Solving and Estimating Models of Financial Crises: An...
-
Upload
nguyenthuan -
Category
Documents
-
view
220 -
download
3
Transcript of Solving and Estimating Models of Financial Crises: An...
Solving and Estimating Models of Financial Crises: AnEndogenous Regime Switching Approach
Gianluca Benigno1 Andrew Foerster2 Christopher Otrok3
Alessandro Rebucci4
1London School of Economics
2Federal Reserve Bank of Kansas City
3University of MissouriFederal Reserve Bank of St Louis
4Johns Hopkins University
BFOR Endogenous Switching 1 / 59
Introduction
Motivation
Global financial crisis proved very costly to resolve
Long history of financial crises in emerging marketsI Strong and volatile capital flows in and out of emerging economies
BFOR Endogenous Switching 2 / 59
Introduction
Motivation (Cont.): Recent Theoretical Issues
Large debate on the role of policy for financial stabilityI Consensus before the crisis: intervene only during crises (e.g., Bailouts)I Current consensus: intervene before crises to the extent possible (e.g.,
Macro-prudential policies to alleviate borrowing externalities)
Key question from the theoretical literature:I When should policy makers intervene?I Which policy tools should they use?
BFOR Endogenous Switching 3 / 59
Introduction
Benigno, Chen, Otrok, Rebucci, Young, 2016
Ex ante and ex post policies interact and depend on the set of toolsavailable
If ex post interventions are effective, no need for ex ante ones:I With sufficient instruments (two in their model), only use ex post
interventionsI With insufficient instruments, both ex ante and ex post interventions
are necessary:
These are theoretical results in small models that highlight theimportance of collateral constraints but abstract away from theshocks that drive crises and richer propagation mechanisms. So theseare not yet implementable policy prescriptions.
BFOR Endogenous Switching 4 / 59
Introduction
Missing piece in this literature
Quantitative analysis of financial crises in estimated modelsI Which shocks drive crises? Are they the same that drive normal cycles?I Is there time variation in the importance of those shocks?I How do the dynamic responses to shocks change when collateral
constraints bind?
One can then return to the theoretical questions of when shouldpolicy makers intervene and with which instruments?
I Does it matter what shocks drive crises?I Which instruments best address which shocks?
BFOR Endogenous Switching 5 / 59
Introduction
Pre-Crisis and Post-Crisis Consensus on Methodology
Pre-crisis: Medium scale estimated linear DSGE modelsI Estimate importance of shocks and frictionsI Analyze policy questions in this fully specified empirical frameworkI Non-linearity restricted to 2nd order solution of model
Post-Crisis: Events studies with quantitative models featuringnon-linear dynamics
I Non-linearity often in the form of occasionally binding borrowingconstraints
This paper bridges the two approaches by providing an empiricalframework that allows for estimation of shocks and frictions while atthe same time incorporating the nonlinearities identified in the crisisliterature
BFOR Endogenous Switching 6 / 59
Introduction
Pre-Crisis and Post-Crisis Consensus on Methodology
Pre-crisis: Medium scale estimated linear DSGE modelsI Estimate importance of shocks and frictionsI Analyze policy questions in this fully specified empirical frameworkI Non-linearity restricted to 2nd order solution of model
Post-Crisis: Events studies with quantitative models featuringnon-linear dynamics
I Non-linearity often in the form of occasionally binding borrowingconstraints
This paper bridges the two approaches by providing an empiricalframework that allows for estimation of shocks and frictions while atthe same time incorporating the nonlinearities identified in the crisisliterature
BFOR Endogenous Switching 6 / 59
Introduction
Pre-Crisis and Post-Crisis Consensus on Methodology
Pre-crisis: Medium scale estimated linear DSGE modelsI Estimate importance of shocks and frictionsI Analyze policy questions in this fully specified empirical frameworkI Non-linearity restricted to 2nd order solution of model
Post-Crisis: Events studies with quantitative models featuringnon-linear dynamics
I Non-linearity often in the form of occasionally binding borrowingconstraints
This paper bridges the two approaches by providing an empiricalframework that allows for estimation of shocks and frictions while atthe same time incorporating the nonlinearities identified in the crisisliterature
BFOR Endogenous Switching 6 / 59
Introduction
Overview
New approach to specifying, solving and estimating models offinancial crises
I Financial crises are rare but large events → model must be non-linearI Non-linearity poses computational problemsI We provide a tractable formulation of collateral constraint and then
develop methods to solve and estimate them
We set up a model with a Kiyotaki-Moore type of collateral constraint
I The constraint limits total debt to a fraction of the market value ofphysical capital (it is a limit on leverage)
I Constraint imposed on the agents as in Kiyotaki and Moore (1997),Aiyagari and Gertler (1999), Kocherlakota (2000) and Mendoza (2010)
I Constraint is not derived from an optimal contract, but is motivated bythe optimal contracting literature
BFOR Endogenous Switching 7 / 59
Introduction
Overview
New approach to specifying, solving and estimating models offinancial crises
I Financial crises are rare but large events → model must be non-linearI Non-linearity poses computational problemsI We provide a tractable formulation of collateral constraint and then
develop methods to solve and estimate them
We set up a model with a Kiyotaki-Moore type of collateral constraint
I The constraint limits total debt to a fraction of the market value ofphysical capital (it is a limit on leverage)
I Constraint imposed on the agents as in Kiyotaki and Moore (1997),Aiyagari and Gertler (1999), Kocherlakota (2000) and Mendoza (2010)
I Constraint is not derived from an optimal contract, but is motivated bythe optimal contracting literature
BFOR Endogenous Switching 7 / 59
Introduction
Overview–Cont.
We propose a new specification of such a collateral constraintI We model the movement from unconstrained state of the world to
constrained state as a stochastic function of the LTV ratio (orcollateral constraint)
F We can then write constraint as a regime switching processF One regime in which the constraint binds (a crisis regime)F One regime in which it does not bind (normal regime)
I Probability of the collateral constraint binds rises with leverageF This captures the fact that the likelihood of a crisis raises with
leverage, without requiring a crisis to occurF Agents in the model know that higher leverage levels (and lower
collateral values) increase the probability of a financial crisis
BFOR Endogenous Switching 8 / 59
Introduction
Overview–cont.
Our constraint specification can be represented as an endogenousregime switching model
We develop a solution method for endogenous regime switchingmodels
I The solution will be an approximate solution around a steady statewhich is the average of the deterministic steady state in the tworegimes, weighting with ergodic probabilities
Some features of our solution methodI We solve with perturbation methods: we can handle multiple state
variables and many shocksI Second order approximation: Capture impact of risk on decision rulesI Fast solution method → non-linear filters can be used to calculate the
likelihood functionI Structural model allows us to perform policy counterfactuals (future
work)
BFOR Endogenous Switching 9 / 59
Introduction Related Literature
DSGE Models with Occasionally Binding Collateral Constraints
Now large literature on DSGE models with occasionally bindingborrowing constraints
I These are models with endogenous financial crises nested in regularbusiness cycles
I Mendoza (2010) uses it for positive analysis of cycles and crisis in atypical emerging market economy (Mexico)
The collateral constraint gives rise to amplification of regular businesscycle and pecuniary externalities (Lorenzoni, 2008; Korinek, 2010;Bianchi, 2011; Benigno et al, 2013 and 2016, Jeanne and Korinek2011, Schmitt-Grohe and Uribe 2015)
A large subsequent literature uses variants of these models to asknormative questions
BFOR Endogenous Switching 10 / 59
Introduction Outline
Outline of Talk
Model
Collateral Constraint Formulation
Mapping into Regime Switching
Solution
Properties of Model Solution
Estimation Procedure
Empirical Results
BFOR Endogenous Switching 11 / 59
Model
Structure and Utility
Small open economyI Bianchi and Mendoza (2015) and Huo and Rios-Rull (2016) apply this
setup even to the United States
Model very similar to Mendoza (2010), but different specification ofthe borrowing constraint and set of shocks considered
Consumer Utility
U ≡ E0
∞∑t=0
{βt
1
1− ρ
(Ct −
Hωt
ω
)1−ρ}
GHH preferences
BFOR Endogenous Switching 12 / 59
Model
Production and Budget Budget
Production uses capital, labor and imported intermediate goods
Yt = AtKηt−1H
αt V
1−α−ηt
There is a working capital requirement
φrt (WtHt + PtVt)
Investment subject to adjustment costs
It = δKt−1 + (Kt − Kt−1)
(1 +
ι
2
(Kt − Kt−1
Kt−1
))Budget constraint
Ct + It = Yt − PtVt − φrt (WtHt + PtVt)−1
(1 + rt)Bt + Bt−1
Bt < 0 denotes the debt position at the end of period t
BFOR Endogenous Switching 13 / 59
Model Collateral Constraint
Collateral Constraint
The agent faces a regime specific collateral constraint
In regime 1 (the crisis regime) the constraint binds, and totalborrowing is equal to a fraction of the value of collateral:
1
(1 + rt)Bt − φ (1 + rt) (WtHt + PtVt) = −κtqtKt
I Both debt and and working capital are restrictedI Collateral in the model is defined over the value of capitalI The price and quantity of collateral are endogenous, since the price is
in the constraint the pecuniary eternality emphasized in the literature ispresent
BFOR Endogenous Switching 14 / 59
Model Collateral Constraint
Collateral Constraint
In regime 0 (the normal regime) the constraint is slack, and collateralvalue is sufficient for international lenders to finance all desiredborrowing
I There is no explicit constraint on borrowing in this regimeI A small debt elastic interest rate premium prevents infinite debt
The likelihood of a borrowing constraint binding in the future dependson both endogenous choices for debt and working capital borrowing:
1
(1 + rt)Bt − φ (1 + rt) (WtHt + PtVt)
The likelihood also depends on the endogenous choice for capital andthe endogenous price of capital: −κqtKt
The fact that high debt today leads to a future binding borrowingconstraint also limits borrowing
BFOR Endogenous Switching 15 / 59
Model Collateral Constraint
Collateral Constraint
Define a new variable (the ”borrowing cushion”) which measures thedistance between debt and the value of collateral in the constraint
B∗t =1
(1 + rt)Bt − φ (1 + rt) (WtHt + PtVt) + κtqtKt
When the borrowing cushion is small the leverage ratio is high
In regime 0 (non-binding) the probability that constraint binds thenext period depends the borrowing cushion
Pr (st+1 = 1|st = 0) =exp (−γ0,1B∗t )
1 + exp (−γ0,1B∗t )
The logistic function reformulates the Kiyotaki-Moore idea thatincreased leverage leads to binding collateral constraints as aprobabilistic statement
The transition probability from regime 0 to regime 1 is a function ofall endogenous variables in B∗t
BFOR Endogenous Switching 16 / 59
Model Collateral Constraint
Collateral Constraint
Define a new variable (the ”borrowing cushion”) which measures thedistance between debt and the value of collateral in the constraint
B∗t =1
(1 + rt)Bt − φ (1 + rt) (WtHt + PtVt) + κtqtKt
When the borrowing cushion is small the leverage ratio is high
In regime 0 (non-binding) the probability that constraint binds thenext period depends the borrowing cushion
Pr (st+1 = 1|st = 0) =exp (−γ0,1B∗t )
1 + exp (−γ0,1B∗t )
The logistic function reformulates the Kiyotaki-Moore idea thatincreased leverage leads to binding collateral constraints as aprobabilistic statement
The transition probability from regime 0 to regime 1 is a function ofall endogenous variables in B∗t
BFOR Endogenous Switching 16 / 59
Model Collateral Constraint
What is the key difference with respect to the the literature?
The typical specification of the constraint in this class of models is:
1
(1 + rt)Bt − φ (1 + rt) (WtHt + PtVt) ≥ κtqtKt
When the left hand side is greater than the right (B∗t > 0 in ournotation) the constraint is slack
When the left hand side is exactly equal to the right (B∗t = 0 in ournotation) the constraint binds
Our assumptions turn the deterministic relationship betweenborrowing and collateral into a stochastic one
High leverage leads to a crisis, but with some uncertainty rather thanin a deterministic manner at a given and fixed LTV ratio
BFOR Endogenous Switching 17 / 59
Model Collateral Constraint
As parameter γ0,1 gets large we recover the deterministic step function of the
literature as a special case
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
1
B*t
Prob(st+1
=0|st=0,B*
t)
γ = 1000γ = 100
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
1
Prob(st+1
=1|st=1,λ
t)
λt
BFOR Endogenous Switching 18 / 59
Model Collateral Constraint
Empirical motivation for our formulation
Borrowing constraints don’t bind at any particular LTV ratio in thereal world, they are are stochastic functions of LTV ratios
When a borrower hits the LTV limit, expenditure is adjusted graduallybecause other source of financing such as cash, precautionary creditlines, asset sales, etc. can be tapped into
I Capello, Graham, and Harvey (JFE, 2010) Survey information onbehavior of financially constrained firms
I Ivashina and Scharfstein (JFE, 2010) loan level data show creditorigination dropped during the crisis because firms drew down frompre-existing credit lines in order to satisfy their liquidity
BFOR Endogenous Switching 19 / 59
Model Collateral Constraint
A Theoretical motivation for our formulation
Take the standard borrowing constraint in the literature and add astochastic monitoring (or enforcement) shock εMt .
1
(1 + rt)Bt − φ (1 + rt) (WtHt + PtVt) ≥ −κtqtKt + εMt
Shock has two interpretations, based on the sign of the shockI Negative shock: The LHS is then greater than the value of collateral
but the lender monitors and decides to impose a borrowing constraintI Positive shock: The LHS is then less than the value of collateral but
the constraint does not bind because the lender does not audit
Distribution of εMt is such that when borrowing is much less than thevalue of collateral the probability of drawing a monitoring shock thatleads to a binding constraint is 0. When borrowing exceeds the valueof collateral by a large amount the probability of drawing a monitoringshock is such that the probability the lender audits goes to 1.
BFOR Endogenous Switching 20 / 59
Model Collateral Constraint
Collateral Constraint
In the binding Regime 1 the Lagrange multiplier λt , associated withthe constraint is strictly positive. The transition probability to goback to regime 0 is given by
Pr (st+1 = 0|st = 1) =exp (−γ1,1λt)
1 + exp (−γ1,1λt)
As the multiplier approaches 0, the probability of transitioning backto the non-binding state rises
BFOR Endogenous Switching 21 / 59
Model Shocks
Shocks
There are four sources of shocks: world interest rate, leverage,productivity, and terms of trade
Interest rate process has a small risk premium component and astochastic shock
rt = r∗ + ψr
(eB−Bt − 1
)+ σwεw ,t
The TFP and TOT processes:
logAt = (1− ρA)a (st) + ρA (st) logAt−1 + σA (st) εA,t
logPt = (1− ρp)p (st) + ρP (st) logPt−1 + σP (st) εP,t
I The means and persistence of these shocks are regime dependent
BFOR Endogenous Switching 22 / 59
Model Shocks
Leverage Shocks
Restrictions on leverage are stochastic, and may depend on regime
Binding regime
1
(1 + rt)Bt − φ (1 + rt) (WtHt + PtVt) = −κtqtKt
Nonbinding regime
B∗t =1
(1 + rt)Bt − φ (1 + rt) (WtHt + PtVt) + κtqtKt
Whereκt = κ (st) + ρκ (st)κt + σκ (st) εκ,t
Questions: What is the role of leverage shocks as a cause of crises?Do they play different roles in and out of the crisis?
BFOR Endogenous Switching 23 / 59
Solution
Solution
The FOCs, constraints and shocks yield 15 equilibrium conditionsI This is the full set of structural equations of the model
The model as written is a nonlinear model similar to the literatureand can in principle be solved with global solution methods
We compute an approximate solution by solving the model around asteady state
I This steady state is between the steady states associated with the tworegimes
I The perturbation solution includes a term that corrects for the fact thatthe switching model is either above or below this approximation point
BFOR Endogenous Switching 24 / 59
Solution
Markov Switching DSGE Papers with Exogenous Switching
Large literature on Markov-switching linear rational expectationsmodels (MSLREs) with exogenous switching
I Leeper and Zha (2003), Davig and Leeper (2007), Farmer, Waggoner,and Zha (2009)
I A MSLRE approach allows for nonlinearity in the form of discretechanges in model parameters across regimes
I MSLREs provide a tractable way to model how agents formexpectations over discrete changes in the economy
Foerster et al (2016) developed a perturbation method forconstructing first and second order solution of exogenousMarkov-switching DSGE models (MSDSGE)
I A key innovation in their paper is to work with the original MSDSGEmodel directly
BFOR Endogenous Switching 25 / 59
Solution
Related Markov Switching DSGE literature
Small literature on endogenous switching DSGE modelsI Davig and Leeper (2006)I Barthelemy and Marx (2016)I Lind (2014)
BFOR Endogenous Switching 26 / 59
Solution Regime Switching
Regime Switching: Approximation
We introduce two additional variables that allow regime switching toaffect both the slope and intercept of the decision rules:
I The variables ϕ (st) = γ (st) = st turn ”on” or ”off” the collateralconstraint, depending on regime
The borrowing constraint for the two regimes can then be written:
ϕ (st)B∗ss+γ (st) (B∗t − B∗ss) = (1− ϕ (st))λss+(1− γ (st)) (λt − λss)
Where SS denotes a steady stateWith this function:
I When st = 0, then ϕ (0) = γ (0) = 0 and the equation simplifies to
λt = 0
I When st = 1, then ϕ (1) = γ (1) = 1 and the equation simplifies to
B∗t = 0
BFOR Endogenous Switching 27 / 59
Solution Regime Switching
Regime Switching: Approximation
Approximate constraint again:
ϕ (st)B∗ss+γ (st) (B∗t − B∗ss) = (1− ϕ (st))λss+(1− γ (st)) (λt − λss)
This equation also pins down the correct steady state valuesI Following FRWZ (2014) only the switching variable ϕ (st) is perturbedI The steady state slackness condition then satisfies
ϕB∗ss = (1− ϕ)λss
I ϕ is the ergodic mean of ϕ (st)I If only the non-binding regime occurs, then ϕ = 0 and
λss = 0
I If only the binding regime occurs then ϕ = 1 and
B∗ss = 0
BFOR Endogenous Switching 28 / 59
Solution Regime Switching
Regime Switching: Endogenous Probabilities
The transition matrix is then
Pt =
[p00,t p01,tp10,t p11,t
]=
[1− exp(−γ0,1B∗t )
1+exp(−γ0,1B∗t )exp(−γ0,1B∗t )
1+exp(−γ0,1B∗t )exp(−γ1,1λt)
1+exp(−γ1,1λt) 1− exp(−γ1,1λt)1+exp(−γ1,1λt)
]
BFOR Endogenous Switching 29 / 59
Solution Equilibrium
Regime Switching: Equilibrium
The model has 15 equilibrium conditions, in vector form:
Et f (yt+1, yt , xt , xt−1, χεt+1, εt , θt+1, θt) = 0
Variables are separated into the predetermined variables xt−1 andnon-predetermined variables ytPredetermined variables:
xt−1 = [Kt−1,Bt−1, κt−1,At−1,Pt−1]
Non-predetermined variables:
yt = [Ct ,Ht ,Vt , It , kt , rt , qt ,Wt , µt , λt ,B∗t ]
4 shocks:εt = [εκ,t , εw ,t , εA,t , εP,t ]
BFOR Endogenous Switching 30 / 59
Solution Equilibrium
Regime Switching: Equilibrium
To solve the model we use/assume the the following functional forms:
θ1,t+1 = θ1 + χθ1 (st+1)
θ1,t = θ1 + χθ1 (st)
θ2,t+1 = θ2 (st+1)
θ2,t = θ2 (st)
xt = hst (xt−1, εt , χ)
yt = gst (xt−1, εt , χ)
yt+1 = gst+1 (xt , χεt+1, χ)
pst ,st+1,t = πst ,st+1 (yt)
BFOR Endogenous Switching 31 / 59
Solution Equilibrium
Regime Switching: Equilibrium
Substituting these in the model’s equilibrium conditions, and given(xt−1, εt , χ) and st , we have:
0 = Fst (xt−1, εt , χ) =
∫ 1∑s′=0
πst ,s′ (gst (xt−1, εt , χ)) f
gst+1 (hst (xt−1, εt , χ) , χε′, χ) ,
gst (xt−1, εt , χ) ,hst (xt−1, εt , χ) ,
xt−1, χε′, εt ,
θ + χθ (s ′) , θ + χθ (st)
dµε′
Stacking these conditions for each regime gives
F (xt−1, εt , χ) =
[Fst=1 (xt−1, εt , χ)Fst=2 (xt−1, εt , χ)
]BFOR Endogenous Switching 32 / 59
Solution Steady State
Regime Switching: Steady State
The switching in parameters ϕ (st), a (st), κ (st), and p (st) all affectthe steady state of the economy and hence its level
In steady state, the transition matrix satisfies
Pss =
[1− exp(−γ0,1B∗ss)
1+exp(−γ0,1B∗ss)exp(−γ0,1B∗ss)
1+exp(−γ0,1B∗ss)exp(−γ1,1λss)
1+exp(−γ1,1λss) 1− exp(−γ1,1λss)1+exp(−γ1,1λss)
]
Let ξ = [ξ0, ξ1] denote the ergodic vector of Pss . Then define theergodic means of the switching parameters as
ϕ = ξ0ϕ (0) + ξ1ϕ (1)
a = ξ0a (0) + ξ1a (1)
p = ξ0p (0) + ξ1p (1)
κ = ξ0κ (0) + ξ1κ (1)
BFOR Endogenous Switching 33 / 59
Solution Steady State
Regime Switching: Steady State
The steady state of the economy depends on these ergodic means,and satisfies the steady state versions of the 15 model equationsabove
Our solution method is a perturbation method around a steady stateof the model
BFOR Endogenous Switching 34 / 59
Solution Perturbation
Regime Switching: Perturbation Solution
The perturbation solution takes the stacked equilibrium conditionsF (xt−1, εt , χ), and differentiate with respect to (xt−1, εt , χ)
The first-order derivative with respect to xt−1 produces a complicatedpolynomial system denoted
Fx (xss , 0, 0) = 0
We solve this polynomial system by finding a fixed point of asequence of eigenvalue problems
This procedure finds a single solution, but does not guaranteeuniqueness
BFOR Endogenous Switching 35 / 59
Solution Perturbation
Regime Switching: Perturbation Solution
After solving the eigenvalue problem, the other systems to solve are
Fε (xss , 0, 0) = 0
Fχ (xss , 0, 0) = 0
If desired second order systems of the form can also be solved
Fi,j (xss , 0, 0) = 0, i, j ∈{x, ε,χ}
The decision rules have the form
xt = hst (xt−1, εt , χ)
yt = gst (xt−1, εt , χ)
BFOR Endogenous Switching 36 / 59
Solution Perturbation
Regime Switching: Second Order Solution
The second-order approximation takes the form
xt ≈ xss + H(1)st St +
1
2H
(2)st (St ⊗ St)
yt ≈ yss + G(1)st St +
1
2G
(2)st (St ⊗ St)
where St =[
(xt−1 − xss)′ ε′t 1]′
Second order solution is critical for endogenous switching model:I We show that the first order solution of the endogenous switching is
identical to the first order solution of an exogenous switching modelI Interpretation: precautionary behavior in the second order solution is
critical for endogenous switching to matter
BFOR Endogenous Switching 37 / 59
Properties of Solution
Solution Results
IRF to shocks for different parameterizations of the model
We compute solution for endogenous switching, exogenous switchingand no switching
I Does endogenous switching matter?
IRF is computed assuming we stay in the state (binding or not)
BFOR Endogenous Switching 38 / 59
Properties of Solution
Probability Functions and IRF to TFP shock
0 10 20−0.01
−0.005
0K
0 10 200
0.005
0.01B (level)
0 10 20−2
0
2x 10
−17 P
0 10 20−0.02
−0.01
0A
0 10 20−0.02
−0.01
0C
0 10 20−0.01
−0.005
0H
0 10 20−0.02
−0.01
0V
0 10 20−0.1
−0.05
0I
0 10 20−0.01
−0.005
0k
0 10 20
−4
−2
0x 10
−4 r (pp)
0 10 20−5
0
5x 10
−3 Q
0 10 20−0.01
−0.005
0W
0 10 200
0.01
0.02µ
0 10 20−1
0
1λ (level)
0 10 20−0.02
−0.01
0B* (level)
0 10 20−0.02
−0.01
0Y
0 10 20−0.02
−0.01
0B/Y (level)
Steeper Probability (γ = 1000)Flatter Probability (γ = 100)
0 10 20−0.01
−0.005
0K
0 10 200
0.005
0.01B (level)
0 10 20−2
0
2x 10
−17 P
0 10 20−0.02
−0.01
0A
0 10 20−0.02
−0.01
0C
0 10 20−0.01
−0.005
0H
0 10 20−0.02
−0.01
0V
0 10 20−0.1
−0.05
0I
0 10 20−0.01
−0.005
0k
0 10 20
−4
−2
0x 10
−4 r (pp)
0 10 20−5
0
5x 10
−3 Q
0 10 20−0.01
−0.005
0W
0 10 200
0.01
0.02µ
0 10 20−1
0
1λ (level)
0 10 20−0.02
−0.01
0B* (level)
0 10 20−0.02
−0.01
0Y
0 10 20−0.02
−0.01
0B/Y (level)
Steeper Probability (γ = 1000)Flatter Probability (γ = 100)
0 10 20−0.01
−0.005
0K
0 10 200
0.005
0.01B (level)
0 10 20−2
0
2x 10
−17 P
0 10 20−0.02
−0.01
0A
0 10 20−0.02
−0.01
0C
0 10 20−0.01
−0.005
0H
0 10 20−0.02
−0.01
0V
0 10 20−0.1
−0.05
0I
0 10 20−0.01
−0.005
0k
0 10 20
−4
−2
0x 10
−4 r (pp)
0 10 20−5
0
5x 10
−3 Q
0 10 20−0.01
−0.005
0W
0 10 200
0.01
0.02µ
0 10 20−1
0
1λ (level)
0 10 20−0.02
−0.01
0B* (level)
0 10 20−0.02
−0.01
0Y
0 10 20−0.02
−0.01
0B/Y (level)
Steeper Probability (γ = 1000)Flatter Probability (γ = 100)
BFOR Endogenous Switching 39 / 59
Properties of Solution
IRF in Non-binding state: large versus small crisis
0 10 20−0.01
−0.005
0K
0 10 20−0.05
0
0.05B (level)
0 10 20−5
0
5x 10
−17 P
0 10 20−0.02
−0.01
0A
0 10 20−0.02
−0.01
0C
0 10 20−0.01
−0.005
0H
0 10 20−0.02
−0.01
0V
0 10 20−0.1
−0.05
0I
0 10 20−0.01
−0.005
0k
0 10 20−2
0
2x 10
−3 r (pp)
0 10 20−5
0
5x 10
−3 Q
0 10 20−0.01
−0.005
0W
0 10 200
0.01
0.02µ
0 10 20−1
0
1λ (level)
0 10 20−0.04
−0.02
0B* (level)
0 10 20−0.02
−0.01
0Y
0 10 20−0.02
−0.01
0B/Y (level)
Smaller CrisisLarger Crisis
BFOR Endogenous Switching 40 / 59
Properties of Solution
IRF in Non-binding state: large versus small crisis (1st order solution)
0 10 20−0.01
−0.005
0K
0 10 200
0.01
0.02B (level)
0 10 200
2
4x 10
−18 P
0 10 20−0.02
−0.01
0A
0 10 20−0.02
−0.01
0C
0 10 20−0.01
−0.005
0H
0 10 20−0.02
−0.01
0V
0 10 20−0.1
−0.05
0I
0 10 20−0.01
−0.005
0k
0 10 20−1
−0.5
0x 10
−3 r (pp)
0 10 20−5
0
5x 10
−3 Q
0 10 20−0.01
−0.005
0W
0 10 200
0.01
0.02µ
0 10 20−1
0
1λ (level)
0 10 20−4
−2
0x 10
−3 B* (level)
0 10 20−0.02
−0.01
0Y
0 10 20−0.02
0
0.02B/Y (level)
Smaller CrisisLarger Crisis
BFOR Endogenous Switching 41 / 59
Properties of Solution
Impulse Response Function in Binding state to technology shock
0 10 200
0.5
1K
0 10 20−2
0
2B (level)
0 10 20−1
0
1P
0 10 200
0.5
1A
0 10 200
1
2C
0 10 200
0.5
1H
0 10 200
1
2V
0 10 20−10
0
10I
0 10 200
0.5
1k
0 10 20−0.1
0
0.1r (pp)
0 10 20−0.5
0
0.5Q
0 10 200
0.5
1W
0 10 20−4
−2
0µ
0 10 20−0.5
0
0.5λ (level)
0 10 20−1
0
1B* (level)
0 10 200
0.5
1Y
0 10 20−0.2
0
0.2B/Y (level)
EndogenousExogenousNo Switching
BFOR Endogenous Switching 42 / 59
Properties of Solution
Calibrated Model
The model matches the ergodic means as in Mendoza
The model can also get second moments to match the data as inMendoza
BFOR Endogenous Switching 43 / 59
Estimation
Estimation: Bayesian Full Information Likelihood Methods
Bianchi (2014) uses Gibbs sampler to estimate MSLRE with first ordersolution of New Keynesian model with exogenous probability switches
I We use second order solution with endogenous probabilities
Bocola (2015) estimates sovereign default model using globalsolutions and nonlinear filter in two step procedure
I Step 1: Estimate non linear model assuming no defaultI Step 2: Estimate exogenous default probability conditional on step 1
model estimate
Aruoba, Cuba-Borda, Schorfheide (2014) estimate a ZLB model withsunspot switches between equilibria
I Step 1: Estimate model with 2nd order solution/particle filter fornon-zlb model (using non-zlb data period)
I Step 2: Estimate sunspot switches with a sequential monte carlo filterconditional on step 1 estimates
BFOR Endogenous Switching 44 / 59
Estimation
Estimation: Bayesian Full Information Likelihood Methods
We cannot assume that the parameters in one regime are independentof parameters of the other regime → two step procedures areinappropriate
I Agents in the model fully understand that a crisis may occur, andadjust their behavior accordingly
I Our estimated model is useful for normative analysis precisely becauseof this feature of the model solution/estimation
We need a procedure for simultaneous estimation of regime switchand parameters in each regime
Second order solution needed
BFOR Endogenous Switching 45 / 59
Estimation
Estimation: Bayesian Full Information Likelihood Methods
We use a Metropolis-in-Gibbs Sampling procedureI Conditional on regimes, draw parameters using the standard MH
algorithmF Given parameters, regimes, data, the value of the likelihood function is
computed with a Sigma Point FilterF The Sigma Point Filter is used in conjunction with the second order
solution of the modelF The value of the posterior is then computed after evaluating priors
I Conditional on parameters, data, draw regimes
BFOR Endogenous Switching 46 / 59
Estimation Results
Estimation from 1981.Q1 to 2016.Q1
Real GDP, Investment growth
Interest rate: (EMBI Global + world interest rate (3month T-Bill rate- US expected inflation)).
Trade Balance / Output
BFOR Endogenous Switching 47 / 59
Estimation Results
Basic Calibration
Table: Basic Calibration
Parameter Calibrated Value
Discount Factor β 0.97959Risk Aversion ρ 2Labor Share α 0.592
Capital Share η 0.306Wage Elasticity of Labor Supply ω 1.846
Capital Depreciation δ 0.022766
Debt to Output Ratio BY -0.86
Interest Rate Elasticity ψr 0.001Mean of TFP Process, Normal Regime a(0) 0
Mean of Import Price Process, Normal Regime p(0) 0Mean of Leverage Process, Normal Regime κ(0) 0.15
BFOR Endogenous Switching 48 / 59
Estimation Results
Prior and Posterior: Preliminary Estimation Results
Table: Estimation Results
Parameter Prior Posterior mean q5 q95σw (0) Uniform(0,1) 0.0007 0.0001 0.0015σw (1) Uniform(0,1) 0.0438 0.0332 0.0496σa(0) Uniform(0,1) 0.0056 0.0043 0.0068σa(1) Uniform(0,1) 0.0091 0.0062 0.0123σp(0) Uniform(0,1) 0.0401 0.0338 0.0478σp(1) Uniform(0,1) 0.0487 0.0218 0.0766σκ(0) Uniform(0,1) 0.0012 0.0001 0.0030σκ(1) Uniform(0,1) 0.0248 0.0072 0.0419
BFOR Endogenous Switching 49 / 59
Estimation Results
Interpretation
Shock std dev for TFP and TOT are basically the same in and out ofa crisis (posteriors overlap)
Shock std dev for financial shocks much larger in a crisis
BFOR Endogenous Switching 50 / 59
Estimation Results
Prior and Posterior
Table: Estimation Results
Parameter Prior Posterior mean q5 q95γ0,1 Uniform(0,1000) 89.0076 73.2143 108.1845γ1,1 Uniform(0,1000) 1.9676 0.0892 5.8921ρa(0) Uniform(0,1) 0.8134 0.7208 0.8843ρp(0) Uniform(0,1) 0.9637 0.9340 0.9876ρκ(0) Uniform(0,1) 0.6656 0.4152 0.8946ρa(1) Uniform(0,1) 0.7746 0.5543 0.8968ρp(1) Uniform(0,1) 0.9260 0.8258 0.9941ρκ(1) Uniform(0,1) 0.7804 0.6728 0.8872a(1) Uniform(-10,0) -0.0059 -0.0072 -0.0047p(1) Uniform(0,10) 0.0005 0.0000 0.0013κ(1) Uniform(0,1) 0.2305 0.2203 0.2440ι Uniform(0,100) 2.8233 2.8144 2.8360φ Uniform(0,100) 0.3036 0.2697 0.3217
BFOR Endogenous Switching 51 / 59
Estimation Results
Interpretation
Probability function for entering the crisis is a function of the statevariables but is not very sharp
Persistence of shocks not very different in and out of crisis
Mean of the distribution for TFP and leverage is shifted in a crisis
BFOR Endogenous Switching 52 / 59
Estimation Results
Probability of a Binding Regime
1981.1 1986.1 1991.1 1996.1 2001.1 2006.1 2011.1 2016.10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Smoothed Probability of Binding
Smoothed Probability of Binding
Figure: Smoothed Probability of Binding
BFOR Endogenous Switching 53 / 59
Estimation Results
Binding Regime Estimates
Mexico had financial crises in 1982, 1994, both of which show up asbinding regimes
The results show that collateral constraints in the 1980s bindedoutside the crisis
We find no crisis for Mexico in 2007
Recession does not mean binding collateral constraint
BFOR Endogenous Switching 54 / 59
Estimation Results
Importance of Shocks
Table: Variance Decomposition
C I r Y
Interest Rate εw ,t Non-Binding 0.0001 0.0128 0.0066 0.0000Technology εa,t Non-Binding 0.3087 0.2670 0.6390 0.3158Import Price εp,t Non-Binding 0.6817 0.3777 0.1971 0.6814Leverage εκ,t Non-Binding 0.0095 0.3424 0.1572 0.0027
Interest Rate εw ,t Binding 0.0074 0.0044 0.3701 0.0145Technology εa,t Binding 0.0106 0.0003 0.0004 0.0705Import Price εp,t Binding 0.0124 0.0002 0.0003 0.0630Leverage εκ,t Binding 0.9696 0.9951 0.6291 0.8520
BFOR Endogenous Switching 55 / 59
Estimation Results
Interpretation
Outside of a crisis we find that the usual shocks matter (TFP, TOT)
Outside of a crisis the 2 financial shocks are not very important
In a crisis leverage shocks are very important
BFOR Endogenous Switching 56 / 59
Estimation Results
Capital Controls in Our Model
To implement capital controls we assume a simple function of anobservable variable in the model/data
I The debt to output ratio is one common policy target Bt
YtI We assume a simple functional form
τBt = aτ
(Bt
Yt
)in non-binding regime
τBt = 0 in binding regime
I A robust result of the theoretical literature is that if capital controls areyour only instrument you should use them in a macroprudential waybefore a crisis, and not use them once a crisis occurs
I Parameters can be chosen to optimize welfare
The important question is whether or not a capital control wouldhave eliminated an actual crisis, but in our simulations on thecalibrated model the capita control prevents the crisis from occurring
BFOR Endogenous Switching 57 / 59
Estimation Results
Capital controls and optimal behavior
0 10 20−5
0
5K
0 10 20−50
0
50B (level)
0 10 20−1
0
1P
0 10 200
1
2A
0 10 200
2
4C
0 10 20−1
0
1H
0 10 20−2
0
2V
0 10 20−20
0
20I
0 10 20−5
0
5k
0 10 20−2
0
2r (pp)
0 10 20−1
0
1Q
0 10 20−1
0
1W
0 10 20−10
−5
0µ
0 10 20−1
0
1λ (level)
0 10 20−20
0
20B* (level)
0 10 20−1
0
1Y
0 10 200
2
4B/Y (level)
EndogenousExogenousNo Switching
BFOR Endogenous Switching 58 / 59
Conclusion
Conclusion
A new approach to specifying, solving, and estimating models offinancial crises nested in regular business cycles
Probability of a change in regime depends on the state of theeconomy
For the occasionally binding constraint model we find:I The endogenous nature of the regime switch impacts in a qualitative
and quantitative manner the decisions of agents in the economyI A second order solution is needed for endogenous switching to matter
economicallyI Leverage shocks drive fluctuations during financial crises
Conditional policy counterfactuals are future work
BFOR Endogenous Switching 59 / 59