Models Stochastic Models STAT 34534453P004, P453

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Models - Stochastic Models (STAT 3453/4453/P004,
P453)

• Shane Whelan
• L527

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Introductory Remarks
3
Introduction

• Timetable
• Tues. 11 Q014 Lecture
• Wed. 11 Q014 Lecture
• Wed. 2 Q014 Lecture
• Thurs. 11 TBA Tutorial
• (Tutotial shared with Models Survival Models)
• Comprising 3 lectures and 1 tutorial
• Exam in December 2005.

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Stochastic Models

• Course Objectives
• To provide a grounding in stochastic modelling,
  especially in actuarial applications.
• To gain exemption from CT4(part 103) in the
  Faculty Institute of Actuaries.
• Textbook/Reading Material
• Part of the Core Reading of Faculty Institute
  of Actuaries for Subject CT4 Modelsthat part
  designated CT4(103).
• Course Syllabus
• Actuarial Modelling?Fundamental Concepts in
  Stochastic processes ?Markov Chains ?Markov Jump
  Processes ? Simulation ?

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Stochastic Models Overview

• Percentage of Course
• Chapter 1 Introduction to (Actuarial)
  Modelling 10
• Chapter 2 Foundational concepts in Stochastic
  Processes 12
• Chapter 3 Markov Chains 30
• Transition Probabilities Chapman-Kolmogorov
  Equations Time-homogeneous Markov chains the
  Long-Term Distribution of a Markov Chain the
  Long-Term Behaviour of Markov Chains.
• Chapter 4 Markov Jump Processes 35
• Markov Jump Processes Kolmogorovs Forward
  Equations Kolmogorovs Backward Equations
  Time-homogeneous Markov Jump Process The Time
  Inhomogeneous Case The Integrated Form of
  Kolmogorovs Backward Forward Equations
  Applications
• Chapter 5 Simulation (of Stochastic
  Processes) 13
• Monte Carlo Simulation Pseudo-random numbers
  Linear Congruential Generators Generation of
  random variates from a given distribution -
  Inverse Transform Method, Acceptance-Rejection
  Method Special Algorithms -Box-Muller, Polar
  Generation of sets of correlated normal random
  variates How many simulations should we do?

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Course Notes

• Following each lecture, the slides are put up on
  web
• As are problem sheets, etc.
• My full website is at
• http//www.ucd.ie/statdept/staff/swhelan.html
• My boardwork (including all proofs and
  supplementary examples) are part of the course
  (i.e., examinable)
• Copy these down

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Chapter 1

• Introduction to (Actuarial) Modelling

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The Modelling Problem

• I have yet to see any problem, however
  complicated, which, when you looked atit in the
  right way, did not become still more
  complicated.
•  
•  
•  
• Poul Anderson, science fiction writer, in New
  Scientist. (London, September 25, 1969).

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Modelling

• Model a simple, stylised imitation of a real
  world system or process.
• Used to predict how process might respond to
  given changes enabling results of possible
  actions to be assessed or simply to understand
  how system will evolve in the future.
• Other methods being too slow, too risky, or too
  expensive.
• Objective of Model is paramount
• we need to know what is best model and this is
  generally not the most accurate model need to
  balance cost with benefits.
• e.g., macroeconometric model of economy
• Price of share at each future date
• Model life office

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Classifying Models

• Deterministic Model Unique output for given set
  of inputs. The output or inputs are not random
  variables.
• Stochastic Model Output is a random variable.
  Perhaps some inputs are also random variables.
• A deterministic model can be seen as a special
  case of a stochastic model.

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Stochastic Analysis
Stochastic Model of a System
Future Period
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Stochastic Analysis
Stochastic Model of a System
Future Period
10
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Components of Model

• Structural Part sets out the relationship
  between the parameters modelled (inputs) so as to
  determine the functioning of the system
  (outputs).
• Relationships are generally expressed in logical
  or mathematical terms.
• Complexity of model is determined by the number
  of parameters modelled and the form of
  relationship posited between them.
• Parameters the value of the inputs.
• Often estimated from past data, using statistical
  techniques.
• Also current observation, subjective assessment,
  etc.

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Building a Model

• Anything Goes

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Introduction to Real Modelling

• Perspective we attempt to get, well captured in
• Real Life Mathematics, Bernard Beauzamy, Irish
  Math. Soc. Bulletin 48 (Summer 2002), 4346.
• Available on Web from
• http//www.maths.tcd.ie/pub/ims/bull48/M4801.pdf
• Repays the 20 minute read!

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Quotes from Real Life Mathematics

• It is always our duty to put the problem in
  mathematical terms, and this part of the work
  represents often one half of the total work
• My concern is, primarily, to find people who are
  able and willing to discuss with our clients,
  trying to understand what they mean and what they
  want. This requires diplomacy, persistence, sense
  of contact, and many other human qualities.
• Since our problem is real life, it never fits
  with the existing academic tools, so we have to
  create our own tools. The primary concern for
  these new tools is the robustness.

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Building a Model 10 Helpful Steps

• Set well-defined objectives for model.
• Plan how model is to be validated
• i.e., the diagnostic tests to ensure it meets
  objectives
• Define the essence of the structural model the
  1st order approximation. Refinement and details
  can come later.
• Collect analyse data for model (and any other
  parameters)
• Involve experts on the real world system to get
  feedback on conceptual model.

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Building a Model 10 Helpful Steps

• Decide how to implement model
• e.g. C, Excel, some statistical package. Often
  random number generator needed.
• 7. Write and debug program.
• Test the reasonableness of the output from the
  model and otherwise analyse output.
• Does it replicate historic episodes reasonably
  well?
• 9. Test sensitivity of output to input parameters
• i.e., ensure small change to inputs has small
  affect on output.
• We do not want a chaotic system in actuarial
  applications.
• Communicate and document results and the model.
• 10.a Review and update in the light of new data
  and other changes.

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Advantages of Modelling

• Modelling can claim all the advantages of the
  scientific programme over any other logical,
  critical ,and evidence-based study of phenomenon
  that builds, often incrementally, to a body of
  knowledge.
• Complex systems, including stochastic systems,
  that are otherwise not tractable mathematically
  (in closed form) can be studied.
• It is quicker (system studied in compressed
  time), and less expensive than alternatives.
• Consequences of different policy actions can be
  assessed, so option can be selected that
  optimizes output.
• We can reduce variance of model as we can better
  control experimental conditions.

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Drawbacks of Modelling (that must be guarded
against)

• Requires considerable investment of time and
  expertise..not free.
• Often time-consuming to use many simulations
  needed and results analysed.
• Not especially good at optimising outputs (better
  at comparing results of input variations)
• Impressive-looking models (especially complex
  ones) can lead to overconfidence in model.
• Model only as good as parameter inputs quality
  and credibility of data.
• Must understand limitations of model (i.e., its
  proper use)
• Must recognise that a model will become obsolete
  change in circumstances.
• Sometimes difficult to interpret output.

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Quotes from Real Life Mathematics

• Most current mathematical research, since the
  1960s, is devoted to fancy situations it
  brings solutions which nobody understands to
  questions nobody asked. Nevertheless, those who
  bring these solutions are called distinguished
  by the academic community. This word by itself
  gives a measure of the social distance real life
  mathematics do not require distinguished
  mathematicians. On the contrary, it requires
  barbarians people willing to fight, to conquer,
  to build, to understand, with no predetermined
  idea about which tool should be used.

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Computers Modelling

• First generation of civiliation computers (say
  the UNIVAC computer) were used as calculators
  performing repetitive calculations
• First was bought by the Census Bureau, second by
  A.C. Nielson Market Research and the third by the
  Prudential Insurance Company.
• Second generation of computers (say the IBM 360
  series) were used as real time databases
• airline reservations processing inventory
  control insurance industry semi-automated its
  back office
• Subsequent generations have been used for, inter
  alia, design (CAD) or, put another way, modelling
  
• Cars and airplanes designed without wind-tunnels
  the next generations of computer chips model
  life offices, etc.

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Computers Modelling

• And since the PC and, argubly, the invention of
  the spreadsheet (first Visicalc, then Lotus
  1-2-3, and finally Microsoft Excel), we all have
  access to a user-friendly aid to modelling.
• The computer is revolutionising modelling.

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Classifying Models Another Look

• Deterministic Model Unique output for given set
  of inputs. The output is not a random variable.
• Gives one scenario.
• Simple systems can be solved for explicitly
  that is solution (output) is known in closed form
  a simple function, f(.).
• But often need numerical methods to solve.
• Stochastic Model Output is a random variable.
  Perhaps some inputs are also random variables.
• Gives multiple scenarios, weighted by
  probability.
• If possible, attempt at least a partial analytic
  solution it simplifies the modelling
  considerably.
• In Monte Carlo simulation a single random drawing
  for each input random variable (and a realisation
  of each randomiser in model) is taken to give an
  input and the process repeated a large no. of
  times - equally likely deterministic models. This
  build up a picture of the output random variable.
  It is a very general and powerful technique but
  its precision depends, inter alia, on the no. of
  simulations.

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Discrete Continuous Time and States

• Consider the system (stochastic process) ltxigt,
  i?T
• State of a model/process a set of variables
  describing the system at time t, i.e. xt
• State space the set of all possible values for
  the process, xt, ?t
• State space is either continous or discrete
  (finite or countable number of possibilities).
• Time can also be considered continous or
  discrete.
• Hence discrete time stochastic process
  continuous time stochastic process.
• Note 1 Whether one employs discrete or
  continuous time or state space depends on the
  objectives of the modelling not solely on the
  underlying reality.
• Note 2 Discrete systems lend themselves for
  easily to simulation. However, sometimes assuming
  continuity can lead to closed form solutions
  making them more tactable mathematically.

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Evaluation of Suitability of a Model

• Evaluate in context of objectives and purpose to
  which it is put.
• Consider data and techniques used to calibate
  model, especially estimation errors. Assess the
  credibility of the inputs.
• Consider correlation structure between variables
  driving the model.
• Consider correlation structure of model outputs.
• Continued relevance of model (if past model).
• Credibility of outputs.
• Dangers of spurious accuracy.
• Ease of use and how results can be communicated.

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Further Considerations in Modelling

• Short and long run properties of model
• are the coded relationships stable over time?
• should we factor in relationships that are second
  order in the short-term but manifest over
  long-term?
• Analysing the output
• generally by statistical sampling techniquesbut
  beware as observations are, in general
  correlated. IID assumption never, in general,
  valid.
• Use failure in Turing-type (or Working) test to
  better model.
• Sensitivity Testing
• Check small changes to inputs produce small
  changes to outputs. Check results robust to
  statistical distribution of inputs.
• Monitor and, perhaps expand on key sensitivities
  in model.
• Use optimistic, best estimate, and pessimistic
  assumptions.

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Further Considerations in Modelling

• Communication documentation of results
• Take account of knowledge and background of
  audience.
• Build confidence in model so seen as useful tool.
• Outline limitations of models.

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Macro-Econometric Modelling A Case Study
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Macro-Economics -V- Macro-Econometrics
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Macro-Economics -V- Macro-Econometrics
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Macro-Economic Management with Large Econometric
Models UK
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Modelling in Early 1970s (UK)

• Four Major Econometric Models
• Bank of England
• Treasury
• NIESR (the National Institute of Economic and
  Social Research)
• London Business School
• Consisting of 500-1,000 equations
• Modelling whole economy
• So complex that, in effect, Black Boxes.

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Macro-Economic Management with Large Econometric
Models UK
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Macro-Economic Management with Large Econometric
Models UK
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Modelling Errors

• General Uncertainty error term in model.
• Parameter misestimation the form of the model
  is right but the parameters are not.
• Model misspecification the form of the model is
  wrong.

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What went wrong in UK models

• First thought to be parameter misestimation
• Exchange rate floats in 1973 but no data to
  estimate what will happen so ignored it.
• Oil shock nothing like it seen before so
  pushing models to extreme.
• But the real problem was...
• Model Misspecification the graph simply could
  not go in that way under Keynesian Theory

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Opinion on Models, Early 1980s

• Treasury forecasters in 1980 were predicting
  the worst economic downturn since the Great Slump
  of 1929-1931. Yet they expected no fall in
  inflation at all. This clearly was absurd and
  underlined the inadequacies of the model.
•   Nigel Lawson, The View from No. 11.
• Modelling was seen as a second-rate activity
  done by people who were not good enough to get
  proper academic jobs.
• Earlier expectations of what models might
  achieve had evidently been set too high, with
  unrealistic claims about their reliability and
  scope.
•  
• Quoted from Economic Models and Policy-Making.
  Bank of England, Quarterly Bulletin, May 1997.

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Modelling from Mid-1980s to Date

• Must satisfy four criteria
• Models and their outputs can be explained in a
  way consistent with basic economic analysis.
• The judgement part of the process is made
  explicit.
• The models produce results consistent with
  relevant historic episodes.
• Results are consistent over time (e.g., the
  parameters are not sensitive to period studied)
• Leads to small scale models.
• Simple, parsimonious in parameters, designed for
  purpose on hand.

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Macro-Economic Management with Large Econometric
Models UK
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Macro-Economic Management with Large Econometric
Models UK
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Macro-Econometric Modelling Lessons Learned

• Be limited in our expectations of what can
  reasonably be achieved.
• Build disposal models.
• Small, stylised, parsimonious models are
  beautiful.

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Completes Case Study
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Have Modest Ambitions when modellingModelling
Orders of Complexity

• Level 1 - Two body problem
• e.g., gravity, light through prism, etc.
• Level 2 - N-identical body with local interaction
• e.g., Maxwell-Boltzmanns thermodynamics
• Ising model of ferromagnetism
• Level 3 - N-identical body with long-range
  interaction
• Level 4 - N-non-identical body with
  multi-interactions
• Modelling Markets
• Modelling economics systems generally
• General actuarial modelling
• The History of Science gives us no example of a
  complex problem of Level 3 or 4 being adequately
  modelled

From Roehner, B.M., Patterns of Speculation A
Study in Observational Econophysics, CUP 2002
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Final Word

• One thing I have learned in a long life that
  all our science, measured against reality, is
  primitive and childlike and yet it is the most
  precious thing we have.

Albert Einstein
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Completes Chapter 1