Solving and Estimating Models of Financial Crises: An ...macro.soc.uoc.gr/docs/Year/2017/papers/Otrok.pdfSolving and Estimating Models of Financial Crises: An Endogenous Regime Switching

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


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?


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.


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?


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


Introduction







Introduction







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


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


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


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)


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


Introduction Outline

Outline of Talk

Model

Collateral Constraint Formulation

Mapping into Regime Switching

Solution

Properties of Model Solution

Estimation Procedure

Empirical Results


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


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


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




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




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




Define a new variable (the ”borrowing cushion”) which measures thedistance between debt and the value of collateral in the constraint

B∗t =1


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



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



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



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



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.




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


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


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


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?


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


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


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)


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



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



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)

]


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 ]




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)




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)


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


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



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)



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


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)



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)


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)




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



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



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



Calibrated Model

The model matches the ergodic means as in Mendoza

The model can also get second moments to match the data as inMendoza


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


Estimation


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


Estimation


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Documents

Solving and Estimating Models of Financial Crises: An ...macro.soc.uoc.gr/docs/Year/2017/papers/Otrok.pdfSolving and Estimating Models of Financial Crises: An Endogenous Regime Switching