Modelling 'Animal Spirits' and Network Effects in Macroeconomics and Financial Markets Thomas Lux

Transcription

1 Modelling 'Animal Sirits' and Network Effects in Macroeconomics and Financial Markets Kiel Institute for the World Economy & Banco de Esaña Chair of Comutational Economics University of Castellón GSDP Agent-based Modeling Worksho Université Paris I, 8 0 Seember 20

2 The Kiel Institute for the World Economy

3 Two arts Estimation of (simle) agent-based models covering sychological effects and dynamic social interaction in economic data (estimation of a micro model with macro data) Analysis of interaction on the micro level: Identification of network structure among agents

4 Agent-based models with social interactions insired by statistical hysics (a time-honoured legacy) Föllmer (974): Random economies with interacting agents Weidlich and Haag (983): Quantitative Sociology Schelling: Micromotives and Macrobehavior Many behavioral finance models Brock and Durlauf (200): Discrete choice with social interactions

5 Available models are successful in exlaining emirical regularities, but have not been really imlemented emirically Missing is: estimation of the underlying arameters, comarison of models, goodness of fit.

6 Missing is a general aroach to arameter estimation of agent-based models Missing is both a theoretical and emirical methodology for sychological effects in macroeconomics Here we introduce a general rigorous methodology for arameter estimation Illustration: estimation of Weidlich model for economic survey data

7 Background on Survey Data Survey data are mostly used to forecast key economic quantities (GDP, IP, stock rices), but are seldomly treated as endogenous variables What drives exectations exressed in surveys? Rational exectations vs. animal sirits RE tests tyically negative, recent revival of interest in animal sirits (Akerlof and Shiller) Research question: Can we identify animal sirits at work? Can we identify the influence of social interaction on oinion formation?

8 In the following, an agent-based model for oinion formation via social interaction will be introduced (the Weidlich model) We develo a general rigorous methodology for arameter estimation on the base of aggregate data (i.e. survey data) from a microscoic model Illustration: estimation of arameters of Weidlich model for economic survey data (ZEW survey)

9 Alication I: Macro Sentiment ZEW Index of Economic Sentiment, , Monthly data, index = #ositive - # negative, ca. 350 resondents

10 A Canonical Interaction Model à la Weidlich Two oinions, strategies etc: + and A fixed number of agents: 2N Agents switch between grous according to some transition robabilities w and w v: frequency of switches, U: function that governs switches α 0, α : arameters v * ex( U ) w w v * ex( U x x n 0 n 2 N U ) Sentiment index

11 Remarks The model is designed as a continuous-time framework, i.e. w and w are Poisson rates (um Markov rocess) The canonical model allows for interaction (via α ) and a bias towards one oinion (via α 0 ), but could easily be extended by including arbitrary exogenous variables in U The framework corresonds closely to that of discrete choice with social interactions, it formalizes non-equilibrium dynamics, while DCSI only considers RE equilibria

12 Theoretical Results For α : uni-modal stationary distribution with maximum x* =, >,< 0 (for α 0 =,>,< 0) For α > and α 0 not too large: bi-modality (symmetric around 0 if α 0 = 0, asymmetric otherwise) If α 0 gets too large: return to uni-modality (with maximum x* >,< 0 for α 0 >,< 0)

13 Some simulations

14

15 How to Arrive at Analytical Results? Our system is quite comlex: 2N couled Markov um rocesses with state-deendent, non-linear transition rates Solution via Master equation: full characterization of time develoment of df: can be integrated numerically, but is too comutation intensive with large oulation More ractical: Fokker-Planck equation as aroximation to transient density

16 How to Arrive at Analytical Results? Different formalisms for um Markov models of interactions: Master equation: full characterization of time develoment of df: can be integrated numerically, but is too comutation intensive with large oulation Fokker-Planck equation: dp(x, t) wp(x dt (w w )P(x, t), t) w P(x t x (x) x x in stes of /N N x x 2 2 g(x) N x, t) drift:-μ Diffusion: 2*g

17 Fokker-Planck-Equation: exanding the ste-oerators for x in Taylor series u to the second order and neglecting the terms o(δx 2 ), we end u with the following FPE: t x (x) x x x 2 2 g(x) z n n ( x ) w ( x ) w( x ) 2N 2N n n g( x ) ( w ( x ) w 2N 2N 2N 2 x )) (

18 Estimation: for a time series of discrete observations X s of our canonical rocess, the likelihood function reads with discrete observations X s, the Master or FP equations are the exact or aroximate laws of motion for the transient density and allow to evaluate log f(x s+ X s,θ) and, therefore, to estimate the arameter vector θ (θ = (v, α 0,α ) )!

19 Imlementation Usually no analytical solution for transient dfs from Master or FP equations Numerical solution of Master equation too comutation intensive if there are many states x (i.e., articularly with large N) Numerical solutions of FP equation is less comutation intensive, various methods available for discretization of stochastic differential equations

20 Finite Difference Aroximation x 2 2 x x g(x) x (x) x t 2 i i i i i i i h g 2g g h k 2 i i i i i i i h g 2g g h k 2 i i i i i i i h g 2g g h k 2 i i i i i i i h g 2g g h k forward difference backward difference Sace-time grid: x min + h, t 0 + ik

21 Numerical Solution of FPE Forward and backward aroximations are of first-order accuracy: combining them yields Crank-Nicolson scheme with second-order accuracy -> solution at intermediate oints (i+/2)k and (+/2)h This allows to control the accuracy of ML estimation: estimates are consistent, asymtotically normal and asymtotically equivalent to comlete ML estimates (Poulsen, 999)

22 Observation Xs, aroximated by shar Normal distr. Evaluation of Lkl of observation Xs+ Time interval [s, s+]

23 Monte Carlo Exeriments Does the method work in our case of a otentially bi-modal distribution, is it efficient for small samles? YES, IT DOES (AS IT SHOULD) Do we have to go at such ains for the ML estimation? Couldn t we do it with a simler aroach (Euler aroximation)? YES, WE HAVE TO

24 Monte Carlo Study of MLE with Crank-Nicolson Aroximation v =3, α 0 = 0, α = 0.8 α 0 = 0.2, α = 0.8 α 0 = 0, α =.2 α 0 = 0.2, α =.2

25 Emirical Alication The framework of the canonical model is close to what is reorted in various business climate indices Germany: ZEW Indicator of Economic Sentiment, Ifo Business Climate Index US: Michigan Consumer Sentiment Index, Conference Board Index...

26 ZEW Index of Economic Sentiment, , Monthly data, index = #ositive - # negative, ca. 350 resondents

27 Extensions of Baseline Model introduction of exogenous variables (industrial roduction, interest rates, unemloyment, olitical variables, ) momentum effect endogenous N: effective number of indeendent agents Ut 0 xt 2IP 3( xt xt )

28 v α 0 α α 2 α 3 N ML AIC Model (baseline) 0.78 (0.06) 0.0 (0.0).9 (0.0) official: 350/ Model (end. N) (0.07) (0.06) (0.4) (9.87) Model (feedback from IP) (0.06) (0.07) (0.6) (2.53) (8.78) Model (moment.) (0.05) (0.06) (0.4) (0.76) (9.63) Model (mom. + IP) (0.05) (0.06) (0.6) (.65) (0.8) (8.95)

29 ... a few simulations of model V (identical starting value of x, identical influence from IP

30 For comarison: simulations of model I (identical starting value of x) -> no similarity

31 Secification tests: Mean and 95% confidence interval from model 3 (conditional on initial condition and influence form IP)

32 Mean and 95% confidence interval from model

33 are the large shifts of oinion in harmony with the estimated model? 95% confidence interval from eriod-byeriod iterations (model V) (conditional on revious realization and influence form IP)

34 AC F 0,9 0,8 0,7 0,6 0,5 0,4 0,3 Model Model 2 Model 3 Model 4 Model 5 Data 0,2 0, lag Autocorrelations: Data vs Simulated Models (average of 000 simulations)

35 Interim Conclusions evidence for interaction effects in ZEW index (α ) effective system size < nominal size (degree of comlexity) some (limited) evidence of interaction with macro data interaction effects are dominant art of the model we can identify the formation of animal sirits and track their develoment similar finding for various other Euroean business climate indices (Ghonghadze and Lux, 20), and investors sentiment in stock market (Lux, 20)

36 New question: Can we identify network structures that exlain the low effective no. of agents? Are some agents more imortant than others for the survey outcome, do they dominate the rocess of social oinion formation (core agenst vs. erihery)? These questions can be addressed using the micro data of the ZEW survey Identification of a Core-Perihery Structure Among Particiants of a Business Climate Survey (with U. Stolzenburg), Euroean Physical Journal (in ress)

37 Selected Survey Questions Business climate, current situation, Germany 7 Business climate, 6 month exectation, Germany 3 Inflation, Germany 9 Short run interest rates, Germany 25 Long run interest rates, Germany 3 DAX 32 DOW JONES 37 Exchange rate Euro - US$ The correlation atterns in individual answers are used as inut for our identification of a core-erihery network

38 A look at resonse atterns: Q7 on 6 months exectations vertical axes: 372 exerts who articiated at least 60 times horizontal axes: 96 months, 2/99 03/2008 Green: +, Yellow: 0 Red: -, Blue: no articiation -> months, 2/99 03/2008 Data set reduction: deleting those resonents with only limited temoral artciation leaves a total of 86 resondents

39 Basic information: Correlations between exerts resonses to Q7 over 96 months

40 Core-Perihery Network Analysis I Discrete model: c i = core membershi of agent i, c i = 0 or In order to determine the core, we solve the following roblem (cf. Borgetti and Everett, Soc. Networks 999) maxcorr( c vec( A ),vec( P )) P Data matrix: emirical correlations Pattern matrix: P = [δ i ] = c*c δ i = if c i = and c =, 0 otherwise

41 Otimization through genetic algorithms Resulting ordering with 23 core agents

42 Variation: Introduction of enalty term allows to maniulate the size of the core

43 How robust are the cores? Correlations of otimized binary core membershi vectors between different survey questions Qu ** 0.29** ** * ** 0.75* 0.329** 0.68* * 0.74* ** ** 0.55* * 37 *: Close to critical values of bootstra distribution; roughly significant at 5% **: Far beyond critical values; highly significant -> tyically large overla in cores

44 Core-Perihery Network Analysis II Continuous model: c i = real-valued coreness ( core roximity ) of agent i In order to determine the core, we again solve the following roblem: maxcorr( c vec( A ),vec( P )) Data matrix: emirical correlations Pattern matrix: P = [δ i ] = c*c

45 Otimization via Nelder-Mead algorithms Resulting ordering of agents according to coreness

46 How robust are the cores? Correlations of otimized continuous coreness vectors between different survey questions Qu ** 0.268** 0.256** 0.280** ** 0.273* 0.509** 0.30* * 0.35* * ** ** * 37 tyically large correlation in degrees of core membershi results are very close, even quantitatively, to those for discrete core

47 Are core members different? Reconstructed indices using coreness weights

48 A bootstra test:,000 random relications of business climate with 40 randomly drawn core or erihery members Distributions show correlations between reconstructed indices and raw data fro erihery (blue) and core (red) Null of equal means is reected with t-statistics > 60

49 Forecast accuracy: How well did core/erihery members forecast direction of business conditions?

50 Binary (+, -) success rates

51 Success rates for agents considered for network analysis

52 Success rates for core/erihery Null of equal means is reected with t-statistics > 6

53 Summary and Conclusions we have conducted a data-analytical study of hyothetical core-erihery network sructures among survey articiants both discrete and continuous C/P models yield lausible core structures cores or coreness vectors are strongly correlated across questions core members more highly correlated with overall index, erihery adds noise some indication of better forecasting erformance of core

54 Conclusions and Outlook some evidence for existence of core grou of survey articiants interretation: are the core members better informed or are they ust oinion leaders? Future research: can one imrove macro forecasts by concentrating on core agents and reduce noise by eliminating the erihery?