Honours Finance Advanced Topics in Finance: Nonlinear Analysis

1
Honours Finance (Advanced Topics in Finance
Nonlinear Analysis)

• Lecture 6 Dual Price Level Hypothesis continued
• Introduction to Chaos

2
Government and Reflation

• Real model with government confirms Minsky on
  Big Government
• Anti-cyclical spending and taxation of government
  enables debts to be repaid
• Renewal of cycle once debt levels reduced
• What about the full Monty?
• Cyclical economy
• Debt
• Prices
• Government
• A more complex picture still...

3
Price dynamics

• One essential aspect of Keyness analysis wage
  bargain set in nominal wage
• Easily introduced into Goodwin model
• Phillips curve is now with respect to money
  wage

• Income distribution dynamic now includes
  inflation

• Workers share of output will grow if wage
  demands exceed the sum of inflation and
  productivity growth

4
Price dynamics

• Next issue how to model prices
• Price equals marginal cost logic is out!
• Simplest alternative Kaleckian markup pricing

• But other aspects of reality worth introducing
• Time lags in price setting (Kydland Prescott,
  Blinder et al., estimate at 1 year)
• Counter-cyclical markups (unexpected Kydland
  Prescott finding)
• Covering time lags
• Same logic as shown in Advanced Political Economy
  lectures on modelling Circuitist theory
• Exponential decay form with length of lag shown
  as inverse

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Price dynamics

• Time lag tp indicates how long before prices
  react
• Expression in brackets gives level once rate of
  change of prices equals zero

• In Goodwin-Minsky system, can be reduced to

• So inflation p now is

6
Government and Reflation with Prices

• A still more complex six dimensional model

• Model basically replicates the post-WWII economic
  record
• No Depression, but
• cycles, inflationary surges
• growing government deficit

7
Employment and Wages Share

• Complex aperiodic cycles
• Period 10-20 years, vs 4 in non-price model

8
Debt Ratio and Price Level

• Inflation counteracts tendency to accumulate debt

• Deflation avoided by countercyclical government,
  but...

9
Net Government

• Government runs an increasing deficit to restrain
  tendencies towards debt-deflation
• (assuming governments maintain a Keynesian
  approach!)

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From Chaotic Limit Cycle to Strange Attractor
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From Chaotic Limit Cycle to Strange Attractor
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From Chaotic Limit Cycle to Strange Attractor
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System Dynamics...
Increasing employment
Falling employment
Rising employment
Rising wages share
Falling debt ratio
Rising price level
And rising debt ratio
Deficit falls tocounterbalance
Deficit rises tocounterbalance
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Conclusion

• Dual Price Level Hypothesis
• Explains
• 19th century trade cycle and frequent Depressions
• 20th century avoidance of Depressions but
  finance-driven cycles and countervailing
  government deficits
• Counters conventional analysis
• Finance (via debt) has macroeconomic impact
• Government countercyclical policies needed to
  counter tendency to debt-deflation
• but system still highly unstable
• Inflation may contribute to avoidance of
  Depression
• Interest-rate based stabilisation policy may
  destabilise via debt amplification effects

15
Complex behaviour from simple rules

• Complex behaviour of model contradicts
  conventional economics/finance beliefs about
  modelling
• Aperiodic cycles can be explained by
  deterministic system
• Long run of model need not be equilibrium
• Model exhibits complexity (modern alternative
  term for chaotic)
• Some basics on complexity...

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And three means chaos

• Original Goodwin two dimensional system had
  peculiar characteristic Conservation law
• we start with

• Work out dl/dw

• Put in separable form

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And three means chaos

• Now integrate

• Constant difference between two integrals
  conserved by system
• In practice means system constrained to a closed
  loop in (l,w) space or phase plane
• Special case of general property of 2-dimensional
  systems
• No time paths of solution to deterministic
  ODE/system of ODEs can intersect
• Systems either diverge or converge to/from point
  or limit cycle. One of most complex examples Van
  der Pols equation

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Nonlinearity (but not Chaos)

• Van der Pols equation

explains cycles in electric current in vacuum
tubes which the linearised equations
cant explain
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And three means chaos

• But behaviour of trajectories constrained by
  2-dimensional, horizontal plane on which dynamics
  occur
• If trajectories cant intersect, then most
  complex behaviour which can occur is limit cycle
• However with three or more dimensions, trajectory
  in 3D space can be infinitely complex
• chaotic long-term solutions, periodic orbits
  (limit cycles in 3D), point attractors
  (equilibria), divergence
• For discontinuous systems (difference equations),
  chaos can occur even in 1D for non-invertible
  maps
• All high order ODEs and some second order can be
  converted into systems of 3 first order ODEs

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Brief introduction to Chaos/Complexity

• An infinitely complex field (though not the same
  as Complex Analysis! this is the mathematics of
  complex numbers aib)
• see Gleik 87, Ott 93, Rosser 91, Lorenz 93,
  Brock 93
• Essential characteristics
• Analytically insoluble differential/difference
  equations
• Though many theorems about behaviour of solutions
  proven
• Analytic techniques applied to equilibria, global
  stability etc.
• Sensitive dependence on initial conditions and
  apparently random behaviour
• accuracy of prediction from current conditions
  degrades rapidly
• Far from equilibrium dynamic behaviour in long
  run

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Brief introduction to Chaos/Complexity

• Two broad classes of Chaotic/Complex systems
• Conservative (or Hamiltonian)
• Some physical attribute conserved in some sense
• System can be characterised geometrically as
  space-distorting while still maintaining the
  volume enclosed by a space
• Many physical process models conservative
• Dissipative (or non-Hamiltonian)
• No physical attribute conserved
• Geometric analogy no longer possible
• Most biological/economic models dissipative
• Dissipative models (unlike conservative ones)
  have attractors which can be stable/unstable,
  points/regions of space, etc.

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Brief introduction to Chaos/Complexity

• Much mathematical analysis of complexity done by
  converting continuous time systems (like DPL
  model) into discrete time by
• sampling at regular intervals
• mapping onto 2D surface
• Analysis of chaotic time series often done by
  mapping time series onto itself with a time lag
  (embedding)
• Time series generated by dissipative chaotic
  process has fractal (non-integer) dimension

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Fractal Dimension

• Dimensionality a crucial aspect of an entity
• How many dimensions do you occupy?
• (If you answer three, then youre saying you are
• dead...
• and not decaying)
• Most mathematical abstractions have integer
  dimensions
• Point 0
• Line 1
• Square 2, Cube 3, etc.
• How well do such abstractions describe real-world
  objects?

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Fractal Dimension

• Path followed by an ant
• Dimension 1, a line? Hardly at best numerous
  line segments
• Dimension 2, a square? No doesnt traverse every
  point in region between origin, end and
  side-deviations
• A mountain
• Dimension 3, a cube? No at best the sum of
  numerous mini-cubes
• Natural objects have fractional dimensions
• ants path between 1 and 2
• mountain between 2 and 3

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Fractal Dimension

• Dimensions of pure mathematical abstractions easy
• Dimensions of real world objects requires a
  concept of measurement
• what is the minimum size of something needed to
  completely describe the coverage of an object?
• One method the box-counting dimension (Ott
  69-71)
• How many squares of length e does it take to
  completely cover an object as e0?
• Formula is

• Consider for 2 points line solid shape
  (standard mathematical abstractions)

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Fractal Dimension

• For two points

Relatively tinynumber as e0compared to 1/e

• For a line (length l)

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Fractal Dimension

• For a closed curve (Area A)

Relatively tinynumber as e0compared to 1/e

• How about a non-integer object?
• Example the Cantor set
• Take line length 1 and remove middle third
• repeat process with each new line so created
• How many (1-dimensional) boxes needed to cover
  it?

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Fractal Dimension

• Cantor Set

• At each stage (n) of construction
• box of length (1/3)n needed to contain each
  segment
• 2n boxes needed

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Fractal Dimension

• Working this out

• So Cantor Set has fractional (Fractal) dimension
• Similar analysis applied to Finance data
• if generated by deterministic (as well as
  stochastic) processes then will have fractional
  dimension

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Nonlinear analysis of complex systems

• To date weve shown
• trade cycle can be modelled as nonlinear
  dynamical system
• nonlinear dynamical systems can generate time
  series which appear random
• actual time series will have both deterministic
  and stochastic processes behind them
• It can be argued therefore that financial time
  series are (in part) driven by deterministic
  nonlinear (in part) self-referential processes
• Can we (approximately) recreate the nonlinear
  process behind the time series, and thus (for a
  short while) predict its future course?

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Nonlinear analysis of complex systems

• Yes (sort of) but
• very difficult to distinguish deterministic chaos
  from random noise
• reconstruction not a proper model as in DPL
• or even regression equation as in (linear)
  econometrics

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Uncovering determinism in randomness

• Recapping
• trade cycle can be modelled as nonlinear
  dynamical system
• nonlinear dynamical systems can generate time
  series which appear random
• actual time series will have both deterministic
  and stochastic processes behind them
• It can be argued therefore that financial time
  series are (in part) driven by deterministic
  nonlinear (in part) self-referential processes
• Can we (approximately) recreate the nonlinear
  process behind the time series, and thus (for a
  short while) predict its future course?

33
Uncovering determinism in randomness

• Yes, with reservations
• difficulty of distinguishing randomness from
  chaos
• random numbers generated by computers are in
  fact produced using deterministic programs
• same seed number generates identical stream of
  numbers
• real data from any given dynamic system an
  overlay of endogenous (mainly deterministic) and
  exogenous (stochastic) forces
• any social system will have input of human
  decision-making
• rapid loss of information in any complex/chaotic
  system (over to logistic_error.mcd)...

34
Techniques to uncover structure

• Two key methods in vogue (but also infancy)
• Neural networks
• Genetic Algorithms
• Neural networks
• based on analogy to structure and (presumed)
  function and methodology of brain
• Human brain has about 100,000,000,000 neurones
• Each neurone has about 1000 connections with
  other neurones
• Inputs from 1000 input neurones determine whether
  and how much a neurone will fire
• Brain a complex network of neural structures...

35
Neural Networks

• Computer analogy to brains neural network
  incredibly simplistic by comparison
• one set of inputs, one input per piece of sensory
  data
• one hidden layer of neurones, the brain of
  network
• one layer of outputs, one per relevant output
• (normally) each node in each layer connected to
  every node in next layer
• connection via nonlinear weighting functions
  (typically logistic curve) where weights (values
  of parameters for function) alterable by network

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Neural Networks
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Neural Networks

• Neural network trained on historical data (for
  time series prediction), object images (for
  recognition e.g. OCR), etc.
• Training alters parameter values of weights
  functions for all inputs
• Training continues until weights stabilise,
  predictions of network approximate to actual data
• Network then used in practice
• Predict next value of Dow Jones
• Recognise previously unseen fonts/handwriting
• Control movements of robotic welding arm

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Neural network behaviour

• Basic pattern is
• Network trained on existing data
• Shown combination of input data and desired
  output (numerically coded if necessary)
• Initially random parameters on weights functions
  generate incorrect outputs
• Errors used (in modified least squares manner)
  to alter weights
• Process continues until errors minimise/stabilise
• Network used to analyse new data

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The importance of being nonlinear

• Earliest form of neural net was perceptron
• Same basic architecture as today, but used linear
  weights functions
• Initial enthusiasm for concept waned when shown
  that perceptron could not solve exclusive OR
  (XOR) problem
• Given two binary inputs, produce an output of 1
  when one but not both of the inputs are also 1
• In truth table form

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The importance of being nonlinear

• Basic structure of perceptron is

• Inputs (1 or 0) are multiplied by weights to
  produce output
• Weights adjusted using least squares until output
  clearly distinguishes between input patterns
• Weights adjustment effectively moves location of
  straight line on a plane
• Consider example of AND

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The importance of being nonlinear

• Truth table is

• Perceptron problem is to find values for
  parameters for a line such that cases 1 to 3 can
  be distinguished from case 4.
• Straight line equation is

• Can we find values for a and b so that, for an
  arbitrary c, cases 1 to 3 lie on one side of the
  line, and case 4 on the other?
• No problem

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The importance of being nonlinear
Many lines will give valid result
This side produces output of 1
(1,1)
(0,1)
This side produces output of 0
(0,0)
(1,0)
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The importance of being nonlinear

• Lets look at XOR problem the same way

This side produces output of 1
(1,1)
(0,1)
No matter where you draw the line, you cant get
both red dots on one sideand both greendots on
the other
This side produces output of 0
(0,0)
(1,0)
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The importance of being nonlinear
This side produces output of 0
Nonlinear shapecan easilydistinguish theinputs
correctly
(1,1)
(0,1)
This side produces output of 1
(0,0)
(1,0)
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Genetic Programming

• Based on an analogy to evolution
• Evolution has produced most complex problem
  solving systems of all using simple operations
• crossover (sexual reproduction at DNA level)
• mutation
• under pressure of environmental selection
• where environment is in large measure a product
  of evolution itself (e.g., without impact of
  carbon dioxide to oxygen-fixing bacteria, Earths
  atmosphere would be mainly carbon dioxide)

46
Genetic Programming

• Computer analogy of evolution
• produces population of randomly varying
  programming fragments (If, While, And, Or etc.)
• Assesses ability of each program to replicate key
  feature of data
• e.g., take economic data and reproduce change in
  exchange rate
• Least Fit programs deleted
• Fittest programs reproduced with code-swapping
  (crossover) and random alteration (mutation)
• System run for many generations to select best
  program(s)

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Genetic Programming

• Hypothetical program fragment

• Many such fragments tested against data
• Those with better fit selected for breeding
• crossover of one half of one program with another
• random alteration to program
• Runs continue till fits to data stabilise