The term structure of cross-sectional dispersion of expectations in a Learning-to-Forecast Experiment

Abstract

In this paper, we present the results of a Learning-to-Forecast Experiment (LtFE) where we eliciting short- as well as long-run expectations regarding the future price dynamics in markets with positive and negative expectations feedback. Comparing our results on short-run expectations with the LtFE literature, we prove that eliciting long-run expectations has no impact on the price dynamics nor on short-run expectations formation. In particular, we confirm that the Rational Expectation Equilibrium (REE) is a good benchmark only for the markets with negative feedback. Interestingly, our data show that while the term structure of the cross-sectional dispersion of expectations is convex in positive feedback markets, it is concave in negative feedback markets. Differences in the slope of the term structure stem from diverse degrees of uncertainty regarding the evolution of prices in the two feedback systems: (1) in the negative feedback system, the convergence of the price to the REE reflects a tendency for coordination of long-run expectations around the fundamental value; (2) conversely, oscillatory price dynamics observed in the positive feedback system is responsible for the diverging pattern of long-run expectations. Finally, we propose a new measure of heterogeneity of expectations based on the scaling of the dispersion of expectations over the forecasting horizon.

Appendices

Term structure of cross-sectional dispersion of expectations

In this appendix we provide some simple examples of how a given forecasting rule affects the term structure of cross-sectional dispersion of expectations.

Let us assume that the subjects, when forming their long run expectations, linearly extrapolate the past price change with a coefficient $m_i$ depending on each individual subject. Considering Eq. (4), it is possible to compute the average expectation across subjects:

$$\begin{aligned} \mathrm {E}[{}_ip^e_{t,t+k}]=p_{t-1}+\mathrm {E}[m_i] \ (k+1) (p_{t-1}-p_{t-2}). \end{aligned}$$

(16)

The variance of the expectations is therefore:

$$\begin{aligned} \mathrm {Var}[{}_ip^e_{t,t+k}]=\mathrm {Var}[m_i] \ (k+1)^2 (p_{t-1}-p_{t-2})^2. \end{aligned}$$

(17)

Consequently, given that:

$$\begin{aligned} \mathrm {Var}[{}_ip^e_{t,t}]=\mathrm {Var}[m_i] \ (p_{t-1}-p_{t-2})^2 , \end{aligned}$$

(18)

and plugging it into Eq. (17), the term structure is:

$$\begin{aligned} \mathrm {Var}[{}_ip^e_{t,t+h+1}]= (k+1)^2 \ \mathrm {Var}[{}_ip^e_{t,t}]. \end{aligned}$$

(19)

An alternative to the linear forecasting rule is a “random walk rule”, whose starting value is the realized price in the previous period:

$$\begin{aligned} {}_ip^e_{t,t+k}=p_{t-1}+\sum _{k=0}^{K} {}_i\eta _{t+k}, \end{aligned}$$

(20)

where ${}_i\eta _{t+k}$ are iid random variables with $\mathrm {E}[{}_i\eta _{t+k}]=0$ for each k and i. Under the rule of Eq. (20), it is easy to show that the term structure of the variance is linear in the forecasting horizon:

$$\begin{aligned} \mathrm {Var}[{}_ip^e_{t,t+h+1}]= (k+1) \ \mathrm {Var}[{}_ip^e_{t,t}]. \end{aligned}$$

(21)

An additional possible forecasting rule for long-run predictions is the following:

$$\begin{aligned} {}_ip^e_{t,t+k}=p_{t-1} + {}_i\xi _{t+k}, \end{aligned}$$

(22)

where $\xi _{t}$ are iid random variables with $\mathrm {E}[{}_i\xi _{t}]=0$ for each t and i. The variance of the subjects’ predictions will not depend on the forecasting horizon H:

$$\begin{aligned} \mathrm {Var}[{}_ip^e_{t,t+k+1}]= (k+1)^0 \ \mathrm {Var}[{}_ip^e_{t,t}]= \mathrm {Var}[{}_ip^e_{t,t}]. \end{aligned}$$

(23)

A more general term structure for the dispersion of the predictions exhibits a curvature which depends on the shape parameter $\alpha$:

$$\begin{aligned} \mathrm {Var}[{}_ip^e_{t,t+h+1}]= (k+1 )^\alpha \ \mathrm {Var}[{}_ip^e_{t,t}], \end{aligned}$$

(24)

which nests all the previous specific cases.

These simple calculations show that the term structure of the variance of subjects’ predictions can be extremely informative about the possible underlying rules for the formation of long-run expectations.

Instructions and screenshot

1.1 Translated instructions (from the original in Spanish)

[General instructions]

Welcome to the Laboratory of Experimental Economics! You are participating in an experiment in which you will take decisions in a financial market. The instructions are very simple but, please, read them carefully.

During the whole experiment you will play with experimental currency unit (ECU) and, at the end of the experiment, your final profit, which will be added to a show-up fee of 3 euros, will be converted into euro according to the following exchange rate: 1 Euro = 500 ECU. The total amount will be paid at the end of the experiment in cash.

[Only in the positive feedback treatment]

You are a financial advisor to a pension fund that wants to invest an amount of money to buy an asset. The pension fund has to choose between investing its money in a bank account which pays fixed interest rate and a risky asset that pays dividends. The allocation depends on your forecast for the evolution of the asset price. When making your predictions remember that the asset price in each period is affected as follows: positively by the dividend, negatively by the interest rate and positively by the investors’ expectations of the asset price in that period.

Your task is to predict the price for 20 periods. In each period (t) you will predict the price for all the remaining $20-t$ periods, i.e. in period 1 you will submit 20 predictions starting from the prediction about the price at the end of period 1, in period 2 you will submit 19 predictions and so on. Your predictions must be between 0 and 100.

In period 1 you will submit predictions with information only about the interest rate and the average dividend. From period 2 onwards, you will have more information: a graph with the time series of your past predictions and the time series of the market prices. The green dots represent the time series of your 1-step-ahead predictions, while the blue dots represent the asset price in each period. Additionally, you will see the values of these time series and the time series of all your past predictions. Remember that in any period you will see the information about the asset price in the previous periods.

The interest rate will be equal to $5\%$ and the mean dividend will be equal to 3.25 (or 3.5 depending on the session).

[Only in the negative feedback treatment]

You are an advisor to a firm that wants to buy a certain amount of a particular good. In each period, the manager of the firm decides how many units of this particular good she wants to buy with the aim of selling it in the next period. To take an optimal decision, the manager needs a good prediction of the market price in the next period. The evolution of the market price will be as follows: if the demand for the good is higher than the supply, the price will rise. Conversely, if the supply is higher than the demand, the price will decrease. The manager will take his/her decision based on your predictions about the market price in given period: the higher (lower) the prediction is, the higher (lower) the demand will be and, as a consequence, the market price will fall (rise).

Your task is to predict the price for 20 periods. In each period (t) you will predict the price for all the remaining 20-t periods, i.e. in period 1 you will submit 20 predictions starting from the prediction about the price at the end of period 1, in period 2 you will submit 19 predictions and so on. Your predictions must be between 0 and 100.

In period 1 you will submit predictions without any information about past prices. From period 2 onwards, you will see a graph with the time series of your past predictions and the time series of the prices. The green dots represent the time series of your 1-step-ahead predictions, while the blue dots represent the market price in each period. Additionally, you will see the values of those time series and the time series of all your past predictions. Remember that in any period you will see the information about the market price of the previous periods.

[General instructions]

Once each subject has submitted his/her prediction for each period, the price will be computed and it will be shown at the beginning of period 2. The same mechanism will be used for subsequent periods. Once you and the other subjects submit the predictions, the price as well as the profits will be computed according to the forecasting accuracy.

Remember that your profit depends on your forecasting accuracy. The lower your forecasting error (the distance between your predictions and the price in a given period), the higher your profit will be. Your profit will be computed at the end of each period. In addition to the profit for your 1-step-ahead prediction each period, you will receive an extra profit for your past predictions about the price for each period. This extra profit will be computed according to the following table:

Difference between price of period t and your prediction for period t	ECU
$\pm 5$	25
$\pm 10$	12
$\pm 1$5	5

At the beginning of each period you will see the profit for all the predictions and the cumulative gains.

1.2 Screenshot

See Fig. 11.

Fig. 11

Screen-shot of the experiment

Individual prediction strategies

Following Heemeijer et al. (2009), we estimate the linerar forecasting rule for each subject:

$$\begin{aligned} {}_ip^e_{t,t} = \alpha _1 p_{t-1} + \alpha _2 \ {}_ip^e_{t-1,t-1} + (1 -\alpha _1 - \alpha _2) p_f + \beta (p_{t-1} - p_{t-2}) + \epsilon _t \end{aligned}$$

(25)

The heuristic described in Eq. (25) can be successfully estimated for 72 out of 90 subjects. Using the estimates obtained we can classify subjects as:

• (i) Fundamentalist in the negative feedback treatment: $\alpha _1 + \alpha _2 \approx 0$ and $\beta =0$

• (ii) Adaptive: $\alpha _1 + \alpha _2 \ne 0$ and $\beta =0$

• (iii) Naïve: $\alpha _1 \approx 1$ and $\alpha _2 =\beta =0$;

• (iv) Naïve and fundamentalist: $\alpha _1 < 1$ and $\alpha _2 =\beta =0$

• (v) Adaptive trend follower in the positive feedback treatment: $\alpha _1 \ne 0, \alpha _2 \ne 0$ and $\beta \ne 0$

• (vi) Naïve trend follower in the positive feedback treatment: $\alpha _1 \ne 0, \alpha _2 = 0$ and $\beta \ne 0$

• (vii) Trend follower in the positive feedback treatment: $\alpha _1 = 0, \alpha _2 = 0$

Tables 4 and 5 summarize our results and compare them with those reported in Heemeijer et al. (2009).

Table 4 Positive feedback treatment: individual prediction strategies according to the estimated parameters in Eq. (25)

Table 5 Negative feedback treatment: individual prediction strategies according to the estimated parameters in Eq. (25)