Lecture Notes 2. The Hilbert Space Approach to Time Series

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1 Time Series Seven N. Durlauf Universiy of Wisconsin. Basic ideas Lecure Noes. The Hilber Space Approach o Time Series The Hilber space framework provides a very powerful language for discussing he relaionship beween various random variables. Collecions of random variables are called sochasic processes; in common usage sochasic processes are usually undersood o refer o collecions of random variables whose elemens are indeed by ime. We focus on he case of a scalar sochasic process { } where is an ineger as i is convenien o hink of ime sreching from o. We assume ha he process is zero-mean and second-order saionary. Second-order saionariy, also known as weak saionariy, means ha he auocovariances beween on. Formally, assume i) = 0 E and ii) σ and E = <. do no depend For random variables such as he elemens of he sochasic process, he naural meric, i.e. noion of lengh, for a random variable is sandard deviaion, ( ) = E (.) wih covariance as he associaed noion of inner produc, ( ), = E (.) One can generae a Hilber space around he sequence. {,, }. Wha his means is ha one forms a space by aking hese elemens, adding all

2 linear combinaions of he elemens, all limis of he linear combinaions, ec. We denoe his Hilber space as H. The enire hisory of he sochasic process from o generaes H ( ). By consrucion, H H. The general properies of Hilber spaces described in Lecure Noes allow one o characerize he linear srucure of H imporan in macroeconomics. decomposiion heorem, one can decompose H in ways ha are very Firs, observe ha by he Hilber space so ha H = H G (.3) where G is anoher Hilber space. The dimension of his Hilber space is eiher 0 or. This is so because he Hilber space ( ) random variable pro H ( ) where ono H. To say he space ( ( )) i.e. G mus be spanned by he single ( ) pro H is he proecion of H, G has dimension 0 means ha var pro H = 0. If one again applies he Hilber space decomposiion heorem, i is he case ha H = H G G (.4) Here G is spanned by pro H. One can repea his decomposiion any number of imes. The G spaces are by consrucion muually orhogonal. This may be repeaed an arbirary number of imes. Noice ha i is no he necessarily he case ha, if his decomposiion is done an infinie number of imes, he H. G s may be used o reconsruc The reason for his is each space is consiued by elemens ha appear in he Technically, we are working wih he smalles Hilber space ha conains he

3 space H ( ) bu no in he space H every member of he sequence H ( ), H he G s. Elemens ha are common o all of he H well. Formally, his space is defined as ; if here are elemens ha appear in, hey will no appear in any of s form a Hilber space as H = H (.5) = The Hilber space generaed by curren and pas s can herefore be decomposed as = H G G... H (.6). Wold decomposiion heorems Equaion (.6) is he basis for wo fundamenal heorems in ime series analysis, each due o Herman Wold; his 948 aricle is sill worh reading. Rozanov (967) is a deep reamen. I find Ash and Gardner s (967) discussion o be especially insighful. Theorem.. Wold decomposiion heorem I Any zero-mean, finie variance, second-order saionary may be decomposed as = + (.7) elemens of he sochasic process. 3

4 where G G G (.8)... and H (.9) In his decomposiion, is called he indeerminisic componen and he deerminisic componen of. The erms refer o wheher or no he componens may be perfecly (linearly) prediced from he pas. When a ime series conains a nonrivial indeerminisic componen, he ime series iself is said o be indeerminisic. If he process does no conain a deerminisic componen, i is purely indeerminisic. The erm may be perfecly prediced from informaion in he arbirarily disan pas. One eample of deerminisic componen is a seasonal. Consider he sochasic process cos( ω + θ) where θ is uniformly disribued on [ ππ] π Since ( ω + θ) d θ = and π cos 0 π depend on, ( ω + θ) π cos is a candidae for. ( + ) cos( + ),. cos ω θ ω θ d θ does no The second Wold Theorem characerizes he linear srucure of he indeerminisic par of a ime series. Theorem.. Wold decomposiion heorem II This follows from he ideniy cos( ω( ) + θ) cos( ω + θ) = ( cos( ω) + cos( ω + ω + θ) ). 4

5 If is a purely indeerminisic, zero-mean, finie variance, second-order saionary process, hen here eiss a represenaion of he process of he form = αε, α 0 = (.0) = 0 where ε G and σε = σ ε. = 0 αε is known as he fundamenal moving average (MA) represenaion of and is unique. H = H G, by consrucion Pf. Since G is a Hilber space of maimum dimension ; he space is spanned by ε which is a scalar random variable. If he dimension is zero (which would mean ha var ( ε ) = 0 ), hen H = H ( ) which conradics he assumpion ha he process is purely indeerminisic. Since he process is indeerminisic, one may find an elemen ε in he proecion of ono G such ha G is ε ; his is nohing more han a choice of ais for he one dimensional space. For he spaces G ( > 0), one can find an elemen in each of hem, denoed as ε, whose variance equals ha of ε ; each ε spans is respecive space. Since is purely indeerminisic, H = G G... Leing ( ) by he Hilber space proecion heorem pro G denoe he proecion of ono G, ( ) αε = pro G = (.) = 0 = 0 where he second equaliy follows from he definiion of he ε s. This verifies he heorem, ecep for uniqueness. 5

6 To prove uniqueness, suppose ha here eised anoher MA represenaion = βε. For his o be he case, he variance of βε αε = 0 = 0 = 0 mus equal zero since by assumpion he pars of he epression are he same. The variance of his epression equals σε ( α β) α β = 0. = 0, which equals zero iff An immediae implicaion of he second Wold heorem is ha he fundamenal moving average coefficiens are square summable, i.e. α = 0 <. This follows from he fac ha var = var = = ε assumpion ha ( 0) σ <. αε σ α and he = 0 0 Wha is mean by he erm fundamenal in he descripion of he moving average represenaion? I is a way of designaing a paricular orhogonal basis for H H. There eis an uncounable infiniy of differen orhogonal bases for, us as here are for spaces such as k R. As far as I know he erm is aken from Rozanov (967); i was popularized by Chrisopher Sims. There is an equivalence beween he Hilber space generaed around he sochasic process innovaions ε and he Hilber space generaed around he fundamenal Theorem.3. Equivalence beween he Hilber space of a ime series and is associaed fundamenal innovaions. Le ( ε ) H denoe he Hilber space generaed by ε ε,..., he fundamenal moving average componens of a zero-mean, weakly saionary process. Then ( ε ) = H H. 6

7 Pf. This is lef as an eercise; he proof amouns o showing ha each of he wo Hilber spaces is a subse of he oher. 3. Predicion Finally, we consider he quesion of how o opimally predic a ime series given is hisory. Le denoe he proecion of ono H. This proecion is imporan in ha i is also he soluion o he linear predicion problem for H. 3 relaive o he informaion se Theorem.4. Opimal linear predicor. The proecion is he soluion o min E( ξ ) ξ H Pf. Le ξ solve he minimizaion problem. The predicion error equals ε + ξ. The variance of his erm will equal σ ε + σ ξ, since ε is orhogonal o ξ. This variance mus eceed σ ε unless ξ is zero. Uniqueness of he proecion hen verifies he resul. This heorem implies ha ε is he forecas error associaed wih he opimal (in a minimum variance sense) linear forecas of given he informaion se H. The erm linear means ha he forecas has o be an elemen of he space, and so is eiher a linear combinaion of,, or he limi of such a 3 This heorem a special case of he general resul on he relaionship beween Hilber space proecions and cerain minimizaion problems described in Lecure Noes. 7

8 sequence. Hence, one can hink of a ime series as a weighed average of curren and pas forecas errors. This is inuiive since hese forecas errors reveal aspecs of he process ha are realized each ime period. The opimal linear predicor heorem provides insigh ino he naure of informaion ses and associaed predicions. By consrucion, + H and G G... G. The fundamenal innovaions ε,..., ε + represen he par of ha is revealed afer he forecas. The fundamenal innovaions can be hough of as informaion incremens beween ime periods. From he perspecive of he Wold heorems, he moving average represenaion of a ime series can hus be seen as a naural way of hinking abou is underlying linear srucure; much of ime series analysis is based on his perspecive, which is ofen referred o as he ime domain approach. Tha said, nohing in hese derivaions rules ou he presence of nonlinear srucure in he sochasic process. 8

9 References Ash, R. and M. Gardner, (975), Topics in Sochasic Processes. New York: Academic Press. Rozanov, Y., (967), Saionary Time Series, San Francisco: Holden Day. Wold, H., (948), On Predicion in Saionary Time Series, Annals of Mahemaical Saisics, 9, 4,

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