Financial Time Series Analysis Week 5

Transcription

1 Financial Time Series Analysis Week 5 25 Estimation in AR moels Central Limit Theorem for µ in AR() Moel Recall : If X N(µ, σ 2 ), normal istribute ranom variable with mean µ an variance σ 2, then X µ σ N(0, ) Recall 2: Let X, X 2,, be a sequence of IID(µ, σ 2 ) Let X n be the sample mean with sample size n, given by X n n X i Note that E[ X n ] µ an V ar( X n ) σ 2 /n Central limit theorem (CLT) says that where In other wors, X n µ σ/ i N(0, ) inicates the convergence in istribution: ( ) Xn µ P σ/ x Φ(x) ( Xn µ) x N(0, σ 2 ) 2π e x2 2 x The CLT is use to establish hypothesis test or confience interval Now we consier stationary AR() moel: X t µ φ(x t µ) + ɛ t where {ɛ t } IID(0, σ 2 ɛ ) an φ < We recall that the AR() moel can be expresse as a linear time series as follows: X t µ + φ j ɛ t j Let ˆµ X n n X t, an estimator of mean µ Our goal is to establish a CLT for µ Theorem 23 In the weakly stationary AR() moel X t µ φ(x t µ) + ɛ t with mean µ where {ɛ t } IID(0, σɛ 2 ), sample mean ˆµ follows asymptotic normality: as n, ( σ 2 ) n(ˆµ µ) ɛ N 0, ( φ) 2

2 Proof: Let b j kj+ φk Then b j b j φ k kj Define a new linear time series Y t b jɛ t j CHECK: b 2 j since φ < b2 j < kj+ φ k 2 ( ) φ j+ 2 φ kj+ φ 2 ( φ) 2 φ k φ j φ 2j φ 2 ( φ) 2 φ 2 < Thus V ar(y t ) σ 2 ɛ b2 j < Also note that EY t 0 an Cov(Y t, Y t+h ) σ 2 ɛ b jb j+h, an thus {Y t } is stationary Now we express X t in terms of {Y t } X t µ + φj ɛ t j µ + ɛ t + j φj ɛ t j µ + ɛ t + j (b j b j )ɛ t j µ + ( φj )ɛ t ( j φj )ɛ t + j b j ɛ t j j b jɛ t j µ + ( φj )ɛ t b 0 ɛ t j b jɛ t j + j b j ɛ t j (letting k j, j k + ) µ + φj )ɛ t b jɛ t j + k0 b kɛ t k ( ) µ + φj ɛ t Y t + Y t Thus Take n X t µ to both sies: X t µ n φ j n φ j ɛ t Y t + Y t ɛ t n Y t + n Y t ˆµ µ ( ) φ n ɛ t n (Y n Y 0 ) ( ) n(ˆµ µ) n φ Since {ɛ t } IID(0, σ 2 ɛ ), by the CLT of IID sequence, we know ɛ t ( ɛ n 0) 2 ɛ t (Y n Y 0 ) N(0, σ 2 ɛ )

3 where ɛ n n ɛ t, sample mean of {ɛ t } Thus, we have ( ) ( n ɛ t N 0, φ Note that Y n p 0, Y 0 σ 2 ɛ ( φ) 2 ) p 0 as n Hence, we have (ˆµ µ) ( σ 2 ) ɛ N 0, ( φ) 2 Yule-Walker Estimator in AR(p) Moel Consier a stationary AR(p) moel with mean zero: where ɛ t W N(0, σ 2 ɛ ) X t φ X t + φ 2 X t 2 + φ p X t p + ɛ t Our goal is to estimate unknown coefficients φ, φ 2,, φ p of AR(p) moel from the ata Suppose the financial log return ata are observe: r, r 2,, r T Recall: Autocorrelation function ρ(h) is estimate by sample autocorrelation function where ˆρ(h) ˆγ(h) ˆγ(0) ˆγ(h) T h (r t r)(r t+h r) with r T h T T r t Using the sample autocorrelation function ˆρ(h), we will estimate the coefficients φ, φ 2,, φ p In orer to construct Yule-Walker Equation, we multiply both sies of AR(p) moel by X t j, for j, 2,, p, an take expectation to obtain E[X t X t j ] E[φ X t X t j + φ 2 X t 2 X t j + + φ p X t p X t j + ɛ t X t j ] γ(j) φ γ(j ) + φ 2 γ(j 2) + + φ p γ(j p) + 0 This equation is calle Yule-Walker Equation of AR(p) moel Divie by γ(0) to obtain γ(j) γ(0) φ γ(j ) γ(j 2) γ(j p) + φ φ p γ(0) γ(0) γ(0) ρ(j) φ ρ(j ) + φ 2 ρ(j 2) + + φ p ρ(j p) 3

4 Note that ρ( h) ρ(h) below ρ() φ ρ(0) + φ 2 ρ() + φ 3 ρ(2) + + φ p ρ(p ) if j ρ(2) φ ρ() + φ 2 ρ(0) + φ 3 ρ() + + φ p ρ(p 2) if j 2 ρ(3) φ ρ(2) + φ 2 ρ() + φ 3 ρ(0) + + φ p ρ(p 3) if j 3 ρ(p) φ ρ(p ) + φ 2 ρ(p 2) + + φ p ρ(0) if j p Its matrix form is given by ρ() ρ(2) ρ(3) ρ(p) ρ(0) ρ() ρ(2) ρ(p ) ρ() ρ(0) ρ() ρ(p 2) ρ(2) ρ() ρ(0) ρ(p 3) ρ(p ) ρ(p 2) ρ(p 3) ρ(0) φ φ 2 φ 3 φ p Denote it by B AX where B is p column vector, A is p p matrix, an X is p column vector with unknown coefficients φ, φ 2,, φ p to be estimate Thus X A B, if A is invertible That is, φ φ 2 φ 3 φ p ρ(0) ρ() ρ(2) ρ(p ) ρ() ρ(0) ρ() ρ(p 2) ρ(2) ρ() ρ(0) ρ(p 3) ρ(p ) ρ(p 2) ρ(p 3) ρ(0) ρ() ρ(2) ρ(3) ρ(p) Therefore, using ˆρ(h) in place of ρ(h), we obtain Yule-Walker estimator ( ˆφ, ˆφ 2,, ˆφ p ) of unknown coefficients (φ, φ 2,, φ p ) : ˆφ ˆφ 2 ˆφ 3 ˆφ p ˆρ() ˆρ(2) ˆρ(p ) ˆρ() ˆρ() ˆρ(p 2) ˆρ(2) ˆρ() ˆρ(p 3) ˆρ(p ) ˆρ(p 2) ˆρ(p 3) ˆρ() ˆρ(2) ˆρ(3) ˆρ(p) Example : In a stationary AR() moel, Yule-Walker equation is given by γ() φ γ(0) 4

5 thus ρ() φ an hence Yule-Walker estimator of φ is given by ˆφ ˆρ() Example 2: In a stationary AR(2) moel, Yule-Walker equation is given by γ(j) φ γ(j ) + φ 2 γ(j 2) for j, 2 Thus for j, 2, ρ(j) φ ρ(j ) + φ 2 ρ(j 2) That is, ρ() φ ρ(0) + φ 2 ρ() ρ(2) φ ρ() + φ 2 ρ(0) Its matrix from is ρ() ρ(2) ρ(0) ρ() ρ() ρ(0) φ φ 2 an hence Yule-Walker estimators of φ, φ 2 are given by ˆφ ˆρ() ˆρ() ˆφ 2 ˆρ() ˆρ(2) Recall: Inverse matrix of 2 2 matrix: a b c b a bc c a Therefore, Yule-Walker estimators of φ, φ 2 are given by ˆφ ˆρ() ˆφ ˆρ() 2 ˆρ() 2 ˆρ() ˆρ(2) ˆρ() 2 ˆρ() ˆρ()ρ(2) ˆρ(2) ˆρ() 2 ˆφ ˆρ() ˆρ()ρ(2) ˆρ() 2, ˆφ2 ˆρ(2) ˆρ()2 ˆρ() 2 Regression Moel Before we start the least square estimator of AR moels, we recall regression moel an its least square estimator Let (X, Y ) (X n, Y n ) be observe We wish to know a linear relation between X t an Y t for each t Y t α + βx t + ɛ t {ɛ t } IID(0, σɛ 2 ) () 5

6 We will estimate unknown coefficients α, β Moel () is calle a regression moel To estimate, we use least square metho The least square metho minimizes the sum of square error (SSE): ɛ 2 t (Y t α βx t ) 2 We will fin α an β so that (Y t α βx t ) 2 is minimize Let (ˆα, ˆβ) arg min (α,β) SSE Let f(α, β) (Y t α βx t ) 2, which is a function of unknown (α, β) β f(α, β) 2 α f(α, β) 2 (Y t α βx t )( X t ) 0 (Y t α βx t )( ) 0 (X ty t αx t βx 2 t ) 0 (Y t α βx t ) 0 From the last equation, we obtain ( α Y t β n Thus, X ty t α X t β X2 t 0 Y t αn β X t 0 X t Y t (Ȳ β X) ) X t Ȳ β X X t β X t Y t n XȲ + βn X 2 β Xt 2 0 Xt 2 0 Therefore ˆβ X ty t n XȲ n, ˆα Ȳ ˆβ X X2 t n X 2 (ˆα, ˆβ) is calle orinary least squares (OLS) estimator of (α, β) Least Squares Estimator of AR() Moel Now we will fin the OLS estimator of the coefficient φ of AR() moel with mean zero (If the mean is µ 0, then consier AR() moel {X t µ : t, 2, }) X t φx t + ɛ t, {ɛ t } IID(0, σɛ 2 ) 6

7 Sum of square errors from ata X, X 2,, X n is given by S(φ) : (X t φx t ) 2 We will fin φ so that S(φ) is minimize Let Thus ˆφ arg min S(φ) φ φ S(φ) 2 (X t φx t )( X t ) 0 X t X t φ Xt 2 0 ˆφ X tx t n X2 t ˆφ is calle OLS estimator of φ in AR() moel Central Limit Theorem of OLS Estimator ˆφ Theorem 24 In AR() moel, X t φx t + ɛ t, with {ɛ t } IID(0, σ 2 ɛ ), the OLS estimator ˆφ follows the asymptotic normality as n, ( ˆφ φ) N ( 0, σ 2 ɛ (EX 2 ) ) Proof: X t φx t + ɛ t Divie by X2 t Thus X t X t φxt 2 + ɛ t X t X t X t φ Xt 2 + ɛ t X t in both sies to obtain ˆφ φ + ɛ tx t n X2 t ( ˆφ φ) ɛ tx t X2 t ɛ tx t n X2 t We will see the limits of numerator an enominator, respectively [] First, we observe the numerator n ɛ tx t : Its mean is E[ɛ t X t ] 0 7

8 Its variance is ( ) V ar ɛ t X t n n (n )V ar(ɛ tx t ) V ar(ɛ t X t ) E[ɛ t 2 X 2 t ] as n, (since E[ɛ t X t ] 0), Thus E[ɛ 2 t ]E[X 2 t ] σ 2 ɛ EX 2, by inepenence of ɛ t, X t an by the stationarity of X t ɛ t X t N ( 0, σ 2 ɛ EX 2 ) (Its etaile proof will be omitte since it is a grauate level A key point is that the sequence {ɛ t X t : t 2, 3,, n} are inepenent an we can apply by the CLT to the sequence) [2] Seconly, we observe the enominator n X2 t : X 2 p t E[X 2 n t ] EX 2 Therefore, by [] an [2], we have ( ˆφ φ) EX 2 N ( ( 0, σɛ 2 EX 2 ) N 0, σ 2 ɛ EX 2 ) where we use the fact that V ar(ax) a 2 V ar(x) Forecasting in AR() Moel Recall: (i) Conitional expectation Y given X : E[Y X] is a function of X, because E[Y X x] yf(y x)y which is a function of x (ii) Conitional expectation Y given X,, X t : E[Y X, X 2 X t ] is a function of (X, X 2,, X t ) (iii) E[g(X, X 2,, X n ) X,, X n ] g(x, X 2,, X n ) where g is a function from R n to R We stuy forecasting future values of a stationary AR() moel with mean µ X t µ + φ(x t µ) + ɛ t, φ < Mean µ an coefficient φ can be estimate from ata, an so we may assume that µ an φ are known: If these are unknown, we use estimators; sample mean ˆµ an Yule-Walker, or OLS estimator ˆφ, respectively: µ ˆµ sample mean 8

9 φ ˆφ Yule-Walker estimator or OLS estimator which are efine from ata X,, X t observe at times, 2, t Now we will forecast future ata Let t be the present time Suppose that we have information X, X 2,, X t an mean µ, coefficient φ One-step ahea forecast: Let ˆX t+ ˆX t () be one-step ahea forecast of AR() moel ˆX t+ ˆX t () E[X t+ X,, X t ] E[µ + φ(x t µ) + ɛ t+ X,, X t ] µ + φ(x t µ) + E[ɛ t+ X,, X t ] µ + φ(x t µ) Two-step ahea forecast: Let ˆX t+2 ˆX t (2) be two-step ahea forecast of AR() moel ˆX t+2 ˆX t (2) E[X t+2 X,, X t ] E[µ + φ(x t+ µ) + ɛ t+2 X,, X t ] µ + φ(e[x t+ X X t ] µ) + E[ɛ t+2 X X t ] µ + φ( ˆX t () µ) + 0 µ + φ(µ + φ(x t µ) µ) µ + φ 2 (X t µ) By the mathematical inuction, we may assume ˆX t+l ˆX t (l ) µ + φ( ˆX t (l 2) µ) ˆX t (l ) µ φ( ˆX t (l 2) µ), an for each k 2, 3,, l, ˆX t (k) µ φ( ˆX t (k ) µ) l-step ahea forecast: 9

10 Let ˆX t+l ˆX t (l) be l-step ahea forecast of AR() moel ˆX t+l ˆX t (l) E[X t+l X,, X t ] E[µ + φ(x t+l µ) + ɛ t+l X,, X t ] µ + φ( ˆX t (l ) µ) since ˆXt (l ) µ + φ( ˆX t (l 2) µ) µ + φ 2 ( ˆX t (l 2) µ) µ + φ l ( ˆX t () µ) µ + φ l (X t µ) Since φ < φ l 0 as l Thus ˆX t (l) µ as l It means that future values of stationary AR() moel approaches to the mean as time goes to infinity 0

11 Homework 3 Consier the following log-return financial time series ata: time t log return r t Compute sample autocovariance ˆγ(h) an sample autocorrelation ˆρ(h) for h, 2 2 Assume that we apply the ata to AR() moel an compute the Yule-Walker estimate of φ 3 Assume that we apply the ata to AR(2) moel an compute the Yule-Walker estimates of φ an φ 2 4 Assume that we apply the ata to AR() moel an compute the OLS estimate of φ 5 Forecast the future values of log-returns at time an time 2 by using the Yule-Walker estimate in problem 2 6 Forecast the future values of log-returns at time an time 2 by using the OLS estimate in problem 4 Warning: In some theories stuie in class, the moel has mean zero Thus when you apply the theory, you shoul use the ata with mean zero, compute from the ata in table above