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STAT 6104 - Financial Time Series Chapter 2 - Probability Models Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 34

Agenda 1 Introduction 2 Stochastic Process Definition 1 Stochastic Definition 2 Finite dimensional distribution function Definition 3 Strictly Stationary Definition 4 Weakly Stationary Definition 5 Auto covariance function 3 Strictly and Weakly Stationary 4 Examples 5 Sample Correlation Function 6 Summary Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 2 / 34

Introduction Assume that we have done the preliminary analysis to remove trend T t and seasonality S t N t = X t T t S t We use basic probability theories and construct models to describe the microscopic component {N t } Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 3 / 34

Stochastic Process - Definition 1 Stochastic A collection of random variable {X t : t R} is called a stochastic process Continuous {X(t) : 0 t < } Discrete {X t : t = 1, 2,..., n} Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 6 / 34

Stochastic Process - Definition 1 Stochastic Understand the meaning of stochastic: {X t : t = 1, 2,..., n} can be regarded as a function of time t We assume that there is a sample space Ω (a set of all possible scenario) that generate the randomness Ω = (ω 1, ω 2, ω 3,......) ω 1 X t (ω 1 ), ω 2 X t (ω 2 ) For a fixed ω, X t (ω) is called a sample function/ realization/ sample path In practice, we can only observe one sample path (i.e. the situation is generated by one particular ω ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 7 / 34

Stochastic Process - Definition 2 Finite dimensional distribution function Finite dimensional distribution functions: F t (x) = P (X t1 x 1,..., X tn x n ) for all possible t = (t 1, t 2,..., t n ) where, 1 t 1 < t 2 < < t n and (X 1, X 2,..., X n ) R n, n = 1, 2,... Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 9 / 34

Stochastic Process - Definition 3 Strictly Stationary Strictly stationary - distribution of a process does not change {X t } is said to be strictly stationary if 1. for all n 2. for all (t 1, t 2,..., t n ) 3. for all τ (X t1,..., X tn ) d = (X t1 +τ,..., X tn+τ ) where d = means equal in distribution Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 11 / 34

Stochastic Process - Definition 4 Weakly Stationary Weakly Stationary: same mean same variance same covariance {X t } is said to be weakly stationary/ second order stationary/ wide-sense stationary if 1) E(X t ) = µ constant mean 2) Cov(X t, X t+τ ) = γ(τ) for all t and all τ (auto-covariance only depends on time lag τ, but not time t) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 13 / 34

Stochastic Process - Definition 5 Auto covariance function Auto covariance function(acvf) γ(τ) = Cov(X t, X t+τ) ) Auto correlation function(acf) ρ(τ) = Cov(X t,x t+τ ) Var(Xt )Var(X t+τ ) = γ(τ) γ(0) γ(0) = Var(X t ) Stationary = Var(X t+τ ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 15 / 34

Strictly vs Weakly Stationary Strictly: distribution of the process unchange Weakly: mean/covariance unchange Strictly stationary implies weakly stationary whenever E(X 2 t ) < Weakly stationary implies strictly stationary when {X t } is normal,but in general not true Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 17 / 34

Strictly vs Weakly Stationary 1. X 1, X 2,..., X n iid t 2 {X t } : Strictly stationary But E(X 2 t ) =, {X t } : NOT weakly stationary 2. X 1, X 2,..., X n iid t 3 Now E(X 2 t ) <, {X t } : Strictly and Weakly stationary 3. X 1 exp(1) 1, X 2, X 3,... iid N (0, 1) {X t } : Weakly stationary, E(X i ) = 0, Var(X i ) = 1, Cov(X i, X i+k ) = 0 {X t } : NOT Strictly stationary, since distribution of X 1,X 2 are not equal (i.e.f X1 (x 1 ) F Xt (x 2 )) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 18 / 34

Example 1 {X t } : i.i.d. random variables ρ(τ) = { 1, τ = 0 0, τ 1 (1) If a time series satisfies (1), it is called a white noise sequence i.i.d. white noise white noise i.i.d. (e.g.x 1 exp(1), X k N (0, 1), k 2) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 20 / 34

Example 2 If {Y t } satisfies {Y 1 = Y 2 = = Y }, then ρ(τ) = Cov(Y, Y ) = 1 for all τ Essentially there is only one observation (all the observations are equal) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 21 / 34

Example 3 X t = A cos θt + B sin θt where A and B are i.i.d. random variables with mean 0 and variance σ 2 Cov(X t+h, X t ) E(X t ) = E(A) cos θt + E(B) sin θt = 0 = Cov(A cos θ(t + h) + B sin θ(t + h), A cos θt + B sin θt) = cos θtcos θ(t + h)var(a) + sin θtsin θ(t + h)var(b) = σ 2 cos(θ(t + h) θt) ( cos(a B) = cos A cos B + sin A sin B) = σ 2 cos θh X t is weakly stationary! Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 22 / 34

ACVF and ACF ACVF : γ(τ) = Cov(X t, X t+τ ) ACF : ρ(τ) = Cov(X t,x t+τ ) = γ(τ) Var(X t )Var(X t+τ ) γ(0) Both ACVF and ACF are population quantities. It means that they involve the true distribution of the process {X t } If we have a dataset X 1, X 2,..., X n but we do not know the true probability distribution of the time series, we can estimate the ACF/ACVF by Sample ACVF: C k = 1 n Sample ACF : r k = C k /C 0 n k t=1 (X t X)(X t+k X) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 24 / 34

Correlogram - ACF plot Correlogram - ACF plot: A graph to summarize r k Example 1 Example 2 Use k up to k = n/3, otherwise r k is not accurate By definition, r 0 = c 0 c 0 = 1 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 25 / 34

Sample Correlation Function Questions need to address: Is Corr(X t, X t+k ) = 0? Is r k significantly different from 0? Solution: r k N (0, 1 n ) for each fixed k when n is large Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 26 / 34

Short Memory vs Long Memory Short Memory = Short term correlation Long Memory = long term correlation Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 27 / 34

Examples of ACF function Y t = Z t, {Z t } are i.i.d. Y t = 0.9Y t 1 + Z t Y t = 0.4Y t 1 + Z t Y t = 0.5Y t 1 + Z t Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 28 / 34

Examples of ACF function Y t = a cos(tω), where a, ω are constants Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 29 / 34

Examples of ACF function Y t = Y t 1 + Z t Y t = at + Z t 1. For both cases, ACF decay very slowly (Non-stationary) 2. In practice, we need to detrend and study Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 30 / 34

Summary of Chapter 2 ACVF : γ(τ) = Cov(X t, X t+τ ) ACF : ρ(τ) = Cov(X t, X t+τ ) Var(X t )Var(X t+τ ) = γ(τ) γ(0) Sample ACVF : C k = 1 n n k Sample ACF : r k = C k /C 0 t=1 (X t X)(X t+k X) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 32 / 34

Summary of Chapter 2 ACF plot (ACF r k against lag k): r k N (0, 1 n ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 33 / 34

Summary of Chapter 2 Short memory Long memory Non-stationary Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 34 / 34