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Long-run expectations in a learning-to-forecast experiment: a simulation approach

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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.

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Notes

  1. For a comprehensive survey of the macroeconomic experiments on expectations, see Assenza et al. (2014).

  2. 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.

  3. 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).

  4. 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.

  5. We used a pay-off mechanism similar to Haruvy et al. (2007).

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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.

  11. Except for the first period, the long-run expectations that we considered lie always in the chosen interval.

  12. 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.

  13. The MSE reported in the those papers is approximately 0.019 per period.

  14. Details on the estimation results and comparative analysis are available from the authors upon request.

  15. 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).

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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.

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Correspondence to Eva Camacho-Cuena.

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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

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Screen-shot of the experiment

Appendix B: Individual long term predictions

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Individual long-run predictions of Group 1. The black dots indicate the realized price, the grey lines the individual forecasts and the dashed line the fundamental value

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Individual long-run predictions of Group 2. The black dots indicate the realized price, the grey lines the individual forecasts and the dashed line the fundamental value

Fig. 24
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Individual long-run predictions of Group 3. The black dots indicate the realized price, the grey lines the individual forecasts and the dashed line the fundamental value

Fig. 25
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Individual long-run predictions of Group 4. The black dots indicate the realized price, the grey lines the individual forecasts and the dashed line the fundamental value

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Individual long-run predictions of Group 5. The black dots indicate the realized price, the grey lines the individual forecasts and the dashed line the fundamental value

Fig. 27
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Individual long-run predictions of Group 6. The black dots indicate the realized price, the grey lines the individual forecasts and the dashed line the fundamental value

Fig. 28
figure 28

Individual long-run predictions of Group 7. The black dots indicate the realized price, the grey lines the individual forecasts and the dashed line the fundamental value

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:

$$ {~}_{i}a_{jt}=y=p_{t-1}+\lambda \ {\Lambda}_{t-1} \frac{J-\frac{N}{2}}{\frac{N}{2}}\qquad J = 0,1,..,N , $$
(13)

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− 1λ Λt− 1, pt− 1 + λ Λt− 1]. We can rewrite (3) as follows:

$$ {~}_{i}V_{jt}=-\gamma_{i} (1+{~}_{i}\phi_{s}) \left( y -\frac{p_{t-1}+ {~}_{i}\phi_{s} \ p^{e}_{t-1,t-1}}{1+{~}_{i}\phi_{s}} \right)^{2}+\text{const} . $$
(14)

We approximate the distribution of Eq. (5) by a Gaussian distribution:

$$ {~}_{i}P_{jt}\approx \frac{1}{\sigma_{i}\sqrt{2\pi}}\exp \left\{-\frac{1}{2} \left( \frac{ y -\mu_{t-1}}{\sigma_{i}} \right)^{2} \right\} \frac{2\lambda \ {\Lambda}_{t-1}}{N} , $$
(15)

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:

$$\begin{array}{@{}rcl@{}} \ln P[{~}_{i}p^{e}_{t,t}|p_{t-1},{~}_{i}p^{e}_{t-1,t-1};{~}_{i}\phi_{s},\gamma_{i}]& = &(T-4)\ln(\lambda {\Lambda} _{t-1})+\frac{(T-4)}{2}\ln(\gamma_{i})\\ &&+\frac{(T - 4)}{2}\ln(1+{~}_{i}\phi_{s})\\ &&-\gamma_{i}(1 + {~}_{i}\phi_{s})\sum\limits_{t = 4}^{T} ({~}_{i}p^{e}_{t,t} - \mu_{t-1})^{2} + \text{const} . \end{array} $$
(16)

It is then straightforward to compute the estimators for iϕs and γi:

$$ \hat{{~}_{i}\phi_{s}} = \frac{ {\sum}_{t = 4}^{T}({~}_{i}p^{e}_{t,t}- p_{t-1}) \cdot (p_{t-1} - {~}_{i}p^{e}_{t-1,t-1})}{ {\sum}_{t = 4}^{T}({~}_{i}p^{e}_{t,t} - {~}_{i}p^{e}_{t-1,t-1}) \cdot ({~}_{i}p^{e}_{t-1,t-1}- p_{t-1}) } . $$
(17)
$$ \hat{\gamma_{i}} = \frac{T-4}{2 (1 + \hat{{~}_{i}\phi_{s}}){\sum}_{t = 4}^{T} (p^{e}_{t,t} - \hat{\mu}_{t-1})^{2} } , $$
(18)

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).

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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

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