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Tutorial Financial Econometrics/Statistics. 2005 SAMSI program on Financial Mathematics, Statistics, and Econometrics. Goal. At the index level. Part I: Modeling. ... in which we see what basic properties of stock prices/indices we want to capture. Contents.
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TutorialFinancial Econometrics/Statistics 2005 SAMSI program on Financial Mathematics, Statistics, and Econometrics
Part I: Modeling ... in which we see what basic properties of stock prices/indices we want to capture
Contents • Returns and their (static) properties • Pricing models • Time series properties of returns
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
Which returns? • Two type of returns can be defined • Discrete compounding • Continuous compounding
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
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
Empirical properties of returns Data period: July 1962- December 2004; daily frequency
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?)
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:
Pricing models 2 • As a result, pricing models model expected returns ... • ... in terms of known quantities or a few ‘almost known’ quantities
Capital Asset Pricing Model • One of the best known pricing models • The theorem/model states
Black-Scholes • Also Black-Scholes is a pricing model • (Exact) contemporaneous relation between asset prices/returns
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
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
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
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
The ARCH model • First model to capture this effect • No mean effects for simplicity • ARCH in mean
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
The GARCH model • Generalized ARCH • Beware of time indices ...
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
Stochastic volatility models • Use latent volatility process
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
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
Consistency • The volatility clustering and other empirical evidence is consistent with appropriate continuous time models • A simple continuous time stochastic volatility model
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
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 • ...
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 • ...
Contents • Parametric inference for ARCH-type models • Rank based inference
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 • ...
Moment estimation (GMM) • Consider an ARCH-type model • We suppose that can be calculated on the basis of observations if is known • Moment condition
Moment estimation - 2 • The estimator now is taken to solve • In case of “underidentification”: use instruments • In case of “overidentification”: minimize distance-to-zero
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
Doing the math ... • Maximizing the log-likelihood boils down to solving with
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
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
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”
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)
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
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
Rank based inference ... in which we discuss the main ideas of rank based inference
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
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
Example: Linear regression • Linear regression model(with observations ) • Innovation density and cdf
Example ARCH(1) • Consider the standard ARCH(1) model • Innovation density and cdf