Neil Shephard Nuffield College, Oxford, UK

Transcription

1 Lecture 1: some statistical and econometric problems Neil Shephard Nuffield College, Oxford, UK September 21 Computationally intensive statistics: inference for some stochastic processes 1

2 1 An econometric problem: stochastic volatility 1.1 Time series of speculative assets Daily exchange rates Dollar against six currencies from 26 July 1985 to 28th July 2. The currencies are the Canadian, DM, FF, Swiss Franc, Y and Pound. Throughout each rate is denominated in Dollars. Rates increase the Dollar has strenghtened. Advent of the Euro in 1st of January 1999 in effect froze the cross rate between the FF and DM. Log-exchange rate y (s). Let the interval be, then we write the changes as y t = y ( s) y ((s 1) ), s =1, 2,... (1) Not economic returns as ignore differential investment returns (e.g. interest baring accounts). 2

3 Canadian Dollar

4 1.1.2 Equity data Indexes recorded in their domestic currency from 7th December 1993 to 8th October 2. DAX 3, FTSE 1, S&P 5 Composite and the Nikkei 5. Throughout we use the opening prices of the index, recorded by DATASTREAM. No major market crash occurred. Biggest movement in DAX which moves down by around nine percent in a single day. FTSE is the most unusual of these series for it displays no really large movements up or down. Each of the indexes looks like it moves more in the second half of the dataset. 4

5 5 DAX 3 FTSE S&P 5 Composite Nikkei Returns in domestic currency, not taking into account interest rates. 5

6 1.1.3 Stylised facts of time series Canadian Dollar German Mark French Franc Fitted GH model Non-parametric estimator -6 Fitted GH model Non-parametric estimator -7.5 Fitted GH model Non-parametric estimator -1 1 Swiss Franc Japanese Yen UK Sterling Fitted GH model Non-parametric estimator -1 Fitted GH model Non-parametric estimator -6 Fitted GH model Non-parametric estimator Marginal distributions of changes drawn is the log-density kernel and then taking logs. Bandwidth is optimal in the Gaussian case. 6

7 1.1.4 Temporal aggregation 5 minute return data on the United States Dollar/ German Deutsche Mark series covers the ten year period from 1st December 1986 until 3th November Records the most recent quote to appear on the Reuters screen. It has been kindly supplied to us by Olsen and Associates in Zurich. 7

8 2.5 5 minutes minutes 1 hour Fitted GH model Non-parametric estimator hours -5-2 Fitted GH model Non-parametric estimator day Fitted GH model Non-parametric estimator days Fitted GH model Non-parametric estimator -4-6 Fitted GH model Non-parametric estimator Fitted GH model Non-parametric estimator

9 Mean Variance Skewness Kurtosis 5minutes minutes hour hours day week

10 1.2 Serial dependence in changes in prices Changes are nearly weak white noise Correlogram for daily exchange rate changes Correlogram for 5 minute changes in Olsen data.6 Canadian Dollar French Franc Japanese Yen German Mark Swiss Franc UK Sterling

11 Squares of changes are autocorrelated.14 Canadian Dollar French Franc Japanese Yen German Mark Swiss Franc UK Sterling.175 Canadian Dollar French Franc Japanese Yen German Mark Swiss Franc UK Sterling

12 Stylised facts Little autocorrelation in levels Important (with long lags) autocorrelation amoungst absolute and squares Marginally has fat tails Aggregational Gaussianity Some negative skews static and dynamic 12

13 1.3 Interests Derivatives. Black-Scholes formula misprices many European options suggests risk neutral process has fatter tails and importance dynamic skews. Although this does not matter much in European market, for exotics this means Brownian motion is a poor working rule. Asset allocation (multivariate). Risk assessment (tails, multivariate). Under the physical measure non-normality may matter greatly. 13

14 2 Models 2.1 Discrete time SV model Simplest model is the log-normal SV model (Taylor (1982)) y t = βe ht/2 ε t,t 1 h t+1 = µ + φ(h t µ)+σ η η t,t 2 h 1 Surveys include Taylor (1994) N µ, σ 2 1 φ 2 and Ghysels, Harvey, and Renault (1996) Shephard (1996) β, µ, φ, σ η and ρ (set β =1). 14 ε t η t NID, 1 ρ ρ 1 (2)

15 If γ 1 < 1with: y t = βe ht/2 ε t,t 1 h t+1 = µ + φ(h t µ)+σ η η t,t 2 h 1 N µ, σ 2 1 φ 2. (3) µ h = E(h t )= γ, σh 2 =Var(h t )= σ2 η 1 γ 1 1 γ1 2. As ε t is always stationary, y t will be stationary iff h t is stationary. Using log-normal distribution E(y 4 t )/(σ 2 y 2)2 =3exp(σ 2 h) 3. 15

16 y t = βe ht/2 ε t,t 1 h t+1 = µ + φ(h t µ)+σ η η t,t 2 h 1 N µ, As ε t is iid, y t is a MD and is WN if γ 1 < 1. As h t is a Gaussian AR(1), and so σ 2 1 φ 2. (4) Cov(y 2 t,y 2 t s) = exp{2µ h + σ 2 h(1 + γ s 1)} {E(y 2 t )} 2 = exp(2µ h + σ 2 h){exp(σ 2 hγ s 1) 1} ρ y 2 t (s) =Cov(yt 2 yt s)/var(y 2 t 2 )= exp(σ2 hγ1) s 1 3exp(σh) 2 1 exp(σ2 h) 1 3exp(σh) 2 1 γs 1. (5) This is the autocorrelation function of an ARMA(1, 1) process. Thus the SV model behaves in a manner similar to the GARCH(1, 1) model. 16

17 2.1.1 Superposition model y t = βe h t/2 ε t,t 1 h t = M j=1 h t,j, where h t,j are independent Gaussian autoregressions. e.g. M = 2 and where φ 1 >φ 2. h t+1,1 = µ + φ 1 (h t,1 µ 1 )+σ η,1 η t,1, h t+1,2 = φ 2 h t,2 + σ η,2 η t,2 17

18 2.2 Discretely observed diffusions SDEs of the form dy(t) =a {y(t),t,θ} dt + b {y(t),t; θ} dw(t), (6) where a{y(t),t,θ} and b{y(t),t,θ} are the non-anticipative drift and volatility functions. Bedrock of much of traditional finance. e.g. option pricing formula and interest rate theory. Observe these processes discretely, how to do inference? Literature includes the indirect inference method of Smith (1993) & Gourieroux, Monfort, and Renault (1993). the efficient method of moments estimator of Gallant and Tauchen (1996) and Gallant and Long (1997). 18

19 dy(t) =a {y(t),t,θ} dt + b {y(t),t; θ} dw(t), the non-parametric approaches of Ait-Sahalia (1996a), Ait-Sahalia (1996b) and Jiang and Knight (1997). estimating functions, see Keller and Sørensen (1999), Sørensen (1997), Florens- Zmirou (1989), Hansen and Scheinkman (1996) the likelihood based method of Pedersen (1995). 19

20 2.3 The illustion of data In financial economics we have trade by trade and quote by quote data. Thus we have a continuous record of all trading on the markets recording the times at which trading occurs. The above problem looks like it has gone away we observe the process in continuous time. But the model is misspecified and so one should worry about using ultra high frequency data. The bias could swamp efficiency gains made by using more data. 2

21 2.4 Continuous time SV models Then the log-price y (s) follows the solution to the SDE dy (s) = { µ + βσ 2 (s) } ds + σ(s)dw(s), (7) where σ 2 (s), the instantaneous or spot volatility, is going to be assumed to (almost surely) have locally square integrable sample paths, while being stochastically independent of the standard Brownian motion w(s). Over an interval of time of length > returns are defined as y t = y ( t) y ((t 1) ), t =1, 2,... (8) which implies that whatever the model for σ 2, it follows that where y t σ 2 t N(µ +βσ 2 t,σ 2 t ). σ 2 t = σ 2 (t ) σ 2 {(t 1) }, and σ 2 (s) = s σ2 (u)du. 21

22 2.5 Spot volatility model Two classes of processes which have this property. The first is the constant elasticity of variance (CEV) process which is the solution to the SDE dσ 2 (s) = λ { σ 2 (s) ξ } ds + ωσ(s) η db(λs), η [1, 2], where b(s) is standard Brownian motion uncorrelated with w(s). Of course the special cases of η = 1 delivers the square root process, while when η = 2 we have Nelson s GARCH diffusion. These models have been heavily favoured by Meddahi and Renault (2) in this context. The second process is the non-gaussian Ornstein- Uhlenbeck, or OU type for short, process which is the solution to the SDE dσ 2 (s) = λσ 2 (s)ds +dz(λs), (9) where z(s) is a Lévy process with non-negative increments. These models have been developed in this context by Barndorff-Nielsen and Shephard (21). 22

23 3 Statistical models Many of the above models are just special cases of non-gaussian, non-linear state space models. Let us write the state α t and observations as y t. States are Markov and the observations are conditionally independent given current state. Model specified through f(y t α t ) and f(α t+1 α t ). Unfortunately this structure is hard to handle outside Gaussian, linear structure. Numerical integration rules Kitagawa (1987) (high dimensions) MCMC Carlin, Polson, and Stoffer (1992), Carter and Kohn (1994), Fruhwirth-Schnatter (1994), Shephard (1994), Shephard and Pitt (1997) etc (filtering) Particle filters Gordon, Salmond, and Smith (1993), Pitt and Shephard (1999), Doucet, de Freitas, and Gordon (21) (likelihood). 23

24 4 Classes of state space models: MCMC design Unstructured: Markov random field structure only. Carlin, Polson, and Stoffer (1992) Conditional Gaussian: y s is a Gaussian SSF. Carter and Kohn (1994) (s is Markov and discrete), Shephard (1994) (s is Markov). Non-Gaussian measurement SSF: ie. α t+1 α t Gaussian but f(y t α t ) non- Gaussian. Shephard and Pitt (1997). 24

25 5 Outline of lectures 1. MCMC methodology for state space models 2. SV inference (a) Univariate: MCMC & particle filters (b) Multivariate: MCMC & particle filters 3. Inference for diffusion based models 25

26 References Ait-Sahalia, Y. (1996a). Nonparametric pricing of interest rate derivative securities. Econometrica 64, Ait-Sahalia, Y. (1996b). Testing continuous-time models of the spot interest rate. Review of Financial Studies 9, Barndorff-Nielsen, O. E. and N. Shephard (21). Non-Gaussian Ornstein Uhlenbeck-based models and some of their uses in financial economics (with discussion). Journal of the Royal Statistical Society, Series B 63, Carlin, B. P., N. G. Polson, and D. Stoffer (1992). A Monte Carlo approach to nonnormal and nonlinear state-space modelling. Journal of the American Statistical Association 87, Carter, C. K. and R. Kohn (1994). On Gibbs sampling for state space models. Biometrika 81, Doucet, A., N. de Freitas, and N. Gordon (21). Sequential Monte Carlo Methods in Practice. New York: Springer-Verlag. Florens-Zmirou, D. (1989). Approximate discrete-time schemes for statistics of diffusion processes. Statistics 2,

27 Fruhwirth-Schnatter, S. (1994). Data augmentation and dynamic linear models. Journal of Time Series Analysis 15, Gallant, A. R. and J. R. Long (1997). Estimating stochastic differential equations efficiently by minimum chi-square. Biometrika 84, Gallant, A. R. and G. Tauchen (1996). Which moments to match. Econometric Theory 12, Ghysels, E., A. C. Harvey, and E. Renault (1996). Stochastic volatility. In C. R. Rao and G. S. Maddala (Eds.), Statistical Methods in Finance, pp Amsterdam: North-Holland. Gordon, N. J., D. J. Salmond, and A. F. M. Smith (1993). A novel approach to nonlinear and non-gaussian Bayesian state estimation. IEE-Proceedings F14, Gourieroux, C., A. Monfort, and E. Renault (1993). Indirect inference. Journal of Applied Econometrics 8, S85 S118. Hansen, L. P. and J. A. Scheinkman (1996). Back to the future: generating moment implications for continuous-time models. Econometrica 64,

28 Jiang, G. L. and J. L. Knight (1997). A nonparametric approach to the estimation of diffusion processes, with an application to a short-term interest-rate models. Econometric Theory 13, Keller, M. and M. Sørensen (1999). Estimating equations based on eigenfunctions for a discretely observed diffusion process. Bernoulli 5, Kitagawa, G. (1987). Non-Gaussian state space modelling of non-stationary time series. Journal of the American Statistical Association 82, Meddahi, N. and E. Renault (2). Temporal aggregation of volatility models. Unpublished paper: CIRANO, Montreal. Pedersen, A. R. (1995). A new approach to maximum likelihood estimation for stochastic differential equations on discrete observations. Scandinavian Journal of Statistics 27, Pitt, M. K. and N. Shephard (1999). Filtering via simulation: auxiliary particle filter. Journal of the American Statistical Association 94, Shephard, N. (1994). Partial non-gaussian state space. Biometrika 81,

29 Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility. In D. R. Cox, D. V. Hinkley, and O. E. Barndorff-Nielsen (Eds.), Time Series Models in Econometrics, Finance and Other Fields, pp London: Chapman & Hall. Shephard, N. and M. K. Pitt (1997). Likelihood analysis of non-gaussian measurement time series. Biometrika 84, Smith, A. A. (1993). Estimating nonlinear time series models using simulated vector autoregressions. Journal of Applied Econometrics 8, S63 S84. Sørensen, M. (1997). Estimating equations for discretely observed diffusions: a review. In I. V. Basawa, V. P. Godambe, and R. L. Taylor (Eds.), Selected Proceedings of the Symposium on Estimating Functions, pp IMS Lecture Notes. Monograph Series. Taylor, S. J. (1982). Financial returns modelled by the product of two stochastic processes a study of daily sugar prices In O. D. Anderson (Ed.), Time Series Analysis: Theory and Practice, 1, pp Amsterdam: North-Holland. Taylor, S. J. (1994). Modelling stochastic volatility. Mathematical Finance 4,