Learning to Forecast with Genetic Algorithms Mikhail Anufriev 1 Cars - - PowerPoint PPT Presentation
Introduction Model Results Conclusion Extra Learning to Forecast with Genetic Algorithms Mikhail Anufriev 1 Cars Hommes 2 , 3 Tomasz Makarewicz 2 , 3 1 EDG, University of Technology, Sydney 2 CeNDEF, University of Amsterdam 3 Tinbergen
Introduction Model Results Conclusion Extra
Mikhail Anufriev1 Cars Hommes2,3 Tomasz Makarewicz2,3
1EDG, University of Technology, Sydney 2CeNDEF, University of Amsterdam 3Tinbergen Institute
Computation in Economics and Finance Taipei 21 June 2015
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
Computational Markov model explaining experimental data: Agents use Genetic Algorithms to optimize a linear forecasting rule. Contribution:
1
Alternative to the RE model of expectation formation motivated by the experimental evidence
2
Generalization of the existing theoretical model, Heuristic Switching Model (Anufriev, Hommes, 2012, AEJ-Micro);
3
Realism: heterogeneous agents using individual learning;
4
Model replicates both aggregate and individual characteristics of data from four Learning-to-Forecast experiments.
Focus of the presentation: experimental data from Bao et al, 2012, JEDC (but also a bit from Heemeijer et al, 2009, JEDC).
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
Price/predictions feedback – cornerstone of any dynamic model. Classical framework: ‘as-if’ perfect rationality leads to rational expectations (RE), i.e., model-consistent predictions. Problems: (i) model is silent on how the agents get RE; (ii) data do not confirm RE hypothesis and its implications
Surveys: housing market (Case, Schiller, Thompson, 2012); inflation expectations of consumers (Malmendier and Nagel, 2009) and professional forecasters (Nunes, 2010). Financial market examples: bubbles and crashes (dot-com bubble of 1997 − 2000; US bear market of 2007 − 09). Experiments: macro-level bubbles in Smith et al. experiments; micro-level expectations in Learning-to-Forecast experiments (Hommes, 2011).
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
A mechanism of bubbles: self-fulfilling trend-following type of price expectations (Brock, Hommes, 1997, 1998) LtF Experiments find individual evidence in favor of this mechanism in the market with positive feedback (Hommes et al, 2005) Heuristic Switching Model: agents switch between simple forecasting heuristics like adaptive vs. trend following expectations. HSM fits well the experimental aggregate data for different types of markets (Anufriev, Hommes 2012; Anufriev et al, 2013). Problem: heuristics in HSM should be specified, but how and how many? Genetic Algorithms (Hommes, Lux, 2013) This Paper: micro-foundations for HSM; model of endogenous learning.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Learning-to-Forecast
Focus on a market of a specific commodity, e.g., financial asset (demand-driven market with positive feedback) or agricultural good (supply-driven market with negative feedback). Subjects play a role of price forecasters to computer agents. The submit forecasts during 50 periods. Computer agents trade rationally given the submitted forecasts. Price is determined from the market clearing condition. Feedback between subject price predictions and prices through
Subjects are rewarded for their forecasting accuracy.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Learning-to-Forecast
earnings per period: et,h = max
49(pt − pe t,h)2, 0
2 euro
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Learning-to-Forecast
1 Heemeijer et al. (2009, JEDC): simple linear framework.
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
t − 60
pt = 60 + 20
21
t − 60
2 Bao et al. (2012, JEDC): large breaks in fundamental price. 3 vd Velden (2001) and Hommes et al. (2007, MD): nonlinear
cobweb producers economy.
4 Hommes et al. (2005, RFS): two-period ahead nonlinear asset
pricing economy.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Learning-to-Forecast
Producer market (negative feedback between forecasts and prices) quickly converges to RE. Asset pricing market (positive feedback) may converge slowly, but typically oscillates. With non-linear feedback more oscillations and instability is observed. Large between-treatment heterogeneity in forecasting and prediction rules; small within-treatment heterogeneity in forecasts (coordination).
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Learning-to-Forecast
Negative feedback
20 40 60 80 10 20 30 40 50 Price Time
Positive feedback
20 40 60 80 10 20 30 40 50 Price Time
Prices in all groups, 6 for negative feedback and 6 for positive feedback (Heemeijer et al., 2009).
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Learning-to-Forecast
Negative feedback
20 40 60 80 100 10 20 30 40 50 60
Positive feedback
20 40 60 80 100 10 20 30 40 50 60
Individual predictions (green) and prices (black) from the selected groups (Bao et al, 2012).
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra GA model
Market with I = 6 artificial agents. Agent i uses a simple forecasting heuristic (adaptive expectations + trend extrapolation) to predict pt: pe
i,t = αipt−1 + (1 − αi)pe i,t−1 + βi(pt−1 − pt−2).
The rule requires specific parameters, e.g., rule may extrapolate trend stronger or weaker. General constraint: αi ∈ [0, 1], βi ∈ [−1.1, 1.1]. Active rule is picked from H = 20 specifications pe
i,h,t = αi,hpt−1 + (1 − αi,h)pe i,t−1 + βi,h(pt−1 − pt−2).
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra GA model
In period t: Each agent has a pool of 20 heuristics, i.e., different (αi,h, βi,h). Forecast is generated by the active rule taken with probability exp(−MSEi,h) 20
k=1 exp(−MSEi,k)
, MSEi,h = (pe
i,h,t−1 − pt−1)2
Between periods t and t + 1: individual learning agents independently update the set of heuristics {(αi,h, βi,h)} with Genetic Algorithms evolutionary operators (Haupt and Haupt, 2004):
each heuristic {(αi,h, βi,h)} is encoded as a binary ‘chromosome’. GA operators: procreation, crossover, mutation, election update the pool of heuristics
We use GA, because it is efficient and simple. Heuristic parameters evolve with time as the price series unfolds!
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra GA model
Between periods t and t + 1 agent i updates the pool of heuristics Hi,t = {(αi,h, βi,h)}20
h=1 in order to get the new pool Hi,t+1
Procreation Agent samples 20 heuristics from the pool with probabilities based on the logit function of MSEi,h = −(pe
i,h,t − pt)2
Mutation Each bit reverses its value with probability δm = 0.01 Crossover Every pair of heuristics swap their α’s entries with probability δc = 0.9 Election So generated 20 ‘new’ heuristics are compared pairwise with the 20 ‘old’ heuristics composing the previous pool Hi,t. New heuristic takes place in the new pool, if its MSE is strictly larger than of the old.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra GA model
Fix the experimental environment (i.e., pricing equation) Initialization
Take as many agents as subjects in the experiment. Initialize 20 heuristics per agent randomly Sample the first forecasts randomly from exogenous distribution. Next time draw randomly one heuristic to form forecast.
Loop of one iteration
1
Agents observe the price and evaluate heuristics’ hypothetical MSE.
2
Every agent update own pool of heuristics with GA.
3
Every agent stochastically picks one heuristic from the new pool based
price forecast.
4
New price is generated from the individual predictions.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Model fitness
Negative feedback
20 40 60 80 100 10 20 30 40 50 20 40 60 80 100 10 20 30 40 50
Positive feedback
20 40 60 80 100 10 20 30 40 50 20 40 60 80 100 10 20 30 40 50
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Model fitness
Negative feedback
20 40 60 80 100 10 20 30 40 50 60 20 40 60 80 100 10 20 30 40 50 60
Positive feedback
20 40 60 80 100 10 20 30 40 50 60 20 40 60 80 100 10 20 30 40 50 60
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Model fitness
1 Monte Carlo study – run the model 1000 times with different learning
and initial predictions: Does the GA model match the stylized facts from the experiments?
2 50-period ahead simulations – take experimental initial predictions
and run the model 1000 times with different learning: How far is the model from the data in the long-run?
3 One-period ahead simulations – take data available to experimental
subjects until period t and predict period t + 1. (Sequential Monte Carlo analysis.) How far is the model from the data in the short-run? Result: GA model beats simple homogeneous models, RE and HSM.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Model fitness
Negative feedback Positive feedback MSE for HSTV12 Prices Forecasts Prices Forecasts Trend extrapolation 2736 1289 101.3 113.3 Adaptive 3.629 10.75 55 62.14 Contrarian 6.984 14.45 58.46 65.95 Naive 94.44 110.9 46.62 52.9 RE 13.871 20.923 55.133 60.859 HSM 73.57 87.86 90.8 101.8 GA-S1: β ∈ [−1.1, 1.1] 8.01 21.97 43.49 49.44 GA-S2: β ∈ [0, 1.1] 6.333 17.39 43.49 49.64
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Model fitness
Negative feedback Positive feedback MSE for HSTV12 Prices Forecasts Prices Forecasts Trend extrapolation 114.061 121.329 1.183 2.165 Adaptive 3.689 10.332 3.776 4.618 Contrarian 5.92 12.534 4.737 5.559 Naive 9.979 16.81 2.411 3.286 RE 13.871 20.923 55.133 60.859 HSM 38.309 45.679 0.9996 2.024 Genetic Algorithm model: Sequential Monte Carlo GA-S1: β ∈ [−1.1, 1.1] 10.247 21.464 0.342 2.059 GA-S2: β ∈ [0, 1.1] 4.208 15.267 0.341 2.036
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Individual learning in the model
What drives the difference between the treatments? In short: Reinforcement between price-predictions feedback and the realized learning. Agents learn to extrapolate the trend under the positive feedback.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Individual learning in the model
Negative feedback
20 40 60 80 100 10 20 30 40 50 60
Positive feedback
20 40 60 80 100 10 20 30 40 50 60
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Individual learning in the model
Negative feedback
20 40 60 80 100 10 20 30 40 50 60
0.5 1 10 20 30 40 50 60
Positive feedback
20 40 60 80 100 10 20 30 40 50 60
0.5 1 10 20 30 40 50 60
Prices and learned trend extrapolation.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Individual learning in the model
Last period: distribution of trend extrapolation learned by agents.
0.1 0.2 0.3
0.3 0.6 0.9 1.2 0.1 0.2 0.3
0.3 0.6 0.9 1.2
Linear pos. feedback
0.1 0.2 0.3
0.3 0.6 0.9 1.2 0.1 0.2 0.3
0.3 0.6 0.9 1.2
Linear pos. feedback with shocks to the fundamental
0.1 0.2 0.3
0.3 0.6 0.9 1.2 0.1 0.2 0.3
0.3 0.6 0.9 1.2
2-period ahead non-linear feedback Result: the more ‘difficult’ feedback, the more trend extrapolation.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
GA model of individual learning: agents fine-tune forecast heuristics the specific market. The model replicates well how individuals learn to forecast prices in experiments (both stylized facts and individual data). Producers economy is more likely to converge to RE. Asset pricing market: individuals learn to extrapolate price trend, thus reinforcing price oscillations. The more difficult feedback, the more trend chasing. Degree of the learned trend extrapolation positively depends on the complexity of the market itself.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
1
Bao, T., Heemeijer, P., Hommes, C., Sonnemans, J. and Tunistra, J. (2012): ‘Individual expectations, limited rationality and aggregate outcomes’, Journal of Economic Dynamics and Control, in press.
2
Heemeijer, P., Hommes C., 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(5),
3
Hommes, C., Sonnemans, J., Tuinstra, J. and van de Velden, H. (2007): ‘Learning in Cobweb Experiments’, Macroeconomic Dynamics, 11(Supplement S1), 8-33.
4
— , (2005): ‘Coordination of Expectations in Asset Pricing Experiments’, The Review of Financial Studies, 18(3), pp. 955-980.
5
van de Velden, H. (2001): ‘An experimental approach to expectation formation in dynamic economic systems’, Ph.D. thesis, Tinbergen Institute and Universiteit van Amsterdam.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
1
Arifovic, J. (1996): ‘The Behavior of the Exchange Rate in the Genetic Algorithm and Experimental Economies’, Journal of Political Economy 104(3), pp. 510-541
2
Dawid, H. and Kopel, M. (1998): ‘On Economic Applications of the Genetic Algorithm: A Model of the Cobweb Type’, Journal of Evolutionary Economics 8,
3
Haupt, R. and Haupt S. (2004): Practical Genetic Algorithms, John Wiley & Sons, Inc., New Jersey, 2nd edn.
4
Hommes, C.H. and Lux, T. (2011): ‘Individual expectations and aggregate behavior in learning to forecast experiments’, Macroeconomic Dynamics, pp. 1-29
5
Vriend, N. (2000): ‘An illustration of the essential difference between individual and social learning, and its consequences for computational analyses’, Journal of Economic Dynamics and Control 24(1), pp. 1-19
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
Parameter Notation Value Number of agents I 6 Number of heuristics per agent H 20 Allowed α, price weight [αL, αH] [0, 1] Allowed β, trend extrapolation coefficient Specification 1 [βL, βH] [−1.1, 1.1] Specification 2 [βL, βH] [0, 1.1] Number of bites per parameter {L1, L2} {20, 20} Mutation rate δm 0.01 Crossover rate δc 0.9 Fitness measure V (·) exp(−MSE(·))
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra
Consider agent i in period t with heuristic h, i.e., (αi,h,t, βi,h,t).
Hypothetical prediction of this heuristic for the price at period s is pe
i,h,t(s) = αi,h,tps−1 + (1 − αi,h,t)pe i,s−1 + βi,h,t(ps−1 − ps−2)
The fitness measure of this heuristic at time t against its hypothetical performance in period s is exp(−MSEi,h(t, s)) 20
k=1 exp(−MSEi,k(t, s))
, MSEi,h(t, s) = (pe
i,h,t(s) − ps)2 .
To pick an active rule in period t, agent uses MSEi,h(t, t − 1), For individual learning between periods t and t + 1, agent uses MSEi,h(t, t), hypothetical performance against the realized price.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra GA model
For details, see the paper (no time :( ). For the first three experiments (linear feedback, linear feedback with shocks to the fundamental price, cobweb economy) our GA model works better than any other model (RE, homogeneous expectations, HSM). But for the 2005 two-period ahead asset pricing experiment, GA model improves very little over other models.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra 2005 experiment
Hommes, Sonnemans, Tuinstra, vd Velden (2005): nonlinear two-period ahead asset pricing market. Subjects have to predict a price of an asset in the next period. Standard myopic mean-variance optimizing computer traders, but also robotic fundamental traders. ‘Supply equals demand’ gives the law of motion: pt = 1 1 + r
pe
t+1 + ntpf + y + εt
200|pt−1 − pf |
fundamentalists, y is the dividend, pf is the fundamental price, ¯ pe
t+1
is the average prediction and pt is the realized price.
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra 2005 experiment
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra GA model and the experiment
MSE Prices Forecasts Trend extr. 178.2 174.9 Adaptive 96.12 145.9 Contrarian 157 146.8 Naive 95.29 144.6 RE 96.0328 145.998 GA: β ∈ [−1.1, 1.1] 103.9 155.8 GA: β ∈ [0, 1.1] 114.9 169.1
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra GA model and the experiment
MSE Prices Forecasts Trend extr. 17.4527 55.0898 Adaptive 44.125 25.3157 Contrarian 59.3905 30.8646 Naive 31.6864 20.8416 RE 96.0328 145.998 GA: β ∈ [−1.1, 1.1] 19.794 39.226 GA: β ∈ [0, 1.1] 17.899 39.256
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra GA model and the experiment
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
Stylized facts:
1
Negative feedback: slow coordination of the subjects around the fundamental value;
2
Positive feedback: fast coordination of the subjects far from the fundamental value.
To measure the degree of coordination between experimental subjects/GA agents at period t, take the standard deviation of their predictions from that period: σt ≡ 6
i,t − pe t
2 0.5 (1) where pe
t is the average prediction at t
pe
t = 6
pe
i,t.
(2)
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
0.2 0.4 0.6 0.8 1 10 20 30 40 50
Negative feedback
0.2 0.4 0.6 0.8 1 10 20 30 40 50
Positive feedback
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
0.5 1 10 20 30 40 50
Negative feedback
0.5 1 10 20 30 40 50
Positive feedback
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
0.2 0.4 0.6 0.8 1 10 20 30 40 50 60
Negative feedback
0.2 0.4 0.6 0.8 1 10 20 30 40 50 60
Positive feedback
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
0.5 1 10 20 30 40 50 60
Negative feedback
0.5 1 10 20 30 40 50 60
Positive feedback
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
20 40 60 80 100 10 20 30 40 50
Negative feedback
20 40 60 80 100 10 20 30 40 50
Positive feedback
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
20 40 60 80 100 10 20 30 40 50
Negative feedback
20 40 60 80 100 10 20 30 40 50
Positive feedback
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
No learning is possible in the first period. Following Diks, C. and Makarewicz, T. (2012): ‘Initial predictions in the Learning-to-Forecast experiment’, Lecture Notes in Economics and Mathematical Systems, forthcoming, we sample initial predictions from calibrated distributions pe
i,1 =
ε1
i ∼ U(9.546, 50)
with probability 0.45739, 50 with probability 0.30379, ε2
i ∼ U(50, 62.793)
with probability 0.23882 (3) for the 2009 experiment and for the 2012 experiment: pe
i,1 =
ε1
i ∼ U(16.406, 50)
with probability 0.32296, 50 with probability 0.35159, ε2
i ∼ U(50, 70.312)
with probability 0.32296. (4)
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48
Introduction Model Results Conclusion Extra Multiple equilibria
In the second period, learning is already possible (first price is already
We assume that the GA agents use the first order heuristic already in the second period. This requires ‘hypothetical’ (counter-factual) price zero p0. Calibration of the model: the best results are obtained under the assumption that the GA agents behave as-if p0 = p1 → p1 − p0 = 0 (5) Interpretation: agents initially disregard the trend, since it cannot be
Mikhail Anufriev (UTS) LtF with GA Computation in Economics and Finance Taipei / 48