Deep Learning in (and of) Agent-Based Models: A Prospectus
aa r X i v : . [ q -f i n . E C ] J un Deep Learning in (and of ) Agent-Based Models:A Prospectus ⋆ Sander van der Hoog a June 21, 2017
Abstract
A very timely issue for economic agent-based models (ABMs) is their empirical estima-tion. This paper describes a line of research that could resolve the issue by using machinelearning techniques, using multi-layer artificial neural networks (ANNs), or so called DeepNets. The seminal contribution by Hinton et al. (2006) introduced a fast and efficient train-ing algorithm called Deep Learning, and there have been major breakthroughs in machinelearning ever since. Economics has not yet benefited from these developments, and there-fore we believe that now is the right time to apply Deep Learning and multi-layered neuralnetworks to agent-based models in economics.
Key words: Deep Learning, Agent-Based Models, Estimation, Meta-modelling. ⋆ This paper has benefited from discussions with Spyros Kousides, Nan Su, Herbert Dawid and Blake LeBaron.Any remaining errors or omissions are the sole responsibility of the author. Financial support from the Horizon2020 ISIGrowth Project (Innovation-fuelled, Sustainable, Inclusive Growth), under grant no. 649186, is gratefullyacknowledged. a Computational Economics Group, Department of Business Administration and Economics, Chair for Eco-nomic Theory and Computation Economics, Bielefeld University, Universit¨atsstrasse 25, 33615 Bielefeld, Germany.Email: [email protected]. Introduction
Agent-Based Models (ABMs) are becoming a powerful new paradigm for describing complexsocio-economic systems. A very timely issue for such models is their empirical estimation. Theresearch programme described in this paper will use machine learning techniques to approachthe problem, using multi-layer artificial neural networks (ANNs), such as Deep Belief Networksand Restricted Boltzmann Machines. The seminal contribution by Hinton et al. (2006)introduced a fast and efficient training algorithm called Deep Learning, and there have beenmajor breakthroughs in machine learning ever since. Economics has not yet benefited fromthese developments, and therefore we believe that now is the right time to apply DeepLearning and multi-layer neural nets to agent-based models in economics.Economic Science is undergoing its own form of ”climate change” in economic theory as newsubfields such as behavioural, experimental, computational, and complexity economics aregaining in support. Complexity economics brings in new tools and techniques that wereoriginally developed in physics and computer science, such as the theory of networks and newstatistical techniques for the study of many-body dynamics.The agenda of this paper is to briefly sketch the current state-of-the-art in ArtificialIntelligence (AI) and Machine Learning (ML), and apply them to economic decision-makingproblems. We outline a research programme that encompasses co-evolutionary learning,learning from experience, and artificial neural networks (ANN), and connect these toagent-based modelling in economics and policy-making.We begin by tracing back the common heritage of complexity economics and evolutionaryeconomics. A rich body of work by evolutionary economists deals with decision-making by realeconomic agents, in an attempt to capture their routines by automatizing theirdecision-making processes in computer algorithms. This starts with Herbert Simon’s ”ABehavioral Model of Rational Choice” (Simon, 1955), it continues with Cyert and March’s ”ABehavioral Theory of the Firm” (Cyert and March, 1963), and culminates in Simon’s seminalwork on ”The Sciences of the Artificial” (Simon, 1969).Around the same time, a computer science conference on ”The Mechanization of ThoughtProcesses” was held at the National Physical Laboratory (NPL) in 1958(National Physical Laboratory, 1959). The conference proceedings contain many of today’s hottopics in Machine Learning: automatic pattern recognition, automatic speech recognition, andautomatic language translation. At the time, the developments in AI, machine learning andtheories of human decision-making were strongly intertwined.After describing this common heritage, we take stock of the current state-of-the-art in MachineLearning and extrapolate into the not too distant future.Firstly, we propose to emulate artificial agent behaviour by so called surrogate modelling,which could be thought of as a Doppelg¨anger approach, in which one agent is observinganother agent’s behavioural pattern and their performance. It then tries to imitate that agent,and eventually replace it in the simulation. In addition, the concept of an ANN-Policy-Agentis introduced who learns from observations of successful policy actions through reinforcementlearning mechanisms.Secondly, we propose to use ANNs as computational emulators of entire ABMs. The ANNfunctions as a computational approximation of the non-linear, multivariate time seriesgenerated by the ABM. It is a meta-modelling approach using statistical machine learningtechniques. There are various advantages to having such an emulator. It allows for acomputationally tractable solution to the issue of parameter sensitivity analysis, robustnessanalysis, and could also be used for empirical validation and estimation. This is particularly2ppealing for large-scale ABMs that are computationally costly to simulate.The goal is to develop new computational methods to improve the applicability ofmacroeconomic ABMs to economic policy analysis. When successful, we would have drasticallyreduced the complexity and computational load of simulating ABMs, and come up with newmethods to model economic agents behaviour. Linking the time series forecasting capabilitiesof the Deep Learning algorithm to ABMs also allows us to envision the possibility of dockingexperiments between different ABMs: the time series output from one ABM can be fed intothe Deep Learning algorithm, resulting in an artificial neural network. This artificial neuralnetwork can then be used as an agent inside another, larger-scale ABM. This notion leads to ahierarchical modelling scheme, in which ABMs of ABMs would become feasible.Each agent in the larger ABM can have an internal mental model of the world it inhabits, andthose mental models can differ to any degree. On the longer term, this approach would allowthe inclusion of computational cognitive models into economic ABMs, allowing the agents tobe fully aware of their environment, and to consider the social embedding of their interactions.
In many cases, we do not know the correct equations of the economic model, and we mightonly know the behaviour of the artificial economic agents approximately through observationsof the empirical behaviour of their real-world counterparts (e.g., through direct observation ofmarket participants, or through laboratory experiments). Therefore, we do not have access tothe mathematical description of the economic system, and have to resort to computationalmodelling. Once we have constructed a computational model that satisfies certainrequirements (e.g., stock- flow consistency of accounting relationships or dynamic completenessof behavioural repertoires) we usually find that the model is realistic enough to reproduceseveral empirical stylized facts of macrovariables, such as GDP growth rates, inflation rates,and unemployment rates, but all too often it is computationally heavy.There are currently several research efforts under way to construct agent-based macroeconomicmodels (Dawid et al., 2014; Dosi et al., 2014; Grazzini and Delli Gatti, 2013). These modelsaim to compete with standard Dynamic Stochastic General Equilibrium (DSGE) models thatare currently in use by ECOFIN and most Central Banks around the world. Using suchmodels for policy analysis requires that they are calibrated and estimated on empirical data.For this, we need new methods and techniques to estimate such policy-oriented ABMs.Current large-scale agent-based simulation models (e.g., Dawid et al., 2014) require largecomputing systems, such as multi-processor servers, HPCs, or grids of GPUs, in order to runsufficiently many simulations. This not only involves running large numbers of simulations forproducing results for publications, but also to perform rigorous robustness testing, parametersensitivity analyses, and general verification and validation (V&V) procedures to ensure thecorrectness and validity of the computer simulations (cf. Sargent, 2011; Kleijnen, 1995).The issue of computational intractability is ubiquitous. It has been around for a long time inphysics and climate science, where research using many-particle systems and large-scaleclimate models is constantly pushing the frontier of what is feasible from a computationalpoint of view.In this paper we describe how to tackle the problem by taking advantage of machine learningtechniques, in particular recent developments in artificial neural networks, such as DeepLearning, Deep Belief Networks, Recursive Networks and Restricted Boltzmann Machines.The scientific relevance and innovativeness of this line of research is that it tries to solve thegeneric problem of computational tractability of computer simulation models (and more3pecifically, of agent-based economic models), not by resorting to technological solutions (e.g.,parallel computing, or GPU grids), but by using machine learning algorithms in order toreduce the computer simulation to a lighter form, by emulating the models using artificialneural networks, and by then adopting that simulation model to obtain results.In order for agent-based models to be useful for economic policy analysis so called ”what-ifscenarios” are used to test counter-factuals in would-be worlds. It is therefore necessary to usemodels with a sufficiently high resolution in terms of the behavioural and institutional details.The target models that we consider in this paper are large-scale agent-based models where’large-scale’ means on the order of millions of agents. High-resolution may refer to theresolution of time-scales, geographical scales, decision-making scales (number of options toconsider), or other dimensionality of the agents’ characteristics.The advantage of such large-scale, high-resolution, high-fidelity agent-based models is thatthey can be used as virtual laboratories, or as laboratory ”in silico” (Tesfatsion and Judd,2006). The model can be used for testing various economic policies (Dawid and Fagiolo, 2008;Dawid and Neugart, 2011; Fagiolo and Roventini, 2012b,a), that may not be feasible to test inthe real world (e.g., due to ethical objections). Examples include: What happens when thebiggest banks go bankrupt? Or: What happens when a Euro member leaves the Euro?Obviously, these are not things we want to simply test in the real world, considering thedetrimental social consequences and ethical objections. The disadvantage is that suchlarge-scale ABMs are quite heavy from a computational perspective. It is easy to generateoverwhelming amounts of data, and reach the boundaries of what is commonly accepted to becomputationally tractable, in terms of simulation time, number of processors used, and datastorage requirements.If we want to apply such models to perform policy analyses, we have to test the robustness ofthe model, i.e., to test whether the empirical stylized facts are still reproduced for manyparameter settings. This involves performing a global parameter sensitivity analysis and arobustness analysis against small changes in the economic mechanisms, or with respect tochanges in the individual behavioural repertoires of the agents. This usually requires a largenumber of simulations (on the order of thousands), in order to obtain a large enough samplingof the phase space, and to be able to ascertain whether the model is sensitive, stable, robust,or fragile.In the social sciences where computer simulation models are being actively pursued (e.g.,economics, sociology, econophysics) there are many discussions surrounding the empiricalestimation and validation of these types of models (e.g., Werker and Brenner, 2004;Brenner and Werker, 2006; Fagiolo et al., 2007; Grazzini et al., 2012; Grazzini and Richiardi,2013; Grazzini et al., 2013; Yildizoglu and Salle, 2012; Barde, 2015; Lamperti, 2015). However,until now, no clear consensus has appeared how to resolve the the empirical validation problem.In econometric applications, some advances have been made on the estimation of ABMs.Noteworthy are two approaches, one using non-parametric bootstrap methods (Boswijk et al.,2007) and the other using estimation of a master equation derived from the Focker-Planckequation (Alfarano et al., 2005; Aoki and Yoshikawa, 2007; Di Guilmi et al., 2008).Currently, multiple projects are under way to construct agent-based macroeconomic models:the Eurace@Unibi model (Dawid et al., 2014), the Crisis Project (Grazzini and Delli Gatti,2013), and the ”Keynes meeting Schumpeter” models (K+S models, Dosi et al., 2010, 2013,2014). These models consider it a feature, not a vice, to model the agents and theirbehavioural repertoires in great detail, by taking care that all the behavioural assumptions aresupported by empirical evidence. 4
Applying machine learning methods to economic problems
A primary motivation for applying machine learning techniques to economic decision makingproblems is the work by Herbert Simon on bounded rationality and satisficing in ”Administrative Behavior” and ”Sciences of the Artificial” (Simon, 1947, 1955, 1959, 1969).Simon being also one of the founders of modern-day Artificial Intelligence (AI), it seems onlyappropriate that in applying artificial neural networks to economic problems, we rely onvarious aspects of Simon’s path-breaking work.The first aspect we adopt is goal-oriented, adaptive behaviour. In a perfect world agents arenot required to spent time on planning and learning. They already have all the relevantinformation available, and are able to compute with full accuracy the outcome of their actions.However, a substantial amount of time of real decision makers is being spent on planning andlearning about new information. Time constraints are important for making decisions, hencesatisficing with threshold aspiration levels rather then optimizing would be the preferredmethodology.Open, complex systems make it essential to behave in a flexible, adaptive manner, rather thanusing rigid, predetermined rules that prescribe an exact course of action for every contingency.This naturally leads to the use of heuristics, routines, and rules of thumb. Satisfying aspirationlevels instead of optimizing appears to be more appropriate as a model of man, as in the adage’Better to be approximately right, rather than exactly wrong.’ Such an approach would lead todecision makers who realize they are fallible, and in order to achieve their goals they must dothe best they can given the circumstances. They would aim for robust decision makingroutines, rather than precise prescriptions.Such considerations point into the direction that human decision makers are not always able tomake perfect decisions, due to various limitations in their decision making capabilities: (i)Imperfect information gathering, or incomplete observation of outcomes. (ii) Limitations instorage capacity or faulty interpretation of those observations (imperfect recall, baddocumentation of results). (iii) Limits in processing abilities. (iv) Imperfections in foreseeingthe exact consequences of their actions. Even when acting in perfect isolation or when they actin the belief that they have precise control over their actions, unintended consequences ofdeliberate, decisive human action may result from a noisy environment. All such imperfectionsin gathering, storing and processing of information and in foreseeing events are a fact of life forthe human decision maker.A second motivation for applying machine learning techniques to economic problems is theseminal book ”A Behavioral Theory of the Firm” by Cyert and March (1963). This bookdescribes many operating procedures related to real firm decision making. Besides an emphasison organizational processes and decision making routines, a further aim of the theory was tolink empirical data to the models by considering the results of case studies of real firms.A clear assessment of the impact of
A Behavioral Theory of the Firm and its methodologicalstance was given by Argote and Greve (2007, p.339):”The general methodological point was that theory should model organizationalprocesses, and should be generated through systematic observation of processes inactual organizations. One component of this point is that organizational theoryshould not oversimplify. Although parsimony is needed in theory building,parsimony that throws out basic insights – like replacing a process model with amaximization assumption – can be harmful.”In the context of agent-based economic models, this idea has been developed further into the5anagement Science Approach (Dawid and Harting, 2012; Dawid et al., 2014). In thisapproach the economic agents are assumed to use decision making routines that areempirically-grounded in the Management Science literature. The underlying assumption isthat managers of a firm apply the methods and techniques that they have been taught whilstdoing their MBA at Management School. This method can for example be applied to modelthe pricing behaviour of firms, the inventory management problem, the interest rate setting forloans by bank managers, or the hiring and firing practices of a Human Resource Managementdepartment.In our approach, we use a combination of the artificial intelligence methods proposed by Simon(learning appropriate heuristics in order to satisfy certain goals), and the empirically-groundedbehavioural rules as proposed by Cyert and March (actual organizational processes).Another exciting field of research is to include a rich cognitive structure into the agents’behavioral repertoires. The decision making routines, although adaptive, are often still toorigid from a cognitive science point of view. A lack of meta-rules to update the behavioralrules is often seen as a serious drawback of these models, especially when it comes toaddressing the Lucas Critique which states that economic policy has to take into account theadaptive behavioral response by the agents that are subject to the policy.This perceived lack of cognitive modelling in the behavioral routines of economic agents can bealleviated if we would allow each agent in the ABM to have an internal ”mental model” of theworld it inhabits, and those mental models can differ to any degree. On the longer term, thisapproach would allow the inclusion of computational cognitive models into economicagent-based models, allowing the agents to be fully aware of their environment, and possiblyalso to consider the social embedding of their interactions.
Applications of ANNs to time series forecasting problems in economics include: financialmarket forecasting (Trippi and Turban, 1993; Azoff, 1994; Refenes, 1995; Gately, 1996),foreign exchange rates (Weigend et al., 1992; Refenes, 1993; Kuan and Liu, 1995), loaddemand forecasts on electricity markets (Bacha and Meyer, 1992; Srinivasan et al., 1994),commodity prices (Kohzadi et al., 1996), and macroeconomic indices (Maasoumi et al., 1994)A review of applications of ANNs in the field of Management Science and Operations Researchis given by Wilson and Sharda (1992) and Sharda (1994). The M-competition(Makridakis et al., 1982) provides a widely cited data base for comparing the forecastingperformance of ANNs in comparison to traditional statistical methods. The data for theM-competition are mostly from business, economics, and finance, see Kang (1991); Sharda(1994); Tang and Fishwick (1993) for examples. Another comparison is provided by the SantaFe forecasting competition (Weigend and Gershenfeld, 1993) which includes very long timeseries coming from various fields.
According to Gorr (1994), ANNs are very appropriate in the following situations: (i) largedata sets; (ii) problems with nonlinear structure; (iii) multivariate time series forecastingproblems. Important issues that can be addressed include:(1) How do ANNs model the autocorrelated time series data and produce better results thantraditional linear and non-linear statistical methods?6ccording to Bengio et al. (2013), sequential statistical data (a.k.a. time series) sufferfrom the ”difficulty of learning long-term dependencies”. If the past is coded linearly(regardless of how many observations in the past) then the effect of the past of the previous step is diminishing exponentially. If the past is modelled non-linearly, then thenon-linearity is ”composed many times”, leading to a highly non-linear relationship ofpast to present. According to the paper cited, recurrent neural networks (RNN) arebetter in modelling such relationships. However, RNNs suffer from the problem of”diminishing gradients” when using back-propagation for training the weights withStochastic Gradient Descent. In such cases Hessian-Free (HF) optimization methods orMomentum Methods such as Nesterov’s Accelerated Gradient (NAG) seem morepromising (see Section 3, Theme 3 for more details).The paper suggest a number of optimizations for RNNs. We believe one of the mostrelevant for our problem is that of filtered, low-pass filter inputs. These are nodes withself-weights close to 1, (similar to exponential filters) that allow non-linearities to persistlonger and not disappear at the next step. This is coupled with non-linearities modelede.g. as out = max (0 , in ) rather than a sigmoid or tanh function. There is justification forthe approaches and some promising results (although this is by no means a solvedproblem) in that the output of the error function is ”rough” and requires some form ofcontrol for local cliffs that lead to local minima. All of the methods proposed in theliterature are gradient-tracking in one way or the other, and are conservative aboutsudden changes. Hessian-free optimization (Martens and Sutskever, 2011) and thePhD-thesis by Sutskever (2013) show the applicability of such methods in themultivariate time series domain.(2) Given a specific forecasting problem, how do we systematically build an appropriatenetwork that is best suited for the problem?We follow current best-practices as outlined above. We can start from the simplest RNNrepresentation, and try state-of-the-art approaches. The design of good initializations ofthe networks is a good point of entry. If we have domain knowledge about the units thatoperate in the system and their qualities, we can estimate the relative size of each inputnode and the long term effect that actions should have. We then use state-of-the-artparameter estimation techniques, as described in Bengio et al. (2013), for example, inorder to fix the weights on the input nodes.(3) What is the best training method or algorithm for forecasting problems, particularlytime series forecasting problems?This is discussed extensively in Sutskever (2013), noting that optimized StochasticGradient Descent (SGD) may be adequate, if one considers proper initialization of thenetwork. Momentum methods are another option. As described above, we could startwith the simplest method first (SGD), or consider the best practice for problems similarto ours. We should keep in mind, however, that we may have various problems that aredifferent in structure. The doppelganger ANN described in Theme 1 (micro-emulation) isnot an RNN, but is rather an actuator based on the inputs. It can have memory of itsown actions, but it is still distinctively different from an RNN that models a time series.Hence, we should find different best practices for each of our sub-tasks. In the themedescriptions in Section 3 we make initial propositions for each of the theme descriptions.(4) How should we go about designing the sampling scheme, and the pre- and7ost-processing of the data? What are the effects of these factors on the predictiveperformance of ANNs?One of the advantages of ANNs is that they alleviate the need for feature engineering which is the art and science of traditional machine learning. Instead, any real numbergoes through a squashing function (logistic or tanh), resulting in a number between 0and 1 (or -1 and 1). In case of categorical values, one can have a ’softmax layer’, thatassigns a probability distribution over the states. Alternatively, one can have ”ON/OFF”nodes with binary values. The fine art then becomes how to design the structure of thenetwork itself: how many layers, and how many nodes per layer.All these questions are addressed in more detail below. In order to use macroeconomic agent-based models for policy, we need to reduce thecomplexity of the ABM simulation to a less complex, more computationally tractable system.In other words, surrogate models or meta-modelling approaches need to be developed, thatallow us to approximate or ’emulate’ the multi-dimensional nonlinear dynamics of the originalsystem. The entire process of finding such an approximate ’emulator’ for an ABM consists of afour-step procedure.First, we construct an ABM and generate synthetic time series data. Then, a multi-layered,deep neural network is designed and trained on the synthetic data. Third, the trained neuralnetwork should be empirically validated using real-world data. And fourth, we apply thetrained, empirically validated deep neural network to economic policy analysis.According to these four steps, the question how to construct efficient emulators of ABMs couldbe structured along four broad research themes:Theme 1: Micro-emulation of the behaviour of individual agents, creating so called”doppelg¨anger” of each agent.Theme 2: Macro-emulation of an entire ABM simulation, using the multivariate timeseries data.Theme 3: Reduction of the complexity to design ANNs, setting the number of inputnodes, number of hidden layers, and number of nodes in each hidden layer.Theme 4: Reinforcement learning in economic policy design, generating an ANN-policyagent that can be used for policy analysis.In the end, the emulation of agent-based models using artificial neural networks(ANN-emulation) allows us to perform specific types of analysis that appear quite complicatedwith large-scale ABMs. For example, the empirical validation of the ANN-emulator could bean efficient, indirect method to calibrate the parameters of the ABM itself. Also, a globalparameter sensitivity analysis could be performed using the ANN-emulator instead of thebulky ABM. And finally, after a careful internal validation procedure to check that the ANN But not impossible in principle. For example, the global sensitivity analysis of a large-scale ABM such as theEurace@Unibi Model was already performed using HPC clusters. Also, empirical validation is currently beingdone for medium-sized ABMs, and given the exponential increase in computing power is expected to yield resultsin the coming years.
This is a local approach, in the sense of modelling the behaviour of each of the rule-basedagents by ANNs. A neural network is trained to predict the actions of a particular agent in themodel, i.e. the ANN acts as a Doppelganger of that agent. Due to the multitude of instancesand agent types, each with their own set of instructions and constraints, and because of thedynamically changing environment of the ABM, such networks can help us model thebehaviour of our agents and reduce the complexity of the model at the local level. The ANNsmay need extensive training, but are cheap when they run. In the end, the original agents maybe replaced by their Doppelganger, or we may run the hybrid model with both types of agents.Various decision making problems in standard macroeconomics models are formulated asoptimization problems. This theme is dedicated to show that this could be dealt with equallywell using machine learning methods. In each example below, we replace a standardoptimization problem with a heuristic.
1. The firm’s consumer demand estimation problem.
In the current model, demand is estimated by two methods, one is backward-looking, the otheris forward-looking. In the backward-looking method, the firm only relies on past observationsand uses a simple OLS regression on the previous months’ sales revenues to estimate the futuredemand for its product. It estimates the parameters of a linear, quadratic, or cubic fit for theperceived demand function that the firm beliefs to be facing (for more details, see Harting,2014, Ch.1, pp 13-14).In the forward-looking method, the firm uses a market research routine that is commonly usedin practice, namely to hold simulated purchase surveys among its consumers(Nagle and Hogan, 2006). Once a year, the firm considers a sample of households to presentthem with a set of products at differing prices. The survey contains questions regardingconsumers’ preferences and price sensitivities, and asks them how they would react c.q. howthey would alter their consumption pattern when the firm would change its price (assumingthe prices of all its competitors stay the same). In this way, the firm tries to gauge the overallprice sensitivity of consumers, and to estimate its future market share. (see Harting, 2014, Ch.3, pp. 79-81 for more details on the market research method, and references therein).
The firm’s consumer demand estimation problem using artificial neural networks.
The idea of replacing the linear, quadratic, or cubic fitting of the data by a neural network isrelatively straightforward. The ANN-firm would try to estimate the local slope of its demandfunction by way of an ANN, and adjust its arc weights while the simulation is ongoing. Sinceneural networks are a data-driven approach, there is no need to assume any particularstatistical model. Also it is not necessary to rely on linear statistical methods such as OLS,since ANNs are non-linear, non-parameteric time series methods.
2. The consumers’ demand problem.
In the current model, the consumers’ decision to buy a product from a specific firm is derivedfrom a discrete choice model using a multinomial logit function. The selection probability toselect a firm is an increasing function of the consumer’s utility for that firm’s product. Theutility value is decreasing in price, so a firm with a higher price will have a lower selectionprobability, but it is bounded away from zero. By replacing the logit function with a neural9etwork formulation of the same problem, the ANN-consumer can learn how to best achieve itstarget value. In this way, the consumers’ choice problem is redefined as goal-orientedbehaviour, rather than as a stochastic model of choice.
3. The banks’ internal risk model.
Banks’ decision making process involves the problem of setting interest rates for individualloans to the private sector. In order to make a profit, banks should assess their debtors’ creditworthiness, and the likelihood that the borrower will not repay the loan in the near future (i.e.,the probability that they will default on the credit). This includes an evaluation of theprobability of default of the debtor firm, but also the default on individual loans. In order tomake such an assessment, the banks either use an internal risk model, or rely on informationprovided by external rating agencies, or both. Whichever method is being used, they have toassess the various types of risk associated to their assets, including market risk, credit risk, andsystemic risk (Duffie and Singleton, 2003). Market risk refers to changes in the value of theassets on the balance sheet of the bank. Typically, these are fluctuating due to themark-to-market asset valuation and the volatility of prices on the financial asset markets.Credit risk refers to the risk the bank is facing due to the changing values of assets on thebalance sheets of its debtors.In the current agent-based macroeconomic model (Eurace@Unibi), the bank uses a highlysimplified model to determine the probability of default of the firms to which they haveoutstanding loans, or of new firms that make credit requests. The essential aspect of the modelis that the bank’s assessment of the firm’s probability of default is based on balance-sheet dataof the firm, and derived from the firm’s equity and financial ratios such as thedebt-equity-ratio, an indicator of financial fragility.Such a ”structural-model” approach may or may not be in accordance with the actualbehavior of real banks, which would be a matter of empirical study that is beyond the scope ofour current research project. But in fact, many alternative models for evaluating credit defaultrisk exist, as illustrated by the rich overview given by Duffie and Singleton (2003).One such an alternative approach is the ”first-passage model” (Duffie and Singleton, 2003, pp.53), which uses empirical time series collected over a certain time window, to determine theactual default probabilities for a population of firms that have similar risk profiles. Such atime series approach differs substantially from the more theoretical ”reduced-form”approaches, but it would fit quite nicely with the neural network approach.The artificial neural network approach to model the banks’ decision making problem will thusprovide us with a nonlinear, nonparametric, multivariate time series forecasting method. Thebank can be modelled as a goal-oriented entity, that tries to set interest rates based on itsforecasted default probabilities, which are derived from time series that are being generatedonline, i.e. during an ongoing simulation. In the end, this could yield an agent-based model ofthe credit market in which the credit risk models proposed in Duffie and Singleton (2003) havebeen internalized into our agents’ behavioural repertoires.This line of research can distinguish between ”offline training” and ”online learning”. Offlinetraining makes use of time series data being generated by an agent-based model that is”detached” from the agent. One can think of this as an outside-observer-approach, where theagent is not part of the simulation environment, but can observe the actions and outcomes ofother agents. This is similar to how children learn how to behave appropriately in a socialenvironment, before they are accepted as full members of that environment.Online learning, on the other hand, occurs while the agent is itself part of the simulationenvironment, and is observing the time series being generated online. If multiple agents aresimultaneously using online learning in this sense, we can speak of co-evolutionary learning
10y a population of heterogeneous, artificial neural network agents.The main aim of this particular research theme is to focus on the appropriate networkstructure to emulate the multivariate time series data being generated by the target system (inthis case, the particular agents to emulate).The final goals of Theme 1 are to obtain: (i) a model of firm behaviour replaced by an ANNfor the firm’s demand estimation routine; (ii) a model of consumer behaviour replaced by anANN for the consumers’ choice problem; (ii) a model of bank behaviour replaced by an ANNfor the bank’s internal risk evaluation and risk-assessment problem.
Due to the recent breakthrough of Deep Learning techniques for multi-layered networks tomodel non-linearities, it becomes possible to emulate an entire ABM simulation by an ANNgenerating time-series. Contrary to the local approach in Theme 1, this is a global approach.A neural network is trained to predict the probabilistic structure of the macro-level variablesof the model.This is useful for robustness and parameter sensitivity analysis, since it allows a much largerexploration of the parameter space. A second advantage is that by training the ANN on manycounter-factual scenarios, it is expected to perform better than an ANN that has been trainedjust on the empirical, historic data, since this is just a single realization of the empirical datagenerating process.In our aim to make the problem of tuning the parameters of an ABM more tractable, we try toemulate the input/output function of the entire ABM by an (ultimately) less complex andmore tractable Deep Neural Network. Our starting point is that, in an ABM, a multitude ofautonomous agents react to changes in their (economic) environment and, in doing so, alterthat very environment. In a Deep Neural Net, a multitude of nodes at different layers canencode different information structures and decision processes, such that the network as awhole can serve specific functions. Bringing the two together, we aim to train a neural networkthat produces the same output (in terms of time series of macro-economic variables) as theABM.This problem is similar to multi-variate time series forecasting, with the difference that insteadof estimating the future values based on the past values, the input to the ABM are the actionsof the agents in the model. A recurrent neural network (RNN) is the type of network that isthe obvious choice for such a task. A key part of the design is a feedback property, namely thatthe output of the model (the values of the measured macro-economic variables) are fed back tothe input. A second key part is to split the network input layer to represent the ‘present’, andthe ‘past’. This is how the RNN design captures ‘history’: at each step t the network receivesinputs at time t , but also of time t −
1, and possibly further time lags.In terms of integrating the decision processes of the individual agents, a first approximationcould be a multi-layered structure in which the nodes of the first input layer are entire ANNsthat model each individual agent in the agent-based model. Of course, this is not expected tobe any more tractable than the ABM itself. However, it is expected that the ANN will be ableto emulate the ABM with a much smaller number of agents , as the multi-layer structure allowsthe ANN to model increasingly more complex functions of the modelled economy without theneed to fully emulate it. Instead of representing the individual agents one by one, the ANN isa representation of the entire ABM at increasing ”layers of abstraction”. Most importantly,this theme will be informed by other themes, e.g. the modelling in Theme 1 for the individualagent ANNs can inform the initial design of the macro-emulation ANN.11raining data for the macro-emulation ABM will be provided by already collected (synthetic)data from ABM simulations, as well as from new simulations with agents that are themselvesANNs, rather than the current fixed behavioral routines. The big advantage of training theANN on simulated data, in addition to the abundance of such data, is that such a network willnot just learn to forecast a specific realization of some ABM emulation, but it will learn themore general underlying data generating mechanism that is common to all such simulationswhich are seen during the training phase, and, ideally, also for new previously unseensimulations. This provides for an out-of-sample validation stage by using a subset of thesynthetic data that was previously unseen by the ANN, and can be used to test theperformance of the macro-emulation Deep Neural Network.The main aim of this particular research theme is to focus on the appropriate networkstructure to emulate the multivariate time series generated by the macro-ABM as a whole. Asecond aim is to investigate what is the most appropriate learning/optimization technique forthis problem.The final goal of Theme 2 is to obtain a deep layered ANN that is trained on data generatedby an ABM, and that can be usefully applied for empirical validation, and for policy analysis.
The design and training of deep ANNs is a complex task. To guide the design of the ANN interms of the number of nodes and hidden layers, and in order to improve the efficiency of theDeep Learning algorithm, the complexity of the ANN must be reduced.The problem of training deep neural networks is an unconstrained global minimizationproblem, i.e. to find the arc weights such that the training error of the ANN is minimized (thetraining error is the difference between the ANNs performance on the training set and on thetest set).This problem is NP-hard, so the computational costs will increase exponentially with problemsize (given by the number of input nodes and the number of hidden layers). Therefore smartheuristics are needed to approximate the global minimum. Many such heuristics have beendeveloped, but most of these assume that the objective function (the loss function or errorfunction) is differentiable in its arguments. Hence the algorithms make use of the gradient andthe Hessian of the objective function. Example methods include Gradient Descent (GD),Stochastic Gradient Descent (SGD), Momentum methods and Nesterov’s Accelerated Gradient(see Sutskever, 2013 for an overview, and references therein).For convex objective functions, to find the global minimum the gradient methods are globallyconverging, i.e. they will always find the global minimum, but they will just take longer toconverge for worse initializations of the parameters. However, for deep and recurrent networks,the initialization does matter since the objective function of such networks cannot be assumedto be convex. Hence, it is important to design good initializations for the algorithms.The greedy unsupervised pre-training algorithm of Hinton et al. (2006) andHinton and Salakhutdinov (2006) is a good starting point since it greedily trains theparameters of each layer sequentially. Such greedy layerwise pre-training is then followed by a”fine-tuning” algorithm such as the standard Stochastic Gradient Descent method.Another method is the Hessian-Free (HF) Optimization (Martens and Sutskever, 2010, 2012)that is able to train very deep feed-forward networks even without such a pre-training step.HF is a second-order method and therefore rather slow, but it is very powerful. It is thepreferred method of optimization if there is no idea about good initializations of the network.The most recent innovations in this field, described by Sutskever (2013, Ch. 7), are able to12rain very deep neural networks (up to 17 hidden layers) by using aggressive MomentumMethods. Such methods use gradient information to update parameters in a direction that ismore effective then steepest descent by accumulating speed in directions that consistentlyreduce the objective function. The most promising method of this type is Nesterov’sAccelerated Gradient (NAG) method, which is a first-order optimization algorithm that hasbetter convergence properties than Gradient Descent. It has two parameters: a learning rate ε and a momentum constant µ , where (1 − µ ) can be thought of as the friction of the errorsurface. High values of µ implies the algorithm retains gradient information and leads to fastconvergence, while low values imply high friction and a loss of gradient information, leading toslower convergence to the global minimum.Using NAG with very aggressive momentum values ( µ close to 0 .
99) leads to excellent resultsfor problems that previously were deemed unsolvable, such as data sets exhibiting very longtime-dependencies (50-200 time steps).The main aim of this research theme is to focus on Hessian-Free Optimization and MomentumMethods, and possibly adapt these methods to our specific problems. A second aim is tooptimize the choice of network parameters: the number of input nodes, the number of hiddenlayers, and how many nodes in each hidden layer.The final goals from Theme 3 are to design good initializations of the network parameters forthe Deep Learning algorithms, and to develop insights to inform the optimization methods andthe Deep Learning algorithms.
The final theme is to apply a surrogate, or meta-modelling approach to policy decision-making.A government or central bank agent may be given certain goals (e.g., maintaining a stableprice level, or a low unemployment rate, or macrofinancial stability) rather than usinghand-crafted rules to serve those goals (such as a Taylor rule for monetary policy). Usingreinforcement learning techniques, an agent starts with little knowledge of the world, but givena reward function, the agent learns to perform better over time, during a simulation run. TheABM allows us to evolve successful policies not only by using empirical data, but also bylearning from online-generated streaming data. The idea is to have a neural network policyagent (ANN-policy-agent), and this is again a local approach.The objective is to develop a model with an endogenous policy-maker, the ANN-policy-agent,who evolves its decision-making routines endogenously. This agent should adapt its policy inresponse to the behavioural changes of the other agents in the model.Similar to Theme 1, we can again distinguish between ”offline training” and ”online learning”.
Offline training of the ANN-policy-agent.
We train the ANN-policy-agent using pre-generated, historical data from the original ABMsimulation. Its input are (rule-based) policy decisions made during that simulation and timeseries of economic variables, and we use unsupervised layer-by-layer training. Thus, theANN-policy-agent learns (unsupervised) the outcome of policy decisions under specificcircumstances, and it is possible to re-enforce this training with data from multiple ABMsimulations by using a Monte Carlo approach. After this unsupervised pre-training, weperform an additional supervised training phase, in which we reward policy decisions that havedesired outcomes. The trained ANN-policy-agent is then used in ABM simulations as thepolicy-making authority. Depending on the properties and coverage of the training data, thistype of ANN-policy-agent is expected to fare well in the test simulations.13 nline learning by the ANN-policy-agent.
In this setting, the ANN-policy-agent learns from online streaming data. It has to train itsweights while taking actual policy actions during a running ABM economy. This situation isbad from an AI point of view, since training normally takes a long time and occurs in isolationof the actual environment. Therefore the analogy to our setup is a ANN-policy-agent that hasnot been trained before, or has inappropriate weights for the situation. It has to adapt as bestit can, similar to a learning child. However, this setting seems more close to what actualreal-world policy-makers are facing, especially in times of a changing economic environment.In times of crisis, policy-makers have to adjust quickly to changing circumstances, possiblymaking choices that appear suboptimal, but satisfying certain target levels.The main aim of this theme is to focus on which reward functions and what network structuresare most appropriate for the ANN-policy-agent. A second aim is to design the endogenouspolicy setting behaviour for the ANN-policy-agent: which behavioral heuristics are used, whatmeta-rules adapt these heuristics, and what are the parameters for the ANN.The final goal of Theme 4 is to develop a model with a ANN-policy-agent that can set policyendogenously, and is able to adjust its policy response to the behavioral changes of the otheragents in the model economy.
The purpose of this paper is to sketch a line of research in which artificial neural networks(ANNs) are used as computational approximations or as emulators of the nonlinear,multivariate time series dynamics of a pre-existing agent-based model (ABM). In other words,it is a meta-modelling approach using statistical machine learning techniques. There arevarious advantages to having such an emulator. For instance, it allows for a computationallytractable solution to the issue of parameter sensitivity analysis, robustness analysis, and couldalso be used for empirical validation and estimation.The overall goal is to develop new methods and techniques to improve the applicability ofmacroeconomic ABMs to economic policy analysis. For the practical implementation of thisgoal, we need to make advances in two domains:1. Deep Learning: developing new machine learning techniques to represent ABMs byANNs (Themes 1-2).2. Complexity Reduction: developing new complexity-reduction techniques to guide thedesign of ANNs (Theme 3).The work to be done consists of the following broad research themes:Theme 1: Micro-emulation of the behaviour of agents. A neural network is trained to predict theactions of a particular agent in the model, i.e. the ANN acts as a Doppelganger of thatagent.Theme 2: Macro-emulation of an entire ABM simulation. This is a global approach. A neuralnetwork is trained to predict the probabilistic structure on the macro-level, of variablesin the ABM model, based on the initialization parameters. ANNs have proved to be verysuccessful for multivariate time series forecasting. They are much more flexible thantraditional statistical methods since they are nonlinear, nonparametric time seriesapproximation techniques. 14heme 3: Reduction of complexity. The design of the structure of the neural network in terms ofnumbers of input- and output nodes and the number of hidden layers is a complicatedproblem. In order to improve the efficiency of the Deep Learning algorithm, thecomplexity of the ANN must therefore be reduced. This can be done by modelling it interms of a Hamiltonian system, and proceed with describing the time-evolution of theHamiltonian.Theme 4: Reinforcement learning in policy design. A government or central bank agent may begiven certain goals (such as a stable price level, low unemployment rates, ormacrofinancial stability), rather than hand-crafted rules. Using reinforcement learningtechniques, an agent starts with little knowledge of the world, but given a rewardfunction that models those goals, the agent learns to perform better over time.This may lead to more flexible policies and more adaptive behaviour on the part of thepolicy agent, as it allows for a more flexible, discretionary policy setting behavior, ratherthan using a fixed, rule-based policy. As the policy agent learns how to set policiesoptimally, it must adapt to the behavioural changes of the other agents, who mightchange their behaviour in response to the policy. Hence, this policy-feedback-loopaddresses in a very natural way the Lucas Critique.In summary, themes 1 through 4 not only help us to design a strategy how to emulate andestimate agent-based models using artificial neural networks, but it may also contribute to theburgeoning literature on learning in macroeconomics and optimal policy design. Hence, theresearch programme connects both micro- and macroeconomics, and joins both estimation andemulation in machine learning.
When successful, we could apply the new methods to a plethoria of problems. We would havedrastically reduced the complexity and computational load of simulating agent-based models,and come up with new methods to model economic agents’ behaviour. Furthermore, linkingthe time series forecasting capabilities of the Deep Learning algorithm to agent-based modelsalso allows us to envision the possibility of docking experiments between different ABMs: thetime series output from one ABM can be fed into the Deep Learning algorithm, resulting in anartificial neural network. This artificial neural network can then be used as an agent insideanother, larger-scale ABM. This notion leads to a hierarchical modelling scheme, in whichABMs of ABMs would become feasible. Each agent in the larger ABM can have an internal”mental model” of the world it inhabits, and those mental models can differ to any degree. Onthe longer term, this approach would allow the inclusion of computational cognitive modelsinto economic agent-based models, allowing the agents to be fully aware of their environment,and to consider the social embedding of their interactions.
References
Alfarano, S., Lux, T., and Wagner, F. (2005). Estimation of Agent-Based Models: The Case ofan Asymmetric Herding Model.
Computational Economics , 26(1):19–49.Aoki, M. and Yoshikawa, H. (2007).
Reconstructing Macroeconomics . Cambridge Books.Cambridge University Press. 15rgote, L. and Greve, H. R. (2007).
A Behavioral Theory of the Firm – 40 years and counting:Introduction and impact.
Organization Science , 18(3):337–349.Azoff, E. (1994).
Neural Network Time Series Forecasting of Financial Markets . John Wileyand Sons, Chichester.Bacha, H. and Meyer, W. (1992). A neural network architecture for load forecasting. In
Proceedings of the IEEE International Joint Conference on Neural Networks , volume 2,pages 442–447.Barde, S. (2015). A Practical, Universal, Information Criterion over Nth Order MarkovProcesses. Studies in Economics 1504, School of Economics, University of Kent.Bengio, Y., Boulanger-Lewandowski, N., and Pascanu, R. (2013). Advances in optimizingrecurrent networks. In
IEEE International Conference on Acoustics, Speech and SignalProcessing (ICASSP) , pages 8624–8628.Boswijk, H. P., Hommes, C. H., and Manzan, S. (2007). Behavioral heterogeneity in stockprices.
Journal of Economic Dynamics & Control , 31(6):1938–1970.Brenner, T. and Werker, C. (2006). A Practical Guide to Inference in Simulation Models.Papers on Economics and Evolution 2006-02, Philipps University Marburg, Department ofGeography.Cyert, R. M. and March, J. G. (1963).
A Behavioral Theory of the Firm . Prentice Hall,Englewood Cliffs.Dawid, H. and Fagiolo, G. (2008). Agent-based models for economic policy design:Introduction to the special issue.
Journal of Economic Behavior & Organization , 67(2):351 –354. Special issue on Agent-based models for economic policy design.Dawid, H., Gemkow, S., Harting, P., van der Hoog, S., and Neugart, M. (2014). Agent-BasedMacroeconomic Modeling and Policy Analysis: The Eurace@Unibi Model. In Chen, S.-H.and Kaboudan, M., editors,
Handbook on Computational Economics and Finance . OxfordUniversity Press.Dawid, H. and Harting, P. (2012). Capturing firm behavior in agent-based models of industryevolution and macroeconomic dynamics. In B¨unsdorf, G., editor,
Evolution, Organizationand Economic Behavior , chapter 6. Edward Elgar.Dawid, H. and Neugart, M. (2011). Agent-based models for economic policy design.
EasternEconomic Journal , 37.Di Guilmi, C., Gallegati, M., and Landini, S. (2008). Modeling Maximum Entropy andMean-Field Interaction in Macroeconomics. Economics Discussion Papers 2008-36, KielInstitute for the World Economy.Dosi, G., Fagiolo, G., Napoletano, M., and Roventini, A. (2013). Income distribution, creditand fiscal policies in an agent-based Keynesian model.
Journal of Economic Dynamics &Control , 37:1598–1625.Dosi, G., Fagiolo, G., and Roventini, A. (2010). Schumpeter meeting Keynes: A policy-friendlymodel of endogenous growth and business cycles.
Journal of Economic Dynamics & Control ,34:1748–1767. 16osi, G., Napoletano, M., Roventini, A., and Treibich, T. (2014). Micro and Macro Policies inKeynes+Schumpeter Evolutionary Models. LEM Papers Series 2014/21, Laboratory ofEconomics and Management (LEM), Sant’Anna School of Advanced Studies, Pisa, Italy.Duffie, D. and Singleton, K. J. (2003).
Credit Risk: Pricing, Measurement, and Management .Princeton University Press, Princeton.Fagiolo, G., Moneta, A., and Windrum, P. (2007). A Critical Guide to Empirical Validation ofAgent-Based Models in Economics: Methodologies, Procedures, and Open Problems.
Computational Economics , 30(3):195–226.Fagiolo, G. and Roventini, A. (2012a). Macroeconomic policy in DSGE and agent-basedmodels.
Revue de l’OFCE , 124:67–116.Fagiolo, G. and Roventini, A. (2012b). On the scientific status of economic policy: a tale ofalternative paradigms.
The Knowledge Engineering Review , 27:163–185.Gately, E. (1996).
Neural Networks for Financial Forecasting . John Wiley, New York.Gorr, W. L. (1994). Editorial: Research prospective on neural network forecasting.
International Journal of Forecasting , 10(1):1–4.Grazzini, J. and Delli Gatti, D. (2013). Paper on the development of MABM Mark II: Theinput-output network in the CRISIS Macro Agent-Based Model. CRISIS Project DeliverableD3.3, Universit Cattolica del Sacro Cuore, Milano.Grazzini, J., Richiardi, M., and Sella, L. (2012). Indirect estimation of agent-based models: Anapplication to a simple diffusion model. LABORatorio R. Revelli Working Papers Series 118,LABORatorio R. Revelli, Centre for Employment Studies.Grazzini, J. and Richiardi, M. G. (2013). Consistent Estimation of Agent-Based Models bySimulated Minimum Distance. LABORatorio R. Revelli Working Papers Series 130,LABORatorio R. Revelli, Centre for Employment Studies.Grazzini, J., Richiardi, M. G., and Sella, L. (2013). Analysis of Agent-based Models.LABORatorio R. Revelli Working Papers Series 135, LABORatorio R. Revelli, Centre forEmployment Studies.Harting, P. (2014).
Policy design in the presence of technological change – an agent-basedapproach . Ph.D. Thesis, University of Bielefeld.Hinton, G. E., Osindero, S., and Teh, Y.-W. (2006). A fast learning algorithm for deep beliefnets.
Neural Computing , 18(7):1527–1554.Hinton, G. E. and Salakhutdinov, R. (2006). Reducing the dimensionality of data with neuralnetworks.
Science , 313(5786):504–507.Kang, S. (1991).
An Investigation of the Use of Feedforward Neural Networks for Forecasting .PhD thesis, Kent State University.Kleijnen, J. P. C. (1995). Verification and validation of simulation models.
European Journalof Operational Research , 82(1):145–162. 17ohzadi, N., Boyd, M., Kermanshahi, B., and Kaastra, I. (1996). A comparison of artificialneural network and time series models for forecasting commodity prices.
Neurocomputing ,10:169–181.Kuan, C.-M. and Liu, T. (1995). Forecasting exchange rates using feedforward and recurrentneural networks.
Journal of Applied Econometrics , 10(4):347–64.Lamperti, F. (2015). An Information Theoretic Criterion for Empirical Validation of TimeSeries Models. LEM Papers Series 2015/02, Laboratory of Economics and Management(LEM), Sant’Anna School of Advanced Studies, Pisa, Italy.Maasoumi, E., Khotanzad, A., and Abaye, A. (1994). Artificial neural networks for somemacroeconomic series: A first report.
Econometric Reviews , 13:105–122.Makridakis, S., Anderson, A., Carbone, R., Fildes, R., Hibdon, M., and Lewandowski, R.(1982). The accuracy of extrapolation (time series) methods: Results of a forecastingcompetition.
Journal of Forecasting , 1:111–153.Martens, J. and Sutskever, I. (2010). Parallelizable sampling of markov random fields.
Artificial Intelligence and Statistics , pages 517–524.Martens, J. and Sutskever, I. (2011). Learning recurrent neural networks with Hessian-Freeoptimization. In
Proceedings of the 28th International Conference on Machine Learning(ICML) .Martens, J. and Sutskever, I. (2012). Training deep and recurrent networks with Hessian-Freeoptimization. In Montavon, G., Orr, G. B., and M¨uller, K.-R., editors,
Neural Networks:Tricks of the Trade , volume 7700 of
Lecture Notes in Computer Science , pages 479–535.Springer Berlin Heidelberg.Nagle, T. and Hogan, J. (2006).
The Strategy and Tactics of Pricing: A Guide to GrowingMore Profitably . Pearson Prentice Hall, New Jersey.National Physical Laboratory, editor (1959).
Mechanisation of Thought Processes , Proceedingsof a Symposium held at the National Physical Laboratory on the 24th, 25th, 26th and 27thNovember 1958. Her Majesty’s Stationary Office, 1959.Refenes, A. (1993).
Constructive learning and its application to currency exchange rateforecasting , chapter 39, pages 777–806. Probus Publishing Company, Chicago.Refenes, A. (1995).
Neural Networks in the Capital Markets . John Wiley, Chichester.Sargent, R. G. (2011). Verification and validation of simulation models. In
Proceedings of theWinter Simulation Conference , WSC ’11, pages 183 – 198. Winter Simulation Conference.Sharda, R. (1994). Neural networks for the ms/or analyst: An application bibliography.
Interfaces , 24:116–130.Simon, H. A. (1955). A Behavioral Model of Rational Choice.
Quarterly Journal ofEconomics , 69(1):99–118.Simon, H. A. (1959). Theories of Decision-Making in Economics and Behavioral Science.
American Economic Review , 49:253–283. 18imon, H. A. (1996 [1969]).
The Sciences of the Artificial . The MIT Press, Cambridge, MA.(3rd ed.).Simon, H. A. (1997 [1947]).
Administrative Behavior . The Free Press, New York, NY. (4thed.).Srinivasan, D., Liew, A., and Chang, C. (1994). A neural network short-term load forecaster.
Electric Power Systems Research , 28:227–234.Sutskever, I. (2013).
Training Recurrent Neural Networks . PhD thesis, Department ofComputer Science, University of Toronto.Tang, Z. and Fishwick, P. (1993). Feedforward neural nets as models for time seriesforecasting.
ORSA Journal on Computing , 5:374–385.Tesfatsion, L. and Judd, K. E. (2006).
Handbook of Computational Economics II: Agent-BasedComputational Economics . North-Holland.Trippi, R. and Turban, E. (1993).
Neural Networks in Finance and Investment: UsingArtificial Intelligence to Improve Real-world Performance . Probus, Chicago.Weigend, A. and Gershenfeld, N. (1993).
Time Series Prediction: Forecasting the Future andUnderstanding the Past . Addison-Wesley, Reading, MA.Weigend, A., Huberman, B., and Rumelhart, D. (1992).
Predicting sunspots and exchangerates with connectionist networks , pages 395–432. Addison-Wesley, Redwood City, CA.Werker, C. and Brenner, T. (2004). Empirical Calibration of Simulation Models. Papers onEconomics and Evolution 2004-10, Philipps University Marburg, Department of Geography.Wilson, R. and Sharda, R. (1992). Neural networks.