Time Series vs SDEs Diffusions Consider the AR(1) process. It is a - PowerPoint PPT Presentation

Financial econometrics based on stochastic differential equations and the sde package S.M. Iacus (University of Milan) Rennes, useR! 2009, July 8th - 10th 1 / 17

Time Series vs SDEs Diffusions Consider the AR(1) process. It is a discrete-time random process, defined as Exact likelihood Pseudo-likelihood X t = θX t − 1 + ǫ t , X 0 = x 0 , ǫ t : i.i.d. random variables (noise) Simulated likelihood method Hermite expansion 2 / 17

Time Series vs SDEs Diffusions Consider the AR(1) process. It is a discrete-time random process, defined as Exact likelihood Pseudo-likelihood X t = θX t − 1 + ǫ t , X 0 = x 0 , ǫ t : i.i.d. random variables (noise) Simulated likelihood method Hermite expansion Its continuous-time counter part (the Ornstein-Uhlenbeck process), written in differential form, looks like d X t = − θX t d t + d W t , X 0 = x 0 , W t : the Wiener process (noise) 2 / 17

Time Series vs SDEs Diffusions Consider the AR(1) process. It is a discrete-time random process, defined as Exact likelihood Pseudo-likelihood X t = θX t − 1 + ǫ t , X 0 = x 0 , ǫ t : i.i.d. random variables (noise) Simulated likelihood method Hermite expansion Its continuous-time counter part (the Ornstein-Uhlenbeck process), written in differential form, looks like d X t = − θX t d t + d W t , X 0 = x 0 , W t : the Wiener process (noise) A stochastic differential equation models a dynamical system with feedback by adding continuous time shocks d X t = b ( X t )d t + σ ( X t )d W t 2 / 17

Time Series vs SDEs In continuous time models: time between X t and X t +∆ t matters ! The length Diffusions Exact likelihood of ∆ t is crucial as well. Pseudo-likelihood Simulated likelihood method Hermite expansion 3 / 17

Time Series vs SDEs In continuous time models: time between X t and X t +∆ t matters ! The length Diffusions Exact likelihood of ∆ t is crucial as well. Pseudo-likelihood Simulated likelihood method In time-series models: nothing happens (probabilistically) between X t and Hermite expansion X t − 1 3 / 17

Time Series vs SDEs In continuous time models: time between X t and X t +∆ t matters ! The length Diffusions Exact likelihood of ∆ t is crucial as well. Pseudo-likelihood Simulated likelihood method In time-series models: nothing happens (probabilistically) between X t and Hermite expansion X t − 1 Why this matters? An example: according to McCrorie & Chambers (2006, J. of Econ.) and others, “spurious Granger causality [tested with VAR models] is only a consequence of the intervals in which economic data are generated being finer than the econometrician’s sampling interval. ” Conclusions: assume a continuous time model (SDE). Discretize that, build a VAR from the discretized SDE and the spurious Granger causality vanishes! 3 / 17

Time Series vs SDEs In continuous time models: time between X t and X t +∆ t matters ! The length Diffusions Exact likelihood of ∆ t is crucial as well. Pseudo-likelihood Simulated likelihood method In time-series models: nothing happens (probabilistically) between X t and Hermite expansion X t − 1 Why this matters? An example: according to McCrorie & Chambers (2006, J. of Econ.) and others, “spurious Granger causality [tested with VAR models] is only a consequence of the intervals in which economic data are generated being finer than the econometrician’s sampling interval. ” Conclusions: assume a continuous time model (SDE). Discretize that, build a VAR from the discretized SDE and the spurious Granger causality vanishes! Rephrasing: why using a binomial distribution if your underlying model is a Gaussian? 3 / 17

A few examples of SDEs Diffusions � gBm : d X t = µX t d t + σX t d W t Exact likelihood √ X t d W t Pseudo-likelihood � CIR : d X t = ( θ 1 + θ 2 X t )d t + θ 3 Simulated likelihood method Hermite expansion � CKLS : d X t = ( θ 1 + θ 2 X t )d t + θ 3 X θ 4 t d W t � nonlinear mean reversion (A¨ ıt-Sahalia) t )d t + β 1 X ρ d X t = ( α − 1 X − 1 + α 0 + α 1 X t + α 2 X 2 t d W t t � double Well potential (bimodal behaviour, highly nonlinear) d X t = ( X t − X 3 t )d t + d W t � Jacobi diffusion (political polarization): X t − 1 � � � d X t = − θ d t + θX t (1 − X t )d W t 2 � radial Ornstein-Uhlenbeck : d X t = ( θX − 1 − X t )d t + d W t t � � � hyperbolic diffusion : d X t = σ 2 X t √ β − γ d t + σ d W t 2 δ 2 +( X t − µ ) 2 4 / 17

Diffusion processes solutions to SDEs Diffusions From the statistical point of view, we are interested in the parametric family of Exact likelihood diffusion process solutions of the SDE Pseudo-likelihood Simulated likelihood method d X t = b ( X t , θ )d t + σ ( X t , θ )d W t , X 0 = x 0 , t ∈ [0 , T ] Hermite expansion θ = ( α, β ) ∈ Θ α × Θ β = Θ , where Θ α ⊂ R p and Θ β ⊂ R q . Observations always come in discrete time form at some times t i = i ∆ n , i = 0 , 1 , 2 , ..., n, where ∆ n is the length of the steps. We denote the observations by X n := { X i = X t i } 0 ≤ i ≤ n . 5 / 17

Diffusion processes solutions to SDEs Diffusions From the statistical point of view, we are interested in the parametric family of Exact likelihood diffusion process solutions of the SDE Pseudo-likelihood Simulated likelihood method d X t = b ( X t , θ )d t + σ ( X t , θ )d W t , X 0 = x 0 , t ∈ [0 , T ] Hermite expansion θ = ( α, β ) ∈ Θ α × Θ β = Θ , where Θ α ⊂ R p and Θ β ⊂ R q . Observations always come in discrete time form at some times t i = i ∆ n , i = 0 , 1 , 2 , ..., n, where ∆ n is the length of the steps. We denote the observations by X n := { X i = X t i } 0 ≤ i ≤ n . Different sampling schemes, different statistical procedures: 1. Large sample asymptotics: ∆ fixed, T = n ∆ → ∞ as n → ∞ 2. High frequency: T = n ∆ n fixed, ∆ n → 0 as n → ∞ 3. Rapidly increasing design: T = n ∆ → ∞ , ∆ n → 0 as n → ∞ under the additional condition n ∆ k n → 0 for k > 1 5 / 17

Likelihood in discrete time Diffusions By Markov property of diffusion processes, the likelihood has this form Exact likelihood n Pseudo-likelihood Simulated likelihood � L n ( θ ) = p θ (∆ , X i | X i − 1 ) p θ ( X 0 ) method Hermite expansion i =1 Problem: the transition density p θ (∆ , X i | X i − 1 ) is often not available! Only for OU, CIR and gBm 6 / 17

Likelihood in discrete time Diffusions By Markov property of diffusion processes, the likelihood has this form Exact likelihood n Pseudo-likelihood Simulated likelihood � L n ( θ ) = p θ (∆ , X i | X i − 1 ) p θ ( X 0 ) method Hermite expansion i =1 Problem: the transition density p θ (∆ , X i | X i − 1 ) is often not available! Only for OU, CIR and gBm Solutions: � discretization of the SDE (Euler, Milstein, Ozaki, etc) � simulation method � hermite polynomial expansion � partial differential equations � other approximations of the transition density 6 / 17

Local Gaussian Approximation. Euler Scheme. By Euler discretization of the SDE : d X t = b ( X t , θ )d t + σ ( X t , θ )d W t X t +∆ t − X t = b ( X t , θ )∆ t + σ ( X t , θ )( W t +∆ t − W t ) , we get an approximate transition density which is Gaussian. This is widely seen in applied contexts. But is this approximation good or not? In general no! For example, for gBm, the true transition density is a log-normal and the Euler schemes provides only a Gaussian approximation! It is possible to prove that estimators are not even consistent for non negligible ∆ . 7 / 17

Euler, ∆ and bias Diffusions Consider OU model Exact likelihood Pseudo-likelihood d X t = ( θ 1 − θ 2 X t )d t + θ 3 d W t , X 0 = x 0 Simulated likelihood method Hermite expansion Both true and Euler approximation are Gaussian respectively with mean and variance v (∆ , x ) = θ 2 1 − e − 2 θ 2 ∆ � � m (∆ , x ) = xe − θ 2 ∆ + θ 1 � 1 − e − θ 2 ∆ � 3 , , θ 2 2 θ 2 and (Euler) v Euler (∆ , x ) = θ 2 m Euler (∆ , x ) = x (1 − θ 2 ∆) + θ 1 ∆ , 3 ∆ , Only under high-frequency setting, i.e. ∆ → 0 , the approximation is acceptable. 8 / 17

Simulated likelihood method Diffusions Let p θ (∆ , y | x ) be the true transition density of X t +∆ at point y given Exact likelihood X t = x . Consider a δ << ∆ , for example δ = ∆ /N for N large enough, Pseudo-likelihood Simulated likelihood and then use the Chapman-Kolmogorov equation as follows: method Hermite expansion � p θ (∆ , y | x ) = p θ ( δ, y | z ) p θ (∆ − δ, z | x )d z = E z { p θ ( δ, y | z ) | ∆ − δ } , It means that p θ (∆ , y | x ) is seen as the expected value over all possible transitions of the process from time t + (∆ − δ ) to t + ∆ , taking into account that the process was in x at time t . So we need simulations! 9 / 17

What about N ? We need many simulations Example: approximation for the CIR model 25 20 conditional density exact N=2 15 N=5 N=10 10 5 0 0.00 0.05 0.10 0.15 0.20 x 10 / 17

What about N ? We need many simulations Example: approximation for the CIR model 25 20 conditional density exact N=2 15 N=5 N=10 10 5 0 0.00 0.05 0.10 0.15 0.20 x We need many simulations ( N ) for each time points ( X t i , X t i +∆ ). But not all simulation schemes are stable for all models 10 / 17