Tme Seres Seven N. Durlauf Unversy of Wsconsn Lecure Noes 4. Unvarae Forecasng and he Tme Seres Properes of Dynamc Economc Models Ths se of noes presens does hree hngs. Frs, formulas are developed o descrbe he opmal lnear predcon of a me seres gven s hsory. Second, he emprcal mplcaons of he clam ha one me seres s an opmal predcon of anoher one are descrbed. Many macroeconomc models mae heorecal predcons of hs ype. Thrd, he dea of excess volaly s developed va a consderaon of some elemenary properes of forecass. Unless specfed, processes are assumed o be second order saonary and srcly ndeermnsc. 1. The Wener-Kolmogorov Predcon Formulas Recall ha he soluon o he problem mn E ξ H ( x) ( ξ ) 2 x (4.1) s equal o he projecon of x ono ( ) H x, denoed as x. Leng αε j = 0 j j denoe he fundamenal MA represenaon of x, from he equvalence of ( ) and ( ε ) H x H s mmedae ha x may be wren as x = α ε j j (4.2) j = 0 1
The reason for hs s smple: a me, one can condon forecass on ε, ε,.e. ( ε ) H. The componen of x ha s deermned by ε,..., ε,.e. 1 1 j = 0 αε j j s orhogonal o ( ε ) H and so s lnear projecon ono hs space s 0. Pu dfferenly, a me one has no bass forecasng he fuure shocs. To acheve a more parsmonous expresson of expressons such as (4.2), I defne he annhlaon operaor for lag polynomals. Defnon 4.1 Annhlaon operaor Le π ( L ) be a polynomal lag operaor of he form 2 2 π π π π π L L L L The annhlaon operaor, denoed as ( ) 2 0 1 2 elmnaes all lag erms wh negave exponens,.e. ( ( )) 2 π L π π L π L (4.3) 0 1 2 as Usng he annhlaon operaor, one can re-express he opmal predcor x = L α ( L) ε (4.4) When he MA polynomal s nverble, α( ) L x = ε, so ha one can explcly express he predcor as a weghed average of curren and lagged x s, (4.4) mples ( L) α x = α ( L) x L (4.5) 2
Ths formula expresses forecass as weghed averages of observables. Equaons (4.4) and (4.5) are nown as he Wener-Kolmogorov predcon formulas, named afer ndependen dscoverers Norber Wener and Andre Kolmogorov. Example 4.1. MA(1) process. Le x = ε ρε 1. Then he AR Wener-Kolmogorov formula for he opmal lnear predcon of he process s 1 ρl = ( ρ ) 1 x 1 L x L (4.6) whch equals 0 for > 1, as one would expec snce he process s unaffeced by shocs more han one perod n he pas. Example 4.2. AR(1) process. If x = ρx ε, hen he AR Wener-Kolmogorov formula for he opmal lnear 1 projecon s x = ρ ( ρ ) = ρ L 1 L x x (4.7) = 0 Forecass of AR(1) process over a fxed horzon hus possess a very smple srucure: x = ρ x. 1 We now consder a much more complcaed use of he Wener- Kolmogorov formulas. 3
2. The Hansen-Sargen formula for geomercally dscouned dsrbued leads The Wener-Kolmogorov formulas have played a sgnfcan role n he developmen of modern emprcal macroeconomcs. The reason for hs s many dynamc aggregae models mply ha one me seres represens a forecas of some combnaon of oher seres. Lars Hansen and Thomas Sargen developed he machnery o undersand how such expecaons-based relaonshps lead o esable mplcaons of macroeconomc heores. Ths machnery has been of mmense mporance n he developmen of modern emprcal macroeconomcs. I develop he basc Hansen-Sargen formula n he conex of he consan dscoun dvdend soc prce model. Ths s an example of a geomerc dsrbued leads model n ha one varable s a geomercally weghed sum of expeced values of fuure levels of anoher varable Defnng he varables D = dvdends (measured n real erms), P = soc prce (measured n real erms), and β = fxed dscoun rae, he consan dscoun dvdend soc prce assers ha he soc prce seres obeys * ( ) E( ) P = β E D = P (4.8) = 0 where * P = β D (4.9) = 0 Wha s he economc nerpreaon of hs model? The model can be hough of as follows. Suppose agens are rs neural (.e. uly from consumpon a s lnear n he level) and dscoun uly geomercally a rae β. One can hn of 4
ownershp of a un of soc as provdng a sream of real consumpon (va he dvdends) f held forever. The opporuny cos of ownershp s he consumpon foregone oday by buyng a un of soc. The prce formula expresses he prce a whch an expeced uly maxmzer s ndfferen beween buyng a un or no, whch s wha he prce mus do n equlbrum. How can one es wheher soc prces are conssen wh hs heory? Suppose ha dvdends have he fundamenal movng average represenaon. ( ) D = α L ε. (4.10) Ths represenaon s recoverable from he daa. One may use hs represenaon o es he heory by compung he projecon of P ono H ( D ) ha s mpled by he heory, and comparng hs projecon o he drec projecon P ono H ( ) mples ha D. Pu dfferenly, he soc prce model ha has been descrbed j proj( P H( D)) = β proj( D j H( D )) (4.11) j = 0 If one nows he process for he dvdend seres, one can derve he rgh hand sde expresson and compare o he lef hand sde formula. In order o derve he projecon on he rgh hand sde of hs expresson, one proceeds n wo seps. Frs one projecs seres π ( ) P ono ( ε ) L ε and second, one nvers he ε s o generae H o generae a me D s. The dervaon reles on he mplcaon of he model ha P may also be expressed as: ( ( )) β 1 ( ) ( ) E P H D = E P H D D (4.12) 5
whch mples ha ( L) π π ( L) ε = β ε α( L) ε L (4.13) or ( L) π π π ( ) ε β 0 L = ε α( L) ε L L (4.14) Algebrac manpulaon yelds ( ) ( ) ( ) 1 βl 1 π L = α L βπ L 1 (4.15) 0 To manpulae hs expresson n order o express π ( L ) n erms of α ( L ), noce ha yelds L = β, hen π0 = α( β). Subsung no hs expresson and rearrangng π ( L) α = ( L) βα ( β ) 1 βl L (4.16) whch leads o ( ) ( ) 1 ( ) ( ) 1 α L βα β L βα β α L L E( PH ( D) ) = ε = D 1 βl 1 βl (4.17) Ths relaonshp beween π ( L) and α( ) wres L s complcaed. For example, f one 6
E ( PH( D) ) = β ε = 0 L α ( L) (4.18) hen s apparen ha π = β α = (4.19) These ypes of cross equaon resrcons are ubquous n lnear macroeconomc models wh raonal expecaons. 3. Some properes of forecass The Hansen-Sargen machnery provdes he mos heorecally appealng way o undersand and evaluae raonal expecaons models. On he oher hand, he machnery can be complcaed o mplemen. Furher, f he cross equaon resrcons are rejeced, s no obvous how o nerpre he dmensons along whch he model s an emprcal falure. A number of smpler ess of macroeconomc models may be consruced usng properes of opmal lnear forecass. For example, suppose ha a varable x s, f a ceran model holds, he opmal forecas of anoher varable y gven an nformaon se F,.e. ( ) x = proj y F (4.20) Such a clam requres ha he forecas error η = y x s orhogonal o F. Ths provdes a smple way of esng such models. Here are some examples of mplcaons ha have been exploed n he emprcal leraure. 7
. One of he elemens of F s x, so one mplcaon of hs s ha forecass mus be orhogonal o forecas errors.. Suppose ha y s an elemen of F 1. Ths means ha he forecas errors η are uncorrelaed.. Snce var ( ) = var( η ) = var ( ) var ( η ) y x x, when x s a forecas of y, one can es hs relaonshp by evaluang he mplcaon ha he varances of realzaons should be greaer han he varance of raonal forecass of he realzaons,.e. var ( ) > var ( ) y x (4.21) (The wo are equal when forecass errors are always equal o 0; I gnore hs unneresng case.) Evaluaon of wheher hs nequaly holds for a daa seres s nown as an excess volaly es. These ess were ndependenly nroduced by Sephen LeRoy and Rober Shller. One can consruc many forms of excess volaly ess. Suppose ha z F. Then s mmedae ha var ( ) > var ( ) y z x z (4.22) Reurnng o (4.21), noce ha ( η ) ( ) ( η ) ( η ) var y = var x = var x var 2cov x, (4.23) so ha var ( y ) > var ( x ) mples ha var ( η ) > 2cov ( η, x ) rewren as regresson ( η x) ( η ), whch may be cov, 1 <. Excess volaly can hus be esed based on he var 2 8
x = ηb ξ (4.24) Excess volaly requres ha he coeffcen b n hs regresson s less han 1 2. Ths regresson has an unusual form as regresses he forecas agans he forecas errors, raher han he forecas error agans he forecas, whch s he regresson suggesed by he requremen ha he forecas errors generaed by raonal forecass are unpredcable. Ths equvalence developed by Seven Durlauf and Peer Phllps. Therefore, excess volaly mples predcably of forecas errors, alhough predcably of forecas errors does no mply excess volaly. 4. Bubbles Reurnng o he dvdend soc prce model, he basc equaon (4.8) may be rewren ( ) β E( ) P = β E D = D D = 0 = 1 ( ) ( ) = D β β E D = D βe β E D 1 1 1 = 0 = 0 = D βe P 1 (4.25) Ths means ha he excess holdng reurn ξ 1, defned as ξ = βp D P (4.26) 1 1 s unpredcable gven all nformaon avalable a me as s smply a parcular forecas error. A naural queson s wheher esng hs unpredcably propery 9
s equvalen o esng he nonlnear resrcons embedded n he Hansen- Sargen formula whch relaes he level of soc prces o he dvdend process. I urns ou ha he answer s no, he approaches are no equvalen. To see hs, call a prce process ha fulflls (4.8) prce. Suppose one adds a process B o f P, whch means he fundamenal f P, a process wh srucure B = β B µ (4.27) where µ s unpredcable gven nformaon a 1. Ths new process wll volae (4.8) ye he excess holdng reurn wll sll be unpredcable snce f f βp D P = βp D P βµ (4.28) 1 1 The process B s an example of a bubble. Tess of he unpredcably of excess holdng reurns canno deec bubbles as hey do no place any resrcons on he source of he reurns. 1 As such, hey do no es he full se of mplcaons of he dvdend soc prce model we have been sudyng. Noce ha B s an explosve process. When presen, prces wll no have a fne varance and n fac he assumpons requred o consruc a Hlber space around curren and lagged prces are volaed. Projecons of prces ono curren and lagged dvdends wll no longer be defned. Ths has mplcaons for he consrucon of ess for he presence of bubbles and can nvaldae some of hem. On he oher hand, he explosve propery s self a resrcon on daa. 1 In laer pars of Economcs 712, you wll be nroduced o he dsncon beween Euler equaons and ransversaly condons n he descrpon of he soluons o nfne horzon maxmzaon problems. Furher, hese soluons wll be shown o correspond o he equlbra of varous models. Unpredcably of excess holdng reurns s analogous o fulfllmen of an Euler equaon. 10
By approprae choce of he probably densy for µ, one can generae a neresng bubble properes, such as bubbles ha collapse wh probably 1. Olver Blanchard and Mar Wason developed an example of hs ype n whch β B B = wh probably λ, 0 oherwse. (4.29) λ I s sraghforward o verfy ha hs bubble s conssen wh (4.28); he ey mplcaon of hs srucure s ha he suppor of µ depends on he value of B. Smaller values of λ render he bubble more explosve before pops. 11