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Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

(Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Mikhail Anufrieva Cars Hommesb Valentyn Panchenkoc

aUniversity of Technology, Sydney bUniversity of Amsterdam cUniversity of New South Wales

Sant’Anna School of Advanced Studies, Pisa 25 May, 2015

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Outline

Motivation: Heterogeneous Agent Models Learning to Forecast Experiments Heuristic Switching Model Heuristic Switching Hidden Markov Model Conclusion

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Expectations in Economic Theory

◮ economy is an expectation feedback system

◮ expectations affect our decisions and realizations ◮ expectations may be affected by past experience

◮ expectations play the key role in most economic models

30s-60s naive and adaptive expectations 70s-90s rational expectations 90s- models of learning and bounded rationality

◮ adaptive learning (OLS-learning) ◮ belief-based learning ◮ reinforcement learning ◮ heterogeneous expectations and switching

(Heterogeneous Agent Models)

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Example: Model of Financial Market

◮ demand for the risky asset

Dh(pt) = Eh,t[pt+1 + yt+1] − (1 + r)pt a Vh,t[pt+1 + yt+1]

◮ solving market clearing eq. at time t find the equilibrium price

• h Dh(pt) = 0
• pt =

1 1 + r

• h Eh,t[pt+1 + yt+1]

◮ rational expectations

pt = 1 1 + r Et[pt+1+yt+1] (for i.i.d. dividends) pt = ¯ y r

◮ heterogeneous expectations

pt = 1 1 + r

• h

Eh,t

• 1

1 + r

• h′

Eh′,t+1[pt+2 + yt+2] + yt+1

• Anufriev, Hommes, Panchenko

UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Example (ctd): Heterogeneous Agent Model

◮ there are two types of investors

◮ fundamentalists, Ef,t[pt+1] = pf + v(pf − pt−1) with 0 ≤ v < 1 ◮ chartists, Ec,t[pt+1] = pt−1 + g(pt−1 − pt−2) with g > 0

◮ evolution of price

pt = 1 1 + r

• nf,t Ef,t[pt+1] + nc,t Ec,t[pt+1]
• +

¯ y 1 + r

◮ evolution of fractions

nf,t+1 = exp [βπf,t] exp [βπf,t] + exp [βπc,t]

◮ profits πf,t and πc,t are computed as their holdings times return

pt + yt − (1 + r)pt−1 and known to everybody

◮ fundamentalists pay fixed cost C > 0

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Example (ctd): Simulation

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Empirical Validation of HAMs

◮ on financial data: Boswijk, Hommes and Manzan (2007),

Goldbaum and Mizrach (2008), De Jong, Verschoor, and Zwinkels (2009), Kouwenberg and Zwinkels (2010), Franke and Westerhoff (2011), Chiarella, He and Zwinkels (2014);

◮ on survey data: Branch (2004);

Experiments with paid human subjects allow to investigate heuristics and switching in a controlled environment, estimate parameters, test hypotheses.

◮ Learning to Forecast Experiments ◮ Switching Experiments

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Experiments about expectations

Learning-to-forecast experiments: Hommes et al (2005, RFS; 2008, JEBO), Adam (2009, EJ), Heemeijer, Hommes, Sonnemans and Tuinstra (2009, JEDC) Experiment with few (six) students who know only qualitative

• features. Story:

◮ each of you is a forecaster of price working for a financial firm ◮ several firms are in the market ◮ price is affected by demand and supply ◮ higher forecasts lead to a higher demand [or to a higher supply] ◮ your payoff (salary) depends on the forecast precision

Information:

◮ observe past prices, own forecasts and payoffs

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Computer Screen

earnings per period: et,h = max

• 1 − 1

49(pt − pe t,h)2, 0

• × 1

2 euro

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Learning to Forecast Experiments

Two treatments Negative feedback Positive feedback

20 40 60 80 100 120Prediction 20 40 60 80 100 120 Price 20 40 60 80 100 120Prediction 20 40 60 80 100 120 Price

pt = 60 − 20

21

• pe

t − 60

• + εt

pt = 60 + 20

21

• pe

t − 60

• + εt

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Rational Benchmark

If everybody predicts fundamental price pf = 60, then pt = pf + εt

40 45 50 55 60 65 70 10 20 30 40 50 Price Time fundamental price price under rational expectations

• 1
• 0.5

0.5 1 10 20 30 40 50

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Negative Feedback Experiment

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Positive Feedback Experiment

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Negative Feedback Experiment: all sessions

20 40 60 80 10 20 30 40 50 Price Time

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Positive Feedback Experiment: all sessions

20 40 60 80 10 20 30 40 50 Price Time

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Estimation of Individual Predictions

OLS regressions of individual predictions on the lagged variables Identified intuitive behavioural rules: pe

i,t = α1pt−1 + α2pe i,t−1 + (1 − α1 − α2)60 + γ(pt−1 − pt−2) ◮ adaptive heuristic

pe

t+1 = wpt + (1 − w)pe t ◮ trend heuristic

pe

t+1 = pt + γ(pt − pt−1)

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Background

Heuristic Switching Model with heterogeneous expectations

◮ introduced in Anufriev and Hommes (2012, AEJ-Micro) and

Anufriev, Hommes and Philipse (2013, JEE)

◮ applied to several “learning to forecast experiments” ◮ Hommes et al (2005, RFS; 2008, JEBO), Heemeijer et al (2009,

JEDC)

◮ applied in experimental papers Bao et al (2011, JEDC), Assenza

et al (2009, CeNDEF WP) Key features in forecasting agents use simple rules of thumb, heuristics

(Tversky and Kahneman, 1974)

in learning agents switch between different forecasting rules on the basis of their performances

(Brock and Hommes, 1997)

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Evolution of Individual Predictions

40 50 60 70 80 10 20 30 40 50 Price, Predictions Predictions in Negative Feedback Experiment prediction price

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Evolution of Individual Predictions

40 50 60 70 80 10 20 30 40 50 Price, Predictions Predictions in Positive Feedback Experiment prediction price

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Further evidence for switching I

52 54 56 58 60 62 64 66 10 20 30 40 50 Group 6, participant 1 prediction price

from Anufriev and Hommes (2012)

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Further evidence for switching II

48 50 52 54 56 58 60 62 64 66 10 20 30 40 50 Time Group 1, participant 3 prediction price

from Anufriev and Hommes (2012)

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Heuristic Switching Model

Two heuristics:

◮ adaptive rule

ADA pe

1,t = 0.75 pt−1 + 0.25 pe 1,t−1 ◮ trend-following rule

TRF pe

2,t = pt−1 + (pt−1 − pt−2)

price dynamics

◮ negative feedback market:

pt = 60 − 20

21

• n1,tpe

1,t + n2,tpe 2,t + 60

• + εt

◮ positive feedback market:

pt = 60 + 20

21

• n1,tpe

1,t + n2,tpe 2,t − 60

• + εt ,

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Model: time-varying impacts of heuristics

impacts of heuristics ni,t are evolving

◮ performance measure of heuristic i is

Ui,t−1 = −

• pt−1 − pe

i,t−1

2 + η Ui,t−2 parameter η ∈ [0, 1] – the strength of the agents’ memory

◮ discrete choice model with asynchronous updating

ni,t = δ ni,t−1 + (1 − δ) exp(β Ui,t−1) 2

i=1 exp(β Ui,t−1)

parameter δ ∈ [0, 1] – the inertia of the traders parameter β ≥ 0 – the intensity of choice

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Two types of simulations

Learning parameters are fixed: β = 1.5, η = 0.1, δ = 0.1.

◮ deterministic path

◮ initialized with ◮ initial price, p0 = 50 ◮ initial impacts of heuristics, n1,0 = n2,0 = 1 2 ◮ simulated for 49 periods with the same noise process as in the

experiment (and also without noise)

◮ one-period ahead prediction

◮ at any moment the experimental data so far are used to produce ◮ predictions of heuristics ◮ impacts of heuristics Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Deterministic path: negative feedback

Parameters: β = 1.5, η = 0.1, δ = 0.1.

40 50 60 70 80 10 20 30 40 50 Price Time 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Impacts Time adaptive trend

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Deterministic path: positive feedback

Parameters: β = 1.5, η = 0.1, δ = 0.1.

40 50 60 70 80 10 20 30 40 50 Price Time 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Impacts Time adaptive trend

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

MSE for deterministic path of different models

Model Negative Feedback Positive Feedback 12 groups fundamental 2.5712 46.8344 24.7028 adaptive 2.3001 2.9992 2.6497 trend 21.1112 0.9260 11.0186 mixed 7.9798 1.0518 4.5158 HSM 3.2967 0.9065 2.1016

MSEs of 5 different models over 47 time periods of the experiment. The results are averaged over all 6 experimental groups of the negative feedback treatment (2nd column), and over 6 out of 7 experimental groups in the positive feedback treatment (3rd column). The last column displays the joint result for 12 experimental groups.

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

One-period ahead simulation: negative feedback

Parameters: β = 1.5, η = 0.1, δ = 0.1. Session 1, group 1.

40 50 60 70 80 10 20 30 40 50 Price Time simulation experiment 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Impacts Time adaptive trend

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

One-period ahead simulation: positive feedback

Parameters: β = 1.5, η = 0.1, δ = 0.1. Session 4, group 1.

20 30 40 50 60 70 80 10 20 30 40 50 Price Time simulation experiment 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Impacts Time adaptive trend

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

One-period ahead simulation: positive feedback

Parameters: β = 1.5, η = 0.1, δ = 0.1. Session 4, group 4.

40 50 60 70 80 10 20 30 40 50 Price Time simulation experiment 0.2 0.4 0.6 0.8 1 10 20 30 40 50 Impacts Time adaptive trend

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

One period ahead predictive performance

Model Negative Feedback Positive Feedback 12 groups adaptive benchmark MSE 2.3001 2.9992 2.6497 w 0.75 0.75 0.75 AIC 2070.3458 2220.0313 2150.1399 BIC 2070.3458 2220.0313 2150.1399

• ptimized

MSE 2.3001 1.8218 2.3680 ˆ w 0.75 1.00 0.91 AIC 2072.3451 1940.8575 2088.7528 BIC 2076.6802 1945.1926 2093.0879 trend benchmark MSE 21.1112 0.9260 11.0186 γ 1.00 1.00 1.00 AIC 3320.6515 1557.2206 2953.9288 BIC 3320.6515 1557.2206 2953.9288

• ptimized

MSE 1.9237 0.7480 2.2940 ˆ γ −0.33 0.71 −0.16 AIC 1971.5531 1438.7973 2070.8502 BIC 1975.8881 1443.1323 2075.1852 mixed benchmark MSE 7.9798 1.0518 4.5158 (w, γ, n) (0.75,1.00,0.50) (0.75,1.00,0.50) (0.75,1.00,0.50) AIC 2771.9429 1629.0202 2450.8370 BIC 2771.9429 1629.0202 2450.8370

• ptimized

MSE 2.3001 0.7568 2.6276 (w, γ, ˆ n) (0.75,1.00,0.76) (0.75,1.00,0.38) (0.75,1.00,1.00) AIC 2072.3458 1445.4272 2147.4263 BIC 2076.6809 1449.7623 2151.7614 Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

One period ahead predictive performance

Model Negative Feedback Positive Feedback 12 groups HSM benchmark MSE 3.2967 0.9065 2.1016 (β, η, δ) (1.50,0.10,0.10) (1.50,0.10,0.10) (1.50,0.10,0.10) (w, γ) (0.75,1.00) (0.75,1.00) (0.75,1.00) AIC 2273.3741 1545.1924 2019.4464 BIC 2273.3741 1545.1924 2019.4464

• ptimized 1

MSE 2.3001 0.7329 1.5402 (ˆ β, ˆ η, ˆ δ) (1.44,1.00,0.00) (0.01,0.96,0.34) (0.02,0.97,0.00) (w, γ) (0.75,1.00) (0.75,1.00) (0.75,1.00) AIC 2076.3458 1431.2754 1850.1692 BIC 2089.3510 1444.2806 1863.1743

• ptimized 2

MSE 1.9225 0.7088 1.5051 (ˆ β, ˆ η, ˆ δ) (1.46,1.00,0.15) (0.48,0.85,0.00) (0.37,0.87,0.00) (ˆ w, ˆ γ) (0.98,−2.00) (0.83,0.75) (0.75,0.75) AIC 1979.2104 1416.4752 1841.1650 BIC 2000.8857 1438.1504 1862.8403 Fit of 9 different models over 47 time periods of the experiment. The entries show the values of three fitness criteria MSE, AIC and BIC, and the values of corresponding parameters.

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Robustness test of the HSM

0% +10% −10% +20% −20% +30% −30% +40% −40% β 1.5051 1.5052 1.5052 1.5053 1.5055 1.5054 1.5061 1.5056 1.5072 η 1.5051 1.5109 1.5158 1.5156 1.5654 1.5163 1.6037 1.5166 1.6350 δ 1.5051 1.5051 1.5051 1.5051 1.5051 1.5051 1.5051 1.5051 1.5051 β, η 1.5051 1.5114 1.5188 1.5160 1.5745 1.5168 1.6190 1.5171 1.6585 β, δ 1.5051 1.5052 1.5052 1.5053 1.5055 1.5054 1.5061 1.5056 1.5072 η, δ 1.5051 1.5109 1.5158 1.5156 1.5654 1.5163 1.6037 1.5166 1.6350 β, η, δ 1.5051 1.5114 1.5188 1.5160 1.5745 1.5168 1.6190 1.5171 1.6585 MSE 1.5051 1.6556 1.8061 1.9566 2.1072 Robustness test of the heuristics switching model. For one, two or three parameters we report the value of MSE (over 47 time periods in 12 experimental groups) of the model resulting in a given percentage change of every of these parameters. The last row shows the corresponding percentage increase of the benchmark value of MSE, 1.5051.

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Overview and Further Steps

HSM in Anufriev and Hommes (2012) and Anufriev et al. (2013)

◮ good overall fit of the aggregate data (price) ◮ separate estimation/calibration of the individual heuristics and of

the learning parameters

◮ learning parameters are estimated on the aggregate data (price)

What is now?

◮ application to other experiments (Bao et al. 2012, Assenza et

• al. 2015)

◮ model with GA learning fitted to individual data ◮ experiments on switching

This project: Unified structural approach for the estimation of the Heuristic Switching Model using Bayesian Hidden Markov Model on the data from learning-to-forecast experiments

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Hidden Markov Model

◮ system is modeled as a Markov Process whose states are latent

◮ states, x: latent heuristics used by the agents ◮ observables y: submitted forecasts ◮ state transition: switching between heuristics

◮ use observables to estimate parameters of the model

◮ Bayesian estimation using Gibbs sampler and Markov Chain

Monte Carlo method

◮ closely related in economics: Shachat and Wei (2013, WP),

Shachat, Swarthout and Wei (2014, EMetricT)

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Heuristic Switching Hidden Markov Model

• 1. finite set of latent heuristics used by the agents (“states”),

mapping available information to the forecast

◮ heuristics are latent because they are subject to some noise

• 2. exogenous multinomial distribution for initial assignment of

experiment participants to the heuristics

• 3. first order Markov matrix of transition probabilities governing

the switching between rules

◮ probabilities depend on the past performances of strategies ◮ possibility of inertia is included ◮ possibility of random experimentation is included (“outliers”) Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Heuristic Switching Model: three forecasting heuristics - latent states for HMM

◮ naive rule,

NAI pe

1,t = pt−1 + u1,t ◮ adaptive rule, [w]

ADA pe

2,t = w pt−1 + (1 − w) pe 2,t−1 + u2,t ◮ trend-following rule, [γ]

TRF pe

3,t = pt−1 + γ(pt−1 − pt−2) + u3,t ◮ noise ui,t ∼ N(0, Vi).

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Heuristic Switching Model: individual switching behavior - state transition for HMM

heuristics are selected using time-varying probabilities Pi,t

◮ performance measure of heuristic i is

Ui,t = ui,t−1 + ξi,t , where ui,t−1 = −

• pt−1 − pe

i,t−1

2 + ηui,t−2 parameter η ∈ [0, 1] – strength of agents’ memory, ξi,t ∼ N(0, 1/ √ 2).

◮ discrete choice (probit) model with asynchronous updating

P(St = i) = It I(St−1 = i) + (1 − It) P

• β Ui,t ≥ max

k {β Uk,t}

• switching indicator: Bernoulli r.v. It ∈ {0, 1}, P(∆t = 1) = δ

parameter δ ∈ [0, 1] – the agents’ inertia in choice parameter β ≥ 0 – the intensity of choice

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Bayesian inference about the parameters

◮ Goal: posterior joint distribution given prior and observed data

(individual predictions) and marginalization P(w, γ, V2, β, η, δ, π, S1:T,1:N, I1:T,1:N|Pe

1:T,1:N, prior)

◮ w, γ, V2 - parameters of the rules, and variance of rule 2, ◮ β, η, δ - parameters of switching, ◮ π - initial states (rules) distribution, ◮ S1:T,1:N - latent states (rules), ◮ I1:T,1:N - latent indicator of switching.

◮ Posterior ∝ Likelihood × Prior

◮ direct sampling from the posterior is not feasible for most HMM Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

(Block) Gibbs sampler

Idea: P(X1, X2|Y) replace by P(Xℓ

1|Xℓ−1 2

, Y), P(Xℓ

2|Xℓ 1, Y)

• Init. Set starting values for the parameters, states and switch. indicator

Step 1 Sample states P(S1:T,1:N|w, γ, V2, β, η, δ, π, I1:T,1:N, Pe

1:T,1:N)

Step 2 Sample initial states distribution P(π|πprior, S1,1:N) Step 3 Sample switching parameters P(β, η|βprior, ηprior, S1:T,1:N, w, γ, I1:T,1:N, Pe

1:T,1:N)

Step 4 Sample parameters of the rules P(w, γ, V2|wprior, γprior, Vprior

2

, S1:T,1:N, Pe

1:T,1:N)

Step 5 Sample indicators of switching P(I1:T,1:N|δ, S1:T,1:N, Pe

1:T,1:N)

Step 6 Sample probability of switching P(δ|δprior, I1:T,1:N)

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Step 1: Sample states - multi-move sampler

(Carter and Kohn, 1994)

◮ Prediction given the past observations (forward)

P(St|Pe

1:t−1) = 3

• i=1

P(St|St−1)P(St−1 = i|Pe

1:t−1) ◮ Filtering given the current observation (forward)

P(St|Pe

1:t) ∝ P(Pe t |St)P(St|Pe 1:t−1) ◮ Sampling (backward) starting from P(ST|Pe 1:T)

P(S1:T|Pe

1:T) = P(ST|Pe 1:T) T−1

• t=1

P(St|Pe

1:t, St+1)

P(St|Pe

1:t, St+1 = ℓ) ∝ P(St+1 = ℓ|St, Pe 1:t)P(St|Pe 1:t)

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Step 3: Sample switching parameters

Bayesian Probit Gibbs sampling (Albert and Chib, 1993)

◮ Identification conditional on It = 0, i.e., switching is possible ◮ Example - switching (transition) to state 1:

P(St = 1|Pe

1:t−1) = Φ(U∗ 12,t > 0, U∗ 13,t > 0),

where U∗

12,t = βℓ−1(U1,t − U2,t), U∗ 13,t = βℓ−1(U1,t − U3,t) and

Ui,t = −

• pt−1 − pe

i,t−1

2 + ξi,t, ξi,t ∼ N(0, 1/ √ 2).

◮ Conditional on state St inverse operation: sample U∗ from

truncated Normal satisfying the appropriate restrictions

◮ Use linear regression to sample from βℓ, :

U∗

12,t = β

• −
• pt−1 − pe

1,t−1

2 +

• pt−1 − pe

2,t−1

2 +εt, εt ∼ N(0, 1)

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Posterior parameters

δ β w γ V2 mean 0.828 0.204 0.693 0.403 55

• std. dev.

0.066 0.052 0.030 0.003 6

50 100 150 200 0.15 0.20 0.25 0.30 iterations beta 50 100 150 200 0.5 0.6 0.7 0.8 0.9 iterations delta

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Predictions and Rule classification: Negative feedback

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Evolution of Fractions: Negative feedback

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47

Naive Adaptive Trend following

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Predictions and Rule classification: Positive feedback

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Evolution of Fractions: Positive feedback

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47

naive adaptive trend-following Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Conclusion

◮ HSM, the model with evolutionary switching between simple

heuristics

◮ this project offers unified estimation approach ◮ observe relatively good fit ◮ To do: more rules, better treatment of outliers, parameter

heterogeneity

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Learning-to-Forecast Experiments:

◮ Hommes, C.H., Sonnemans, J., Tuinstra, J. and Velden, H. van

de, (2005), Coordination of expectations in asset pricing experiments, Review of Financial Studies 18, 955-980.

◮ Heemeijer, P., Hommes, C.H., Sonnemans, J., and Tuinstra, J.,

(2009), Price stability and volatility in markets with positive and negative expectations feedback: an experimental investigation, Journal of Economic Dynamics and Control 33, 1052-1072.

◮ Bao, T., Hommes, C., Sonnemans, J., and Tuinstra, J., (2012),

Individual expectations, limited rationality and aggregate

• utcomes, Journal of Economic Dynamics and Control 36,

1101-1120.

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Heuristic Switching Model:

◮ Anufriev, M., and Hommes, C.H., (2012), Evolutionary selection

• f individual expectations and aggregate outcomes, American

Economic Journal: Microeconomics 4 (4), 35-64.

◮ Anufriev, M., Hommes, C.H., and Philipse, R.H.S., (2013),

Evolutionary selection of expectations in positive and negative feedback markets, Journal of Evolutionary Economics, in press.

◮ Anufriev, M., Hommes, C.H., and Makarewitz, T. (2015) Simple

Forecasting Heuristics that Make us Smart: Evidence from Different Market Experiments, Working Paper

◮ Anufriev, M., Hommes, C.H., and Panchenko, V. (2015) Simple

Forecasting Heuristics that Make us Smart: Evidence from Different Market Experiments, Work in progress

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data

Heterogeneous Agent Models Experiments Heuristic Switching Model HS-Hidden Markov-M Conclusion

Switching Experiments:

◮ Anufriev, M., Bao, T. and Tuinstra, J. (2015), Microfoundations

for switching behaviour in Heterogeneous Agent Models: an experiment, Working paper.

◮ Anufriev, M., Bao, T., Tuinstra, J., and Sutan, A., (2015). Fee

structure, return chasing and mutual fund choice: an experiment, Working paper.

Anufriev, Hommes, Panchenko UTS, UvA, UNSW (Towards a) Bayesian Estimation of the Heuristic Switching Model using Experimental Data