BEST LINEAR FORECASTS VS. BEST POSSIBLE FORECASTS Opimal ear Forecasig Alhough we have o meioed hem explicily so far i he course, here are geeral saisical priciples for derivig he bes liear forecas, ad he bes possible forecas. Suppose we wa o +h 1, h predic x from x, x,.... The he bes liear forecas (i erms of mea squared error) f is deermied by wo characerisics:, h (1) f is a liear combiaio of x, x,..., 1 +h, h (2) The forecas error x f is ucorrelaed wih all liear combiaios of x, x,... 1. Noe ha codiio (1) simply says ha he forecas has o be liear, ad is oly allowed o use currely available iformaio. Codiio (2) is also sesible, sice if i does o hold, he here mus be some oher liear foreca which is beer ha he origial oe. To see his, suppose ha here is some liear combiaio of he available daa which is correlaed wih he error. The his liear com- biaio would provide a (orivial) liear forecas of he error. Bu if we ca forecas he error, he we ca improve our origial forecas: jus add he origial forecas o he forecas of he error. We have acually used codiio (2) hroughou he course. For example, i liear forecasig for a AR (1) model x =αx +ε where he ε are zero mea whie oise (i Chaper 3, Par II), we wroe 1 x =αx +ε +1 +1, ad he said ha "he opimal [liear] forecas of ε is zero sice ε is ucorrelaed wih all prese liear combiaios of x, x +1 +1 ad pas values of {x }." This is he same as sayig ha he forecas error, ε 1 +1, is ucorrelaed wih all,..., ad herefore is o liearly forecasable, excep by is mea, which is zero. So we were really usig codiio (2) o show ha f =αx is he bes possible liear forecas.,1
-2- Opimal Forecasig ad Codiioal Expecaio As we have see, i is o ecessarily rue ha he bes liear forecas is he bes possible fore- cas. I may be ha some oliear fucio of he observaios gives a smaller mea squared predic- io error ha he bes liear forecas. I ca be show ha he opimal foreca, i erms of mea squared error, is he codiioal expecaio, E [x x, x,... ]. This is a fucio of he prese +h 1 ad pas observaios. Isead of givig a mahemaical defiiio of codiioal expecaio (i is relaed o codiioal probabiliy), we ca hik of i simply as he mea (i.e., expecaio) for he fuure observaio x +h, amog all realizaios (pahs) of our ime series which agree wih he available iformaio, x, x,.... For example, cosider he AR (1) process, ad ake h = 1. If x, x,... are kow, 1 1 i is sesible o "codiio o hem", i.e., o say ha, for he purposes of forecasig x oly ieresed i realizaios of our ime series which pass hrough x, x 1 +1, we are ow,.... All ohers are irreleva o us, sice hey are i coradicio o our available iformaio. If we igore he available iformaio, ad average over all possible pahs, he expecaio for he fuure value x is he ucod +1 +1 +1 iioal mea E [x ] = 0. This would be our bes forecas of x if we did o kow he values of x, x,.... O he oher had, sice for he AR (1) model we have x =αx +ε, we see ha if 1 +1 +1 we cosider oly hose pahs which pass hrough x, x 1,..., he we ca rea x as a (oradom) cosa. We fid ha he codiioal mea is E [x x, x,... ] =αx + E [ε x,... ]. +1 1 +1 1 The quesio as o wheher αx is he bes possible oe-sep predicor for his AR (1) model is he same as he quesio as o wheher he codiioal expecaio of ε, give he available iformaio, +1 is zero or o. If he {ε } are sric whie oise, i.e., if hey are idepede wih zero mea, he he above codiioal expecaio is ideed zero, ad αx is he bes possible forecas. O he oher had, we showed i he hadou o oliear models ha if ε =e +βe e where {e } is sric whie oise 1 2, or. he {ε } is zero mea whie oise, bu i ca be prediced oliearly from is ow pas. The he co diioal expecaio for ε would o be zero, ad hece αx would o be he bes possible predic +1 -
-3- Marigales ad Marigale Differeces A more geeral quesio is: Whe is he bes liear predicor he same as he bes possible pred- ay h. We also assume ha he series has o uderlyig liear red. icor? We will focus o oe-sep predicio, alhough he mai resuls preseed here acually hold for Suppose he ime series has a MA () represeaio, x = a ε,(a = 1), k = 0 k k 0 where {ε } is a zero mea whie oise process. Sice x =ε + a ε +1 +1 k =1 k +1 k, he bes liear predicor is, 1 f = a ε. (1) k = 1 k +1 k The bes possible predicor is f, he codiioal expecaio of x give he available iformaio,1 +1. We ca regard he "available iformaio" as eiher he prese ad pas observaios x, x as he prese ad pas iovaios ε, ε 1,..., sice i priciple hese iovaios ca be deermied from he observaios. So he bes possible predicor is 1,...,or f = E [x ε, ε,... ].,1 +1 1 Bu if we are codiioig o ε, ε ge 1,..., he hese quaiies ca be reaed as cosas, so we f = E [ε ε, ε,... ] + a ε. (2),1 +1 1 k = 1 k +1 k Comparig Equaios (1) ad (2), we see ha he bes liear predicor f he bes possible predicor f,1 if, ad oly if,, 1 will be he same as E [ε +1 ε, ε 1,... ] = 0. (3) Equaio (3) says ha {ε } is o forecasable, eiher liearly or oliearly, so ha he bes possible
-4- forecas of ε +1 is zero. If {x } is a liear process, i.e., if {ε } are idepede, he Equaio (3) holds, so ha he bes liear predicor is also he bes possible predicor i his case. If he {ε } are Gaussia, he sice hey are ucorrelaed hey mus also be idepede, so ha,1 Equaio (3) mus hold. Thus, f = f for ay saioary Gaussia process. Aoher way o see his,1 is ha if he {x } are Gaussia, he (i ca be show ha) he codiioal expecaio E [x x, x,... ] is a liear combiaio of x, x,..., so he bes possible predicor i +1 1 1 s, 1 liear, ad hece f = f.,1 Ay process {ε } saisfyig Equaio (3) is said o be a Marigale differece. So we have show ha he bes possible predicor is he same as he bes liear predicor whe, ad oly whe, he iovaios are a Marigale differece. To udersad he reaso for his ermiology, we have o defie a Marigale. The process {x } is said o be a Marigale if for all, ad for all lead imes h > 0, E [x x, x,... ] = x. (4) +h 1 Equaio (4) says ha he bes possible forecas of x +h give he available iformaio is simply he mos rece observaio, x. Ifx represes he forue of a gambler a each roud of a game, he Equaio (4) says ha he game is fair: The gambler s expeced fuure forue, give he game s hisory, is simply he curre forue. The chage i he forue, x +h x, is o predicable, sice he bes forecas of his chage is E [(x x ) x, x,... ] = E [x x, x,... ] x = x x = 0. +h 1 +h 1 We see from he above (wih h = 1) ha if {x } is a Marigale, he he firs differece ε =x x 1 is a Marigale differece. This explais he reaso for he erm "Marigale differece". I ca be show, usig properies of codiioal expecaios, ha he values of ay Marigale differece series {ε } mus be ucorrelaed, ad mus have zero mea. The Efficie Marke Hypohesis holds ha securiy prices are a Marigale. I herefore says ha he chage i price cao be prediced from prese ad pas prices, by ay mehod, eiher liear or
-5- oliear. Thus, he chage i price is a Marigale differece. I erms of he sock marke, we could say ha, accordig o he Efficie Marke Hypohesis, he curre price reflecs all publicly available iformaio abou he value of he sock, so he bes forecas is jus he curre price. A aleraive (somewha differe) form of he hypohesis is ha he reurs r = (x x 1 )/x 1 are a Marigale differece. I his case, i is he reurs which cao be prediced, i.e., he bes possi- ble forecas of a fuure reur is zero. To see wha his hypohesis says abou he prices x, we use he Taylor series expasio, log x x 1 whe x is close o 1. If x /x ypically be he case for sock prices), we ca wrie he reurs as 1 is reasoably close o 1 (as would r = x /x 1 log (x /x ) = log x log x 1 1 1. Thus, he reur is esseially he firs differece i he log prices. I is o hard o show ha if we iegrae a Marigale differece, he resul is a Marigale. So he hypohesis ha he reurs {r } are a Marigale differece is esseially equivale o he hypohesis ha he log prices {log x } are a Mar- igale. If he raio of he highes o he lowes observed price is large, as i he case of he Dow Joes Idusrial Average observed for log periods or durig periods of high volailiy, he i may be prefer- able o work wih he reurs, or equivalely wih he firs differece of he log prices, ad cosider his aleraive versio of he Efficie Marke Hypohesis. Wha is he relaioship bewee he Efficie Marke Hypohesis ad he Radom Walk Hypohesis, i.e., ha he prices follow a radom walk? Here, we eed o be careful abou our defiiio of a radom walk. Up uil ow, we have simply assumed ha he iovaios ε =x x 1 are a zero mea whie oise process. Tha was eough o guaraee ha he iovaios are o liearly predic- able. From ow o, we will oly use he erm "radom walk" o refer o he case where he iovaios are idepede, ha is, x = x + e, (5) where he {e } are idepede wih zero mea ad equal variaces. I his case, he iovaios are 1 compleely upredicable by ay mehod. I addiio, all fucios of he iovaios are also com-
-6- pleely upredicable uder our prese defiiio of a radom walk. This poi will be impora whe we cosider ARCH models. As defied i Equaio (5), a radom walk is a example of a Marigale, sice E [x e, e,... ] = x + E [e e, e,... ] = x, +1 1 +1 1 ad i is easy o show ha for ay h > 0, E [x x, x,... ] = x as well. Thus, if he Rado +h 1 m Walk Hypohesis holds for a give ime series, he he Efficie Marke Hypohesis mus also hold. However, he Radom Walk Hypohesis is more resricive ha he Efficie Marke Hypohesis. Tha is, here are series which saisfy he Efficie Marke Hypohesis which are o radom walks. To come up wih a example of oe, we simply eed o fid a Marigale which is o a radom walk. This is he same as fidig a series whose firs differeces are a Marigale differece bu are o idepede. As we will see, a cocree example is provided by he series x = x + ε where he {ε } obey a 1 ARCH model.