Discrete time processes

Transcription

1 Discrete time processes Predictions are difficult. Especially about the future Mark Twain. Florian Herzog 2013

2 Modeling observed data When we model observed (realized) data, we encounter usually the following situations: Time-series data of a single time series (D t ), e.g. economic data. Online observations also single time series (D t ) Ensemble data from measurements (experiments) or seasonal data (D j t ) Modeling ensemble data is often easier than single time series, because we have the same stochastic information of a given point in time t with multiple realizations. By D t t = 1,..., T we denote the data observation at time in the case that we have only one single time series observation. By D j t t = 1,..., T j = 1,..., l we denote the time series observation at time t and measurement j. Data D t is recorded in discrete intervals (sampled data) and assumed to be generated by an SDE. Stochastic Systems,

3 Modeling real-world data Stochastic differential equations are can be viewed as stochastic process from two point of view: Stochastic process with a probability distribution p(t θ) across-time, where θ denotes the parameters A stochastic process, where the changes in the resulting time series is the stochastic process, i.e. p(d t D t 1 θ) or p( D t D t 1 D θ) t 1 The first interpretation is help full to describe ensemble data and the second to analyze single time series. In order to deal with discrete data, all SDEs need to be discretized. Stochastic Systems,

4 Discrete-time white noise When we assume a discrete white noise process ε t which identically and independently distributed (i.i.d) with zero expectation and σ 2 variance. Then following statistical properties apply: E[ε t ] = 0 (1) E[ε 2 t ] = σ2 (2) E[ε t ε τ ] = 0 for t τ (3) The last property is derived from the assumption of independence. A discretized Brownian Motion is a discrete-time white noise process with ε t N (0, t) where t is the time discretization. In discrete-time models, a white noise process can be normally distributed (Gaussian white noise) but can be distributed by any other distribution as long as the i.i.d. assumption is valid. Stochastic Systems,

5 Statistical Theory Stationarity: A process is covariance-stationary if and only if (iff): E[D t ] = µ t (4) E[(D t µ)(d t j µ)] = ν j t j. (5) This form of stationarity is often also called weak stationarity. For example the process: D t = µ + ε t where ε t is white noise process with zero mean and σ 2 variance. In this case D t is stationary since: E[D t ] = µ { σt j = 0 E[(D t µ)(d t j µ)] = 0 j 0 Stochastic Systems,

6 Statistical Theory Stationarity: A process is strictly stationary iff for any value of j 1, j 2, j 3,... the joint distribution of D t, D t+j1, D t+j2, D t+j3,.. depend only on the intervals separating the different values of j and not the time t. Ergodicity: A process is called ergodic for the mean iff: A process is said to be ergodic for the second moment iff: lim T 1 lim t E[D t] = µ. (6) T j E[(D t µ)(d t j µ)] p ν j (7) where p denotes convergence in probability. From an ergodic process, we can estimate all necessary data without possing ensemble data. If an process is not ergodic, we have only one observation, we can not identify its parameters. Stochastic Systems,

7 Statistical Theory It can be shown that if the autocovariance of a covariance stationary process satisfies the following: ν j < (8) j=0 then the process is ergodic for the mean. Stochastic Systems,

8 First Order Moving Average Process When ε k be a Gaussian white noise processes with mean zeros and variance σ 2 and consider the process: Y k = µ + ε k + θε k 1, (9) where µ and θ are nay real valued constant. The process is called first order moving average process and ussualy denotes as MA(1). The expectation of Y k is given as: E[Y k ] = E[µ + ε k + θε k 1 ] = µ + E[ε k ] + θe[ε k 1 ] = µ The constant µ is thus the mean of the processes. Stochastic Systems,

9 First Order Moving Average Process The variance of the MA(1) is calculated as Var[Y k ] = E[(Y k µ) 2 ] = E[(ε k + θε k 1 ) 2 ] = E[ε 2 k + θ2 ε 2 k 1 + 2θε kε k 1 ] = σ θ 2 σ 2 = (1 + θ 2 )σ 2 The first autocovariance is computed as E[(Y k µ)(y k 1 µ)] = E[(ε k + θε k 1 )(ε k 1 + θε k 2 )] = E[ε k ε k 1 + θε 2 k 1 + θε k 1ε k θ 2 ε k 1 ε k 2 ] = 0 + θσ = θσ 2 Stochastic Systems,

10 First Order Moving Average Process The higher order autocovariances are all zero: E[(Y k µ)(y k j µ)] = E[(ε k + θε k 1 )(ε k j + θε k j 1 )] = 0. Since the mean and autocovariance are not function of times, an MA(1) process is covariance stationary regardless of the value of θ, since ν j = (1 + θ 2 )σ 2 + θσ 2. (10) j=0 When ε k is a Gaussian white noise process, the MA(1) is ergodic for all moments. Stochastic Systems,

11 First Order Moving Average Process Autocorrelation is defined as ρ 1 = Cov[Y k Y k 1 ]/ V ar[y k ]V ar[y k 1 ] and V ar[y k ] = V ar[y k+1 ] = (1 + θ 2 )σ 2. Therefore we get ρ 1 = θσ 2 (1 + θ 2 )σ 2 = θ 1 + θ 2. (11) All higher order autocorrelations are zero. Positive values of θ induce positive autocorrelation, negative values induces negative autocorrelation. The autocorrelation of an MA(1) process can be between -0.5 and 0.5. For any value of ρ 1, there exists two values of θ, see ρ 1 = (1/θ) 1 + (1/θ) = θ 2 (1/θ) 2 θ 2 (1 + (1/θ) 2 ) = θ 1 + θ 2. (12) Stochastic Systems,

12 Q-th Order Moving Average Process A q-th order moving average process, denotes as MA(q) is given by Y k = µ + ε k + θ 1 ε k 1 + θ 2 ε k θ q ε k q, (13) where µ and θ 1,...θ 1 are real valued constant. The expectation of Y k is given as: E[Y k ] = µ + E[ε k ] + θ 1 E[ε k 1 ] θ q E[ε k q ] = µ The constant µ is thus the mean of the processes. The variance is computed as The variance of the MA(1) is calculated as Var[Y k ] = E[(Y k µ) 2 ] = E[(ε k + θ 1 ε k 1 + θ 2 ε k θ q ε k q ) 2 ]. Since all ε s are uncorrelated, all cross terms ε j ε k are zero. Stochastic Systems,

13 Q-th Order Moving Average Process The variance is therefore Var[Y k ] = σ 2 + θ 2 1 σ θ 2 q σ2 = σ 2 (1 + q θ 2 j ). j=1 The autocovariance is for the i-th lag is computed as ν i = E[(ε k + q θ j ε k j )(ε k i + j=1 q θ j ε k j i )] j=1 = E[θ i ε 2 k i + θ i+1θ 1 ε 2 k i 1 + θ i+2θ 2 ε k i θ q θ q i ε 2 k q ] Stochastic Systems,

14 Q-th Order Moving Average Process The ε s at different dates have been drop since their product is zero. For i > q there are no ε s with common dates in the definition of of ν i. We get for the autocovariance thus ν i = { (θi + θ i+1 θ 1 + θ i+2 θ θ q θ q j )σ 2 i = 1, 2,..., q 0 i > q The autocorrelation is given as ν i = θ i +θ i+1 θ 1 +θ i+2 θ θqθ q j (1+ q j=1 θ2 j ) i = 1, 2,..., q 0 i > q Stochastic Systems,

15 Autoregressive first order Process We analyze an so-called first order autoregressive (AR(1)): Y k+1 = c + ΦY k + ε k Long-term mean: We take the mean on both sides of the equation E[Y k+1 ] = E[c + ΦY k + ε k ] E[Y k+1 ] = c + ΦE[Y k ] µ = = c + Φµ µ = c 1 Φ Stochastic Systems,

16 Autoregressive first order Process µ = c 1 Φ From this formula, we can see that Φ must be smaller than 1, otherwise the mean would be negative when c 0. Also we have assumed that E[Y k ] = E[Y k+1 ] = µ and exists. Second moment: Y k+1 = µ(1 Φ) + ΦY k + ε k Y k+1 µ = Φ(Y k µ) + ε k Stochastic Systems,

17 Autoregressive first order Process E[(Y k+1 µ) 2 ] = E[(Φ(Y k µ) + ε k ) 2 ] = Φ 2 E[(Y k µ)] + 2ΦE[(Y k µ)ε k ] + E[ε 2 k ] E[(Y k+1 µ) 2 ] = Φ 2 E[(Y k µ)] + σ 2 where E[(Y k µ)ε k ] = 0. Assuming covariance stationarity, i.e. E[(Y k+1 µ) 2 ] = ν and E[(Y k µ) 2 ] = ν. We get: ν = Φ 2 ν + σ 2 ν = σ 2 1 Φ 2 Stochastic Systems,

18 Autoregressive first order Process Autocorrelation: E[(Y k+1 µ)(y k µ)] = E[(Φ(Y k µ) + ε k )(Y k µ)] E[(Y k+1 µ)(y k µ)] = ΦE[((Y k µ) 2 ] + E[(ε k )(Y k µ)]. Again E[(ε k )(Y k µ)] = 0 and E[((Y k µ) 2 ] = ν we get: E[(Y k+1 µ)(y k µ)] = Φν = Φ σ2 1 Φ 2. Autocorrelation is defined as Cov[Y k+1 Y k ]/ V ar[y k+1 ]V ar[y k ] and V ar[y k ] = V ar[y k+1 ] = ν. Therefore we get acf(y k+1 Y k ) = ϕ (14) Stochastic Systems,

19 Autoregressive first order Process AR(1) process with Y k+1 = ΦY k + c + ε k hast the following properties: Mean : µ = V ariance : ν = Autocorr. : acf(y k+j Y k ) = ϕ j c 1 Φ σ 2 1 Φ 2 A covariance stationary AR(1) process is only well defined when Φ 1, otherwise the variance is infinite and mean makes no sense. We can check if a process is a AR(1) process by calculating the mean and variance over-time and the autocorrelation. From these three calculations the parameters can be estimated. It is important to notice that the autocorrelation is exactly the coefficient of the autoregression. Stochastic Systems,

20 Example: GBM and mean reversion The two examples have a discrete-time structure as an first-order autoregressive process with: (I) (II) Φ = 1 + µ t Φ = 1 b t When we assume that µ > 0, the first model is instable and we can not use the techniques from auto-regressions to estimate the parameters. We have to deal differently in this case, by using p = ln(x), since then we get an SDE with constant coefficients. When we assume that b > 0, the process is a true mean reverting process that can be estimated by AR(1) methods. In this case, the discrete-time model is also stable. Stochastic Systems,

21 First order difference equation A first order difference equation give by y k+1 = Φy k + w k is of the same type as an AR(1) equation. The can be solved by recursive substitution since y 1 = Φy 0 + w 0, y 2 = Φy 1 + w 1, y 3 = Φy 2 + w When we plug in the results of y 1 into the equation for y 2, we get y 2 = Φ 2 y 0 + Φw 0 + w 1, which in turn we plug in the equation for y 3 and get y 3 = Φ 3 y 0 + Φ 2 w 0 + Φw 1 + w 2. Expanding this further on, we get the general solution y k+1 = Φ k+1 y 0 + k Φ k j w j j=0 Stochastic Systems,

22 p-th order difference equation A p-th order difference equation give by y k+1 = Φ 1 y k + Φ 2 y k Φ p y k p+1 + w k We write this in a vectorized form with: y k Φ 1 Φ 2 Φ 3 Φ p 1 Φ p y k ξ k = y k 2 F = y k p v k = w k and get ξ k+1 = F ξ k + v k. Stochastic Systems,

23 p-th order difference equation The solution is given as: ξ k = F k ξ 0 + F k 1 v 0 + F k 2 v 1 + F k 3 v F v k 1 + v k The p-th order difference equation is stable (or stationary) when the eigenvalues of F are smaller than one. The eigenvalue polynom is given as: λ p Φ 1 λ p 1 Φ 2 λ p 2... Φ p 1 λ Φ p = 0 Stochastic Systems,

24 Lag operators A highly useful operator for time series models is the lag operator. Suppose that we start with an sequence of data x k where k =... and generate a new sequence y k where the value of y for date k is equal to the value x took at time k 1 y k = x k 1. This is described as applying the Lag operator on x k and is represented by symbol L: Lx k = x k 1. Consider applying the lag operator twice to a series: L(Lx k ) = L(x k 1 ) = x k 2, which is denotes as L 2 or L 2 x k = x k 2. Stochastic Systems,

25 Lag operators In general for any integer l L l x k = x k l, The lag operator is linear and can be treated as the multiplation operator and thus L(βx k ) = βlx k = βx k 1 L(x k + w k ) = Lx k + Lw k y k = (a + bl)lx k = (al + bl 2 )x k = ax k 1 + bx k 2 An expression such as (al + bl 2 ) is referred as a polynomial lag operators. Notice if x k = c then Lx k = x k 1 = c Stochastic Systems,

26 First order difference equation and lag operators A first order difference equation give by can be written as y k+1 = Φy k + w k y k+1 = ΦLy k+1 + w k y k+1 ΦLy k+1 = w k y k+1 (1 L) = w k when we multiply both sides with (1 + ΦL + Φ 2 L 2 + Φ 3 L Φ k L k ) we get y k+1 (1 + ΦL + Φ 2 L 2 + Φ 3 L Φ k L k )(1 L) = (1 + ΦL + Φ 2 L 2 + Φ 3 L Φ k L k )w k Stochastic Systems,

27 First order difference equation and lag operators y k+1 (1 Φ k+1 L k+1 ) = (1 + ΦL + Φ 2 L 2 + Φ 3 L Φ k L k )w k y k+1 (1) = Φ k+1 L k+1 y k+1 + (1 + ΦL + Φ 2 L 2 + Φ 3 L Φ k L k )w k y k+1 = Φ k+1 y 0 + w k + Φw k 1 + Φ 2 w k 2 + Φ 3 w k Φ k w 0 which results exactly in the same solution as by the usage of the recursive substitution. Stochastic Systems,

28 P-th order difference equation and lag operators The p-th order difference equation given by y k+1 = Φ 1 y k + Φ 2 y k Φ p y t p+1 + w k, can be written using the lag operator as: y k+1 (1 Φ 1 L Φ 2 L 2 Φ 3 L 3... Φ p L p ) = w k, which results in the same polynomial as in the case of the eigenvalues of the matrix F. Stochastic Systems,

29 P-th order difference equation and lag operators Since (1 Φ 1 z Φ 2 z 2 Φ 3 z 3... Φ p z p ) = (1 λ 1 z)(1 λ 2 z)...(1 λ p z), and by defining z 1 = λ and multiply both sides with z p and we get (λ p Φ 1 λ p 1 Φ 2 λ p 2... Φ p ) = (λ λ 1 )(λ λ 2 z)...(λ λ p z), which is nothing but the root search of the left side polynomial: λ p Φ 1 λ p 1 Φ 2 λ p 2... Φ p 1 λ Φ p = 0 Stochastic Systems,

30 Second order Autoregression A second order autoregressive process (denotes as AR(2)) has the form: written in lag notation: Y k+1 = c + Φ 1 Y k + Φ 2 Y k 1 + ε k Y k+1 (1 Φ 1 L Φ 2 L 2 ) = c + ε k The AR(2) is stable if the roots of (1 Φ 1 z Φ 2 z 2 ) = 0 lie outside the unit circle. The process has the mean value: E[Y k+1 ] = c + Φ 1 E[Y k ] + Φ 2 E[Y k 1 ] + E[ε k ] µ = c + Φ 1 µ + Φ 2 µ µ = c 1 Φ 1 Φ 2 Stochastic Systems,

31 Second order Autoregression To find the second moments of the AR(2), we write Y k+1 = µ(1 Φ 1 Φ 2 ) + Φ 1 Y k + Φ 2 Y k 1 + ε k Y k+1 µ = Φ 1 (Y k µ) + Φ 2 (Y k 1 µ) + ε k. multiplying both sides with Y k+1 j µ we get and taking the expectation yields: ν j = Φ 1 ν j 1 + Φ 2 ν j 2 j = 1, 2, 3.. The auto-covariances follow the same second order difference equation as the AR(2) process. The autocorrelation is given by dividing the last equation by ν 0 we get ρ j = Φ 1 ρ j 1 + Φ 2 ρ j 2 j = 1, 2, 3.. Stochastic Systems,

32 Second order Autoregression For j = 1 we get for the autocorrelation: ρ 1 = Φ 1 + Φ 2 ρ 1 ρ 1 = Φ 1 1 Φ 2, since ρ 0 = 1 and ρ 1 = ρ 1. For j=2 we get ρ 2 = Φ 1 ρ 1 + Φ 2 ρ 2 = Φ Φ 2 + Φ 2. The covariance is computed by E[(Y k+1 µ)(y k+1 µ)] = Φ 1 E[(Y k µ)(y k+1 µ)] + Φ 2 E[(Y k 1 µ)(y k+1 µ)] + E[ε k (Y k+1 µ)], Stochastic Systems,

33 Second order Autoregression The covariance equation can be written as ν 0 = Φ 1 ν 1 + Φ 2 ν 2 + σ 2 j = 1, 2, 3.. The last term from noticing E[ε k (Y k+1 µ)] = E[(ε k )(Φ 1 (Y k µ) +Φ 2 (Y k 1 µ) + ε k )] = Φ Φ σ 2 The covariance equation can be written as ν 0 = Φ 1 ρ 1 ν 0 + Φ 2 ρ 2 ν 0 + σ 2, ν 0 = (1 Φ 2 )σ 2 (1 + Φ 2 )[(1 Φ 2 2 ) Φ2 1 ] Stochastic Systems,

34 P-th order Autoregression A p-th order autoregressive process (denotes as AR(p)) has the form: written in lag notation: Y k+1 = c + Φ 1 Y k + Φ 2 Y k Φ p Y k p+1 + ε k Y k+1 (1 Φ 1 L Φ 2 L 2... Φ p L p ) = c + ε k The AR(2) is stable if the roots of (1 Φ 1 z Φ 2 z 2... Φ p z p ) = 0 lie outside the unit circle. The process has the mean value: E[Y k+1 ] = c + Φ 1 E[Y k ] + Φ 2 E[Y k 1 ] Φ p E[Y k p+1 ] + E[ε k ] µ = c + Φ 1 µ + Φ 2 µ Φ p µ µ = c 1 Φ 1 Φ 2... Φ p Stochastic Systems,

35 P-th order Autoregression The autocovariance is given as ν j = { Φ1 ν j 1 + Φ 2 ν j Φ p ν j p j = 1, 2, 3.. Φ 1 ν 1 + Φ 2 ν Φ p ν p j = 0 The equation for the autocorrelation is given as ρ j = Φ 1 ρ j 1 + Φ 2 ρ j Φ 2 ρ j p j = 1, 2, 3.. which is known as the Yule-Walker equation. The equation is again a p-th order difference equation. Stochastic Systems,

36 Mixed Autoregressive Moving Average Process (ARMA) A ARMA(p,q) process includes autoregressive elements as well as moving average elements Y k+1 = c + Φ 1 Y k + Φ 2 Y k Φ p Y k p+1 The process is written in Lag form as +ε k + θ 1 ε k 1 + θ 2 ε k θ q ε k q. Y k+1 (1 Φ 1 L Φ 2 L 2... Φ p L p ) = +c + ε k (1 + θ 1 L + θ 2 L θ q L q ). Provided that the roots of (1 Φ 1 L Φ 2 L 2... Φ p L p ) = 0 lie outside the unit circle we obtain the following general form for an ARMA(p,q) process: Y k+1 = µ + (1 + θ 1L + θ 2 L θ q L q ) (1 Φ 1 L Φ 2 L 2... Φ p L p ) ε k. Stochastic Systems,

37 Mixed Autoregressive Moving Average Process (ARMA) The mean is given as µ = c (1 Φ 1 L Φ 2 L 2... Φ p L p ). The stationarity of an ARMA process depends only on the autoregressive parameters and not on the moving average parameters. The more convenient form of an ARMA(p,q) process is given as Y k+1 µ = Φ 1 (Y k µ) + Φ 2 (Y k 1 µ) Φ p (Y k p+1 µ) +ε k + θ 1 ε k 1 + θ 2 ε k θ q ε k q. The autocovariance equation for j > q is given as ν j = Φ 1 ν j 1 + Φ 2 ν j Φ p ν j p j = q + 1, q + 2,... Stochastic Systems,

38 Mixed Autoregressive Moving Average Process (ARMA) Thus after q lags, the autocovariance function ν j follows a p-th order difference equation governed by the the autoregressive part. Note that last equation does not hold for j q since the correlation between θ j ε k j and Y t j cannot be neglected. In order to analyze the auto-covariance, we introduce the auto-covariance generating function g Y (z) = j= ν j z j which is constructed as taking the j-th auto-covariance and multiplying with z to the power of j and then summing over all possible values of j Stochastic Systems,

39 Parameter Identification We start to examine the appropriate log-likelihood function with our frame work of the discretized SDEs. Let ψ R d be the vector of unknown parameters, which belongs to the admissible parameter space Ψ, e.g. a, b, etc. The various system matrices of the discrete models, as well as the variances of stochastic processes, given in the previous section, depend on ψ. The likelihood function of the state space model is given by the joint density of the observational data x = (x N, x N 1,..., x 1 ) l(x, ψ) = p(x N, x N 1,..., x 1 ; ψ) (15) which reflects how likely it would have been to have observed the date if ψ were the true values of the parameters. Stochastic Systems,

40 Parameter Identification Using the definition of condition probability and employing Bayes s theorem recursively, we now write the joint density as product of conditional densities l(x, ψ) = p(x N x N 1,..., x 1 ; ψ)... p(x k x k 1,..., x 1 ; ψ)... p(x 1 ; ψ) (16) where we approximate the initial density function p(x 1 ; ψ) by p(x 1 x 0 ; ψ). Furthermore, since we deal with a Markovian system, future values of x l with l > k only depend on (x k, x k 1,..., x 1 ) through the current values of x k, the expression in (16) can be rewritten to depend on only the last observed values, thus it is reduced to l(x, ψ) = p(x N x N 1 ; ψ)... p(x k x k 1 ; ψ)... p(x 1 ; ψ) (17) Stochastic Systems,

41 Parameter Identification A Euler discretized SDE is given as: x k+1 = x k + f(x k, k) t + g(x k, k) tε k (18) where ε k is an i.i.d random variable with normal distribution, zero expectation and unit variance. Since f and g are deterministic functions, x k+1 is conditional (on x k normal distributed with E[x k+1 ] = x k + f(x k, k) t (19) Var[x k+1 ] = g(x k, k)g(x k, k) T t (20) p(x k+1 ) = 1 (2π) n 2 Var[xk+1 ] e 1 2 (x k+1 E[x k+1 ])T Var[x k+1 ] 1 (x k+1 E[x k+1 ]) (21) Stochastic Systems,

42 Parameter Identification The parameter of an SDEs are estimated by maximizing the log-likelihood. To maximize the log-likelihood is equivalent to maximizing the likelihood, since the log function is strictly increasing function. The log likelihood is given as: ln(l(x, ψ)) = ln(p(x N x N 1 ; ψ)... p(y k x k 1 ; ψ)... p(x 1 ; ψ))(22) ln(l(x, ψ)) = N ln(p(x j x j 1 ; ψ) (23) j=1 ln(l(x, ψ)) = N { j=1 ln((2π) n 2 Var[x k+1 ] 1 2) 1 } 2 (x k+1 E[x k+1 ]) T Var[x k+1 ] 1 (x k+1 E[x k+1 ]) (24) Stochastic Systems,

43 Example: Parameter Identification We want to identify the parameters of a AR(1) process without any constant: which has the log-likelihood function: x k+1 = Φx k + σε k (25) ln(l(x, ψ)) = N ln((2π) 1 2σ 1 ) (x k+1 Φx k ) 2. (26) 2σ 2 j=1 The function is maximized by taking the derivatives for Φ and σ. The log-likelihood function has the same from as an so-called ordinary least-square (OLS) estimation. Stochastic Systems,

44 Example: Parameter Identification The solution of the maximum likelihood estimations is: Φ = N j=1 x k+1x k N j=1 x2 k (27) σ = 1 N N (x k+1 Φx k ) 2 (28) j=1 The solution for Φ is the same solution as for an OLS regression. Furthermore, the solution is exactly the definition of autocorrelation with lag one. The estimation for σ is the empirical standard deviation of the estimation of the empirical white noise, i.e. ε k = x k+1 Φx k Stochastic Systems,