Long-run expectations in a learning-to-forecast experiment

ABSTRACT We conduct a Learning to Forecast Experiment using a novel setting in which we elicit subjects’ short- and long-run expectations on the future price of an asset. We find that: (i) the rational expectations equilibrium is not a meaningful description for the whole time spectrum of subjects’ expectations; (ii) they are, instead, better described by an anchor-and-adjustment learning scheme; (iii) subjects exhibit a higher degree of heterogeneity in their long-run expectations vis-à-vis short-run expectations.


I. Introduction
An economic system can be modelled as an expectation-feedback system where individuals form their expectations observing past realizations of economic variables and with their decisions agents influence the future realizations of those variables (see Hommes 2013). In order to design efficient economic policies, then, it is of crucial importance to understand how economic agents form and revise their expectations. In particular, several empirical as well as theoretical contributions have stressed the importance of taking into account the whole time spectrum of agents' expectations when designing economic policies (Gürkaynak, Sack, and Swanson 2005;Coeuré 2013). Since the early 2000s, it has been introduced a new view of monetary policy, based on a higher level of transparency to increase its effectiveness. This new role of transparency, together with the traditional accountability view, goes under the term of 'forward guidance', i.e. the management of the agents' expectations using various communication instruments, such as publishing Central Bank forecasts of macroeconomic variables or announcing an inflation target. Within this framework, the Central Bank can influence the yield curve of interest rates not only using the traditional 'mechanical' instrument of the over-night rate, but also influencing the agents' expectations on the long-term interest rates, which are more relevant for investment decisions and, ultimately, for the overall economic activity. Within the expectationalist view, the channel of transmission of the monetary policy relies on managing agents' expectations at different timescales.
Measuring and estimating expectations is a difficult task since they are not directly observable. Learning to Forecasts Experiments (LtFE), introduced by (Marimon and Sunder 1993), are controlled laboratory experiments used to elicit subjects' expectations in an expectation-feedback environment where subjects perform a forecasting task. Many LtFEs have been conducted to study how agents form their short-run expectations in different environments (see Assenza et al. 2014 for a survey). In this article, we conduct a LtFE in which, differently from the existing settings, subjects submit a prediction for the asset price at different time horizons. The novelty of our experimental design is that it incorporates into the LtFEs the elicitation of long-run expectations, in order to study how expectations form and coevolve with the market price. With our article, we want to fill the gap in the existing LtFE experimental literature in eliciting the whole spectrum of subjects' expectations. To the best of our knowledge, just two experimental contributions (Haruvy et al., 2007;Hanaki, Akiyama, and Ishikawa 2016) elicit long-run expectations in an asset market with bubbles.

II. Experimental design
We modify the LtFE by Heemeijer et al. (2009) to elicit simultaneously subjects' short-and long-run expectations about the future price of an asset. In each of the 7 sessions implemented, 6 subjects play the role of professional forecasters for 20 periods (see the instructions in the supplementary material). At the beginning of period t, subject i submits his/her short-run prediction for the asset price at the end of period t ( i p e t;t ), as well as his/her set of long-run predictions for the price at the end of each one of the 20 À t remaining periods ( i p e t;tþk with 1 k 20 À t). 1 When submitting their predictions, subjects are informed about: (i) the interest rate (r ¼ 0:05) and average dividend (d ¼ 3:25 or d ¼ 3:5 depending on the session), (ii) the time series of realized prices until period t À 1, (iii) all their own past predictions and profits. They are informed that there is a positive feedback between their one-step-ahead predictions and the realized price at the end of the period.
Following Heemeijer et al. (2009), the price generating mechanism is where p f ¼ d r denotes the asset fundamental value, p e t;t ¼ 1 6 P 6 i¼1i p e t;t is the average of the six one-stepahead predictions submitted at the beginning of period t, and ε t ,Nð0; 0:25Þ is an iid Normal shock.
Individual earnings at the end of each period t, denoted as i π t , depend on short-and long-run prediction errors: i π t ¼ i π s t þ i π l t . We denote as i π s t the subject pay-off that depends on his/her shortrun prediction error: and as i π l t the subject pay-off that depends on longrun prediction error. We define i π l t ¼ P tÀ1 j¼1 i π l tÀj;t , where i π l tÀj;t represents the individual profit associated with the accuracy of the prediction submitted by subject i at the beginning of period t À j about the asset price in period t, where 1 j t À 1. It is computed according to the following payment schedule 2 : The final payment of each subject is the sum of pay-offs across all periods. Note that subjects have an immediate feedback about the accuracy of their short-run predictions, while they experience a delay in evaluating the accuracy of their long-run predictions.
The experiment involves 42 undergraduate students and it was conducted in the Laboratory of Experimental Economics at University Jaume I. Each session lasted approximatively 40 min and the average gain was 20 Euros.

III. Theoretical framework
According to Equation 1, if we assume all subjects follow rational expectations, their one-step-ahead predictions in period t have to be equal to the fundamental value p f and the realized price p t will converge to p f with fluctuations proportional to the SD of ε t : Equations 3 and 4 describe the Rational Expectations Equilibrium (REE) in typical LtFE as in Heemeijer et al. (2009). In our experimental setting, we give to the subjects' incentives to reveal also their long-run expectations about the future price at different time horizons. When submitting their long-run predictions at the beginning of period t, a subject should form his/ her expectations about the price at the end of period t þ k, with k > 0. Iterating Equation 1, the price at the end of period t þ k depends on all subjects' short-run predictions submitted at the beginning of period t þ k. Let us introduce E t ½ i p e tþk;tþk , i.e. the expectation in period t of future short-run predictions for period t þ k, meaning what I expect my future short-run prediction will be. Since it is common knowledge that 1 The choice of 20 periods is based on a trade-off between having a sufficiently long time series of observations and avoiding a too demanding task for the subjects. 2 We used a pay-off mechanism similar to Haruvy et al. (2007). We calibrated the parameters of the pay-off functions such that max P 20 t¼1 iπ s t % max P 20 t¼1 iπ l t , in order to give to the subjects similar incentives to provide accurate predictions in the short-as well as in the long-run.
the price-generating mechanism is invariant over time, at a given period t, the REE implies so that all expected short-run predictions are equal to p f . Hence, iterating Equation 1, we obtain p tþk % p f , and therefore, long-run predictions have also to converge to p f for any time horizon: The REE predicts that subjects' short-and long-run expectations will coordinate and converge to the fundamental value. The experimental literature on LtFE has shown that short-run expectations coordinate around the realized price, but the realized price not always converges to the fundamental value, demonstrating that REE does not hold in the short-run (see Bao et al., 2012 andHommes et al., 2005). Our experimental setting allows for a better characterization of the performance of REE at different timescales. If REE does not hold, we can study how past price dynamics is incorporated into the subjects' short-run expectations and whether and how short-run expectations and the past time series of prices influence long-run expectations.

IV. Results
Figures 1 and 2 show the evolution over time of the asset price and individual short-and long-run predictions, respectively. 3 Figure 1 shows that prices do not satisfy Equation 4, since the distance to the fundamental value is at least one order of magnitude higher than the SD of the idiosyncratic shock ε t . Additionally, the pattern of the time series of prices is similar to those observed in the LfFE literature, i.e. an apparent long-term oscillatory or monotonic convergence to the fundamental price. Table 1 shows the results of a Wilcoxon test. Note that the difference between observed prices and the fundamental value is statistically significant for all groups.
To measure how close individual expectations are to the fundamental value, we compute, for each period t and time horizon k, the Root Mean Square Error (RMSE) as the average of the distance between individual predictions and the asset fundamental value: Figure 3 shows that the conditions of Equations 3 and 6 are not satisfied, since short-and long-run predictions do not converge to the fundamental value, given that the value of the RMSE is significantly above the SD of ε t . Our results confirm that the REE is not a good predictor of prices and subjects' individual short-run expectations in the positive feedback environment, and generalizes this results to individual long-run expectations. Given the strong positive feedback between the individual short-run predictions and the market price from Equation 1, each subject has to guesstimate the expectations of the other subjects when submitting his/her short-run predictions. We expect to observe a strong coordination of subjects' shortrun predictions, in line with the literature. However, subjects' coordination motive when submitting their long-run expectations is more complex. They should guesstimate k-periods in advance, and the short-run expectations of the other subjects at the beginning of period t þ k in order to forecast future prices. We expect that the longer is the forecasting horizon the lower is the degree of coordination, assuming a higher level of uncertainty in guesstimating the future short-run behaviour of the other subjects.
As a measure of coordination of expectations, we use the (within-group) SD of subjects' individual predictions submitted in period t for the asset price at different time horizons. From Figure 4, we clearly observe that subjects learn to coordinate their individual short-and long-run expectations, since the average in-group heterogeneity of individual predictions reduces over time. However, we find a lower level of consensus among subjects concerning their long-run predictions as compared to their short-run predictions. In fact, we observe that subjects' longrun expectations are persistently heterogeneous across periods and the heterogeneity appears to increase with the time horizon. An important question arises: which is the origin of such increasing heterogeneity? Since the past price dynamics is common knowledge, we can think that subjects have different interpretations of whether and how past prices influence future prices.
We conjecture that, when forming their expectations, subjects give different weights to the available information depending on the forecasting horizon. In order to formalize and test our conjecture, we estimate a Generalized Method of Moments dynamic panel, regressing short-and long-run individual expectations on past prices and past individual predictions. We distinguish between short-and long-run explanatory variables. As a proxy for the subjects' long-run expectations in period t, we consider i p e t;tþ5 ¼ ð1=5Þ P 5 k¼1i p e t;tþk , i.e. the moving average of the expected prices for the next five periods. As a proxy for the recent past dynamics of prices, we use p tÀ2;tÀ6 ¼ ð1=5Þ P 6 j¼2 p tÀj , which is the moving average of past prices considering a time window of five periods. 4 From the results of our regression in Table 2, we infer that subjects' short-run predictions follow a simple linear adaptive rule anchored in the last realized price, where the adjustment component depends on short-run variables only. After dropping the   i p e t;t ¼ p tÀ1 þα ð i p e tÀ1;tÀ1 À p tÀ1 Þ whereα ¼ 0:38. The value ofα is computed as the average between (a) and 1 À ðbÞ.
Turning to the long-run expectations formation, our results show that subjects follow an adaptive rule anchored in the last realized price, where the adjustment component depends on long-run variables only: i p e t;tþ5 ¼ p tÀ1 þγ ð i p e tÀ1;tþ4 À p tÀ2;tÀ6 Þ whereγ ¼ 0:6 : Similarly to the value ofα in Equation 7, we compute the value ofγ as an average between (c) and the absolute value of coefficient (d), considering that we cannot reject the null hypothesis ðcÞ ¼ ðdÞ j j. The results confirm our previous conjecture: when forming their short-run expectations, subjects do not consider the long-run variables. Conversely, when forming their long-run expectations, short-run expectations do not have a significant effect. Although subjects use an anchor-adjustment learning scheme, the adjustment parameters are significantly different, i.e.γ >α. This asymmetry can be directly connected to the slower coordination of individual long-run expectations (see Figure 4). Our empirical analysis shows that the 'first order heuristic' 5 used by Heemeijer et al. (2009) and in the Heuristic Switching Model (see Anufriev and Hommes, 2012) has a limited validity when considering long-run expectations, since it systematically underestimates the heterogeneity of individual long-run expectations.
Finally, given that we elicit simultaneously short-and long-run expectations, subjects have the incentive to submit short-run predictions following their previous long-run predictions (inter-temporal hedging), potentially distorting the elicitation of expectations. Such effect can be excluded for two reasons: (i) the properties of the short-run expectations are similar to those observed in existing LtFEs; (ii) from Table 2, we see Rmse 1 step Rmse 2 steps Rmse 4 steps Rmse 7 steps Rmse 10 steps Figure 3. Convergence of individual short-and long-run expectations: average across groups of the RMSE measured as the distance between the predictions submitted in period t for the asset price in period t þ k for k ¼ 0; 1; 3; 6; 9 and the fundamental value. Sd 1 step Sd 2 steps Sd 4 steps Sd 7 steps Sd 10 steps Figure 4. Coordination of expectations: for each period t ¼ 1; :::; 20 is displayed the average within-group SD of the subjects' forecasts in period t for the price at the end of period t þ k, where k is 0, 1, 3, 6 and 9. that in the formation of short-run expectations, past long-run expectations do not have a significant effect.

V. Conclusion
We generalize the LtFEs proposing a novel design that allows for eliciting expectations at different timescales. Our results show that subjects' expectations are not consistent with the REE, neither in the short-nor in the long-run. Subjects expectations, instead, can be described using an anchor-adjustment learning scheme with an asymmetric speed of adjustment.
Given that long-run expectations cannot be obtained as simple iterations of the 'first-order heuristic', our future research goes in the direction of modifying the Heuristic Switching Model accounting for different timescales in the formation of expectations.
Furthermore, we consider our experimental setting a first step in developing a tool for studying the impact of different monetary policy measures, from changes of short-term interest rate, to announcements of future interventions, following the expectationalist view of monetary policy (see Woodford, 2001). We consider that our laboratory setting can be used as a test-bed in order to assess the performance of different policy instruments and advise policymakers for more efficient monetary policy measures.