Part 1: White Noise and Moving Average Models


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1 Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical srucure does no evolve wih ime. A saionary series is unlikely o exhibi longerm rends. To see why, we need a beer definiion of rend. Trend is a endency of he series o increase (or decrease) no necessarily for he realizaion ha acually occurred, bu insead for he "ypical" realizaion. Consider he underlying (populaion) mean of he series, E [x ]. This represens he average value we would ge for he series a ime if we zaion, bu he key aspec of saisical hinking is o admi ha he daa series we acually go is jus a could urn back he hands of ime and look a many realizaions of he series. (We only have one realisample realizaion from he many series we migh have goen bu didn.) Since E [x ] is deermined by he underlying saisical srucure of he series, saionariy implies ha E [x ] mus no depend on ime. Bu his means ha he series has no underlying rend. applied. Since saionary ime series are unlikely o exhibi longerm rends, any apparen rend in he daa should firs be removed (e.g. by differencing) if he models discussed here are o be usefully Auocorrelaion A useful measure of linear associaion beween wo random variables X and Y (assumed here o have mean zero) is heir correlaion : Corr (X, Y ) = Covariance (X, Y ) = E [XY ]. Variance (X ) Variance (Y ) E [X ] E [Y ] I is always rue ha 1 Corr (X, Y ) 1. The larger he absolue value of Corr (X, Y ), he sronger he linear associaion beween X and Y. The concep of correlaion plays a key role in ime series analysis. We hink of x
2  ( = 1,,3,... ) as a sequence of random variables, any pair of which may be correlaed. The saionariy assumpion implies ha Corr (x, x since i is a correlaion beween he ime series and is own pas. j ) depends only on he ime separaion j, and no on he ime locaion. Noe ha Corr (x, x ) = 1. We ofen refer o Corr (x, x j )asanauocorrelaion As long as Corr (x, x ) 0 for some j > 0, we can obain a linear forecas of x from j x 1, x,.... To give a simple example of his, suppose ha Corr (x, x 1) =ρ 1, and ha E [x ] = E [x ] = 0. Since ρ is a correlaion, we mus have 1 ρ 1. The larger ρ, he sronger he linear associaion beween x and x 1. The bes linear predicor of x based on x 1 is ρ1x 1. Thus, if ρ >0(posiive auocorrelaion ), he forecas for x increases as x 1 1 increases. Examples include he daily NASDAQ index, daily emperaures, monhly ineres raes. If ρ <0(negaive auocorrelaion ) 1, he forecas for x decreases as x increases. Examples include hours of sleep per nigh, he daily 1 caloric inake of an individual, and oupu from a producion process which is being coninuously adjused o achieve a desired arge oupu. If ρ =0, hen x and x are said o be uncorrelaed,and 1 1 he bes linear forecas of x is zero (or, in general, E [x ]). Examples include he daily reurns on Gen eral Moors sock, and he difference beween 3.5 and he number rolled in independen osses of a fair die. Noe ha, if x and x are uncorrelaed, knowledge of x does no help us o linearly forecas x. 1 1 Whie Noise A saionary ime series ε is said o be whie noise if Corr (ε, ε ) = 0 for all s. s Thus, ε is a sequence of uncorrelaed random variables wih consan variance and consan mean. We will assume ha his consan mean value is zero. Plos of whie noise series exhibi a very erraic, jumpy, unpredicable behavior. Since he ε are uncorrelaed, previous values do no help us o forecas fuure values. The resuls of successive spins of a roulee wheel provide an example of a whie noise series. Whie noise series hemselves are quie unineresing from a forecasing sandpoin (hey are no linearly forecasable), bu hey form he building blocks for more general models. In economic ime series, he whie noise series is ofen hough of as represening innovaions, or
3 3 shocks. Tha is, ε represens hose aspecs of he ime series of ineres which could no have been prediced in advance. Moving Averages A simple moving average is a series x generaed from a whie noise series ε by he rule x =ε +βε 1. Noe ha, unless β=0, x will have a nonrivial correlaion srucure. In fac, we have Corr (x, x 1 var ε ) =β var x, (1) Corr (x, x j ) = 0 (j > 1). () Equaion (1) implies ha if β is posiive, hen adjacen erms of x will be posiively correlaed, so ha an above average (i.e., posiive) x will end o be followed by a furher above average value. For a whie noise series, β=0, an above average value is equally likely o be followed by an above average or a below average value. If β is negaive, hen an above average erm is likely o be followed by a below average erm. We see ha if β is posiive (zero or negaive), hen x will be smooher (as smooh or rougher) han he whie noise series ε. Equaion () implies ha here is no correlaion beween he presen value of x and all previous values apar from he mos recen. Thus, he simple moving average is said o have a shor memory. To prove equaion (1), noe ha E [x x ] = E [(ε + βε )(ε + βε )] = E [ε ε ] +βe [ε ] +β E [ε ε ] +βe [ε ε 1 ] 1 =βe [ε ] s j ince E [ε ε ] = 0 for all j > 0. Since ε is saionary wih zero mean, so is x, and we have 1 E [ε ] = var ε = var ε, var x = var x 1 1.
4 4 Thus, Corr [x, x ] = 1 E [x x 1] = var x var x 1 βe [ε 1] = var x var x 1 β var ε. var x The proof of () is similar. of x Suppose we wan o forecas xn +1 from x 1,...,xn for he simple moving average x =ε +βε 1 wih β known. Since x =ε +βε and since he bes forecas of ε is zero, he opimal forecas n +1 is n +1 n +1 n n+1 f n,1 =βεn. To obain a useful forecas, however, we mus express ε in erms of x,...,x. Since f n 1,1 n 1 n n n 1 =βε and since x =ε +βε, we have n 1 n ε = x βε = x f. n n n 1 n n 1,1 If we (somewha arbirarily) se f = 0, hen we can compue (approximaions o) ε, ε,...,ε b recursively applying he formula 0,1 1 n y ε = x f, = 1,,...,n. 1,1 We can hen evaluae he forecas f =βε, n,1 n a n 1,1 1,1 s well as f,...,f. A measure of forecasabiliy of a series is R = 1 Variance of Forecas Error. Variance of x This analogous o he version of R used in regression analysis, R = 1 SS /SS. Jus as in regres Resid Toal  sion, he forecasing version of R is guaraneed o be beween 0 and 1. If R is close o zero, hen he variance of he forecas error is almos as large as he variance of he series x iself. Thus, he bes linear forecas is no performing much beer han he rivial forecas (zero), so he series is no very forecasable when R is close o zero. On he oher hand, if R is close o 1, hen he forecas error
5 5 from he bes forecas is much less han he forecas error from he rivial forecas, so he series is very forecasable in his case. For he simple moving average, we have var x = var (ε +βε ) 1 = var ε + β var ε + βcov (ε, ε 1 1) = (1 +β ) var ε. The forecas error is e = x f = [ε + βε ] βε =ε. n,1 n +1 n,1 n +1 n n n+1 Thus, R reduces o R = 1 var ε = 1 1 = β. (1 +β )var ε 1 +β 1 +β Forecass for more han one sep ahead are all zero, since for h > 1 we have x =ε +βε, n +h n+h n+h 1 boh erms of which are "unforecasable". In his case, R = 0. The simple moving average is also called a f irs order moving average, denoed by MA(1), because i conains jus one parameer, β. A generalizaion of he MA(1) model is he q h order moving average MA(q), given by x =ε +βε +βε β ε. 1 1 q q I is easily shown ha for he MA(q) series, and hence he forecas f n, h Corr (x, x ) = 0 j > q j is zero for h > q. Since he opimal onesep forecas is x =ε +βε +βε β ε, n +1 n +1 1 n n 1 q n q +1 f n,1 =βε 1 n +βε n βqεn q + 1
6 6 and he onesep forecas error is To acually f orm he opimum forecas, sar wih f e = x f =ε. n,1 n +1 n,1 n +1 0, 1 = 0 and hen form εˆ = x f, = 1,,...,n. 1,1 In general, o form he forecas f for j q, wrie down he expression for x, replace every unk n, j n+j  nowable fuure value of ε by 0 and all he remaining values by he εˆ given above. How migh a moving average model arise in he real world? For example, suppose y is he daily change in price of some produc, and ε +1 is he effec on omorrow s price of unexpeced news. The full impac of his news may no be compleely absorbed by he marke. Suppose his akes wo days o happen. Then we would have y =ε +b ε, where ε is he news which canno be prediced from ime +1, and b ε +1 is he reassessmen of he earlier piece of news. This would lead o an MA (1) model for y. Anoher way ha an MA (1) model can arise is hrough overdif f erencing. Suppose, for example, ha our original ime series is whie noise, x =ε. This is saionary, and should no be differenced. Differencing a series which is already saionary is called overdif f erencing and should be avoided if possible, bu i ofen occurs by acciden. If we (over) difference our whie noise series, we ge y =ε ε, which is an MA (1) series, wih negaive auocorrelaion. Alhough he original series was 1 no linearly predicable, he differenced series will be!
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