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Rational Expectations or Social Opinion Formation:

Rational Expectations or Social Opinion Formation: Identification of Interaction Effects in a Business Climate Index Thomas Lux University of Kiel International Workshop on Nonlinear Economic Dynamics and Financial Market Modelling, Beijing University, 9 – 10 October 2008. Two goals.

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Rational Expectations or Social Opinion Formation:

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  1. Rational Expectations or Social Opinion Formation: Identification of Interaction Effects in a Business Climate Index Thomas Lux University of Kiel International Workshop on Nonlinear Economic Dynamics and Financial Market Modelling, Beijing University, 9 – 10 October 2008 ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  2. Two goals • materially: covering dynamic interaction aspects in economic data • technically: developing an estimation methodology for models of interactive , distributed dynamic systems ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  3. The Importance of Social Interactions • Neighborhood effects like role models, externalities, network effects • Spillovers and externalities in spatial agglomerations • Technology choice (standards, network effects) • Direct influences of others on utility (conformity effects, fads, fashions) -> Föllmer, 1974 • Social pathologies (crime, school absenteeism etc.) -> discrete choice • Opinion formation through mutual influence • Macroeconomics: Animal spirits ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  4. Agent-based models with social interactions inspired by statistical physics (a time-honoured legacy) • Föllmer (1974): “Random economies with interacting agents” • Weidlich and Haag (1983): “Quantitative Sociology” • Schelling: “Micromotives and Macrobehavior” • Many econophysics models • Brock and Durlauf (2001): Discrete choice with social interactions ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  5. Available models are successful in explaining (financial) stylized facts, but have not been really implemented empirically Missing is: • estimation of the underlying parameters, • comparison of models, • goodness of fit. ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  6. Missing is also both a theoretical and empirical methodology for “psychological effects” in macroeconomics: • psychological factors are important in both consumption and investment • animal spirits are certainly not sunspots! ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  7. Empirical Estimation of Agent-Based Models • Gilli and Winker (2003, 2007):Estimation of Kirman model with a heuristic method, Alfarano, Lux and Wagner (2005): stationary ML estimation • Westerhoff and Reitz (2003), Diks and van der Weide (2002): Agent-based model leads to STAR or ARCH dynamics • literature on discrete choice with social interactions • empirical work on animal spirits? ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  8. Missing is a general approach to parameter estimation of agent-based models • Missing is both a theoretical and empirical methodology for “psychological effects” in macroeconomics • Here we introduce a general rigorous methodology for parameter estimation • Illustration: estimation of Weidlich model for economic survey data ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  9. ZEW Index of Economic Sentiment, 1991 – 2006, Monthly data, index = #positive - # negative, ca. 350 respondents ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  10. A Canonical Interaction Model à la Weidlich • Two opinions, strategies etc: + and – • A fixed number of agents: 2N • Agents switch between groups according to some transition probabilities w↓ and w↑ v: frequency of switches, U: function that governs switches α 0, α1: parameters Sentiment index ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  11. Remarks • The model is designed as a continuous-time framework, i.e. w↓ and w↑ are Poisson rates (jump Markov process) • The “canonical” model allows for interaction (via α1) and a bias towards one opinion (via α 0), but could easily be extended by including arbitrary exogenous variables in U • The framework corresponds closely to that of “discrete choice with social interactions”, it formalizes non-equilibrium dynamics, while DCSI only considers RE equilibria ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  12. Theoretical Results • For α1 ≤ 1: uni-modal stationary distribution with maximum x* =, >,< 0 (for α 0 =,>,< 0) • For α1 > 1 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) ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  13. Some simulations ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  14. ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  15. How to Arrive at Analytical Results? • Our system is quite complex: 2N coupled Markov jump processes with state-dependent, non-linear transition rates • Solution via Master equation: full characterization of time development of pdf: can be integrated numerically, but is too computation intensive with large population • More practical: Fokker-Planck equation as approximation to transient density ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  16. How to Arrive at Analytical Results? x in steps of 1/N Different formalisms for jump Markov models of interactions: • Master equation: full characterization of time development of pdf: can be integrated numerically, but is too computation intensive with large population • Fokker-Planck equation: Diffusion: 2*g drift:-μ ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  17. Fokker-Planck-Equation: expanding the step-operators for x in Taylor series up to the second order and neglecting the terms o(Δx2), we end up with the following FPE: ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  18. The Master equation or FPE can be used to derive quasi-deterministic laws of motion for moments, e.g.: Investigating the mean-value dynamics, one recovers the modes of the stationary distribution as stable fix points. ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  19. Estimation: for a time series of discrete observations Xs of our canonical process, the likelihood function reads with discrete observations Xs, the Master or FP equations are the exact or approximate laws of motion for the transient density and allow to evaluate log f(Xs+1|Xs,θ) and , therefore, to estimate the parameter vector θ (θ = (v, α0,α1)‘)! ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  20. Implementation • Usually no analytical solution for transient pdfs from Master or FP equations • Numerical solution of Master equation too computation intensive if there are many states x (i.e., particularly with large N) • Numerical solutions of FP equation is less computation intensive, various methods available for discretization of stochastic differential equations ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  21. Finite Difference Approximation Space-time grid: xmin + jh, t0 + ik forward difference backward difference ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  22. Numerical Solution of FPE • Forward and backward approximations are of first-order accuracy: combining them yields Crank-Nicolson scheme with second-order accuracy -> solution at intermediate points (i+1/2)k and (j+1/2)h • This allows to control the accuracy of ML estimation: estimates are consistent, asymptotically normal and asymptotically equivalent to complete ML estimates (Poulsen, 1999) ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  23. Observation Xs, approximated by sharp Normal distr. Evaluation of Lkl of observation Xs+1 Time interval [s, s+1] ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  24. Monte Carlo Experiments • Does the method work in our case of a potentially bi-modal distribution, is it efficient for small samples? • Do we have to go at such pains for the ML estimation? Couldn‘t we do it with a simpler approach (Euler approximation)? ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  25. Applicability to our famework: check the order of accuracy Denote by v1, v2, v3 the approximation errors from expansions using step sizes k and h, h/2, h/4 First order: the ratio is ~2, Second order: it yields ~4 ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  26. Procedure works as expected, even for bi-modality ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  27. Monte Carlo Study of MLE with Crank-Nicolson Approximation v =3, α0 = 0, α1 = 0.8 α0 = 0.2, α1 = 0.8 α0 = 0, α1 = 1.2 α0 = 0.2, α1 = 1.2  ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  28. Results from Experiments • Crank-Nicolson has expected order of accurary, works well in ML estimation • Modest number of time steps (k) sufficient for high accuracy • Implicit FDs have practically the same peformance as CN • ML estimation also works well with endogenous N ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  29. Empirical Application • The framework of the canonical model is close to what is reported in various business climate indices • Germany: ZEW Indicator of Economic Sentiment, Ifo Business Climate Index • US: Michigan Consumer Sentiment Index, Conference Board Index • .... ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  30. ZEW Index of Economic Sentiment, 1991 – 2006, Monthly data, index = #positive - # negative, ca. 350 respondents ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  31. ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  32. Extensions of Baseline Model • introduction of exogenous variables (industrial production, interest rates, unemployment, political variables,…) • ‘momentum’ effect • endogenous N: ‘effective’ number of independent agents ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  33. ... a few simulations of model V(identical starting value of x, identical influence from IP ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  34. For comparison: simulations of model I(identical starting value of x) -> no similarity ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  35. Specification tests: Mean and 95% confidence interval from model 3(conditional on initial condition and influence form IP) ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  36. Mean and 95% confidence interval from model 1 ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  37. …are the large shifts of opinion in harmony with the estimated model? 95% confidence interval from period-by-period iterations (model V)(conditional on previous realization and influence form IP) ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  38. data models 1 2 3 4 5 mean 0.352 -0.588 0.349 0.324 0.391 0.343 (95%) (-0.631,-0.405) (0.045,0.552) (0.035,0.528) (0.214,0.521) (0.173,0.479) std.dev: 0.370 0.098 0.355 0.384 0.314 0.356 (95%) (0.068,0.185) (0.251,0.484) (0.293,0.506) (0.245,0.391) (0.284,0.424) skewness -0.620 0.615 -0.908 -0.991 -0.947 -0.857 (95%) (0.029,1.685) (-1.880,0.097) (-1.878, 0.053) (-1.620,0.247) (-1.571,-0.181) kurtosis -0.428 0.575 0.591 0.540 0.804 0.207 (95%) (-0.652,3.717) (-1.322,4.347) (-1.283,3.296) (-0.911,3.585) (-1.000,2.315) rel. dev: 0.905 49.705 1.368 1.030 1.766 1.049 (95%) (8.706,82.360) (0.024,4.022) (0.022,2.729) (0.342,3.456) (0.209,2.439) distance 0.952 0.335 0.297 0.233 0.223 (95%) (0.899,0.968) (0.255,0.493) (0.220,0.455) (0.171,0.318) (0.165,0.311) Statistical Evaluation: Unconditional moments from 1.000 Monte Carlo simulations

  39. ~49.7 Moments: Data vs Simulated Models (average of 1000 simulations) ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  40. models ACF data 1 2 3 4 5 1 0.935 0.630 0.923 0.939 0.904 0.930 (95%) (0.456,0.963) (0.845,0.967) (0.908,0.968) (0.844,0.944) (0.890,0.955) 2 0.830 0.404 0.853 0.880 0.796 0.848 (95%) (0.162,0.929) (0.715,0.936) (0.820,0.934) (0.674,0.879) (0.769,0.901) 3 0.709 0.266 0.789 0.819 0.691 0.762 (95%) (0.013,0.890) (0.595,0.907) (0.732,0.900) (0.523,0.811) (0.653,0.845) 4 0.584 0.175 0.729 0.758 0.592 0.675 (95%) (-0.080,0.857) (0.496,0883) (0.652,0.8866) (0.393,0.751) (0.541,0.784) 5 0.465 0.116 0.673 0.696 0.499 0.589 (95%) (-0.133,0.820) (0.398,0.860) (0.566,0.833) (0.266,0.699) (0.432,0.723) 6 0.075 0.620 0.633 0.419 0.508 (95%) (-0.171,0.784) (0.319,0.840) (0.478,0.797) (0.169,0.638) (0.335,0.662) Autocorrelations from 1.000 Monte Carlo Simulations page 1/2

  41. models ACF data 1 2 3 4 5 7 0.272 0.048 0.571 0.571 0.355 0.434 (95%) (-0.188,0.747) (0.241,0.813) (0.392,0.759) (0.092,0.594) (0.250,0.616) 8 0.186 0.032 0.525 0.512 0.302 0.366 (95%) (-0.197,0.703) (0.184,0.793) (0.317,0.722) (0.049,0.553) (0.171,0.565) 9 0.094 0.022 0.482 0.454 0.251 0.298 (95%) (-0.213,0.668) (0.121,0.774) (0.293,0.678) (-0.013,0.516) (0.088,0.514) 10 0.017 0.014 0.442 0.398 0.196 0.228 (95%) (-0.220,0.631) (0.078,0.752) (0.176,0.640) (-0.084,0.468) (0.002,0.463) d 0.553 0.194 0.826 0.923 0.551 0.668 (95%) (-0.343,0.978) (0.338,1.261) (0.455,1.346) (0.027,0.992) (0.202,1.051) Autocorrelations from 1.000 Monte Carlo Simulations page 2/2

  42. Autocorrelations: Data vs Simulated Models (average of 1000 simulations) ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  43. Conclusions • evidence for interaction effects in ZEW index (α1 ≈ 1) • no significant bias, slow development (v small) • effective system size < nominal size (degree of complexity) • some (limited) evidence of interaction with macro data • interaction effects are dominant part of the model • we can identify the formation of animal spirits and track their development ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

  44. Avenues for further research • other time series: in economics, finance (investor sentiment), politics, marketing • estimating combined models with joined dynamics of opinion formation and real economic activity • check for system size effects: correlations in individual behavior (micro data) ? • indirect identification of psychological states from economic data ____________________________________________________________________________________________________________________________________________________________ Thomas Lux Department of Economics University of Kiel

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