SUPPLEMENTARY APPENDIX. A Time Series Model of Interest Rates With the Effective Lower Bound

Transcription

1 SUPPLEMENTARY APPENDIX A Time Series Model of Ineres Raes Wih he Effecive Lower Bound Benjamin K. Johannsen Federal Reserve Board Elmar Merens Bank for Inernaional Selemens April 6, 8 Absrac This appendix conains supplemenary resuls as well as furher descripions of compuaional procedures for our paper. Secion I, describes he MCMC sampler used in esimaing our model. Secion II describes he compuaion of predicive densiies. Secion III repors addiional esimaes of rends and sochasic volailiies well as poserior momens of parameer esimaes from our baseline model. Secion IV repors esimaes from an alernaive version of our model, where he CBO unemploymen rae gap is used as business cycle measure insead of he CBO oupu gap. Secion V repors rend esimaes derived from differen variable orderings in he gap VAR of our model. Secion VI compares he forecasing performance of our model o he performance of he no-change forecas from he random-walk model over a period ha begins in 985 and ends in 7:Q. Secions VII and VIII describe he paricle filering mehods used for he compuaion of marginal daa densiies as well as he impulse responses. The views in his paper do no necessarily represen he views of he Bank for Inernaional Selemens, he Federal Reserve Board, any oher person in he Federal Reserve Sysem or he Federal Open Marke Commiee. Any errors or omissions should be regarded as solely hose of he auhors. For correspondence: Benjamin K. Johannsen, Board of Governors of he Federal Reserve Sysem, Washingon D.C benjamin.k.johannsen@frb.gov. Tel.: +() 5 6.

2 Conens I Priors and Poserior Sampling II Compuaion of Predicive Densiies III Addiional Esimaes III. Laen Variables III. Parameer Esimaes IV Resuls using he Unemploymen Rae Gap V Resuls wih Alernaive Orderings of Gap Variables 9 VI Forecasing Performance vs. a Random Walk since VII Paricle Filering he Likelihood for MDD Compuaions 8 VII.Rao-Blackwellized Paricle Filer VII.Adjusing he Paricle Filer o accoun for he ELB VIII Consrucing IRF wih a Paricle Filer VIII.Paricle Weighs used for Impulse Responses VIII.Accouning for he ELB

3 Lis of Tables A. Shock Loadings B in Gap VAR A. Coefficiens of AR()-SV Processes A. Correlaion beween Gap SV Shocks A. Comparison of Ineres Rae Forecass agains Random Walk (saring 985:Q).. 7 Lis of Figures A. Inflaion Trend A. Sochasic Volailiies for Gap Variables A. Persisence of Gap VAR A. Variance of Shocks o he Trend Real Rae A.5 Average Term Premia A.6 CBO Measures for Oupu Gap and Unemploymen Rae Gap A.7 Shadow Rae Esimaes (w/unemploymen Rae Gap) A.8 The Real Rae in he Long Run (w/unemploymen Rae Gap) A.9 Real-Rae Esimaes: Trend Level and Gap Volailiy (w/unemploymen Rae Gap) A. Inflaion Trend (w/unemploymen Rae Gap) A. Sochasic Volailiies for Gap Variables (w/unemploymen Rae Gap) A. Shor-Term Ineres Rae Responses o Moneary Policy Shock (w/unemploymen Rae Gap) A. Responses o Moneary Policy Shock (w/unemploymen Rae Gap) A. Inflaion Trend Esimaes wih re-ordered Gap VAR A.5 Real Rae Trend Esimaes wih re-ordered Gap VAR A.6 Shadow Rae Esimaes wih re-ordered Gap VAR A.7 Inflaion Trend Esimaes wih re-ordered Gap VAR (w/unemploymen Rae Gap). A.8 Real Rae Trend Esimaes wih re-ordered Gap VAR (w/unemploymen Rae Gap) A.9 Shadow Rae Esimaes wih re-ordered Gap VAR (w/unemploymen Rae Gap).. 5

4 SUPPLEMENTARY APPENDIX (online only) I Priors and Poserior Sampling MCMC esimaes of he model are obained from a Gibbs sampler. The sampler is run muliple imes wih differen saring values and convergence is assessed wih he scale reducion es of Gelman e al. (). For each run,, draws are sored afer a burn-in period of, draws; he pos-burnin draws from each run are hen merged. Our models feaures wo layers of laen saes as well as various parameers; he firs layer of laen saes is given by: apple = apple X = apple X = X X r r r 5 r X... X p+ c s ỹ ỹ 5 ỹ (A.) (A.) (A.) wih r i = r + p i 8 i =, 5,, and where p is he lag lengh of he gap VAR in equaion (8). The second layer of laen variables consiss of he vecor of sochasic log-variances of he gap shocks, h log ( ) in equaion (9) ogeher wih he sochasic log-variance process of shocks o he inflaion rend, h log (,) in equaion (). All old, our model consiss of equaions (), (), (), (), (5), (6), (7), (8) and (9); ou of hese, equaions (), (), (5), (6), (7), (8) are represened as a condiionally linear sae-space given by equaions (), (), as well as he runcaed measuremen equaion () for he nominal policy rae. In our baseline case wih consan-variance shocks in he rend real rae deails of he sae space sysem given by equaions () and (), which are also reproduced below, are given Specifically, for every model, independen runs for he Gibbs sampler were evaluaed; each run iniialized wih differen saring values drawn from he model s prior disribuion. Convergence is deemed saisfacory when he scale reducion saisics for every parameer and laen variable are below.; (values close o indicae good convergence). Sochasic volailiy in shocks o he real rae rend if par of he model specificaion can also be wrapped ino h. As noed already in he main ex, in principle, a separae version of equaion () applies for nominal ineres raes of differen mauriies. However, in esimaing our model, he disincion beween longer-erm yields and heir respecive shadow raes would be moo because he ELB never binds for he oher yields in our daa. Neverheless, when simulaing predicive densiies for longer-erm yields, he runcaion implied by () is, of course, applied.

5 SUPPLEMENTARY APPENDIX (online only) 5 as follows: = A + B / " " N(, I) () X = C C = I A = 6 A A 7 5 I () (A.) (A.5) where A and A are he lag coefficiens marices of he gap VAR in equaion (8) and boldface symbols denoe (sub-)marices. Furhermore, we have B = B / B = 6 B 7 5 (A.6) (A.7) where B is a uni-lower-riangular marix, and he sochasic volailiies as well as he consanvariance parameer r are sacked ino / as follows:

6 SUPPLEMENTARY APPENDIX (online only) 6 / = 6, r, c, s, ỹ, ỹ 5, ỹ,. (A.8) 7 5 The parameer vecor of he model comprises he means, persisence and variance-covariances of shocks o h and h, see equaions () and (9). as well as he variance of shocks o he rend real rae, denoed r, he ransiion coefficiens of he gap VAR in equaion (8), sacked in a vecor a, and he lower diagonal elemens of gap shock loadings B in equaion (8) ha can be sacked in a vecor denoed b. For ease of reference, all parameers are colleced in he vecor. Furhermore, since MCMC esimaes of h T and h T will be obained from he muli-move filer of Kim e al. (998), he use of a se of discree indicaor variables, s T, is required o o approximae log, and log in equaions () and (9), respecively, wih a mixure of normals. For ease of exposiion, we sack he log of he sochasic variances of rend and gap shocks ino he vecor h. Combining equaions () and (9) yields he following vecor sysem: h =(I ) µ + h + N(, I) (A.9) apple where µ = µ µ, is a diagonal marix wih and he diagonal values of on is diagonal and is a block-diagonal marix consising of and. Condiional on draws for he various parameers, ( ), and log-volailiies h T, we can consruc Since rend sochasic volailiies in equaion () are independen of he gap sochasic volailiies in equaion (9) is diagonal and is block-diagonal boh sochasic volailiy blocks can be esimaed in separae Gibbs seps.

7 SUPPLEMENTARY APPENDIX (online only) 7 marices A, B, C. and { / } T = and obain he linear, Gaussian sae space sysem described by equaion () in he main paper. For he iniial values of he laen saes, he following priors were used: N (E( ), ) wih E( )= 5 and = 5 (A.) An uninformaive prior for he iniial gap levels is obained by seing equal o he ergodic variance-covariance marix of he gaps implied by he VAR in equaion (8), evaluaed a he ime zero draws for he sochasic volailiies, encoded in, for every MCMC draw. 5 The prior for he iniial rend levels are se o be consisen wih. r. r N.5, I 6 r 5 7 B6.7 C 5 A.5 r (A.) which implies generally vague prior levels for he various rend componens. The prior for he average levels of he log-variances is normal, wih a mean value of log (.) and variance corresponding o he ergodic disribuion implied by draws for he shock variance and AR() lag coefficiens associaed wih he corresponding log-variance process. For each AR() lag coefficien, he prior is N(.8,. ), as in Clark and Ravazzolo (5). For he variance of shocks o he volailiy of he inflaion rend, he prior is a univariae inverse-wishar disribuion wih 6 degrees of freedom and cenered around a mean of., which coincides wih he fixed coefficien-value of. used by Sock and Wason (7) in heir univariae model for inflaion. 6 5 In he case of a VAR(), he ergodic variance-covariance marix solves = A A + B B for given values of A, B, and. 6 The univariae inverse-wishar disribuion corresponds o an inverse-gamma disribuion, whose shape parameer is ypically expressed in unis ha correspond o half he degrees of freedom of he inverse-wishar.

8 SUPPLEMENTARY APPENDIX (online only) 8 For he vecor of shocks o he gap volailiies vecor, in equaion (9), he prior is inverse Wishar, cenered around a mean of. I and N +degrees of freedom where N is he number of gap variables (N =6in our baseline model). The parameer governing he variabiliy of real-rae rend shocks, r, has a univariae inverse- Wishar disribuion wih degrees of freedom and is cenered around a prior mean of.. The prior mean for r is hus similar o he esimaed value for he corresponding parameer repored by Holson e al. (7) in he conex of heir model; see he value of r =.9 in heir Table. Wih hree degrees of freedom, our prior is only vaguely informaive while also embedding some belief ha rend shocks explain only a small share of variaions in real raes. In addiion, his prior also helps o avoid he pile-up problem known, for example, from Sock and Wason (998) and considered in he conex of esimaing r also by Laubach and Williams () as well as Clark and Kozicki (5) and Holson e al. (7) by keeping poserior parameer draws from zero; see our esimaes shown in Figure A. furher below. A Minnesoa-syle prior (cenered around a mean of zero) is used for he VAR coefficiens a, wih hyperparameers =.5 (own lags) and =. (cross lags). The prior b is mulivariae normal, b N (, I). The Gibbs sampler is iniialized wih values drawn from he prior for h T, and and hen generaes draws from he join poserior disribuion p T, h T, a, b,, µ,, r, s T Z T by ieraing over draws from he following condiional disribuions: 7. Draw from p T h T, a, b,, µ,, r, s T, Z T wih he disurbance smoohing sampler of Durbin and Koopman () and rejecion sampling for he shadow rae when he observed nominal shor-erm rae is a he ELB as described in Appendix A.. Draw from p a T, h T, b,, µ,, r, s T, Z T = p a T, h T, b, a normal conjugae 7 For ease of noaion, h T and h T are sacked ino h T unless when he disincion becomes maerial.

9 SUPPLEMENTARY APPENDIX (online only) 9 poserior for a VAR wih known heeroscedasiciy, wih rejecion sampling o ensure a saionary VAR (Cogley and Sargen, 5; Clark, ). Draw from p b T, h T, a,, µ,, r, s T, Z T via recursive Bayesian regressions wih known heeroscedasiciy o orhogonalize he gap shocks of he VAR in (8).. Draw from he univariae inverse-wishar conjugae poseriors for r : p r T, h T, a, b,,, s T, Z T = p r T 5. Draw from he normal conjugae poserior for : p T, h T, a, b, µ,,, rs T, Z T = p h T, µ, = p h T,µ, p h T, µ, Due o he assumed independence beween he sochasic volailiies affecing shocks o rend inflaion he residuals of he gap VAR, which are specified in equaion (9), his sep can be broken ou ino wo separae Bayesian regression seps. The presence of correlaed shocks in equaion (9) necessiaes a SUR regression o consruc he poserior for. Rejecion sampling is applied o ensure ha all elemens of are inside he uni circle. 6. Draw from he inverse-wishar conjugae poserior for, which can again be broken ou ino wo independen seps: p T, h T, a, b,, µ, r,, st, Z T = p h T, µ, = p h T,µ,, p ht, µ, 7. Draw he mixure indicaors s T from p s T T, h T, a, b,, µ,, Z T. 8. Draw from p h T, µ s T, T, a, b,,, r, s T, Z T by applying he disurbance smoohing

10 SUPPLEMENTARY APPENDIX (online only) sampler of Durbin and Koopman () o a linear sae space for h T, µ as in Kim e al. (998). 8 Sricly speaking, his is no a simple Gibbs sampler consising of seps 8, bu raher a Gibbswihin-Gibbs sampler wih he ouer Gibbs sampler ieraing beween p s T, T, h T, µ, Z T (hus, a block consising of seps hrough 7) and p h T, µ s T, T,, Z T (sep 8), similar o he discussion in Del Negro and Primiceri (5). II Compuaion of Predicive Densiies In order o derive ineres-rae forecass ha conform o he ELB (and oher daa in Z ), we firs proceed by characerizing he predicive densiy for he shadow rae. Forecass for acual raes can hen be compued by inegraing over he censored shadow-rae densiy. The shadow rae is included in he non-censored vecor of variables X described in Appendix A. Apar from handling he runcaion issues relaed o he ELB, our approach is fairly sandard, building, for example, on he work by Geweke and Amisano (), Chrisoffel e al. (), and Warne e al. (5). Given he runcaion issues for ineres raes and he fa ails inroduced ino he predicive densiy by he sochasic volailiy specificaion, we have chosen o compue he predicive densiy based on he mixure of normals ha is implied by he draws from our MCMC sampler, insead of approximaing he predicive densiy solely based on is firs wo momens, reaing he predicive densiy as a normal disribuion, as has been done, for example, by Adolfson e al. (7) in he case of linearized, consan-parameer DSGE models. In order o compue he predicive densiy for Z +h jumping off daa a ime, we firs employ he MCMC sampler described in Appendix I o re-esimae all model parameers and laen vari- 8 The consan µ is embedded in he sae space as a uni roo wihou shocks, which improves he efficiency of he Gibbs sampler by joinly sampling h T and µ.

11 SUPPLEMENTARY APPENDIX (online only) ables (, and h ) condiional on daa available hrough ime. Draws from his MCMC sampler will henceforh be indexed by k. Condiional on draws ( k, h k, k ), i is sraighforward o compue he predicive mean for uncensored variables: E X +h k, h k, k = C k A k h k (A.) and he predicive mean, condiional solely on daa hrough, can hen be approximaed by averaging over he means derived from each MCMC draw: E X +h Z X k E X +h k, h k, k (A.) However, in order o characerize he enire predicive densiy for uncensored variables or even he predicive densiy for ineres raes, which are subjec o censoring due o he ELB consrain, we need o accoun for non-lineariies in he disribuion for fuure :+h arising from he sochasic volailiy shocks in our model. We simulae J =rajecories, each indexed by j, of h k,j :+h as well as shocks o :+h for each draw k from he MCMC sampler. From each draw k,j :+h we consruc X j,k +h ; he ensemble of hese draws across k and j approximaes he predicive densiy for X +h. III III. Addiional Esimaes Laen Variables This secion repors esimaes of addiional laen variables, no shown in he main paper. Specifically, Figure A. displays esimaes of he level of rend inflaion as well as sochasic volailiy affecing shocks o rend inflaion. Figure A. repors he sochasic volailiies affecing residuals of he gap VAR in equaion (8).

12 SUPPLEMENTARY APPENDIX (online only) Figure A.: Inflaion Trend 8 Trend Daa (a) Level of (b) SV of shocks o Noe: Smoohed esimaes using all available observaions from 96:Q hrough 7:Q. Thick dashed lines are poserior medians, shaded areas depic 9% and 5% uncerainy bands ha reflec he join uncerainy abou model parameers and saes.

13 SUPPLEMENTARY APPENDIX (online only) Figure A.: Sochasic Volailiies for Gap Variables (a) (b) c (c) s (d) ỹ (e) ỹ (f) ỹ Noe: Sochasic volailiies affecing he residuals in he gap VAR in equaion (8). Specifically, hese volailiies affec he Choleski residuals of he VAR and hus correspond o he diagonal elemens of / in equaion (8). Smoohed esimaes using all available observaions from 96:Q hrough 7:Q. Thick dashed lines are poserior medians, shaded areas depic 9% and 5% uncerainy bands ha reflec he join uncerainy abou model parameers and saes.

14 SUPPLEMENTARY APPENDIX (online only) III. Parameer Esimaes This appendix repors prior and poserior momens of various model parameers. Figure A. repors esimaes of he persisence in he gap VAR (8) as measured by he larges, absolue eigenvalue of he associaed companion-form ransiion marix. Figure A. repors prior and poserior densiies for r, he variance of shocks o he rend real rae. Figure A.5 depics prior and poserior densiies for average erm premia beween longer-erm yields and he shor-erm shadow rae implied by he model esimaes. Tables A., A. and A. repor prior and poserior momens of he various SV processes as well as he Choleski facorizaion of shocks o he gap VAR in equaion (8).

15 SUPPLEMENTARY APPENDIX (online only) 5 Figure A.: Persisence of Gap VAR 8 6 Poserior Prior Noe: Prior and poserior disribuion of maximum, absolue eigenvalue of he companion-form ransiion marix associaed wih he gap VAR in equaion (8); prior and poserior means are depiced wih horizonal lines. Full-sample esimaes using all available observaions from 96:Q hrough 7:Q. A Minnesoa-syle prior (cenered around a mean of zero) has been used for he VAR coefficiens a, wih hyperparameers =.5 (own lags) and =. (cross lags).

16 SUPPLEMENTARY APPENDIX (online only) 6 Figure A.: Variance of Shocks o he Trend Real Rae 5 5 Poserior Prior Noe: Prior and poserior disribuion of r, he variance of shocks o he rend real rae, see equaion (6); prior and poserior means are depiced wih horizonal lines. Full-sample esimaes using all available observaions from 96:Q hrough 7:Q. The prior for r is a univariae inverse Wishar disribuion wih mean of. =. and hree degrees of freedom. Our prior mean is hus similar o he esimaed value for he corresponding parameer repored by Holson e al. (7) in he conex of heir model; see he value of r =.9 in heir Table.

17 SUPPLEMENTARY APPENDIX (online only) 7 Figure A.5: Average Term Premia.5 Poserior Prior (a) -year rae.5 Poserior Prior (b) 5-year rae.5 Poserior Prior (c) -year rae Noe: Reflecing he assumed coinegraion beween ineres raes of differen mauriies, average erm premia, p i = r i r, are he consan offses beween real rae rends of longer-erm yields, r, i relaive o he rend of he shor-erm (shadow) real rae, r. The prior, implied by equaion (A.), is essenially fla. Poserior means are depiced wih horizonal lines. Full-sample esimaes using all available observaions from 96:Q hrough 7:Q.

18 SUPPLEMENTARY APPENDIX (online only) 8 Table A.: Shock Loadings B in Gap VAR Choleski Residuals of Shocks o... Variables c s ỹ y ỹ 5 ỹ y. c.. [ ] s... [ ] [.66. ] ỹ y [..8 ] [.76. ] [.7.9 ] ỹ [.8.9 ] [..6 ] [ ] [.8.8 ] ỹ y [..97 ] [.. ] [.6.9 ] [ ] [.. ] Noe: Poserior momens for coefficiens of he uni-lower-riangular marix B ha maps shocks o he gap variables in he VAR of equaion (8) o is Choleski residuals (scaled by sochasic volailiy): A(L) X = B / ", where / is a diagonal marix of sochasic volailiies. By definiion, diagonal elemens of B have uni value, and upper-diagonal elemens are zero (indicaed by blank enries above). Bold, lower-diagonal enries repor poserior means; below he means, numbers in square brackes repor 5% and 95% quaniles. Each coefficien has a sandard normal prior disribuion; his wih mean zero, 5% (95%) quanile equal o.65 (.65). Full-sample esimaes using all available observaions from 96:Q hrough 7:Q. Esimaed rajecories of he sochasic volailiies on he diagonal of / are depiced in Figure A..

19 SUPPLEMENTARY APPENDIX (online only) 9 Table A.: Coefficiens of AR()-SV Processes PANEL A: Trend Inflaion SV in... exp (µ /) [.96. ] [ ] [..79 ] PANEL B: Gap Variables SV in... exp ( µ i /) i p ii [ ] [ ] [.99.5 ] c [ ] [ ] [.9.55 ] s [.5.9 ] [ ] [..6 ] ỹy [.6.6 ] [ ] [.8.9 ] ỹ [.75. ] [ ] [.5.96 ] ỹy [.6.95 ] [ ] [.7. ] Noe: Poserior momens of coefficiens of he AR() sochasic volailiy (SV) processes affecing differen variables. Bold numbers are poserior means wih 5% and 95% quaniles repored in square brackes underneah. All esimaes reflec daa on all available observaions from 96:Q hrough 7:Q; esimaed rajecories of he sochasic volailiies on he diagonal of / are depiced in Figure A.. Panel A repors he coefficiens of he SV process affecing shocks o rend inflaion, see equaion (): log, =( )µ + log, +,, N(, ) () Prior disribuions are N(.8,. ), is univariae inverse Wishar wih mean. and degrees of freedom. The (condiional) prior disribuion of µ is normal wih mean equal o log (. ) and variance corresponding o he ergodic mean of he SV process implied by given values of and. Panel B repors coefficiens of he SV processes affecing he orhogonalized shocks in he gap VAR of equaion (8) given by he SUR sysem (9): log =(I ) µ + log + N(, I) (9) In Panel B, µ i and i refer o he ih (diagonal) elemen of µ and, respecively. is he variance-covariance marix of shocks o he gap SV processes, and p ii is he volailiy of shocks o he ih gap SV process. Correlaion coefficiens implied by he off-diagonal elemens of are repored in Table A.. Prior disribuions are i N(.8,. ), µ i N(log (. ), 5 ), and is inverse Wishar wih mean. I and N y + = 7 degrees of freedom.

20 SUPPLEMENTARY APPENDIX (online only) Table A.: Correlaion beween Gap SV Shocks c s ỹ y ỹ 5 ỹ y [ ] [ ] [ ] [ ] [ ] c [..79 ] [ ] [ ] [ ] [ ] s [.6.89 ] [ ] [ ] [ ] [ ] ỹ y [.5.79 ] [..778 ] [..85 ] [ ] [ ] ỹ [ ] [ ] [ ] [ ] [ ] ỹ y [ ] [ ] [.7.87 ] [..75 ] [ ] Noe: Prior and poserior momens of correlaion coefficiens implied by he variance-covariance marix of shocks o he SUR sysem of SV processes in equaion (9) affecing residuals in he gap VAR equaion (8): log =(I ) µ + log + N(, I) (9) Upper-diagonal enires repor prior momens, lower-diagonal enries poserior momens. For each correlaion coefficiens, he able repors he mean as well as 5% and 95% quaniles (in square brackes). The prior disribuion of is inverse Wishar wih mean. I and Ny + = 7 degrees of freedom.

21 SUPPLEMENTARY APPENDIX (online only) IV Resuls using he Unemploymen Rae Gap This secion repors esimaes generaed from an alernaive specificaion where daa for he CBO oupu gap is replaced by daa measuring he CBO unemploymen rae gap. As can be seen on Figure A.6, boh measures convey a broadly similar picure of he business cycle in he U.S.; correspondingly, resuls from our model obained using eiher measure are very similar. The unemploymen rae gap is compued as he difference beween he CBO s measure of he naural long-erm rae of unemploymen for a given quarer and he quarerly average rae of unemploymen. 9 All oher daa series are idenical o hose used in he main paper. 9 Daa for he unemploymen rae and he CBO s esimae of he naural rae of unemploymen in he long run are obained from he FRED daabase, available a hps://fred.slouisfed.org, where hey are labeled UNRATE and NROU, respecively.

22 SUPPLEMENTARY APPENDIX (online only) Figure A.6: CBO Measures for Oupu Gap and Unemploymen Rae Gap 6 Oupu gap Inverse unemploymen rae gap Noe: Shaded areas indicae NBER recessions. The unemploymen rae gap is compued as he log-difference beween he CBO s measure of he naural long-erm rae of unemploymen for a given quarer and he quarerly average rae of unemploymen; shown in he figure is he inverse of he unemploymen rae gap (corresponding o he log-difference beween he naural and he acual rae of unemploymen). The oupu gap is compued as he log difference beween real GDP and he CBO s measure of poenial real GDP for a given quarer. (In order o be comparable o annualized growh raes, for our compuaions he log difference beween acual and poenial GDP is been scaled by a facor of when compuing he oupu gap.) All compuaions are based on he vinage of FRED daa available ha has been available a he end of Ocober 7.

23 SUPPLEMENTARY APPENDIX (online only) Figure A.7: Shadow Rae Esimaes (w/unemploymen Rae Gap) (a) Full-Sample Esimaes and Oher Esimaes (b) Full-Sample and Quasi Real-Time Esimae Noe: Shaded areas indicae 5 and 9 percen uncerainy bands, dashed lines are poserior means. Resuls shown in Panel (a) reflec smoohed esimaes using all available observaions from 96:Q hrough 7:Q. The red line in Panel (b) reflecs mean of he endpoins of sequenially re-esimaing he enire model over growing samples of quarerly observaions saring in 96:Q, hus reflecing filered esimaes of he model s laen variables. Uncerainy bands reflec he join uncerainy abou model parameers and saes. Figure A.8: The Real Rae in he Long Run (w/unemploymen Rae Gap) (a) Smoohed Esimaes (b) Quasi Real-Time Esimae Noe: Shaded areas indicae 5 and 9 percen uncerainy bands, dashed lines are poserior means. Resuls shown in Panel (a) reflec smoohed esimaes using all available observaions from 96:Q hrough 7:Q. Resuls shown in Panel (b) reflec he endpoins of sequenially re-esimaing he enire model over growing samples of quarerly observaions saring in 96:Q, hus reflecing filered esimaes of he model s laen variables. Uncerainy bands reflec he join uncerainy abou model parameers and saes.

24 SUPPLEMENTARY APPENDIX (online only) Figure A.9: Real-Rae Esimaes: Trend Level and Gap Volailiy (w/unemploymen Rae Gap) Acual Trend (a) r (b) Vol ( r ) Noe: Panel (a) depics poserior means as well as 5% and 9% uncerainy bands for he rend real rae as well as for model esimaes of he acual real rae, r = i E +. Panel (b) repors esimaes of he condiional volailiy of shocks o he (shadow) real-rae gap, Vol ( r ) where r = s E +. Boh panels reflec smoohed esimaes compued using all available observaions from 96:Q hrough 7:Q.

25 SUPPLEMENTARY APPENDIX (online only) 5 Figure A.: Inflaion Trend (w/unemploymen Rae Gap) 8 6 Trend Daa (a) Level of (b) SV of shocks o Noe: Smoohed esimaes using all available observaions from 96:Q hrough 7:Q. Thick dashed lines are poserior means, shaded areas depic 9% and 5% uncerainy bands ha reflec he join uncerainy abou model parameers and saes.

26 SUPPLEMENTARY APPENDIX (online only) 6 Figure A.: Sochasic Volailiies for Gap Variables (w/unemploymen Rae Gap) (a) (b) c (c) s (d) ỹ (e) ỹ (f) ỹ Noe: Sochasic volailiies affecing he residuals in he gap VAR (8). Specifically, hese volailiies affec he Choleski residuals of he VAR and hus correspond o he diagonal elemens of / in (8). Smoohed esimaes using all available observaions from 96:Q hrough 7:Q. Thick dashed lines are poserior means, shaded areas depic 9% and 5% uncerainy bands ha reflec he join uncerainy abou model parameers and saes. The business cycle measure c is given by he inverse unemploymen rae gap.

27 SUPPLEMENTARY APPENDIX (online only) 7 Figure A.: Shor-Term Ineres Rae Responses o Moneary Policy Shock (w/unemploymen Rae Gap) (a) Shadow Rae (b) Policy Rae Noe: Responses o moneary policy shocks esimaed for 7:Q, 9:Q, :Q as well as 6:Q; dashed lines indicae responses a imes when he ELB was binding for acual daa. Shocks are scaled o generae a percenage poin drop in he shadow rae on impac. Verical axis unis are in percenage poins. Horizonal axis unis are quarers afer impac of he moneary policy shock, which occurs a quarer zero.

28 SUPPLEMENTARY APPENDIX (online only) 8 Figure A.: Responses o Moneary Policy Shock (w/unemploymen Rae Gap) (a) -year Yield (b) -year Yield (c) Inflaion (d) Inflaion Gap (e) Inverse Unemploymen Rae Gap (f) Spread: - over -year Yield Noe: Responses o moneary policy shocks esimaed for 7:Q, 9:Q, :Q as well as 6:Q; dashed lines indicae responses a imes when he ELB was binding for acual daa. Shocks are scaled o generae a percenage poin drop in he shadow rae on impac. Verical axis unis are in percenage poins. Horizonal axis unis are quarers afer impac of he moneary policy shock, which occurs a quarer zero.

29 SUPPLEMENTARY APPENDIX (online only) 9 V Resuls wih Alernaive Orderings of Gap Variables Our model embeds a VAR for he gap componens wih sochasic-volailiy in is orhogonalized shocks, see equaion (8). In a consan-variance case, esimaion of he VAR would be invarian of he ordering of shocks in he Choleski decomposiion implied by he uni-lower-riangular srucure of B in equaion (8) However, due o he sochasic volailiies, esimaion of he VAR coefficiens is no invarian o he ordering of variables in he sochasic-volailiy case (Primiceri, 5). This appendix documens he robusness of rend and gap esimaes o differen variable orderings.

30 SUPPLEMENTARY APPENDIX (online only) Figure A.: Inflaion Trend Esimaes wih re-ordered Gap VAR (a),c,s,y (baseline) (b),s,c,y (c) c,,s,y (d) s,c,,y (e) y,s,c, Noe: Esimaes generaed from alernaive orderings of gap variables in he VAR described in equaion (8). Panel A.a depics baseline esimaes; alernaive orderings are as indicaed above where y indicaes he block of yield gaps y, y 5, y. Filered esimaes in red, smoohed esimaes in black; boh surrounded by 9% uncerainy bands (Filered esimaes reflec he endpoins of sequenially re-esimaing he enire model over growing samples saring in 96:Q. Smoohed esimaes use all available observaions from 96:Q hrough 7:Q.

31 SUPPLEMENTARY APPENDIX (online only) Figure A.5: Real Rae Trend Esimaes wih re-ordered Gap VAR (a),c,s,y (baseline) (b),s,c,y (c) c,,s,y (d) s,c,,y (e) y,s,c, Noe: Esimaes generaed from alernaive orderings of gap variables in he VAR described in equaion (8). Panel A.5a depics baseline esimaes; alernaive orderings are as indicaed above where y indicaes he block of yield gaps y, y 5, y. Filered esimaes in red, smoohed esimaes in black; boh surrounded by 9% uncerainy bands (Filered esimaes reflec he endpoins of sequenially re-esimaing he enire model over growing samples saring in 96:Q. Smoohed esimaes use all available observaions from 96:Q hrough 7:Q.

32 SUPPLEMENTARY APPENDIX (online only) Figure A.6: Shadow Rae Esimaes wih re-ordered Gap VAR (a),c,s,y (baseline) (b),s,c,y (c) c,,s,y (d) s,c,,y (e) y,s,c, Noe: Esimaes generaed from alernaive orderings of gap variables in he VAR described in equaion (8). Panel A.6a depics baseline esimaes; alernaive orderings are as indicaed above where y indicaes he block of yield gaps y, y 5, y. Filered esimaes in red, smoohed esimaes in black; boh surrounded by 9% uncerainy bands (Filered esimaes reflec he endpoins of sequenially re-esimaing he enire model over growing samples saring in 96:Q. Smoohed esimaes use all available observaions from 96:Q hrough 7:Q.

33 SUPPLEMENTARY APPENDIX (online only) Figure A.7: Inflaion Trend Esimaes wih re-ordered Gap VAR (w/unemploymen Rae Gap) (a),c,s,y (baseline) (b),s,c,y (c) c,,s,y (d) s,c,,y (e) y,s,c, Noe: Esimaes generaed from alernaive orderings of gap variables in he VAR described in equaion (8). Panel A.7a depics baseline esimaes; alernaive orderings are as indicaed above where y indicaes he block of yield gaps y, y 5, y. Filered esimaes in red, smoohed esimaes in black; boh surrounded by 9% uncerainy bands (Filered esimaes reflec he endpoins of sequenially re-esimaing he enire model over growing samples saring in 96:Q. Smoohed esimaes use all available observaions from 96:Q hrough 7:Q.

34 SUPPLEMENTARY APPENDIX (online only) Figure A.8: Real Rae Trend Esimaes wih re-ordered Gap VAR (w/unemploymen Rae Gap) (a),c,s,y (baseline) (b),s,c,y (c) c,,s,y (d) s,c,,y (e) y,s,c, Noe: Esimaes generaed from alernaive orderings of gap variables in he VAR described in equaion (8). Panel A.8a depics baseline esimaes; alernaive orderings are as indicaed above where y indicaes he block of yield gaps y, y 5, y. Filered esimaes in red, smoohed esimaes in black; boh surrounded by 9% uncerainy bands (Filered esimaes reflec he endpoins of sequenially re-esimaing he enire model over growing samples saring in 96:Q. Smoohed esimaes use all available observaions from 96:Q hrough 7:Q.

35 SUPPLEMENTARY APPENDIX (online only) 5 Figure A.9: Shadow Rae Esimaes wih re-ordered Gap VAR (w/unemploymen Rae Gap) (a),c,s,y (baseline) (b),s,c,y (c) c,,s,y (d) s,c,,y (e) y,s,c, Noe: Esimaes generaed from alernaive orderings of gap variables in he VAR described in equaion (8). Panel A.9a depics baseline esimaes; alernaive orderings are as indicaed above where y indicaes he block of yield gaps y, y 5, y. Filered esimaes in red, smoohed esimaes in black; boh surrounded by 9% uncerainy bands (Filered esimaes reflec he endpoins of sequenially re-esimaing he enire model over growing samples saring in 96:Q. Smoohed esimaes use all available observaions from 96:Q hrough 7:Q.

36 SUPPLEMENTARY APPENDIX (online only) 6 VI Forecasing Performance vs. a Random Walk since 985 Table A. compares he forecass from our model wih he no-change forecass from a randomwalk model for he period afer 985. We sar in 985 so ha our model has several periods o use a raining sample before producing ou-of-sample forecass. The saisics for our model are shown as calculaed and he saisics for he random-walk model are shown on a relaive basis o our model. We include resuls from versions of our model where he CBO s measure of he oupu gap is used as our business cycle measure and where he CBO s measure of he unemploymen rae gap is used as our business cycle measure. For boh he shor- and long-erm rae, our model performs abou as well, on balance, as he random-walk model.

37 SUPPLEMENTARY APPENDIX (online only) 7 Table A.: Comparison of Ineres Rae Forecass agains Random Walk (saring 985:Q) Forecas horizon h 5 8 Panel A: Shor-erm ineres rae i +h Model (oupu gap) MAD RMSE RW rel. o model (oupu gap) rel. MAD rel. RMSE Model (unemploymen rae gap) MAD RMSE RW rel. o model (unemploymen rae gap) rel. MAD rel. RMSE Panel B: -year ineres rae y +h Model (oupu gap) MAD RMSE RW rel. o model (oupu gap) rel. MAD rel. RMSE Model (unemploymen rae gap) MAD RMSE RW rel. o model (unemploymen rae gap) rel. MAD rel. RMSE Noe: RMSE are roo-mean-squared errors compued from using he medians of our model s and he no-change forecass from a random-walk model; MAD are mean absolue deviaions obained from using he same forecass. Relaive RMSE and MAD are expressed as raios relaive o he corresponding saisics from he baseline model (values below uniy denoing beer performance han our model). Predicive densiies are re-esimaed over growing samples ha sar in 985:Q for our model. Sars indicae significan differences, relaive o baseline, in squared losses, absolue losses and densiy scores, respecively, as assessed by he es of Diebold and Mariano (995);, and denoe significance a he %, 5% respecively % level.

38 SUPPLEMENTARY APPENDIX (online only) 8 VII Paricle Filering he Likelihood for MDD Compuaions Compuaion of he marginal daa densiies (MDD) in Secion. relies on paricle filer esimaes of our model s likelihood. We esimae he MDD using he harmonic mean esimaor of Geweke (999), as presened by Herbs and Schorfheide (), given by: " p(z) N NX n= # f( n ) p(z n )p( n, (5) ) where N is he number of draws from he poserior disribuion, is a vecor ha collecs all of he esimaed parameers and n is a paricular draw from he poserior disribuion, f is a funcion of he parameer vecor ha inegraes o one, p( n ) is he prior densiy of n, and p(z n ) is he likelihood of he daa, sacked ino he vecor Z. As in Geweke (999), we use he following choice of f: f( ) = ( ) d/ V / exp.5 ( ) V ( ) I n ( ) V ( ) apple F d o ( ) where and V be he mean and variance of he poserior disribuion of, d he lengh of, and F d is he cumulaive disribuion funcion of he disribuion wih d degrees of freedom, and I is he indicaor funcion. We se =.9, and our resuls are robus o oher choices of. Since our model has wo layers of laen variables, he likelihood p(z n ) in equaion (5) canno be compued analyically. Using noaion inroduced in Appendix I, he wo layers of laen sae variables are he rends and gaps sacked in as well as he sochasic volailiies capured by h. Effecively, our model The prior, described in Secion I, can be broken down ino producs of normal and inverse-wishar densiy funcions. For some parameers, like he coefficiens of he gap VAR, denoed a in Secion I, or he AR() lag coefficiens ) in equaion (A.9), he priors are runcaed normals and he necessary rejecion probabiliies are sraighforward o sample.

39 SUPPLEMENTARY APPENDIX (online only) 9 is a non-linear sae-space model wih he following composie sae vecor: 6 S = h 7 5 (A.) For he likelihood compuaion, his composie sae vecor needs o be inegraed ou, which is no provided by he MCMC sampler. However, a paricle filer can be used o approximae he likelihood of such a non-linear model. In fac, apar from he ELB consrain, esimaion of an unobserved componen model wih sochasic volailiy (UC-SV) like our is fairly sraighforward wih a Rao-Blackwellized paricle filer (RB-PF) ha explois he condiionally linear srucure of he model. RB-PFs are surveyed, for example by Creal () and Lopes and Tsay (); see also he applicaions by Carvalho e al. (7), Merens and Nason (7) and Merens (6). Afer describing a sandard RB-PF ha neglecs any ELB issues, his secion urns o our handling of he ELB consrain wihin a RB-PF. VII. Rao-Blackwellized Paricle Filer For ease of exposiion, le us firs describe he RB-PF wihou considering issues arising from he ELB. For now, we hus rea shadow raes, and hus X as observable. The paricle filer approximaes he likelihood p(x ) and filered poserior densiy of he laen sae vecor S X, for given X and parameer values. The priors for he iniial values S are as described in Appendix I. A each poin in ime, indexed by, he filer racks a swarm of M paricles, indexed by i, ha consis of he sochasic volailiies h (i) and Kalman filered esimaes of he linear sae ha Condiional on he sochasic volailiies, he dynamics of are linear, and we will refer o also as linear sae variables. Recall ha he MCMC sampler ieraes only beween generaing smoohed condiional draws of T h T ; and h T T ; as well as generaing appropriae draws of bu does no joinly filer and h. As before, he iniial variance of he gaps componens in, depends on he parameer vecor as well as sochasic volailiies for he gap shocks, ; he gap variance is recompued for each paricle draw of and based on he specific parameer vecor n used when evaluaing he paricle filer.

40 SUPPLEMENTARY APPENDIX (online only) condiion on he paricles hisory of h (i) ; he paricle s Kalman filering disribuion of he linear saes and condiional on daa X is characerized by he mean vecor (i) marix (i). and variance-covariance We uilize an auxiliary paricle filer (APF) as described by Lopes and Tsay (). Firs inroduced by Pi and Shephard (999), he APF is a refinemen of he boosrap filer. While he boosrap filer propagaes paricles from one period o he nex based on heir prior disribuion, he APF seeks o adap new paricles based on he likelihood hey imply for he daa. Before urning o he unique seps of he auxiliary paricle filering seps, we describe he sandard boosrap algorihm. The filer begins by generaing M iniial paricles h (i), (i) and (i) from he heir respecive priors described in Appendix I. For =,...,T, he filer repeas he following seps:. For i =,...,M draw new paricles h (i) based on is prior condiional on h (i) implied by (A.9) and consruc he corresponding diagonal marix of log-volailiies /,(i) B (i) = B /,(i).. For each paricle i, engage he Kalman filer for equaion () o compue: and (i) (i) e (i) = A = C l (i) = (i) A + B (i) B (i) (A.5) (i) C (A.6) = X C A (i) (A.7) log ( ) Nx +log (i) + e (i) + (i) (i) e (A.8) + K (i) = (i) C (i) + (A.9) (i) = (i) (i) = (i) + K(i) e (i) (i) C (i) + C (i) (A.) (A.) In ligh of he possibiliy of missing daa encoded as elemens of X fixed a zero and a rank deficien C noe ha N x is he number of acual observaions in X, corresponding

41 SUPPLEMENTARY APPENDIX (online only) o he number of non-zero rows of C, and + and + denoe he pseudo-deerminan and pseudo-inverse operaors, respecively.. Compue he paricle weighs w (i) = exp l (i) P M i exp l (i). The filered disribuion of h is approximaed by he discree disribuion of paricle draws h (i) using he pdf described by w (i). The associaed filered disribuion of is approximaed by a mixure of normals N (i), (i) wih weighs w (i). (For his purpose, he values for h (i), (i) and (i) are sored before he resampling described in he nex sep.). For <Tprepare he nex ieraion by applying sysemaic resampling o he paricles h (i) (i) and (i) based on he paricle weighs w(i) = exp l (i) P M i exp l (i) The likelihood of he dae observaion, condiional on parameers and he previous hisory of observaions is esimaed by averaging over he likelihoods of each paricle:., p X X, / M MX i= exp l (i) (A.) The log-likelihood, which corresponds also o he log-predicive score for given parameer values (Geweke and Amisano, ; Creal e al., ), is hen given by L X T = TX log p X X,. (A.) = The boosrap filer described above generaes new paricles for iem merely by propagaing sochasic volailiies based on heir prior values, h (i) and he law of moion (A.9) bu wihou regard for he likelihood hey will arac a. The APF seeks o adap new paricle draws h (i) Since paricles ge reweighed a every sep, he simple average is appropriae, see, for example, Creal ().

42 SUPPLEMENTARY APPENDIX (online only) o daa a. To do so, we employ an algorihm described by Lopes and Tsay () for Rao- Blackwellized paricle filers ha performs wo resampling seps. Firs, before simulaing new paricles h (i) from equaion (A.9), ime paricles are resampled based on he paricle weighs implied a when using he auxiliary paricles h (i), = µ + h (i) µ. 5 Denoing he weighs P implied by he auxiliary paricles w,(i), weighs are hen given by w (i) = w (i) / i w(i) where w (i) = l (i) /w,(i) and, as in equaion (A.8), l (i) are he likelihood conribuions bu now generaed by he paricles obained from propagaing h (i) afer he auxiliary reweighing. Furher deails are provided by Lopes and Tsay (). VII. Adjusing he Paricle Filer o accoun for he ELB Neglecing he ELB, he Rao-Blackwellized paricle filer described above relies on he condiionally linear srucure of he UC-SV model so ha, for each paricle, poserior disribuions of he linear saes are normal and can be racked by following he evoluion of heir firs wo momens, (i) and (i). Indeed, for daa hisories where he ELB has no been binding, we have X = Z and can direcly employ he RB-PF described above. However, once he ELB binds, he shadow rae becomes a laen variable, and hus X 6= Z. The shadow rae is a linear combinaion of. Thus, o ensure ha shadow raes are below he ELB when he acual rae is a he ELB, he poserior for mus be characerized by a runcaed disribuion. To accoun for he ELB, we adap he RB-PF as follows: Since a conjugae prior-poserior characerizaion for is no available a he ELB, we abandon he Rao-Blackwellizaion once he ELB binds in he daa and proceed as follows. Insead, we add draws for (i) ha saisfy he runcaion consrain s = c s (i) apple ELB. Consider he case where he ELB has no been binding for Z o he paricle vecor bu binds for Z. In his case, for every paricle, we inheri normal priors for he linear saes characerized by (i) and (i). Analogously o he MCMC algorihm described in Appendix A, hese priors for can be updaed o NC,(i) and NC,(i) by condiioning only on he elemens of Z for which he ELB did no bind. 5 The auxiliary paricles are hus generaed a he means implied by he prior paricles.

43 SUPPLEMENTARY APPENDIX (online only) To accoun for he ELB binding a, we hen sample paricles (i) are consisen wih s (i) = c s (i) apple ELB. from N NC,(i), NC,(i) ha The RB-PF algorihm described in he previous secion above can hen simply be amended by replacing he priors for every paricle a +, originally denoed (i) +, (i) + by he draw (i)) as well as a marix of zeros before evaluaing he ime +Kalman filering seps described above. Effecively, he algorihm proceeds by using a degenerae normal poserior a for ha has all poin mass a (i). While he Rao-Blackwellized paricle filer racks a leas par of he uncerainy abou via paricle-specific condiional disribuions, uncerainy abou becomes enirely capured by he range of values for (i) generaed across paricles when he ELB binds. In addiion o abandoning he Rao-Blackwellizaion, he likelihood conribuions l (i) in (A.8) need o be augmened by adding he probabiliy of he shadow rae being below he ELB condiional on all oher elemens of Z, which is sraighforward o compue based on he normal CDF for s implied by NC,(i), and NC,(i). This add facor accouns for he informaion conained in observing i = ELB by iling paricle weighs owards paricles ha place higher likelihood on he ELB binding. Specifically, (A.8) is replaced by l (i) = log ( ) Nx +log (i) + +log + e (i) + (i) (i) e Prob s apple ELB NC,(i), NC,(i). (A.)

44 SUPPLEMENTARY APPENDIX (online only) VIII Consrucing IRF wih a Paricle Filer Neglecing issues relaed o he ELB, he impulse responses described in Secion 5 of he main paper, are sraighforward o compue from he Rao-Blackwellized paricle filer (RB-PF) described in Secion VII. above. In paricular, compuing responses o one-sandard deviaion shocks, simplifies he compuaion of paricle weighs for he impulse responses as described below. To accoun for he ELB, we employ a mean-variance approximaion o he UC-SV model s innovaions represenaion described in he remainder of his secion. VIII. Paricle Weighs used for Impulse Responses Impulse responses reflec he difference beween a baseline forecas and a revised forecas promped by he hypoheical observaion of an idenified shock. In he conex of he RB-PF, he arrival of new informaion generally promps also a reweighing of paricles reflecing changes in each paricles likelihood given he new daa. To simplify he IRF compuaion, we compue responses o one-sandard deviaion shocks. As such, hese shocks provide informaion abou he direcion of he impulse in he space of innovaions e bu do no lead o a reassessmen of he likelihood of differen values for he sochasic volailiies ha are racked by he differen paricles. As such, he arrival of such a sandardized shock does no affec a change in paricle weighs relaive o he baseline forecas. Paricle weighs for he baseline forecas made a ime, denoed w (i), are compued as described in Secion VII above. When compuing he revised forecas, hese weigh are simply propagaed as w (i) = w(i) and we can compue impulse responses as: IRF k = X i w (i) dm k, (i) wih d m k, (i) = CA k K (i) q m,(i). () Of course, he analysis could be exended o consider also he effecs from reweighing he differen paricles promped by observing shocks of a paricular absolue size. In his case, he impulse responses would reflec an addiional erm, w (i) w (i) E(X X ), ha capures he reevaluaion of he weigh aached o each paricle s baseline forecas in ligh of he incoming shock.

45 SUPPLEMENTARY APPENDIX (online only) 5 VIII. Accouning for he ELB The impulse responses are based on innovaions in he shadow rae and rely on he innovaions represenaion of he UC-SV model wih linear measuremens and Gaussian shocks. As indicaed in Secion 5.. of he main paper, when he ELB binds, we use a mean-variance approximaion of he innovaions represenaion. This approximaion differs from our handling of he ELB for he purpose of compuing he model s likelihood as described in Secion VII. for he MDD compuaion. There, we could simply abandon he Rao-Blackwellizaion a he ELB, albei a he cos of removing any uncerainy abou rend and gap levels per paricle, since each paricle condiions on a specific draw (i) before propagaing when a he ELB. Bu, ime-varying uncerainy abou rend and gap levels is a key driver for ime-variaion in he VMA resuling from he model s innovaions represenaion. Insead of replacing he Rao-Blackwellizaion wih draws of (i) obey s (i) = c s (i) apple ELB when he ELB binds, we have hus chosen o resor o he following mean-variance approximaion for he consrucion of IRFs: When he ELB binds a, we generae N NC,(i), hen approximae he runcaed normal NC,(i) ha as described in Secion VII.. We (i) c s (i) apple ELB,(i),(i) TN E Var NC,(i), (i) NC,(i) c s (i) c s (i) apple ELB (i) c s (i) apple ELB apple ELB (A.5) (A.6) (A.7) wih a normal disribuion N,(i),,(i) ha maches firs and second momens of he rue runcaed disribuion. Esimaes of he model s laen variables obained wih his approximaion are close o his obained from he paricle filer described in Secion VII where he ELB is enforced exacly. Mean and variance coefficiens in (A.6) and (A.7) can be compued analyically as follows: Firs, we derive he condiional disribuion of for given values of he shadow rae. Second, we