Affine Term Structure Models: An Introduction

Transcription

1 / 53 Affine Term Srucure Models: An Inroducion Jens H. E. Chrisensen Federal Reserve Bank of San Francisco Term Srucure Modeling and he Lower Bound Problem Day : Term Srucure Modeling in Normal Times Lecure I. European Universiy Insiue Florence, Sepember 7, 25 The views expressed here are solely he responsibiliy of he auhor and should no be inerpreed as reflecing he views of he Federal Reserve Bank of San Francisco or he Board of Governors of he Federal Reserve Sysem.

2 Overview of Course Day : Term Srucure Modeling in Normal Times Affine erm srucure models and heir esimaion Arbirage-free Nelson-Siegel models Sochasic volailiy Finie-sample bias and oher pracical issues Day 2: The Lower Bound Problem Models ha respec lower bounds for yields How do hey compare o affine models? Which challenges are solved? And which remain? Day 3: Applicaions o Policy Quesions How does QE work? Wha are he poenial coss or risks of QE? How o exrac marke-based inflaion expecaions? 2 / 53

3 Day : Term Srucure Modeling in Normal Times. Lecure: Affine Term Srucure Models: An Inroducion 2. Lecure: The Affine Arbirage-Free Class of Nelson-Siegel Term Srucure Models 3. Lecure: An Arbirage-Free Generalized Nelson-Siegel Term Srucure Model 4. Lecure: Can Spanned Term Srucure Facors Drive Sochasic Volailiy? 5. Lecure: How Efficien is he Kalman Filer a Esimaing Affine Term Srucure Models? 3 / 53

4 Hisory and Background The arbirage-free erm srucure lieraure sared wih founding papers like Vasiček (977) and Cox, Ingersoll, and Ross (985). Affine erm srucure models were hen and remain he workhorse model classes hanks o heir richness and racabiliy. Afer wo decades of sudying one- and wo-facor models, i was clear by he 99s ha more is needed 3 facors a leas. Alhough muli-dimensional affine models come in many varieies, hey can sill be caegorized as shown by Dai and Singleon (2). Wih muli-dimensional models he focus shifs from fi o forecasing and risk premiums. Duffee (22) and Cheridio, Filipović, and Kimmel (27) give us essenially and exended affine risk premiums Maximum affine flexibiliy is achieved! Wih flexibiliy comes esimaion problems. Aenion moves from fi and risk premiums o economeric issues and idenificaion. Mos recenly, lower bounds of yields pose ye anoher challenge. 4 / 53

5 The Ulimae Quesion Can we solve challenges and keep all benefis? Good fi Good forecasing Robus and racable esimaion Respec of lower bounds Goal of course is o provide a posiive answer: Yes! 5 / 53

6 Ouline of Presenaion Inroducion of Affine Models Duffie and Kan (996) Marke Prices of Risk Characerizaion of Canonical Affine Models Dai and Singleon (2) Risk Premiums in Affine Models Compleely affine Dai and Singleon (2) Essenially affine Duffee (22) Exended affine Cheridio, Filipović, and Kimmel (27) Kalman Filer Esimaion of Affine Models Oher Esimaion Schemes for Gaussian Affine Models Joslin, Singleon, and Zhu (2) Hamilon and Wu (22) Conclusion 6 / 53

7 The Class of Affine Term Srucure Models () Define a filered probabiliy space (Ω,F,(F ), Q), where he filraion (F ) = {F : } saisfies he usual condiions (Williams, 997). The sae variables X are assumed o be a Markov process defined on a se M R n ha solves he following sochasic differenial equaion where dx = K Q ()[θ Q () X ]d +Σ()D(X, )dw Q, W Q is a Brownian moion in R n conained in (F ), θ Q : [, T] R n is a bounded, coninuous funcion, K Q : [, T] R n n is a bounded, coninuous funcion, Σ : [, T] R n n is a bounded, coninuous funcion, while D : M [, T] R n n has a diagonal srucure... 7 / 53

8 The Class of Affine Term Srucure Models (2)... he sochasic par of he volailiy marix has a diagonal srucure: γ ()+δ ()X... D(X, ) =.....,... γn ()+δ n ()X where γ () δ ()... δ n() γ() =. and δ() =..... γ n () δ n()... δn n() wih γ : [, T] R n is a bounded, coninuous funcion, δ : [, T] R n n is a bounded, coninuous funcion, δ j () will be used o denoe he jh row of he δ()-marix. 8 / 53

9 The Class of Affine Term Srucure Models (3) Given his srucure we have an affine diffusion process dx = K Q ()[θ Q () X ]d +Σ()D(X, )dw Q. For asse pricing, we will also need he insananeous risk-free rae, which is assumed affine in X : r = ρ ()+ρ () X, where ρ : [, T] R is a bounded, coninuous funcion, ρ : [, T] R n is a bounded, coninuous funcion. Finally, for X o be well defined and admissible, i mus be he case ha for all X M R n. γ j ()+δ j ()X, j n, 9 / 53

10 The Class of Affine Term Srucure Models (4) Noe ha he possible ime dependence of ρ, ρ, K Q, θ Q, Σ, γ, and δ is suppressed in his noaion. / 53 For admissible X, Duffie and Kan (996) show ha arbirage-free bond prices will be exponenial-affine P(, T) = E Q [ ( T exp r u du )] = exp ( B(, T) X + A(, T) ), where B(, T) and A(, T) are soluions o a sysem of ordinary differenial equaions (ODEs) db(, T) = ρ +(K Q ) B(, T) n (Σ B(, T)B(, T) Σ) j,j (δ j ), d 2 da(, T) d = ρ B(, T) K Q θ Q 2 j= n (Σ B(, T)B(, T) Σ) j,j γ j, wih boundary condiions A(T, T) = and B(T, T) =. j=

11 The Class of Affine Term Srucure Models (5) This resul implies ha zero-coupon bond yields are affine: y(, T) = ln P(, T) = T T B(, T) X A(, T). T Key poin: Affine models only characerize he Q-dynamics. X is an affine diffusion process. r is affine in X. Bond yields are affine in X. A(, T) and B(, T) are sraighforward o solve. Noe: The resul is silen abou he P-dynamics of X. For pricing and calibraion exercises, only Q-dynamics are needed. However, for forecasing and erm premium decomposiions, we will need he marke prices of risk. / 53

12 Marke Prices of Risk () 2 / 53 Assume ha X and r are well-defined sochasic processes, bu now formulaed under he objecive P probabiliy measure dx = K P [θ P X ]d +ΣD(X )dw P. A sandard assumpion in finance is ha here exiss a sochasic discoun facor M wih dynamics given by: dm M = r d Λ dw P. M is also known as he pricing kernel or sae price deflaor. Noe ha he process Λ is an n-dimensional vecor.

13 Marke Prices of Risk (2) The pricing of he zero-coupon bond is given by [ P(, T) = E P MT ]. M By Io s lemma, Z = ln M has dynamics dz = r d Λ dw P 2 Λ Λ d. This implies ha ln M T = ln M or, equivalenly, M T M = exp( T T T r s ds r s ds T T T Λ s dw P s 2 Λ sdw P s 2 T T Λ s Λ sds Λ sλ s ds). 3 / 53

14 Marke Prices of Risk (3) Now, consider he process Provided E P [exp( 2 V = exp( T Λ s dw P s 2 Λ s Λ sds). Λ sλ s ds)] < (Novikov s condiion), i is he case ha {V } T is a P-maringale w.r.. F and dq = V T dp dq dp = V T (Radon-Nikodym derivaive) defines a probabiliy measure for which (Girsanov s heorem) dw Q = Λ d + dw P is a Brownian moion under his new Q-probabiliy measure. This has wo implicaions... 4 / 53

15 Marke Prices of Risk (4) Firs, bond prices can be wrien as [ P(, T) = E P MT ] M T = dp(ω) M Ω M T T = exp( r s ds Λ sdws P T Λ Ω 2 sλ s ds)dp(ω) T T = exp( r s ds) exp( Λ sdws P T Λ Ω 2 sλ s ds)dp(ω) T [ T ] = exp( r s ds)dq(ω) = E Q exp( r s ds) Ω as if invesors were risk-neural and jus discoun wih r. Second, under his risk-neural measure, he dynamics are dx dx dx = = K P [θ P X ]d +ΣD(X )dw P = K P [θ P X ]d +ΣD(X )[dw Q Λ d] ) (K P [θ P X ] ΣD(X )Λ d +ΣD(X )dw Q Thus, drif of X is adjused wih ΣD(X )Λ he price of risk! 5 / 53.

16 Risk Premiums in Affine Models To esablish he connecion beween he P- and Q-dynamics, we need he funcional form for he marke prices of risk Λ. Three specificaions dominae he lieraure: Compleely affine risk premiums Dai and Singleon (2). Essenially affine risk premiums Duffee (22). Exended affine risk premiums Cheridio, Filipović, and Kimmel (27). Common rai: P- and Q-dynamics remain affine in X. Noe: Wih risk premiums specified, i is also possible o characerize so-called canonical represenaions of affine models as demonsraed by Dai and Singleon (2). 6 / 53

17 Compleely Affine Risk Premiums () Early on, compleely affine risk premiums were popular where λ is an n vecor. The measure change is dw Q Λ = D(X )λ, = dw P Λ d. Thus, he erm o deduc from he drif of he P-dynamics is ΣD(X )Λ = ΣD(X ) 2 λ γ +δ X... ΣD(X )Λ = Σ γ n +δ n X λ. λ n. Noe: The covariance marix of Λ is affine and X is affine under boh P and Q Model is compleely affine. 7 / 53

18 Compleely Affine Risk Premiums (2) 8 / 53 As emphasized by Duffee (22), compleely affine risk premiums are proporional o yield volailiy. In Gaussian models, his becomes paricularly resricive: Consan volailiy Consan risk premiums K P = K Q. Thus, facor ineracions have o be he same under boh probabiliy measures. Also, he expecaions hypohesis mus hold!

19 Canonical Characerizaion of Affine Models () Dai and Singleon (2) classify affine erm srucure models and describe he canonical represenaion for each model class using compleely affine risk premiums Λ = D(X )λ. They inroduce he noaion of A m (n) model classes: n is he number of facors. Subscrip m is he number of square-roo processes. Problem: Invarian ransformaions imply ha he same model (idenical implicaions for he disribuion of bond yields) can be formulaed in an infinie number of ways. Approach: Find a sandard way of represening A m (n) models. Deermine he maximally flexible admissible specificaion ha is economerically idenifiable. This is he canonical represenaion of A m (n) models. 9 / 53

20 Canonical Characerizaion of Affine Models (2) DS consider admissible affine processes wih Q-dynamics dx = K Q [θ Q X ]d +ΣD(X )dw Q, r = ρ +ρ X. They use compleely affine risk premiums Λ = D(X )λ. This implies ha he P-dynamics are where dx = K Q [θ Q X ]d +ΣD(X )Λ d +ΣD(X )dw P = K P [θ P X ]d +ΣD(X )dw P, K P θ P = K Q Σdiag(λ )δ, = (K P ) [K Q θ Q +Σdiag(λ )γ]. 2 / 53

21 Canonical Characerizaion of Affine Models (3) The canonical represenaion of he A m (n) class is characerized by he following specificaions. ( ) K K P CC m m m (n m) = for m >. K UC (n m) m K UU (n m) (n m) For he Gaussian case (m = ), DS specify K P as riangular, while Singleon (26) and subsequen lieraure impose he riangular propery on K Q. ( ) (θ θ P = P ) C m. (n m) Means of unresriced processes are unidenified and fixed a zero. ( ) ( m m m m (n m) Σ = I, γ =, δ = (n m) B CU (n m) m (n m) (n m) Resricions on volailiy srucure ensure boh admissibiliy and idenificaion. 2 / 53 ).

22 Canonical Characerizaion of Affine Models (3) DS impose addiional parameric resricions. ρ (i), m+ i n. Unresriced processes mus have nonnegaive loadings in he shor rae oherwise heir facor values can swich sign. K P i, θp >, i m. Square-roo processes mus have posiive drif a zero oherwise zero is absorbing sae. K P i,j, i m, i j. Square-roo processes mus be posiively correlaed. θ P i, i m. Square-roo processes mus have nonnegaive drif. δ i,j, m+ i n, j m. Volailiy sensiiviies mus be nonnegaive. 22 / 53

23 Canonical Characerizaion of Affine Models (4) dx dx 2 Example: The canonical A (2) model has P-dynamics ) ( κ P = κ P 2 κ P 22 )[( θ P ) ( X X 2 )] ( X d+ +δ 2 X The compleely affine marke prices of risk are ( )( X ΣD(X )Λ = λ +δ 2X λ 2 Thus, he Q-dynamics are ). dx = K P [θ P X ]d +ΣD(X )dw Q ΣD(X )Λ d ( dx dx 2 ) = + [( κ P )( θ P κ P 2 κ P 22 λ 2/κ P 2 ( X +δ 2X ) ) ( dw,q dw 2,Q ( κ P +λ κ P 2 +δλ 2 2 κ P 22 ). ) ( dw,p dw 2,P )( X X 2 )] d ) Noe he limied flexibiliy beween P- and Q-dynamics! 23 / 53

24 Essenially Affine Risk Premiums () Essenially affine risk premiums were inroduced in Duffee (22). To describe hem, define he diagonal marix: { Dess (γ (X j +δ ) = j X ) /2, if infγ j ()+δ j ()X > ;, oherwise. Noe ha he denominaors in D (X ) already are well-defined processes No exra parameer resricions are required! Now, Duffee (22) considers he following risk premiums where λ 2 is an n n marix. Λ = D(X, )λ + D ess(x )λ 2 X, Noe: For he rows wih in D ess(x ), we will no be able o idenify he corresponding rows in λ 2 They are zero! 24 / 53

25 Essenially Affine Risk Premiums (2) The wo naural exremes are illusraive. A one exreme, we have he Gaussian models. For hose classes of models, Dess(X ) is given by (γ ) /2... Dess(X ) = (γ n ) /2 This means ha D ess(x )λ 2 X adds n n parameers o Λ ha all load on X. By implicaion, here is complee separaion beween he specificaions of K P and K Q in Gaussian models wih essenially affine risk premiums. 25 / 53

26 Essenially Affine Risk Premiums (3) 26 / 53 A he oher exreme, we have he A n (n) models wih square-roo processes only. For hose classes of models, Dess (X ) is given by... Dess (X ) =..... =!... Thus, compleely and essenially affine risk premiums are idenical for A n (n) models!

27 Essenially Affine Risk Premiums (4) dx dx 2 ( Example: The canonical A (2) model has P-dynamics ) ( κ P = κ P 2 κ P 22 )[( θ P ) ( X X 2 )] ( X d+ Since ( Dess (X ) = (+δ 2X ) /2 he essenially affine marke prices of risk are ( X ΣD(X )Λ = +δ 2X Thus, he Q-dynamics are )( λ λ 2 ) + +δ 2 X ), ( )( λ 2 2 λ 2 22 ) ( dw,p dw 2,P )( X X 2 dx = K P [θ P X ]d +ΣD(X )dw Q ΣD(X )Λ d ( dx dx 2 κ P +λ κ P 2 +δλ 2 2 +λ 2 2 κ P 22 +λ 2 22 ) ( κ P = κ P 2 κ P 22 )( X X 2 )( ) d+ θ P λ 2/κ P 2 ( X ) d +δ 2 X ) ( dw,q dw 2,Q ). ). 27 / 53 )

28 Exended Affine Risk Premiums () Cheridio, Filipović, and Kimmel (27) inroduce he exended affine risk premiums. To describe hem, define he following addiional marix: D ex (X ) = { (X j ) /2, if X j > ;, oherwise. Now, CFK consider he following exension of he essenially affine risk premiums Λ = D(X, )λ + Dess (X )λ 2 X + D ex (X )(λ 3 +λ 4 X ), where λ 3 is an n vecor { λ 3 λ 3 = j, if X j > ;, oherwise, while λ 4 is an n n diagonal marix { λ 4 λ 4 = ij, if i j and X i, X j > ;, oherwise. 28 / 53

29 Exended Affine Risk Premiums (2) While for Gaussian A (n) models here is no difference beween essenially and exended affine risk premiums, he exension is paricularly powerful for he A n (n) models: λ 3 = λ 3.. λ 3 n λ λ 4 n and λ 4 λ λ 4 2n = λ 4 n... λ 4 n(n ) Thus, here is a oal of n n exra free parameers in he canonical A n (n) model wih exended affine risk premiums. In principle, his implies ha he P- and Q-dynamics are fully flexible relaive o each oher dx dx = K P [θ P X ]d +ΣD(X )dw P = K Q [θ Q X ]d +ΣD(X )dw Q. However, his is neglecing he sign resricions in λ / 53

30 Exended Affine Risk Premiums (3) Essenially affine risk premiums ruly nes compleely affine risk premiums. CFK claim (p. 29, l. 34) ha he exended affine risk premiums always nes boh essenially and compleely affine risk premiums. A face value, his is rue (more parameers are indeed indispuably allowed!), bu i does no come for free! Under he exended affine risk premiums, he resriced processes have o saisfy Feller condiions under boh probabiliy measures in addiion o all he usual resricions for exisence and admissibiliy: (K P θ P ) j >.5σ 2 jj for j m, (K Q θ Q ) j >.5σ 2 jj for j m. 3 / 53

31 Exended Affine Risk Premiums (4) Problem: We need E P [ exp( T If so, exp( T Λ sdws P 2 Q = exp( Λ s dw P s 2 T T T ] Λ s Λ sds) =. Λ sλ s ds) is a maringale and Λ s dw P s 2 T Λ s Λ sds)p defines an equivalen probabiliy measure and absence of arbirage is ensured. Quesion: If X is a square-roo process, is E P [exp( T Xs dw P s 2 T X s ds)] <? This is abou properies of sochasic exponenials. Answer: Yes provided Feller condiions are saisfied! 3 / 53

32 Exended Affine Risk Premiums (5) 32 / 53 dx dx 2 Example: The canonical A (2) model has P-dynamics ) ( κ P = κ P 2 κ P 22 )[( θ P ) ( X X 2 )] ( X d+ +δ 2 X ) ( dw,p dw 2,P ) Since D ess (X ) = ( (+δ 2 X ) /2 ) ( and D (X ex (X ) = ) /2 ), he exended affine marke prices of risk are ΣD(X )Λ = + ( X +δ 2X ( (X ) /2 )( λ λ 2 )[( λ 3 ) ( + ) + ( )( )]. λ 2 2 λ 2 22 )( X X 2 )

33 Exended Affine Risk Premiums (6) Thus, he Q-dynamics are dx = K P [θ P X ]d +ΣD(X )dw Q ΣD(X )Λ d, which is equivalen o ( ) ( )( ) dx κ P dx 2 = θ P λ 3 κ P 2 κ P /κp 22 λ d 2 /κp 2 ( κ P +λ κ P 2 +δ2 λ 2 +λ2 2 κ P 22 +λ2 22 ( )( X + dw,q +δ 2 X dw 2,Q )( X X 2 ). ) d Noe: The Q-dynamics of X are now compleely deached from is P-dynamics, bu here are Feller condiions κ P θp > 2 σ2 and κ P θp λ 3 > 2 σ2. 33 / 53

34 Esimaion of Affine Models Dai and Singleon (2) use simulaed mehod of momens. Duffee (22) esimaes affine models wih QML and assume n yields observed wihou error. Cheridio, Filipović, and Kimmel (27) use approximae maximum likelihood esimaion and n yields observed wihou error. More recenly, Joslin, Singleon, and Zhu (2) and Hamilon and Wu (22) have offered efficien esimaion algorihms for Gaussian affine models. Throughou he course, he Kalman filer is used for model esimaion, so deails of his esimaion procedure are briefly provided in he following. 34 / 53

35 Kalman Filer Esimaion of Affine Models () 35 / 53 For affine Gaussian models, in general, he condiional mean vecor and he condiional covariance marix are E P [X T F ] = (I exp( K P ))θ P + exp( K P )X, V P [X T F ] = e K P s ΣΣ e (K P ) s ds, where = T is he ime beween observaions. We compue condiional momens of discree observaions and obain he sae ransiion equaion X = (I exp( K P ))θ P + exp( K P )X +ξ.

36 Kalman Filer Esimaion of Affine Models (2) In he sandard Kalman filer, he measuremen equaion is y = A+BX +ε. The assumed error srucure is ( ) [( ξ N ε ) ( Q, H )], where he marix H is assumed diagonal, while he marix Q has he following srucure: Q = e K P s ΣΣ e (K P ) s ds. In addiion, he ransiion and measuremen errors are assumed orhogonal o he iniial sae. Now, Kalman filering is used o evaluae he likelihood funcion. 36 / 53

37 Kalman Filer Esimaion of Affine Models (3) Under a saionariy assumpion he filer can be iniialized a he uncondiional mean and covariance marix: X = θ P, Σ = e K P s ΣΣ e (K P ) s ds. Le Y = (y, y 2,...,y ) be he informaion available a ime, and denoe model parameers by ψ. Consider period and suppose ha he sae updae X and is mean square error marix Σ have been obained. The predicion sep is where Φ X, X = E P [X Y ] = Φ X, (ψ)+φ X, (ψ)x, Σ = Φ X, (ψ)σ Φ X, (ψ) + Q (ψ), (ψ) = (I exp( K P ))θ P, Φ X, (ψ) = exp( K P ), Q (ψ) = e K P s ΣΣ e (K P ) s ds. 37 / 53

38 Kalman Filer Esimaion of Affine Models (4) In he ime- updae sep, X is improved by using he addiional informaion conained in Y : where X = E[X Y ] = X +Σ B(ψ) F v, Σ = Σ Σ B(ψ) F B(ψ)Σ, v = y E[y Y ] = y A(ψ) B(ψ)X, F = cov(v ) = B(ψ)Σ B(ψ) + H(ψ), H(ψ) = diag(σ 2 ε(τ ),...,σ 2 ε(τ N )). A his poin, he Kalman filer has delivered all ingrediens needed o evaluae he Gaussian log likelihood, he predicion-error decomposiion of which is T ( log l(y,...,y T ;ψ) = N 2 log(2π) 2 log F 2 v F v ), = where N is he number of observed yields. 38 / 53

39 Kalman Filer Esimaion of Affine Models (5) I ypically maximize he likelihood wih respec o ψ using he Nelder-Mead simplex algorihm. Upon convergence, we obain sandard errors from he esimaed covariance marix Ω( ψ) = [ T log l ( ψ) log l ( ψ) ], T T ψ ψ = where ψ denoes he esimaed model parameers. Noe: For non-gaussian affine models, he only change is ha Q and Σ are calculaed wih T exp( K P (T s))σd(e P [X s X ])D(E P [X s X ]) Σ exp( (K P ) (T s))ds using formulas from Fisher and Gilles (996). 39 / 53

40 Problems wih Canonical Models 4 / 53 Empirical problems wih canonical models: Since X are laen facors, hey may roae during he esimaion. This leaves muliple maxima wih close o idenical likelihood values bu differen yield decomposiions. Consequence: Two researchers may come up wih very differen esimaion resuls despie he fac ha hey use he same daa, he same model, and he same esimaion mehod. Kim and Orphanides (22) describe hese problems for he specific case of he A (3) model. Duffee (2) also conains an elaborae discussion. Wha o do?

41 Joslin, Singleon, and Zhu (2) () Joslin, Singleon, and Zhu (2) consider a generic discree-ime represenaion of Gaussian affine models X = K P + K P X +Σ X ε P, X = K Q + K Q X +Σ X ε Q, r = ρ +ρ X. Canonical: maximally flexible subjec only o resricions required for economeric idenificaion. Le X be n-dimensional and assume ha we observe J > n bond yields denoed y. For any full-rank marix W R n J, P = Wy represens n porfolios of yields. 4 / 53

42 Joslin, Singleon, and Zhu (2) (2) Assuming ha he n porfolios are priced perfecly, JSZ use invarian affine ransformaions o demonsrae ha any canonical Gaussian affine model has a unique observaionally equivalen represenaion P = K P P + K P P P +Σ P ε P, P = K Q P + K Q P P +Σ P ε Q, r = ρ P +ρ P P, where K Q P, K Q P, ρ P, and ρ P are funcions of (λ Q, k Q,Σ P): λ Q are he eigenvalues of K Q. Σ P Σ P is he covariance marix of innovaions o he porfolios. k Q is long-run mean parameer. 42 / 53

43 Joslin, Singleon, and Zhu (2) (3) Key observaion: Since P are observed facors, he P-condiional likelihood funcion of he observed yields is f(y y ;ψ) = f(y P ;λ Q, k Q,Σ P,Σ ε ) f(p P ; K P P, K P P,Σ P). Criical resul: KP P and K P P can be esimaed by OLS independen of Σ P. This delivers K P P and K P P. In he final sep, λ Q, k Q,Σ P,Σ ε are esimaed by maximizing he likelihood funcion using K P P and K P P as inpu and Σ P Σ P as iniial guess for Σ P Σ P : T f(y y ;ψ) = (2π) (J n) Σ ε = exp ( 2 Σ ε (y o A(λ Q, k Q,Σ P Σ P) B(λ Q, k Q ) P ) 2) f(p P ; K P P, K P P,Σ P). 43 / 53

44 Joslin, Singleon, and Zhu (2) (4) The pricing facors P are linear combinaions of yields. They focus on he firs N principal componens: Their P-dynamics follow an unconsrained VAR. Therefore, no-arbirage assumpions do no improve forecas performance by hemselves. Two conribuions: Compuaionally efficien esimaion of Gaussian affine models. Forecas gains from using a Gaussian affine model relaive o an unconsrained VAR mus come from addiional resricions on he P-dynamics and no from imposing absence of arbirage. Noe: Malab code is available on Ken Singleon s websie. 44 / 53

45 Hamilon and Wu (22) Overview Hamilon and Wu (22, HW) describe anoher novel way o idenify and esimae Gaussian affine models. Unlike oher papers ha emphasize how affine models should be represened, HW s focus is on how o esimae any given Gaussian affine model wihou prejudice. In he paper, HW only consider he popular class of Gaussian affine models where i can be assumed ha M yields are observed wihou error, and M is idenical o he number of pricing facors, i.e., he facors become observable. Implicaion: Their so-called reduced-form represenaion is a resriced vecor auoregression ha can be esimaed wih OLS. Model esimaion involves solving muliple equaions wih as many unknowns, bu presens a more manageable problem han brue force full join ML esimaion of all model parameers. 45 / 53

46 Hamilon and Wu (22) Canonical Model The general Gaussian affine model in discree ime is given by (Eq. (), (2), (3)) X = c +ρx +Σu, X = c Q +ρ Q X +Σu Q, r = δ +δ X. In he case of M = 3, here are 37 free parameers. HW use he following normalizaion: Σ = I M (9 resricions); ρ Q is (upper) riangular (3 resricions); c = (3 resricions); δ. Wih hese 5 resricions he model is canonical wih 22 parameers ha are all economerically idenified. 46 / 53

47 Hamilon and Wu (22) HW Base Model The saring poin for HW s analysis is he model X = ρx + u, wih a shor rae given by r = δ +δ X, while zero-coupon bond yields are given by y = B(ρ Q,δ ) X + A(c Q,δ,ρ Q,δ ), where c Q is an M vecor; ρ Q is an upper riangular M M marix; δ is an M vecor; δ is a consan. 47 / 53

48 Hamilon and Wu (22) Reduced-Form Model () Since M linear combinaions of yields are priced wihou error, i is possible o presen he model in he following reduced-form represenaion ( Y Y 2 ) = ( A A 2 wih u e i.i.d.n(, I N M ). ) ( B + B 2 No Pricing error X becomes observable: ) ( X + Σ e ) u e, Y = A + B X X = B (Y A ). This defines an invarian affine ransformaion A + B X = A + B ρb B X + B u = A B ρb A + B ρb (A A + B X )+B u. This is equivalen o Y = A +φ Y + u. 48 / 53

49 Hamilon and Wu (22) Reduced-Form Model (2) The reduced-from model is where Y = A +φ Y + u, A φ u and u i.i.d.n(, B B ). = A B ρb A, = B ρb, = B u Poin: Reduced-form model is a VAR() esimaed wih OLS. This gives us Â, φ, and he covariance marix of u : Ω = T (Y T Â φ Y )(Y Â φ Y ) = = B B. 49 / 53

50 Hamilon and Wu (22) Reduced-Form Model (3) 5 / 53 The equaions for he N M yields wih errors ake he form Y 2 = A 2 +φ 2Y + u 2. This equaion can also be esimaed wih OLS o yield  2, φ 2, and he covariance marix of u 2 Ω 2 = T T = (Y 2  2 φ 2Y )(Y 2  2 φ 2Y ). This was he easy par. The remaining seps are a lile more compuaionally involved o ge back o (δ,δ,ρ, c Q,ρ Q ).

51 Hamilon and Wu (22) B(ρ Q,δ ) Parameers HW consider he following se of equaions: φ 2 Ω = B 2 B B B = B 2B, Ω = B B. The firs line conains M equaions, while he second line conains M(M + )/2 equaions. Combined hey mach he number of unknowns: (ρ Q,δ ). Guess (ρ Q,δ ), calculae B(ρ Q,δ ), and define π = (vec( φ 2 Ω ), vech( Ω )), g(ρ Q,δ ) = (vec(b 2 B ), vech(b B )). Keep adjusing (ρ Q,δ ) o solve ( ρ Q, δ ) = argmin ρ Q,δ ( π g(ρq,δ )) ( π g(ρ Q,δ )). Noe: If N M >, here are more equaions han unknowns - how o weigh deviaions? For N M =, problem is exacly idenified. 5 / 53

52 Hamilon and Wu (22) ρ and A(c Q,δ,ρ Q,δ ) Parameers Wih ( ρ Q, δ ) in hand, we can now recall ha φ = B ρ B ρ = B φ B. Finally, we exrac he las M + parameers, δ and c Q, from: Â = (I B ρ B )Â, Â 2 = Â2 B 2 B Â. Noe: If N M >, here are again more equaions han unknowns - weighs? For N M =, problem is exacly idenified. Procedure: Guess (c Q,δ ), calculae A(c Q,δ, ρ Q, δ ), and compue sum of squared differences in he equaions. Sop once minimized! This is minimum-chi-square esimaion. 52 / 53

53 Limiaions of JSZ and HW 53 / 53 A shor lis of some limiaions: JSZ and HW only apply o Gaussian affine models. They rely on observable sae variables. Hence, missing observaions or irregular daa frequencies are problemaic o handle. No easy exension o nonlinear measuremen equaions. No exensions o non-gaussian dynamics. Forunaely, here exiss an alernaive approach ha is able o overcome hese limiaions...