LECTURE 4 : ARMA PROCESSES

Transcript

1 LECTURE 4 : ARMA PROCESSES Movng-Average Processes The MA(q) process, s defned by (53) y(t) =µ ε(t)+µ 1 ε(t 1) + +µ q ε(t q) =µ(l)ε(t), where µ(l) =µ +µ 1 L+ +µ q L q and where ε(t) s whte nose An MA model should be nvertble such that µ 1 (L)y(t) =ε(t) Ths AR( ) representaton s s avalable f and only f all the roots of µ(z) =le outsde the unt crcle Example Consder the MA(1) process (54) y(t) =ε(t) θε(t 1)=(1 θl)ε(t) Provded that θ < 1, ths can be wrtten n autoregressve form as (55) ε(t) =(1 θl) 1 y(t) = { y(t)+θy(t 1) + θ y(t ) + } Imagne that θ > 1 nstead Then we have to wrte (56) y(t +1)=ε(t+1) θε(t) = θ(1 L 1 /θ)ε(t), where L 1 ε(t) =ε(t+ 1) Ths gves (57) ε(t) = θ 1 (1 L 1 /θ) 1 y(t +1) = θ 1{ y(t+1)/θ + y(t +)/θ + y(t 3)/θ 3 + } Normally, an expresson such as ths, whch embodes future values of y(t), would have no reasonable meanng 1

2 DSG POLLOCK : LECTURES IN THE CITY 4 The Autocovarances of an MA Process Consder (58) γ τ = E(y t y t τ ) { } = E µ ε t µ ε t τ = µ µ E(ε t ε t τ ) Snce ε(t) s whte nose, t follows that {, f τ + ; (59) E(ε t ε t τ )= σε, f = τ + Therefore (51) γ τ = σε µ µ +τ Now let τ =,1,,q Ths gves (511) γ = σε(µ + µ 1 + +µ q), γ 1 =σε(µ µ 1 +µ 1 µ + +µ q 1 µ q ), γ q =σ εµ µ q Also, γ τ = for all τ>q Example The MA(1) process y(t) = ε(t) θε(t 1) has (51) γ = σε(1 + θ ), γ 1 = σεθ, γ τ = f τ>1 Thus the dsperson matrx of y =[y 1,y,,y T ] s 1+θ θ θ 1+θ θ (513) D(y) =σε θ 1+θ 1+θ

3 LECTURE 4 : ARMA PROCESSES Autocovarance Generatng Functon Ths s denoted by (514) γ(z) = τ γ τ z τ ; wth τ = {, ±1, ±,} and γ τ = γ τ To fnd the autocovarance generatng functon of the MA(q) process, consder µ(z)µ(z 1 )= µ z µ z (515) From (1) t follows that = = τ µ µ z ( µ µ +τ )z τ, τ = (516) γ(z) =σ εµ(z)µ(z 1 ) 3

4 DSG POLLOCK : LECTURES IN THE CITY 4 Autoregressve Processes The AR(p) process, s defned by (517) α y(t)+α 1 y(t 1) + +α p y(t p)=ε(t) Ths can be wrtten as α(l)y(t) =ε(t), where α(l) =α +α 1 L+ +α p L p For the process to be statonary, the roots of α(z) = must le outsde the unt crcle In that case the AR process can be wrtten as an MA( ) process: y(t) =α 1 (L)ε(t) The autocovarance generatng functon for the AR(p) process s (53) γ(z) = σ ε α(z)α(z 1 ) Example Consder the AR(1) process defned by ε(t) =y(t) φy(t 1) (518) =(1 φl)y(t) Provded that φ < 1, ths can be represented n MA form as y(t) =(1 φl) 1 ε(t) (519) = { ε(t)+φε(t 1) + φ ε(t ) + } The autocovarances of the AR(1) process can be obtaned va the formula (1) for the autocovarances of an MA process Thus (5) and t follows from (9) that γ τ = E(y t y t τ ) { } = E φ ε t φ ε t τ = φ φ E(ε t ε t τ ); γ τ = σε φ φ +τ (51) = σ εφ τ 1 φ The dsperson matrx of y =[y 1,y,,y T ] s (5) D(y) = σ ε 1 φ 1 φ φ φ T 1 φ 1 φ φ T φ φ 1 φ T 3 φ T 1 φ T φ T 3 1 4

5 The Yule-Walker Equatons LECTURE 4 : ARMA PROCESSES For an alternatve way of fndng the AR autocovarances, consder multplyng α y t = ε t by y t τ and takng expectatons to gve (54) α E(y t y t τ )=E(ε t y t τ ) Gven that α = 1, t follows that { σ ε, f τ =; (55) E(ε t y t τ )=, f τ> Therefore, on settng E(y t y t τ )=γ τ, equaton (4) gves { σ ε, f τ =; (56) α γ τ =, f τ> The second of these s a homogeneous dfference equaton whch enables us to generate the sequence {γ p,γ p+1,} once p startng values γ,γ 1,,γ p 1 are known By lettng τ =,1,,p n (6), we generate a set of p + 1 equatons whch can be arrayed n matrx form as follows: (57) γ γ 1 γ γ p 1 γ 1 γ γ 1 γ p 1 α 1 γ γ 1 γ γ p α = γ p γ p 1 γ p γ α p These are called the Yule Walker equatons, and they can be used ether for generatng the values γ,γ 1,,γ p from the values α 1,,α p,σ ε or vce versa Example Consder the second-order autoregressve process We have γ γ 1 γ γ 1 γ γ 1 α α 1 = α γ α 1 α γ α α 1 α 1 γ γ γ 1 γ α α α 1 α γ 1 (58) γ = α α 1 α α 1 α +α γ γ 1 = σ ε α α 1 α γ Gven α = 1 and the values for γ,γ 1,γ, we can fnd σ ε and α 1,α Conversely, gven α,α 1,α and σ ε, we can fnd γ,γ 1,γ 5 σ ε

6 DSG POLLOCK : LECTURES IN THE CITY 4 The Partal Autocorrelaton Functon Let α r(r) be the coeffcent assocated wth y(t r) n an autoregressve process of order r whose parameters correspond to the autocovarances γ,γ 1,,γ r Then the sequence {α r(r) ; r =1,,}of such coeffcents, whose ndex corresponds to models of ncreasng orders, consttutes the partal autocorrelaton functon In effect, α r(r) ndcates the role n explanng the varance of y(t) whch s due to y(t r) when y(t 1),,y(t r+ 1) are also taken nto account The sequence of partal autocorrelatons may be computed effcently va the recursve Durbn Levnson Algorthm whch uses the coeffcents of the AR model of order r as the bass for calculatng the coeffcents of the model of order r +1 Imagne that we already have the values α (r) =1,α 1(r),,α r(r) Then, by extendng the set of rth-order Yule Walker equatons to whch these values correspond, we can derve the system γ γ 1 γ r γ r+1 1 σ (r) γ 1 γ γ r 1 γ r (59) α 1(r) = γ r γ r 1 γ γ 1 α r(r), γ r+1 γ r γ 1 γ g wheren (53) g = r α (r) γ r+1 wth α (r) =1 = The system can also be wrtten as γ γ 1 γ r γ r+1 γ 1 γ γ r 1 γ r (531) α r(r) γ r γ r 1 γ γ 1 α 1(r) γ r+1 γ r γ 1 γ 1 = g σ (r) The two systems of equatons (9) and (31) can be combned to gve γ γ 1 γ r γ r+1 1 σ(r) γ 1 γ γ r 1 γ r (53) α 1(r) + cα r(r) + cg = γ r γ r 1 γ γ 1 α r(r) + cα 1(r) γ r+1 γ r γ 1 γ c g + cσ(r) 6

7 LECTURE 4 : ARMA PROCESSES If we take the coeffcent of the combnaton to be (533) c = g σ(r), then the fnal element n the vector on the RHS becomes zero and the system becomes the set of Yule Walker equatons of order r + 1 The soluton of the equatons, from the last element α r+1(r+1) = c through to the varance term σ(r+1) s gven by (534) α r+1(r+1) = 1 { r } σ(r) α (r) γ r+1 = α 1(r+1) α r(r+1) = α 1(r) α r(r) + α r+1(r+1) α r(r) α 1(r) σ (r+1) = σ (r){ 1 (αr+1(r+1) ) } Thus the soluton of the Yule Walker system of order r + 1 s easly derved from the soluton of the system of order r, and there s scope for devsng a recursve procedure The startng values for the recurson are (535) α 1(1) = γ 1 /γ and σ (1) = γ { 1 (α1(1) ) } 7

8 DSG POLLOCK : LECTURES IN THE CITY 4 Autoregressve Movng Average Processes The ARMA(p, q) process, s defned by (536) α y(t)+α 1 y(t 1) + +α p y(t p) =µ ε(t)+µ 1 ε(t 1) + +µ q ε(t q) Ths can also be wrtten as α(l)y(t) =µ(l)ε(t) If the roots of α(z) = le outsde the unt crcle, then the process has an MA( ) form: y(t) = α 1 (L)µ(L)ε(t) If the roots of µ(z) = le outsde the unt crcle, then t has an AR( ) form: µ 1 (L)α(L)y(t) =ε(t) The autocovarance generatng functon for the ARMA process s (537) γ(z) =σ ε µ(z)µ(z 1 ) α(z)α(z 1 ) To fnd the autocovarances n practce, consder multplyng the equaton α y t = µ ε t by y t τ and takng expectatons Ths gves (538) α γ τ = µ δ τ, where γ τ = E(y t τ y t ) and δ τ = E(y t τ ε t ) Snce ε t s uncorrelated wth y t τ whenever t s subsequent to the latter, t follows that δ τ =f τ> Snce the ndex n the RHS of the equaton (38) runs from to q, t follows that (539) α γ τ = f τ>q Gven the q+1 nonzero values δ,δ 1,,δ q, and p ntal values γ,γ 1,,γ p 1, the equatons can be solved recursvely for {γ p,γ p+1,} To fnd the requste values δ,δ 1,,δ q, consder multplyng the equaton α y t = µ ε t by ε t τ and takng expectatons Ths gves (54) α δ τ = µ τ σε, where δ τ = E(y t ε t τ ) The equaton may be rewrtten as (541) δ τ = 1 ( µ τ σε δ τ ), α =1 and, by settng τ =,1,,q, we can generate recursvely the requred values δ,δ 1,,δ q 8

9 LECTURE 4 : ARMA PROCESSES Example Consder the ARMA(, ) model whch gves the equaton (54) α y t + α 1 y t 1 + α y t = µ ε t + µ 1 ε t 1 + µ ε t Multplyng by y t, y t 1 and y t and takng expectatons gves (543) γ γ 1 γ γ 1 γ γ 1 α α 1 = δ δ 1 δ δ δ 1 µ µ 1 γ γ 1 γ α δ µ Multplyng by ε t, ε t 1 and ε t and takng expectatons gves (544) δ δ 1 δ α α 1 = σ ε σε µ µ 1 δ δ 1 δ α σε µ When the latter equatons are wrtten as (545) α α 1 α δ δ 1 = σ µ ε µ 1, α α 1 α δ µ they can be solved recursvely for δ, δ 1 and δ on the assumpton that that the values of α, α 1, α and σε are known Notce that, when we adopt the normalsaton α = µ = 1, we get δ = σε When the equatons (43) are rewrtten as (546) α α 1 α α 1 α + α γ γ 1 = µ µ 1 µ µ 1 µ δ δ 1, α α 1 α γ µ δ they can be solved for γ, γ 1 and γ Thus the startng values are obtaned whch enable the equaton (547) α γ τ + α 1 γ τ 1 + α γ τ =; τ> to be solved recursvely to generate the succeedng values {γ 3, γ 4,} of the autocovarances 9