Forecasting dynamics and convergence to market fundamentals

Journal of Economic Behavior and Organization 22 (1993) 269-284. North-Holland

Steven

eterson

Received May 199 1. final vcrsisn receive

Subjects trade shares of a ktitious asset in a computerized double-auction market and submit one-period ahead forecasts of the rice. On average, the predictions of the Rational Expecta?ions Hypothesis a and forecasts tend to be both biased and inconsistent with the speci ation of the process which would generate prices in a Ra;ional Expectations Equilibrium. evertheless. an empirical analysis of agents’ brecasting dynamics indicates that learning does occur, and that agents’ forecasts evolve in a direction consistent gith a Rational Expectations Equilibrium.

In the context of efficient asset pricing theory [Fama (197U, 1976)], the so-called rational expectations hypothesis (REH) assumes that agents make efficient use of the relevant information set and that expectations are unbiased, or equivalent to actual outcomes up to a mean-zero independent random error. By now a substantial body of evidence has accumulated against this hypothesis.’ it indicates that forecasting biases occur quite generally regardless of whether that bias is measured relative to a Nash (1950) or Muth (1960) notion of rationality.3 Furthermore, Williams (1987, p. 11) in a series of double auction markets could find ‘. . . no consistent Correspondence CO: S.P. Peterson, Virginia Commonwealth University, 1015 Floyd Avenue. Richmond, VA 23284, USA. “Comments b y Pravin Trivedi, Arlington Williams, ar, anonymous referee, and the editor are gratefully acknowledged. Funding for the experiments was provided through the National Science Foundatior t,o Vernon L. Smith (University of Arizona) and to Arlington W. Williams (Indiana University). 2Lovell (198 6 ) reviews survey-based studies, while Williams (198?) contains a review of the experimental work on the formation of price expectations. Keane and Runkel ( 1990) analyze the statistical problems associated with previous studies based on panel data. 31n the context of Nash rationality, bias implies that expectations are not supported by outcomes while for Muth, the same conclusion holds with the added proviso that outcomes, in turn, do not support the predictions of the underlying theory. Smith, Suchanek and Williams ( 1988) elaborate on this point.

167-268 l/93/$06.

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.V. All rights resewed

270

S.P. Peterson, Forecasting dynamics and convergence

evidence that forecasts were becoming “more rational” over time’. Despite this fact, various studies have shown that in experimental markets prices converge to competitive equilibria.4 Smith, Suchanek, and Williams (hereafter SSW) suggest that this seen=;ngly contradictory result of rational expectations equilibrium (REE) prices and adaptive expectations may be part of a transient (learning) process that culminates in a rational expectations equilibrium (p. 1144). Though they do not pursue that topic, it does suggest that the adaptive process of price expectation formation which appears to govern actual tehavior can be thought of as a learning mechanism which EH. This paper presents an may approacn behavior consistent with experimental investigation of this possibility. The premise here is that adaptive expectations can be modelled as part of a learning process and that ;L c~w:-,i .g+ loward rational expectations 2s the market 2. : ;\d CT\b7t.2i&ions Converges tow2i :.brational expectations equilibrium. This finding woufd help to reso”r+ethe’aforementioned cnntradiction of REE prices and biased expectations. The experimc?tai design used for this analysis is the asset-double auction market described in SSW and was chosen because of the institutional similarities it shares with naturally occurring asset markets as well as the fact that agents whose forecasting behavior we analyze also do the trading. However, while SSW show that price bubbles e:dc:itually burst they could not present any evidence supporting sustained equilibrium prices. This was partly because their markets consisted of relatively short trading horizons (single fifteen period experiments). In order to enhance the possibility that price expectations would converge to equilibrium values and that sufficiently long time-series on prices and forecasts would be available, the experiments conducted in this paper recruited subjects to participate in sequences of three consecutive experiments. To maintain the control variables, each experiment had identical parameters. In particular, the dividend probability distribution remained the same (altough the actual draws changed). The investigation itself focuses on whether the structural relationship between agents’ forecasts and information sets (to be described below) converges to the specification consistent with REM. This is achieved by fitting the dynamics in agents’ forecasts to an error-learning mechanism which nests the REE specification, and wh ._.ere the information contained in the regressors are generated via competitive market trading. The coefficients are estimated recursively and arc permitted to vary over time to accommo4F~r example, -!e Miller. Plott and Smith (1977) Williams (1979), Plott and Agha (1982), and Williams and SmLl (1984) for double auction experiments characterized by some form of asset trading and Forsythe, Palfry and Plott d1982), Plott and Sunder (1982), Friedman, Harrison and Salmon (1984), and Smith, Suchanek and Williams (1988) for designs which involved trading in an asset proper (i.e., the market pays per unit dividends to shareholders at the close of each trading period).

S.P. Peterson, Forecasting dynamics and convergence

271

date the feedback from expectations to outcomes that would presumably occur under incomplete learning, e.g., see Pesaran (1987, p. 255). The recursive coefficients are then plotted and compared to the values &rich would be consistent with the REE specification of the model. Tests of whether changes in the estimated coefficients are significant, indicating that agents are changing the structural relationship between their forecasts and information sets (learning), are conducted using the Brown, Durbin, and Evans (1975) CUSUM and CUSUM-of-squares statistics. Section three develops these tests in more detail. The empirical results and some concluding remarks are presented in sections four and five respectively. Section two contains a brief description of the experiments.

The experimental design is identical to the PLATO computerized doubleaction procedure described in SSW. -What foiiowshere is a brief des&ptirsn of the parameters of the experiments conducted for this paper. In the instruction module preceeding the start of each experiment, each trader5 was informed that the dividend revealed at the end of each period is chaser; at random from a discrete uniform distribution over ($0.00, $O.M, $0.28, $0.60). Each dividend has a probability of I j4, the expected dividend is $0.24, and a share held for the duration of the experiment (15, 14, 13, . . . periods) would on average, earn an amount equal to the accumulated expected dividend stream. This is merely the product of the expected dividend and the number of periods the share is held, e.g., ($3.60, $3.36, $3.12 . . .}_ Traders were also told that this was a common dividend paid on all shares regardless of the holder. At the close of each period each trader’s dividend earnings were recorded automatically on his or her screen display! Each experiment consisted of a sequence of fifteen trading periods with each trading period lasting four minutes. Traders could end the period by unanimous vote. Consequently, traders were not pressured to accept illadvised contracts nor were they hindered from learning. At the end of each period, each trader was asked to submit a one periodahead forecast of the mean contract price. As a forecasting aid each trader could access a table displaying the maximum, minimum, and mean contract price up to that time, the dividend history, and the history of their private forecasts and forecast errors. Each was also informed in the instructional ‘Subjects were recruited from principles and intermediate-level economics courses at Indiana University. 6The screen display slso automatically adjusts a subject’s ‘working capital’, which differs from his or her initial endowment, by the amount of capital gains (losses) accumulated as a result of market trading.

S.P. Peterson, Forecasting dynaml’cs and convergence

272

module that a $1 forecasting reward would be awarded to the trader with the lowest accumulated absolute forecaSt error.’ To examine possible differences in learning and price dynamics which could occur due to varying experience levels, three independent groups of subjects were recruited for a total of nine experiments (three sequential experiments per group). Each experiment could accommodate up to nine subjects. Group I had the greatest level of trading experience. Each member, with two exceptions, had participated in at least one previous double auction experiment (some as many as five). Of these nine, seven participated in all three experiments and the remaining two slots were filled by different subjects.* Group II also consisted of nine subjects, all of whom participated in each experiment in that sequence. -Although each subject had part:-ipated in one previous experiment, none had any double auction experie-c< * Ience, these subjb -.-“5 were clearly less experienced than the subjects ‘.? :oup I. I ‘B.lally, Group III consisted of eight subjcs;;; ;one dropped out after the -?CXO~ -i experiment), none of whom had any pre:vious trading experience wh 19oever. The c- ->%LaCzn is that learning and agents’ forecasting behavior -x11 b more errs*&.r;with less experience and that this will manifest itself in “: SC ‘;-. i’._ds in market prices. 3. Zearning and snuctura~ stability In standard efficient asset pricing theory expected share value would be &i;ual to the expected value of the finite discounted dividend stream. Since the expected value of this stream diminishes by $0.24 in each succeeding trading period, then in equilibrium, the movement of the mean price level pt could be represented by the following reduced-form specification: Pt =

-0.24+p,-

1

(1)

which is the risk-neutral Muthian-rational (REM) spedp~?~~;.‘;qn given by SSW [eq. (5.1)]. Hence, forecasts based upon this spt Ps .+;: )n are also consistent with the predictions of the relevant theory. A. J. agents share complete information then according to REH, each (risk-neutral) agent’s forecasts would be: F, = -0.24+p,-

1 +pt

(2)

where pt is assumed -io be a normally distributed, mean zero, finite variance random disturbance which captures unobservable individual effects. ‘Williams (1987) argues that tk award appears to be sufficiently high to induce agents to submit their best forecast without motivating them to attempt to manipulate the mean price in order to win the forecasting award. ‘The analysis of the forecasting dynamics is for the seven common subjects.

S.P. Peterson, Forecasting dynamics and convergence

273

Notice that forecasts which are consistent with this specification would be Nash-rational (REN) as well as Muthian-rational. The reason is that forecasts would be consistent with the predictions of the theory (REM) while prices and forecasts are also mutually supportive.g The distinction is made because a test of rationality based upon (2) implies that expectations and -- _ outcomes are not only mutually supportive in the REN sense, but satisfy a REE. It is the aspect of equilibrium convergence that the empirical works presented below focuses on. Subtracting the lagged forecast from both sides of eq. (2) yields a dynamic specification of forecasts equivalent to adaptive expectations:

In general, innovations in forecasts are allowed to adapt to previous forecast errors (to accommodate 1earnr:ng).Risk-neutral, REM rational be5aviot can *be tested via the imcjosition of the set of restrictions (OC, 8) =( -0 Xl). Failure to reject these restrictions lmrply that (3) is equivalent to (2) The analysis which follows focuses on the evolution of the values of the coefficients relative to their rationally restricted values. The information set necessary for (1) to hold is public, but each agent also possesses private information. This includes individual forecasts, forecast errors, and unobservable prior beliefs. Hence, the specification given in (1) does not describe the time path of prices out of equilibrium and Iikewise, the restricted form of eq. (3) does not describe the forecasting dynamics out of ~:ruilibrium. However, if forecasts converge to ratiorral expectations, the /? e:iust converge to unity with differences in the value of the intercept from $0.24 reflecting deviations from risk- i,eutrality. If agents learn to value the asset intrinsically, then the estimated value of /I would approach unity arid outcomes would in turn support REE prices. With this in mind, the recursive least squares (RLS) estimate of B was obtained for each trader adding observations as they became available with the passage of each trading period. The final RLS estimates are equivalent to OLS applied to the entire sample (for each trader) and are presented in tables 2-4. Time plots of the individual coefllcients of adaptation fl along with the CUSUM and CUSUM-of-squares plots for selected traders in each group are presented in figs. 2-d . lo Plots for the remaining traders are included in an appendix which is available from the author upon request. The evolution of /3 alone, does not provide a test of whether the changes 9A test of REN is provided by way of a regression of prices on forecasts and an intercept but imposes none of the REE restrictions on the process which is actually generating prices. Hence, tests of REN are weaker in the sense that prices and forecasts are not required to be in a REE. See eq. (4) for a test of REN. “These plots do not include the RLS estimates of the intercept due to difficulty in interpreting changes in this value over time.

274

S.P. Peterson, Forecasting dynamics and convergence

agents are actually changing the structure! relationship giv, I rat) GIS forecasting dynamics depicted in (3). A formal test of the structural statl.ifg, of the learning model can be constr dized rti::ursi~:eprediction er upon the Stan (3). The details of these te estimation of squares) are now standard textbook fare, e. be taken up here.’ ’ The intuition, however, is straig either do not learn on the basis of previous forecasting already completed the learning process then the regression relationship would disturbances would ha This in t “V r=mncrPAnctic. ~*V~~~“““VUIO” recursive prediction errors are zero. If, on the dynamic specification given by eq. (3) then we will tend to over or ict the change in the forecast de or decreased. Thus, if an a information contained in his lagged forecasting error in u period forecasts, fl has fallen and we will begin his forecasting dynamics. This will manifest itself in a positive Similarly, if an agent makes a sudden and large change in the structural relationship between lagged forecasting errors and the current forecast change, then this will impact largely on the squared recursive pre errors and therefore, the CUSU -of-squares plot. The idea is t bts along with the plots of the adaptive coeficients as indicator learning is actually occurring. in /I are significant, 8.c.

that

egins with a few basic observations concerning the overall havior of these markets before tying together more specific results. In general, we start by highlighting the dynamics and then procee sectional comparisons.

ewpcted to persist. the convergence of the in

ecture 12 supporte

S.P. Peterson,

Group

Forecasting

dynamics and convergewe

Devlatlons

I

from

275

REE pwes

6-

O-

a2-

O-

-2 5

15

25

trms

35

d2o-

trme

Fig. 1

Table 1

Disequilibrium prices.

____Group

p

I II III

Experiment I 0.865 1.805 0.888 2.243 0.768 2.311

Experiment Z - 0.043 0.834 0.863 0.6 13 0.967 0.866

Experiment 3 -0.063 0.314 0.616 0.658 0.2 17 0.359

1[

Experiment J-3 0.253 1.218 0.789 1.368

Experiment I-Z 0.411 1.247 0.876 1.615

Eweriment 2-3 -0.053 0.619 0.739 0.637

III

0.65 1

I

d

1.443

P -

0.868

d

1.717

P

0.592

0

0.755

Measured as the difference, each trading period, of the mean price level from the value of the expected dividend stream. Statistics reported below are the mean and standard deviation of the price bubble in each experiment, arranged across experiments.

276

S.P. Peterson, Forecasting dynamics and convergence

Table 2 Adaptive coefficients-Group

I

F,-F,_,=a+B(p,_,-F,_,)+&. Trader 1 2 3 4 i 6 7

B

R2

DW

-0.116 (0.059) -0.132 (0.055) -0.041 (0.049) -0.108 (O.tul) 0.026

1.016 (0.104) 0.979 (0.106) 1.157 (0.087) i.130 (0.078) 0.999

0.718

1.32”

0.694

1.21”

0.826

1.67

0.849

1.75

0.766

1.28”

0.146 (0.064) -0.144 (0.03cr)

0.713 (0.079) G.955 (0.058)

0.73 1

1.83

0.876

2.02

z

--“Rejection of no positive serial correlation at the SF/, level of significance. Standard errors in parentheses.

gence of either prices or forecasts to REE values cannot occur independently of the other. Thus, given the existence of the price bubbles, one would not expect the RE restricions on (3) to hold on average. The OLS estimates of eq. (3) which are presented in tables 2-4 appear to confirm this; O!does not even have the correct sign for most agents. Furthermore, incidences of serial correlation in the residuals which are common for inexperienced traders, would bias inferences. ence, if learning dominates the sample then OLS will tend to reject individu ith regard io cross-sectional comparisons, the sk of the bubble is expected to be inversely related to the level of aggregate group trading experience. The evidence shows that prices in markets involving more experienced traders (Group I) are less erratic and closer to REE prices. ar to converge to a REE relatively faster for Group EE price by the close of the second experiment) whereas the existence of a pric elatively more pronounced for the less experienced gro t. The experience factor can be extended as rences in the a tive coefficient Pooling /I across time ano individuals and measuring the within-group mean difference from unity and standard deviation yields: mean Group I Group II

- 0.0233 0.0706 0.1258

standard deviation 0.3530 0.9652 e.0230

S.P. Peterson,

I orecasting

dynamics and convergence

Table 3 Adaptive coefkients-Group Trader

II R2

DW

0.729

1.69

0.235 (0.M)

0.?28

I.

0.311

.776

1.52

1

1.63

r 0.036 40.079)

0.780 40.078)

0.896

0.170 40.073) 0.156

0.729

1.61”

0.862

1.47

0.

aala

0.841

0-w

0.82 E

1.18”

(0.078)

--

1.1 (0.084)

-Rejection at &he 5y0 level of si errors in parentheses.

caxe.

Standard

Table 4 Adaptive coefIicients-Group Trader

r 0.265 (0.150) 0.066 (0.021 j 0.151 (0.050) 0.022 (0.029) -0.045 (0.034) - 0.053 (0.068 j -0.164 (0.054) -0.048 (0.065)

B 1.098

(0.159 0.999 (0.038) 1.033 (0.099) 1.025 (0.059) 1.096 (0.078) 0.883 (0.139) 1.397 (0.105) 1.609 (0.166)

III

R2

DW

0.563

2.07

0.949

1.06”

0.744

0.87”

0.888

1.58

0.839

1.60

0.519

1.63

0.825

1.24b

0.715

0.41”

‘Rejection at the 5”/(, level of significance. Standard errors in parentheses.

277

218

S.P. *%terson, Forecasting dynamics and convergence

Obviously, mean differences from the rationally restricted value of the adaptive coefficient increase with inexperience. In general, these observations would appear to suggest that the learning process for Group I converges relatively earlier and that overall, prices tend to deviate less from equilibrium. Nevertheless, pooled OLS estimation (by group) of the regression

shops that expectations are biased for all groups despite the convergence of this result extends to individual tests _ price toward REE values. l3 In general, o? forecast unbiasedness (see table 5) although less so for the relatively more experienced traders in Group I. Nevertheless, positive serial correlation is a W-;WS pro’Ultiri regarding these results and would not only tend to confuse ;k ir&erentiat legitimacy of the F-tests presented in column two of table 5 bt!t would also indicate that forecasts lack the efficiency required of REH. Once again, conventional tests of rationality will have a tendency to be confounded by the learning process. The interesting question then is if forecasts are biased, what are the underlying forecasting dynamics and do forecasts tend to converge to predictions consistent with the REH? The time plots for the CUSUM and CUSUM-of-squares statistics regarding the coefficient of adaptation /? are uti to address this question. Recall that the coefficient of adaptation fl is estimated recursively from a RLS regression of (3) and that the CUSUM and CUSUM-of-squares statistics are based upon the recursive residuals from those regressions. Positive (negative) values for CUSUM imply that agents are increasing (decreasing) the weight they place on previous forecasting errors in updating their forecasts. On the other hand, relatively large changes in the CUSUkf-of-squares plots indicate sudden, or erratic, movements in /I. Violations of the CUSUM (and CUSUM-of-squares) plot accross their respective bound;aries &dicate that these changes in /3 are significant at the 57; level. Nevertheless, the informational content of these plots in not entirely diminished in the absence of such violations. A diszucqion of all of these plots for each of the twenty-four traders would be impractical due to space limitations. Howevep, several representative examples regarding each group of traders are presented in figs. 2-4 for discussion. The remaining plots are generalizations of these and will be summz+zed verbally. 13This is a test of REN. The F statistics for the joint test that Ho: (Q,/?) =(O, 1) are: Group I = 9.00, Group II = 47.04, Group III = 32.23 with sample sizes Ib’= 273, 351, 312 respectively. These statistics are suficientiy high to reject unbiasedness at any conventional level of significance. Furthermore,poolingthe data for eachgroupby experimentand retestingHo does not indicate that forecasts are becoming any less biased over time.

279 Table 5 Forecast bias. Joint test of the restrictions Ho: r=O, /I= 1 for the REN specification giveu by pI=a+#W,+~-

Group I

Trader 1 2 3 4 5 6 7

m 37) 0.359 1.539 1.634 0.652 4.mb 7.538b C.09S

DW 1.31’ 1.77 1.41” 1.48 1.14” 0.61” l.lP

Group II

1 2 3 4 5 6 7 8 9

0993 1.584 7.669” 3.662b 12.869” 7.703b 4.497b 4.362b 11.533”

219 1.61 1.72 1.60 1.76 1.97 207 1.99 1.49

Group III

1 2 3 4 5 6 7 8

2.339 9.959b 3.466b 8.267b 2.029 1.671 3.236b 2-635

1.53 1.66 1.79 1.52 1.67 1.80 1.55 1.87

‘Rejection of no positive serial correlation at the 5y0 level of significance. bRejection of Ho at the 5% level of significance.

Each figure con+sts oi ivlro VC~ Lical arrays representing two typical traders. The top, middle, lnd bottom graphs show the evolutions of the coefficient of adaptation, CUWM, and CUSUM-of-squares statistics respectively. As in fig. 1, ths two vertical lines separate the three experiments. Hence, in fig. 2 for example, t’.re coefficient of adaptation for Trader 3 - Group I peaks in value at abcut 1.25 at the beginning of the second experiment and subsequently converges very slowly toward unity. The same is true for Trader 4 - Group I. In both cases, the CUSUM statistic displays no protracted negative or positive runs indicating that t e evolution of B is statistically insignificant. Finally, the CUSUM-of-squares plots do not appear to support any sudden or erratic shift in this parameter value. This behavior is typical of the experienced traders who comprise Group I. Changes in the

S. P. PQterson,

Group

Fwecasting dynamics and convergence

6roup I-rratserI

i-rraacr 3 20 10 0 -10 -20

GPOUQ 1

-Traoer 3

srOUQ 1-1rrJ*r . 15

15 I

*

5

5

0

0

-5

-5 0

10

20 time

30

40

0

10

20

30

40

Fig. 2

coeffkient of adaptation occur early and are rather small and statistically wever, the opposite behavior is typical of inexperienced traders. In fig. 3, the coefficient of adaptation has a much higher variance, usually peaking in value very igh relative to Group I. CUSU tends to display runs diction of B indicating indicating prot cted periods of over or under changes in the coefficient of adaptation. Furthermore, this evolution is in the direction Df the

S.P. Peterson, Forecasting dykaamirsand convergence

I

25 f 75

30

t0

10

2

0

a

t

-to

-PO

-20

?5

P

5

0

-5

0

.Q

20 ame

Fig. 3

12s 8 75

20

10

0

-10

-20

Fig. 4

282

S. P. Petmon,

Forecasting dynamics and convergence

periods of over and underprediction of changes in the a&$ive coeficicnt provide additional evidence, though less obvious. of learning. Since fl usually peaks in value well above unity, interest might center on when the value of /3 begins to fall and henceforth converge towarc unity. That is, when CUSUM become negative? For instance, during the first experi CUSUM is positive and rising for six of seven agents in Group I. The val of /3 is generally r&n,e as well. By the second experiment however, CUSU is negative or falling and the coefficient of adaptation has either stabilized or is converging towards unity (with the exception of subject six who incidently, also had the highest mean forecasting e r). For the ine ienced subjects ly to revert to in Groups II and III, values of CUSU are much less their mean zero values. In fact CUSUM has a endency to display s positive (and negative) runs in these zases. T s would imply that le for inexperienced traders and also has a wider range of values (supporting the pooled results for fl ‘ven earlier in this section). Given the relative magnitudes and the variabi y of price bubbles across experience levels, it appears that agents’ expectations of price bu or correlated with, rising values of both the CUS coefficients of adaptation. Nevertheless, learning does occur as evidenced by both the convergence and stability of b toward unity and the convergence of prices to intrinsic value. It is also clear that there are often sudden shifts in the value of fl inexperienced traders as evidenced by the rather large changes in CUSU of-squares for Groups II and III. The implication here is that behavior in markets comprised of inexperienced traders may display relatively ‘unstable’ episodes regarding convergence to equilibrium values. Nevertheless, though the forecasting dynamics are sometimes erratic for inexperienced traders, structural changes are relatively more significant and frequent. And though inexperienced agents’ expectations tend to display more bias, their forecasing dynamics appear to change quite suddenly (rather than monotonically) at some point toward values consistent with the REH. Upon reflection, this observation may t be so surprising. Among inexperienced traders, :here may be less in ay of certain shared fundamentals concerning the operation of the market. Learning in these cases may be more of a catharsis than deliberate and syste ot assumed here that experienced traders rs regarding the rket, they do have in common some fundamental notions t operates and thus, for these individuals, changing may tend to be relatively more systematic rather than xistence of relatively lar ay be closely linked to 1s p&t. of the plots and not tested for in

d the sudden subsequent zrashing of these Nevertheless, the point is ased on visual inspection

S.P. Peirr~rc,

I’~~ecasttng dynamics unJ convergence

283

This study investigated the possibility that agents learn to form rational expectations by analyzing the structural stability of an adaptive expectations learning model. The evidence indicated that, in general, the cc&icient of sdagtation in the learning model evolves in a direction consistent with the REH as prices converge to ard REE values. Less ienced traders tend to submit forecasts which a the learning model for less tent with the strict R structural changes in cant, indicating that the model more o consistent with the ese rest&s suggest that markets dominatei by new an to display protracted e size of the bubbl learn intrinsic vak.

Blume, L.E.. Bray and D. Easley. 1982. Introduction to the stabuity of rational expectations equilibrium,‘Journal of Economic Theory 26. 3 13-3 f 7. Blume, L.E. and D. Easley, 1982, Learnin, %tto be rational, Journal of Economic Theory 26, no. 2, 340-351. NJ., 1983. Convergence to rational expectations equilibrium, in: Roman Frydman and E. elps, eds., Individual forecasting and aggregate outcomes (Cambridge University Press) 123-133. Bray, M. and D. Kreps, 1986, Rational learning and rational expectations, in. W. Heller, R. Starr and D. Starrett, eds., Essays in honour of K.J. Arrow (Cambridge University Pre$. Bray, M. and NE. Savin, 1986, Rational expectations equilibria, learning and model specification, Econometrica 54, 1129-l 160. Brown, R.L., J. Durbin and J.M. Evans, 1975, Technique for testing ;Se eo&ancy of regression relationships over time, Journal of the Royal Statistical Association B37, 149-192. Cyert, R.M. and M.H. De Groot, 1974, IUionai expectations and bayesian analysis, Journal of Political Economy 82, 521-536. DeCanio, Stephen J., 1979, Rational expectations and learning from experience. Quarterly Journal GCEconomics. Feb., 47-57. DuFour, Jean-Marie, 1982, Recursive stability analysis of linear regression relationshrps. Joumai of Econometrics 19, 3 l-76. Forsythe, Robert, Thomas Palfrey and Charies Plott, I!%& Ass~;. valuation in an experimental market, Econometrica 50, 537-567. Fourgeaud, C., C. Gourieroux and J. Pradel, 1986, Learning procedures and convergence to rationaiity, Econometrica 54, 845-868. Friedman, Daniel, Glenn Harrison and Jon Salmon, 1984, The informational efficiency of experimental asset markets, Journal of Political Economy 92, 349-348. Frydman, Roman, 1982, Towards an understanding of market processes: Individual expectations, learning and convergence to rational expectations equilib;-ium, American Economic Review 72,652-658.

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Forecasting

dynan;ics cand convergence

Frydman, Roman, 1983, Individual rationality, decentralization and the rational expectation; _hypothesis, in: Frydman and Phelps, eds., Individual forecasting and aggregate outcomes (Cambridge University Press). Harvey, A.C., 1981, The econometric analysis of time series (Philip Allan, London;. Keane, Michael P. and David E. Runkle, 1990, Testing the rationality of price forecasts: New evidence from panel data, American Economic Review 80,714-735. Lovell, Michael C., 1986, Tests of the rational expectations hypothesis, American Economic Review 76, 110-124. Miller, Ross M., Charles R. Plott and Vernon L. Smith, 1977, I~terte equilibrium: An empirical study of speculation, Quarterly Journal of Economics 91, 599-624. Muth, John F., 19.50, Optimal properties of exponentially weighted forecasts. Journal of the Amerian Statistical Association 55, 299-306. Muth. John F., 1961, Rational expectations and the theory of price movements, Econometrica 29,3 1S-335. Pesaran, M.H., 1987, The limits to rational expectations (3asiI Blackwell, Oxford). Plott, Charles R. and Gul &$I~ X%’w “intertemporal speculation with a random demand in an experimental market, in: R Tietz, ed., Aspiration levels in bargaining and economic decision ger-Verlag, Berlin). security markets wit and Shyam Sunder, 1982, Efficiency of ex~~rne~ta insider trading, Journal of Political Economy 90. 663-698. Smith, V.L., G.L. Suchanek and A.W. Williams, 1988, I3ubbles, crashes, and endogenous expectations in experimental spot assets markets, Econometrica 56, I1 ‘N-1 151. Taylor, John B., 1975, Monetary policy during a transition to rational expectations, Journal of Political Economy 83, IOO9-1021. Williams, Arlington W., I987, The formation of price forecasts in experhental markets, Journal of Money. Credit and Banking 19, 1-18. Williams, Arlington W., 1979, Intertemporal competitive equilibrium: On further experimental results, in: V.L. Smith, ed., I&search in experimental economics, 1 (JAI Press, Greenwich Conn.). William:% Arlington W. and Vernon L. Smith, 1984, Cyclical double-auction markets w?h and without speculators, Journal of Business 57, l-33.