Description

Junior Maih
junior.maih@norges-bank.no
Dynare Workshop
Agenda
I The DSGE model with anticipated information
I Conditional forecasting
I Occasionally-binding constraints with anticipated shocks
The DSGE model with anticipated information
The generic problem
Ef (xt+1,xt,xt−1,ηt) = 0
I xt is the vector of endogenous variables
I f (.) is vector of potentially nonlinear functions of their arguments
I ηt is the vector of shocks with, without loss of generality,
ηt ∼ N(0,I)
|{z} nη×1
Solution with News shocks I
We modify the model to include ”news shocks”

The solution takes the form

Solution with News shocks II
without loss of generality, the problem and the solution can be re-written as
Ef (xt+1,xt,xt−1,ηt) = 0
xt = T (xt−1,ηt)
by defining

and
ηt ≡ εt
Solution under anticipated shocks
xt = T (xt−1,ηt,ηt+1,…,ηt+h) Finding the solution T amounts to solving the problem

or

We do not modify the model

I
Abstracting from constant terms the problem to solve is
A+Ext+1 + A0xt + A−xt−1 + Bηt = 0
and its (exact) solution is
xt = Txxt−1 + Tη,0ηt + Tη,1ηt+1+,…,+Tη,hηt+h
with special case
xt = Txxt−1 + Tη,0ηt
which applies when Etηt+s = 0 for s > 0
II
The general solution implies
0 =
III
or

I Hence the system can be solve in a recursive fashion.
I Once we have solved for Tx (using Sims, Klein, AIM, Uhlig, etc.) we can solve for Tη,0, then Tη,1, …, then Tη,h
IV
In particular,
Tη,0 = −
Tη,1 = −
Tη,2 = −
···

V
I The signs of the impacts alternate in powers of
I A discounting occurs if the eigenvalues of are inside the unit circle:
I future shocks have lower impact than current ones.
I Or alternatively, future shocks have to be big for them to matter for today’s decisions.

Conditional forecasting
I
Consider the system k periods ahead
xt+1 = Txxt + Tη,0ηt+1 + Tη,1ηt+2+,…,+Tη,hηt+1+h
xt+2 = Txxt+1 + Tη,0ηt+2 + Tη,1ηt+3+,…,+Tη,hηt+2+h
k h
xt+k = Txkxt + XXTxk−jTη,sηt+j+s
j=1 s=0
II
Stacking all future data

III
or
X = X¯ + Φη

if h = 0 IV
k
xt+k = Txkxt + XTxk−jTη,0ηt+j
j=1
 xt+1
 xt+2  …


xt+k where 
  =

  Tx   Φ1,0 0 ···
 Tx2   Φ2,0 Φ2,1
 … xt +  … … … 
Txk Φk,0 Φk,1 ··· 0
0
… 



 ηt+1 ηt+2
… 




Φk,k ηt+k+1
 Φ1,0
 Φ2,0  …


Φk,0 0
Φ2,1
…
Φk,1 ··· 0   Tη,0
0   TxTη,0
… …  =  … 0 ··· 0
Tη,0 0
… … … 




··· Φk,k
V
We have

I
Suppose we are given the restriction
DX ∼ TN(µ,Ω,[L,H])
I In practice, D is usually a selection matrix but it can also be a set of linear combinations on the rows of X.
I Maih (2010) develops algorithms for finding solutions with anticipated shocks in linear(ized) rational expectations models. I Juillard and Maih (2010) use these algorithms to estimate models with observed real-time expectations data. The restriction implies that

where rectangular matrix R is of size q × (h + k + 1)nη and of rank q ≤ mη ≡ (h + k + 1)nη
II
Consider the decomposition
η = M1γ1 + M2γ2 (2)
with

where
I M1 is chosen to be an orthonormal basis for the null space of
mη×(mη−q)
R
I and M2 either an orthonormal basis for the null space of or an
mη×q orthonormal basis for the column space of R0
III
Then

with RM2 invertible!!!

Hard conditions

and the shocks that are needed to satisfy the restrictions are
η = M1γ1 + M2γ2
Occasionally-binding constraints
with anticipated shocks
Principle
Apply conditional forecasting whenever a restriction is violated
Shock selection and the ZLB
Not all shocks need to have the same anticipation horizon e.g.
I the government can announce future taxes,
I the central bank can announce interest rates can remain low for an extended period of time

I
I The original forward-back shooting algorithm by Hebden et al. (2011) is based on news shocks.
II
I The steps for any particular simulation period t
1. Simulate the system
2. if the ZLB is not violated move to t+1, go to step 1
3. if the ZLB is violated let n =1
3.1 assume it is going to last n periods and find the anticipated monetary policy shocks that are necessary for keeping the interest rate at the ZLB for n periods
3.2 if period t+n−1 does not violate go to step 1 with t set to t+n−1
3.3 if t+n−1 violates, set n = n+1 and go to 3a
III
I At each step, the number of anticipated shocks is equal to the number of violations.
I In this particular case, M1 = 0 and M2 = I, i.e. there is a unique combination of shocks that satisfy the restrictions

Sign reversals and forward guidance
I Implicit in the max operator is the idea that interest rates cannot go below the floor when they should. This would translate into positive monetary policy shocks at the ZLB.
I Does Sign reversals imply forward guidance?
Comparison to OCCBIN
I The piece-wise linear strategy considers a sequence of time-varying matrices
I No special consideration for the explicit role of policy
I No possibility of modeling green shoots (e.g. staying longer at the ZLB)
I Poor handling of complementary slackness conditions (e.g. sign of lagrange multipliers)
I No (direct) possibility of multiple regimes (e.g. high, medium, low)
I It would be equivalent to using all shocks in the system to satisfy the restrictions without having a control over the exact combination of those shocks
Handling multiple constraints
I The theory for conditional forecasting applies to multiple constraints I Hence it is straightforward to apply it to multiple occasionally binding constraints.
References I
Hebden, J., Linde, J., and Svensson, L. E. O. (2011). Optimal Monetary Policy in the Hybrid New Keynesian Model under the Zero Lower Bound. Mimeo, Federal Reserve Board.
Juillard, M. and Maih, J. (2010). Estimating DSGE Models with Observed Real-Time Expectation Data.
Maih, J. (2010). Conditional forecasts in DSGE models. Working Paper 2010/07, Norges Bank.