# SPECULATIVE OVERPRICING IN ASSET MARKETS WITH INFORMATION FLOWS

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10 146 T. R. PALFREY AND S. W. WANG We refer to traders with 0 θ<1asskeptical types. At one extreme is the θ = 0 type. Traders of this type believe that the signals are just noise, as if the signal distribution were independent of the state. They do not update their prior after either signal a or signal b. Such a type s probabilistic belief that A is the state of the world remains unchanged for any sequence of signals. That is, ρ s t+1=a it+1 (0) = p t s 1 s t We refer to traders with θ = 1 as the Bayesian type. Traders of this type believe they are receiving signals of strength q, so their posterior probabilities are equivalent to those of a Bayesian: ρ s t+1=a qρ it it+1 (1) = qρ it + (1 q)(1 ρ it ) We refer to θ>1asgullible types. Traders of this type update as if the informativeness of signals is higher than q. For extremely high values of θ, fickle traders treat a signal as nearly a full revelation of the state. For example, if p = 0 5, q = 0 7, and θ i = 10, then, after the first signal, if s 1 = a, traderi s posterior is ρ s 0 7 1=a 10 i1 (10) = = This does not imply that gullible types beliefs move immediately to 0 or 1 and stay there. In fact, exactly the opposite is the case. In the above example, if s 2 = b, then i s beliefs go back to ρ s 1=a s 2 =b i2 = 0 5, and then if s 3 = b again, the trader s belief would be ρ s 1=a s 2 =b s 3 =b i3 = Thus, gullible types have relatively volatile beliefs, while skeptical types have relatively sticky beliefs Equilibrium Prices We maintain the assumptions of no short sale (implemented in the experimental design) and sufficient liquidity so that any trader can hold all units of the risky asset for any price less than or equal to 1 Under these assumptions, we can apply arguments similar to those used to solve the HK model and characterize the equilibrium price dynamics in our model. For the remainder of the paper, we will assume p = For any fixed p, the updating process depends only on q, the number of a signals, which we denote by α, and the number of b signals, which we denote by β t α. Hence, in the baseline case of homogeneous Bayesian beliefs (θ i = 1 i), the equilibrium 6 The model extends in a straightforward way to the more general case of p 0 5.

17 SPECULATIVE OVERPRICING IN ASSET MARKETS 153 TABLE I SUMMARY OF EXPERIMENTAL SESSIONS Session Type Signal # Mkts # Traders Period (sec) 1 One asset 2 One asset 3 One asset 4 One asset 5 One asset 6 One asset 7 Complete 8 Complete Shortsale 10 Short sale 11 Short sale RESULTS We present the results of our experiment in the following order. First, we analyze whether there is significant overpricing in the data from the baseline sessions, and to what extent this can be attributed to a speculative premium. We next test two related implications of the model: asymmetric reaction to good news versus bad news, and the horizon effect. Third, we report results from our two alternative market designs, which allow inframarginal traders, who believe prices are too high relative to their beliefs, to engage in arbitrage in ways that were not available in the baseline design. This includes completing the market by opening up markets for state contingent claims in both states, and explicitly allowing short sales. Finally, we examine the dynamics of individual asset ownership from period to period as information is gradually revealed, and compare it to the theoretical predictions about ownership dynamics. The hypotheses generated by the model concern the trajectory of transaction prices in the markets. The central hypotheses concern overpricing relative to the Bayesian benchmark, which, as we showed in Section 3, isdrivenby two different phenomena: the valuation component, which derives from heterogeneous current hold-to-maturity valuations, and a speculative component, which derives from heterogeneous future hold-to-maturity valuations. Because we cannot control for, or directly observe, each individual trader s θ i, measuring these two components of overpricing in our data means that the distribution of θ i s has to be estimated from the observed transaction prices. Before proceeding with the estimation, we first ask simply whether, in our markets, the combined effect of the two components produces prices that are in excess of the prices that would arise if traders had homogeneous, Bayesian beliefs.

19 SPECULATIVE OVERPRICING IN ASSET MARKETS 155 TABLE II MEDIAN PRICES BY INFORMATION REVEALED (q = 5 ) δ Median Price (N) Bayesian Price (θ = 1) (4) (4) (24) (52) (113) (182) (initial period) 53 8 (136) (148) (138) (58) (24) (35) (7) 75 3 signed-rank test (p<0 0001). 13 This suggests that although the traders are receiving informative signals about the state of the world, they may be using non- FIGURE 1. Median prices versus Bayesian predictions (q = 5 ). 13 The Wilcoxon test assumes independence across observations. To the extent that there is some correlation across observations in our data, the p-values we report for these tests should be treated as a lower bound. Alternative tests with lower power (e.g., binomial test) also yield highly significant test statistics.

20 156 T. R. PALFREY AND S. W. WANG Bayesian updating heuristics. To the extent that there is heterogeneity of these heuristics, asset prices will deviate systematically from those predicted under the assumption of perfect Bayesian updating and lead to overpricing according to the multiple-θ model of speculation. The homogeneous beliefs model predicts the price to be 50 in all periods where there are equal pieces of good and bad news, δ = 0. A Wilcoxon signedrank test reveals that the median prices in these periods are significantly higher than 50 (p<0 0001). Next, we turn to the price in the initial period when there have been no news announcements. Under the null model, the price in the initial period of each market should be 50, reflecting the flat prior. This prediction does not hinge upon any assumptions about the belief updating process. In fact, the median price is above 50 in all 18 initial periods (Wilcoxon signedrank: p = ). These transaction prices in the initial periods may offer the clearest evidence of speculative trading. Since no information has been revealed, if the prices are above 50, at least some traders must be trading based on speculation about price changes in future periods. Signal Strength: q = 6. Table III and Figure 2 display the aggregate median prices for the q = 6 sessions. We find that observed prices follow the trajectory of predicted Bayesian prices more closely than in the q = 5 sessions. However, the observed prices are still greater than the predicted prices for nearly all δ, and these differences are significant according to the Wilcoxon signed-rank test (p<0 0001). TABLE III MEDIAN PRICES BY INFORMATION REVEALED (q = 6 ) δ Median Price (N) Bayesian Price (θ = 1) 7 0 (6) (8) (2) (12) (30) (57) (80) (initial period) 58 1 (110) (8) (146) (88) (53) (3) (5) (8) (2) (1) 6

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