Abstract
In this paper, we elicit short-run as well as long-run expectations on the evolution of the price of a financial asset in a Learning-to-Forecast Experiment (LtFE). Subjects, in each period, have to forecast the the asset price for each one of the remaining periods. The aim of this paper is twofold: first, we fill the gap in the experimental literature of LtFEs where great effort has been devoted to investigate short-run expectations, i.e. one step-ahead predictions, while there are no contributions that elicit long-run expectations. Second, we propose a new computational algorithm to replicate the main properties of short and long-run expectations observed in the experiment. This learning algorithm, called Exploration-Exploitation Algorithm, is based on the idea that agents anchor their expectations around the last realized price rather than on the fundamental value, with a range proportional to the past observed price volatility. When compared to the Heuristic Switching Model, our algorithm performs equally well in describing the dynamics of short-run expectations and the realized price dynamics. The EEA, additionally, is able to reproduce the dynamics long-run expectations.
Similar content being viewed by others
Notes
For a comprehensive survey of the macroeconomic experiments on expectations, see Assenza et al. (2014).
We use the term “prediction” referring to the forecasts submitted by the subjects during the experiment. We assume that subjects submit their predictions based on their expectations, which are not observable. Therefore, across the paper, we use the word “prediction” and “expectation” as (almost) interchangeable.
According to this principle, the strategy for estimating unknown quantities is to start with information one does know, i.e. an anchor, and then adjust until an acceptable value is reached; see Tversky and Kahneman (1974).
We use different fundamental values to prevent communication among subjects between sessions conducted in the same day, as it could affect their decisions. We claim that the small difference in the fundamental values does not impact the results of the experiment.
We used a pay-off mechanism similar to Haruvy et al. (2007).
If we implement instead a smooth pay-off function equal to Eq. 2 for long-run predictions, we could probably discourage subjects from providing accurate forecasts, since they would perceive it as too a difficult task to gain profits.
A Wilcoxon test shows that the difference between observed prices and the fundamental value is statistical significant, except for group 5 with a p-value of 0.12.
The range of the actions turns out to be influential in the dynamics of the EEA if it is sufficiently wide. The range of actions will be determined by the parameters ϕs and γ. See in Appendix C the comments on the estimators of those parameters.
Despite the fact that, in the experiment, we elicit the expectations for the whole time horizon, we replicate the individual expectations up to four-step-ahead. Our choice represents a good compromise between considering the whole time-span and having sufficient statistics to analyze the properties of the EEA as a function of the time horizon and comparing them to the experimental data.
Note that, following the payment schedule used to reward the subjects’ long-run expectations, if the absolute difference between the price and the long-run prediction is higher than 15, the profit is equal to zero.
Except for the first period, the long-run expectations that we considered lie always in the chosen interval.
We have eliminated the estimates of two subjects because of the estimated values of iϕs are out of range due to an error in typing their predictions.
The MSE reported in the those papers is approximately 0.019 per period.
Details on the estimation results and comparative analysis are available from the authors upon request.
A sum-rank test shows that the difference between EEA and the experimental data is not statistically significant at 5% level in 17 out of 21 cases, whereas, in the case of EEA(pf), this ratio falls to three out of 21 cases (essentially group 6).
References
Anufriev M, Hommes CH (2012) Evolutionary selection of individual expectations and aggregate outcomes in asset pricing experiments. American Economic Journal: Microeconomics 4(4):35–64
Anufriev M, Assenza T, Hommes CH, Massaro D (2013a) Interest rate rules and macroeconomic stability under heterogeneous expectations. Macroecon Dyn 17(8):1574–1604
Anufriev M, Hommes CH, Philipse RH (2013) Evolutionary selection of expectations in positive and negative feedback markets. J Evol Econ 23(3):663–688
Arifovic J, Masson P (2004) Heterogeneity and evolution of expectations in a model of currency crisis. Nonlinear Dynamics Psychol Life Sci. 8(2):231–58
Ashiya M (2003) Testing the rationality of Japanese gdp forecasts: the sign of forecast revision matters. J Econ Behav Organ 50(2):263–269
Assenza T, Heemeijer P, Hommes CH, Massaro D (2011) Individual expectations and aggregate macro behavior. Technical report ceNDEF Working Papers 11-01, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance
Assenza T, Bao T, Hommes CH, Massaro D (2014) Experiments on expectations in Macroeconomics and Finance in John Duffy (ed.) Experiments in Macroeconomics (Research in Experimental Economics, Volume 17) Emerald Group Publishing Limited, pp 11 – 70
Auer P, Cesa-Bianchi N, Fischer P (2002) Finite-time analysis of the multiarmed bandit problem. Mach Learn 47(2-3):235–256
Bao T, Ding L (2016) Non-recourse mortgage and housing price boom, bust, and rebound. Real Estate Econ 44(3):584–605
Bao T, Hommes CH, Sonnemans J, Tuinstra J (2012) Individual expectations, limited rationality and aggregate outcomes. J Econ Dyn Control 36 (8):1101–1120
Bao T, Duffy J, Hommes CH (2013) Learning, forecasting and optimizing: an experimental study. Eur Econ Rev 61:186–204
Brock WA, Hommes CH (1997) A rational route to randomness. Econometrica 65(5):1059–1096
Brock WA, Hommes CH (1998) Heterogeneous beliefs and routes to chaos in a simple asset pricing model. J Econ Dyn Control 22(8-9):1235–1274
Campbell SD, Sharpe SA (2009) Anchoring bias in consensus forecasts and its effect on market prices. J Financ Quant Anal 44(02):369–390
Cœuré B (2013) Monetary policy in the crisis – confronting short-run challenges while anchoring long-run expectations. Tech. rep., Speech by Benoit cœuré, Member of the Executive Board of the ECB at the Journées de l’ AFSE 2013
Colasante A, Alfarano S, Camacho-Cuena E, Gallegati M (2018) Long-run expectations in a learning-to-forecast experiment. Appl Econ Lett 25(10):681–687
Cornand C, Kader M’baye C (2013) Does inflation targeting matter? An experimental investigation. Macroeconomic Dynamics, forthcoming
Diks C, Van Der Weide R (2005) Herding, a-synchronous updating and heterogeneity in memory in a CBS. J Econ Dyn Control 29(4):741–763
Draghi M (2008) Monetary policy, expectations and financial markets, Speech delivered at the Central Bank Whitaker Lecture in Dublin 18
Fujiwara I, Ichiue H, Nakazono Y, Shigemi Y (2013) Financial markets forecasts revisited: Are they rational, stubborn or jumpy? Econ Lett 118(3):526–530
Galati G, Heemeijer P, Moessner R (2011) How do inflation expectations form? New insights from a high-frequency survey. BIS Working Papers No 349
Gurkaynak RS, Sack B, Swanson E (2005) The sensitivity of long-term interest rates to economic news: evidence and implications for macroeconomic models. The American Economic Review 95(1):425–436
Hanaki N, Akiyama E, Ishikawa R (2016) A methodological note on eliciting price forecasts in asset market experiments. Working paper halshs-01263661
Haruvy E, Lahav Y, Noussair CN (2007) Traders’ expectations in asset markets: experimental evidence. The American Economic Review 97(5):1901–1920
Heemeijer P, Hommes CH, Sonnemans J, Tuinstra J (2009) Price stability and volatility in markets with positive and negative expectations feedback: an experimental investigation. J Econ Dyn Control 33(5):1052–1072
Hommes CH (2001) Financial markets as nonlinear adaptive evolutionary systems. Quant Finan 1(1):149–167
Hommes CH, Huang H, Wang D (2005a) A robust rational route to randomness in a simple asset pricing model. J Econ Dyn Control 29(6):1043–1072
Hommes CH, Sonnemans J, Tuinstra J, van de Velden H (2005b) A strategy experiment in dynamic asset pricing. J Econ Dyn Control 29(4):823–843
Hommes CH (2013) Behavioral rationality and heterogeneous expectations in complex economic systems. Cambridge University Press
Hommes CH, Lux T (2013) Individual expectations and aggregate behavior in learning-to-forecast experiments. Macroecon Dyn 17(2):373–401
Joyce M, Relleen J, Sorensen S (2008) Measuring monetary policy expectations from financial market instruments. Bank of England working papers 356, Bank of England
Koulouriotis DE, Xanthopoulos A (2008) Reinforcement learning and evolutionary algorithms for non-stationary multi-armed bandit problems. Appl Math Comput 196(2):913–922
Lucas RE Jr (1978) Asset prices in an exchange economy. Econometrica 46 (6):1429–1445
Marimon R, Sunder S (1993) Indeterminacy of equilibria in a hyperinflationary world: Experimental evidence. Econometrica 61(5):1073–1107
Manski CF (2004) Measuring expectations. Econometrica 72:1329–1376
Nakazono Y (2012) Heterogeneity and anchoring in financial markets. Appl Financ Econ 22(21):1821–1826
Tversky A, Kahneman D (1974) Judgment under uncertainty: Heuristics and biases. Science 185(4157):1124–1131
Woodford M (2001) Monetary policy in the information economy. In: Economic Policy for the Information Economy. Kansas City: Federal Reserve Bank of Kansas City, pp 97–370
Acknowledgments
The authors are grateful for funding the Universitat Jaume I under the project P11B2015-63 and the Spanish Ministry Science and Technology under the project ECO2015-68469-R. We thank the anonymous reviewers for their careful reading of our manuscript and their insightful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Funding: This study is funded by the Universitat Jaume I under the project P11B2015-63 and the Spanish Ministry Science and Technology under the project ECO2015-68469-R. The authors declare that they have no conflict of interest.
Appendices
Appendix A: Screenshot and Instructions
Figure 21
Appendix B: Individual long term predictions
Appendix C: Estimation procedure
In order to illustrate the ML procedure to compute the individual values of ϕs and γ, let us introduce a parametrization for the action as a function of the discrete index J:
where N = 101, are the number of steps into which the range of actions is divided and λ = 5. The width of the step is \(\frac {2\lambda {\Lambda }_{t-1}}{N}\). We approximate the discrete variable ajt, by a continuous variable y, where y ∈ [pt− 1 − λ Λt− 1, pt− 1 + λ Λt− 1]. We can rewrite (3) as follows:
We approximate the distribution of Eq. (5) by a Gaussian distribution:
where μt− 1 and σi are given by Eqs. (10) and (11), respectively. Essentially, the conditioned probability of realization of \({~}_{i}p^{e}_{t,t}\) can be approximated by a Gaussian distribution multiplied by the width of the step of the range of the actions. The log-likelihood function can be easily computed:
It is then straightforward to compute the estimators for iϕs and γi:
The probability of the actions from Eq. (15) depends linearly on the width of the range. Note, however, that the individual estimators of γ and ϕs do not depend on the range of the actions. Therefore, under the Gaussian approximation, we do not have to worry about the choice of the range. On the other hand, it is important to check that the experimental expectations submitted by the subjects lie in the range of actions, i.e. \(p^{e}_{t,t}\) can be expressed in terms of the continuous variable y. In our implementation of the EEA, this condition always hold, for short- as well as long-run expectations. In order to estimate the parameters governing the long-run expectations, we make use of a numerical optimization algorithm computing the likelihood using Eq. (6). The main problem for arriving at a closed-form solution for the individual estimator for ϕl is the non-analyticity of the absolute value in Eq. (4).
Rights and permissions
About this article
Cite this article
Colasante, A., Alfarano, S., Camacho-Cuena, E. et al. Long-run expectations in a learning-to-forecast experiment: a simulation approach. J Evol Econ 30, 75–116 (2020). https://doi.org/10.1007/s00191-018-0585-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00191-018-0585-1