Expectations and learning in lab experiments and in macro-economic modeling

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EXPECTATIONS AND LEARNING IN LAB

EXPERIMENTS AND IN MACROECONOMIC

MODELING

Word count: +-25000

Ruben van Eupen

student number: 01513523

Supervisor: dr. Ewoud Quaghebeur

Master’s Dissertation submitted to obtain the degree of

Master of Science in Economics

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Preamble

As this thesis is a survey of the relevant scientific literature, the corona measures had no impact on the proceedings of this thesis. The thesis was carried out as originally planned, except that consultations between student and promotor took place via e-mail only instead of also via meet-ups. This preamble is drawn up in consultation between the student and the supervisor and is approved by both.

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Voorwoord

Voor het schrijven van deze thesis heb ik uitgebreid mijn tijd genomen, wat leidde tot een zekere graad van ongemak voor enkele mensen die dicht bij mij staan. Daarom wil ik vooreerst mijn ongelooflijk fantastische levenspartner, Lise, bedanken, die ondanks alles bleef geloven, en zelfs durfde te verwachten, dat ik ooit effectief eens ging afstuderen. Hoewel deze zaak met dit schrijfsel mogelijks wel in kannen en kruiken komt, gezien enig voorzichtig optimisme dat opborrelde bij mijn promotor, Ewoud Quaghebeur, tijdens het schrijfproces van deze thesis, had het even goed anders kunnen lopen. Inderdaad, verwachtingen kunnen een zichzelf vervullende voorspelling blijken te zijn, en eenmaal daardoor een zekere weg ingeslaan in het leven, kunnen andere initiële mogelijke uitkomsten opeens onbereikbaar blijken. Daarom dus heel veel dankbaarheid om in mij te blijven geloven, Lise, en mij altijd op het juiste pad te helpen. Mijn financiële voorspellingen bleken dan weer minder accuraat, daarom bedank ik graag mijn ouders voor de erg vrijgevige financiële injectie tijdens de laatste loodjes, zodat ik mijn thesis kon afwerken badend in alle decadente luxe die we in onze Westerse wereld gewend zijn. Uiteraard had deze thesis er niet kunnen komen zonder de nodige ontspanning nu en dan, die altijd leuker is in het gezelschap van vrienden en familie. Dus, bedankt allemaal voor de fijne momenten. Een extra bedankje verdienen mijn collega FPV piloten: het leven is niet hetzelfde zonder regelmatig de vreugde van een geslaagde mattyflip te kunnen delen. Graag bedank ik ook Ewoud Quaghebeur voor de goede begeleiding van mijn thesis. Het was gemakkelijk afspraken te maken en er kwam altijd ten gepaste tijde de nodige feedback. Sommige mensen namen de moeite om delen van mijn thesis na te lezen: Ben ‘GOE uitgelegd!’ C., Inge ‘gortdroog’ S., Daphne, Steventje, Lise, de mama, Jakob ‘da cunning linguist’ S., Ruben VdM en Jelle ‘de verhevene’ S. verdienen een eervolle vermelding. Inukshuk ontwierp het gratis te gebruiken https://anystyle.io/, wat mij toeliet het opstellen van mijn bibliografie grotendeels te automatiseren. En Politie Gent zorgde regelmatig voor de nodige rust in mijn straat om lang genoeg in dromenland te kunnen vertoeven. Hoewel initieel dit thesisonderwerp mij niet zo een heet hangijzer leek, begon ik het gaandeweg steeds interessanter en leuker te vinden. Uiteindelijk heb ik met plezier aan deze thesis geschreven, al ben ik uiteraard ook blij dat ze nu afgerond is.

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List of abbreviations

2-type HSM HSM of Anufriev, Hommes and Philipse (2013) 4-type HSM HSM of Anufriev and Hommes (2012)

AE adaptive expectations ALM actual law of motion

sBH-HSM simplified Brock and Hommes heuristic switching model

BHST12 research paper of Bao, Hommes, Sonnemans and Tuinstra (2012) DSGE dynamic stochastic general equilibrium

eq. equation

FN-HSM fundamental vs. naive agent heuristic switching model GA-P2 learning model of Anufriev, Hommes and Makarewicz (2019)

HHST09 research paper of Heemeijer, Hommes, Sonnemans and Tuinstra (2009) HSM heuristic switching model

HSTV05 research paper of Hommes, Sonnemans, Tuinstra and van de Velden (2005) iid independent and identically distributed

OLS ordinary least squares

LtFE learning-to-forecast experiment LtOE learning-to-optimize experiment

NK New Keynesian

OLG overlapping generations PLM perceived law of motion RE rational expectations

REE rational expectations equilibrium RPE restricted perceptions equilibrium

SCEE stochastic consistent expectations equilibrium

SE squared error

TFP total factor productivity

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Permission

I declare that the content of this Master’s Dissertation may be consulted and/or reproduced, provided that the source is referenced.

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Introduction

This thesis addresses the question of how individuals form expectations about macro-level economic variables and adjust, update or select their forecasting heuristics in several different economic contexts. Furthermore, the implications of boundedly rational expectations for the properties of the business cycle, as well as for the conduct of monetary and fiscal policy are discussed.

This boundedly rational expectations approach is contrasted with the influential hypothesis of rational expectations, which has become mainstream in economics since the 1970s. The benefits and limitations of rational expectations are laid out in Chapter 1. In Chapter 2, we look at boundedly rational expectations, focusing on an adaptive learning approach, in which economic agents are assumed to behave like statisticians, forming expectations based on simple heuristics, adjusting either the type of heuristic or its coefficients, or both, over time, as new data becomes available.

Many different theories of learning have been proposed that tried to discipline the wilderness of bounded rationality by empirically validating these theories in various experimental setups, such as a learning-to-forecast design. First, a survey of field data studies and experiments on expectation formation is presented, followed by a discussion of several influential theories of learning that were developed based on these findings. Next, research papers building macro-economic models that adopted some form of adaptive learning are discussed. General properties, as well as monetary and fiscal policy implications of these behavioural macro models are examined, and compared with macro models assuming rational expectations. In Chapter 3, a few non-learning approaches are briefly mentioned.

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Chapter 1. Rational Expectations

1.1. History

“You can fool some of the people all of the time, and all of the people some of the time, but you cannot fool all of the people all of the time.”

quote usually attributed to Abraham Lincoln

Virtually all microeconomic and macroeconomic decisions depend on expectations concerning uncertain future outcomes. For example, investors continuously make net present value calculations, conditional on expected future interest rates and prices. Consumer spending today depends partially on their expected future income. Likewise, an unemployed worker who turns down a job offer because he considers the earnings inadequate, entertains the expectation that there are preferable employment opportunities available elsewhere.

The earliest indications of our understanding of the crucial role of expectations for the economy date back to the Old Testament (Genesis 41-47) and narratives about Thales of Miletus (c. 636–c. 546 B.C.) as described in Aristotle’s Politics, when forecasting good or bad harvests could make you a profit if you timely reserved the oil presses or stocked up a few additional grain silos (Evans and Honkapohja, 2001, p. 6).

In contemporary macroeconomics, the economy is understood as an expectation feedback system: individual expectations about the future state of the economy will affect the current state of the aggregate economy and thus its future path. In turn, past and current states of the economy will influence individuals’ expectations about its future (Anufriev & Hommes, 2012).

The development of formal expectation modeling has come a long way since Adam Smith give birth to economics as a separate discipline. For the classical economists, whose ideas flourished in the late 18th and early-to-mid 19th century, the economy was thought of as a sequence of static equilibria. As Evans and Honkapohja (2001, p.6) wrote: “a part of this classical interpretation was the notion of perfect foresight, so that expectations were equated with actual outcomes. This downplayed the significance of expectations”.

Between the end of the Second World War, and the mid 1960s, a Keynesian consensus dominated both macroeconomic theory and policy making, practically everywhere in the non-communist world. Expectations were a central theme in Keynes’ work. For example, in his Treatise on Probability (1973a), he investigated the possibility of acting rationally under uncertainty. Keynes’ concepts of precautionary savings, liquidity preference and conventional behaviour, are all particular manifestations of the attempt to get protection against the losses that could result due to uncertainty (Carvalho, 2015). Keynes used the felicitous “animal spirits” expression (waves of optimism and pessimism) to refer to “a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative beliefs multiplied by quantitative probabilities” (Keynes, 1936, p. 161). However, this concept is extraneous to his analytical

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developments, as Keynes’ effective demand model had the perfect information assumption as its cornerstone (De Vroey, 2016, p. 17) and expectations were considered as exogenous variables (Dovern, 2018). That is, expectations are conceptualized as not determined by the parameters of the economic system.

Adaptive expectations, introduced by Fisher (1911), and popularized by Cagan (1956) and Friedman (1957), played a prominent role in (Keynesian) macroeconomics in the 1960s and 1970s. For example, in this era, inflation expectations were modeled adaptively in the analysis of the expectations augmented Phillips curve, during the monetarist “counterrevolution”. The hypothesis of adaptive expectations postulated that individuals used information on past forecasting errors to revise current expectations (Sent, 1998). The formula for adaptive expectations is given by 𝐸𝑡[𝑥𝑡+1] = 𝐸𝑡−1[𝑥𝑡] + 𝜆(𝑥𝑡− 𝐸𝑡−1[𝑥𝑡]) (Gallego,

n.d.). There are two limiting cases: when 𝜆 = 1 , this equation reduces to naive or static expectations and with 𝜆 = 0 the formula simplifies to constant expectations. However, no widely accepted economic theory was offered to explain the magnitude of the adjustment parameter 𝜆 (Sent, 1998, p.7). In the jargon of control theory, this is an example of a constant gain algorithm, proposed at the time as a plausible and empirically meaningful way to track an unknown time-varying system (Evans and Ramey, 2005).

The limitation of purely adaptive expectations is that economic agents are assumed to be purely backward looking. As such, current expectations about the future are not taken into account. This means that neither anticipated policy changes nor anticipated demand, supply or other shocks will directly influence adaptive expectations made for the following period. Expectations will therefore be persistently biased, especially if the underlying variable follows a trend (Dovern, 2018). The adaptive expectation approach was subject to the ‘‘Lucas critique’’, which showed that expectation parameters (and endogenous variable dynamics) depend on policy parameters (but see Evans and Ramey (2005) for a more detailed treatment) (Lucas, 1976). Backward-looking models of expectations suggested a constant rigidity in economic models that, in theory, allowed policymakers to systematically affect the macro-economy (De Vroey, 2016).

This “Lucas critique” was resolved by the rational expectations hypothesis (Sargent, 1987). The basic idea of rational expectations is that agents are forward looking, and that it is rational to use all available information (Muth, 1961; Lucas, 1976). The rational expectations hypothesis states that individual expectations about the future state of the economy might still be wrong, but that the errors would be random. For if errors followed a pattern, they held information that could be used to make more accurate forecasts (Sent, 1998). Moreover, individual forecasting errors are considered to cancel out at the aggregate level. An important underpinning of the rational expectations approach comes from an early evolutionary argument made by Friedman (1953), that “irrational” agents will not survive competition and will be driven out of the market by rational agents, who will trade against them and earn higher profits (Hommes, 2013). We will later see this is not necessarily always true in an economic system with boundedly rational agents.

Since the seminal works of Muth (1961) and Lucas (1972a, 1972b), dynamic stochastic equilibrium models (DSGE), featuring homogeneous agents with rational expectations, have become the mainstream tool of analysis in macroeconomics (Hommes, 2011). This early work by Lucas lead to a Real Business Cycle (RBC) baseline model. The RBC approach was kickstarted by the seminal contributions by Kydland and Prescott (1982) and Long and Plosser (1983) (De Vroey, 2016, p. 262). Still within the same framework,

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the DSGE methodology eventually lead to a class of models which De Vroey (2016) refers to as ‘second-generation new Keynesian modeling’. De Vroey (2016, p. 307) identifies two important modifications. First, the replacement of the initial perfect-competition framework with flexible prices with a framework with monopolistic-competition with rigid prices. Second, the return to the forefront of the monetary side of the economy. However, even these advanced contemporaneous New Keynesian DSGE models do not seem to be able to easily explain certain macroeconomic phenomena while assuming rational expectations. This will be discussed in more detail in Chapter 2.

1.2. Defining Features and Implications

The rational expectations (RE) hypothesis of the 1970s places individual optimization and expectation formation at the forefront of macroeconomic research (Branch et al., 2012). According to Thomas J. Sargent (n.d.), rational expectations should rather be understood as a modeling technique, not as a school of thought. As such, it can be part of a wide array of economic models ranging from RBC models, over New Keynesian DSGE models, to even boundedly rational models, such as, for example, rational inattention models.

The basic idea of the rational expectations (RE) concept is that agents use all relevant information in forming expectations, possibly including probabilistic knowledge about future events. Therefore, the rational expectation hypothesis postulates that agents’ expectations are no longer systematically biased, as with adaptive expectations, so that errors are random. This does not necessary imply individual rationality, because individual expectations may still be wrong, but will be correct on average (Kallianiotis, 2013, p.78). Rational expectations usually imply purely forward looking forecasts, unless the underlying model postulates that the to-be forecasted variables have a particular structural relationship to their past realizations, as in markow switching models (Caraiani, 2018).

In a typical rational expectations framework, all agents are the same and forecast rationally (Anufriev & Hommes, 2012). Usually, but not necessarily, also full information is assumed in this context. Rational expectations are formulated as the optimal conditional expectation given all available information, and all agents take as their subjective expectation of future variables the objective prediction by the economic model they are part of (Hommes, 2013). Together, these two assumptions ensure that expectations are model consistent and coincide, on average, with realizations, without systematic forecasting errors. Because, on average, expectations and realizations coincide, the RE framework provides an elegant “fixed point” solution to an economic expectations feedback system (Hommes, 2013). In other words, rational expectations constitute an equilibrium in the feedback loop between aggregate economic outcomes and individual expectations, which closes the self-referential model (Evans & Honkapohja, 2001) and insures internal consistency (Sargent, 1987b). This implies that each rational agent knows the true law of motion of the model economy (Evans & Honkapohja, 2001) and can also identify the type of exogenous shock, when it occurs (De Grauwe, 2012). Moreover, the assumption of rational expectations also implies that agents know the true statistical distribution of all shocks hitting the economy (De Grauwe, 2012). Moreover,

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agents in a RE framework are able to calculate the equilibrium of the model, and presume that all other agents will choose the same equilibrium strategies, in essence implying a perfect coordination of expectations (Driscoll & Holden, 2014).

In the absence of exogenous shocks, rational expectations imply that agents have perfect foresight and make no mistakes. In the RE framework, there is no room for market psychology and “irrational” herding behaviour, instead, it postulates that expectations are in equilibrium and perfectly self-fulfilling (Hommes, 2013).

1.3. Achievements and Limitations

The rational expectations approach has important advantages: it is simple, elegant and puts a strong discipline on individuals’ forecasting behavior, minimizing the number of free parameters (Anufriev & Hommes, 2012). In one word, it creates parsimony (Mallard, 2015). According to Mirjam Sent (1998, p. 7), some economists believe that the success of rational expectations may be attributed to its ability to fight the threat of indeterminacy (see, e.g., Lucas and Sargent 1979). This indeterminacy followed from the realization that it was possible to formulate predictions that are either self-falsifying or self-fulfilling (Merton, 1948). This led some to believe that economic models could produce so many outcomes that they were useless as instruments for generating predictions (Sent, 1998). Rational expectations, however, is a powerful hypothesis for restricting the range of possible outcomes since it usually (see e.g. Gauthier, 2004) leads to an equilibrium that is a fixed point of a particular mapping from believed laws of motion to the actual laws of motion for the economy (Sent, 1998). Harstad and Selten (2013) present a more detailed discussion of the benefits of the rational expectation framework.

The rational expectations approach assumes that economic agents have learned all there ever is to learn from past mistakes. As such, it addresses the critique of adaptive expectations, embodied in the latter part of Lincoln’s quote (see section 1.1). However, it completely ignores the first two statements. This approach also implicitly assumes that the law of motion of the economy is known to all agents. This in turn requires each agent to know what all other agents will forecast and decide, which seems only a reasonable assumption if one also assumes homogeneous agents. In other words, agents are considered to know the prevailing equilibrium (Rubinstein, 2002).

The rational expectations hypothesis also implicitly assumes that the law of motion of the economy does not change (when structural parameters are fixed), which actually is a self-fulfilling assumption as long as all agents behave rationally, leading to “expectations in equilibrium” (Hommes, 2013). However, as we will see in Chapter 2, this is not longer true in models that introduce boundedly rational agents (Hommes, 2013, p27).

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As early as 1957, Simon realized that rationality requires extreme assumptions concerning agents’ information gathering and computing abilities. As Hommes (2013) notes:

rational agents are typically assumed to have perfect information about economic fundamentals and perfect knowledge about underlying market equilibrium equations. This assumption seems unrealistically strong, especially since the “law of motion” of the economy depends on the expectations of all other agents. even if such information and knowledge were available, typically in a nonlinear market equilibrium model it would be very hard, or even impossible, to derive the rational expectations forecast analytically, and it would require quite an effort to do it computationally. (p.7)

Proponents of the rational expectation hypothesis sometimes defend it by arguing that agents, although they do not forecast rationally, act as if they do. The rational expectation hypothesis therefore makes up a good approximation. Moreover, agents could learn optima through experience, and the rational expectation hypothesis therefore acts as a shortcut to the eventual outcome. Indeed, many research focused exactly on the question under which conditions (Conlisk, 1996) rational expectations are learnable (Evans & Honkapohja 2001). Peseran (1988) argues that learning will not in general converge on rational expectations equilibria. Cases where this has been found to occur are special ones (Pesaran, 1988). Therefore, the as if statement has become conditional on a principle known as the E-stability principle (see Evans & Honkapohja, 2001), and bounded rationality should be investigated.

Indeed, one can wonder how people can form rational expectations about how the economy responds to extreme shocks when most people have never experienced this (Heylen, 2019), nor studied economic history. Similarly, the shock can be of a different nature than all previous ones, in an economy that might have structurally changed since the last big shock (Stiglitz, 2011). At the macro level, frequently overheating economies leading to asset price bubbles, and followed by deep lasting downturns are hard to reconcile with the behaviour of fully rational, homogeneous agents. For example, overly optimistic expectations about the economic outlook may have exaggerated the excessive growth in housing prices in 2000–2008, while an overly pessimistic outlook by the public may have amplified the Great Recession of 2008 and deepened the following economic crisis (Hommes, 2011). In addition, empirical investigations of time series of asset prices show a number of market phenomena that are hard to reconcile with rational agents, including bubbles, crashes, short-run momentum and long-run mean reversion (Dieci & He, 2018). Similarly, Pagan (1996) and Lux (2009b) discuss several stylized facts for asset prices: excess volatility, excess skewness, fat tails, volatility clustering, long range dependence in volatility, and various power-law behaviors.

The rational expectations assumption has been employed in a wide variety of models. Contemporaneous state-of-the-art models assuming rational expectations often make use of the dynamic stochastic general equilibrium (DSGE, e.g. Smets & Wouters, 2007) or the overlapping generations (OLG, e.g. Auerbach & Kotlikoff, 1987) paradigm. Several authors have questioned these type of models on several grounds (e.g. Blanchard, 2018). Only a few of the issues raised in the literature regarding the DSGE or OLG methodology can be addressed by reformulating the way they model expectation formation. In the

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remainder of this section, I limit the discussion to these issues specifically. In Chapter 2, we will discuss how successful the boundedly rational expectation approach is in tackling these issues.

According to Stiglitz (2011), it is important that macroeconomic theory focusses on the right questions. Not only should macroeconomic models be able to explain small “normal” variations in aggregate variables, they should also be able to explain the occurrence of endogenous, periodic bubbles and deep downturns, and provide guidance as to how to recover quickly from these large recessions. Moreover, according to Stiglitz (2011), these new models should focus on why market economies amplify shocks, why the effects of shocks are persistent and why recovery is slow. Stiglitz (2011) claims that most important disturbances are endogenous, not small exogenous shocks, and DSGE models typically can only deal with the latter. As we will see in Chapter 2, the boundedly rational expectations literature does provide relieve for these concerns.

Typically, in New Keynesian DSGE models, a set of distortions and extensions are added to the pure “RBC DSGE model” in order to better match the empirical evidence on business cycle data and the impact of various policies: nominal price and wage rigidities (e.g. Calvo pricing), “hand to mouth” consumers, monopolistic competition, information problems, allowing for investment and capital accumulation and financial intermediation. However, according to Blanchard (2016, p.2), many of these added distortions are at odds with the empirical evidence. In the same paper, Blanchard argues the same is true for the value of many calibrated parameters.

Korinek (2015) points out that if DSGE models need to employ fundamental parameter values or assumptions that are at odds with empirical estimates at the micro level, in order to replicate certain aggregate summary statistics of the economy, then the model is not actually capturing the true microeconomic incentives faced by economic agents, but is ‘bent’ to fit the data. As we have discussed extensively, this is especially true for the assumption of rational expectations. Even more so, because of this oversimplifying RE assumption, other extra model features need to be included that are at odds with microfoundational evidence as well. Examples are utility functions that exhibit strong habit persistence in the New Keynesian literature so as to fit the behaviour of the inflation rate, or the elasticity of labor supply, which is typically assumed to be an order of magnitude higher than what is observed in micro data, so as to fit the observed response of employment in recessions (Korinek, 2015). According to Korinek, tractability concerns leading to very particular and simplifying assumptions, biases results towards very particular specifications in which markets are stable and efficient. Therefore, he argues, key questions of interest are, in effect, answered by assumption”. In Chapter 2, we will see how boundedly rational expectations can provide a simpler, more elegant modeling solution, while explaining various macroeconomic phenomena better than models simultaneously postulating homogeneous rational agents and various counterfactual microfoundational fixes to match the data.

The broader point of the Lucas critique is that useful predictions and policy recommendations depend on including the relevant microfoundations in a macroeconomic model for the research question at hand. In the 1970s, this led to the RE revolution, which pointed out that monetary policy cannot permanently increase output since if economic agents have rational expectations, they can foresee that permanently expansive monetary policy only leads to inflation (Korinek, 2015). However, some key observations like

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excess volatility in macroeconomic data, and certain macroeconomic co-movements after shocks, cannot be easily explained by models using rational expectations. We will discuss in Chapter 2 to which extent boundedly rational expectations can provide an elegant explanation for these phenomena.

Chari, Kehoe, and McGrattan (2008) argue that, for a model to be useful for policy analysis, it needs enough microfoundations consistent with the data, such that both the model’s shocks and parameters become structural, in that they can reasonably be argued to be invariant to the policy shocks the model is set-up to investigate. Moreover, these shocks should have a clear, primitive, interpretation. They then go on to argue that for a modern state-of-the-art New Keynesian DSGE model, the Smets and Wouters (2007) DSGE model, this is not the case. The Smets and Wouters DSGE model uses shocks with an AR(1) process, which exhibit very persistent autocorrelation. Moreover, Slobodyan & Wouters (2012) found that when adaptive learning was introduced into the Smets-Wouters DSGE model, using an AR(2) heuristic, the data were fit better, while autocorrelations in the shocks could be reduced (Hommes, 2018).

A realistic model of the macroeconomy should allow for the possibility of a crisis other than through large exogenous shocks (Hommes, 2018). As we will see in Chapter 2, boundedly rational expectations can allow for endogenous waves of optimism and pessimism, which causes excess volatility and can lead to booms unrelated to economic fundamentals, as well as to deep and protracted recessions.

Vines & Wills (2018), argue that four main changes to the standard DSGE RE model in macro-economics are recommended: first, to emphasize financial frictions, second, to place a limit on the operation of rational expectations, third, to include heterogeneous agents, and fourth, to devise more appropriate microfoundations. The literature on boundedly rational expectations addresses all latter three issues, as we will see in Chapter 2.

In the Chapter 2, first, research that looks at how the rational expectation hypothesis holds up at the microeconomic level is reviewed, focusing mainly on laboratory experiments. Subsequently, I discuss how these findings got integrated in macroeconomic modeling techniques that incorporated some form of boundedly rational expectation formation. Lastly, general model properties and monetary and fiscal policy implications of the rational versus boundedly rational approach to expectation formation are compared.

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Chapter 2. Boundedly Rational Expectations

2.1. Rationale

Allowing for cross-sectional differences in expectations is a simple matter, because their aggregate effect is negligible as long as the deviation from the rational forecast for an individual firm is not strongly correlated with those of the others. Modifications are necessary only if the correlation of the errors is large and depends systematically on other explanatory variables. (Muth, 1961, p.321)

As early as the mid 1950s, Herbert Simon (1955) strongly argued for the use of boundedly rational expectations in economic models (Friedman & Rubinstein, 1998). At the time, Simon’s reasoning was overshadowed by the rational expectations revolution in the 1970s (Hommes, 2013, p. 7). The rational expectations revolution in economics took place before the time that the irregular behaviour and complexity of nonlinear dynamics were widely known among economists (Hommes, 2013, p. 6). However, the complexity view, or more broadly, the boundedly rational expectations view, has again attracted more and more interest since the late 1980s (Hommes, 2013).

In one of the leading interpretations of the boundedly rational expectations approach, agents are conceived as ‘intuitive statisticians’, forming expectations based upon time series observations, using a simple statistical model for their perceptions about the motion of the economy (Salehnejad, 2007). As such, agents are not assumed to know the actual law of motion of the economy (Hommes, 2013), as with rational expectations, but instead construct a perceived law of motion, based on simple heuristics. The actual law of motion, given these expectations held by the agents, then becomes a mapping from these perceptions to a temporary equilibrium of the economy (Friedman & Rubinstein, 1998). This so-called adaptive learning approach, postulates that agents, at each moment in time, make forecasts formulated on the basis of available data (Evans & Honkapohja, 2001). They do this by updating the parameters of their perceived law of motion according to some learning scheme that minimizes a particular objective function of the agent (Hommes, 2013). For example, this objective function can be based on the forecast error of their expectations. These expectations are revised over time as new data become available, usually through recursive ordinary least squares (Hommes, 2018) or a constant gain method. In most macroeconomic models that incorporate some form of adaptive learning, economic agents are still assumed to maximize utility and/or profit given their forecasts at each moment in time (Hommes, 2013).

The rationale for this view comes from the information and deliberation costs involved in coming up with a rational solution for a complex problem by an agent that is constrained both in terms of information gathering and in computing abilities. These costs are reduced considerably when decisions are made based on simple heuristics (Conlisk, 1996). In good economics, all relevant costs should be taken into account.

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Another feature that is overlooked in a homogeneous rational agent approach of modeling the macroeconomy, is the possibility of heterogeneity of expectations among agents. In section 2.3, this approach will be discussed in detail. According to Hommes (2018), the interaction of both heterogeneity and nonlinear feedback in the economy can lead to the existence of multiple equilibria, chaos and bifurcations. Such a complex economic system may then shift from a desirable steady state to an undesirable steady state, e.g. a deep recession, due to a small change in the interactions at the micro level. In such a system, recovery may not be automatic, nor easy, as is illustrated by Scheffer’s (2009) concept of critical transitions or tipping points.

The learning literature originally focused on the conditions under which an (RE) equilibrium would be stable when rational expectations are replaced with an adaptive learning rule. It also offers a way of selecting among multiple equilibria, which is a major conundrum for many rational expectations models (Evans and Honkapohja, 2001).More recently, there has been a shift in focus to transitional or persistent learning dynamics that have the potential for explaining phenomena that remained ill-explained in a RE framework. As noted by Branch, Evans and McGough (2012), in the early literature, adaptive learning was applied either to ad-hoc or finite horizon models. Later, the first attempts at modeling adaptive learning in infinite horizon DSGE models employed so called “reduced-form learning,” in which RE are replaced in the equilibrium conditions with a boundedly rational expectations operator and the stability of the equilibrium is then studied (see, e.g. Evans & Honkapohja (2001) and Bullard & Mitra (2002). However, the ad-hoc nature of reduced-form learning was at odds with the requirement for microfoundatations in modern macroeconomic models (Branch et al. 2012). This issue was addressed by the development of Euler equation learning by Honkapohja et al. (2002) and Evans and Honkapohja (2006). Evans and Honkapohja (2006) also showed that in a New Keynesian (NK) model, Euler-equation learning is equivalent to reduced form learning (Branch et al. (2012). Euler-equation learning identifies agents as two-period planners: they make decisions today based on their forecasts of tomorrow. Under rationality, this type of behaviour is optimal: forecasts of tomorrow contain all the information needed to make the best possible decision today. If agents are boundedly rational, however, it is less clear that a two-period planning horizon is optimal (Branch et al. 2012). However, a longer planning horizon would not only be at odds with agents’ supposedly limited cognitive abilities, but also postulate that agents’ behaviour is predicated upon the assumption that their beliefs are correct, which is a-priory unlikely in a context where agents, or at least the economists specifying the model, know agents’ beliefs are likely to be inaccurate and will need adjustment over time. For a more detailed historic overview of boundedly rational expectations research, I refer to Branch et al. (2012), upon whose work this and the next paragraph is mainly based.

For some issues, expectations at longer horizons do make sense in a learning context. For example, for fiscal policy analysis, one may want to take into account beliefs about future effects of how a growing government deficit will be financed, or assess the effects of temporary increases in public expenditures on future expected taxes (Hommes, Massaro & Salle, 2019). To this end, Marcet and Sargent (1989) and also Preston (2005) introduced an alternative approach, namely, “infinite horizon learning,” in which agents use forecasts of the whole time path of future variables to make current economic decisions. Yet another approach, shadow price learning, was developed in Evans and McGough (2010) and assumes that agents

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are two-period planners who make choices conditional on the perceived value of additional future state variables. They show that, under certain circumstances, shadow price learning reduces to Euler equation learning (Branch et al., 2012). To the best of my knowledge, so far, no experimental study implemented these learning concepts developed for longer horizons.

In section 2.2, I review the microeconomic evidence regarding expectation formation by economic agents and address the question what behaviour arises at the macro level when heterogeneous boundedly rational agents interact. In section 2.3, several theories of learning are presented. These are either rooted in the experimental work that is set out in section 2.2, or are influential techniques used in macroeconomic modeling (section 2.4). Although laboratory experiments and their corresponding theories of learning are usually published within the same paper, these are discussed more or less separately in this thesis in order to provide a clear overview of the literature. However, for the layman reader, it may be necessary to take a look at section 2.3.1 while reading section 2.2., in order to clarify some concepts. In section 2.4, we look at how boundedly rational expectations got incorporated in macro-economic models. Here, also, policy implications of these behavioural models are addressed. In chapter 2, I focus mainly on the learning approach. In Chapter 3, I touch upon a few alternative approaches that have been used to relax the rational, full information expectation assumption in macroeconomic models.

2.2. Evidence

2.2.1. Introduction

The literature on bounds of rationality in individuals shows many deviations from rationality (Conlisk, 1996). In this chapter, we address the question which of these deviations matter for the typical rational expectation formation process that is often assumed in macroeconomic models and assess the implications of bounded rationality at the aggregate level. Model-based inference of expectation formation faces the difficult task of identifying unknown expectations from an equally unknown set of other model properties (Kryvtsov & Petersen, 2013). Different combinations of particular expectation formation mechanisms and particular other model features might explain the data equally well, without a definite procedure to select the most appropriate combination. Therefore, many studies investigating expectation formation look at survey data, or use an experimental approach.

Laboratory experiments in economics that focus on the issue whether agents are able to take actions that are in line with dynamic stochastic intertemporal optimization behaviour are broader than the topic of expectation formation alone. Some authors focus on optimal consumption-savings decisions, or on time (in)consistency of preferences (reviewed by Duffy, 2014). In this master thesis, I only focus on research, mainly laboratory experiments, investigating expectation formation.

Evans and Honkapohja (1992, 2001) show that for a broad class of macro models, the asymptotic stability of the rational expectations equilibrium (REE) under adaptive learning depends on the

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expectational stability principle, as long as the agent’s estimated perceived law of motion nests the rational expectations equilibrium under consideration. If, in contrast, agents use a misspecified, underparametrized expectational rule, or don’t use all available information, Evans and Honkapohja (2001) show that full convergence under learning to this rational expectations equilibrium is impossible. An adaptive learning rule for an agent in a New Keynesian (NK) macro model economy is typically misspecified (Hommes, 2018, p. 17). However, this does not preclude convergence of the estimators to a some form of misspecification equilibrium. This could be either a restricted perceptions equilibrium (Branch & Evans, 2010), a behavioural learning equilibrium (Hommes & Zhu, 2014), a limited information learning equilibrium (Chung & Xiao, 2014), an exuberance equilibrium (Bullard, Evans & Honkapohja, 2008) or a stochastic consistent expectations equilibrium (Hommes and Sorger, 1998; Hommes et al., 2013). In the latter, agents learn the optimal parameters of a simple, parsimonious AR(1) rule. If agents are aware of their possibly misspecified forecast rule, they may choose to adopt an expectational rule with a gain sequence bounded above zero. With a positive, non-zero sensitivity of forecast rules to new data points, agents are able to track an economic structure which is evolving over time, at the disadvantage of randomly fluctuating forecasts (non-convergence) in the limit (Evans & Honkapohja, 2001). This contrasts with, for example, recursive least squares learning, where the gain decreases to zero in the limit (Evans & Honkapohja, 2001). As argued by Hommes (2018), constant gain learning models provide a better fit to macroeconomic and financial data and are able to generate observed stylized facts in time series data, such as high persistence, excess volatility and clustered volatility (Evans & Honkapohja (2001), p.49; Sargent (1993), Milani (2007, 2011); Branch & Evans (2010), De Grauwe & Ji (2017). Constant gain learning is especially useful when the economy goes through structural changes from time to time, as it assures perpetual learning. Also, from a behavioural point of view, discounting of past data can be seen as a way to formalize finite-memory forecasting by the agents (Quaghebeur, 2017).

Another use of the adaptive learning approach is to select and provide justification for a particular rational expectations equilibrium for models that have non-unique solutions (Duffy, 2016). Especially when expectations become boundedly rational, multiple equilibria models become more common, and lab experiments probably are the best method to provide a selection mechanism to indicate the most plausible, learnable, solutions, as economic theory cannot answer this question (Lucas, 1986). As such, an analysis of the stability under adaptive learning offers a check on the robustness of equilibria with respect to expectational errors (Evans & Honkapohja, 2001). Furthermore, when a sound theory of expectations is missing, empirical tests of dynamic macroeconomic models face the difficulty of testing joint hypotheses. When a model’s predictions are at odds with the empirical data, it is not clear whether it is because of an incorrect assumption regarding expectation formation or another misspecification of the model (Assenza, Bao, Hommes and Massaro, 2014).

Hommes (2013, p. 37) shows that even in a simple nonlinear cobweb model, the replacement of rational expectation rules by simple adaptive expectation heuristics may lead to chaotic price fluctuations. An issue that has received much attention in the literature is whether expectation formulations based on adaptive learning converge asymptotically to the fully rational expectation equilibria, making the latter more plausible as a long run description of the economy (Evans & Honkapohja, 2001, p. 22). As argued by Duffy

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(2016), in many cases, it has been found that adaptive learning does not converge to rational expectations equilibria. Hommes (2018) lists the following examples: “learning equilibria in overlapping generations models (Bullard 1994; Grandmont, 1985; 1998), learning to believe in chaos (Schönhofer, 1999), the consistent expectations equilibria in nonlinear cobweb models (Hommes & Sorger 1998), the learning to believe in sunspots (Woodford, 1990) and exuberance equilibria (Bullard et al., 2008)”. As we will see more extensively in section 2.2.2, adaptive learning could also lead to bubble-bust phenomena, for example, when the evolutionary selection of forecasting rules leads to agents converging on trend extrapolation forecasting strategies (Anufriev & Hommes, 2012; Brock & Hommes, 1998). These findings stress the importance of what Hommes (2018) calls the complex systems approach.

2.2.2. Non-Experimental Studies

2.2.2.1. Introduction

As the main focus of this thesis is on experimental research, this survey of evidence for boundedly rational expectations in non-experimental work will be rather brief. As discussed earlier, within the boundedly rational expectations literature, an influential approach is adaptive learning with or without heuristic switching between a set of expectations (heterogeneous expectations approach). We will focus on this approach in this section, as opposed to non-learning approaches such as rational inattention, “near-rational” beliefs or eductive approaches. The latter approaches will be discussed briefly in the remainder of this thesis (Chapter 3).

Apart from experimental work, Anufriev, Hommes and Makarewicz (2019) note that empirical evidence for a heuristic switching learning model can be found in survey data (e.g. Branch, 2004), estimated financial models (e.g. Boswijk, Hommes and Manzan, 2007), estimated DSGE models (e.g. Cornea-Madeira, Hommes and Massaro, 2017) and housing market models (Bolt, Demertzis, Diks, Hommes and Van der Leij, 2014). While many tests of rational expectations have been conducted using survey data, (e.g. Frankel and Froot, 1987), these tests do not always include a self-referential aspect and are beset by problems of interpretation. These difficulties result from uncontrolled variations in the underlying fundamental factors, or from the limited incentives of forecasters to provide accurate forecasts, or from disagreement about the true underlying model or data generating process (Duffy, 2016). Despite these difficulties, several researchers investigated expectation formation, and its effect at the macro-level, through survey data. In this section, we focus on tests of rational expectations, and in section 2.2.2.3, studies that investigated the relation between the macro level and expectations, based on survey data, are reviewed.

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2.2.2.2. Tests of Rational Expectations

Early work based on survey data pointed out that generally, consumers, professional economists and other market participants show that forecast errors for macro-economic variables have a non-zero mean, are correlated with other observable information, and follow an adaptive process (Camerer, 1995, pp. 609– 611). For example, prolonged misevaluation of the future value of assets was observed in several contexts (Goodman & Ittner, 1992; Reinhart & Rogoff, 2009; Kindleberger & Aliber, 2011). Case et al. (2012) conducted a survey of households’ expectations regarding their home value and rejected the RE hypothesis. They conclude that people’s expectations are consistent with trend-extrapolation and that people systematically misjudge the long-term value of their houses (Anufriev et al., 2019). As noted by Assenza, Bao et al. (2014), Shiller (1990) and Case, Shiller, and Thompson (2012) find that during the housing market boom, investors in the US housing market expected housing prices to grow at an extremely high rate that cannot be supported by reasonable estimates of fundamental factors. Investors also expected an even higher long-run growth rate, although the growth could not reasonable be expected to be sustainable, and indeed the market collapsed soon afterwards.

Other studies used surveys of inflation expectations. For example, Malmendier and Nagel (2009) study inflation expectations in the Reuters/Michigan Survey of Consumers. They find that differences in life-time experiences strongly predict differences in subjective inflation expectations. Additionally, their evidence supports perpetual learning, e.g. via a constant gain mechanism, as memory fades gradually over time, which leads to the underweighting of past memories. In contrast, Mankiw, Reis, and Wolfers (2003) found evidence for substantial disagreement among both consumers and professional economists regarding inflation expectations in the Michigan Survey of Consumers. They state that the data are inconsistent with rational or adaptive expectations, but may be consistent with a sticky information model. They report that this disagreement shows substantial variation through time, moving with the level of inflation, the absolute value of the change in inflation, and relative price variability. Capistran and Timmermann (2009) report similar results, with heterogeneity of inflation expectations of professional forecasters varying over time, depending on the level and the variance of recent inflation. Branch (2004) presents the results for three competing forecasting models of expectation formation estimated based on Michigan survey data on inflation expectations. He finds that about 48% of agents use a VAR predictor, which in this context can be interpreted as an almost rational forecast. More than half of the agents behave non-rational: 44% of agents seemed to behave adaptively, while 7% seemed to use naive forecasting rules (see also Carroll (2003), Nunes (2010), and Mankiw, Reis and Wolfers (2003), for similar results). In a follow-up paper, Branch (2007) shows that the heuristic switching model explains this survey data better than a static sticky information model. Based on the same dataset, evidence in favor of heterogeneity is also presented in Mankiw et al. (2003). Pfajfar and Santoro (2010) measure the degree of heterogeneity in private agents' inflation forecasts. They show that heterogeneity in inflation expectations is pervasive, and identify three different forecasting heuristics that seem to have been frequently used: either static or highly autoregressive heuristics, nearly rational expectations or an adaptive learning rule with sticky information forecasting.

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Support for heterogeneous expectations has also been found using survey data on exchange rates expectations (e.g. Frankel and Froot, 1987; Allen & Taylor, 1990). As noted by Pfajfar and Zakelj (2014), also price expectation formation has been studied. For example, Chavas (2000) estimated that in the US beef market, about 82% of agents forecast boundedly rational using simple models to forecast prices. The other 18% were found to make rational forecasts. In contrast, Baak (1999) estimated the proportion of rational agents in the same market at two thirds, with the remaining fraction of agents behaving boundedly rational.

2.2.2.3. Expectations and the Macro Level

Milani (2011) explores whether expectational shocks may affect business cycle fluctuations. He exploited survey data on expectations to estimate a New Keynesian (NK) macro-economic model which allows for learning by economic agents. He finds that expectational shocks explain roughly half of business cycle movements in this model, whereas structural demand, supply, and policy shocks explain the other half. In this model with exogenous expectational shocks, Milani (2011) finds that the adjustment of the economy after demand shocks is much faster than commonly implied by standard monetary dynamic stochastic general equilibrium (DGSE) models. In contrast, expectational shocks cause a substantially more persistent adjustment. The effect of expectational shocks on output is larger, delayed and more long-lived than the corresponding effect provoked by structural demand shocks. It was also found that fluctuations in inﬂation were also mostly driven by expectational shocks related to future real activity and future inﬂationary pressures.

Using a similar methodology, Ormeno (2009) used data from the Survey of Professional Forecasters in the estimation of a dynamic stochastic general equilibrium model (DSGE) under learning in a two-step approach. First, he exploited the information provided by surveys to determine the forecasting model used by agents. Second, he used the survey data in order to estimate the DSGE model. He finds that once survey data are included, learning not only matches actual expectations but also emerges as a key determinant of inflation persistence, explaining around 30 percent of inflation persistence. Also, Eusepi and Preston (2011) develop a real business cycle model with learning dynamics that can replicate patterns in forecast errors over the business cycle implied by data from the Survey of Professional Forecasters.

Beaudry, Nam and Wang (2011) find that mood swings account for over 50% of business fluctuations in hours worked and output, and that these moods swings are strongly associated with long-run (2-3 years) movements in total factor productivity (TFP). The paper lists three possible explanations for these findings. At one extreme, there is the view that such mood swings are entirely rational because of a self-fulfilling positive feedback loop (see e.g. Benhabib and Farmer, 1994). Closely related to this view is the “news” view of mood swings (see e.g. Jaimovich and Rebelo, 2009). In this view, optimism arises when agents learn about forces that will positively affect future fundamentals, so periods of optimism precede positive changes in fundamentals but do not cause them. Finally, there is a third view suggesting that macroeconomic mood swings are only driven by psychological factors and therefore are not directly related to future developments of fundamentals (see e.g. Akerlof and Shiller, 2009). Although Beaudry et al.’s (2011) identification methodology, based on survey data, cannot distinguish whether the mood swings are

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a reflection of the future growth, or cause the future growth (Driscoll & Holden, 2014), the third view is rejected by their findings.

In a similar effort by Barsky and Sims (2011), little evidence of a strong causal channel from autonomous movements in sentiment to economic outcomes is found. Rather, their findings support an alternative hypothesis that the surprise movements in confidence reflect information about future economic prospects. They conclude that confidence innovations are best characterized as noisy measures of changes in expected productivity growth over a relatively long horizon.

2.2.2.4. Conclusive Remarks

To summarize, there is ample empirical evidence documenting that the forecasting of macro-economic aggregates, when proxied by surveys, is characterized by a significant fraction of agents that forecast non-rationally. Heterogeneity of expectations has been confirmed in many studies based on survey data. These studies indicate that several simple forecasting heuristics are used by economic agents, consistent with an adaptive learning or a sticky information approach, or a combination of both. Estimates of the type of rules that were used by agents, as well as the fraction of agents using them, varies widely over papers and in function of the variables to forecast. The distribution of heterogeneity of forecasting rules has been found to vary over time, with the type of forecasting rules used by a certain fraction of agents depending on the level and the variance of recent realisations of aggregate macro-economic variables. Moreover, boundedly rational expectations have been found to be highly correlated with business cycle fluctuations, although causal relations cannot be inferred from survey data.

2.2.3. Laboratory Experiments

2.2.3.1. Introduction

Assenza, Bao et al. (2014) lists three advantages of laboratory experiments over survey data. First, expectations are properly rewarded, such that subjects are incentivized to produce accurate forecasts. Second, the true model used in the experiment is known and controlled by the experimenter, including the self-referential nature of economic systems. And third, there is the increased ease of high frequency data gathering.

Experiments that elicit expectations can be classified using their experimental design. In one type of experiments, there is no feedback from subjects’ expectations to the generated series. These experiments make use of field data or generated random processes. The evidence from field data studies (e.g. Schmalensee, 1976; Bernasconi, Kirchkamp, & Paruolo, 2009) seems to point in the direction of adaptive models. Note however, that, as the data generating process of the real world time series data is not known, the definition of rational expectations is not obvious in this context (Assenza, Bao et al., 2014). This remark does not apply to studies in which experimenters ask to predict an exogenously generated random process (e.g. Hey, 1994; Kelley and Friedman, 2002). For these type of studies, the evidence seems to point in the

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direction of deviations from rationality in various degrees, depending on the exact data generating process that was employed. For example, using a simple random walk, in which case rational expectations coincide with naive expectations, Dwyer et al. (1993) found subjects’ expectations to be in line with “rational” (naive) expectations. But in Hey (1994), making use of a stochastic AR(1) process, evidence for adaptive expectations was found. Note however, that these studies use rather simple data generating processes compared to even very simplified macroeconomic models, and do not encompass expectation feedback.

Learning-to-forecast (LtFEs) and learning-to-optimize (LtOEs) experiments do incorporate expectation feedback into their design, and are therefore more relevant for assessing the degree of bounded rationality in real world economic expectation formation.

Learning-to-optimize (LtOE) refers to a design where subjects are asked to submit their economic quantity decisions regarding consumption, trading or production, without direct elicitation of their forecasts of market aggregate outcomes like prices, output, or inflation. (Assanza, Bao et al., 2014). These market outcomes are calculated for the subjects by the experimenters. All individual quantity decisions by the subjects are fed into the model macro economy that was adopted in the experiment, in order to obtain the aggregate outcome.

In learning-to-forecast experiments (LtFEs), subjects are asked to directly forecast the aggregate variable of interest, like for example the output gap, inflation, or asset prices. These individual forecasts are fed into the equations for the macro model economy adopted in the experiment. The resulting value for the aggregate variable of interest is then calculated and made available to each subject. All other agents’ decisions such as consumption, production, investment etc., leading to this outcome, are calculated by the experimenters, usually based on optimal, rational assumptions (Hommes, 2018), making use of utility and profit optimization techniques (Evans and Honkapohja, 2001). As such, subject’s trading actions are considered rational, conditional on the submitted forecast (Anufriev et al., 2019).

Subjects are thus forecasting in a dynamic self-referential system, with market realisations depending endogenously on subject’s average forecasts. Realisations which in turn influence subjects’ forecasts. Separating the quantity choice from direct forecast elicitation in a LtOE versus a LtFE respectively, is a way of decomposing the problem faced by agents in complex macroeconomic settings, so that it does not involve a joint test of rationality in both optimization and expectation formation (Duffy, 2016). These macro experiments provide data at the micro level as well as at the macro level, which can be used to formulate and test theories of learning, as well as assess the time series properties of aggregate economic variables like output and inflation.

A lot of influential early work on bounded rationality makes use of rather simple models, like the cobweb model (e.g. Brock & Hommes, 1997) or an asset pricing model (e.g. Brock & Hommes, 1998). More recently, some learning-to-forecast studies also studied learning in a New Keynesian framework (e.g. Assenza, Heemeijer, Hommes & Massaro, 2014a). The LtFE and LtOE design was actually pioneered in the context of overlapping generation (OLG) economies by Marimon and Sunder (1993, 1994, 1995) and Marimon, Spear and Sunder (1993). For a more extensive review, we refer to Assenza, Bao et al. (2014, pp.35-p40). We note here that also in an OLG framework, evidence for adaptive expectations was found (e.g. Marimon and Sunder, 1993; 1994), although not necessarily for first order adaptive rules (Bernasconi

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& Kirchkamp, 2000). Heemeijer, Hommes, Sonnemans and Tuinstra (2012) also found evidence for adaptive constant gain learning, and heuristic switching on the basis of heuristics’ forecasting performance. From LtFEs and LtOEs, theories of learning were developed, which in later work got incorporated into macroeconomic models to replace the full information, rational expectation assumption. Macroeconomic models incorporating adaptive learning and heterogeneous expectations are reviewed in section 2.4.1 and 2.4.2 (discussing monetary and fiscal policy, respectively). Chapter 3 discusses some alternative approaches that each strip particular properties from the perfect information, rational expectations assumption to replace it with some form of bounded rationality. In this section (2.2.3), experiments in asset price and New Keynesian frameworks are discussed. Section 2.3 discusses the theories of learning that were developed to fit to these experimental data. Because other review papers (e.g. Hommes, 2011; Duffy; 2016; Assenza, Bao et al, 2014; Hommes 2013) already extensively surveyed the early cobweb experimental results, here, the discussion is limited to laboratory experiments that had a direct and major influence on theories of learning that received the most attention in the literature. For reasons of space restrictions, results in OLG frameworks are not discussed in this thesis.

2.2.3.2. Learning-to-Forecast Experiments

In LtFEs, subjects typically only have qualitative information about the model economy or market setup used in the experiment (Hommes, 2013). They know that there is expectation feedback: the economic variable to forecast depends on an aggregation of their individual forecasts. In an asset market framework, the aggregate variable’s value is derived from equilibrium between demand and supply, while in a New Keynesian framework it depends on an IS-curve, a (NK-)Phillips curve, a monetary policy rule, and the type of exogenous shocks (Assenza, Bao et al., 2014). Subjects are typically able to infer the type of expectations feedback, positive or negative. Positive (negative) feedback means that an increase of the average individual forecasts leads to a higher (lower) value for the to be forecasted variable (Hommes, 2013). Subjects in the LtFEs are usually given an overview of past aggregate realisations and their own past forecasts and earnings, typically presented in table as well as in graphic form (Hommes, 2013). Subjects, however, do not know the forecasts of other participants, the exact underlying model’s equations, the benchmark rational expectations equilibrium, nor the exact number of other subjects participating in the experiment.

Two closely related LtFEs by Hommes, Sonnemans, Tuinstra and van de Velden (2005;2008) make us of a dynamic asset pricing model (e.g. Campbell et al., 1997). There are two assets: a risk free asset paying a fixed rate of return and a risky asset paying an uncertain dividend. Under some assumptions, specific to the asset pricing model, the rational market clearing price, or fundamental price, of the risky asset is calculated, and serves as a benchmark to match against the outcomes of the experiment. The realized asset price in the experiment is derived by market clearing using all subjects’ forecasts. Some small independent and identically (iid) normally distributed shocks are added to the pricing equation, representing a small fraction of noise traders. Each experimental session consisted of 50 periods and six subjects, whose only task was to submit a two period ahead point prediction in every period for the price of

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the risky asset. Subjects’ earnings were inversely related to their prediction errors. In principle, participants could compute the rational price, as they were given the values of the fundamental parameters: the fixed interest rate for the risk free asset and the mean dividend of the risky asset, even although the exact underlaying model was unknow to them. In addition, the information set of the participants also entailed past prices, past own predictions and past own earnings.

The key difference between both studies, is that in Hommes et al. (2005), a variable fraction of computerized fundamental robot traders is present, and Hommes et al. (2008), they are not. These traders always forecast the fundamental price, exerting a stabilizing, negative feedback force. The fraction of robot traders was set to increase, the further the price deviates from the rational expectations equilibrium price.

Hommes et al. (2015; 2018) found that in most sessions, participants were unable to learn the rational, fundamental price, but rather, prices’ time series exhibit expectations driven bubbles and crashes. Only during some sessions, individual predictions moved slowly in the direction of the fundamental price towards the end of the experiment. For exactly the same experimental setup, three different price patterns were observed: first, slow, (almost) monotonic convergence. Second, persistent price oscillations with almost constant amplitude. Third, large initial oscillations dampening slowly towards the end of the experiment (Hommes, 2011). Similar large and long lasting bubbles have been observed in larger groups of up to 32 subjects (Bao et al. 2016) and even for groups up to a hundred subjects (Hommes, Kopanyi-Peuker, & Sonnemans, 2018).

It was found that despite huge fluctuations in market clearing prices, already after a short period (3 to 5), participants were able to coordinate their forecasting activity, submitting forecasts similar to all other subjects’ forecasts in every period. This led to almost self-fulfilling equilibria. This coordination of individual forecasts has been achieved in the absence of any communication between subjects other than through the observed realized price (Hommes, 2011). In fact, it was found that the bubbles are driven by a strong coordination of individual expectations on trend-following behaviour. This pattern was even more extreme in the experiment of Hommes et al. (2018), which did not include fundamental robot traders.

Positive versus Negative Feedback

In a similar LtFE, Heemeijer, Hommes, Sonnemans and Tuinstra (2009), investigate how the expectations feedback structure affects individual forecasting behavior and aggregate market outcomes by considering market environments that only differ in the sign of the expectations feedback, but are equivalent along all other dimensions. Negative feedback occurs in supply driven markets, where producers face production lags, like for example in a hog cycle model. A higher expected price will incentivize producers to produce more, driving prices down. Positive feedback is prominent in, for example, a stock exchange market: investors with optimistic beliefs will buy more stock and this increased demand in turn drives the price up, probably creating even more optimism, and thus, more demand, up to some point, possibly creating a self-fulfilling prophecy (Anufriev, Hommes & Makarewicz, 2019, p.1544). In this study, it was found that with small iid shocks, in negative feedback markets, as in commodity markets, prices converge

Expectations and learning in lab experiments and in macro-economic modeling