty: decision makers in uncertain environments can be without clues about the future or follow their “animal spirits”.
This work investigates the relationship between the two main approaches to expectation modeling in macroeconomics.2 Rational and extrapolative expectations are compared both from an empirical and theoretical (for the most part) point of view. Specifically, we investigate the role of expectations in the Keynesian meeting Schumpeter agent-based model [2020b]. We aim to shed lights on the effects of a wide range of expectation formation processes on the individual and aggregate level performance of the Economy. The underlying objective is to develop a new framework for expectations and agents’ behavior by means of agent-based mod-eling. To this end a new form of expectations is advanced, which can parsimoniously be taken as a way to implement rational expectations in Muth’s sense [1961] in agent-based models. Agents endowed with the “true model” of the world perform one-step ahead forecasts. There-fore, agents are rational as their subjective expectations equal the objective mathematical expectations and ,consequently, their forecasts are correct on average (given the agents form expectations by using the true-data-generating process). In this respect, the novel expectation rule allows to implement a Pesudo version of rational expectations, that is a building block of mainstream macro models, in agent-based frameworks, thus aiding comparability of results by gaining a better understanding of the fundamental differences between the two theoretical approaches.
The investigation of expectations can shed light on economic fluctuations. Indeed, expec-tations play a key role in explaining business cycle, as usually suggested by the macro theory. The 1999-2001 boom-bust cycle can be seen as an example of expectations-driven business cycle. According to a common interpretation, the optimism towards the future contributed to the high growth rates of 1999 and 2000, while a reversion of expectations caused the downturn of 2001.
Beandry and Portier [2006] find that changes in expectations are important determinants of business cycle fluctuations. Specifically, in a VAR analysis they find that productivity growth anticipated by economic agents lead to expectation-driven booms.
Furthermore, understanding expectations is crucial for the design of policy institutions, as monetary policymakers may be able to affect economic agents’ beliefs. The recent long-lasting coincidence of low inflation and persistent decline in real interest rate have forced to alter the standard conduction of fiscal and monetary policy. In particular, the management of expectations have become an important tool among the monetary policy instruments of central banks. First, in response to the Coronavirus pandemic the ECB launched a new asset purchase, namely the Pandemic Emergency Purchase Programme (PEPP), in order to limit the risks associated to the monetary policy transmission mechanism in the proximity of the zero lower bound. Indeed, the launch of PEPP affected considerably market expectations. For example, sovereign spreads in Greece had fallen by more than 150 basis points even before the purchase actually started.3 Second, another wave of the Targeted Long-Term Refinancing Operations (TLTRO) have been issued by the ECB, in order to offer banks long-term funding at attractive conditions. Third, especially after the 2008 financial crisis the forward guidance has become a common tool to manage expectations in the financial market. Explicit statements of the central bank about the outlook for future policy and its announcements about immediate policy actions that is undertaking have become a viable strategy, as discussed by Woodford [2012].
As pointed out, these unconventional monetary policies have direct and indirect effects on the expectations of economic agents. In this respect, modeling expectation formation processes and evaluating their impact on the functioning of the Economy will aid to assess how agents’ expectations affect the efficacy of fiscal and monetary policies and how monetary policymakers may affect agents’ expectations.
Specifically, forward-looking expectations will enable to formally analyze the effects of an-nouncements of policy changes made by central banks in agent-based frameworks, whereas in the presence of adaptive expectations only there is no room for forward-guidance at all, since agents’ expectations are function of the past only and current news will not affect agents’ expectations. In general, agent-based models are exposed to the Lucas Critique, as heuristic-guided agents do not react to announcements to policy changes. On the one hand assuming that all agents or a representative one perfectly react to all relevant news is totally unreal-istic, on the other modeling an Economy where no one reacts to explicit statements about policy is too simplistic. Several approaches have been adopted to implement forward-looking behaviors in agent-based models, such as genetic algorithms (Catullo et al.[2020a]), recursive learning (Dosi et al.[2020b]), and more broadly machine learning techniques (Georges and Pereira [2020d]). Differently from all such approaches, the Pseudo Rational expectations
pro-3More than 150 basis points of the Greece versus Germany 10 years government bonds from the
vide the way to implement the extreme modeling case, in which agents, or a few of them,4 always perform perfect forecasts on the future state of the Economy by exploiting all relevant information.
The thesis is oganized as follows, Chapter 1 briefly introduces rational and extrapola-tive expectations and the empirical evidence on the testable implications of such models is reviewed. Chapter 2 describes the Keynes meeting Schumpeter model by motivating the suit-ability of a complex evolving system to study the role of expectations, then adaptive learning “rational expectations” (in the form of RLS) is compared to heuristics in the model. Chapter 3 increases further the degree of agents’ sophistication by introducing the new Pseudo Ra-tional Agent-Based Expectations. Again, the effects of “raRa-tional expectations” on the micro and macro performance is analyzed by comparing them with static na¨ıve expectations, taken as the benchmark for simple heuristics. Chapter 4 concludes.
CHAPTER
1
EMPIRICAL EVIDENCE ON EXPECTATIONS
1.1
Rational and extrapolative expectations
1.1.1
Rational Expectations Hypothesis
Mainstream macro models generally assume that agents have rational expectations (Muth, [1961]). Agents expectations correspond to the mathematical conditional expectations im-plied by the model; consequently, individuals do not commit systematic mistakes and the mean error is null. According to the dominant interpretation, agents act “as if” they know how the true model of the Economy works and have access to all relevant information. More-over, each agent knows that other agents know (common knowledge assumption). In other terms, events occur according to a specified probability distribution and agents behave as if they all knew the true data-generating process governing economic phenomena.
In order to introduce the rational expectations hypothesis (REH) formally, let us define individual expectations in the statistical framework. Denote individual i’s point expectations of a k dimensional vector of future variables, xt+1, formed with respect to the information set
Ii,t by Ei[xt+1|Ii,t]. Analogously, denote individual i’s density expectations by fi(xt+1|Ii,t) so
that
Ei[xt+1|Ii,t] =
Z
xt+1fi(xt+1|Ii,t)dxt (1.1)
For a formal representation of the REH the individual specific information set Ii,t is
decom-posed into a public information set ψt, and an individual specific information set φi,t such
that
Ii,t = ψt∪iφi,t, ∀i (1.2)
Further, we assume that the “objective” probability density function is given by f (xt+1|ψt).
The REH postulates that private information plays no role in the expectations formation process, and information are efficient considering only public information, ψt. The optimality
independence
E[εt+1|St] = 0 (1.3)
where εt+1 is the expectation error:
εt+1= xt+1− E[xt+1|ψt], (1.4)
and St⊆ ψt, is a subset of ψt. Therefore, if conditional mean independence holds, individual
expectations coincide with the conditional expectation function (CEF), which is the minimum mean squared error predictor of xt+1 given the information set St. The “orthogonality”
con-dition, thus, implies that expectations errors have zero mean and are serially uncorrelated. Notice, instead, that conditional homoscedasticity is not necessary for the rational expecta-tions hypothesis to hold. According to the “orthogonality” condition, only random errors may appear in individual forecasts and agent do not make systematic mistakes. Indeed, the mean independence requirement is often used to test the informational efficiency of survey expectations, but it is neither necessary nor sufficient for the rationality of expectations if individual expectations are formed as optimal forecasts with respect to general cost functions under incomplete learning. In this approach full rationality is relaxed by assuming imperfect and/or incomplete information.
Indeed, the common knowledge assumptions in the muthian sense has been relaxed in the literature. Different notions of rational expectations equilibria are defined and implemented under asymmetric information; see, for instance, Radner [1979], Grossman and Stiglitz [1980c], just to mention some early contributions.
The rational expectations hypothesis does not deny heterogeneity in expectations formation, on the contrary, Muth [1961] was aware of the importance of allowing cross-section hetero-geneity in expectations. Indeed, one of his objectives in advancing the REH was to explain the following stylized facts:
• Averages of expectations in an industry are more accurate than naive models and as accurate as elaborate equation system, although there are consider-able cross-sectional differences in opinions.
• Reported expectations generally underestimate the extent of changes that actually take place.
(Muth, [1961])[p.316]
Now, the considerable differences in individual forecasts of economic variables indicated by surveys 1 does not violate the REH. In Muth’s pioneering work,“[In a linear cowbeb model ] the rational expectations hypothesis states that, in the aggregate, the expected price is an unbiased predictor of the actual price”. It is not necessary that individual expectations of the variables of interest are correct, rather than, in the aggregate, expectations are unbiased. That is to say that individuals do commit mistakes and their expectations can be heterogeneous, but the error εt+1 is just random noise that is washed out, as the sample size increases.
Individuals make mistakes in forecasting, but they do not do so systematically.
ex-heterogeneous expectations and asymmetric information. Early attempts in dealing with im-perfect information, in particular, include Lucas [1972] and Townsend [1978] . As we shall see in chapter 3, heterogeneous expectations models incur in the “infinite regress in expectations” problem, which arises as agents need to forecast the forecasts of others. A few solution strate-gies have been proposed to limit this problem, for example Binder & Pesaren [2014b] assume that each agent bases her forecasts of others on the common knowledge only, ψt. In this way,
the fact that agents, who do not have access to private information sets of others, are not able to update the expected forecasts of others (private knowledge), and the circularity between expectations is broken.
1.1.2
Extrapolative expectations
Several alternatives to the REH have been advanced with different degrees of informational requirements. Most of these expectations formation models can be included in an “extrap-olative” category, where expectations are determined by past realizations. Along these lines, the economic agent collects data and she bases the forecasts on such past data. A general specification for extrapolative expectations is given by:
Ei[xt+1|Ii,t] = ∞
X
j=0
φi,jxt−j, (1.5)
where the coefficient matrices, φi,j, are assumed to be absolute summable subject to the
adding up condition
∞
X
j=0
φi,j= Ik (1.6)
this condition guarantees that unconditional expectations and observations have the same means.
Static na¨ıve expectations
The simplest form of extrapolative expectations is the static expectations considered by Keynes [1936], according to which the expectations for the future variables equal to the current variables:
Ei[xt+1|Ii,t] = E[xt+1|St] = xt (1.7)
in particular, this predictor is optimal, i.e. it is the minimum mean squared error predictor of xt+1, if xt follows a random walk process.
A more recent alternative version of static expectations is the difference version of them, given by
E[xt+1|St] = xt+ ∆xt−1 (1.8)
which instead is optimal, when ∆xt+1 follows a random walk. This latter specification has
the main advantage that if the process xt is integrated of order 1, xt∼ I(1), then ∆xt∼ I(0).
Adaptive expectations
The most usual form of extrapolative expectations is the adaptive one:
E[xt+1|St] − E[xt+1|St−1] = Γ[xt− E[xt+1|St−1]] (1.9)
where xt− E[xt+1|St−1 is the forecast error made at time period t. Thus, agents adopting
adaptive expectations form their forecasts on the base of past forecast errors. Notice that adaptive expectations can be seen as a partcular case of the general extrapolative expectations, by setting
φj = Γ(Ik− Γ)j, j = 0, 1, ... (1.10)
and assuming that the invertibility condition holds, i.e. the eigenvalues of Ik− Γ are inside
the unit circle.
Adaptive expectations do not need to be informationally efficient, and errors could be seri-ally correlated. The adaptive expectations, indeed, were introduced as a rule of thumb for updating and revising expectations, without claiming that they are optimal. In this respect, Muth [1960] showed that adaptive expectations are optimal only if the time series process xt has a MA representation, where the shocks are independently distributed with mean zero
and variance σ2. Therefore, adaptive expectations can perform particularly poorly when the underlying processes are subject to structural breaks.
Many other approaches to expectations modelling fall in the category of extrapolative ex-pectations, such as return-to normality and error-learning expectations. Basically, infinitely many specifications describing the expectations formation can be conceived. This problem leads to Sims’ metaphor of the “Wilderness of bounded rationality”: if we discard the Ratio-nal Expectations Hypothesis, we have to deal with infinite ways of how agents make mistakes. To be more precise, the parameters of the stucture of the expectations formation process can not be identified without imposing apriori restrictions.
In what follows we provide an overview on the empirical evidence about expectations, confining ourselves to the rational expectations hypothesis and its relationship with alternative models of expectations formation. Section 2 investigates the evidence from survey data, Section 3 analyzes laboratory experimental evidence.
1.2
Survey data
Estimating expectations is not an easy task, as they are unobservable variables, which can not be measured directly. What individuals report in surveys may differ from what individuals actually expect and the data could be subject to relevant measurement errors, specific to the methodology adopted in the works reviewed in this Section. Nevertheless, the empirical in-vestigation of expectations has to be preferred to merely theoretical speculation, as discussed by Mansky [2004b]:
persons respond informatively to questions eliciting probabilistic expectations for personally significant event. We have learned enough for me to recommend, with some confidence, that economists should abandon their antipathy to measure-ment of expectations. The unattractive alternative to measuremeasure-ment is to make unsubstantiated assumptions.
Mansky [2004b][p.1370]
Using survey data to proxy otherwise unobservable expectations facilitates investigating the expectations formation process and its statistical properties. We will focus on the econometric developments in testing rational expectations and the main finding coming from this source of data from the 1970s onward. Specifically, this section focuses on unbiasedness tests.
The early literature
After the Rational Expectations Revolution, during the 1970s and 1980s, the empirical va-lidity of the REH has largely been investigated in a linear regression framework. Typically, the mean of the forecasts was taken as a dependent variable in a level-level regression and hypothesis testing was performed to establish whether the REH holds in the aggregate. A general specification of the linear model is:
xt+1= α + βxet+1+ ut+1 (1.11)
where xet+1 is an aggregate measure of expectation of the variable x at time t+1, typically denoted as “consensus” prediction xet+1= N1 PN
i=1xei,t+1, N is the number of respondents in
a survey, and xt+1 is the actual realized variable at time t+1. The “consensus” prediction
aims to test the rationality of expectations at the aggregate level, according to the genuine statement of the REH by Muth [1961], as seen in Section 1. Expectations are rational in the muthian sense, if subjective expectations correspond to the mathematical conditional expectations operator. Hence, a key property is the unbiasedness. Recalling that the linear model is a model for the conditional expected value of the independent variable, namely a model for E[y/x], where x is the vector of regressors, unbiasedness is tested with the null hypothesis being:
H0 : (α, β) = (0, 1) (1.12)
However, the validity of the test above relies on the orthogonality between the errors ut+1
and the “consensus” expectation xe
t+1. If the coefficients are consistently estimated and
E[xt+1|xet+1], then expectations (second term) coincides with the mathematical conditional
expectations (first term). 2
In these studies β is substantially lower than unity and the REH is generally rejected; see , Pesando [1975], Friedman [1980b], Brown and Maital [1981], Lovell [1986]. However, many macroeconomic variables have unit roots. Realizations and forecasts typically share a common stochastic trend, rational forecasts will be integrated and cointegrated with target series. Thus, conventional OLS should give a slope coefficient biased towards one.
2Further implications of the rational expectations hypothesis have been tested in this stream of
lit-erature. Examples of these tests include: test of efficiency, as rational expectations should incorporate efficiently all available information; tests of consistency; tests of orthogonality to available information; tests on the variance, as under rational expectations the variance of actual realizations should exceed the variance of forecasts.
Unbiasedness tests in returns regressions
To directly tackle the inferential problem due to the lack of stability of the time series used, in the second generation of regression a differenced model is taken by subtracting current realization from the forecast and from the future realization, by replacing level-level regressions with “returns” regressions:
xt+1− xt= α + β(xet+1− xt) + ut+1 (1.13)
In this specification of stationary variables unbiasedness is still tested as the null (α, β) = (0, 1), but the regression is interpreted as cointegrating regression, with conventional t-statistics following nonstandard distributions which depend on nuisance parameters. An example of this approach can be found in Urich and Wachtel [1981b], in which they tested unbiasedness for the money supply, by using a weekly survey on financial market participants. They argue that using “consensus” prediction is not suitable to test the muthian hypothesis. Specifically, when aggregating individual predictions it could be that individuals biases are off-set and, consequently, the rational expectations hypothesis should be tested at the individual level instead. Muth stated that the rationality of expectations should hold at the aggregate level, so a doubt arises in this interpretation. Is the REH satisfied, if individual forecasts are biased but they are such that by aggregating them we have an unbiased predictor? On the one hand we could conclude that the unbiasedness requirement is satisfied simply because the aggregate predictor is unbiased, on the other the existence of agents that systematically do forecast mistakes violate the muthian hypothesis. According to the dominant interpretation, if individual expectations are biased, the REH is violated.
Indeed, it is plausible to think that Muth was stating the hypothesis “at the aggregate”, because it is not even possible to define asymptotic stochastic properties for a single indi-vidual forecast. Taking a large sample of forecasts make it possible to take the probability limit of the sample statistic. However, the “aggregate” does not necessarily coincide with the whole population. Let us imagine subsetting the whole set of forecasters and take a sample of upward biased forecasters, assuming one exists, and this sample large being enough to take the probability limit of the sample mean. In this sub-sample the aggregate forecast would be biased, then the REH is violated.
We conclude that the first problem of the “consensus” approach is that aggregation may mask systematic individual differences, which are individual deviations from rationality. An example of this can be found in Keane and Runkle [1990c].
A second problem is related to the “common knowledge” assumption. If forecasts are rational but based on different private information sets, they could result in a biased aggregate forecast (again see Figlewsky and Wachtel, [1981a]).
The evidence from “returns regressions” is unclear with both rejections of the REH , and lack of rejections, such as Urich and wachel [1981b] for expectations on the money supply.
Panel methods and heterogeneity of expectations
Since the mean predictor is unsuitable to test rationality, an alternative approach is to use panel methods with individual specific coefficients, using specification of the kind:
xt+1− ¯xei,t+1= αi+ βi(xei,t+1) + ui,t+1 (1.14)
where ¯xei,t+1 = T1xei,t+1, is individual i’s time average of the one period ahead forecast. Now, “consensus” regressions is a model for the average individual, but an individual systematic deviation from rationality is sufficient for the violation of the REH, as we have pointed out. Thus, results of an aggregate regression are based on the restriction αi = α, βi = β ∀i. This
restriction, also known as micro-homogeneity, should not be imposed without being tested.3 Therefore, the focus has shifted from “consensus” regressions towards testing rationality at the individual level.
First, the application of this approach robustly finds that expectations are heterogeneous. Second, it finds that individual differ in a systematic way, which, also, implies the violation of the rational expectations hypothesis. For example, Ito [1990b] looking at the foreign exchange rates and using a survey from the Japan Centre for International Finance finds clear evidence for individual specific effects. Specifically, respondents tend to be optimistic, as exporters tend to expect a yen depreciation, while importers expect an appreciation. The phenomenon is described as “wishful thinking”.
Bonham and Cohem [2001a] develop a GMM approach to the issue, using the survey of pro-fessional forecasters. They reject the micro-homogeneity in most cases and conclude that the REH needs to be tested at the individual level.
An original alternative approach to investigate the heterogeneity in expectations is taken by Branch [2004a], who uses inflation expectations from the Michgan survey.4 Branch uses a model in which forecasters choose their expectation rule among static na¨ıve expectations, adaptive expectations (with coefficients estimated by least squares) and a forecast rule based on a vector autoregression. The model fits the data quite well and , although it can not be considered a test for rational expectations, it shows that heterogeneity itself does not contra-dict the REH, since forecasters can “rationally” choose different expectations rules. On the contrary, the genuine rational expectations are not present in the set of agents’ expectations. Moreover respondents could have faced many other model choices, not considered by Branch. Additionally, past performance as a selection criterion does not necessarily correspond to agents’ rationality in the sense discussed so far. The fact that the forecasting rule, which was more accurate than others in the past, does not imply that it will also be so in the future. Summing up, neither aggregating forecasts nor pooling panel of survey expectations are suit-able approaches in testing the REH. On the one hand “consensus” level-level regressions can lead to biased results given the lack of stationarity in the time series of macroeconomic vari-ables, on the other hand “returns regressions” (as level-level regression) are based on the
3This implies the well-known problems related to pretesting.
4The same modelling approach has been taken by Anufriev and Hommes [2012a], and building on
crucial assumption of micro-homogeneity. There exist systematic differences in individual expectations. Hence, they should be tested at the individual level, allowing for forecasters heterogeneity in the estimation methods. Nevertheless, the rational expectations hypothesis is typically rejected both in the aggregate and at the individual level, by using survey data.
1.3
Laboratory experimental data
A relatively more recent stream of literature investigates expectations by means of laboratory experiments. Generally, subjects participate in a financial market and their expectations are either inferred or asked for directly by the experiment designer. The existing body of literature could be classified in 3 categories, according to the features characterizing the methodological approach. The first consists of laboratory auction markets, in which players buy and sell assets. This was proposed by Smith et al. [1998b]; other examples can be found in Noussair et al. [2007b], Kircheler et al. [2009b], which aim to explain the emergence of bubbles in asset markets, as a deviation from the rational expectations hypothesis. The main drawback of this approach is that expectations are inferred and not directly observed. This, in turn, exposes the conclusions made on the forecasting process of individuals to an additional inferential step.
The second, in which experimental subjects predict the time series of price exogenously generated and not affected by the realizations of the time series itself. For example, Bloomfield and Hales [2002] propose an experiment, where the time series is generated by the most simple Gaussian-Markov process (random walk), while Hey [1994] proposes an auto-regressive process to generate the time series of prices, also with the occurrence of structural breaks at pre-specified time periods.
The third, also known as learning-to-forecast experiments further increases the complex-ity of the environment the experimental subjects face. It relaxes the independence of the expectations and the price dynamics. To be more precise, the price in the market is assumed to be function of individual forecasts, thus introducing expectations feedback mechanisms. Literature provides numerous contributions in this field with both positive (Hommes et al. [2005b], Bottazzi et al. [2011a], Anufriev and Hommes [2011a] and negative feedback mecha-nisms (Hommes et al., [2009a]).
Evidence from the controlled experiments approach suggests that individual behavior de-viates from rationality in a considerable way. In this respect, auction market experiments, such as Noussair et al. [2001c], and Kirchler [2009b] find that forecasters follow an adaptive rule to form forecasting, rather than forming rational expectations, and bubbles frequently emerge contrarily to the rationality of financial market participants.
Since individuals’ rationality is robustly rejected, the consequent logical step is to capture the actual features of the forecasting process adopted by experimental participants, as Hey [1994] has clearly pointed out:
So, our statistical tests reject the detailed rational expectations hypothesis, though the general flavour of the data support the general notion that subjects were
general model of the Data Generating Process in their minds which they were using in a broadly sensible fashion. [...] The following questions remain: how did they form their “models of the world”; how might these models change if the Data Generating Process changes?; and how was their model of the world translated into a prediction strategy? [...] What we must now do is investigate these questions - to see how “models of the world” are formed and changed - and to see how these models are translated into strategies.[Hey][p.348]
The step indicated by Hey has, indeed, been the focus of this stream of literature for a long period of time. Laboratory experiments have shown that individual decisions under uncertainty are much better described by simple heuristics, which sometimes lead to persistent biases. Heuristics may be thought of as the alternative “model” through which economic agents perceive the world. Generally, they are guidelines and behavioral orientations adopted to face problem solving, in this context broad strategies to forecast uncertain future events. The concept of heuristic can be described by a deterministic function of available information (Anufriev and Hommes, [2012a]).
Specifically, in Learning-to-forecast experiments the time series of prices are endogenously affected by individual forecasters, who usually do not know neither the forecasts of others, nor the data-generating-process, as they have imperfect information on the market in which they play. Then, the forecasting process followed by the participants (if any) is tried to be inferred, to explain the emergence of statistical regularities in the actual realized time series. These studies suggest that: first, simple linear adaptive expectations rules fit the series well, second, heterogeneity in expectations is crucial to describe individual forecasting and aggregate price behavior.
An influential work in this category is Anufriev and Hommes ( [2012a], [2012b], [2019]), which provides a more realistic microfoundation of agents’ expectations by means of labora-tory experiments.6 For this study learning-to-forecast experiments have been performed at the University of Amsterdam, in order to analyze individual forecasting behavior (see Hommes [2011d] for an overview). In these experiments participants were advisors to large pension funds and had to submit point forecasts for the price of a risky asset. The RE benchmark is not a good explanation of such laboratory experiments in terms of individual forecasting and aggregate behavior. Conversely, simple linear forecasting rules have been estimated by using the experimental data and individual forecasts and the data are remarkably well explained by such heuristics. These rules typically have R2 higher than 0.80.7 Additionally to a good fit of the data, these heuristics ,when implemented in homogeneous heuristic models, explained the observed patterns or stylized facts of the price dynamics in the HSTV05 experiments:
• Constant convergence; • Constant oscillations; • Dampened oscillations.
6The implementation of expectations in the K+S model in Chapter 2 builds upon this work 7Building on previous literature on bounded rationality they estimated 3 classes of linear heuristics:
adaptive, trend-following and anchoring and adjustment heuristics (more on this in the Section 2 of Chapter 3).
However, homogeneous expectations models did not explain the emergence of different patterns in the aggregate behavior in different experimental sessions. Individual heterogene-ity in expectations plays a key role in explaining the phenomena observed in the experiments at the aggregate level, indeed participants seem to learn which forecast rule to use by switch-ing between heuristics based upon past performance.
To conclude the rationality is typically rejected by using survey data. The rejection is robust to a wide variety of specifications, but we have argued that the existence of one agent only that systematically commits mistakes is sufficient to violate the rational expectations hypothesis. Furthermore, the experimental evidence is better explained by extrapolative models of expectation formation. However, it could be that some economic agents, or at least one of them, perform forecasts that are on average correct, even though we should reject the hypothesis by applying the standard tests reviewed.
CHAPTER
2
ADAPTIVE LEARNING IN KEYNES MEETING
SCHUMPETER MODEL
2.1
Introduction
Is it rational to have rational expectations? This is the provoking question posed by Alan Kirman [2014c]. The arguments against rational expectations take place in 3 different layers. First, a large empirical literature rejects the implications of the rational expectation hy-pothesis. Surveys indicate a considerable difference in the individual forecasts of economic variables (it is typical to reject the key implications of rational expectations, as we have seen in Chapter 2). Second, methodologically it is assumed that agents are rational as a consequence of an aspiration to internal consistency in one’s model, but it has been shown that individuals who believe in a wrong model can produce outcomes which confirm those beliefs ( see Bray [1982], Kirman [1975], [1983]). Internal consistency could be guaranteed even without rational expectations, simply if expectations are met.
Third, it would be irrational for agents, in an evolving world differently from stationary settings of standard economic modelling, to have rational expectations even were they capable of understanding the evolution of the Economy. The paradoxical claim on the irrationality of rational expectations is based on their polysemy in economic literature: the relatively recent meaning of the term is that agents’ expectations of the future should be correct (as introduced in Chapter 1), originally rationality meant conveying the idea the agents act in their own best interest. Given rational expectations are not empirically supported, they are not necessary for internal coherence, are they at least convenient to be adopted for a self-interested agent?
Hendry and Mizon [2014a] argue that unpredictability is a usual feature of economic phenomena. The non-negligible probability of ‘extreme draws’ or outliers (instance dictability) and unpredictable shifts in the distribution of a random variable (extrinsic unpre-dictability) imply serious complications in forecasting. If a random variable is unpredictable in mean due to unanticipated shifts in its conditional distribution neither the economic agent neither the economist can have rational expectations.
Additionally and related to that, according to Stiglitz ([2011e]; [2018b]) major economic downturns are sui generis, such as the 2008 financial crisis, and policymaking is an art that entails judgments about how to shape current behavior. Radical changes in policy can not be taken into account by a rational agent.
According to a standard evolutionary argument in favor of the rational expectation hy-pothesis, “irrational agents” will not survive competition and will be driven out of the market by rational agents, who on the ground of their accurate expectations have higher pay-offs (Friedman, [1953]). On the contrary, Brocks and Hommes [1997] have shown that irrational agents can coexist with rational ones in a linear Cobweb demand-supply model, where agents can choose between rational and static na¨ıve expectations.
The objective of the analysis that follows is twofold: on the one hand it aims to investigate the role of hetereogeneous expectations in macroeconomic models, on the other it aims to establish the validity of the standard evolutionary argument in favor of rational expectations, according to which self-interested agents rationally choose to adopt rational expectations. Would agents choose to be rational in complex evolving worlds characterized by structural breaks and non-trivial endogenous shocks?
In this Chapter we investigate the performance of “rational expectations” in complex evolving systems by means of Agent-Based Modeling. Specifically, adaptive learning rules [2012a] are implemented in the Keynes meeting Schumpeter model and we compare simple heuristics with such more “rational ” expectation rules by evaluating their individual and collective performance in the simulated economic system.
To be more precise, adaptive learning does not necessarily coincide with the rational ex-pectations, as defined in Chapter 1. Recursively applying a forecasting model based on Time Series data does not always provide unbiased forecasts (as we shall see), but adaptive learning can be interpreted as an expectation form characterized by “a higher degree of rationality” with respect to simple heuristics, since adaptive learners enlarge their information sets (re-gardless of the actual forecasting performance).1
Section 2.2 provides an exposition of the Keynes meeting Schumpter. In Section 2.3 adaptive learning and heuristics are introduced in the model. In Section 2.4 the heteroge-neous expectation switching model by Anufriev and Hommes [2012a] is implemented in the expectations extended K+S model.
1Instead, agents with the Pseudo Rational Expectations, introduced in Chapter 3, have perfect
2.2
Keynes meeting Schumpeter Model
The following exposition of the model focuses on the sources of uncertainty, which con-tribute to overall unpredictability due to structural breaks, fat-tailed distributions (with negligible probability of extreme draws), lack of stationarity, and the emergence of non-linearities in the system dynamics.
2.2.1
Structure of the K+S model
The Schumpeter meeting Keynes model (Dosi et al. [2020b]) is an extended version of previ-ous evolutionary agent-based models (Dosi et al., [2013]; [2015a]). Its distinguishing feature is that it combines Schumpeterian innovation with typical keynesian features of demand gen-eration.
The structure of the K+S, which will be outlined in the following section, together with its emergent properties make the model a suitable setting to analyze expectations, mainly due to the “radical uncertainty” characterizing the Economy. The keynesian meeting Keynes model is an agent based model populated by heterogeneous interacting agents with endogenous tech-nical change, imperfect information and coordination hurdles. The data-generating-process is evolving overtime, as the outcomes of the model influence the actions taken by the agents and ,conversely, the actions of the agent are function of the outcomes. The simultaneity between agents’ decisions and the dynamics of the model complicate performing forecasting, as we shall see.
The model is populated by heterogeneous interacting agents, who are not committed to a mutual coordination ex-ante. In this disequilibrium setting the concept of Rational Expec-tations Equilibrium does not apply, an equilibrium towards which the system may or may not converge does not exist. On the contrary, coordination failures are the norm, rather than the exemption. The Schumpeterian innovation is endogenous, and it occurs in the form of stochastic shocks at the micro level. The innovation process is unpredictable (see Hendry and Mizon, [2014a]). For example, there are intrinsic shifts in the probability distribution of innovation, since the probability of success to innovate depends on the financial resources devoted to R&D by each firm, and it changes over time accordingly.
All in all, the model is characterized by path-dependent processes, such that the data-generating process evolves in a complex and unpredictable way; both stationarity and er-godicity do not hold. Again, agents can not easily learn the underlying structure of the world, as it evolves in a non-trivial way.
The demand side of the model is made of the workers, who do not invest but allocate their whole income on consumption, this holds both for employed workers that spend their wage and unemployed ones that spend their subsidy.
The production side is composed of a capital-good industry, where capital-good firms perform R&D activity by investing a share of past sales and sell machines to the consumption-good industry, where consumption-good firms produce final goods by using labor and machines. In the financial sector there are the Central Bank and other banks which stock deposits,
lend to firms and own government bonds.
Finally, the government levies taxes, gives unemployment subsidies and bails failed banks. A detailed description of the model is provided in Dosi et al. [2015a].
Figure 2.1: K+S model structure
The microeconomic decisions taken at each time period follow a precise temporal sequence, the timeline of the events is the following:
1. Policy variables are fixed (banks’s capital requirement, tax rate, Central Bank interest rate, etc.).
2. Banks fix the maximum credit supply.
3. Capital good firms perform R&D, innovate and imitate their competitors.
4. Consumption good firms decide how much to produce and invest according to different expectation rules. They apply for bank credit if their internal funds are not enough. 5. The capital good market opens. Given the presence of imperfect information,
capital-good firms advertise their products to an evolving subset of consumption-capital-good firms, which in turn choose their supplier.
7. The imperfect consumption good market opens. Imperfect information implies that the market shares of firms evolve according to their price competitiveness(even though the goods are homogeneous).
8. The firms compute their profits and pay back their loans.
9. Entry and exit takes place. In both sectors, firms with very low market share or negative net liquid assets exit and are replaced by the ones.
10. Banks compute their profits and net worth. If the latter is negative they fail and are bailed out by the Government.
11. The government computes its surplus or deficit, the latter being financed by sovereign debt which is bought by banks.
12. Machines ordered at the beginning of the period are delivered and become part of the capital stock of consumption-good firms at time (t+1).
At the end of each time step, aggregate variables are computed by summing the corresponding microeconomic variables.
2.2.2
The model
The agents populating the Economy are divided in F1machine-producing firms, F2
consumption-good firms, Ls consumers/workers, B commercial banks, a central bank, and a public sector.
Endogenous innovation: the capital-good industry
On the supply side firms of the capital-good industry produce heterogeneous machines by using labor as the only input. The technology of the firm is given by (Aτi, Bτi), where the former coefficient is the labor productivity of the machines manufactured by firm i for the consumption-good industry, while the latter coefficient is the labor productivity of the produc-tion technique employed. τ denotes the current technology vintage. They sell their machine-tools at a price defined by a fixed mark-up over the unit costs of production. Finally, as capital good firms produce by using the cash advanced by their customers, they do not need external funding.
The endogenous innovation affects the current technology represented by the coefficients above. Specifically, they innovate and imitate their competitors to increase labor productivity or to reduce costs, by investing part of the revenues from their sales. In order to innovate and imitate, they invest in R&D a fraction of their past sales
RDi(t) = vSi(t − 1), (2.1)
with 0 < v < 1. R&D expenditures are employed to hire researchers paying the market wage w(t). Firms split their R&D resources between innovation (IN ) and imitation (IM ) according to the parameter ξ ∈ [0, 1]:
INi(t) = ξRDi(t)
IMi(t) = (1 − ξ)RDi(t)
(2.2) Innovation is modeled as a two step stochastic process. The first step determines whether they get or not access to the innovation , through a draw from a Bernoulli distribution, with parameter θi(t) given by
θini (t) = 1 − e−ξ1INi(t), (2.3)
with 0 < ξ1≤ 1. According to (4.3) the probability of the successful event positively depends
on the resources invested in innovation activity, thus the innovative discovery is more likely if a firm invest more on R&D. Notice that the the stochastic process on which innovation is based is endogenous, as θIN
i depends on R&D. In the second step, if a firm innovates it may
draw a new machine with coefficient (AINi , BiIN) from a Beta distribution AINi (t) = Ai(t) 1 + Xia(t)
BiIN(t) = Bi(t) 1 + Xib(t)
(2.4)
where Xa
i and Xibare independent draws from a Beta(α1, β1) over the support [x1, ¯x1] with
x1belonging to the interval [−1, 0] and ¯x1 to [0, 1]. Notice that the technological opportunities
are captured by the supports of the Beta distribution and its shape, characterized by the parameters above. For example, with low technological opportunities the largest probability density falls on the region such that x is lower than 1, where the innovation is worse than the technology currently in use.
Analogously to innovation search, imitation follows a two steps procedure. Again, access to imitation is determined by the draw from a Bernoulli with θIMi (t):
θIMi (t) = 1 − e−ξ2IMi(t), (2.5)
with 0 < ξ2 < 1. Firms with a success at the first stage are able to copy the technology of the
competitors (AIMi , BiIM). An Euclidean metric is used to compute the technological distance between every pair of firms to weight imitation probabilities. The closer the technology of the competitor is, the more likely it is imitated. Once a firm has drawn a potential innovation and imitation, they have to put this into production or keep to produce according to the current technology, they choose according to the best trade-off between price and efficiency. Once the type of machine is chosen, each firm sends a brochure containing information on the price and the productivity of its offered machine to a subset of consumption-good firms. Thus, the capital good market is characterized by imperfect information. The clients, namely the consumption-good firms, have partial information of the machine supplied to the market, and they “learn” what is offered on the market by interacting locally with the capital good firms.
Demand expectations: the consumption-good industry
Expectations are formed in the consumption-good sector, where firms have expectations on future demand. Consumption-good firms produce homogeneous goods, taking as inputs
cap-The production is constrained by their capital stock (Kj), if the desired capital stock,
which is determined by the production plan, is higher than the actual stock, firms invest to expand their production capacity:
EIjd(t) = Kjd(t) − Kj(t) (2.6)
Among the assets of the firms there are heterogeneous machines with specific productivity. Firms scrap machines by following a payback period routine, such that technical change and equipment prices affect the replacement investments of consumption-good firms. In the capital good market under imperfect information consumption-good firms choose their suppliers by comparing the productivity of the subset of machines they know. Consumption-good firm compare “brochures” describing the machines of the suppliers they are aware of. They choose the machines with the lowest price and unit costs of production and order the machines accordingly. The gross investment is the sum of expansion and replacement investment. Summing up the investment of all consumption-good firms, we get aggregate investment (I).Contrary to capital-good firms, consumption-good firms have to finance their investments and production, as they advance worker wages. Firms use liquid assets (N W ), as the internal source of finance, but if the liquid assets are not sufficient to cover the costs, they borrow the remaining part at the interest rate r (set by the central bank) up to a maximum debt/sales ratio. Only firms that are not production rationed can try to fulfill their investment plans in this way. Prices are set by applying a mark-up on unit costs of production:
pj(t) = (1 + µj(t))cj(t) (2.7)
The mark-up dynamics is determined by the evolution of the market shares (fj):
µj(t) = µj(t − 1) 1 + νfj(t − 1) − fj(t − 2) fj(t − 2) , (2.8)
with 0 ≤ ν ≤ 1. Analogously to the capital-good market, the consumption-good market is imperfect and consumers do not switch towards better products instantaneously, but they do so in a time-consuming process, according to the competitiveness of the producers of the final goods (Ej). The competitiveness level is determined by the price and the unfulfilled demand
inherited from previous period:
Ej(t) = −ω1pj(t) − ω2Ij(t) (2.9)
The average competitiveness is the weighted average of the competitiveness of all consumption-good firm in the industry, weighted by the past market shares (fj):
¯ E(t) = N X j=1 Ej(t)fj(t − 1) (2.10)
The market shares evolve according to their competitiveness relative to average level, following a “quasi” replicator dynamics:
fj(t) = fj(t − 1) 1 + χEj(t − 1) − ¯¯ Ej(t − 2) Ej(t − 2) , (2.11)
with χ > 0. The profits (Πj) are given by the difference of sales and firms’ cost, which include
costs of production and interests on debt:
Πj(t) = Sj(t) − cj(t)Qj(t) − rDebj(t) (2.12)
The investment choices determine the evolution of liquid assets (N Wj):
N Wj(t) = N Wj(t − 1) + Πj(t) − cIj(t) (2.13)
where cIj are the internal funds devoted to investment.
At the end of each time period firms with almost zero market shares or with negative net assets die and new firms enter the market. As entrants are on average smaller than incumbents, the stock of capital (consumption-good firms only) and the stock of liquid assets (both kinds of firms) are a fraction of the average stock of incumbents.
Expectations in the production of consumption-good firms
Expectations are formed in the consumption good sector, where firms have expectations on the future demand analogously to the keynesian apparatus. The desired production Qdj is based on the expected demand of consumers and desired increases in inventories Nje.
Qdj(t) = Dje(t) + Nje(t) (2.14)
In previous versions of the K+S demand expectations (Dje) were adaptive and given by Dej(t) = f (Dj(t − 1), Dj(t − 2), Dej(t − 1), Y (t − 1)), (2.15)
where Dj is the j-th firm’s demand and Y is the gross domestic product. Notice that prices
and quantities are set in different stages and subject to different adjustment processes. The desired increase in Inventories is given by the difference between the desired level of inventories (Nd) and their stock.
Nje(t) = Njd(t) − Nj(t − 1) (2.16)
The desired level of inventories is just a share of expected demand Nje(t) = iDje(t), with i ∈ [0, 1] identical for all firms.
Finally, the desired level of production is the sum of the expected demand and the desired increase in inventories (14).
Given the actual stock of inventories, if the capital stock constrains the production plans of the firm, it invests in new machines. Thus, firms’ investment choices are affected by their demand expectations too.
The role of expectations in the model
The role of expectations will be investigated in the rest of this Chapter and in Chapter 3 as well, here a few channels, through which the effects of expectations propagate, are stylized. As pointed out in the exposition of the model above, demand expectations have a crucial
either directly or indirectly .
First, according to their investment plans consumption-good firms buy machines from capital-good firms. Firms, which produce the final capital-goods, invest to acquire state of the art technologies from the capital good firms: they replace old machines when it is profitable to do so. Thus, the demand expected by the former, in turn, influences the orders received by their suppliers. Second, based on the production plans consumption-good firms eventually produce and hire workers. Consequently, the labor demand and the number of workers employed depend on expectations, although indirectly.
Third, the costs firms have to face are influenced by their production. Now, when the liquid assets are insufficient to cover such costs, they can get credit from banks at an interest rate r. They may use external sources of finance to fund their production and investment plans. In this way, the profits of the banks may be affected by firms’ decisions, which are influenced by the expectations they have.
The channels mentioned above illustrate that expectations directly affect consumption-good firms’ decisions, but also their impact propagates to the capital-good industry, consumers2 and the financial sector.
2.3
Bounded rational expectations and heuristics
Expectation rules: Heuristics and RLS learning
In line with the experimental evidence from Anufriev and Hommes [2012a], a set of behavioral rules, or heuristics, have been implemented in the model.3
First, firms may adopt static demand expectations, or na¨ıve (NA), according to which the future demand will be equal to the present:
Dnaivee (t) = Dj(t − 1) (2.17)
This is the benchmark case with respect to which the results will be evaluated.
Second, under adaptive expectations (ADA) firms correct their past demand forecast on the basis of their past mistake:
Deada,j(t) = Dje(t − 1) + wada(Dj(t − 1) − Dje(t − 1)) (2.18)
The weight wada is set to 0.65 in accordance with the empirical evidence in Hommes [2001b]
and Anufriev and Hommes [2012a].
Third, in the weak (WTR) and strong (STR) trend expectation rules, firms behave like “chartists” traders, basically assuming that the demand trend is constant through time, i.e. the current change in demand will be in the same direction as in the past.
Dewtr,j(t) = Dj(t − 1) + wwtr(Dj(t − 1) − Dj(t − 2)) (2.19)
Dstr,je (t) = Dj(t − 1) + wstr(Dj(t − 1) − Dj(t − 2)) (2.20) 2Recall that workers consume their entire income
The difference between the two is the value of the parameter weighing past demand changes, that is wwtr = 0.4 and wstr = 1.3.
Finally, firms may follow the “anchor and adjustment” expectation rule (AA), according to which they react to both their demand pattern and to some aggregate “anchor”, the GDP.
Recursive least squares learning has been introduced in the ADA specification.4 Expecta-tions are based on the recursive procedure, according to which firms estimate the parameter (wada) of the ADA heuristic, that varies cross-sectionally and over time, given the updated
time series (wbrls,j,t).5
Sophisticated firms estimate the weighing parameter by RLS, i.e. they estimate the following equation:
Dj(t − 1) − Dej(t − 2) = const + wrls,j(Dj(t − 2) − Dje(t − 2)) + ε(t) (2.21)
They estimate the equation by OLS by using a sliding window of data that ranges from Tmin rls
to Trlsmax time period observations.
Table 2.1: Set of Expectation Rules
Expectation Rules Description NA Static expectations
ADA Firms adjust the past expected value in the direction of the past forecast error
TR Firms use the demand trend by assuming that is constant though time
AA Firms react to demand pattern and GDP anchor
ADA + RLS Adaptive specification in which firms recursively estimate the parameter by OLS
Note: All expectation rules take linear specifications. Coefficient estimated in lab-oratory experiments by Anufriev and Hommes [2012a]. Notice instead that Naive expectations were not present in the heterogeneous expectations model in Anufriev and Hommes
2.3.1
RLS learning and adaptive expectations scenario
The first experiment consists of comparing agents that recursively learn the parameter by OLS with heuristic-guided agents that adopt expectation rules with fixed parameters.
4As a robustness check, it has been implemented in the Trend Rule too, but we shall present results
for the ADA specification only.
obser-In this expectation scenario heuristic-driven and sophisticated firms coexist, since a mini-mum number of observations is required to perform least squares estimation of the parameter of interest. In particular, new entrants do not have memory of the past, simply because they were not yet in the market. Consequently, young firms follow the adaptive heuristics with fixed parameter (wada = 0.65). Only when the firm is old enough, that is it has collected
sufficient time period observations (Trlsmin) to estimate the parameter of interest wrls, does it
becomes sophisticated by forecasting the future demand via RLS. Therefore, the Economy is populated by heterogeneous firms with heterogeneous expectation rules. The share of sophis-ticated firms is affected by the entry and exit processes, as when a sophissophis-ticated firm exits the market it will be replaced by a heuristic-guided firm. Additionally, the relative share depends on the minimum number required to perform OLS, as the lower Trlsmin is, the higher the number of sophisticated firms will be.
2.3.2
Simulation results
In what follows the simulations results of the expectations-enhanced K+S model are presented. We shall analyze the forecasting accuracy of the expectation rules by computing the mean squared forecast error (MSFE). Additionally, we study the macroeconomic performance, in terms of GDP growth, GDP volatility (standard deviation of GDP growth), unemployment rate, and likelihood of crises (given by the number of crises occurred over the total time periods6Each expectations rule is evaluated with respect to na¨ıve expectations. The model is set according to the baseline parametrization in Dosi et al. [2015a], see table 6. All estimates presented below are the Monte Carlo averages for 50 simulation runs, and considering all time periods (t = 1, ..., 600). The series plots, for the sake of illustration, represent a “typical” single Monte Carlo run (when not otherwise specified).
Forecasting accuracy
We study the forecast mistakes of the firms, measured as follows: Errorj(t) = D j(t) − (Dej(t) + Nje(t)) De j(t) + Nje(t) (2.22) which includes expected inventories and we compute the mean squared forecast error by aggregating firms’ errors over time and for all Monte Carlo simulation runs. In Table ?? the forecast errors of the sophisticated firms are compared with the errors of heuristic-driven firms. On the one hand, we compare RLS firms to the heuristic-guided firms within the same simulation scenario. On the other, the sophisticated firms’ errors are evaluated with respect to na¨ıve and adaptive agents in homogeneous expectation scenarios. In both cases sophisti-cated firms are worse at forecasting than firms adopting simple heuristics. For example, they commit errors 10 times as large as the adaptive firms in the homogeneous scenario and 8.3 times as large as the simple adaptive firms in the same setting.
6A crisis occurs, when the unemployment rate is greater than 2 times its standard deviation. That
