PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES

Transcription

1 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN Preliinary and incoplete Abtract. Taking a Bayeian approach, we focu on the inforation content in two data et: (1) the o-called Lothian Taylor data and (2) pot-war data for countrie that adopted the Euro. We find the two data et have iilar iplication for the exitence of unit root: Both data et produce evidence goe againt a unit root. 1. Introduction We exaine two data et. We refer to the firt data et a the Lothian Taylor data. The bae currency-country for the Lothian Taylor data i the U.K. (pound terling). The data are annual. For the dollar/terling exchange rate there 11 obervation ( ) and for the franc/terling exchange rate there are 169 obervation (1 1973). We refer to the econd data et a the Euro-related data. The are ten exchange rate; the baecurrency country for the Euro-related data i Gerany; thee data are onthly. There are 5 obervation for all ten exchange rate (January 1957 through Deceber 25). Fro a frequentit/claical perpective that focue on tet tatitic, it appear that the Lothian Taylor data tell a different tory fro the Euro-related data. By contrat, fro a Bayeian perpective the two data et appear to tell eentially the ae tory. Taking the Bayeian approach, it i natural to focu on the the inforation content a ebodied in the likelihood (rather than tet tatitic per e). We preent a odel that allow learning acro exchange-rate regie. The odel involve a hierarchical prior with hyperparaeter. The inforation flow fro one regie to another via the hyperparaeter. We preent the poterior ditribution of the paraeter of interet baed on both dataet cobined. Thee poterior ditribution how that it i unlikely that the root i near one. But we alo think it i intereting to addre the following quetion: Given a prior baed on the Lothian Taylor data, doe the Euro-related data increae or decreae the odd in favor of a unit root? Perhap urpriingly, the anwer i that the Euro-related data decreae the odd in favor of a unit root. 2. The data and the likelihood We adopt a iple odel for the dynaic of a real exchange rate regie. Let y i = (y i,1,..., y i,ti ) denote the real exchange rate between two countrie and let y = (y 1,..., y n ) Date: Noveber 17, 2. The view expreed herein are the author and do not necearily reflect thoe of the Federal Reerve Bank of Atlanta or the Federal Reerve Syte. 1

2 2 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN denote a collection of n exchange rate regie. Let the data generating proce for y i be an AR(p) proce: p y it = α i + γ ij y i,t j + ε it, (2.1) j=1 where ε it iid N(, σ 2 i ). Let τ i denote the frequency of the data (in year) for regie i, o that τ i = 1 for annual data and τ i = 1/12 for onthly data. It i convenient to reparaeterize (2.1) a follow: 1 p 1 y it = α i + (β i ) τ i y i,t 1 + ψ ij y i,t j + ε it, (2.2) ( p ) where β i := j=1 γ 1/τi ij and ψij := p k=j+1 γ ik. The paraeter of interet i β i. 2 The proce i aid to have a unit root if β i = 1. Let θ i = (α i, β i, σ i, {ψ ij }). Then where p({y it } T i t=p+1 {y it} p t=1, θ i) = j=1 T i i=p+1 ( p(y it {y i } t 1 =t p, θ i) = N y it α i + (β i ) τ i y i,t 1 + The log-likelihood function for θ i i given by Define p(y it {y i } t 1 =t p, θ i), (2.3) p 1 ψ ij y i,t j, σi 2 j=1 ). (2.) l(θ i ) := log ( p({y it } T i t=p+1 {y it} p t=1, θ i) ). (2.5) ( ) 2 1 θ i := arg ax l(θ i ) and Σθi := θ i θ θ l(θ i), (2.6) θi = θ i and let ŷ i := ( β i, σ βi ) where β i := ( θ i ) 2 and σ βi := ( Σ θi ) 22. Define L(β i ŷ i ) := N(β i β i, σ 2 β i ). (2.7) In effect, we treat ( β i, σ βi ) a ufficient tatitic for y i a far a β i i concerned. Note that N(β i β i, σ βi ) i iply the functional for for the likelihood and not a apling ditribution. 3 See Table 1 for the ufficient tatitic for both data et. For a plot of the likelihood, ee Figure 1. See alo Figure 2, which how that each of the β i are within 2 σ βi of the weighted average. Table 2 how that the lag-length p ha little effect on the average value of β i and σ βi for the Euro-related data. Therefore, we conduct the analyi with p = 1. 1 yi,t j := y i,t j y i,t j 1. 2 The paraetrization (βi) τ i calibrate β i to an annual rate. 3 In the appendix we derive (2.7) by integrating out the nuiance paraeter.

3 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES 3 3. Bayeian inference and neted hypothei teting We provide a brief review of Bayeian inference and neted hypothei teting. poterior ditribution for β i i given by Baye rule: p(β i ŷ i ) = L(β i ŷ i ) p(β i ) p(ŷ i ) The L(β i ŷ i ) p(β i ), (3.1) where p(ŷ i ) = L(β i ŷ i ) p(β i ) dβ i i the likelihood of the data according to the odel. For future reference, a rearrangeent of Baye rule (3.1) produce p(β i ŷ i ) p(β i ) = L(β i ŷ i ) L(βi ŷ i ) p(β i ) dβ i. (3.2) Two odel of β i. Conider two odel, one odel where the probability that β i < 1 i one and another odel where the probability β i = 1 i one. The firt odel i characterized by β i [, 1]. Although thi odel include 1 in the pace for β i, the probability that β i (, 1) i one. The econd odel i characterized by β i = 1. The econd odel can be obtained fro the firt odel by iply conditioning on β i = 1; thu we refer to the firt odel a the bae odel and the econd odel a the neted odel. Let A denote the bae odel and let A 1 denote the neted odel. The poterior probability of odel A ( {, 1}) i given by an application of Baye rule: p(a ŷ i ) L(A ŷ i ) p(a ), (3.3) where L(A ŷ i ) i the likelihood of odel A and P (A ) i the prior probability of odel A. Uing (3.3), the poterior odd ratio can be expreed the product of the prior odd ratio and the Baye factor: p(a 1 ŷ i ) p(a ŷ i ) = L(A 1 ŷ i ) L(A ŷ i ) p(a 1) p(a ), (3.) where L(A 1 ŷ i )/L(A ŷ i ) i the Baye factor. The Baye factor expree how the evidence change the prior odd ratio into the poterior ratio. Let p(β i ) denote the prior for β i for the bae odel. The likelihood of the oberved data given the bae odel i L(A ŷ i ) = 1 L(β i ŷ i ) p(β i ) dβ i. The likelihood of the oberved data given the neted odel i L(A 1 ŷ i ) = L(β i = 1 ŷ i ) = N(1 β i, σ 2 β i ). The Baye factor in favor of the neted odel relative to the bae odel i 5 Given (3.2), we can expre the Baye factor a 6 L(β i = 1 ŷ i ) B(β i = 1) = 1 L(β. (3.5) i ŷ i ) p(β i ) dβ i B(β i = 1) = p(β i = 1 ŷ i ) p(β i = 1). (3.6) We aue the prior for βi in thi odel ha a denity with repect to Lebegue eaure. 5 In due coure, we will be intereted in Baye factor for different odel coparion, and therefore we will ue notation that ake explicit the coparion being ade. 6 The right-hand ide of (3.6) i known a the Savage Dickey denity ratio.

4 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN A ore data are acquired, the poterior ditribution for β i in the bae odel will becoe ore concentrated on the true value of β i (auing the true value i in the unit interval). Conequently, if β i = 1 i true, the bae odel will becoe ore like the neted odel, and in the liit the two odel will be inditinguihable. On the other hand, if β i < 1 i true, the bae odel will eventually becoe quite ditinct fro the neted odel. Referring to (3.6), one ay ee li B(β i = 1) = li T i T i p(β i = 1 ŷ i ) p(β i = 1) = { β i = 1 β i < 1. Thi involve a equence of Baye factor, each of which ue the ae prior p(β i ). 7. Senitivity to prior (3.7) The poterior ditribution p(β i ŷ i ) and the Baye factor B(β i = 1) both depend on the prior p(β i ). Therefore, an invetigation of the enitivity of the poterior and the Baye factor to different prior i appropriate. Truncated unifor prior. To begin, conider a unifor ditribution truncated to the interval [a, 1] where a [, 1): U(x a) = 1 [a,1](x) 1 a. (.1) Note the ean and tandard deviation of x U(x a) are (1 + a)/2 and (1 a)/ 12, repectively. Let p(β i a) = U(β i a), where a indexe the bae odel. 9 The poterior for β i i given by p(β i ŷ i, a) 1 [a,1] (β i ) L(β i ŷ i ). The arginal likelihood of the bae odel (indexed by a) i 1 ( ) ( ) 1 1 a L(a ŷ i ) := L(β i ŷ i ) p(β i a) dβ i = N(β i β i, σ β 2 i ) dβ i Φ 1 βi σ βi Φ a βi σ βi =. (.2) 1 a 1 a Define L(a = 1 ŷ i ) := li a 1 L(a ŷ i ) = N(1 β i, σ 2 β i ) and note L(a = 1 ŷ i ) = L(β i = 1 ŷ i ). The Baye factor in favor of β i = 1 (conditioning on the bae odel indexed by a) i given by B(β i = 1 a) := L(β i = 1 ŷ i ) L(a ŷ i ) = L(a = 1 ŷ i). (.3) L(a ŷ i ) Conequently, we ee B(β i = 1 a = 1) = 1. Figure 3 how the Baye factor B(β i = 1 a) for a [, 1] for each of the twelve exchange-rate regie. 7 By contrat, if βi = 1 and we update our prior a each additional obervation i obtained, then li Ti B(β i = 1) = 1. { 1 x A The indicator function i defined a 1A(x) = x A. 9 p(βi a) i horthand notation for p(β i A (a)) where the latter notation ake explicit the fact that thi i the prior for the bae odel (which itelf depend on the paraeter a). 1 L(a ŷi ) i horthand notation for L(A (a) ŷ i ).

5 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES 5 Truncated Gauian prior. Next, conider a truncated Gauian ditribution: Ñ(x µ, σ 2 ) := 1 [,1](x) N(x µ, σ 2 ) 1 N(z µ, σ2 ) dz. (.) Let p(β i µ, σ) = Ñ(β i µ, σ 2 ). 11 Then the poterior i alo a truncated Gauian: where and p(β i ŷ i, µ, σ) = Ñ(β i µ i, σ 2 i ), (.5) µ i = α i βi + (1 α i ) µ and σ 2 i = α i σ 2 β i (.6) α i = σ 2 σ 2 + σ 2 β i. (.7) Note (1) li σ α i = 1 and (2) li σ α i =. In the firt cae the liiting prior i flat and the poterior i proportional to the likelihood (over the unit interval); in the econd cae the liiting prior put all it weight on µ and conequently o doe the poterior. Before we exaine the Baye factor, it i convenient to copute the ean and tandard deviation of the the truncated Gauian ditribution. Define the tranforation F : (µ, σ) (, ) a follow: 1 ( 1 1/2 := Ñ(x µ, σ 2 ) x dx and := Ñ(x µ, σ 2 ) (x ) dx) 2. (.) The range of F i the haded region R hown in Figure, which can be characterized by 1 and g(), where g() equal the tandard deviation of a truncated exponential ditribution with the ean. 12 Now let 13 p(β i, ) = Ñ(β i µ, σ) (µ,σ)=f 1 (,). (.9) The poterior ean and tandard deviation are given by ( i, i ) = F (µ i, σ i ). The arginal likelihood of the bae odel (indexed by and ) i given by 1 1 L(, ŷ i ) := L(β i ŷ i ) p(β i, ) dβ i ( 1 ) = N(β i β i, σ β 2 i ) Ñ(β i µ, σ 2 ) dβ i. (µ,σ)=f 1 (,) (.1) Note L(, = ŷ i ) := li L(, ŷ i ) = L(β i = ŷ i ) for (, 1) and L( = 1, = ŷ i ) := li 1 L(, = ŷ i ) = L(β i = 1 ŷ i ). The Baye factor in favor of β i = 1 relative to the bae odel (indexed by and ) i B(β i = 1, ) := L(β i = 1 ŷ i ) L(β = L( = 1, = ŷ i). (.11) i ŷ i ) p(β i, ) dβ i L(, ŷ i ) 1 11 Again, p(βi µ, σ) i horthand notation for p(β i A (µ, σ)). 12 A truncated exponential ditribution i proportional to h(x) = 1[,1] (x) e λ x for oe λ (, ). 13 Although (, ) = F (µ, σ) ha a cloed-for olution, (µ, σ) = F 1 (, ) ut be olved nuerically. 1 There i a cloed-for expreion for the integral in the econd line of (.1).

6 6 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN Therefore, B(β i = 1 = 1, = ) = 1. In addition, it i intereting to note that the greater the likelihood of [the odel indexed by] (, ), the aller the evidence in favor of β i = 1. We can obtain the axiu evidence againt β i = 1 with a prior that i a pecial cae of the truncated Gauian prior. Let p(β i, = ) := li p(β i, ) = δ(β i ), where δ( ) i the Dirac delta function. Thi prior put all it weight on and conequently ( i, i ) = (, ). A we have noted above, L(, = ŷ i ) = L(β i = ŷ i ) and thu B(β i = 1, = ) = L(β i = 1 ŷ i )/L(β i = ŷ i ). Therefore (auing β i [, 1]), the axiu evidence againt β i = 1 i obtained with the prior p(β i = β i, = ), which iniize the Baye factor in favor of β i = 1: Bi := B(β i = 1 = β i, = ) = L(β i = 1 ŷ i ) L(β i = β 1. (.12) i ŷ i ) All other prior of any for (a long a the upport i liited to β i [, 1]) produce Baye factor ore favorable to β i = 1. Therefore we refer to 1/Bi a the axiu evidence againt β i = If 1/Bi i all, then we can be confident that the data contain little evidence againt β i = 1 regardle of one prior view. By contrat, if 1/Bi i large, then the evidence againt β i = 1 ay well depend on one prior view. The axiu evidence againt β i = 1 i hown in Table 1. Graphically, one can ee the axiu evidence againt in Figure 1 a the ratio of the axiu likelihood to the value of the likelihood at β i = 1. See Figure 5 for B(β i = 1, ) for (, ) R for each of the twelve exchange-rate regie. We ee that Baye factor that favor β i = 1 can be produced by chooing and ufficiently all (which ha the effect of producing a poterior that ha a low ean). 5. A odel for learning acro regie The preceding etup doe not provide a foral way to draw inference fro one exchangerate regie to another. In thi ection, we adopt a hierarchical prior that allow u to do jut that. In a nuthell, we for a joint prior over all the coefficient {β i } and treat (, ) a unknown hyperparaeter. Thi induce correlation acro the β i coefficient in the prior, thereby allowing inforation to flow fro β i to β j via the hyperparaeter. A hierarchical prior. We adopt the following hierarchical prior: n p(β,, z) = p(, z) p(β, ) = p(, z) p(β i, ), (5.1) where β := (β 1,..., β n ) and where p(β i, ) i given in (.9). The prior for (, ) R depend on a ingle paraeter, z. Conider the following prior for (, ): p(, λ) 1 R (, ) h( λ), (5.2) where h( λ) 1 [,1] () e λ i the truncated exponential ditribution. By contruction, p( =, λ) = h( λ) and p(, λ)/ =. We chooe value of λ a follow. Define 15 Note log(1/b i ) = 1 2 ( 1 βi σ βi ) 2. i=1

7 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES 7 H(λ) := E[ =, λ] = 1 h( λ) d. Given a value z (, 1), let λ = H 1 (z). Then define p(, z) := p(, λ) λ=h 1 (z). (5.3) Thu we index the bae odel by the paraeter z. 16 To illutrate the effect of varying the prior for (, ), we let z take on the value.5,., and.95. A plot of p(, z =.) i hown in Figure 6. The arginal prior for β i i a ixture, given by 17 p(β i z) = p(β i, ) p(, z) d d. (5.) Plot of p(β i z) for three value of z are hown in Figure 7. Likelihood and poterior. Let the joint likelihood for β be given by n L(β ŷ) = L(β i ŷ i ), (5.5) i=1 where ŷ := (ŷ 1,..., ŷ n ). Then the joint poterior i p(β,, ŷ, z) L(β ŷ) p(β,, ) p(, z). (5.6) A we will ee hortly, the arginal poterior ditribution for (, ) play a central role in learning acro regie. The (arginal) poterior ditribution for (, ) i where L(, ŷ) = p(, ŷ, z) L(, ŷ) p(, z), (5.7) L(β ŷ) p(β, ) dβ = n L(, ŷ i ), (5.) and where L(, ŷ i ) i given in (.1). The arginal poterior for β i i a ixture of the conditional ditribution: 1 p(β i ŷ, z) = p(β i ŷ i,, ) p(, ŷ, z) d d. (5.9) We can ue (5.) and (5.9) to expre the Baye factor in favor of β i = 1 relative the bae odel indexed by z: 19 B(β i = 1 z) = p(β i = 1 ŷ, z) p(β i = 1 z). (5.1) The arginal poterior for β n+1 (for a regie for which we have no data but would have included if we had) i 2 p(β n+1 ŷ, z) = p(β n+1, ) p(, ŷ, z) d d. (5.11) i=1 16 In the liit a z 1, p(, z) put all it weight on (, ) = (1, ). 17 See the appendix for a derivation of (5.). 1 See the Appendix for a derivation of (5.9). 19 It can be hown that liz 1 B(β i = 1 z) = 1. 2 See the Appendix for a derivation of (5.11).

8 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN In (5.11) we ee that it i the hyperparaeter that carry the inforation acro regie. Finally, we can ue (5.) and (5.11) to expre the Baye factor in favor of β n+1 = 1: B(β n+1 = 1 z) = p(β n+1 = 1 ŷ, z) p(β n+1 = 1 z). (5.12) Two pecial cae of thi hierarchical prior erit ention. Firt, if p(, ) place all it weight on a ingle point, ay (, ), then we have a odel in which there i no learning. In other word, we learn nothing about β j fro β i. Second, if =, then we have a odel in which β i = for all i. In thi latter cae, what we learn about β i tranfer directly to β j without attenuation. For future reference it i convenient to define two poterior oent: β i := β i p(β i ŷ) dβ i and σ 2 β i := (β i β i ) 2 p(β i ŷ) dβ i. (5.13) Collection of regie. Let ŷ All := (ŷ 1,..., ŷ 12 ), and partition ŷ All into ŷ LT := (ŷ 1, ŷ 2 ) and ŷ Euro := (ŷ 3,..., ŷ 12 ). The poterior baed on all the data (ŷ All ) can be obtained either directly or equentially: p(β,, ŷ All, z) p(β,, z) L(β ŷ All ) = p(β,, z) L(β ŷ LT ) L(β ŷ Euro ) p(β,, ŷ LT, z) L(β ŷ Euro ). (5.1) In the lat line of (5.1), p(β,, ŷ LT, z) play the role of the prior; it incorporate the Lothian Taylor data. Figure how β i ± 2 σ βi coputed fro p(β i ŷ All, z =.). Copare with Figure 2. Figure 9 plot β i veru β i to how the hrinkage. The Baye factor in favor of β i = 1 that correpond to the direct route can be decopoed into the product of two Baye factor that correpond the the equential route: p(β i = 1 ŷ All, z) p(β i = 1 z) = p(β i = 1 ŷ LT, z) p(β i = 1 z) Of the three Baye factor in (5.15), the one we focu on i p(β i = 1 ŷ All, z) p(β i = 1 ŷ LT, z). (5.15) B(β i = 1 ŷ LT, z) := p(β i = 1 ŷ All, z) p(β i = 1 ŷ LT, z), (5.16) which expree the reviion in the odd ratio in favor of a unit root given the inforative prior baed on the Lothian Taylor data. Note that B(β i = 1 ŷ LT, z) applie to i = 1,..., 13. For i = 3,..., 12, it expree the reviion in the odd ratio for the Euro-related exchange rate. For i = 1, 2, it expree the reviion in the odd ratio for the Lothian-Taylor exchange rate that coe about fro eeing the Euro-related data (above and beyond having een the Lothian Taylor data itelf). Finally, for i = 13, it expree the reviion in the odd ratio for an exchange-rate regie for which we do not have data, but would have included if we did. See Figure 1 for the arginal poterior for p(β 3 ŷ LT, z) for three value of z. Figure 11 how B(β i = 1 ŷ LT, z =.) a defined in (5.16) for i = 3,..., 13. [Need to do thi for i = 1, 2 a well.]

9 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES 9 Table 3 diplay 1/B(β i = 1 ŷ LT, z) [the Baye factor againt β i = 1] for three value of z. In addition, Table 3 how the nuerical tandard deviation for the Baye factor, which are roughly 1/1 the ize of the correponding Baye factor. 21 Are they all the ae? According to the hierarchical odel, if =, then β i = for all i. The Baye factor in favor of = i given by B( = z) = p( = ŷ, z) p( = z) = 1 1 p(, = ŷ, z) d. (5.17) p(, = z) d Again, we chooe to ue the inforative prior baed on the Lothian Taylor data to copute thi Baye factor: B( = ŷ LT, z) := p( = ŷ All, z) p( = ŷ LT, z). (5.1) In thi cae, we fine B( = ŷ LT, z =.) 2.6. Thu, the evidence favor =. Figure 12 how the poterior p( ŷ All, =, z =.). Appendix A Derivation of L(β i ŷ i ). We can obtain (2.7) via an alternative route. The arginal likelihood for the regreion coefficient can be obtained a follow. Adopt a non-inforative prior p(σ i ) σi 1 and integrate out σ i : where L(α i, β i, {ψ ij } y i ) = L(θ i y i ) = p({y it } T i t=p+1 {y it} p t=1, θ i). L(θ i y i ) σ i dσ i. (A.1) (A.2) It can be hown that L(α i, β i, {ψ ij } y o i ) i proportional to a t ditribution with T i (p + 1) degree of freedo. (Detail below.) The arginal likelihood for β i, L(β i y i ), i alo proportional to a t ditribution witht i (p + 1) degree of freedo, which can be coputed directly fro L(α i, β i, {ψ ij } y o i ). Define l(β i ) := log ( L(β i y o i )). Since T i (p + 1) > 1 for our data et, we adopt the Gauian approxiation for the likelihood: L(β i β i, σ βi ) = N(β i β i, σ 2 β i ), where Show t ditribution. β i = arg ax β i l(β i ) and σ βi = ( l ( β i ) ) 1/2. (A.3) 21 Thee nuerical tandard deviation are coputed a follow. Firt, we apply the forula for coputing the nuerical tandard deviation of an expectation uing iportance apling given in Geweke (25, pp ) and apply the to the coputation of p(β i ŷ All, z) and p(β i ŷ LT, z) via Rao Blackwellization. Next, we conduct a Monte Carlo iulation of the Baye factor by drawing fro the the two ditribution and taking the ratio. The tandard deviation fro the Monte Carlo are reported in the table.

10 1 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN Marginal ditribution for β i in odel with learning. Let β i = β \ {β i }. The arginal prior for β i (5.) i a ixture of p(β i, ): p(β i ) = p(β,, ) dβ i d d ( ) = p(β, ) dβ i p(, ) d d (A.) = p(β i, ) p(, ) d d, where the third equality follow fro p(β, ) = n j=1 p(β j, ) and n j=1 p(β j, ) dβ i = p(β i, ) n j=1 j i p(β j, ) dβ j = p(β i, ). The arginal poterior ditribution for β i (5.9) i a ixture of p(β i ŷ i,, ): p(β i ŷ) = p(β,, ŷ) dβ i d d ( ) = p(β ŷ,, ) dβ i p(, ŷ) d d = p(β i ŷ i,, ) p(, ŷ) d d, (A.5) (A.6) where the third equality follow fro p(β ŷ,, ) = n j=1 p(β i ŷ i,, ) and the ae tep a in (A.5). The arginal poterior for β n+1 (5.11) can be obtained a a pecial cae of (5.9) by adopting the forali L(β n+1 ŷ n+1 ) = 1, o that p(β n+1 ŷ n+1,, ) = p(β n+1, ). Truncated exponential ditribution. The truncated exponential ditribution i given by The ean, a a function of λ, i given by H(λ) := h(x λ) = 1 [,1] (x) λ e λ x 1. (A.7) 1 e λ h(x λ) x dx = 1 λ e λ (A.) and the tandard deviation i 1 G(λ) := h(x λ) (x H(λ)) 2 1 dx = λ (1 coh(λ)). (A.9) Then g() := G(λ) λ=h 1 ().

11 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES 11 Weighted boottrap (iportance apling). Here i an outline of how to copute the poterior for the hierarchical odel p(β ŷ Both, z) with the prior p(, z) via the weighted boottrap. (1) For r = 1,..., R (a) Draw ( (r), (r) ) fro p(, z) (b) Copute (µ (r), σ (r) ) = F 1 ( (r), (r) ) (c) For i = 1,..., 13 (i) Draw β (r) i fro p(β i µ (r), σ (r) ) (ii) Copute v (r) i (d) Copute v (r) All = 13 i=1 v(r) i (2) Copute w (r) All = v(r) All / R r=1 v(r) (3) Reaple {(β (r) () Done 1,..., β(r) = L(β (r) i β i, σ βi ) (note: v (r) )}R r=1 All (for r = 1,..., R) uing {w(r) All }R r=1 = 1 for all r) a probabilitie To copute the poterior baed on the LT data only, copute w (r) LT = v(r) LT / R r=1 v(r) LT where v (r) LT = 2 i=1 v(r) i and reaple {(β (r) 1,..., β(r) 13 )}R r=1 uing {w(r) LT }R r=1 a probabilitie. (The poterior for {β i } 13 i=3 will all be the ae.) The denity p(β i = 1 ŷ All, z) can be coputed via Rao Blackwellization a follow. Firt, analogou to (5.9), we can expre p(β i ŷ All, z) a: p(β i ŷ All, z) = p(β i ŷ i, µ, σ) p(µ, σ ŷ All, z) dµ dσ. (A.1) Equation (.5) deliver a analytical expreion for p(β i ŷ i, µ, σ) and we can obtain draw of (µ, σ) fro p(µ, σ ŷ All, z) by reapling {(µ (r), σ (r) } R r=1 uing {w(r) All }R r=1 a probabilitie. Therefore, we can copute p(β i = 1 ŷ All, z) 1 R R p(β i = 1 ŷ i, µ (r ), σ (r ) ), r =1 (A.11) where {(µ (r ), σ (r ) )} R r =1 are the reapled value. If we want p(β i = 1 ŷ LT, z), then we reaple {(µ (r), σ (r) } R r=1 uing {w(r) LT }R r=1 a probabilitie intead. To copute the poterior uing the prior p(, z ) intead of p(, z), firt copute q (r) := p((r), (r) z ) p( (r), (r) z) (A.12) and then reaple {(β (r) 1,..., β(r) where w (r) All = 13 )}R r=1 q (r) v (r) All R r=1 q(r) v (r) All uing either { w(r) All }R r=1 or { w(r) LT }R r=1 a probabilitie, and w (r) LT = q (r) v (r) LT R r=1 q(r) v (r) LT. (A.13)

12 12 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN 12 U.S France Autria Belgiu Finland France Greece Ireland Italy Netherland Portugal Spain Figure 1. The Lothian Taylor likelihood are hown in the firt row, and the Euro-related likelihood are hown in the reaining row.

13 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES 13 Reference Geweke, J. (25). Conteporary Bayeian econoetric and tatitic. John Wiley & Son. (Fiher) Federal Reerve Bank of Atlanta, Reearch Departent, 1 Peachtree Street N.E., Atlanta, GA E-ail addre: ark.fiher@atl.frb.org URL:

14 1 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN Table 1. Sufficient tatitic for Lothian Taylor data and for Euro-related data. The lag-length i one (p = 1). Alo hown i 1/Bi, the axiu evidence againt β i = 1. Dataet i βi σ βi 1/Bi Country LT U.S France Average.3.5 Euro Autria Belgiu Finland France Greece Ireland Italy Netherland Portugal Spain Average.6.65 Table 2. Average of { β i } 12 i=3 and { σ β i } 12 i=3 (Euro-related data) by lag length. p β σβ p β σβ p β σβ p β σβ

15 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES 15 Table 3. Baye factor againt β i = 1, uing three LT-baed prior indexed by z. (Nuerical tandard error hown beneath.) z i i z

16 16 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN Figure 2. Error bar how β i ± 2 σ βi. The weighted ean i indicated, 12 i=1 w i β i =.6, where the weight are proportional to the preciion: w i = σ 2 β i / 12 j=1 σ 2 β j.

17 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES Β 5 Β 7 Β 1 Β 9 Β Β Β 2 Β 6 Β Β 12 Β Β Figure 3. Baye factor B(β i = 1 a) for a [, 1]. [See equation (.3).] Value greater than one favor β i = 1. The location of β i i indicated.

18 1 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN Figure. Doain for (, ): R = {(, ) : 1 g()}.

19 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES Figure 5. Baye factor B(β i = 1, ) = L(β i = 1 ŷ i )/L(, ŷ i ) for (, ) R. Contour run fro 1 2 to 1 2 in power of 1. Shaded region favor β i = 1. The location ( β i, ) that produce 1/Bi (the axiu evidence againt β i = 1) i indicated.

20 2 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN Figure 6. Prior p(, z =.) over R i hown with evenly paced contour. Marginal prior p Β i z derived fro prior for p, z 6 z.5 z. z Β i Figure 7. Marginal prior p(β i z) for three value of p(, z), where z = E[ = ]. The ean of the prior for β i are.5,.7, and.9.

21 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES Figure. Poterior ean and tandard deviation: β i ± 2 σ βi fro p(β i ŷ Both, z =.). coputed

22 22 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN.95.9 Β i.5 lope Β i Figure 9. Poterior ean β i veru β i how hrinkage.

23 PPP AND UNIT ROOTS: LEARNING ACROSS REGIMES 23 7 Marginal poterior forβ 3 for 3 value of z 6 5 z.5 z. z Β Figure 1. Marginal poterior p(β 3 ŷ, z) for three value of p(, z), where z = E[ = ]. The ean of the poterior are.7,.2, and.7.

24 2 GERALD P. DYWER, MARK FISHER, THOMAS J. FLAVIN, AND JAMES R. LOTHIAN Baye factor in favor of a unit root for Euro data Figure 11. Baye factor B(β i = 1 z =.). 25 Poterior forβ i given 2 Probability denity Β i Figure 12. The poterior ditribution p( ŷ Both, =, z =.).