Tutorial Financial Econometrics/Statistics

1. TutorialFinancial Econometrics/Statistics 2005 SAMSI program on Financial Mathematics, Statistics, and Econometrics

2. Goal

3. At the index level

4. Part I: Modeling ... in which we see what basic properties of stock prices/indices we want to capture

5. Contents • Returns and their (static) properties • Pricing models • Time series properties of returns

6. Why returns? • Prices are generally found to be non-stationary • Makes life difficult (or simpler...) • Traditional statistics prefers stationary data • Returns are found to be stationary

7. Which returns? • Two type of returns can be defined • Discrete compounding • Continuous compounding

8. Discrete compounding • If you make 10% on half of your money and 5% on the other half, you have in total 7.5% • Discrete compounding is additive over portfolio formation

9. Continuous compounding • If you made 3% during the first half year and 2% during the second part of the year, you made (exactly) 5% in total • Continuous compounding is additive over time

10. Empirical properties of returns Data period: July 1962- December 2004; daily frequency

11. Stylized facts • Expected returns difficult to assess • What’s the ‘equity premium’? • Index volatility < individual stock volatility • Negative skewness • Crash risk • Large kurtosis • Fat tails (thus EVT analysis?)

12. Pricing models • Finance considers the final value of an asset to be ‘known’ • as a random variable , that is • In such a setting, finding the price of an asset is equivalent to finding its expected return:

13. Pricing models 2 • As a result, pricing models model expected returns ... • ... in terms of known quantities or a few ‘almost known’ quantities

14. Capital Asset Pricing Model • One of the best known pricing models • The theorem/model states

15. Black-Scholes • Also Black-Scholes is a pricing model • (Exact) contemporaneous relation between asset prices/returns

16. Time series properties of returns • Traditionally model fitting exercise without much finance • mostly univariate time series and, thus, less scope for tor the ‘traditional’ cross-sectional pricing models • lately more finance theory is integrated • Focuses on the dynamics/dependence in returns

17. Random walk hypothesis • Standard paradigm in the 1960-1970 • Prices follow a random walk • Returns are i.i.d. • Normality often imposed as well • Compare Black-Scholes assumptions

18. Box-Jenkins analysis

19. Linear time series analysis • Box-Jenkins analysis generally identifies a white noise • This has been taken long as support for the random walk hypothesis • Recent developments • Some autocorrelation effects in ‘momentum’ • Some (linear) predictability • Largely academic discussion

20. Higher moments and risk

21. Risk predictability • There is strong evidence for autocorrelation in squared returns • also holds for other powers • ‘volatility clustering’ • While direction of change is difficult to predict, (absolute) size of change is • risk is predictable

22. The ARCH model • First model to capture this effect • No mean effects for simplicity • ARCH in mean

23. ARCH properties • Uncorrelated returns • martingale difference returns • Correlated squared returns • with limited set of possible patterns • Symmetric distribution if innovations are symmetric • Fat tailed distribution, even if innovations are not

24. The GARCH model • Generalized ARCH • Beware of time indices ...

25. GARCH model • Parsimonious way to describe various correlation patterns • for squared returns • Higher-order extension trivial • Math-stat analysis not that trivial • See inference section later

26. Stochastic volatility models • Use latent volatility process

27. Stochastic volatility models • Also SV models lead to volatility clustering • Leverage • Negative innovation correlation means that volatility increases and price decreases go together • Negative return/volatility correlation • (One) structural story: default risk

28. Continuous time modeling • Mathematical finance uses continuous time, mainly for ‘simplicity’ • Compare asymptotic statistics as approximation theory • Empirical finance (at least originally) focused on discrete time models

29. Consistency • The volatility clustering and other empirical evidence is consistent with appropriate continuous time models • A simple continuous time stochastic volatility model

30. Approximation theory • There is a large literature that deals with the approximation of continuous time stochastic volatility models with discrete time models • Important applications • Inference • Simulation • Pricing

31. Other asset classes • So far we only discussed stock(indices) • Stock derivatives can be studied using a derivative pricing models • Financial econometrics also deals with many other asset classes • Term structure (including credit risk) • Commodities • Mutual funds • Energy markets • ...

32. Term structure modeling • Model a complete curve at a single point in time • There exist models • in discrete/continuous time • descriptive/pricing • for standard interest rates/derivatives • ...

33. Part 2: Inference

34. Contents • Parametric inference for ARCH-type models • Rank based inference

35. Analogy principle • The classical approach to estimation is based on the analogy principle • if you want to estimate an expectation, take an average • if you want to estimate a probability, take a frequency • ...

36. Moment estimation (GMM) • Consider an ARCH-type model • We suppose that can be calculated on the basis of observations if is known • Moment condition

37. Moment estimation - 2 • The estimator now is taken to solve • In case of “underidentification”: use instruments • In case of “overidentification”: minimize distance-to-zero

38. Likelihood estimation • In case the density of the innovations is known, say it is , one can write down the density/likelihood of observed returns • Estimator: maximize this

39. Doing the math ... • Maximizing the log-likelihood boils down to solving with

40. Efficiency consideration • Which of the above estimators is “better”? • Analysis using Hájek-Le Cam theory of asymptotic statistics • Approximate complicated statistical experiment with very simple ones • Something which works well in the approximating experiment, will also do well in the original one

41. Quasi MLE • In order for maximum likelihood to work, one needs the density of the innovations • If this is not know, one can guess a density (e.g., the normal) • This is known as • ML under non-standard conditions (Huber) • Quasi maximum likelihood • Pseudo maximum likelihood

42. Will it work? • For ARCH-type models, postulating the Gaussian density can be shown to lead to consistent estimates • There is a large theory on when this works or not • We say “for ARCH-type models the Gaussian distribution has the QMLE property”

43. The QMLE pitfall • One often sees people referring to Gaussian MLE • Then, they remark that we know financial innovations are fat-tailed ... • ... and they switch to t-distributions • The t-distribution does not possess the QMLE property (but, see later)

44. How to deal with SV-models? • The SV models look the same • But now, is a latent process and hence not observed • Likelihood estimation still works “in principle”, but unobserved variances have to be integrated out

45. Inference for continuous time models • Continuous time inference can, in theory, be based on • continuous record observations • discretely sampled observations • Essentially all known approaches are based on approximating discrete time models

46. Rank based inference ... in which we discuss the main ideas of rank based inference

47. The statistical model • Consider a model where ‘somewhere’ there exist i.i.d. random errors • The observations are • The parameter of interest is some • We denote the density of the errors by

48. Formal model • We have an outcome space , with the number of observations and the dimension of • Take standard Borel sigma-fields • Model for sample size : • Asymptotics refer to

49. Example: Linear regression • Linear regression model(with observations ) • Innovation density and cdf

50. Example ARCH(1) • Consider the standard ARCH(1) model • Innovation density and cdf