Perturbation in Macroeconomics -...
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Perturbation in Macroeconomics
A Short Course for the 2011 CRC 649 Annual Conference
Hong LanHumboldt-Universität zu Berlin
Alexander Meyer-GohdeHumboldt-Universität zu Berlin
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Motivation 2 | 87
Economic Risk
Risk and uncertainty [...] influence microeconomic decisionswhich ultimately sum up to macroeconomic outcomes.
—Wolfgang Härdle & Michael C. Burda, “About the CRC 649”
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Third-Order Approximation
Response of consumption to a persistent increase in volatility
0 20 40 60 80−4
−3
−2
−1
0
1
2x 10
−7D
evia
tions
Periods since Shock Realization
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Linear Approximation
Response of consumption to a persistent increase in volatility
0 20 40 60 80−4
−3
−2
−1
0
1
2x 10
−7D
evia
tions
Periods since Shock Realization
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Motivation | Consequences 5 | 87
Limitations of Linear Methods
I Linearization eliminates many phenomena of interest
. Precautionary behavior, risk sensitivity, asymmetries, large shocks
I Nonlinear methods needed to capture these components of themacroeconomic consequences of economic risk.
This short course will review one such method for the study of DSGE
models.
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Outline | 6 | 87
Outline
I Part I: An introduction to perturbation methods
I Part II: A State-space approach
I Part III: A Nonlinear MA approach
I Part IV: Applications
I Part V: Frontiers of Current Research
I Appendix: Detailed derivations
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Intro | 7 | 87
Part I: An introduction to perturbation methods
I Perturbation: The basic idea
I Linear methods
I Perturbation methods
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Intro | 8 | 87
The basic idea of perturbation
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Intro | Perturbation: Basic idea 9 | 87
Background
I Macroeconomists are frequently faced with a functional equation
of the form
f (y) = 0
for an unknown function y(x).I Many solution methods have been developed
. Global methods : projection, numerical dynamic programming...
. Local methods : perturbation, linear methods...
I Focus of this presentation: Perturbation. Pioneered by Fleming (1970) for continuous-time control problems
. Applied by Judd and Guu (1997) and Judd (1998) to a specific
DSGE model.
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Intro | Perturbation: Basic idea 10 | 87
Basic idea of perturbation methods
I Perturbation solves the functional problem by specifying
y [n](x) =n∑
i=0
θi(x − x0)i
I Implicit function theorem pins down θi ’s.
I A local approximation, but perform far better than purely local(i.e., linear) methods.
I Perturbation is a generalization of traditional linear methods.
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Intro | Linear Methods 11 | 87
A review of linear methods
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Intro | Linear Methods 12 | 87
Model Setup
I Throughout the presentation, we analyze a system of dynamic,discrete-time rational expectations equations
0 = Et [f (yt−1, yt , yt+1, εt)]
where εs ∼ iid Ψ(z), s > t
I f is an (ny × 1) function, sufficiently smooth in all its arguments;
I yt : (ny × 1) endogenous variables;
I εt : (ne × 1) exogenous stochastic process;
I The time-invariant solution takes the form
yt = g(yt−1, εt), and
yt+1 = g(yt , εt+1) = g(g(yt−1, εt), εt+1)
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Intro | Linear Methods 13 | 87
Linear Methods
1. Solve for the non-stochastic steady state
y = g(y , 0), and y satisfies 0 = f (y , y , y , 0)
2. Then linearize the model by around the non-stochastic steadystate and rearrange
A
[Et yt+1
yt
]= B
[yt
yt−1
]+ Cεt where yt = yt − y
3. The solution (e.g., Blanchard and Kahn (1980), Uhlig (1999),
Klein(2000)) takes the form
yt = αyt−1 + βεt
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Intro | Linear Methods 14 | 87
Limitation of the Linear Methods
I Spurious welfare reversal: Tesar (1995) and Kim & Kim (2003)
I The effects of volatility shocks: Fernandez-Villaverde et. al (RES,2007; AER, forthcoming)
I Neglects precautionary behavior: Schmitt-Grohé & Uribe (JEDC2004), Fernandez-Villaverde et. al (JEDC, 2005)
I Notably poor for analysis of asset pricing: Rudebusch &Swanson (JME 2008)
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Intro | Perturbation 15 | 87
Perturbation Methods
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Intro | Perturbation 16 | 87
Perturbation Parameter σ
I σ ∈ [0, 1] denotes an auxiliary, ”scaling” parameter for the
distribution of stochastic shocks
I Scales uncertainty, i.e., stochastic shocks in period t +1 and later
I Effectively, consider a continuum of auxiliary models
parameterized by σ
0 = Et [f (yt−1, yt , yt+1, εt)], εs ∼ iid Ψ(z/σ), s > t
I with a family of solutions indexed by σ
yt = g(yt−1, εt , σ)
yt+1 = g(yt , σεt+1, σ) = g(g(yt−1, εt , σ), σεt+1, σ)
σ = 1—original, stochastic model; σ = 0—deterministic model
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Intro | Perturbation 17 | 87
State Space Solution
I The policy function
yt = g(yt−1, εt , σ)
defines the state vector to be (yt−1, εt) and σ;
I Often, this policy function is referred to as the state spacesolution of the model;
I Collard and Juillard (2001), Schmitt-Grohé and Uribe (2004) andothers construct a second-order Taylor expansion of the state
space solution
I Anderson et al. (2006), Dynare++ and others construct a
higher-order Taylor expansion of the state space solution
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Intro | Perturbation 18 | 87
Nonlinear Moving Average Solution
I The policy function
yt = y(σ; εt , εt−1, . . .)
defines the state vector to be σ and the infinite sequence of pastshock realizations;
I Extension of linear MA methods of Muth (1961), Whiteman(1983), Taylor (1984) and others
I We are developing methods based on this form of the policyfunction
I Allows, e.g., natural extension of the IRF property of linear MA tononlinear spaces
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State Space | 19 | 87
Part II : State Space Solution
I Intuition via the stochastic growth model
I Numerical expansion
I Limitation of the state space solution
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State Space | Intuition 20 | 87
The Stochastic Growth Model
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State Space | Intuition 21 | 87
The Stochastic Growth Model
I An economy is populated by infinitely-lived agents;
I A representative agent maximizes the discounted sum of herexpected utility
I given a resource constraint;
I The state of the economy evolves according to a Markov
process.
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State Space | Intuition 22 | 87
The Stochastic Growth Model-Cont.
I The consumption Euler equation characterizes the agent’s
utility-maximization behavior
c−γ
t = βEt [c−γ
t+1(αezt+1 kα−1t + 1 − δ)]
I under following constraints
yt = ct + it
yt = ezt kα
t−1
it = kt − (1 − δ)kt−1
I The state of the economy evolves according to
zt = ρzt−1 + εt
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State Space | Intuition 23 | 87
The Stochastic Growth Model
I From the state-space perspective, we seek the policy function
kt = k(kt−1, εt)
I The model has a closed-form solution when γ = δ = 1 (The
Brock-Mirman model)
ln(kt ) = (1 − α) ln(k ) + αln(kt−1) + ρzt−1 + εt
I Otherwise, we resort to perturbation methods: kt = k(kt−1, εt , σ)
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State Space | Expansion 24 | 87
Numerical expansion
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State Space | Expansion 25 | 87
Setting Up the Expansion
I Insert the solution function into the model
yt = g(yt−1, εt , σ)
yt+1 = g(y(yt−1, εt , σ), εt+1, σ);where εt+1 ≡ σεt+1
→0 = Et [f (yt−1, g(yt−1, εt , σ), g(g(yt−1, εt , σ), εt+1, σ), εt )]
I Successive differentiation wrt yt−1, εt , σ
I evaluated atyt−1 = y , εt = σ = 0 identifies coefficients in approx.
The zeroth order expansion identifies the non-stochastic steady state
f (y , y , y , 0) = 0,where y = y(y , 0, 0)
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State Space | Expansion 26 | 87
First Order Expansion
I We aim to construct
yt = y + gy (yt − y) + gεεt + gσσ
I Using the implicit function theorem, we can pin down all thecoefficients in the Taylor expansion;
I In particular
gσ = 0
I The first order expansion is a certainty equivalent solution;
I First order perturbation cannot capture the effect of uncertainty
in the model!
I gy and gε as in traditional linearizations
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State Space | Expansion 27 | 87
Second Order Expansion
I We aim to construct (where yt abbreviates yt − y )
yt =gy yt−1 + gεεt + gσσ +12
gy2(yt−1 ⊗ yt−1) + gεy (yt−1 ⊗ εt)
+ gσy yt−1σ +12
gε2(εt ⊗ εt) + gσεσεt +12
gσσσ2
I In particular
gσy = 0 gσε = 0, and gσ = 0
I Thus, up to second order, uncertainty only affects the constantterm σ2 in the expansion of the policy function;
I This reflects that the second order expansion depends on the
size of the variance of the exogenous shocks!
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State Space | Expansion 28 | 87
Third Order Expansion
I Analogous to before
I In particular
gσy2 = gσε2 = gσyε = 0
but, in general gσ2y 6= gσ2ε 6= 0
I Third order captures time-varying correction for uncertainty (e.g.,
risk-sensitive dynamics);
I This reflects that the second order expansion depends on the
size of the variance of the exogenous shocks!
I gσ3 the shape (skewness) of the distribution matters.
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Policy Function: Stoch. Growth
-4 -2 2 4k[t-1]
-2
-1
1
2
3
4
Consumption
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State Space | Expansion 30 | 87
State-Space Policy Function
I Provides an approximation accurate up to the order of approx.
I For the mapping yt−1, εt , σ, 7→ yt
I Often interested in a different mapping
I IRF’s and simulations: ..., εt−1, εt , σ, 7→ yt
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State Space | Limitations 31 | 87
Limitations of the state space solution
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State Space | Limitations 32 | 87
Iterating a State-Space Perturbation
From Kim et al. (2008): Consider a 2nd-order solution
yt = ρyt−1 + αy2t−1 + εt , |ρ| < 1 and α > 0
|ρ| < 1 follows from stability of linear solution.
I Iterating forward generates spurious higher order terms;
yt+1 = α3y4t−1 + 2α2ρy3
t−1 + 2α2yt−1εt + (αρ2 + αρ)y2t−1 + ρ2yt−1 + . . .
I Iterating again yields sixth and fifth order terms
I ...
Obviously not a 2nd-order simulation or impulse response.
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State Space | Limitations 33 | 87
What Goes Wrong
Moreover: Examine an impulse response with
I yt−1 = 0
I and a single shock of size εt = (1 − ρ)/α
yt =1 − ρ
α
yt+1 = ρ1 − ρ
α+ α
(1 − ρ
α
)2
=1 − ρ
α
...
I yt+s is constant: εt = (1 − ρ)/α is a threshold
I if εt > (1 − ρ)/α: yt+s → ∞!
Potentially explosive paths despite stability of linear solution.
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Source of explosive paths
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
1
45 line
true Þxed point
truth
additional undesirable
Þxed point
second-order
approximation
Notes: This �gure plots the function f(x�1) described in Section 1 and its second-order
Taylor-series approximation.
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State Space | Limitations 35 | 87
Pruning?
Kim et al.(2008) suggest “pruning” the solution
I Let the first-order solution be yFt : yF
t = ρyFt−1 + εt
I The pruned second-order solution is ySt = ρyS
t−1 + α(yFt−1)
2 + εt
I Intuition: Replace ySt−1 with yF
t−1 for the quadratic terms;
I Repeat throughout simulation/impulse responses to maintain
desired order of approximation;
I Discard all spurious higher order terms.
I Pruning is an ad-hoc procedure, Lombardo (2010), and
I does not represent a valid perturbation approximation, Den Haan
& De Wind (2010).
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Nonlinear MA | 36 | 87
Part III : Nonlinear MA Solution
I Overview: Nonlinear MA solution
I An equivalent solution in Brock-Mirman model
I Numerical expansion
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Nonlinear MA | Overview: Nonlinear MA 37 | 87
Overview: Nonlinear MA solution
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Nonlinear MA | Overview: Nonlinear MA 38 | 87
Nonlinear MA Solution
I The Nonlinear MA policy function takes the form
yt = y(σ; εt , εt−1, . . .)
I Nonlinear MA policy function is a direct mapping:
. . . , εt−1, εt , σ, 7→ yt ,
I explicitly taking the history of the exogenous shocks intoconsideration;
I Stability of first-order approx. carries over to higher-order approx.
I It avoids any ” pruning” in generating simulation and impulse
responses, and
I demonstrate the specific contribution of the different moments ofthe exogenous shocks to the impulse responses.
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Nonlinear MA | Special Case: Brock-Mirman 39 | 87
An equivalent solution in Brock-Mirman model
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Nonlinear MA | Special Case: Brock-Mirman 40 | 87
Nonlinear MA solution of Brock-Mirman model
I Recall the closed-form state space solution of Brock-Mirman
model
ln(kt ) = (1 − α) ln(k ) + αln(kt−1) + ρzt−1 + εt
I This is is equivalent to the following nonlinear MA solution
ln (kt)− ln(k)=
∞∑
i=0
kiεt−i
I with k = (αβ)1
1−α , ki = αki−1 + ρi , with k−1 = 0
I Generally, though, numerical approximation unavoidable.
I The one-to-one mapping btw. state space and MA breaks down
under approx.
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Nonlinear MA | Expansion 41 | 87
Solution Form
The time-invariant solution takes the form
yt = y(σ; εt , εt−1, . . .)
yt−1 = y−(σ; εt−1, εt−2, . . .)
yt+1 = y+(σ; εt+1, εt , εt−1, . . .) where εt+1 ≡ σεt+1
Inserting into the model
0 = Et [f (y−(σ; εt−1, εt−2, . . .), y(σ; εt , εt−1, . . .), y+(σ; εt+1, εt , εt−1, . . .), εt)]
Differentiate with respect to σ, εt , εt−1, ... to identify Taylor series
coefficients.Zeroth-order solution the same as before: non stochastic steady
state.
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Nonlinear MA | Expansion 42 | 87
First Order Expansion
I The first order expansion of the policy function takes the form
yt = y + yσσ +
∞∑
i=0
yiεt−i , i = 0, 1, 2, . . .
I Using the implicit function theorem, we pin down the coefficientsvia a recursion
yi = αyi−1 + β1ui , y−1 = 0
I Additionally, like the state space solution
yσ = 0
I The first-order expansion is still a certainty equivalent solution
I Equivalent to state-space solution (recursion in coefficients notvariables)
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Nonlinear MA | Expansion 43 | 87
Second Order Expansion
I Uncertainty again only affects the constant term up to second
order
yiσ = 0
I thus
yt = y +12
yσ2σ2 +
∞∑
i=0
yiεt−i +12
∞∑
j=0
∞∑
i=0
yj,i(εt−j ⊗ εt−i)
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Nonlinear MA | Expansion 44 | 87
Third Order Expansion
I The uncertainty affects the policy function not only through theconstant term, but also through a time-variant term!
yjiσ = 0, but, in general yiσ2 6= 0
I Third order expansion can capture time-varying precautionarybehavior.
I Third order expansion writes
yt =y +12
yσ2σ2 +
∞∑
i=0
(yi +
12
yσ2,iσ2)εt−i +
12
∞∑
j=0
∞∑
i=0
yj,i(εt−j ⊗ εt−i)
+16
∞∑
k=0
∞∑
j=0
∞∑
i=0
yk ,j,i(εt−k ⊗ εt−j ⊗ εt−i)
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Nonlinear MA | Expansion 45 | 87
Discussion of Nonlinear MA
I Work in progress (look for an SFB Discussion Paper soon!)
I Alternate state basis (infinite history of shocks) for policy function
I Straightforward approach to impulse responses and simulations
I With stability properties inherited from first-order
I Numerical methods: Don’t use a knife to turn a screw...
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Examples | 46 | 87
Part IV : Nonlinear MA Solution
I RBC with stochastic volatility
I Int’l RBC with real interest rate risk
I Nominal Asset Pricing with risk-sensitive preferences
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Examples | RBC: Stochastic Volatility 47 | 87
Basic RBC with Stochastic Volatilty
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Examples | RBC: Stochastic Volatility 48 | 87
Basic RBC with Stochastic Volatility
I Add disutility from working
I yt = ezt l1−αkα
t−1
I to stochastic growth model above
I Standard RBC (e.g., Hansen (198))
I Add time varying volatility to tech shock
I ln(εt ) is now AR(1)
I mean as as in constant volatility case
I persistence and stand. dev.: post-war US (Fernandez-Villaverde(RES 2007))
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IRF: Technology Shock
0 10 20 30 40
0
2
4
6
8
10
x 10−3
Periods since Shock Realization
Dev
iatio
ns
Y L Y/L C K
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IRF: Volatility Shock
0 10 20 30 40−4
−2
0
2
4
x 10−7
Dev
iatio
ns
Periods since Shock Realization
KL Y Y/L C
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Examples | RBC: Stochastic Volatility 51 | 87
Consequences of Stochastic Volatility
I Weakens the positive correlation of consumption and labor
productivity with output
I But note the scale: tech. shocks overwhelm vol. shocks
I Additional fundamental nonlinearities required for vol. shocks tocontribute significantly
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Examples | Int’l RBC: Interest Rate Volatility 52 | 87
Int’l RBC with real interest rate riskFernández-Villaverde et. al (AER forthcoming)
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Examples | Int’l RBC: Interest Rate Volatility 53 | 87
Fernández-Villaverde et. al (AER forthcoming)
I Small open economy populated by infinitely-lived agents;
I A representative household maximizes the discounted sum ofher expected utility w.r.t some constraint;
I The household can invest in the stock of physical capital and aninternationally-traded bond;
I The real interest rate in international markets follows
rt = r + εtb,t + εr ,t
where εr ,t = ρrεr ,t−1 + eσr,t ur ,t
and σr ,t = (1 − ρσr ) + ρσrσr ,t−1 + ηr uσr ,t
I Consider a positive shock to uσr ,t .
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0 20 40−0.1
−0.05
0
0.05
0.1Consumption
0 20 40−0.4
−0.3
−0.2
−0.1
0Investment
0 20 40−0.03
−0.02
−0.01
0
0.01Output
0 20 40−2
0
2
4x 10
−4 Hours
0 20 40−1
−0.5
0
0.5
1Real Interest Rate
0 20 40−4
−3
−2
−1
0Debt
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Examples | Int’l RBC: Interest Rate Volatility 55 | 87
Fernández-Villaverde et. al (AER forthcoming)
I Increase in riskiness of financing debt in int’l capital mkt’s
I causes a quantitatively significant and protracted contraction
I induced by a precautionary winding down of exposure to int’lfinancing
Compared to simple RBC above
I quantitatively significant impact of volatility shocks
I linear methods miss this “important force behind the
I distinctive size and pattern of business cycle fluctuations inemerging economies”
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Examples | Nominal Asset Pricing with risk-sensitive preferences 56 | 87
Nominal Asset Pricing with risk-sensitive preferencesRudebusch & Swanson (AEJ forthcoming)
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Examples | Nominal Asset Pricing with risk-sensitive preferences 57 | 87
Rudebusch & Swanson (AEJ forthcoming)
I Representative agent model with Calvo sticky prices
I Epstein-Zin preferences (separate RRA from IES)
I SDF mt,t+1 =
(Vt+1
(Et V1−α
t+1 )1
1−α
)α
βUc,t+1
Uc,t
1πt+1
I α = 0 standard exp. utility framework
I Avg. term premium 10 year zero coupon bond (US postwar):
100 bp
I Linear Methods: 0 bp, Exp. util: 4 bp, Epstein-Zin: 10´0 bp
I DSGE can reproduce the upward sloping yield curve.
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Examples | Nominal Asset Pricing with risk-sensitive preferences 58 | 87
Rudebusch & Swanson (AEJ forthcoming)
I Backus-Gregory-Zin (1989), Den Haan (1995)
I Low interest rates in recession imply increasing bond prices
I Thus, bonds pay out high when consumption is low
I Implies a negative premium and downward sloping yield curve
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Monetary Policy Shock
0 5 10 15 20−0.1
−0.05
0
0.05
0.1C
Dev
iatio
ns
0 5 10 15
0
0.1
0.2
0.3
termprem
Dev
iatio
ns
Periods since Shock Realization
0 5 10 15 20−0.3
−0.2
−0.1
0
0.1pi
Dev
iatio
ns
0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
0.2bond price
Dev
iatio
ns
Periods since Shock Realization
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Examples | Nominal Asset Pricing with risk-sensitive preferences 60 | 87
Rudebusch & Swanson (AEJ forthcoming)
I Structural interpretation
I Tech. shocks main contributor of variance
I Negative tech. shock recession produces increase in inflation
I Causing nominal bond prices to fall
I Positive premium and upward sloping yield curve!
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Tech. Shock
0 5 10 15 20−5
−4
−3
−2
−1
0termprem
Dev
iatio
ns
Periods since Shock Realization
0 5 10 15 200
1
2
3
4bond price
Dev
iatio
ns
Periods since Shock Realization
3rd Order 2nd Order 1st Order0 5 10 15 20
−1.5
−1
−0.5
0pi
Dev
iatio
ns
0 5 10 15 200
0.05
0.1
0.15
0.2C
Dev
iatio
ns
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Frontiers | 62 | 87
Part V : Frontiers
I Measuring the quality of approximation
I Estimation
I Generalized transfer functions
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Frontiers | Measuring the quality of approximation 63 | 87
Measuring the quality of approximation
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Frontiers | Measuring the quality of approximation 64 | 87
Euler Equation Error
I Currently, only one accepted method to measure quality ofapproximation
I insert approximation back into functional
I measure residuals over range in state space of particular interest
I In simple stoch. growth model: the functional is the Euler
equation
I Interpretation: one-period optimization error given current state
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Euler equation errors
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Frontiers | Measuring the quality of approximation 66 | 87
Euler Equation Error
For the stochastic growth model
I Error on the magnitude of 1E-6
I implies a $1 mistake
I for every $1,000,000 of expenses
I Judd & Guu (1997): “Few economists would seriously argue that
real-world agents do better than this.”
Nonlinear MA has an infinite dimensional state space...what to put on
the x-axis?Alternative measures? How to interpret Euler equation errors in
multi-state DSGE model?
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Frontiers | Estimation 67 | 87
Estimation
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Frontiers | Estimation 68 | 87
Estimation
I Can apply Kalman filter for ML to estimate Gaussian model
I With nonlinear models, observables no longer inherit Gaussian
distribution from shocksI Hence, no comp. advantage to Gaussianity in the first place
So
I How do we estimate fully parameterized, nonlinear time seriesmodels
I where the mapping from parameters to reduced form time series
model is highly nonlinear and available only numerically?I Current cutting edge: Particle filter (extends Kalman filter idea of
tracking conditional distributions to evaluate likelihood function)I Alternative approaches?
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Frontiers | Generalized transfer functions 69 | 87
Generalized transfer functions
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Frontiers | Generalized transfer functions 70 | 87
Generalized transfer functions
I Breakdown of superposition
I History of shocks impacts current response nonlinearly
I What is an impulse response?
I Initial point: stoch. steady state, non stoch. steady state, ergodicmean?
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0 50 100 150 200 250 300 350 400−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3Second Order Contributions to Response of k to 2 100 Std. Dev. Shocks in e
Periods
Dev
iatio
ns
Sum of individual second−order contributionsTotal second−order contributions
0 200 4000
0.1
0.2
0.3
0.42nd−Ord Contr. of 1st Shock
Dev
iatio
ns
Periods0 200 400
0
0.1
0.2
0.3
0.4
Indiv. 2nd−Ord Contr. of 2nd Shock
Dev
iatio
ns
Periods0 200 400
−0.1
−0.05
0
Cross Correction to 2nd−Ord Contr. of 2st Shock
Dev
iatio
ns
Periods
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Frontiers | Generalized transfer functions 73 | 87
Thank you very much for your attention!
Perturbation in Macroeconomics
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Detailed Derivations | 74 | 87
Part V : The detailed derivations
I State space solution
I Nonlinear MA solution
Perturbation in Macroeconomics
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Detailed Derivations | 75 | 87
State Space Solution
Perturbation in Macroeconomics
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Detailed Derivations | 76 | 87
First Order Expansion
I To determine gy , Et
[DyT
t−1f∣∣∣y
]= 0
fy− + fy gy + fy+gy gy = 0
I This is a version of Blanchard and Kahn (1980), Anderson andMoore (1985), Uhlig (1999), Klein (2000) saddle-point problem,
and can be solved using, i.e., the QZ algorithm proposed byKlein (2000).
Perturbation in Macroeconomics
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Detailed Derivations | 77 | 87
First Order Expansion-Cont.
I To determine gε, Et
[D
εTtf∣∣∣y
]= 0
fε + fy gε + fy+gy gε = 0
I Therefore
gε = −(fy + fy+gy )−1fε
I To determine gσ, Et
[Dσf
∣∣∣y
]= 0
fy gσ + fy+gy gσ = 0
I Therefore
gσ = 0
Perturbation in Macroeconomics
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Detailed Derivations | 78 | 87
Second Order Expansion
I To determine gy2 , Et
[DyT
t−1yTt−1
f∣∣∣y
]= 0
(fy+ + fy )gy2 + fy+gy2(gy ⊗ gy ) = B
I This is a specific Sylvester equation studied in, and solution
methods proposed by Kamenik (2005).
Perturbation in Macroeconomics
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Detailed Derivations | 79 | 87
Second Order Expansion-Cont.
I To determine gεy , gσy , gσε and gε2
Et
[D
εTt yT
t−1f∣∣∣y
]= 0 : fy+(gy2(gy ⊗ gε) + gy gεy ) + fy gεy = B
⇒ gεy = (fy+gy + fy )−1(B − fy+gy2(gy ⊗ gε))
Et
[D
σyTt−1
f∣∣∣y
]= 0 : fy+gy gσy + fy gσy = 0 ⇒ gσy = 0
Et
[D
σεTtf∣∣∣y
]= 0 : fy+gy gσε + fy gσε = 0 ⇒ gσε = 0
Et
[D
εTt ε
Ttf∣∣∣y
]= 0 : fy+(gy2(gε ⊗ gε) + gy gεε) + fy gεε = B
⇒ gεε = (fy+gy + fy )−1(B − fy+gy2(gε ⊗ gε))
Perturbation in Macroeconomics
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Detailed Derivations | 80 | 87
Second Order Expansion-Cont.
I To determine gσσ, Et
[Dσσf
∣∣∣y
]= 0
fy+(gσσ + gy gσσ) + fy gσσ + (fy+2(gε ⊗ gε) + fy+gεε)Et (σ2εt ⊗ εt) = 0
I Therefore
gσσ = −(fy+(I + gy ) + fy )−1(fy+2(gε ⊗ gε) + fy+gεε)Et (σ2εt ⊗ εt)
Perturbation in Macroeconomics
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Detailed Derivations | 81 | 87
Nonlinear MA Solution
Perturbation in Macroeconomics
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Detailed Derivations | 82 | 87
First Order Expansion
I To determine yi , evaluate Et
[D
εTt−i
f∣∣∣y
]= 0
fy−yi−1 + fy yi + fy+yi+1 + fuui = 0
I This is an inhomogeneous version of Blanchard and Kahn
(1980), Anderson and Moore (1985), Uhlig (1999), Klein (2000)saddle-point problem, solved in detail by Anderson (2010).
I Anderson (2010) method can be applied under our assumption,and it delivers a convergent inhomogeneous solution in the form
yi = αyi−1 + β1ui
Perturbation in Macroeconomics
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Detailed Derivations | 83 | 87
First Order Expansion-Cont.
I To determine yσ, Et
[Dσf
∣∣∣y
]= 0
(fy− + fy + fy+)yσ = 0
I From our no-unit-roots assumption, it follows that
det(fy− + fy + fy+) 6= 0
and hence yσ = 0.
The first order expansion of the policy function therefore takes theform
yt = y +
∞∑
i=0
yiεt−i , i = 0, 1, 2, . . .
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Second Order Expansion
I To determine yj,i , evaluate Et
[D2
εTt−jε
Tt−i
f∣∣∣y
]= 0
fy−yj−1,i−1 + fy yj,i + fy+yj+1,i+1 + fx2(xj ⊗ xi) = 0
I The inhomogeneous part xj ⊗ xi , by construction, is
xj ⊗ xi = (γ1 ⊗ γ1)(Sj ⊗ Si)
where Si =
[yi−1
ui
]
and Si has the 1st order Markov representation: Si+1 = δ1Si .
I Therefore, yj,i will take the form
yj,i = αyj−1,i−1 + β2(Sj ⊗ Si )
I β2 solves the following
(fy + fy+α)β2 + fy+β2(δ1 ⊗ δ1) = −fx2(γ1 ⊗ γ1)
I The foregoing is a specific Sylvester equation studied in and the
solution method developed by Kamenik (2005).
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Second Order Expansion-Cont.
I To determine yiσ, evaluate Et
[D2
σεTt−i
f∣∣∣y
]= 0
(fy− + fy + fy+)yiσ = 0
With no-unit-roots, the forgoing delivers: yiσ = 0.
I To determine yσ2 , evaluate Et
[D2
σ2 f∣∣∣y
]= 0
[fy+y02 + fy+2(y0 ⊗ y0)]Et (εt+1 ⊗ εt+1)− (fy− + fy + fy+)yσ2 = 0
hence yσ2 can be recovered as follows
yσ2 = (fy− + fy + fy+)−1[fy+y02 + fy+2(y0 ⊗ y0)]Et (εt+1 ⊗ εt+1)
Therefore the 2nd order expansion of the policy function writes
yt = y +12
yσ2σ2 +∞∑
i=0
yiεt−i +12
∞∑
j=0
∞∑
i=0
yj,i(εt−j ⊗ εt−i)
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Detailed Derivations | 86 | 87
Third Order Expansion
I To determine yk ,j,i , evaluate Et
[D3
εTt−kε
Tt−jε
Tt−i
f∣∣∣y
]= 0
fy−yk−1,j−1,i−1 + fy yk ,j,i + fy+yk+1,j+1,i+1
+ fx3(xk ⊗ xj ⊗ xi) + fx2(xk ⊗ xj,i) + fx2(xk ,j ⊗ xi)
+ fx2(xj ⊗ xk ,i)Kne,ne2 (Ine ⊗ Kne,ne) = 0
I The solution will take the form
yk ,j,i = αyk−1,j−1,i−1 + β3Sk ,j,i
I β3 solves the following Sylvester equation
(fy + fy+α)β3 + fy+β3δ3 = −[fx3 fx2 fx2 fx2
]γ3
Perturbation in Macroeconomics
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Third Order Expansion-Cont.
I The state space Sk ,j,i contains not only the triple Kroneckerproduct of the first order state spaces Sk ⊗ Sj ⊗ Si , but also the
combinations of both first and second order state spaces, i.e.,
Sk ⊗ Sj,i , Sk ,j ⊗ Si . . .
I Solving Et
[D3
σεTt−jε
Tt−i
f∣∣∣y
]= 0 leads to yσ,j,i = 0, whereas
Et
[D3
σ2εTt−i
f∣∣∣y
]= 0 leads to
yσ2,i = αyσ2,i−1 + β3Sσ2,i
I yσ3 = 0 if the exogenous variables are normally distributed.
The 3rd order expansion of the policy function writes
yt =y +12
yσ2σ2 +∞∑
i=0
(yi +
12
yσ2,iσ2)εt−i +
12
∞∑
j=0
∞∑
i=0
yj,i(εt−j ⊗ εt−i)
+16
∞∑
k=0
∞∑
j=0
∞∑
i=0
yk ,j,i(εt−k ⊗ εt−j ⊗ εt−i)