Chaos and Complexity Courses for Spring 2001, UW-Madison


Remark 1:  This is a major revamping of 606 over past  606's  except  possibly
last year's.  Hence, if you have had 606 before, and you would  like  to  take
this course for credit, even  though  you may have "taken it before,"  I  will
sign the necessary paper  to  do  this  for you.

Remark 2:  The topics below seem disparate and unrelated.  I shall show how  a
few analytical principles organize the lot.  There  will  be  some  similarity
between the material of previous 606 courses but 606 S2001 will  contain  much
more work on dynamical  systems  approaches  to  learning  and  to  design  of
experiments    following    recent    work    by     CeNDEF     in     Holland
(,  as  well  as  recent  work  on  systems  with
multiple time scales and multiple "spatial" ("space"  is  widely  interpreted)
scales, as well as a detailed contrast and comparison of different methods  of
presenting "stylized facts."  For example in much of  natural  science  it  is
popular to present facts in the form of "scaling laws" but in  social  science
it is  popular  to  present  facts  in  the  form  of  conditional  predictive
     We shall also discuss complex systems approaches to the analysis of  time
series data and of panel data in an attempt to separate  "spurious"  "spatial"
and temporal dependencies from "true" dependencies that have  some  notion  of
"multiplier."  There will also be  more  attention  paid  to the  relationship
between dynamical systems phenomena such as bifurcations and jumps  caused  by
presence  of  "multipliers"  to  assist  identification  of  such  "endogenous
interactions" than recent work on identification of self selection effects and
treatment effects which was treated in years before.  This will be a  blending
of stochastic dynamical systems approaches with the review for the HANDBOOK OF
ECONOMETRICS by Brock and  Durlauf.   Researchers  such  as  N.  Bockstael  of
University of Maryland,  E.  Irwin  of  Ohio  State  (see  the  dramatic  maps
generated by simulated urban/suburban interacting systems models compared with
actual on E. Irwin's website) have recently taken interactive  systems  models
towards exciting empirical applications.  We shall  cover  some  of  this  new

Remark 3:  The topics below have become very  popular,  not  only  because  of
their  intrinsic  interest,  but  also  because  of  the   recent   entry   of
"establishment" figures.  The popularity is projected to  increase  even  more
due to recent empirical applications to issues of high political salience such
as the control of urban sprawl.  The purpose of this course is  to  bring  our
students to the research frontier as well as to inform our students of  recent
empirical applications as well as to suggest open research problems.


     Since much of the material that I have taught before in  this  course  is
available elsewhere (Examples:  Dynamic Programming is taught  in  first  year
macroeconomics, Stochastic Calculus and Stochastic Optimal Control  Theory  is
taught in the Business School and the Math Department, Game Theory  is  taught
by other courses here) I keep revamping this course to  teach   material  that
is not so easily available elsewhere  on   campus.    I   will   teach   newer
methods that have become popular in recent years.  I list  these  methods  and
topics below.  The emphasis  in  teaching  the  methods  will  be  to  isolate
potential PhD thesis topics.  More will be said about this below.
     Some major writers and/or sources are  included  in  parentheses  beneath
each topic.  However, these  will  be  very  incomplete  because,  during  the
course, I shall make up lists of current papers and their website locations in
order to build the good research habit of drawing up a list of  high  priority
websites and the havit of continuously monitoring these in order to keep up in
today's fast moving research environment.
     This course will be accessible to students with preparation at the  level
of "mature" first year graduate students in economics and business.  Hence,  I
am choosing a mathematical level to widen  the  accessibility  of  the  course
compared to the past.  I will also attempt to make the  course  accessible  to
students in other disciplines such as statistics, physics,  biology,  ecology,
limnology, etc.
     Since much of the material for the course  consists  of  current  working
papers as well as many published papers, I will make much use of websites  and
other internet resources.  I shall  teach  my  own  favorite  internet  search
methods for framing a research topic and projecting potential  value-added  of
the proposed topic before investing time on it.  While this might seem  banal,
it is surprising how many people fail to do this and end up  re-inventing  the
wheel.  Students here have too many time committments to have  their  research
time wasted on projects that have already been done by someone else.
     This course will represent a good hunting  ground  for  potential  thesis
topics because the new methods can be  applied  to  many  different  areas  of
economics and finance.  I shall try to outline  potential  topics  during  the
lectures.  For other students, the course will give a tour of some interesting
scenery on the research frontier of economics.  The course is also designed to
be useful to advanced undergraduate students that are  contemplating  academic
research careers.
     The unifying concepts and tools of the course  will  be:  (i)  stochastic
dynamical systems theory, (ii) self-organization  theories  of  the  Santa  Fe
Institute variety, and some of the  tools  being  used  by  papers  posted  on
websites such as "econophysics," "the New England Complex Systems  Institute,"
and  many  more,  (iii)  econometric  methods   that   stress   heterogeneity.
Interesting material may be found by simply doing a  netscape  search  on  the
words "econophysics,"  "complex  systems,"  "genetic  algorithms,"  and  other
jargon words from "complexity theory."
     The Santa Fe Institute website

is a good place to start looking at complex systems materials.
     For applications of complex adaptive systems ideas and related  ideas  to
ecological economics, see Carpenter, S., Brock, W.,  Hanson,  P.,  "Ecological
and Social Dynamics in Simple Models of Ecosystem Management," SSRI W.P. 9905,
for ecology.  For some mathematical tools and their applications, see,  Brock,
W., "Some Mathematical Tools for Analyzing  Complex-Nonlinear  Systems,"  SSRI
W.P. 2020.  SSRI Working Papers are available on the Sixth Floor in  the  SSRI


I.  Recent  econometric  and  theoretical  modelling  of  Increasing  Returns,
Threshold Effects, Interaction Effects.

(Anderson, Arrow, Pines, (1988), THE ECONOMY AS AN  EVOLVING  COMPLEX  SYSTEM,
Addison Wesley: Redwood City, CA.  Brock, W., (1993), "Pathways to  Randomness
in the Economy: Emergent Nonlinearity and Chaos  in  Economics  and  Finance,"
SSRI Reprint 410; Brock, W., (1991), "Understanding Macroeconomic Time  Series
using  Complex  Systems  Theory,"  SSRI  Reprint  392.  Manski,  C.,   (1993),
"Identification Problems in the Social Sciences," SSRI  Reprint  409.  Manski,
C., "Dynamic Choice in Social Settings," SSRI Reprint 408.  A main  source  of
recent work is Arthur, Durlauf, and Lane, eds.,  (1997),  THE  ECONOMY  AS  AN
EVOLVING COMPLEX SYSTEM II, Addison Wesley: Redwood  City,  CA.).   Brock  and
Durlauf "Interactions-Based Models," SSRI W.P. 9910.)

     This material will be taught and used in  a  rather  different  way  this
year, than in past years.  See the discussion below.

II.  Neural Nets, Connectionist Networks, Bootstrapping,  Surrogate  Data  and
their relationship to other received methods in econometrics such as nonlinear
least squares.

(Sullivan, Timmerman, and White,  (1998),  "Data-Snooping,  Technical  Trading
Rule Performance and the Bootstrap," Department of  Economics,  UCSD  and  LSE
Finance),  Casdagli,  M.,  Eubank,  S.,  (1990),   NONLINEAR   MODELLING   AND
FORECASTING, Addison-Welsey: Redwood City, CA. Work of Halbert White and   his
students at UCSD on estimating neural nets  using  "Robbins-Monro"  procedures
and  nonlinear  least squares.  Similar  methods  are  used  in  the  adaptive
learning literature below.)

     This year's  606  will  also  discuss  the  problem  of  controlling  for
data-snooping in  econometric  methodology  in  general  as  well  as  in  the
application of bootstrap-based specification tests (cf.  Sullivan,  Timmerman,
and White).  See also the discussion below on how this material will be taught
and used this year.

III.  Self Organized Criticality Models

(Bak,  P.,  Chen,  K.,  Scheinkman, J.,  Woodford,  M.,   (1993),    RICHERCHE
ECONOMICHE.  Krugman, P.,  (1995),  THE  SELF  ORGANIZING  ECONOMY.   Purpose:
Attempt to explain the evidence for long dependence in economic and  financial
data stressed by Mandelbrot and others.   Recent  articles  on  SOC  that  are
mentioned in Per Bak's new book, HOW NATURE WORKS, Springer 1996 will also  be
covered.  See also the ECONOPHYSICS website for much material on  SOC  applied
to economics.  See  especially  the  joint  work  of  Martin  Shubik  of  Yale
Economics with Per Bak of Physics.)

     This year's emphasis will be to review high points of the work posted  on
ECONOPHYSICS websites as well as other related websites such as the  Santa  Fe
Institute from the point of view of assisting the construction of  econometric
structures as discussed below.

IV.  Econometric and Theoretical issues raised by  the  possible  presence  of
chaos and other forms of deep nonlinearity in  economic  and  financial  data.
The Problem of Detecting "Spurious" Nonlinearity in Data.

ECONOMETRICS,  December,  1992.   Barnett,  W.,  Gallant,  A.,  Hinich,    M.,
Jungeilges, J., Kaplan, D., Jensen, M., "A Single-Blind Controlled Competition
between Tests for Nonlinearity and Chaos," Washington  University,  St.  Louis
working paper.  See William Barnett's website at  Washington  University,  St.
Louis for many interesting papers and well as useful links.  Benhabib J., ed.,
(1992), CYCLES AND CHAOS IN ECONOMIC EQUILIBRIUM, Princeton University  Press:
Princeton, NJ.  Brock, W., Hsieh, D., LeBaron, B., (1991), NONLINEAR DYNAMICS,
Press:  Cambridge,  MA.   Granger,   C.,  Terasvirta,  T.,  (1993),  MODELLING
NONLINEAR  ECONOMIC  RELATIONSHIPS,  Oxford  University  Press:   Oxford.   De
Grauwe,  P.,  Dewachter,  H.,  Embrechts,  M., (1993), EXCHANGE  RATE  THEORY:
CHAOTIC MODELS  OF  FOREIGN  EXCHANGE  RATES,   Basil  Blackwell:  Oxford.   A
challenge to this literature is posed by Bickel and Buhlmann, (1996) "What  is
a Linear Process?"  PROC.NAT.  ACAD.  SCI.  USA,  Vol.  93,  pp.  12128-12131,
December.  BB argue that the closure of the set of  ARMA  processes  "under  a
suitable metric" is "unexpectedly large"  (Caution:   This  is  NOT  the  Wold
representation).  Further work on this  problem  should  be  at  Peter  Bickel
(Berkeley Statistics) and Peter Buhlmann's websites.)

     This area has grown rapidly.  I shall pick highlights, teach the  basics,
and show what still needs to be done.  New  work  that  has  become  available
recently will be covered.  The emphasis will be to  inform  students  on  what
research problems are still open in this area and how it relates to the recent
surge of interest in modelling "bounded rationality" and "process  approaches"
to economics rather than "equilibrium" approaches.

V.  Complex Systems Modelling and Scaling "Laws"

(Stein,  D.,  (1988),  ed.,  LECTURES   ON   THE   SCIENCES   OF   COMPLEXITY,
Addison-Wesley: Redwood  City,  CA.   Brock,  W.,  "Scaling  in  Economics:  A
Reader's Guide,"  SSRI Reprint. Blake LeBaron's website at Brandeis


LeBaron, B., 1999,  "Volatility  Persistence  and  Apparent  Scaling  Laws  in
Finance,"  (available  at  LeBaron's  website).  ECONOPHYSICS   website   (see
especially the links to "minority games."))

     Examples of Scaling "Laws" in economics and finance: (i) Gibrat's Law  of
firm size distribution, (ii) logistic "laws" of growth  and  diffusion,  (iii)
Pareto's  Law  of  income  distribution,  (iv)  Mandelbrot's  "self   similar"
stochastic processes and "1/f" scaling  in  economics  and  finance,  (v)  the
stylized facts of  finance  such  as  autocorrelation  structure  of  returns,
volatility measures, and volume measures across individual stocks and indices,
(vi)  the  stylized  autocorrelation  and  cross  correlation   structure   of
aggregative and less aggregated macroeconomic time series.
     An attempt will be made to show what useful insights can be learned  from
locating  scaling  laws  and  how  to  correct  for  improper   treatment   of
heterogeneity.  In particular we will stress  how  "spurious"  "unconditional"
scaling "laws" can easily be produced from a system of  individual  stochastic
processes relaxing to different stochastic  steady  states  (even  though  the
relaxation rate is the same for each process).  This exercise will stress  the
importance of correctly controlling for heterogeneity.   Scaling  laws  appear
also in ecology and we will teach some of this material and draw lessons  from
it for econometric practice.

VI.  Adaptive Learning

(T. Sargent, (1993), BOUNDED RATIONALITY IN MACROECONOMICS, Oxford  University
Press.  Chen, X., White, H., (1994),  "Nonparametric  Adaptive  Learning  with
Feedback," UCSD Working Paper.  Holland, J., (1992), ADAPTATION IN NATURAL AND
ARTIFICIAL  SYSTEMS,  MIT  Press:  Cambridge,  MA.   CeNDEF   experiments   on
expectation formation as well as other CeNDEF research on bounded rationality.
Fudenberg/Levine's book,  THE  THEORY  OF  LEARNING  IN  GAMES  (1998),  Larry
Samuelson's book on evolutionary games, Peyton Young's book  on  evolution  of
conventions in games,  ECONOPHYSICS  website  (see  especially  the  links  to
"minority games"), R.  Selten's  lab  on  strategy  experiments  in  oligopoly
theory, CeNDEF work on strategy experiments in other types of games.)

     The basic first year courses  say  little  about  dynamics  and  adaptive
learning towards a notion of "equilibrium."  For example, Selten's lab at Bonn
has recently shown that optimization  appears  to  play  no  role  at  all  in
repeated oligopoly games (i.e. finite horizon supergames) with  small  numbers
of players. Rather something somewhat like Axelrod's  TIT-FOR-TAT  emerges  as
players evolve "ideal points" and induce play towards  them  by  "measure  for
     First year courses say even less about any kind of  socially  interactive
learning on any  kind  of  network or Selten-like behavior of  players  trying
to "train" each other towards a more cooperative outcome.
     Since, much of economics is based on equilibrium  concepts  which  impose
restrictions   on   data  which  can  be  tested  and  since  introduction  of
"disequilibrium"  concepts  such as adaptive learning introduces  extra  "free
parameters," this imposes an  even higher priority to discipline theorizing by
data than usual.
     Researchers here, at the Santa Fe Institute, and other  research  centers
are trying to carry out this kind of research program consistent with observed
"scaling laws" and observed estimated conditional distributions  in  economics
and finance.  We shall cover the basic methods  and  highlights  of  this  new
literature.  We shall also review experimental results.   For  example  CeNDEF
has been using  strategy  experiments  (originating  from  Selten's  work)  to
produce a set  of  stylized  regularities  about  the  expectations  formation
process which is separated from other aspects of the game  (such  as  strategy
involved via sharing a market as in oligopoly games) via a special  design  of
the experiment to "control-out" all other expects of the game except  for  the
expectation formation process itself.
     Research on the  "El  Farol"  problem  (called  the  "minority  game"  by
physicists) has documented a "phase transition" and a "scaling law" (cf.  work
on the ECONOPHYSICS website by Robert Savit of the University of Michigan  and
many others).  The parameters are "s" the  "size  of  brain"  of  each  player
(measured by the size of the strategy set available to each player), the "size
of the universal brain" (measured by the size of the universal set Omega(m) of
potential strategies that could be played) and memory  "m"  (measured  by  the
number of lagged observations allowed to be in each prediction function  which
describes each forecasting strategy).  The focus of the CeNDEF group is on the
dynamic evolution of adaptive forecasting systems whereas the focus  of  Savit
et al. is  on  uncovering  "scaling"  relationships  and  evidence  of  "phase
transitions" via computational experiments.  There  is  also  analytical  work
reported on the ECONOPHYSICS website.
     We shall spend  some  time  comparing  and  contrasting  these  different
approaches to the modelling of adaptive learning as well as learning  what  we
can from results reported from laboratory experiments around the  world.   The
emphasis of this part of the course will be  to  develop  model  systems  that
replicate experimental results,  but  at  the  same  time  develop  analytical
methods for general use in this area.
     Development of methods from  natural  science  in  searching  for  useful
"order parameters" to uncover  "phase  transitions"  and  "scaling  laws"  and
relating these to "scaling laws" from sampling theory in statistics  (such  as
central limit  theorems,  Edgeworth  expansions,  large  deviations  "scaling"
relations, breakdowns of  central  limit  theorems  due  to  series  of  cross
correlations diverging) will be stressed.
     We will stress the incentive  differences  inherent  in  "small  numbers"
adaptive (or other) "learning" situations of  repeated  play  in  contrast  to
"large numbers" situations  of  repeated  play.   Selten's  lab  stressed  the
inherent incentives of repeated "small numbers" play to "train" each other  to
reach a cooperative outcome.  Such incentives will get smaller as  the  number
of players increases because  each  player  will  be  increasingly  unable  to
capture the benefits of her own "training efforts"  onto  the  other  players.
This relates to work on "evolution of norms and conventions" in Peyton Young's
     As one varies  the  number  of  players,  the  memory  allowed  in  their
strategies, the size of their individual strategy sets and  the  size  of  the
size of all potential strategies of fixed memory as well as other quantifiable
aspects of the game the "order parameter" approach suggests  looking  for  key
"order parameters" such that when an order  parameter  increases,  the  system
goes through an abrupt change in  dynamical  behavior  (a  "phase  transition"
and/or a "bifurcation").  Analysis of continuous state space dynamical systems
of increasing size creates a demand for analytic  results  on  eigenvalues  of
dynamical systems of increasing size.  Alan  Edelman's  website  at  MIT  Math
contains very nice papers on this problem  (e.g.  "circular  laws")  which  we
shall discuss.
     Since we will be on new ground here, this should be an exciting  part  of
the course.

VII.  Cellular Automata, Ising Models, Spin Glass Models

(Mezard, M., Parisi, G., Virasoro, (1987), SPIN GLASS THEORY AND BEYOND, World
Scientific.  Durlauf, S.  (1993),  REVIEW  OF  ECONOMIC  STUDIES  and  various
working papers.  Mitchell, M., Crutchfield, J., Hraber, P., (1994),  "Evolving
Cellular Automata  to  Perform  Computations:   Mechanisms  and  Impediments,"
PHYSICA D, 75, 361-391.  Doyon, B., Cessac,  B.,  Quoy,  M.,  Samuelides,  M.,
(1993), "Control of the Transition to Chaos in  Neural  Networks  with  Random
279-291. Material on eigenvalues of large systems from Alan Edelman's  website
at MIT Math.  Material from Jim  Crutchfield,  Melanie  Mitchell,  and  others
available by linking from the Santa Fe Institute's website.)

     This material will give math modules from which we can  build  models  of
adaptive interaction and parse out the components due to socially  interactive
learning from "plain vanilla" adaptive expectations formation and other kinds
of "individualistic" adaptation.  Much in the style of Peyton  Young's  book's
approach to recovering results from  "common  knowledge  ultra  rationalistic"
game theory via adaptation we shall take  a  related  approach  to  recovering
results from rational expectations  theory.   Our  posture  will  be  somewhat
different however.  It will be guided by a  desire  to  formulate  econometric
frameworks where tools like  the  Efficient  Method  of  Moments  (cf.  George
Tauchen's website at Duke) and Computational Bayes (cf. John Geweke's  website
and his paper,  "Computational  Experiments  and  Reality"  available  at  his
websites at Minnesota and Iowa along with software  available  there)  can  be
used to measure the "statistical significance" of the "extra free  parameters"
brought by adaptive theory.  Emphasis will also be placed upon econometrically
separating interactive effects from empirically similar looking effects due to
correlated unobservables and other phenomena.

VIII.  Information Contagion,  Polya  Processes,  Cascades,  Self  Reinforcing
Mechanisms, Magnification Mechanisms of Income and Wealth.

(Arthur, W., (1988), "Self Reinforcing Mechanisms  in  Economics,"  in  Arrow,
Anderson, Pines,  op.cit.   Bikhchandani,  S.,  Hirshleifer,  D.,  Welch,  I.,
(1992),  "A  Theory  of  Fads,  Fashion,  Custom,  and  Cultural   Change   as
Informational Cascades," JOURNAL OF POLITICAL ECONOMY, 100 (5): 992-1026.   De
Vany, A., and Walls,  W.,  (1994),  "Information,  Adaptive  Contracting,  and
Distributional Dynamics:  Bose-Einstein Statistics and the Movies," University
of California, Irvine, Working Paper, recently appeared in  ECONOMIC  JOURNAL.
Rosen, S., "The  Economics  of  Superstars,"  AMERICAN  ECONOMIC  REVIEW,  71:

     This material relates naturally to the above  discussions  in  the  sense
that it lays out a variety of channels through which interaction  may  operate
in a dynamically evolving social system  as  an  economy.   Emphasis  will  be
placed on econometric identification of the  different  "observable  empirical
signatures" produced by each of these very different mechanisms of interaction
that may look the same to an econometric  exercise  if  it  is  not  carefully
formulated.  Formulation of econometric exercises to  differentiate  different
channels of interaction including "social learning," "informational cascades,"
"positional reward structures"  (e.g.  "tournament"  payoff  structures),  and
other related channels of possible interaction will take a very high  priority
in this year's 606.

IX.  "Process vs. Equilibrium"

     A common theme thoughout the above materials is moving thinking about the
economy away from "equilibrium" (even that of the stochastic process RBC  type
modelling) towards a view more like Artificial Life and John Holland's Complex
Adaptive Systems (cf. Leigh Tesfatsion's website, Tom Ray's TIERRA, The  Santa
Fe Artificial Stock Market, "Sugarscape" and other ALife frameworks) where the
system never settles down.  This kind of approach to economics can  be  viewed
as a modern form of Austrianism.  We  shall  try  to  develop  some  analytics
(rather like large system limits over a hierarchy of  "spatial"  and  temporal
scales) to complement the exciting computational work in this area.

X.   Bayesian  Model  Averaging  and  other  methods  of   appraising   "Model
Uncertainty".  Econometrics and Decision Theory:  New Approaches.

     The paper "Growth Economics and Reality" by Brock and Durlauf  (available
at the SSRI office and website) reviews this area and  applies  it  to  growth
econometrics.   We  will  discuss  some  of  these   methods   and   potential
applications for them in the course.

Chaos and Complex Systems Seminar Page