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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 ModellingFundamental 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 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 0 2 4 6 8 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 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 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 0 1 2 3 4 5 6 7 8 9 10 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>, iT 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 7 6 5 4 3 2 1 0 1 6 11 16 21 26 31 36 41 46 Macro-Economics -V- Macro-Econometrics 7 6 5 4 3 2 1 0 1 6 11 16 21 26 31 36 41 46 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