13 : CHAPTER. Linear Stochastic Models. Stationary Stochastic processes

Transcription

1 13 : CHAPTER Lnear Stochastc Models Statonary Stochastc processes A temporal stochastc process s smply a sequence of random varables ndexed by a tme subscrpt Such a process can be denoted by x(t) The element of the sequence at the pont t = τ s x τ = x(τ) Let x(τ) =[x τ+1,x τ+2,,x τ+n ] denote n consecutve elements of the sequence Then the process s sad to be strctly statonary f the ont probablty dstrbuton of the elements does not depend on τ regardless of the sze of n Ths means that any two segments of the sequence of equal length have dentcal probablty densty functons In consequence, the decson on where to place the tme orgn s arbtrary; and the argument τ can be omtted Some further mplcatons of statonarty are that (1) E(x t )=µ< for all t and C(x τ+t,x τ+s )=γ t s The latter condton means that the covarance of any two elements depends only on ther temporal separaton t s Notce that, f the elements of the sequence are normally dstrbuted, then the two condtons are suffcent to establsh strct statonarty On ther own, they consttute the condtons of weak or 2nd-order statonarty The condton on the covarances mples that the dsperson matrx of the vector x =[x 1,x 2,,x n ] s a bsymmetrc Laurent matrx of the form (2) D(x) =E { [x E(x)][x E(x)] } γ γ 1 γ 2 γ n 1 γ 1 γ γ 1 γ n 2 = γ 2 γ 1 γ γ n 3 γ n 1 γ n 2 γ n 3 γ Gven that a sequence of observatons of a tme seres represents only a segment of a sngle realsaton of a stochastc process, one mght magne that 1

2 DSG POLLOCK : ECONOMETRIC THEORY there s lttle chance of makng vald nferences about the parameters of the process However, provded that the process x(t) s statonary and provded that the statstcal dependences between wdely separated elements of the sequence are weak, t s possble to estmate consstently those parameters of the process whch express the dependence of proxmate elements of the sequence If one s prepared to make suffcently strong assumptons about the nature of the process, then a knowledge of such parameters may be all that s needed for a complete charactersaton of the process Movng Average Processes The qth-order movng average process, or MA(q) process, s defned by the equaton (3) y(t) =µ ε(t)+µ 1 ε(t 1) + +µ q ε(t q), where ε(t), whch has E{x(t)} =, s a whte-nose process consstng of a sequence of ndependently and dentcally dstrbuted random varables wth zero expectatons The equaton s normalsed ether by settng µ =1orby settng V {ε(t)} = σε 2 = 1 The equaton can be wrtten n summary notaton as y(t) =µ(l)ε(t), where µ(l) =µ +µ 1 L+ +µ q L q s a polynomal n the lag operator A movng-average process s clearly statonary snce any two elements y t and y s represent the same functon of dentcally dstrbuted vectors ε t = [ε t,ε t 1,,ε t q ] and ε s =[ε s,ε s 1,,ε s q ] In addton to the condton of statonarty, t s usually requred that a movng-average process should be nvertble such that t can be expressed n the form of µ 1 (L)y(t) =ε(t) where the LHS embodes a convergent sum of past values of y(t) Ths s an nfnte-order autoregressve representaton of the process The representaton s avalable only f all the roots of the equaton µ(z) =µ +µ 1 z+ +µ q z q = le outsde the unt crcle Ths concluson follows from our dscusson of partal fractons As an example, let us consder the frst-order movng-average process whch s defned by (4) y(t) =ε(t) θε(t 1)=(1 θl)ε(t) Provded that θ < 1, ths can be wrtten n autoregressve form as ε(t) =(1 θl) 1 y(t) (5) = { y(t)+θy(t 1) + θ 2 y(t 2) + } Imagne that θ > 1 nstead Then, to obtan a convergent seres, we have to wrte (6) y(t +1)=ε(t+1) θε(t) = θ(1 L 1 /θ)ε(t), 2

3 where L 1 ε(t) =ε(t+ 1) Ths gves LINEAR STOCHASTIC MODELS (7) ε(t) = θ 1 (1 L 1 /θ) 1 y(t +1) = θ 1{ y(t+1)/θ + y(t +2)/θ 2 + y(t 3)/θ 3 + } Normally, an expresson such as ths, whch embodes future values of y(t), would have no reasonable meanng It s straghtforward to generate the sequence of autocovarances from a knowledge of the parameters of the movng-average process and of the varance of the whte-nose process Consder (8) 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 a sequence of ndependently and dentcally dstrbuted random varables wth zero expectatons, t follows that {, f τ + ; (9) E(ε t ε t τ )= σε, 2 f = τ + Therefore (1) γ τ = σε 2 µ µ +τ Now let τ =,1,,q Ths gves (11) γ = σε(µ µ µ 2 q), γ 1 =σε(µ 2 µ 1 +µ 1 µ 2 + +µ q 1 µ q ), γ q =σεµ 2 µ q Also, γ τ = for all τ>q The frst-order movng-average process y(t) = ε(t) θε(t 1) has the followng autocovarances: (12) γ = σε(1 2 + θ 2 ), γ 1 = σεθ, 2 γ τ = f τ>1 3

4 DSG POLLOCK : ECONOMETRIC THEORY Thus, for a vector y =[y 1,y 2,,y T ] of T consecutve elements from a frstorder movng-average process, the dsperson matrx s 1+θ 2 θ θ 1+θ 2 θ (13) D(y) =σε 2 θ 1+θ 2 1+θ 2 In general, the dsperson matrx of a qth-order movng-average process has q subdagonal and q supradagonal bands of nonzero elements and zero elements elsewhere It s also helpful to defne an autocovarance generatng functon whch s a power seres whose coeffcents are the autocovarances γ τ for successve values of τ Ths s denoted by (14) γ(z) = τ γ τ z τ ; wth τ = {, ±1, ±2,} and γ τ = γ τ The generatng functon s also called the z-transform of the autocovarance functon The autocovarance generatng functon of the qth-order movng-average process can be found qute readly Consder the convoluton (15) µ(z)µ(z 1 )= = = τ µ z µ z µ µ z ( µ µ +τ )z τ, τ = By referrng to the expresson for the autocovarance of lag τ of a movngaverage process gven under (1), t can be seen that the autocovarance generatng functon s ust (16) γ(z) =σ 2 εµ(z)µ(z 1 ) Autoregressve Processes The pth-order autoregressve process, or AR(p) process, s defned by the equaton (17) α y(t)+α 1 y(t 1) + +α p y(t p)=ε(t) 4

5 LINEAR STOCHASTIC MODELS Ths equaton s nvarably normalsed by settng α = 1, although t would be possble to set σ 2 ε = 1 nstead The equaton can be wrtten n summary notaton 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 the equaton α(z) =α +α 1 z+ + α p z p = must le outsde the unt crcle Ths condton enables us to wrte the autoregressve process as an nfnte-order movng-average process n the form of y(t) =α 1 (L)ε(t) As an example, let us consder the frst-order autoregressve process whch s defned by (18) ε(t) =y(t) φy(t 1) =(1 φl)y(t) Provded that the process s statonary wth φ < 1, t can be represented n movng-average form as (19) y(t) =(1 φl) 1 ε(t) = { ε(t)+φε(t 1) + φ 2 ε(t 2) + } The autocovarances of the process can be found by usng the formula of (1) whch s applcable to movng-average process of fnte or nfnte order Thus (2) γ τ = E(y t y t τ ) { } = E φ ε t φ ε t τ = φ φ E(ε t ε t τ ); and the result under (9) ndcates that γ τ = σε 2 φ φ +τ (21) = σ2 εφ τ 1 φ 2 For a vector y =[y 1,y 2,,y T ] of T consecutve elements from a frst-order autoregressve process, the dsperson matrx has the form 1 φ φ 2 φ T 1 φ 1 φ φ T 2 (22) D(y) = σ2 ε φ 2 φ 1 φ T 3 1 φ 2 φ T 1 φ T 2 φ T 3 1 5

6 DSG POLLOCK : ECONOMETRIC THEORY To fnd the autocovarance generatng functon for the general pth-order autoregressve process, we may consder agan the functon α(z) = α z Snce an autoregressve process may be treated as an nfnte-order movngaverage process, t follows that σε 2 (23) γ(z) = α(z)α(z 1 ) For an alternatve way of fndng the autocovarances of the pth-order process, consder multplyng α y t = ε t by y t τ and takng expectatons to gve (24) α E(y t y t τ )=E(ε t y t τ ) Takng account of the normalsaton α = 1, we fnd that { σ 2 ε, f τ =; (25) E(ε t y t τ )=, f τ> Therefore, on settng E(y t y t τ )=γ τ, equaton (24) gves { σ 2 ε, f τ =; (26) 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 (26), we generate a set of p + 1 equatons whch can be arrayed n matrx form as follows: γ γ 1 γ 2 γ p 1 σ 2 ε γ 1 γ γ 1 γ p 1 α 1 (27) γ 2 γ 1 γ γ p 2 α 2 = γ p γ p 1 γ p 2 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,σε 2 or vce versa Example To llustrate the two uses of the Yule Walker equatons, let us consder the second-order autoregressve process In that case, we have γ γ 1 γ 2 γ 1 γ γ 1 α α 1 = α γ 2 2 α 1 α γ 1 α 2 α 1 α γ γ 2 γ 1 γ α 2 α 2 α 1 α γ 1 (28) γ 2 = α α 1 α 2 α 1 α +α 2 γ γ 1 = σ2 ε α 2 α 1 α γ 2 6

7 LINEAR STOCHASTIC MODELS Gven α = 1 and the values for γ,γ 1,γ 2, we can fnd σ 2 ε and α 1,α 2 Conversely, gven α,α 1,α 2 and σ 2 ε, we can fnd γ,γ 1,γ 2 It s worth recallng at ths uncture that the normalsaton σ 2 ε = 1 mght have been chosen nstead of α = 1 Ths would have rendered the equatons more easly ntellgble Notce also how the matrx followng the frst equalty s folded across the axs whch dvdes t vertcally to gve the matrx whch follows the second equalty Pleasng effects of ths sort often arse n tme-seres analyss Autoregressve Movng Average Processes The autoregressve movng-average process of orders p and q, whch s referred to as the ARMA(p, q) process, s defned by the equaton (29) 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) The equaton s normalsed by settng α = 1 and by settng ether µ =1 or σ 2 ε = 1 A more summary expresson for the equaton s α(l)y(t) =µ(l)ε(t) Provded that the roots of the equaton α(z) = le outsde the unt crcle, the process can be represented by the equaton y(t) =α 1 (L)µ(L)ε(t) whch corresponds to an nfnte-order movng-average process Conversely, provded the roots of the equaton µ(z) = le outsde the unt crcle, the process can be represented by the equaton µ 1 (L)α(L)y(t) =ε(t) whch corresponds to an nfnte-order autoregressve process By consderng the movng-average form of the process, and by notng the form of the autocovarance generatng functon for such a process whch s gven by equaton (16), t can be seen that the autocovarance generatng functon for the autoregressve movng-average process s (3) γ(z) =σε 2 µ(z)µ(z 1 ) α(z)α(z 1 ) Ths generatng functon, whch s of some theoretcal nterest, does not provde a practcal means of fndng the autocovarances To fnd these, let us consder multplyng the equaton α y t = µ ε t by y t τ and takng expectatons Ths gves (31) α γ τ = µ δ τ, where γ τ = E(y t y t τ ) and δ τ = E(ε t y t τ ) Snce ε t s uncorrelated wth y t τ whenever t s subsequent to the latter, t follows that δ τ =f 7

8 DSG POLLOCK : ECONOMETRIC THEORY τ> Snce the ndex n the RHS of the equaton (31) runs from to q, t follows that (32) 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 for the autocovarances, the equatons can be solved recursvely to obtan the subsequent values {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 (33) α δ τ = µ τ σε, 2 where δ τ = E(y t ε t τ ) The equaton may be rewrtten as (34) δ τ = 1 ( µ τ σε 2 δ τ ), α =1 and, by settng τ =,1,,q, we can generate recursvely the requred values δ,δ 1,,δ q Example Consder the ARMA(2, 2) model whch gves the equaton (35) α y t + α 1 y t 1 + α 2 y t 2 = µ ε t + µ 1 ε t 1 + µ 2 ε t 2 Multplyng by y t, y t 1 and y t 2 and takng expectatons gves (36) γ γ 1 γ 2 γ 1 γ γ 1 α α 1 = δ δ 1 δ 2 δ δ 1 µ µ 1 γ 2 γ 1 γ α 2 δ µ 2 Multplyng by ε t, ε t 1 and ε t 2 and takng expectatons gves (37) δ δ 1 δ α α 1 = σ2 ε σε 2 µ µ 1 δ 2 δ 1 δ α 2 σε 2 µ 2 When the latter equatons are wrtten as (38) α α 1 α δ δ 1 = σ 2 µ ε µ 1, α 2 α 1 α δ 2 µ 2 8

9 LINEAR STOCHASTIC MODELS they can be solved recursvely for δ, δ 1 and δ 2 on the assumpton that that the values of α, α 1, α 2 and σε 2 are known Notce that, when we adopt the normalsaton α = µ = 1, we get δ = σε 2 When the equatons (36) are rewrtten as (39) α α 1 α 2 α 1 α + α 2 γ γ 1 = µ µ 1 µ 2 µ 1 µ 2 δ δ 1, α 2 α 1 α γ 2 µ 2 δ 2 they can be solved for γ, γ 1 and γ 2 Thus the startng values are obtaned whch enable the equaton (4) α γ τ + α 1 γ τ 1 + α 2 γ τ 2 =; τ>2 to be solved recursvely to generate the succeedng values {γ 3, γ 4,} of the autocovarances Mnmum Mean-Square Error Predcton Imagne that y(t) s a statonary stochastc process wth E{y(t)} = We may be nterested n predctng values of ths process several perods nto the future on the bass of ts observed hstory Ths hstory s contaned n our so-called nformaton set In practce, the latter s always a fnte set {y t,y t 1,,y t p } representng the recent past Nevertheless, n developng the theory of predcton, t s also useful to consder an nfnte nformaton set {y t,y t 1,,y t p,} representng the entre past We shall denote the predcton of y t+m whch s made at the tme t by ŷ t+m t or by ŷ t+m when t s clear that we a predctng m steps ahead The crteron by whch we usually udge the performance of an estmator or predctor ŷ of a random varable y s ts mean-square error defned by E{(y ŷ) 2 } If all of the avalable nformaton on y s summarsed n ts margnal dstrbuton, then the mnmum mean-square error predcton s smply the expected value E(y) However, f y s statstcally related to another random varable x whose value we can observe, and f we know the form of the ont dstrbuton of x and y, then the mnmum mean-square error predcton of y s the condtonal expectaton E(y x) We may state ths proposton formally: (41) Let ŷ =ŷ(x) be the condtonal expectaton of y gven x whch s also expressed as ŷ = E(y x) Then we have E{(y ŷ) 2 } E{(y π) 2 }, where π = π(x) s any other functon of x Proof Consder (42) E{(y π) 2 } = E[{(y ŷ)+(ŷ π)} 2 ] =E{(y ŷ) 2 }+2E{(y ŷ)(ŷ π)} + E{(ŷ π) 2 } 9

10 DSG POLLOCK : ECONOMETRIC THEORY In the second term, we have E{(y ŷ)(ŷ π)} = (y ŷ)(ŷ π)f(x, y) y x x y { } (43) = (y ŷ)f(y x) y (ŷ π)f(x) x x y = Here the second equalty depends upon the factorsaton f(x, y) = f(y x)f(x) whch expresses the ont probablty densty functon of x and y as the product of the condtonal densty functon of y gven x and the margnal densty functon of x The fnal equalty depends upon the fact that (y ŷ)f(y x) y = E(y x) E(y x) = Therefore E{(y π) 2 } = E{(y ŷ) 2 } + E{(ŷ π) 2 } E{(y ŷ) 2 }and our asserton s proved We mght note that the defnton of the condtonal expectaton mples that E(xy) = xyf(x, y) y x x y { } (44) = x yf(y x) y f(x) x x y = E(xŷ) When the equaton E(xy) = E(xŷ) s rewrtten as (45) E {x(y ŷ)} =, t may be descrbed as an orthogonalty condton Ths condton ndcates that the predcton error y ŷ s uncorrelated wth x The result s ntutvely appealng; for, f the error were correlated wth x, we should not usng the nformaton of x effcently n formng ŷ The proposton of (41) s readly generalsed to accommodate the case where, n place of the scalar x, we have a vector x =[x 1,,x p ] Ths generalsaton ndcates that the mnmum-mean-square-error predcton of y t+m gven the nformaton n {y t,y t 1,,y t p } s the condtonal expectaton E(y t+m y t,y t 1,,y t p ) In order to determne the condtonal expectaton of y t+m gven {y t,y t 1,,y t p }, we need to known the functonal form of the ont probablty densty functon all of these varables In leu of precse knowledge, we are often prepared to assume that the dstrbuton s normal In that case, t follows that the condtonal expectaton of y t+m s a lnear functon of {y t,y t 1,,y t p }; 1

11 LINEAR STOCHASTIC MODELS and so the problem of predctng y t+m becomes a matter of formng a lnear regresson Even f we are not prepared to assume that the ont dstrbuton of the varables n normal, we may be prepared, nevertheless, to base our predcton of y upon a lnear functon of {y t,y t 1,,y t p } In that case, we satsfy the crteron of mnmum mean-square error lnear predcton by formng ŷ t+m = φ y t +1 from the values φ 1,,φ p+1 whch mnmse (46) E { ( (y t+m ŷ t+m ) 2} p+1 ) } 2 = E{ y t+m φ y t +1 = γ 2 =1 φ γ m+ 1 + φ φ γ Ths s a lnear least-squares regresson problem whch leads to a set of p +1 orthogonalty condtons descrbed as the normal equatons: (47) p E{(y t+m ŷ t+m )y t +1 } = γ m+ 1 φ γ =1 = ; =1,,p+1 In matrx terms, we have (48) γ γ 1 γ p γ 1 γ γ p 1 γ p γ p 1 γ φ 1 φ 2 φ p+1 = γ m γ m+1 γ m+p Notce that, for the one-step-ahead predcton of y t+1, these are nothng but the Yule Walker equatons Forecastng wth ARMA Models So far, we have avoded makng any specfc assumptons about the nature of the process y(t) other than that t can be represented by an nfnte-order movng average We are greatly asssted n the busness of developng practcal forecastng procedures f we can assume that y(t) s generated by an ARMA process such that (49) y(t) = µ(l) α(l) ε(t)=ψ(l)ε(t) We shall contnue to assume, for the sake of smplcty, that the forecasts are based on the nformaton contaned n the nfnte set {y t,y t 1,,y t p,} 11

12 DSG POLLOCK : ECONOMETRIC THEORY comprsng all values that have been taken by the varable up to the present tme t Knowng the parameters n ψ(l) enables us to recover the sequence {ε t,ε t 1,ε t 2,} from the sequence {y t,y t 1,y t 2,} and vce versa; so ether of these can be regarded as our nformaton set Let us wrte the realsatons of equaton (49) as (5) y t+m = m 1 = ψ ε t+m + ψ ε t+m Here the frst term on the RHS embodes dsturbances subsequent to the tme t when the forecast s made, and the second term embodes dsturbances whch are wthn the nformaton set {ε t,ε t 1,ε t 2,} Let us now defne a forecastng functon, based on the nformaton set, whch takes the form of (51) ŷ t+m = =m ρ ε t+m =m Then, gven that ε(t) s a whte-nose process, t follows that the mean square of the error n the forecast m perods ahead s gven by (52) E{(y t+m ŷ t+m ) 2 } = σ 2 ε m 1 = ψ 2 + σ 2 ε (ψ ρ ) 2 Clearly, the mean-square error s mnmsed by settng ρ = ψ ; and so the optmal forecast s gven by (53) ŷ t+m = ψ ε t+m =m Ths mght have been derved from the the equaton y(t + m) =ψ(l)ε(t+m), whch generates the the true value of y t+m, smply by puttng zeros n place of the unobserved dsturbances ε t+1,ε t+2,,ε t+m whch le n the future when the forecast s made Notce that, as the lead tme m of the forecast ncreases, the mean-square error of the forecast tends to the value of =m (54) V {y(t)} = σ 2 ε ψ 2 whch s nothng but the varance of the process y(t) We can also derve the optmal forecast of (45) by specfyng that the forecast error should be uncorrelated wth the dsturbances up to the tme of makng the forecast For, f the the forecast errors were correlated wth some of the elements of our nformaton set, then, as we have noted before, we would 12

13 LINEAR STOCHASTIC MODELS not be usng the nformaton effcently, and we could not be generatng optmal forecasts To demonstrate ths result anew, let us consder the covarance between the forecast error and the dsturbance ε t : (55) m E{(y t+m ŷ t+m )ε t } = ψ m k E(ε t+k ε t ) k=1 + (ψ m+ ρ m+ )E(ε t ε t ) = = σ 2 ε(ψ m+ ρ m+ ) Here the fnal equalty follows from the fact that { σ 2 ε, f =, (56) E(ε t ε t )=, f If the covarance n (55) s to be equal to zero for all values of, then we must have ρ = ψ for all, whch means that our forecastng functon must be the one that we have already specfed under (53) It s helpful, sometmes, to have a functonal notaton for descrbng the process whch generates the m-steps-ahead forecast The notaton provded by Whttle (1963) s wdely used To derve ths, let us begn by wrtng (57) y(t + m t) = { L m ψ(l) } ε(t) On the LHS, we have not only the lagged sequences ε(t),ε(t 1), but also the sequences ε(t + m) =L m ε(t),,ε(t +1) = L 1 ε(t), all of whch are assocated wth negatve powers of L Let {L m ψ(l)} + be defned as the part of the operator contanng only postve powers of L Then we can express the forecastng functon as (58) ŷ(t + m t) ={L m ψ(l)} + ε(t) { } ψ(l) 1 = L m ψ(l) y(t) + The Forecasts as Condtonal Expectatons We have already seen that we can regard the optmal (mnmum meansquare error) forecast of y t+m as the condtonal expectaton of y t+m gven the values of {ε t,ε t 1,ε t 2,} or {y t,y t 1,y t 2,} Let us denote the forecast by ŷ t+m = E t (y t+m ) where the subscrpt on the operator s to ndcate that 13

14 DSG POLLOCK : ECONOMETRIC THEORY the expectaton s condtonal upon the nformaton avalable at tme t On applyng the operator to the sequences y(t) and ε(t), we fnd that (59) E t (y t+k )=ŷ t+k ; k> E t (y t )=y t ; E t (ε t+k )= ; k> E t (ε t )=ε t ; In ths notaton, the forecast m perods ahead s (6) m E t (y t+m )= ψ m k E t (ε t+k )+ ψ m+ E t (ε t ) k=1 = = ψ m+ ε t = In practce, we may generate the forecasts usng a recurson based on the equaton (61) y(t) = {α 1 y(t 1) + α 2 y(t 2) + +α p y(t p)} +µ ε(t)+µ 1 ε(t 1) + +µ q ε(t q) By takng the condtonal expectaton of ths functon, we get (62) ŷ t+m = {α 1 ŷ t+m 1 + +α p y t+m p } +µ m ε t + +mu q ε t+m q when <m p, q, (63) ŷ t+m = {α 1 ŷ t+m 1 + +α p y t+m p } f q<m p, (64) ŷ t+m = {α 1 ŷ t+m 1 + +α p ŷ t+m p } +µ m ε t + +µ q ε t+m q f p<m q, and (65) ŷ t+m = {α 1 ŷ t+m 1 + +α p ŷ t+m p } when p, q < m We can see from (65) that, for m>p,q, the forecastng functon becomes a pth-order homogeneous dfference equaton n y The p values of y(t) from t = r = max(p, q) tot=r p+ 1 serve as the startng values for the equaton 14

15 LINEAR STOCHASTIC MODELS The behavour of the forecast functon beyond the reach of the startng values can be charactersed n terms of the roots of the autoregressve operator We can assume that none of the roots of α(l) = le nsde the unt crcle If all of the roots are less than unty, then ŷ t+m wll converge to zero as m ncreases If one of the roots of α(l) = s unty, then we have and ARIMA(p, 1,q) model; and the general soluton of the homogeneous equaton of (65) wll nclude a constant term whch represents the product of the unt root wth an coeffcent whch s determned by the startng values Hence the the forecast wll tend to a nonzero constant If two of the roots are unty, then the the general soluton wll embody a lnear tme trend whch s the asymptote to whch the forecasts wll tend In general, f d of the roots are unty, then the general soluton wll comprse a polynomal n t of order d 1 The forecasts can be updated easly once the coeffcents n the expanson of ψ(l) = µ(l)/α(l) have been obtaned Consder (66) ŷ (t+1)+m = {ψ m ε t+1 + ψ m+1 ε t + ψ m+2 ε t 1 + } ŷ t+(m+1) = {ψ m+1 ε t + ψ m+2 ε t 1 + } and The frst of these s the forecast for m perods ahead made at tme t + 1 whlst the second s the forecast for m + 1 perods ahead made at tme t We can easly see that (67) ŷ (t+1)+m =ŷ t+(m+1) + ψ m ε t+1, where ε t+1 =ŷ t+1 y t+1 s the current dsturbance at tme t + 1 The later s also the predcton error of the one-step-ahead forecast made at tme t Example Consder the AR(41) process We have (68) y(t + m) =φy(t + m 1) + ε(t + m) On applyng the operator E t to ths equaton we obtan the followng: (69) ŷ(t +1)=φy(t) ŷ(t +2)=φŷ(t+1)=φ 2 y(t) ŷ(t+m)=φŷ(t+m 1) = φ m y(t) Gven that y(t) =ε(t)/(1 φl) ={ε(t)+φε(t 1)+φ 2 ε(t 2)+ }, t follows from (44) that { (7) E{(y t+m ŷ t+m ) 2 } = σε 2 1+φ 2 +φ 4 + +φ 2(m 1)} 15

16 DSG POLLOCK : ECONOMETRIC THEORY As m, ths tends to σ 2 ε/(1 φ 2 ) whch s ust the varance of the AR(41) process Example(Exponental Smoothng) A common forecastng procedure s exponental smoothng Ths depends upon takng a weghted average of past values of the tme seres wth the weghts followng a geometrcally declnng pattern The functon generatng the one-step-ahead forecasts can be wrtten as (71) (1 θ) ŷ(t +1)= 1 θl y(t) =(1 θ) { y(t)+θy(t 1) + θ 2 y(t +2)+ } On multplyng both sdes of ths equaton by 1 θl and rearrangng, we get (72) ŷ(t +1)=θŷ(t)+(1 θ)y(t), whch shows that the current forecast for one step ahead s a convex combnaton of the prevous forecast and the value that actually transpred It s possble to show that the method of exponental smoothng corresponds to the optmal forecastng procedure for the ARIMA(, 1, 1) model (1 L)y(t) =(1 θl)ε(t) To see ths, let us consder the ARMA(1, 1) model y(t) φy(t 1) = ε(t) θε(t 1) Ths gves (73) ŷ(t +1)=φy(t) θε(t) (1 φl) = φy(t) θ 1 θl y(t) {(1 θl)φ (1 φl)θ} = y(t) 1 θl (φ θ) = 1 θl y(t) On settng φ = 1, whch converts the ARMA(1, 1) model to an ARIMA(, 1, 1) model, we obtan precsely the forecastng functon of (6) 16