Introduction to Time Series Analysis. Lecture 6.

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1 Introduction to Time Series Analysis. Lecture 6. Peter Bartlett bartlett/courses/153-fall2010 Last lecture: 1. Causality 2. Invertibility 3. AR(p) models 4. ARMA(p,q) models 1

2 Introduction to Time Series Analysis. Lecture 6. Peter Bartlett bartlett/courses/153-fall ARMA(p,q) models 2. Stationarity, causality and invertibility 3. The linear process representation of ARMA processes: ψ. 4. Autocovariance of an ARMA process. 5. Homogeneous linear difference equations. 2

3 Review: Causality A linear process {X t } is causal (strictly, a causal function of {W t }) if there is a ψ(b) = ψ 0 + ψ 1 B + ψ 2 B 2 + with ψ j < j=0 and X t = ψ(b)w t. 3

4 Review: Invertibility A linear process {X t } is invertible (strictly, an invertible function of {W t }) if there is a π(b) = π 0 + π 1 B + π 2 B 2 + with π j < j=0 and W t = π(b)x t. 4

5 Review: AR(p), Autoregressive models of order p An AR(p) process {X t } is a stationary process that satisfies X t φ 1 X t 1 φ p X t p = W t, where {W t } WN(0, σ 2 ). Equivalently, φ(b)x t = W t, where φ(b) = 1 φ 1 B φ p B p. 5

6 Review: AR(p), Autoregressive models of order p Theorem: A (unique) stationary solution to φ(b)x t = W t exists iff the roots of φ(z) avoid the unit circle: z = 1 φ(z) = 1 φ 1 z φ p z p 0. This AR(p) process is causal iff the roots of φ(z) are outside the unit circle: z 1 φ(z) = 1 φ 1 z φ p z p 0. 6

7 Reminder: Polynomials of a complex variable Every degree p polynomial a(z) can be factorized as a(z) = a 0 + a 1 z + + a p z p = a p (z z 1 )(z z 2 ) (z z p ), where z 1,...,z p C are the roots of a(z). If the coefficients a 0, a 1,...,a p are all real, then the roots are all either real or come in complex conjugate pairs, z i = z j. Example: z + z 3 = z(1 + z 2 ) = (z 0)(z i)(z + i), that is, z 1 = 0, z 2 = i, z 3 = i. So z 1 R; z 2, z 3 R; z 2 = z 3. Recall notation: A complex number z = a + ib has Re(z) = a, Im(z) = b, z = a ib, z = a 2 + b 2, arg(z) = tan 1 (b/a) ( π, π]. 7

8 Review: Calculating ψ for an AR(p): general case φ(b)x t = W t, X t = ψ(b)w t so 1 = ψ(b)φ(b) 1 = (ψ 0 + ψ 1 B + )(1 φ 1 B φ p B p ) 1 = ψ 0, 0 = ψ j (j < 0), 0 = φ(b)ψ j (j > 0). We can solve these linear difference equations in several ways: numerically, or by guessing the form of a solution and using an inductive proof, or by using the theory of linear difference equations. 8

9 Introduction to Time Series Analysis. Lecture Review: Causality, invertibility, AR(p) models 2. ARMA(p,q) models 3. Stationarity, causality and invertibility 4. The linear process representation of ARMA processes: ψ. 5. Autocovariance of an ARMA process. 6. Homogeneous linear difference equations. 9

10 ARMA(p,q): Autoregressive moving average models An ARMA(p,q) process {X t } is a stationary process that satisfies X t φ 1 X t 1 φ p X t p = W t +θ 1 W t 1 + +θ q W t q, where {W t } WN(0, σ 2 ). AR(p) = ARMA(p,0): θ(b) = 1. MA(q) = ARMA(0,q): φ(b) = 1. 10

11 ARMA(p,q): Autoregressive moving average models An ARMA(p,q) process {X t } is a stationary process that satisfies X t φ 1 X t 1 φ p X t p = W t +θ 1 W t 1 + +θ q W t q, where {W t } WN(0, σ 2 ). Usually, we insist that φ p, θ q 0 and that the polynomials φ(z) = 1 φ 1 z φ p z p, θ(z) = 1 + θ 1 z + + θ q z q have no common factors. This implies it is not a lower order ARMA model. 11

12 ARMA(p,q): An example of parameter redundancy Consider a white noise process W t. We can write X t = W t X t X t X t 2 = W t W t W t 2 (1 B B 2 )X t = (1 B B 2 )W t This is in the form of an ARMA(2,2) process, with φ(b) = 1 B B 2, θ(b) = 1 B B 2. But it is white noise. 12

13 ARMA(p,q): An example of parameter redundancy ARMA model: φ(b)x t = θ(b)w t, with φ(b) = 1 B B 2, θ(b) = 1 B B 2 X t = ψ(b)w t ψ(b) = θ(b) φ(b) = 1. i.e., X t = W t. 13

14 Introduction to Time Series Analysis. Lecture Review: Causality, invertibility, AR(p) models 2. ARMA(p,q) models 3. Stationarity, causality and invertibility 4. The linear process representation of ARMA processes: ψ. 5. Autocovariance of an ARMA process. 6. Homogeneous linear difference equations. 14

15 Recall: Causality and Invertibility A linear process {X t } is causal if there is a ψ(b) = ψ 0 + ψ 1 B + ψ 2 B 2 + with ψ j < and X t = ψ(b)w t. j=0 It is invertible if there is a π(b) = π 0 + π 1 B + π 2 B 2 + with π j < and W t = π(b)x t. j=0 15

16 ARMA(p,q): Stationarity, causality, and invertibility Theorem: If φ and θ have no common factors, a (unique) stationary solution to φ(b)x t = θ(b)w t exists iff the roots of φ(z) avoid the unit circle: z = 1 φ(z) = 1 φ 1 z φ p z p 0. This ARMA(p,q) process is causal iff the roots of φ(z) are outside the unit circle: z 1 φ(z) = 1 φ 1 z φ p z p 0. It is invertible iff the roots of θ(z) are outside the unit circle: z 1 θ(z) = 1 + θ 1 z + + θ q z q 0. 16

17 ARMA(p,q): Stationarity, causality, and invertibility Example: (1 1.5B)X t = ( B)W t. φ(z) = 1 1.5z = 3 2 ( z 2 ), 3 θ(z) = z = 1 5 (z + 5). 1. φ and θ have no common factors, and φ s root is at 2/3, which is not on the unit circle, so {X t } is an ARMA(1,1) process. 2. φ s root (at 2/3) is inside the unit circle, so {X t } is not causal. 3. θ s root is at 5, which is outside the unit circle, so {X t } is invertible. 17

18 ARMA(p,q): Stationarity, causality, and invertibility Example: ( B 2 )X t = (1 + 2B)W t. φ(z) = z 2 = 1 ( z ) = 1 (z + 2i)(z 2i), 4 ( 4 θ(z) = 1 + 2z = 2 z + 1 ) φ and θ have no common factors, and φ s roots are at ±2i, which is not on the unit circle, so {X t } is an ARMA(2,1) process. 2. φ s roots (at ±2i) are outside the unit circle, so {X t } is causal. 3. θ s root (at 1/2) is inside the unit circle, so {X t } is not invertible. 18

19 Causality and Invertibility Theorem: Let {X t } be an ARMA process defined by φ(b)x t = θ(b)w t. If all z = 1 have θ(z) 0, then there are polynomials φ and θ and a white noise sequence W t such that {X t } satisfies φ(b)x t = θ(b) W t, and this is a causal, invertible ARMA process. So we ll stick to causal, invertible ARMA processes. 19

20 Introduction to Time Series Analysis. Lecture Review: Causality, invertibility, AR(p) models 2. ARMA(p,q) models 3. Stationarity, causality and invertibility 4. The linear process representation of ARMA processes: ψ. 5. Autocovariance of an ARMA process. 6. Homogeneous linear difference equations. 20

21 Calculating ψ for an ARMA(p,q): matching coefficients Example: X t = ψ(b)w t ( B 2 )X t = ( B)W t, so B = ( B 2 )ψ(b) B = ( B 2 )(ψ 0 + ψ 1 B + ψ 2 B 2 + ) 1 = ψ 0, 0.2 = ψ 1, 0 = ψ ψ 0, 0 = ψ ψ 1,. 21

22 Calculating ψ for an ARMA(p,q): example 1 = ψ 0, 0.2 = ψ 1, 0 = ψ j ψ j 2 (j 2). We can think of this as θ j = φ(b)ψ j, with θ 0 = 1, θ j = 0 for j < 0, j > q. This is a first order difference equation in the ψ j s. We can use the θ j s to give the initial conditions and solve it using the theory of homogeneous difference equations. ψ j = ( 1, 1 5, 1 4, 1 20, 1 16, 1 80, 1 64, 1 320,...). 22

23 Calculating ψ for an ARMA(p,q): general case so φ(b)x t = θ(b)w t, X t = ψ(b)w t θ(b) = ψ(b)φ(b) 1 + θ 1 B + + θ q B q = (ψ 0 + ψ 1 B + )(1 φ 1 B φ p B p ) 1 = ψ 0, θ 1 = ψ 1 φ 1 ψ 0, θ 2 = ψ 2 φ 1 ψ 1 φ 2 ψ 0,. This is equivalent to θ j = φ(b)ψ j, with θ 0 = 1, θ j = 0 for j < 0, j > q. 23

24 Introduction to Time Series Analysis. Lecture Review: Causality, invertibility, AR(p) models 2. ARMA(p,q) models 3. Stationarity, causality and invertibility 4. The linear process representation of ARMA processes: ψ. 5. Autocovariance of an ARMA process. 6. Homogeneous linear difference equations. 24

25 Autocovariance functions of linear processes Consider a (mean 0) linear process {X t } defined by X t = ψ(b)w t. γ(h) = E (X t X t+h ) = E (ψ 0 W t + ψ 1 W t 1 + ψ 2 W t 2 + ) (ψ 0 W t+h + ψ 1 W t+h 1 + ψ 2 W t+h 2 + ) = σw 2 (ψ 0 ψ h + ψ 1 ψ h+1 + ψ 2 ψ h+2 + ). 25

26 Autocovariance functions of MA processes Consider an MA(q) process {X t } defined by X t = θ(b)w t. σ 2 q h w j=0 γ(h) = θ jθ j+h if h q, 0 if h > q. 26

27 Autocovariance functions of ARMA processes ARMA process: φ(b)x t = θ(b)w t. To compute γ, we can compute ψ, and then use γ(h) = σ 2 w (ψ 0 ψ h + ψ 1 ψ h+1 + ψ 2 ψ h+2 + ). 27

28 Autocovariance functions of ARMA processes An alternative approach: X t φ 1 X t 1 φ p X t p = W t + θ 1 W t θ q W t q, so E((X t φ 1 X t 1 φ p X t p ) X t h ) = E((W t + θ 1 W t θ q W t q )X t h ), that is, γ(h) φ 1 γ(h 1) φ p γ(h p) = E(θ h W t h X t h + + θ q W t q X t h ) = σ 2 w q h j=0 This is a linear difference equation. θ h+j ψ j. (Write θ 0 = 1). 28

29 Autocovariance functions of ARMA processes: Example ( B 2 )X t = ( B)W t, X t = ψ(b)w t, ( ψ j = 1, 1 5, 1 4, 1 20, 1 16, 1 80, 1 64, 1 ) 320,.... γ(h) φ 1 γ(h 1) φ 2 γ(h 2) = σ 2 w γ(h) γ(h 2) = q h j=0 θ h+j ψ j σ 2 w (ψ ψ 1 ) if h = 0, 0.2σ 2 wψ 0 if h = 1, 0 otherwise. 29

30 Autocovariance functions of ARMA processes: Example We have the homogeneous linear difference equation for h 2, with initial conditions γ(h) γ(h 2) = 0 γ(0) γ( 2) = σ 2 w (1 + 1/25) γ(1) γ( 1) = σ 2 w/5. We can solve these linear equations to determine γ. Or we can use the theory of linear difference equations... 30

31 Introduction to Time Series Analysis. Lecture 6. Peter Bartlett bartlett/courses/153-fall ARMA(p,q) models 2. Stationarity, causality and invertibility 3. The linear process representation of ARMA processes: ψ. 4. Autocovariance of an ARMA process. 5. Homogeneous linear difference equations. 31

32 Difference equations Examples: x t 3x t 1 = 0 x t x t 1 x t 2 = 0 x t + 2x t 1 x 2 t 3 = 0 (first order, linear) (2nd order, nonlinear) (3rd order, nonlinear) 32

33 Homogeneous linear diff eqns with constant coefficients a 0 x t + a 1 x t a k x t k = 0 ( a0 + a 1 B + + a k B k) x t = 0 a(b)x t = 0 auxiliary equation: a 0 + a 1 z + + a k z k = 0 (z z 1 )(z z 2 ) (z z k ) = 0 where z 1, z 2,...,z k C are the roots of this characteristic polynomial. Thus, a(b)x t = 0 (B z 1 )(B z 2 ) (B z k )x t = 0. 33

34 Homogeneous linear diff eqns with constant coefficients a(b)x t = 0 (B z 1 )(B z 2 ) (B z k )x t = 0. So any {x t } satisfying (B z i )x t = 0 for some i also satisfies a(b)x t = 0. Three cases: 1. The z i are real and distinct. 2. The z i are complex and distinct. 3. Some z i are repeated. 34

35 Homogeneous linear diff eqns with constant coefficients 1. The z i are real and distinct. a(b)x t = 0 (B z 1 )(B z 2 ) (B z k )x t = 0 for some constants c 1,...,c k. x t is a linear combination of solutions to (B z 1 )x t = 0, (B z 2 )x t = 0,..., (B z k )x t = 0 x t = c 1 z t 1 + c 2 z t c k z t k, 35

36 Homogeneous linear diff eqns with constant coefficients 1. The z i are real and distinct. e.g., z 1 = 1.2, z 2 = c 1 z 1 t + c2 z 2 t c 1 =1, c 2 =0 c 1 =0, c 2 =1 c 1 = 0.8, c 2 =

37 Reminder: Complex exponentials a + ib = re iθ = r(cosθ + isinθ), where r = a + ib = a 2 + b 2 ( ) b θ = tan 1 ( π, π]. a Thus, r 1 e iθ 1 r 2 e iθ 2 = (r 1 r 2 )e i(θ 1+θ 2 ), z z = z 2. 37

38 Homogeneous linear diff eqns with constant coefficients 2. The z i are complex and distinct. As before, a(b)x t = 0 x t = c 1 z t 1 + c 2 z t c k z t k. If z 1 R, since a 1,...,a k are real, we must have the complex conjugate root, z j = z 1. And for x t to be real, we must have c j = c 1. For example: x t = c z t 1 + c z 1 t = r e iθ z 1 t e iωt + r e iθ z 1 t e iωt = r z 1 t ( e i(θ ωt) + e i(θ ωt)) = 2r z 1 t cos(ωt θ) where z 1 = z 1 e iω and c = re iθ. 38

39 Homogeneous linear diff eqns with constant coefficients 2. The z i are complex and distinct. e.g., z 1 = i, z 2 = 1.2 i c 1 z 1 t + c2 z 2 t c= i c= i c= i

40 Homogeneous linear diff eqns with constant coefficients 2. The z i are complex and distinct. e.g., z 1 = i, z 2 = 1 0.1i c 1 z 1 t + c2 z 2 t c= i c= i c= i

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