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					Models - Stochastic Models
STAT 30090 & STAT 40210




          Dr Shane Whelan, FFA, FSAI, FSA
                                     L400
Introductory Remarks
Introduction
Timetable:
    Mon.             2 pm    Theatre R, Arts        Lecture
    Mon.             3 pm    Theatre P, Arts        Lecture
    Wed.             4 pm    Theatre P, Arts        Lecture
    TBA                      TBC                    Tutorial

    Total of 30 lectures and about 6 or more tutorials

   Exam in December 2007.
Stochastic Models
 Course Objectives
       To provide a grounding in stochastic modelling, especially in
        actuarial applications.
       To gain credit towards 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 Models…that part designated CT4(103).
 Course Syllabus
       Actuarial ModellingFundamental Concepts in Stochastic
        processes  Simulation Markov Chains Markov Jump
        Processes 
Stochastic Models: Overview
                                                                      Percentage of Course
                                                                           (circa)
  Chapter 1: Introduction to (Actuarial) Modelling                         10%
  Chapter 2: Foundational concepts in Stochastic Processes                 10%
  Chapter 3: Simulation (of Stochastic Processes)                          15%
      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?
  Chapter 4: 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 5: Markov Jump Processes                                         35%
      Markov Jump Processes; Kolmogorov’s Forward Equations; Kolmogorov’s Backward
       Equations; Time-homogeneous Markov Jump Process; The Time Inhomogeneous
       Case; The Integrated Form of Kolmogorov’s Backward & Forward Equations;
       Applications
Course Notes
 The slides are put up on web, together with Chapter-by-
  chapter fully written up notes.
    Click here: http://www.ucd.ie/statdept/shanewhelan/shanecp4.html
 Problems are set in the Chapters. These will be gone over in
  tutorials.
 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
Chapter 1:

    Introduction to (Actuarial)
    Modelling
The Modelling Problem
 “I have yet to see any problem, however
  complicated, which, when you looked at
  it in the right way, did not become still more
  complicated.”




  Poul Anderson, science fiction writer, in New Scientist.
  (London, September 25, 1969).
Modelling
 Model – a simplification of a real system, facilitating
   understanding, prediction and perhaps control of the real
   system.
        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
Model of Salary Progession
 Salary (t )  €25000e.05t

Salary Level
  € 250,000

  € 225,000

  € 200,000

  € 175,000

  € 150,000

  € 125,000

  € 100,000

   € 75,000

   € 50,000

   € 25,000

        €0
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                                        Years since Graduation
Classifying Models
  Deterministic Model – Unique output for given set of inputs.
   The output is not a random variable.

  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.
Stochastic Model of Salary Progession
                           Salary (t )  €25000e.05t  X t
                           where
                           Xt        N (0, (500t ) 2 )
  Salary Level – Some Possible Paths
   € 250,000

   € 225,000

   € 200,000

   € 175,000

   € 150,000

   € 125,000

   € 100,000

    € 75,000

    € 50,000

    € 25,000

         €0
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                                           Years since Graduation
                  Stochastic
Stochastic Model of a System
                    Analysis




                     Future Period
     0   1   2   3   4    5     6    7   8   9   10
                  Stochastic
Stochastic Model of a System
                    Analysis




                     Future Period
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Components of Model
 Structural Part – sets out the relationship between
  the variables 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 variables
        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.
Building a Model



            Anything Goes
Introduction to Real Modelling

 Perspective we attempt to get, well captured
  in
   Real Life Mathematics, Bernard Beauzamy, Irish
    Math. Soc. Bulletin 48 (Summer 2002), 43–46.
   Available on Web from:
    http://www.maths.tcd.ie/pub/ims/bull48/M4801.pdf
   Repays the 20 minute read!
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.”
Building a Model – 10 Helpful Steps

 1.   Set well-defined objectives for model.
 2.   Plan how model is to be validated
         i.e., the diagnostic tests to ensure it meets objectives
 3.   Define the essence of the structural model – the 1st order
      approximation. Refinement and details can come later.
 4.   Collect & analyse data for model (and any other
      parameters)
 5.   Involve experts on the real world system to get feedback
      on conceptual model.
Building a Model – 10 Helpful Steps
 6.    Decide how to implement model
       e.g. C, Excel, some statistical package. Often random number generator
       needed.
 7.    Write and debug program.
 8.    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.]
 10.   Communicate and document results and the model.
 10.a Review and update in the light of new data and other changes.
Advantages of Modelling
1.   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.
2.   Complex systems, including stochastic systems, that are otherwise not
     tractable mathematically (in closed form) can be studied.
3.   It is quicker (system studied in compressed time), and less expensive than
     alternatives.
4.   Consequences of different policy actions can be assessed, so option can
     be selected that optimizes output.
5.   We can reduce variance of model as we can better control experimental
     conditions.
Drawbacks of Modelling (that must be
guarded against)
 1.   Requires considerable investment of time and expertise..not free.
 2.   Often time-consuming to use – many simulations needed and results
      analysed.
 3.   Not especially good at optimising outputs (better at comparing results of
      input variations)
 4.   Impressive-looking models (especially complex ones) can lead to
      overconfidence in model.
 5.   Model only as good as parameter inputs – quality and credibility of data.
 6.   Must understand limitations of model (i.e., its proper use)
 7.   Must recognise that a model will become obsolete – change in
      circumstances.
 8.   Sometimes difficult to interpret output.
Quotes from Real Life Mathematics

  “Most current mathematical research, since the [19]60’s, 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.”
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.
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.
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.
Discrete & Continuous Time and States

 Consider the system (stochastic process) <xi>, 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 continuous or discrete (finite or countable number of
   possibilities).
  Time can also be considered continuous 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 more easily to simulation. However,
    sometimes assuming continuity can lead to closed form solutions – making them
    more tractable mathematically.
Evaluation of Suitability of a Model

 1.   Evaluate in context of objectives and purpose to which it is put.
 2.   Consider data and techniques used to calibrate model, especially
      estimation errors. Assess the credibility of the inputs.
 3.   Consider correlation structure between variables driving the model.
 4.   Consider correlation structure of model outputs.
 5.   Continued relevance of model (if past model).
 6.   Credibility of outputs.
 7.   Dangers of spurious accuracy.
 8.   Ease of use and how results can be communicated.
Further Considerations in Modelling
 1)     Short and long run properties of model
      i. are the coded relationships stable over time?
      ii. should we factor in relationships that are second order in the
            short-term but manifest over long-term?
 2)     Analysing the output –
      i. generally by statistical sampling techniques…but beware as
            observations are, in general correlated. IID assumption never, in
            general, valid.
      ii. Use failure in Turing-type (or Working) test to better model.
 3)      Sensitivity Testing
      i. Check small changes to inputs produce small changes to outputs.
            Check results robust to statistical distribution of inputs.
      ii. Monitor and, perhaps expand on key sensitivities in model.
      iii. Use optimistic, best estimate, and pessimistic assumptions.
Further Considerations in Modelling

 4)      Communication & documentation of results
      i. Take account of knowledge and background of
           audience.
      ii. Build confidence in model so seen as useful tool.
      iii. Outline limitations of models.
Macro-Econometric Modelling

       A Case Study
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
30.0%


25.0%


20.0%


15.0%


10.0%


 5.0%


 0.0%
        1949 1952 1955 1958 1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997

-5.0%

                                     UK Real GDP Growth   UK Inflation
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.
Macro-Economic Management with Large
Econometric Models :UK
30.0%


25.0%


20.0%


15.0%


10.0%


 5.0%


 0.0%
        1949 1952 1955 1958 1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997

-5.0%

                                     UK Real GDP Growth   UK Inflation
Macro-Economic Management with Large
Econometric Models :UK
30.0%


25.0%


20.0%


15.0%


10.0%


 5.0%


 0.0%
        1949 1952 1955 1958 1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997

-5.0%

                                     UK Real GDP Growth   UK Inflation
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.
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
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.
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.
Macro-Economic Management with Large
Econometric Models :UK
30.0%


25.0%


20.0%


15.0%


10.0%


 5.0%


 0.0%
        1949 1952 1955 1958 1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997

-5.0%

                                     UK Real GDP Growth   UK Inflation
Macro-Economic Management with Large
Econometric Models :UK
30.0%


25.0%


20.0%


15.0%


10.0%


 5.0%


 0.0%
        1949 1952 1955 1958 1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997

-5.0%

                                     UK Real GDP Growth   UK Inflation
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.
Completes Case Study
 Have Modest Ambitions when modelling
 Modelling: 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-Boltzmann’s 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
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
Completes Chapter 1

				
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