Ideas para el futuro

At the beginning of iterationn we have: sp_{n 1}; ss_{n 1}; vo_{n 1}; ST

n 1; sp;T n 1; vo;T n 1; H sp n 1; Hnm 1;and Hvo n 1:

Step 1: Sampling the Markov-switching states sp;T_n and vo;T_n

Conditional on the DSGE parameters and on ST

n 1, we have a Markov-switching VAR with known hyperparameters:

St = T ( spt )St 1+R( spt ) t (15)

t N(0; Q( vot )); Q( vot ) =diag( vo( vot )) (16)

Hsp(; i) D(asp_ii; asp_ij); Hvo(; i) D(avo_ii; avo_ij) (17)

Therefore, for givenH_nsp₁ andHvo

n 1, Bayesian updating can be used to derive the …ltered probabilities of the di¤erent regimes. Then, the multimove Gibbs-sampling of Carter and Kohn (1994) can be used to draw sp;Tn and vo;Tn (see step 4 for a description of method).

Step 2: Sampling the transition matrices (Hsp

n and Hnvo)

Given the draws for the MS state variables sp;T_n and vo;T_n , the transition probabilities are independent ofST

n 1 and the other parameters of the model and have a Dirichlet distribution. For each column ofHsp

n and Hnvo the posterior distribution is given by

H_nsp(; i) D(a_iisp+ sp_ii; asp_ij + sp_ij)

H_nvo(; i) D(a_iivo+ vo_ii; avo_ij + er_ij)

where sp_ij and vo

ij denote respectively the numbers of transitions from state isp to state jsp

and from stateivo_{to statej}vo_{and a}sp

ii; a

sp

ij; avoii; avoij are the parameters describing the prior.

Step 3.a: Sampling the DSGE parameters ( n=f spn; von ; ssng)

Start drawing a new set of parameters from the proposal distribution:#sp_n N sp_{n 1}; csp sp _;

#vo_n N vo_{n 1}; cvo vo _{; #}oe

n N oen 1; coe

oe

(if a block optimization algorithm has been used to …nd the posterior mode) or vec(#) N n 1; c . Here is the inverse of the Hessian computed at the posterior mode and cis a scale factor. Ifn = 1, set n 1 = +c , where is the posterior mode estimate of the DSGE parameters. A Metropolis-Hastings algorithm is used to accept/reject #. Conditional on sp;T_n and vo;T_n there is no uncertainty

around the hyperparameters characterizing the state space form model:

yt = D( ss) +ZSt+vt (18)

St = T( spt )St 1+R( spt ) t (19)

t N(0; Q( vot )); Q( vot ) =diag( vo( vot )) (20)

vt N(0; U); U =diag 2y; 2; 2F (21)

Therefore, the Kalman …lter can be used to evaluate the conditional likelihood according to n 1, the old set of parameters, and #, the proposed set of parameters. Then the condi- tional likelihood is combined with the prior distributions of the DSGE parameters. Compute

cut=minf1; rg where

r = ` # sp ; #vo; #ssjYT<sub>;</sub> sp;T n ; vo;T n ; ::: p(# sp ; #vo; #ss) ` sp_{n 1}; vo_{n 1}; ss_{n 1}jYT_; sp;T n ; vo;Tn ; ::: p sp n 1; von 1; ssn 1

Draw a random numberdfrom an uniform distribution de…ned over the interval [0;1]. If

d < r; ( sp_n; ss_n; vo_n) = (#sp; #vo; #ss), otherwise set ( sp_n; ss_n; vo_n) = sp_{n 1}; ss_{n 1}; vo_{n 1} .

Step 3.b: Sampling the transition matrix used by agents Hm

n

Start drawing a new set of values for the columns of Hm _{using a Dirichlet distribu-}

tion: Hem_{(; i) D(b}m

ii;n 1; bmij;n 1), where bmii;n 1 and bmii;n 1 depend on the columns of Hnm1. This step de…nes the transition probability q Hem_jHm

n 1 . Then, use a Metropolis-Hastings algorithm to accept/rejectHem_{. Compute cut=minf1; rg where}

r=

` Hem_jYT_;

n; sp;Tn ; ::: p Hem q Hnm1jHem

` Hm

n 1jYT; n; sp;Tn ; ::: p Hnm1 q HemjHnm1

Draw a random numberdfrom an uniform distribution de…ned over the interval [0;1]. If

d < r; Hm

n =Hem, otherwise set Hnm =Hnm1.

Step 4: Sampling the DSGE state vector ST

n

For a given set of DSGE parameters and MS states, (18)-(21) form a state-space model with known hyperparameters. Step 3 returns a …ltered estimate of the state variable: ST

njYT.

The multimove Gibbs-sampling of Carter and Kohn (1994) can be used to draw the whole vector of ST n. Note that: p S_nTjYT =p ST;njYT T_Y1 t=1 p StjSt+1; YT

Therefore, the whole vector ST

njYT can be obtained drawing ST;n from p ST;njYT and

then using a backward algorithm to drawSt;n,t= 1:::T 1. Note that the state space model

(18)-(21) is linear and Gaussian. It follows that:

ST;njYT N ST;njT; PT;njT StjYT; St+1 N St;njt;St+1;; Pt;njt;St+1 where ST;njT = E ST;njYT (22) PT;njT = Cov ST;njYT (23) St;njt;St+1 = E StjY T_{; S} t+1 (24) Pt;njt;St+1 = Cov StjY T_{; S} t+1 (25)

Step 3 returns ST;njT and PT;njT , while St;njt;St+1 and Pt;njt;St+1 can be obtained updat-

ing the estimate of St;n combining St;njt, the …ltered estimate from step 3, with the new

information contained in St+1;n. See Kim and Nelson (1999) for further details.

Step 5

If n < nsim, go back to 1, otherwise stop, wherensim is the desired number of iterations.

Step 1, step 2 and step 3.b when Hm _=Hsp _=Hsp;m

In this case we cannot draw Hsp

n simply counting the number of transitions across the

MS states, because a change in the transition matrix implies also a change in the law of motion of the DSGE states. Instead, we can apply a Metropolis-Hastings algorithm treating

ST

n 1 as observed data and using the Hamilton …lter to evaluate the likelihood. In this case, de…ne cut=minf1; rg where

r = ` Hesp;m<sub>jS</sub>T n 1; n 1; ::: p Hesp;m q Hnsp;m1 jHesp;m ` H_nsp;m₁jST n 1; n 1; ::: p Hnsp;m1 q Hesp;mjH sp;m n 1

As a side product, we obtain …ltered estimates for the MS states and we can use them to draw sp;T_n and vo;T_n with the usual backward drawing algorithm. Finally, Hvo _{can be drawn}

jointly with Hsp;m _{or according to the standard procedure described above, given that the}