Exchange Rate Expectatons: Evdence from an Artfcal Economy Phlp S. Marey Abstract Survey studes on exchange rate expectatons tend to reject the ratonal expectatons hypothess for longer horzons. Extrapolatve, adaptve and regressve expectatons have been tested as alternatves, usually rejectng statc expectatons. The purpose of ths paper s to nvestgate the plausblty of these alternatve exchange rate expectatons mechansms n an artfcal economy wth traders whch are heterogeneous n ntal endowments, rsk averson and use of nformaton. Artfcal markets whch consst of ether extrapolatve or adaptve expectatons traders fal to reproduce statstcal propertes that are characterstc for emprcal quarterly exchange rate seres, whle artfcal markets wth regressve expectatons traders often succeed. Adaptve expectatons markets are rarely weakform effcent. Extrapolatve expectatons markets may be weak-form effcen but generate too many extreme returns to be emprcally plausble. Regressve expectatons markets often produce exchange rate seres that are smlar to emprcal data. The (perceved) exstence of an anchor seems to play an mportant role n the functonng of foregn exchange markets, determnng the frequency of extreme exchange rate returns. (JEL F3) ROA, Maastrcht Unversty P.O. Box 66 6200 MD Maastrch The Netherlands. Telephone: +3 43 3883 287 Telefax: +3 43 320 94 E-mal: p.marey@roa.unmaas.nl
. Exchange rate expectatons Survey data studes ndcate that long term exchange rate expectatons are heterogeneous (Taylor & Allen (992), Ito (990)) and are not adequately descrbed by ratonal expectatons (Domnguez (986), Frankel & Froot (987a), Froot & Frankel (989), Ito (990), Cavagla, Verschoor & Wolff (993)). Extrapolatve, adaptve and regressve expectatons have been tested as alternatves, aganst whch statc expectatons are usually rejected (Frankel & Froot (987a) and (987b), Bank of Japan (989), Froot & Frankel (990), Cavagla, Verschoor & Wolff (993)). The objectve of ths paper s to nvestgate the plausblty of these alternatve exchange rate expectatons mechansms n an artfcal economy wth traders whch are heterogeneous n ntal endowments, rsk averson and use of nformaton. We wll focus on three months ahead exchange rate expectatons, as survey evdence s most elaborate for ths horzon. The alternatve expectatons schemes (extrapolatve, adaptve and regressve expectatons) can be summarzed (Frankel & Froot (987a)) as: E (s t+ ) = (-α ) s t + α z () where s t s the natural logarthm of the spot exchange rate S t n perod t and E (.) denotes trader s perod t (not necessarly mathematcal) expectaton wth respect to the varable between brackets. If z = s t- we speak of extrapolatve expectatons. We dstngush between three cases: α < 0 (bandwagon expectatons), α = 0 (statc expectatons) and α > 0 (dstrbuted lag expectatons). Another scheme that s techncal or chartst n nature s adaptve expectatons: z = E t-, (s t ). It s straghtforward to show that such traders use the entre hstory of the exchange rate and that the forecast follows from a geometrc seres. 2
If z = q s the natural logarthm of some fundamental or long run equlbrum exchange rate Q, we speak of regressve expectatons. The paper s organzed as follows. In secton 2 we ntroduce the artfcal economy approach that wll be used to perform controlled experments wth exchange rate expectatons. In secton 3 we descrbe a theoretcal model of foregn exchange between heterogeneous traders wth extrapolatve, adaptve and regressve expectatons. In secton 4 we assgn emprcally plausble parameter and ntal values to the theoretcal model, whch s thus transformed nto an artfcal economy. In secton 5 we use ths artfcal economy as a foregn exchange laboratory: we perform smulatons for dfferent specfcatons of exchange rate expectatons. In secton 6 we draw conclusons from the controlled experments n secton 5. 2. The artfcal economy approach An mportant objecton aganst the use of survey data s the uncertanty concernng the truthfulness of the partcpants reportng. As Frankel & Froot (987a) put t: Economsts generally dstrust survey data. It s a cornerstone of postve economcs that we learn more by observng what people do n the marketplace than what they say. A related objecton may be that the expectatons of a partcpant at the tme that he answers the survey, may dffer from hs expectatons at the tme at whch he performs a market transacton. Especally when we are nterested n the relatonshp between exchange rate expectatons and the 3
actual exchange rate, ths nonsynchroncty wth market data may lmt the approprate use of survey data. A more controlled envronment could be obtaned wth an expermental foregn exchange market. We could then solve the nonsynchroncty problem by askng partcpants about ther expectatons at the moment that they perform a transacton. However, we would stll be uncertan about the truthfulness of the partcpants reportng. In other words, wth real people a perfectly controlled experment s not feasble when expectatons, perceptons and preferences are varables that we are nterested n. Ths problem can be solved by performng experments wth artfcal people,.e. by performng computer smulatons wth agents whose behavor exactly follows the mathematcal descrpton that we mpose va the program. In ths way we know wth certanty what the expectatons of the partcpants are at the tme of transacton. The dea of ths artfcal economy approach s to smulate a theoretcal model for emprcally plausble parameter and ntal values and study whch theoretcal assumptons lead or do not lead to statstcal regulartes n the generated tme seres that match emprcally observed regulartes. We have depcted ths n fgure : 4
Fgure : Artfcal economy methodology emprcally plausble parameter and ntal values nput theoretcal model output (emprcally observed?) statstcal regulartes Backus, Gregory & Telmer (993) formally defne the concept of an artfcal economy:... we buld what has come to be called an artfcal economy - a numercal representaton of the theory whose propertes can be compared to those observed n the data. Notce that not all numercal representatons of theoretcal models are artfcal economes: only f the assgned numercal values are emprcally plausble, do we speak of an artfcal economy. Artfcal economes based on the representatve agent model of asset prcng - such as Backus, Gregory & Zn (989), Macklem (99) and Backus, Gregory & Telmer (993) - have not (yet) been very successful n reproducng emprcally plausble tme seres. One way to nvestgate the 5
plausblty of alternatve exchange rate expectatons mechansms s to relax the ratonal expectatons assumpton of the representatve agent model. However, the survey evdence that we mentoned n secton suggests that exchange rate expectatons are nether ratonal nor homogeneous. Hence we wll also relax the representatve agent assumpton and buld a model of foregn exchange between heterogeneous traders that do not necessarly form ratonal expectatons. Snce we are nterested n the plausblty of alternatve exchange rate expectatons mechansms, we would lke to have a model that has the flexblty to ncorporate all three alternatves (extrapolatve, adaptve and regressve). We can then perform smulatons wth dfferent versons of the model, correspondng wth dfferent specfcatons of the expectatons mechansm. Wth the valdty of the model as a mantaned hypothess, we can nterpret the (n)ablty of a model verson to generate emprcally plausble output as evdence of the (m)plausblty of the correspondng exchange rate expectatons mechansm. The artfcal economy can be used as a laboratory of foregn exchange that allows us to perform controlled experments wth exchange rate expectatons. 3. A model of foregn exchange between heterogeneous traders 3. Traders, currency postons and nterest rates Consder an nternatonal bankng system that organzes a one perod domestc money marke a one perod foregn money market and a spot foregn exchange market that allows traders to exchange domestc currency money market accounts for foregn currency money market accounts and vce versa. Suppose domestc trader (foregn trader j) starts the tradng sesson n perod t wth an 6
account of A (A j ) domestc currency unts and B (B j ) foregn currency unts. Tradng allows hm to change hs dle account (A, B ) ( (A j,b j ) ) nto an nterest-bearng account (X, Y ) ( (X j,y j ) ). In perod t+ the domestc currency wealth of domestc trader wll be W t+, = ( + I t ) X + ( + I t ) Y S t+ (2) The perod t+ foregn currency wealth of foregn trader j wll be W t+,j = ( + I t ) Y + ( + I t ) X j /S t+ (3) The partcpaton of domestc trader n the foregn exchange market wll have to satsfy X + S t Y = A + S t B (4) Therefore we may rewrte the domestc currency wealth of domestc trader n perod t+ as follows W t+, = [ ( + I t ) S t+ - ( + I t ) S t ] Y + ( + I t ) ( A + S t B ) (5) Analogously, the exchange of foregn trader j has to obey X j /S t + Y j = A j /S t + B j (6) We can therefore rewrte hs foregn currency wealth n perod t+ as 7
W t +, j + It I = + St + S t t A X j + ( + I t ) B j + S j t (7) Before tradng the nternatonal bankng system as a whole (we assume that there are m domestc traders and n foregn traders) has a domestc currency balance A = A + A t m n j = j= (8) and a foregn currency balance m n t t, t, j = j= B = B + B (9) After tradng the nternatonal bankng system as a whole has a domestc currency balance X = X + X t m n j = j= (0) and a foregn currency balance m n t t, t, j = j= Y = Y + Y () 8
The nternatonal bankng system guarantees ts account holders that they wll be able to attan any nterest-bearng domestc and foregn currency account balance they lke. Ths requres that X t = A t Y t = B t (2) For smplcty we assume that domestc and foregn nterest rates are exogeneously gven. Hence the exchange rate S t smultaneously clears the two money markets and the foregn exchange market f t ensures that both condtons n (2) are met. Snce the domestc and foregn currency postons of domestc and foregn traders are nterdependent va ther respectve budget equatons, Walras Law tells us that we only have to solve S t from one of the two condtons. We assume that the nternatonal bankng system knows the foregn currency supply and demand schedules of all ts clents and s thus able to act as a Walrasan auctoneer by announcng a clearng exchange rate S t. 3.2 Currency endowments For smplcty we assume that for all t > : A = (+I t- )X t-, (3) B j = (+I t- )Y t-,j (4) 9
A j = (+I t- )X t-,j (5) B = (+I t- )Y t-, (6) where A,, B,j, A,j, B, are exogeneously gven ntal values. It follows from the market clearng condtons that between perod t- and t the domestc and foregn currency stocks n the nternatonal bankng system evolve accordng to X t = (+I t- )X t- (7) Y t = (+I t- )Y t- (8) The nternatonal bankng system creates domestc (foregn) money at a rate that equals the domestc (foregn) nterest rate. 3.3 Trader preferences and demand functons Let E (S t+ ) denote the expectaton of domestc trader n perod t concernng the yet unknown future value S t+ and let Var (S t+ ) be the varance of S t+ that domestc trader predcts n perod t. Gven a certan value of exchange rate S t, t follows from (5) that domestc trader expects hs domestc currency wealth n perod t+ to be 0
E (W t+, ) = [ ( + I t ) E (S t+ ) - ( + I t ) S t ] Y + (+ I t ) ( A + S t B ) (9) and he expects the varance of hs perod t+ domestc currency wealth to be Var (W t+, ) = (+I t ) 2 Var (S t+ ) Y 2 (20) Assume that domestc traders are only nterested n ther one perod ahead domestc currency wealth. The one perod horzon can be nduced by an overlappng generatons structure n the sprt of De Long, Shlefer, Summers & Waldmann (990): Such an overlappng generatons structure may be a frutful way of modelng the effects on prces of a number of nsttutonal features, such as frequent evaluatons of money managers performance, that may lead ratonal, longlved market partcpants to care about short-term rather than long-term performance. The evaluaton of wealth n terms of domestc currency may be justfed by assumng that domestc traders only consume domestc goods. A convenent specfcaton of the expected utlty functon (Newbery (988)) s t+, t+, 2 t+, E ( U ) = E ( W ) γ Var ( W ) (2) where γ s the parameter of absolute rsk averson. If we substtute (9) and (20) nto expected utlty functon (2), we obtan
E (U t+, ) = [ (+I t ) E (S t+ ) - ( + I t ) S t ] Y + (+I t )(A + S t B ) + - /2 γ (+I t ) 2 Var (S t+ ) Y 2 (22) Hence the optmal foregn currency poston of domestc trader s Y = t t + t t 2 t t+ ( + I ) E ( S ) ( + I ) S ( + I ) γ Var ( S ) (23) provded that γ > 0 (second order condton). Let E ( S ) j t+ denote the subjectve expectaton of foregn trader j n perod t wth respect to the unknown future exchange rate /S t+ and let Var ( S ) j t+ be the varance of /S t+ that foregn trader j expects n perod t. Gven a certan value of exchange rate S t, t follows from (7) that foregn trader j expects hs foregn currency wealth n perod t+ to be I A t j E j( Wt+, j) = ( + It ) E j( St + ) + X j + ( + I t ) + B j S t S (24) t and he expects the varance of hs perod t+ foregn currency wealth to be 2 j t +, j t j t 2 + j Var ( W ) = ( + I ) Var ( S ) X (25) 2
Denote foregn trader j s absolute rsk averson n perod t as γ j. We assume that foregn traders are only nterested n ther one perod ahead foregn currency wealth and that they maxmze the followng expected utlty functon: j t+, j j t+, j 2 j j t +, j E ( U ) = E ( W ) γ Var ( W ) (26) If we substtute (24) and (25) nto (26), we get I A t j E j( U t+, j) = ( + It) E j( S t + ) + X j + ( + It ) + B S t S t j + 2 2 2 γ t, j ( + I t ) Var t, j ( S t + ) X t, j (27) The optmal domestc currency poston of foregn trader j s X j = I ( + It ) E j( St ) + + S t t 2 j j t + t ( + I ) γ Var ( S ) (28) provded that γ j > 0 (second order condton). The foregn trader s demand for foregn currency can now be derved by substtutng (28) nto budget equaton (6): 3
Y j It A ( + I t) E j( S t ) + + j St = B j + S 2 S ( + I ) γ Var ( S ) t t t j j t+ (29) From (23) and (29) we obtan the aggregate excess demand for foregn currency: Φ( S ) = t m ( + I ) E, ( S ) ( I ) S m t t t + + t t 2 = ( + I t ) γ Var ( S t+ ) = B + n n + + + ( I t) E S S I S j( t ) t ( t ) t 2 j t j= 2 2 ( + It) γ jvar j ( S t+ ) + A S j= (30) 3.4 Exchange rate expectatons Assume the followng specfcaton of expectatons formaton: α α t + t E S S t, ( ), = Z t (3) α α j t + t j t, j E ( S ) = S Z t, j (32) t+ = 2 t Var ( S ) ω S (33) 4
j t+ j t 2 Var ( S ) = ω S (34) The varance specfcatons (33)-(34) reflect a varance of S t+ /S t whch - condtonal on all nformaton n perod t - does not change over tme. Ths s approprate f one perod s nterpreted as one month or longer (for our purposes: three months): Balle & Bollerslev (989) found that ARCH effects n daly and weekly exchange rates seem to dsappear for monthly exchange rates. Defnton (Heterogeneous traders market) A heterogeneous traders market s a market consstng of m ( ) domestc traders descrbed by (2), (4), (5), (3), (6), (9), (20)-(23), (3), (33), n ( ) foregn traders descrbed by (3), (6), (7), (4), (5), (24)-(29), (32), (34), and whch has aggregate propertes (8)-(2), (7), (8), (30). 3.5 The equlbrum exchange rate If we substtute (3)-(34) nto aggregate demand functon (30), we obtan Φ( S ) = t α α m ( + It ) S t Z ( + It ) S m t 2 2 = ( + I t ) γ ω St = B + 5
α, n α, 2 ( + I S Z S + I S t) t j t ( t ) t j t j= 2 2 ( + It ) γ jω j S t n t j t j + A S j= (35) After rearrangng terms we have Proposton (Equlbrum exchange rate) In a heterogeneous traders market an equlbrum exchange rate s any real postve soluton S t to m = α t, j Z n α Z t, j S t ( + I t ) γ ω j= ( + I ) γ ω α t j j S α t t, j + n + A j j= + m I n t + + I t S 2 t = ( + I t ) γ ω j= ( + I ) γ, ω 2 t t j j m B = = 0 An equlbrum soluton s a root of a polynomal n m+n+ factors of S t wth a constant. The polynomal of proposton thanks ts convenent form to the varance specfcatons (33) and (34). Obvously, the polynomal may have several real-valued roots. In secton 3. we already mentoned that the nternatonal bankng system - knowng the foregn exchange supply and demand schedules of all traders - acts as a Walrasan auctoneer to clear the foregn exchange market. We can now be more specfc: n perod t the Walrasan auctoneer uses the Newton-Raphson algorthm wth ntal value S t- to compute the nearest real-valued root of the polynomal of proposton. Although more 6
than one equlbrum exchange rate may exs we choose the one that s selected by the Walrasan auctoneer. For each perod we can supply emprcally plausble numercal values for I t, I t, Z, Z j, α, α j, γ, γ j, ω, ω j, A j, B and compute the equlbrum exchange rate S t from proposton. Ths wll transform the theoretcal model nto an artfcal economy. 4. The artfcal economy 4. Number of traders and perods We wll assume that there are 200 traders: m = 00 domestc traders and n = 00 foregn traders. The length of the experment s related to the tme nterpretaton. Exchange rates have been floatng snce March 973, hence by the end of 997 there were 00 quarterly exchange rate observatons under free float. Wth the use of statstcal tables (Dckey-Fuller tests) n mnd, we wll set the length of the experment at 00 perods. 4.2 Intal wealth The ntal currency endowments are obtaned n two stages. Frst we draw domestc and foregn currency endowments from a unform dstrbuton. Then we multply wth scale parameters λ and λ. Ther relatve sze reflects the relatve szes of the domestc and foregn economes. These scale parameters are also drawn from a unform dstrbuton: 7
λ~u(0,) (36) λ~u(0,) (37) The ntal currency endowments of domestc traders are descrbed by: A, ~λ.0 5.U[0,] (38) B, ~λ.0 5.U[0,] (39) The ntal currency endowments of foregn traders are: A,j ~λ.0 5.U[0,] (40) B,j ~λ.0 5.U[0,] (4) 4.3 Interest rates We let the domestc nterest rate follow a random walk: I t = I t- + η t (42) 8
wth the (for quarterly nterest rates) emprcally plausble η t ~ N(0,0-5 ) (43) and an ntal value drawn from a unform dstrbuton, such that the annualzed quarterly nterest rate starts at a value between 2% and 0%: 00{(+I ) 4 -} ~ U[2,0] (44) Smlarly for foregn nterest rates: I t = I t- + η t (45) η t ~ N(0,0-5 ) (46) 00{(+I ) 4 -} ~ U[2,0] (47) 9
4.4 Rsk averson update The measure of rsk averson n our model s the absolute rsk averson parameter γ. However, Frend & Blume (975) and Landskroner (977) found emprcal evdence of constant relatve rsk averson (hence decreasng absolute rsk averson). In order to approxmate constant relatve rsk averson we wll use the update formula γ = κ W t, (48) wth W 0, = A, + S 0 B, (S 0 = ). If W t-, > 0, κ s the target value of the relatve rsk averson. Snce W t-, s only an approxmaton of E (W t+, ), the actual relatve rsk averson κ = E (W t+, ) γ (49) wll hover around κ. If W t-, < 0, update formula (48) ensures a postve absolute rsk averson γ, gven a postve κ. In secton 3.3 we mentoned that ths s requred for the second order condton for the optmal foregn currency poston of domestc trader. Notce that ths update mples that the more negatve a trader s wealth becomes, the more rsk he s gong to take: n ths case we are dealng wth rogue traders. In ths subsecton, we have dscussed only the domestc traders, but smlar statements hold for foregn traders. 20
4.5 Renterpretaton of rsk averson and predcted varance In each equaton the absolute rsk averson parameter and the predcted varance parameter appear as a product. For domestc traders ths rsk product s γ ω and for foregn traders t s γ j /ω j. Hence for the smulaton output t s rrelevant whether the rsk-adjustment of currency postons s caused by rsk averson or predcted exchange rate varance. We can explot ths property to neutralze the effects of second moment expectatons Var (S t+ ) and Var j (/S t+ ). After each experment we can renterpret the rsk product components such that all traders have ratonal second moment expectatons. Suppose that a specfc experment generates a sample varance Var ( ) = ~ ω (50) s t Then the predcted varance of domestc trader concdes wth the sample varance f ω = ω~ (5) However, n general ths wll not hold for the value of ω that we put nto the experment. We can remedy ths by redefnng ths parameter after the smulaton s completed. We have to keep the rsk product of domestc trader the same: γ ω = γ~ ω~ (52) Ths mples that we also have to redefne the absolute rsk averson parameter: 2
~ γ ω = ~, ω γ t (53) In turn ths mples a renterpretaton of the relatve rsk averson parameter. Snce κ = E (W t+, ) γ (54) and ~ κ = E ( W ) ~ + γ (55) t, t holds that ~ κ ~ γ = κ (56) γ and equvalently ~ κ ω = ~, ω κ t (57) The mportance of ths post-experment renterpretaton of rsk s that t makes t mpossble to attrbute output results to rratonalty n the predcted varance (assumng that the generated exchange 22
rate seres does not exhbt volatlty clusterng). Havng neutralzed the second moment expectatons Var (S t+ ) and Var j (/S t+ ), we can attrbute all smulaton results to the frst moment expectatons E (S t+ ) and E j (/S t+ ), whch are after all the focus of our experments. Landskroner (977) estmated κ between 2.4 and 8.2. Hansen & Sngleton (983) found values between 0 and 2. Dunn & Sngleton (986) found values between.2 and 3.5. To be safe, we assume that relatve rsk averson s between 0 and 5. We wll set the predcted varance at the emprcally observed value (see table 2) ω = 4.0-3 (58) and draw κ from a unform dstrbuton: κ ~ U[,5] (59) After a smulaton s completed, we renterpret the predcted varance and rsk averson parameters accordng to (5), (53) and (57). If the average value of ~ κ t, over all traders and over all perods exceeds 5, we dscard the smulaton on the bass of emprcally mplausble hgh relatve rsk averson. Although we set the lower boundary of κ at (for numercal reasons), the lower boundary for ~ κ t, s 0 (as suggested by the emprcal evdence). In ths subsecton, we concentrated on the domestc traders, but analogous statements hold for foregn traders. 23
4.6 Exchange rate expectatons, trader types and market types We dstngush between 4 types of ndvdual traders, as summarzed n table. For smplcty, the descrptons are stated for domestc traders. The correspondng descrptons for foregn traders are analogous. Table : Types of traders trader type type of nformaton use of nformaton α expectatons mechansm Z E ( s t+ ) dstrbuted lag expectatons S t- > 0 - α s t+ bandwagon expectatons S t- < 0 - α s t+ adaptve expectatons E t-, (S t ) > 0 α (E t-, (s t ) - s t ) regressve expectatons Q > 0 α (q - s t ) We defne S 0 = E -, [S 0 ] = and we use a random walk to represent the natural logarthm q t of the fundamental exchange rate Q t : q t = q t- + υ t (60) υ t ~ N(0,ω) (6) q = 0 (62) 24
Parameter ω s set at the value ndcated by equaton (58). We wll assume that each trader has an expectatons scheme that s fxed over tme: Z = Z and α = α. Based on the 4 types of ndvdual traders we wll consder 5 types of markets (n defntons 2-6 the symbol α refers to both domestc and foregn traders): Defnton 2 (Bandwagon expectatons market) A bandwagon expectatons market s a heterogeneous traders market whch conssts of bandwagon expectatons traders wth α ~ α.ln(-0.25,0.25) and α < 0. 2 2, Wth y ~ LN(µ,σ 2 ) we mean that y = exp(x) wth x ~ N(µ,σ 2 ). It holds that E[ y] = exp( µ + σ ) hence E[ y] exp( ) α = α µ + 2σ 2 = α f σ 2 = - 2µ. In other words: f we set σ 2 = - 2µ, the average alpha s gven by α. We have chosen µ = -0.25 and σ 2 = 0.25 after nspecton of the results wth a random number generator. Large spreads mply mplausble ndvdual values for alpha, whle small spreads exhbt mplausbly lttle heterogenety. The spread of ndvdual values of alpha wth µ = -0.25 and σ 2 = 0.25 seemed approprate. Defnton 3 (Dstrbuted Lag expectatons market) A dstrbuted lag expectatons market s a heterogeneous traders market whch conssts of dstrbuted lag expectatons traders wth α ~ α.ln(-0.25,0.25) and α > 0. 25
Defnton 4 (Extrapolatve expectatons market) An extrapolatve expectatons market s a heterogeneous traders market whch conssts of bandwagon and dstrbuted lag expectatons traders wth α ~ N(α,0.25). The varance parameter 0.25 was agan selected on the bass of nspecton of random number generator results. Defnton 5 (Adaptve expectatons market) An adaptve expectatons market s a heterogeneous traders market whch conssts of adaptve expectatons traders wth α ~ α.ln(-0.25,0.25) and α > 0. Defnton 6 (Regressve expectatons market) A regressve expectatons market s a heterogeneous traders market whch conssts of regressve expectatons traders wth α ~ α.ln(-0.25,0.25) and α > 0. In each of the markets descrbed by defntons 2-6, the traders use the same type of nformaton Z t to form exchange rate expectatons E (S t+ ). However, they are heterogeneous n the way that they use the nformaton. The ndvdual expectatons parameter α, whch ndcates the weght that s beng gven to the nformaton Z t, s drawn from a dstrbuton around α, whch n turn ndcates how the average trader uses the nformaton Z t. 26
5. The experments 5. Expermental desgn In secton 4 we transformed the theoretcal model of secton 3 nto an artfcal economy wth a flexble specfcaton of the exchange rate expectatons mechansms of ndvdual traders. We defned 5 market types, correspondng wth dfferent exchange rate expectatons mechansms. Wth each market type, we wll perform experments that can be depcted n fgure 2: Fgure 2: The experment pre-exchange currency postons: A, B, A j, B j nterest rate tme seres: I t, I t rsk averson parameters: γ,γ j predcted varance parameters: ω, ω j expectatons formaton: α, z, α j, z j nput model of foregn exchange between heterogeneous traders output exchange rate tme seres: s t exchange rate expectatons: E (s t+ ), E j (s t+ ) 27
We descrbed the nput of the experment n secton 4 and the model of foregn exchange between heterogeneous traders n secton 3. In ths secton we wll dscuss the output and nvestgate f there s a relatonshp wth specfc nput. 5.2 Emprcally plausble output We want to evaluate the output of the experments, n partcular the artfcal exchange rate seres, on ther emprcal plausblty. Ths requres nvestgatng the stylzed facts of emprcally observed quarterly exchange rates. Apart from the frst 4 moments of s t, we look at 4 test statstcs that are nspred by the (weak-form) effcent market hypothess and the statstcal phenomena of volatlty clusters and fat tals. Under weak-form market effcency, the (log) exchange rate s t reflects all nformaton contaned n the lagged exchange rates s t-, s t-2,... (see for example Balle & McMahon (989)). Consder the unvarate model s t = λ s t- + ε t (63) where ε t s an error term. If we take expectatons condtonal on the lagged exchange rates, we obtan E t- [s t ] = λ s t- + E t- [ε t ] (64) 28
or equvalently E t- [ s t ] = (λ-) s t- + E t- [ε t ] (65) Under weak-form market effcency, t should hold that E t- [ s t ] = 0. Ths mples that λ = and that ε t should not contan seral correlaton. In other words, s t should follow a martngale process. These propertes can be tested wth a Dckey-Fuller unt root test on s t and a Box-Perce Q-test for seral correlaton n s t. Among the stylzed facts of hgh frequency exchange rates are volatlty clusters and fat tals (de Vres (995)). We verfy whether these propertes are retaned for low frequency data such as our quarterly exchange rates. We wll test for volatlty clusters by performng a Box-Perce Q-test for seral correlaton n ( s t ) 2. We test for fat tals wth a Jarque-Bera test for normalty of s t. The results are descrbed n table 2: 29
Table 2: Summary statstcs of emprcal quarterly exchange rates 973-995 mean( s) x0-3 $/AS 7.878 4.083-0.68 2.47 -.29 3.3 (0.05) var( s) skew( s) kurt( s) DF(s) BP 8 ( s) BP 8 ( s 2 ) JB( s) x0-3 $/BFr 3.400 4.240-0.324 2.647-0.57 3.64 (0.096) $/Can$ - 3.432 0.476 0.06 3.49 0.6 9.7 (0.286) $/DKr.237 3.779-0.53 2.503-0.33.88 (0.568) $/FKr -.307 2.666-0.306 2.863 0.07 20.66 (0.008) $/FFr - 0.836 3.779-0.69 2.565-0.05 0.55 (0.2287) $/DM 7.507 4.258-0.47 2.629 -.45 3.6 (0.066) $/IP - 4.768 3.408-0.063 2.50 -.54 22.4 (0.0042) $/Lra -.000 3.576-0.556 3.474.67 8.2 (0.428) $/- 0.437 3.74 0.23 2.79 -.65 2.74 (0.22) $// 6.673 4.084-0.240 2.504 -.29 4.9 (0.077) $/NKr - 0.745 2.843-0.68 3.907-0.0 6.65 (0.0340) $/esc -9.50 3.848-0.45 3.309 2.66 4.63 (0.0667) $/pta - 8.042 3.323-0.509 3.69.22 9.95 (0.2685) $/SFr.369 5.239 0.23 2.83-2.02 8.80 (0.3596) $/, - 5.55 3.65-0.228 2.884 -.45 8.5 (0.020) 4.67 (0.799) 5.39 (0.749) 2.68 (0.9526) 3.38 (0.9080) 7.7 (0.585) 4.85 (0.7738) 3.74 (0.880) 3.47 (0.904) 2.84 (0.9440) 4.9 (0.8393) 3.56 (0.8944) 3.86 (0.8693) 7.30 (0.505) 6.30 (0.636) 7.55 (0.478) 4.42 (0.870).7 (0.4243) 0.94 (0.6259) 3.5 (0.2065) 0.92 (0.6327) 2.28 (0.3206).20 (0.5478).0 (0.577) 2.0 (0.3507) 20.5 (0.0000) 7.57 (0.0227).5 (0.5637) 2.2 (0.0022) 24.48 (0.0000) 3.67 (0.00) 5.2 (0.0772) 3.84 (0.465) var(x) s varance of x, skew(x) s skewness of x, kurt(x) s kurtoss of x DF(x) s Dckey-Fuller test statstc (H 0 : x contans unt root) whch has tabulated crtcal values BP 8 (x) s Box-Perce test statstc (H 0 : frst 8 autocorrelatons of x are zero) whch s χ 2 (8) under H 0 JB(x) s Jarque-Bera test statstc (H 0 : x s normally dstrbuted) whch s χ 2 (2) under H 0 p-values of BP- and JB-test statstcs between parentheses Data source: Man Economc Indcators, varous ssues Frst consder the results on weak-form market effcency. The Dckey-Fuller test ndcates a unt root for 5 out of the 6 exchange rates at the 5% sgnfcance level and for all exchange rates at the 2.5% sgnfcance level (see Fuller (976), table 8.5.2). The Box-Perce Q-tests ndcate the absence 30
of seral correlaton n s t for only 9 out of the 6 exchange rates at the usual 0% sgnfcance level, 2 out of 6 at the 5% level and 4 out of 6 at the % level. Snce the sgnfcance level ndcates the probablty of mstakenly acceptng the presence of seral correlaton, strong belevers n weakform market effcency may stll nterpret the DF(s) and BP 8 ( s) results as evdence of weak-form market effcency. However, t seems more sensble to nterpret the results as an ndcaton that quarterly exchange rates are close to a martngale process, whle exhbtng some seral correlaton. For our purposes,.e. the comparson of statstcal propertes of artfcal data wth those of emprcal data, ths dscusson s not very mportant: we are more nterested n the range of values that the test statstcs assume. Therefore we wll call DF(s) > -.95 (the usual 5% sgnfcance level) and BP 8 ( s) < 22.55 (correspondng wth a nce round p-value of 4.0-3 ) emprcally plausble output. Now we look at the results on volatlty clusterng. The BP 8 ( s 2 ) tests ndcate the absence of volatlty clusterng for all exchange rates. Ths result confrms the results of Balle & Bollerslev (989), who found that ARCH effects n daly and weekly exchange rates seem to dsappear for monthly exchange rates. Hence for emprcally plausble output we requre BP 8 ( s 2 ) < 3.36, the usual 0% sgnfcance level. Fnally we consder the results on normalty: the JB( s) test statstcs ndcate normalty for only out of 6 exchange rates at the 5% sgnfcance level and 3 out of 6 at the % sgnfcance level. Hence normally dstrbuted exchange rate returns do not seem to be a stylzed fact even for quarterly data. The hghest JB( s) value s 24.48. Therefore for emprcally plausble output we requre JB( s) < 24.86, whch corresponds wth a p-value of 4.0-6. We summarze the results n defnton 7: 3
Defnton 7 (Emprcally plausble output) An artfcal economy generates emprcally plausble outpu f the smulated (log) exchange rate seres s t has the followng statstcal propertes: DF(s) > -.95 BP 8 ( s) < 22.55 BP 8 ( s 2 ) < 3.36 JB( s) < 24.86 The frst two propertes of defnton 4 correspond wth a process that approxmates a martngale: an exact martngale s characterzed by BP 8 ( s) < 3.36. The economc nterpretaton of such a process s approxmate weak-form effcency. The thrd property adds the restrcton that volatlty clusterng s absent. The fourth property reflects an upper lmt for the fatness of the tals that we have found for emprcally observed quarterly exchange rates. The tal fatness s postvely related to the occurrence of extreme exchange rate returns n the sample. In the next subsecton we wll use ths crteron to evaluate the expermental output. 32
5.3 Smulaton results We smulate all market types wth m = 00 domestc traders and n = 00 foregn traders. Each market type s characterzed by the type of nformaton Z and the use of nformaton α. Wth respect to the type of nformaton, we dstngush between adaptve expectatons markets, bandwagon expectatons markets, dstrbuted lag expectatons markets, extrapolatve expectatons markets and regressve expectatons markets. At the same tme, we consder dfferent values for the average use of nformaton α. For each market type we perform 00 smulatons of each 00 perods and we count the number of tme seres whch can be dentfed as martngales, martngales wthout volatlty clusterng, random walks and emprcally plausble tme seres. 33
For the adaptve expectatons marke we consder the followng values of the average α: α = 0.05, 0.0, 0.5, 0.20, 0.25, 0.30. Emprcal estmates of α range from 0.07 to 0.9 (Frankel & Froot (987a) and (987b), Cavagla, Verschoor & Wolff (993)). The smulaton results are reported n table 3: Table 3: Adaptve expectatons markets (00 smulatons) average α martngales martngales wthout random walks emprcally plausble volatlty clusterng DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 22.55 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 JB( s t ) < 5.99 JB( s t ) < 24.86 0.05 0 0 0 0 0.0 0 0 0 0 0.5 0 0 0 0 0.20 5 4 0 0 0.25 8 8 0 0 0.30 6 6 0 0 The results ndcate that adaptve expectatons markets wth emprcally plausble values of α are ncapable of generatng emprcally plausble exchange rates. In a small number of cases the adaptve expectatons market s able to produce a martngale process. However, n these cases the Jarque- Bera test statstcs are far too hgh (not lower than 2026.36) n comparson wth emprcal results. Although adaptve expectatons markets are sometmes weak-form effcent and exhbt no volatltyclusterng, they generate too many extreme returns to be emprcally plausble. 34
For the bandwagon expectatons marke we consder the followng values of the average α: α = -0.25, -0.20, -0.5, -0.0, -0.05. Emprcal estmates of α for extrapolatve expectatons range from -0.07 to 0.58 (Frankel & Froot (987a) and (987b), Bank of Japan (989), Froot & Frankel (990), Cavagla, Verschoor & Wolff (993)). The results are reported n table 4: Table 4: Bandwagon expectatons markets (00 smulatons) average α martngales martngales wthout random walks emprcally plausble volatlty clusterng DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 22.55 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 JB( s t ) < 5.99 JB( s t ) < 24.86-0.25 0 0 0 0-0.20 0 0 0 0-0.5 0 0 0 0-0.0 0 0 0 0-0.05 0 0 0 0 The bandwagon expectatons market performs even worse than the adaptve expectatons market. For each of the 5 parameter values, there s not a sngle smulaton that generates an exchange rate seres that passes the four tests of emprcal plausblty. The man defcency of bandwagon expectatons markets s that they are not weak-form effcen whch means that the most recent exchange rate return contans nformaton on the drecton and sze of the next exchange rate change. 35
For the dstrbuted lag expectatons marke we consder 6 values of the average α: α = 0.0, 0.20, 0.30, 0.40, 0.50, 0.60. We just mentoned that the emprcal estmates of α for extrapolatve expectatons range from -0.07 to 0.58. The results are reported n table 5: Table 5: Dstrbuted Lag expectatons markets (00 smulatons) average α martngales martngales wthout random walks emprcally plausble volatlty clusterng DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t )] < 22.55 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 JB( s t ) < 5.99 JB( s t ) < 24.86 0.0 0 0 0 0 0.20 0 0 0 0 0.30 0 0 0 0 0.40 0 0 0 0 0.50 0 0 0 0 0.60 0 0 0 0 The dstrbuted lag expectatons market suffers from the same problem as the bandwagon expectatons market: the exchange rate process s predctable (n a lnear sense). Notce that these two expectatons mechansms are mrror mages of each other. 36
The extrapolatve expectatons market contans both bandwagon and dstrbuted lag expectatons traders. For ths marke we consder 0 values of the average α: α = -0.20, -0.0, 0.00, 0.0, 0.20, 0.30, 0.40 0.50, 0.60, 0.70. The emprcally observed range [- 0.07,0.58] s contaned n ths spectrum. The results are reported n table 6: Table 6: Extrapolatve expectatons markets (00 smulatons) average α martngales martngales wthout random walks emprcally plausble volatlty clusterng DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 22.55 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 JB( s t ) < 5.99 JB( s t ) < 24.86-0.20 40 38 0 0-0.0 56 53 0 0 0.00 66 6 0 0 0.0 66 62 0 0 0.20 62 56 0 0 0.30 54 54 0 0 0.40 47 4 0 0 0.50 43 40 0 0 0.60 43 4 0 0 0.70 27 25 0 0 The presence of both types of traders consderably mproves weak-form market effcency, especally n the [-0.,0.3] regon, where more than half of the smulatons follow a martngale process wthout volatlty clusters. Ths performance s also much better than for adaptve expectatons markets. However, the smulated extrapolatve expectatons markets whch exhbt weak-form market effcency generate too many extreme exchange rate returns to be emprcally 37
plausble. As a matter of fac only % of all smulatons have a Jarque-Bera test statstc below 24.86. For the regressve expectatons marke we consder 0 values of the average α: α = 0.0, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.0. The emprcally observed nonnegatve parameters α for regressve expectatons are 0.02 and 0.09 (Frankel & Froot (987a) and (987b), Bank of Japan (989), Froot & Frankel (990)). The results are reported n table 7: Table 7: Regressve expectatons markets (00 smulatons) average α martngales martngales wthout random walks emprcally plausble volatlty clusterng DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 DF(s t ) > -.95 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 3.36 BP 8 ( s t ) < 22.55 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 BP 8 ( s 2 t ) < 3.36 JB( s t ) < 5.99 JB( s t ) < 24.86 0.0 52 36 27 43 0.02 59 47 38 54 0.03 67 50 36 54 0.04 59 5 44 58 0.05 67 56 44 63 0.06 76 64 50 64 0.07 59 45 33 49 0.08 62 48 39 57 0.09 6 47 38 54 0.0 55 39 29 42 The regressve expectatons market s the only market that s able to generate emprcally plausble exchange rate seres. The performance s best for α = 0.05-0.06, whch s n the mddle area of the 38
range of emprcal estmates. For these average expectatons parameter values, regressve expectatons markets generate emprcally plausble exchange rate seres n more than 60 of the 00 smulatons. In other words, for about 60% of the states of the world (as represented by the artfcal dstrbutons of ntal currency endowments, nterest rates, rsk averson and expectatons parameters) the regressve expectatons market seems to be a plausble representaton of the foregn exchange market. Notce that the regressve expectatons market s only margnally better than the extrapolatve expectatons market n generatng exchange rate processes whch are weak-form effcent and free of volatlty clusters: the averages are 48.3 and 47. respectvely. The crucal dfference s that the tals of the dstrbuton of exchange rate returns are consderably less fat n case of the regressve expectatons market than n case of the extrapolatve expectatons market. In other words, the occurrence of extreme exchange rate returns n the regressve expectatons market s comparable wth emprcal markets. It should be kept n mnd that regressve expectatons may take several forms. The survey data studes consdered purchasng power party, some constant long run exchange rate level and a movng average of the exchange rate. For the artfcal economes we smply generated a random walk and presented t as the fundamental exchange rate. Of course, ths s nterestng n tself. Even f ths random walk has no relatonshp at all wth the fundamentals n the economy, the artfcal foregn exchange market exhbts the same statstcal behavor as emprcal currency markets. In other words: the relevant ssue s that traders beleve n a fundamental exchange rate, t does not seem to 39
matter what the fundamental exchange rate s based on. The fundamental exchange rate may be as fcttous as the perceved patterns n the charts. 6. Concluson A clear pcture emerges from the smulatons: artfcal economes wth techncal expectatons schemes (adaptve, bandwagon, dstrbuted lag, extrapolatve) do not generate exchange rate seres wth statstcal propertes that match those of emprcally observed exchange rates. Bandwagon, dstrbuted lag and adaptve expectatons markets are rarely weak-form effcent. Extrapolatve expectatons markets are frequently weak-form effcent and free of volatlty clusters, but generate too many extreme returns compared wth emprcal foregn exchange markets. The artfcal economes wth fundamentalst expectatons schemes perform consderably better. Regressve expectatons markets generate emprcally plausble exchange rate seres n about 40 to 60 of the 00 smulatons, dependng on the average α. The smulatons wth the artfcal foregn exchange market therefore suggest that regressve expectatons are the most plausble representaton of quarterly exchange rate expectatons, when compared wth extrapolatve, bandwagon, dstrbuted lag and adaptve expectatons. The (perceved) presence of an anchor seems to play an mportant role n the functonng of foregn exchange markets, determnng the frequency of extreme exchange rate returns. In ths paper we performed experments wth artfcal markets consstng of traders whch are heterogeneous wth respect to ntal endowments and rsk averson. However, n each smulated 40
market all traders base ther exchange rate expectatons on the same type of nformaton z. The expectatons are heterogeneous because of ndvdual dfferences n the use of nformaton, as expressed by the weght α. In other words, the artfcal markets exhbt weak-form heterogenety wth respect to expectatons. Ths form of heterogenety s approprate for a comparson wth survey data studes on exchange rate expectatons, such as Frankel & Froot (987a). It allowed us to nvestgate the plausblty of adaptve, bandwagon, dstrbuted lag, extrapolatve and regressve expectatons markets. The next step s to perform experments wth markets consstng of, for example, both chartsts (extrapolatve traders) and fundamentalsts (regressve expectatons traders). References Backus, D.K., A.W. Gregory & C.I. Telmer, Accountng for forward rates n markets for foregn currency, Journal of Fnance 48 (993): 887-908 Backus, D.K., A.W. Gregory & S.E. Zn, Rsk premums n the term structure: evdence from artfcal economes, Journal of Monetary Economcs 24 (989): 37-399 Balle, R.T. & T. Bollerslev, The message n daly exchange rates: a condtonal-varance tale, Journal of Busness and Economc Statstcs 7 (989): 297-305 4
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