Models of Investor Forecasting Behavior - Experimental Evidence

Investors’ forecasting behavior affects their trading decisions and the resulting asset prices. It has been shown in the literature how different, apparently reasonable investor forecasting behaviors can lead to qualitatively different asset price trajectories. For example, investors’ confidence or panic in relying on observed asset prices to forecast future prices can lead to price bubbles, price crashes, and unpredictable price cycles. In this paper, we report the results of behavioral experiments to study investors’ forecast formation. We present the design of an experiment consisting of multiple investors that participate in a market for a virtual asset. We conducted three experimental sessions with different participants in each session. Unlike most other market experiments with known finite durations and predictable decreasing prices toward the end of the experiment, our experimental sessions had random durations chosen to emulate an infinite horizon market with discounting. Asset price bubbles and cycles occurred in all three sessions. We fit different models of forecast formation to the observed data. There is strong evidence that the investors forecast future prices by extrapolating past prices, even when they know the fundamental value exactly and the extrapolated forecasts differ significantly from the fundamental value. The rational expectations hypothesis seems inconsistent with the observed forecasts. There is some evidence for adjustment of forecasts towards the fundamental value as the extrapolated forecasts deviate from the fundamental value. The calibrated forecasting models of all participants were consistent with dynamical systems that exhibit price bubbles and cycles.

in which forecasters know the rational expectations value of the variable to be forecasted, and they also observe data of the variable to be forecasted, where the individual data values are allowed to deviate from the rational expectations value. We conducted experiments in which the participants were asked to forecast future prices, in which the actual prices were endogenous to the experiment through the trading decisions of the participants, and in which the rational expectations value of future prices (fundamental value of the traded asset) was clear and known to all the participants. One question is whether such participants place more emphasis on the rational expectations value or on the observed data to make their forecasts.
If investors' forecasts were consistent with the rational expectations hypothesis, then the volatility in forecasts and prices would be caused by "external" surprises and not by the "internal" forecasting behavior itself, and the forecasts and resulting prices would be less volatile than the earnings produced by the assets. However, LeRoy and Porter [13], and Shiller [17,18], and later Shiller [19], Campbell and Shiller [6,7], West [22,23], and LeRoy [12], presented empirical evidence that stock prices are typically much more volatile than the earnings produced by the stocks. Thus empirical evidence indicates that the rational expectations hypothesis does not provide an accurate model of investor forecasts and asset prices. Hence we are interested in studying alternative models of forecasting behavior.
Models of boundedly rational forecasting behavior have shown that small changes in investors' forecasting behavior can qualitatively change price trajectories. For example, Cheriyan and Kleywegt [8] showed that a small change in the weight that investors' forecasts place on recent data relative to older data can change price trajectories from convergent trajectories to cyclical trajectories consisting of repeated bubbles and crashes, and the instability is affected by investors' overconfidence in the information contained in observed price data. Also, it was shown that if investors' forecasts exhibit behavior called panic, then the price cycles can become unpredictable. Thus, an understanding of forecasting behavior contributes to an understanding of the qualitative behavior of price trajectories. Therefore, our interest in investor forecasting behavior lead us to the following questions: Are there simple models that can describe investors' forecasts accurately? How accurately does the rational expectations hypothesis describe investors' forecasts? What does observed forecasting behavior imply regarding the resulting behavior of price trajectories, that is, whether price trajectories converge, or predictable price cycles occur, or trajectories are unpredictable?
Questions such as these have been asked by both researchers as well as policy makers who are interested in studying, and sometimes containing or controlling, asset price dynamics.
In this paper we report experiments conducted to study investor forecasting behavior. We designed a market for a virtual asset in which the experimental investors report their price forecasts to the experimenter, and enter their buy and sell orders in the market, repeatedly over time. The market clearing prices are determined by the investors' buy and sell orders, and are therefore endogenous, and the market clearing prices are reported to the investors in real time.
Before each experimental session, the investors were reminded of the theory of the fundamental value of an asset, and they were told how the earnings of the asset would be determined and what the resulting fundamental value of the asset was. Nevertheless, the resulting price trajectories exhibited cycles. We calibrated and compared various models of investor forecast formation. 2 The experimental results indicate that: 1. The rational expectations hypothesis does not provide an accurate model of investor forecasting behavior, even when the earnings rate and fundamental value of the asset are known with certainty by investors.
2. A simple one-parameter exponential smoothing model of investor forecasting behavior is remarkably accurate. Models with a larger number of parameters provide only a slightly better fit.
3. There is strong evidence of investor overconfidence in the information content of observed price data. In the experiment, the fundamental value of the asset was known to the investors and therefore the observed prices contained no additional information regarding the fundamental value of the asset. Nevertheless, investor forecasts relied mostly on observed price data. 4. Except for a few experimental investors, the price forecasts were increasing in the extrapolated prices. That is, most experimental investors did not exhibit evidence of panic, as defined in Cheriyan and Kleywegt [8], in their forecasts.

Definition of Key Terms
A typical experimental setting consists of a number of participants buying and selling units of a virtual asset with virtual currency in a market. Usually, the participants can buy and sell assets. In some experiments, the participants can also be forecasters who only generate forecasts and do not trade.
A virtual asset market implements a specified trading mechanism for trading units of a virtual asset and virtual currency . All virtual asset markets have a finite duration. The fundamental value of the asset may be specified exogenously or it can be computed from the dividend payoffs. The asset may provide either periodic dividends, or only a terminal dividend.
Also, it may or may not have a salvage value.
An experiment consists of one or more sessions. A session involves a group of participants participating in the asset market. Each session consists of one or more trading runs. Two or more trading runs with common participants are called repeated markets by some authors.
At the beginning of each trading run, the investors are given an initial endowment of virtual currency and units of the virtual asset. The investors use the trading mechanism to participate in trades. At the end of the trading run, the participants are rewarded, typically as a function of their final portfolio of virtual currency earned and virtual asset in possession. In some experiments, there is also a reward associated with forecast accuracy.
The trading mechanism can be a continuous-time trading mechanism such as a doubleauction, or a discrete-time trading mechanism such as a call market mechanism. For a market with a discrete-time trading mechanism, the trading run is divided into multiple (trading) periods. During a period, traders enter buy and sell orders. At the end of the trading period, the price of the asset is determined by market clearing and the trades are executed based on the market clearing price. For a market with a continuous-time trading mechanism, the trading run consists of only one trading period. For example, in a double auction market, the traders may post buy and sell offers, or accept existing open offers resulting in trade; in this case the instantaneous price varies throughout the trading period. 3

Literature Review
Several researchers such as Smith et al. [20] have conducted experiments to study behaviors that may result in asset price bubbles. The subsequent literature in this area is vast. Here we focus on the body of work closely related to the topic of this paper. In particular, we focus on experimental work that study asset price expectation formation. We classify the experiments in this area based on the following properties: endogeneity of supply and demand, finiteness of the time horizon, market mechanism, reward mechanism, and purpose of the experimental work.

Exogenous and Endogenous Supply and Demand
The supply and demand of the asset are exogenously specified in experiments that study the effectiveness of market mechanisms in facilitating the convergence of prices to the equilibrium price. In such a setup, in each trading period some of the participants are given the role of buyers and the rest are given the role of sellers. Each buyer and seller is given an initial endowment of currency or the asset, respectively. Moreover, the value of the asset for each buyer and seller is specified privately to the buyer and seller. Thus, there are a priori supply and demand curves and a market clearing price in each trading run. Different market mechanisms, such as the double auction mechanism, are used to generate a price trajectory for the asset and this price trajectory is compared with the market clearing price. For example, Smith et al. [20], Ball and Holt [2] conducted such studies.
Most of the papers that our current work builds on use endogenous supply and demand [5,11,10,14]. In this setup, at the beginning of a trading run, every participant is given an initial endowment. Thereafter, in each period, each participant can enter buy and sell orders.
Presumably, they make these decisions by taking into account the market clearing prices, their own forecasts of the future prices, as well as the revenue stream associated with owning the asset.
Thus, in each period the supply and demand of the asset are endogenous to the experimental market. Our experimental setup also follows this endogenous supply and demand approach.

Finite Time Horizon and Learning
Many of the experiments reported in the literature have a pre-announced number of periods in each trading run. Typically, it is 9, 12, or 15 periods. For example, Haruvy et al. [10] studied price forecasts in trading runs consisting of 15 periods each. Lugovskyy et al. [14] also used trading runs consisting of 15-period-long trading runs to study the effect of tâtonnement mechanism on asset price bubbles.
In many experiments reported in the literature the asset is worthless at the end of the trading run. For example, Haruvy et al. [10] and Lugovskyy et al. [14] considered an asset that paid a fixed expected dividend d at the end of each period t = 1, . . . , 15 and that had no salvage value. Assuming no discounting, the fundamental value of the asset at the beginning of period t wasp t = (16 − t)d. Thus, the fundamental value decreased linearly throughout the life of the asset and the participants knew ex ante that the asset would be worthless at the end of the trading run. Consequently, prices fell towards the end of the trading run. Since the dividend process was pre-announced, the participants could anticipate this price trajectory. In repeated 4 markets as in Haruvy et al. [10], one reasonable estimator of when the price process peaks is the peak time period in the previous trading run. Consequently, in each trading run the peak occurs earlier than in the previous trading run, and at the same time the size of the bubble decreases as the fundamental value is also larger in earlier periods. This observation leads to the conclusion that learning causes the bubbles to disappear.
One question that arises naturally is what happens when investors do not have the opportunity to form expectations using backward induction -e.g. reasoning such as "if it is certain that the price is 0 at the end of period 15, it may not be very high at the end of period 14".
Hirota and Sunder [11] considered a setup which makes such backward induction difficult. In their setup, the asset paid a known terminal dividend only at maturity (at the end of 15 or 30 trading periods). In some trading runs, the duration was 15 periods and the asset matured at the end of 15 periods; in this setting, backward induction was possible. In other trading runs, the asset matured at the end of 30 periods; however the participants were told that the experiment would end at an unknown time most probably much earlier than the life of the asset.
If the trading run lasted for 30 periods, they would receive the known dividend, otherwise they would receive a payout of the average price forecast for the asset when the trading run ends.
In this setting, the apparently random end of the market removed the anchor that the terminal dividend provided for backward induction. Their results suggest that the latter setup is more conducive to formation of asset price bubbles.
Another approach to eliminate backward induction is to make the number of periods in the trading run random. In this kind of probabilistic stopping, each period the trading run ends with a pre-specified probability. It can be shown that for an asset paying constant dividends, the fundamental value under probabilistic stopping with probability p is equal to the fundamental value under an infinitely-long trading run with discount factor of 1 − p (see Lemma 1). Camerer and Weigelt [5] used trading runs with probabilistic stopping to study convergence of prices to fundamental value and to test if prices are rational forecasts of the fundamental value. Ball and Holt [2] used a variation of this in which each individual asset can expire with a pre-specified probability. However, in their experiment the total length of the trading run was pre-announced (10 periods), thus the probabilistic stopping only created an effective discounting but did not eliminate the end of horizon effect.
In the current work, we use probabilistic stopping; in each period the trading run can end with a pre-announced stopping probability similar to Camerer and Weigelt [5]. Thus the participants cannot use backward induction to value the assets. Moreover, we chose the total duration of the experimental sessions to be about three and a half hours, which is atypical in this literature. Our reasoning was that even with probabilistic stopping, towards the end of the time slot for the session, the participants may reason that even if the trading run does not end, the session will end soon; consequently they may want to apply backward-induction thinking and offer to sell their assets for lower prices. A long duration for the experimental sessions helps to reduce the effect of such thinking at least in the earlier trading periods.

Market Mechanism
Depending on the objectives of their studies, different researchers use different market mechanisms. For example, Ball and Holt [2], Camerer and Weigelt [5], Hirota and Sunder [11] used 5 a continuous double auction mechanism. In this mechanism, participants announced their bids and asks; at any time any participant could accept a bid or an ask, and the price associated was recorded as the current transaction price. In these studies, the average of the transaction prices was used as a proxy for the market clearing price for that period. On the other hand, researchers such as Haruvy et al. [10] used a call market mechanism. In this mechanism, in each trading period, each participant entered buy and/or sell orders. At the end of the period, all orders were aggregated into market supply and demand curves. An equilibrium price was determined that cleared the market. All feasible trades (bids above the equilibrium price, asks below the equilibrium price) were executed. Bids and asks at the equilibrium price might be only partially executed (in which case, some tie breaking or apportioning mechanism was used).
Other authors have used other market mechanisms. For example, Lugovskyy et al. [14] used the tâtonnement mechanism. In this mechanism, in each period the market maker announced a suggested price and the participants placed their tentative buy and sell orders at that price.
If the aggregate supply equaled the aggregate demand then the suggested price became the market clearing price and all trades were executed; otherwise, the market maker announced an updated suggested price and the participants again placed their buy and sell orders at the new suggested price. The process repeated until the market cleared or a particular exit criterion was met (e.g. if the number of iterations reached a maximum).
In our setup, we use a slight modification of the call market mechanism. In each period, each participant can place multiple buy and sell orders, effectively specifying their individual supply and demand curves. These are aggregated into market supply and demand curves. Their intersection determines the market clearing price. Trades are executed at the market clearing price -all buy orders above the market clearing price and all sell orders below the market clearing price are fully executed. Orders at the market clearing price are filled completely if the aggregate supply and demand match exactly; in general, the maximum number of orders at the market clearing price are filled, and they are filled in the temporal sequence in which the orders were entered (i.e. on a first-come first-served basis). Some orders can be partially filled.
We chose to use this market clearing mechanism primarily to make sure that each period results in a unique equilibrium price. The prices in prior periods are common knowledge to all participants, and we hypothesize that the participants use this information to form price forecasts.

Reward Mechanism
The reward mechanisms found in the literature are quite varied. Most experiments pay a combination of a fixed participation fee and a reward proportional to the participant's performance in the experiment. The participant's performance is evaluated based on factors like the total virtual wealth at the end of the experiment and the forecast accuracy (if the experiment collected forecast data). Some researchers used a fixed exchange rate (i.e. a pre-announced virtual currency to real currency exchange rate), a pre-announced payment schedule (e.g. for accuracy of predictions), or they divided a pot of money in proportion to the total virtual wealth at the end of the experiment (e.g. Hirota and Sunder [11]). In Hirota and Sunder [11] participants who were price predictors were paid based on the accuracy of their predicted prices; in Haruvy et al. [10] the participants also received payments for accurate predictions. 6 Ball and Holt [2] argued against rewarding only the participant with the highest earnings as it may induce indifference or extreme risk seeking in people who are relatively behind in terms of their earnings.
In our experiment, rather than provide a fixed participation fee to each participant, three prizes were awarded at the end of each experimental session. Two of these were a function of their total wealth at the end of the experiment and one was for the best forecast accuracy. The details and rationale for these prizes are given in Section 4.4.

Purpose of the Experiments
Many of the experimental studies focus on convergence of asset prices to the theoretical fundamental value. For example, Ball and Holt [2] compared the prices resulting from trading with the fundamental value.
Some of the researchers went further to study price bubbles in repeated markets with the same set of participants. Haruvy et al. [10] found that the magnitude of price bubbles decreased as the same investors participated in successive trading runs.
Some of the experimental studies addressed belief or expectation formation among participants. For example, Haruvy et al. [10] explicitly captured participants' price expectations for future periods and concluded that prices converged to the fundamental value ahead of beliefs. Hirota and Sunder [11] studied the effect of first and higher-order beliefs by designing experiments with a publicly announced range of dividends that is bigger than the privately communicated actual range of dividends.
There are also other specialized objectives to some of these studies. Hirota and Sunder [11] studied the effect of investors' decision horizon on the presence of bubbles and they found that shorter decision horizons (compared with the maturity of the asset) can lead to larger bubbles.
Lugovskyy et al. [14] studied the effect of the tâtonnement trading institution on price bubbles and concluded that the participants were able to learn about supply and demand during the tâtonnement process, thereby reducing price bubbles.
The goals of our experiments were two-fold: One goal was to collect investors' price forecasts so that various models of investor forecast formation could be calibrated and compared. Another goal was to determine which of the qualitative regimes of asset price trajectories identified in models (convergence of asset prices, cycling of asset prices, or unpredictable asset price trajectories) observed forecasting behavior corresponds to. We conducted these experiments in a setting with endogenous supply and demand, while minimizing the propensity of the participants to sell their assets for lower prices when they think that the experiment is near to completion.

Some Models of Price Forecast Formation
In this section we briefly review a class of models of asset price forecasts and the resulting market clearing prices. Time is indexed by t = 1, 2, . . .. Each unit of asset pays a dividend d t at time t. Let p t denote the price of the asset at time t before the dividend d t has been paid.
Letp t+1 denote the investors' forecast at time t of the price at time t + 1. Suppose that the investors in the market are indifferent between investing and not investing in the asset if they expect a rate of return ofr, that does not depend on time. In each time period, investors can rebalance their portfolios without transaction cost, and they forecast only one period into the future. Then the indifference price at time t is In the experimental sessions we conducted, the dividend process was revealed to everyone, enabling the participants to compute E[d t ] explicitly. Along with specification of a model to computep t+1 , and appropriate initial conditions, equation (1) defines a discrete time dynamical system model of the asset price process.
For the 'fundamentalist' investor, the price forecastp t+1 is simply the fundamental valuē (1 +r) i assuming the infinite sum is convergent. (That is, the expected growth rate of the forecasted dividends is less thanr.) However, in general, forecasts are also influenced by observed data, especially the most recent data. Different assumptions about how past data is used gives rise to different models for forecast formation.

Extrapolation-Correction Models of Forecast Formation
The model given here is an adaptation of the more generic model given in Cheriyan and Kleywegt [8]. We consider forecasts that depend on both observed data as well as the fundamental value, as explained in the next four subsections.
It is convenient to consider the prices and forecasts scaled by the fundamental value, as follows: Let y t denote the growth rate of scaled prices in period t, that is, The extrapolation forecastŷ t of the price growth rate is given bŷ whereŷ 2 is an appropriate initial value. Thus, the extrapolation forecast is given by the exponential smoothing forecast. Note that α corresponds to the weight the investor gives to the most recent price ratio. If the forecaster used only the extrapolation forecast, then the corresponding scaled forecast for the price at time t + 1 would bê Note that at time t, (p 1 , . . . , p t−1 ) are known to the investor. The price p t is yet to be realized and will depend on the buy and sell orders of the investor.

Price Forecast
We consider a price forecast that is a function H of the extrapolation forecast and the fundamental value. Although the price forecast depends on both the extrapolation forecast and the fundamental value, it is convenient to use notation that shows only the scaled extrapolation forecast as an argument of H. Thus, the scaled price forecastπ t+1 is given bŷ where H : R + → R + is a function that captures some behavioral characteristics of the investor.
A fairly general class of H functions can be considered. The only property we require here is that H(1) = 1. That is, if the extrapolation forecast is equal to the fundamental value, then the price forecast should just equal the fundamental value. As an example, H(θ) ≡ 1 corresponds to rational expectations, while H(θ) ≡ θ corresponds to extrapolation-only price forecast.
Generally, in a neighborhood of θ = 1, H is nondecreasing, that is the price forecast increases with the extrapolation forecast. For larger values of θ, H may increase, perhaps at a decreasing rate. If, on the other hand, H increases and then decreases, we say that the investor exhibits panicking behavior . This corresponds to the investor losing confidence in the extrapolation forecast and adjusting the price forecast closer to the fundamental value. Such functions H will be called non-mononotic.
Substituting the scaled price and forecast into (1), the market clearing equation becomes Finally, in our experiments, E[d t+1 ],p t andp t+1 can be computed explicitly from the dividend process and the salvage value of the asset. These computations are given in Appendix A.

Linear Model of Forecast Formation
We also fit the data with a linear model given in Brock and Hommes [3]. According to this model, the investors price forecast for next period,p t+1 is given bŷ wherep t is the fundamental value at time t, and b 0 and b 1 are constants. Thus, the forecasts are linear functions of the deviations of the past price from the fundamental value.

Design of the Behavioral Experiment
Our experiment consisted of three sessions of a virtual asset market with a discrete-time trading mechanism, as given in Process Flow 1.
Process Flow 1 Process flow for an asset market with a discrete-time trading mechanism. 1. t ← 0 2. t ← t + 1 (New trading period begins) 3. Investors generate forecasts 4. Investors enter buy and sell orders 5. Market clears according to the specified market clearing algorithm. Price p t is determined 6. Trades are executed based on the market clearing price p t 7. Dividend d t is paid for each unit of asset held 8. Check termination of trading run. If not ended, go to step 2; otherwise compute rewards and debrief.

Approaches to Reduce End of Horizon Effect
In most market experiments the asset becomes worthless at the end of the last period.
Therefore, if the total number of periods in the trading run is known in advance, the asset price decreases to zero towards the end of the trading run. If there are repeated trading runs in an experimental session, the decreasing trend of prices during a trading run eventually reduces the occurrence of price bubbles in the later trading runs.
To alleviate this effect caused by a known time horizon, we used probabilistic stopping. At the end of each period the trading run can end with a stopping probability p = 0.02. Thus, the total number of periods in each trading run is a geometric random variable. This was explained to all participants, including the value of the stopping probability, in advance of each trading run.
Unlike traditional classroom experiments that last 1 or 2 hours, we set up our sessions to last for three and a half to four hours. Thus, at least during the initial part of the session, the participants did not consider the end of the session when making their trading decisions.
We also instituted a salvage value of 100 Experimental Currency Units (ECUs) for each unit of asset held when the trading run ended. Theoretically, this just adds a constant to the fundamental value of the asset. We decided to add the salvage value after an initial trial run of the experiment (the data from the trial run were not used) to reduce the fixation of some participants on the possibility that the asset may become worthless at any time.

Dividend Structure
In each period, a dividend was paid for each unit of asset held. The dividend was added to the cash-on-hand of each participant.
For the first two sessions, the dividend was fixed to 10 ECUs per unit of asset in each period.
For the third session, the dividends were the same for all participants but were random for each period. In each period the market was in a state X t ∈ {low, high}. The state transition followed a discrete time Markov chain with the transition matrix The initial state, X 1 , was chosen at random with equal probability. The dividend D t for each period had a distribution depending on X t as given in Table 1.

Experimental Sessions
The experiment consisted of three sessions, each with a new group of participants. Each session lasted for about four hours in total and consisted of an overview, training trading run, actual trading runs, and post-session debriefing.
The overview included an explanation of the Process Flow 1, probabilistic stopping of trading runs including the memoryless property of the geometric distribution, the dividend process, the concept of fundamental value and its calculation, the process for placing buy and sell orders, and the market clearing process to determine the market clearing price and market clearing trades. The stopping probability for all trading runs was 1/50 and this was announced to the participants.
The training trading run lasted four periods. The purpose of the training trading run was to familiarize the participants with the market simulation software, by entering forecasts, placing buy and sell orders, and answering questions that tested their understanding of the experiment.
It was made clear that the number of periods was fixed only for the training trading run. The earnings in the training trading run were not taken into account for the determination of the eventual compensation.
After the training trading run, it was announced that a real trading run was about to start and that the total number of periods would be random as explained. The experiment started with a number of questions for each participant to test participants' understanding of the experiment. Each participant was endowed with an initial portfolio of 5000 Experimental Currency Units (ECUs) and 50 units of the virtual asset.
Thereafter, period 1 started. In each period each participant was shown a screen displaying the participant's current cash and asset balance, a chart of the previous prices and a table of previous prices, the dividend history, and previously executed buy and sell orders for the participant, input boxes to answer questions described below, an input box to enter the participant's price forecast, and input boxes to enter buy and sell orders. For the third session with Markov dividends, additional information was displayed as described in the next section.
In each period each participant had to enter answers to the following questions: • The expected number of time periods remaining in the trading run.
• The expected total dividends paid by one unit of asset from the current period until the end of the trading run.
• The total of the expected dividend from the current period until the end of the trading run and the salvage value for one unit of asset held until the end of the trading run.
• The participant's forecast of the price per unit of asset.
• The current state of the market (only session 3 -Markov dividend case).
For the third session with Markov dividend process, the order entry screen also displayed the estimated (based on realized past dividends) probability of the current market state being high or low, the conditional expected dividend for the current time period and fundamental value for the next time period if the market continues. This information, however, was not common knowledge. At the beginning of the experiment, it was announced that only a randomly chosen subset of participants would be provided with this additional information. In the actual experiment, all participant screens had this information. This was done to create the perception of asymmetry in information among the participants. In particular, the participants may have believed that they had privileged information, inaccessible to many of the other participants.
The current market state itself was not revealed to the participants.
After they entered the feedback and forecast information, the order entry portion of the screen was enabled and they could enter multiple buy or sell orders. Participants could also not trade in a period, however, the software design enforced that there would be at least one order entered among all participants in every period. This was required to ensure that there is an equilibrium price in every period.
Once a participant made her entries, a wait screen was displayed. After all participants made their entries, the market was cleared based on the orders and the new equilibrium price was determined.
Subsequent periods followed the same pattern until the computer determined that the trading run was over.
At the end of the trading run, the participants' portfolios were converted into virtual currency. Then, the winners were determined and were announced.
Finally, there was a short debrief session where we gathered feedback from the participants about the session.
The specific details of the experimental sessions are given in Table 2.

Incentive Mechanism
Rather than provide a fixed participation fee to each participant, three prizes were awarded at the end of each experimental session.
1. Two randomly selected players split $300 in proportion to the value of their final amount of virtual currency, after payment of the salvage value.
2. The participant with the largest final amount of virtual currency received a prize of $100.
3. The participant with the smallest mean square forecast error received a prize of $100.
The motivation for this incentive scheme is as follows. Other behavioral experiments have shown that people are more willing to perform a task when either it has no remuneration (i.e. it is considered as a help) or when the remuneration is significantly large. In our case, $18.75 (= $300/16) would not have been a lucrative enough participation fee for four hours' time.
On the other hand, splitting the $300 among two randomly selected participants makes the incentive more lucrative. This is supported by the framing effect and pseudocertainty effect reported by Kahneman and Tversky (Tversky and Kahneman [21]) wherein when evaluating such conditional situations, people evaluate the options assuming that the selection process has already happened. Thus, in our case, the participants would be looking at an amount of the order of $150, which is lucrative. Also, random selection of two participants helped avoid participants dropping out of the experiment if they are doing poorly.
The prize for the largest portfolio incentivized playing the game strategically and thoughtfully and the prize for the smallest forecast error incentivized careful forecasting and reporting of forecasts.

Participant Demographics
The participants were students belonging to the Ph.D. (ISyE, Math) and Masters (Quantitative and Computational Finance -QCF) programs. We chose to include only Masters or Ph.D. students to ensure that they had sufficient background knowledge of basic probability.

Summary of Observations
Though the first session had two trading runs, the second trading run did not complete within the time allotted for that session. So we used only the completed trading run for data fitting. There was only one trading run per session for the remaining two sessions. Therefore, from here on we will use the terms session and trading run interchangeably.

Equilibrium Price
Since in each period, the trading run could end with a stopping probability of 1/50, the number of time periods T was Geometric with parameter p = 1/50. The expected total returns for one unit of asset can be computed as where d t is the dividend at period t. For sessions 1 and 2, the fundamental value was constant throughout and equaled 600 ECUs. For session 3, the computation of fundamental value is given in Appendix A. For session 3, the fundamental value at the beginning of the trading run was also 600 ECUs.
The dark blue line in Figure 1 shows the realized equilibrium prices for the three sessions.
Sessions 1 and 2 had prominent price cycles whereas session 3 had milder price cycles.
In all three sessions, the prices started from well below the fundamental value and started increasing thereafter. This phenomenon has been observed in many other market experiments, see for example Camerer and Weigelt [5], Haruvy et al. [10].
In the third experimental session, there was a sudden downward spike in period 56. It was revealed to have been caused by one participant who panicked for some unknown reason and 13 wanted to get rid of all his assets. So he offered to sell them for 0.01 ECU each. The market rebounded the very next period. (The data from this spike onwards were excluded from the calculations that follow.) From the graphs of the price process, it appears that the participants had a perceived value that was lower than the fundamental value of 600 ECUs. In other words, they seemed to have made most buy-sell decisions as if the fundamental value was somehow smaller than the true fundamental value.

Price Forecasts
In each period, each participant was required to enter a price forecast for that period. Recall that the market clearing price in a period was determined only after the buy and sell orders were processed. Figure 1 shows a plot of the price forecasts for each session by each participant along with the market clearing price (dark blue line). It can be seen that the price forecasts usually followed the market clearing price.
The forecasts in the initial few periods are interesting. It seems that some of the participants started with a near-rational forecast and adjusted them down as the trading run progressed. On the other hand, some "skeptics" started with a very low forecast and adjusted them upwards as the trading run progressed.

Earning Forecasts
As part of the initial survey, each participant was asked the following question: Given the compensation rules and the number of participants, how much money do you think you will earn in this experiment?

Data Used
The data used for model calibration are the equilibrium prices and the participants' forecasts from the three sessions 15 First, we cleaned the data by fixing obvious typographical errors. Typically, the error was omission of the decimal point.
In the initial few periods of each of the three sessions, the equilibrium price increased from a value much lower than the fundamental value. A possible explanation is that the participants were trying to learn how the rest of the participants would behave. However, the models that we wanted to fit with the data were not intended to capture such initial learning or adaptation.
Consequently, we used a subset of the data when the initial effect had passed. Moreover, we wanted to choose a subset of the data in a way that is endogenous to the data itself and not dependent on the model that we fit. To achieve this, we used the following approach: In the equilibrium price chart, let p 0 be the price in the first price trough. We discarded the data from the initial periods during which the price was smaller than p 0 . According to this approach, the number of dropped periods for the three sessions was between 7 and 11. Finally, to use the same number for all three sessions, we simply discarded the data from the first 10 periods in each session.
We also checked this approach using the calibrated models with the quadratic H function given in Section 6.3.5. We calibrated a sequence of models by successively dropping more initial periods from the data set. We observed that the fitted parameter values stabilized by the time we dropped the first 10 periods' data.
In addition, for session 3, we dropped the data from period 56 onwards. This was done to avoid the effects of the spike that occurred in period 56.

Parameter Estimation Method
Parameter estimation was done by least squares fitting.
To compare the different models that were fitted, we use the Leave-One-Out-Cross-Validation (LOOCV) approach. According to this approach, the observations in the data set are partitioned into subsets. One fits the model using all but one subset, and then use the fitted model to calculate forecasts for the omitted subset. The forecast error is recorded. This step is repeated with each subset of the data set left out in turn. Then the root mean square error (RMSE) of all the recorded forecast errors is calculated. The LOOCV RMSE of different models are compared. We perform the LOOCV validation with two types of subsets: (1) leave-one-period-out LOOCV, in which each period's data for each participant is a subset, and (2) leave-one-sessionout LOOCV, in which each session's data is a subset.
The leave-one-period-out LOOCV root mean square error is denoted by RM SE P ψ and the leave-one-session-out LOOCV RMSE by RM SE S ψ . The corresponding coefficient of variation is obtained by scaling the RMSE by the fundamental value: Appendix B gives formulas for computing these quantities.

Parameter Fits for Various Models
We fit various models with the data. The individual models are covered in subsequent sections; the key parameters of these models are given in Table 4. belong to the family of extrapolation-correction models proposed in Cheriyan and Kleywegt [8], with 3 and 4 parameters respectively.

Rational Base Case
For comparison, we take as base case the rational expectations forecast of a fully informed participant (BASE). That is Thus, there are no parameters to fit for this model. For the Markov dividend case in session 3, we used the reference fundamental value of 600 for the model. Table 5 gives the leave-one-period-out LOOCV RMSE for each session.
Also, the leave-one-session-out LOOCV RMSE is RM SE S BASE = 150.34 CV S BASE = 25.06%

Modified Rational Expectations
Model F is similar to BASE, except that a parameter representing "perceived fundamental value" is now fitted with data.
wheref is the parameter to be fitted. Table 6 gives the leave-one-period-out LOOCV RMSE for each session.   The leave-one-session-out LOOCV RMSE is given by

Simple Exponential Smoothing
Model ES uses simple exponential smoothing with initial price ratio of 1.
. . , n y 2 := 1 Table 7 gives the leave-one-period-out LOOCV RMSE for each session.  The leave-one-session-out LOOCV RMSE is given by

Brock and Hommes Model (BH)
We adapt the model of expectation formation used in Brock and Hommes [4] given in (2): Note that the forecast in period t + 1 is a linear function of the price in period t − 1. However, in our experiment, at the beginning of period t, the participants entered the price forecast for that period. Therefore, we modify the BH model tô  The leave-one-session-out RMSE is RM SE S BH = 23.97 CV S BH = 3.99%

Exponential Smoothing with Quadratic H (ESH2)
The correction function is given by In this case, the model function ψ j is given by There are four parameters: α = exponential smoothing parameter y 2 = initial price ratio to start the exponential smoothing ρ = parameter for the H function f = fundamental value 20 Initial numerical results showed that the objective function was very flat with respect to y 2 . This is because the effect of y 2 on the forecast goes down exponentially at rate 1 − α, and the data of the first 10 periods are used to prime the exponential smoothing method, but as discussed before, the data of the first 10 periods are not used in the squared error calculations.
So unless α is very close to 0, y 2 has very little effect on the objective. Therefore, y 2 was fixed at 1 and the remaining three parameters were estimated by minimizing the sum of squared errors. Table 9 gives the leave-one-period-out LOOCV RMSE for each session.  Table 9: Forecast errors of the exponential smoothing with quadratic H model (ESH2) for the three sessions.
The leave-one-session-out LOOCV RMSE is

Exponential Smoothing with Non-Mononotic H (ESHE)
The correction function is given by The structure of the H function was chosen so that when ρ ∈ [0, 1], then H is increasing and when ρ < 0, then H is non-monotonic. Thus, a fitted value of ρ that is negative would indicate that the participant exhibits panicking behavior. Also, the parameter η is such that H (1) = 2η.  The model function ψ j is given by There are five parameters: α = exponential smoothing parameter y 2 = initial price ratio to start the exponential smoothing ρ = monotonicity parameter for the Hfunction η = slope paramter for the Hfunction f = fundamental value In order to be consistent with fitting the ESH2 model, the y 2 parameter was fixed to 1.
We used a trust region type method to fit the four parameters. Unfortunately the objective function for the parameter fitting problem has multiple local minima. We started from 10 random starting points and picked the solution that gave the best objective value. (For session 3, participant 13, we started from 100 random starting points as the algorithm terminated with a local solution for only some of the starting points.) Table 10 gives the leave-one-period-out LOOCV RMSE for each session.  The leave-one-session-out LOOCV RMSE is It is also interesting to note that fitting a single set of parameters to all the participants in all the sessions gave an RM SE = 16.07 with the corresponding CV of 2.68%. Thus, the behavior of the group can be described fairly well by a single set of parameters.  The leave-one-session-out LOOCV RMSE of the various models are given in Table 11. As is the case for the leave-one-period-out LOOCV RMSE, the rational expectations model has the highest leave-one-session-out LOOCV RMSE, and model F has slightly smaller leave-

Interpretation and Implications of Results
In Cheriyan and Kleywegt [8] the dynamical system associated with the price process was studied both analytically and numerically. It was shown there that the qualitative behavior of the trajectories of this dynamical system is determined by the parameter α and the nature of the H function. These can also be related to the investors' behavioral characteristics. Although the purpose of our experiment was to calibrate models of investor forecasting behavior and not to calibrate the dynamical system itself, in this section we make observations regarding the fitted models of investor forecasting behavior and the qualitative behavior of the dynamical system.
The parameter α captures the investors' memory -the weight that they put on the most recent observed price ratio. Given an H function and a fundamental dividend to price ratio δ (or price-earnings ratio 1/δ), a critical value of α is given by  [8] show that a price cycle appears that attracts all trajectories. This bifurcation process is continuous in the sense that the price cycles are small if α is slightly larger than α * , and the price cycles gradually grow larger as α increases from α * .
For α not too far from α * , the price cycles appear to be predictable, that is, the price trajectories converge to a smooth curve, and any two trajectories that start close to each other will remain close to each other at all times. If the H function is monotonic, this behavior persists for α > α * . However, if H is non-monotonic, then for larger values of α, the price trajectories can be non-predictable. This means that the trajectories do not converge to a smooth curve, and they exhibit sensitive dependence on initial conditions, that is, two trajectories that start close to each other will grow apart exponentially fast. A non-monotonic H captures what we called panicking behavior of the investor -as the extrapolation forecast increases beyond the fundamental value, the price forecast initially increases (confidence or exuberance), but beyond some value the price forecast decreases and moves closer to the fundamental value (panic). Table 12 shows the fitted parameters for the ESH2 model.      Brock and Hommes [4] calls b 0 and b 1 the "bias" and "trend" parameters respectively. In particular, if b 1 < 0, then the investor is called a "contrarian", and if b 1 > 0, then the investor is called a "trend chaser". If b 0 = 0 and b 1 = 0, then the investor is a "fundamentalist". The fitted parameter values indicate that all the participants in our experiment were trend chasers.

Conclusions
We designed an experiment to study investors' price forecast formation in the context of a market for an investment asset. Many experiments in the literature use trading runs with a pre-announced finite number of periods. However, the known end-of-horizon seems to affect participants' forecasts, apparently via reasoning involving backward induction from the end of the horizon, that is not representative of forecasting in actual asset markets. In our experiment, we emulated an infinite horizon with discounting by stopping the trading run in each period with a pre-announced stopping probability. We conducted three experimental sessions with one trading run each. The equilibrium prices in all three trading runs exhibited cycles.
We fit a number of models of expectation formation to the data.
Proof. Let ξ t be the random variable that is 1 if the market is running in period t and is 0 otherwise. Then {ξ t } ∞ t=1 is a sequence of iid Bernouilli(1 − p) random variables. Let D t be the dividend in period t. We have that Note that if any of the ξ i 's are zero, then the entire expression is zero, this automatically captures the fact that if the market stops in period i, then for t ≥ i, D t = 0. Also, for the present period, we know that the dividend is certain to be d, that is The dividend stream at period t is given by Therefore, Now, Therefore, which is independent of t. Since a unit of asset held for ever necessarily will result in a final payout of s, and since the discount factor is 0, the fundamental value of a unit of asset is given by Remark. Equation (A.1) says that, in the case of deterministic dividends, probabilistic stopping of the market with stopping probability p is equivalent to an infinitely lived marked with discount factor 1 − p.

A.2. Markov Dividends
In the case of Markov Dividends, the support of the dividend distribution in a period t depends on the market state in that period. The market states X t ∈ {1, 2} evolve according to a Markov chain with transition matrix P . Let π denote the stationary distribution corresponding to transition matrix P . In our experiment, the initial state X 0 was drawn from the distribution π.
Let the matrix Q denote the conditional p.m.f. for the dividend q ij = P (d t = j|X t = i) Algorithm 2 gives the details of the market algorithm in the case of Markov dividends.
Algorithm 2 Market algorithm with deterministic constant dividend. Given: • p, the stopping probability • π, the initial probability distribution for market state • P = [p ij ], the state transition matrix for the Markov chain; p ij = P (X t = j|X t−1 = i) • Q = [q ij ], the dividend distribution; q ij = P (d t = j|X t = i) Algorithm: 1. t ← 1 2. For each participant, initialize account with initial asset and initial cash 3. Begin period t 4. Generate X t . IF t = 1, generate X 1 ∼ π, ELSE generate X t ∼ p X t−1 5. Capture for forecast data for period 6. Conduct trades (a) Participants enter buy and sell orders (b) Perform market clearing to determine market clearing price (c) Execute trades and update cash and asset accounts for each participant 7. Generate dividends d t ∼ q Xt 8. For each participant, distribute dividend d t for each unit of asset 9. Generate ξ t+1 ∼ Bern(p) (a) IF ξ t+1 = 1 i. For each participant, distribute s for each unit of asset ii. STOP (b) ELSE i. End period t. t ← t + 1 ii. Proceed to step 3 Participants have complete information about the parameters of the Markov Chain (P, Q, π) and but they do not know the underlying state process {X t }. They know that the market started in the steady state π = [π 1 , π 2 ], that is X 0 is chosen according to π.
They observe the prices p t−1 , p t−2 , . . . and the dividends d t−1 , d t−2 , . . . Let d t−1 = (d 1 , . . . , d t−1 ) denote the history of dividends up to and including period t − 1. Letν x,t (d t ) = P (X t+1 = x|d t ). That is,ν x,t (d t ) is the estimate at the end of period t (i.e. beginning of period t + 1) that the probability of the state X t+1 will be x ∈ {1, 2, }. Let ν t (d t ) = [ν 1,t (d t ),ν 2,t (d t )]. In each period, the investor updatesv t and uses it to compute the fundamental value for the next period.
The estimate of the probability distribution of the state is given bŷ Then, the fundamental value at period t can be computed as follows: We use the following notation. σ = 1, 2, 3 session index n σ = number of periods in session σ N σ = number of participants in session σ X σ,j = equilibrium price in period j Y σ,j,u = price forecast reported by participant u in session σ in period j (n 3 = 55, as discussed above.) Consider a family of models of forecast formation represented as Y σ,j,u = ψ j (X σ,1 , X σ,2 , . . . , X σ,j−1 ; γ) + ε j where γ denotes the vector of parameters of the model and ε j are i.i.d. N (0, σ 2 ) random variables.