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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
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”
Perturbation in Macroeconomics
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
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
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.
Perturbation in Macroeconomics
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.
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
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)
Perturbation in Macroeconomics
State Space | Expansion 24 | 87
Numerical expansion
Perturbation in Macroeconomics
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)
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
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!
Perturbation in Macroeconomics
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.
Perturbation in Macroeconomics
Policy Function: Stoch. Growth
-4 -2 2 4k[t-1]
-2
-1
1
2
3
4
Consumption
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
Perturbation in Macroeconomics
State Space | Limitations 31 | 87
Limitations of the state space solution
Perturbation in Macroeconomics
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.
Perturbation in Macroeconomics
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.
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).
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
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.
Perturbation in Macroeconomics
Nonlinear MA | Special Case: Brock-Mirman 39 | 87
An equivalent solution in Brock-Mirman model
Perturbation in Macroeconomics
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.
Perturbation in Macroeconomics
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.
Perturbation in Macroeconomics
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)
Perturbation in Macroeconomics
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)
Perturbation in Macroeconomics
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)
Perturbation in Macroeconomics
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...
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
Examples | RBC: Stochastic Volatility 47 | 87
Basic RBC with Stochastic Volatilty
Perturbation in Macroeconomics
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))
Perturbation in Macroeconomics
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
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
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)
Perturbation in Macroeconomics
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 .
Perturbation in Macroeconomics
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
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”
Perturbation in Macroeconomics
Examples | Nominal Asset Pricing with risk-sensitive preferences 56 | 87
Nominal Asset Pricing with risk-sensitive preferencesRudebusch & Swanson (AEJ forthcoming)
Perturbation in Macroeconomics
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.
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
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
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!
Perturbation in Macroeconomics
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
Frontiers | 62 | 87
Part V : Frontiers
I Measuring the quality of approximation
I Estimation
I Generalized transfer functions
Perturbation in Macroeconomics
Frontiers | Measuring the quality of approximation 63 | 87
Measuring the quality of approximation
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
Euler equation errors
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?
Perturbation in Macroeconomics
Frontiers | Estimation 67 | 87
Estimation
Perturbation in Macroeconomics
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
Perturbation in Macroeconomics
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?
Perturbation in Macroeconomics
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
Frontiers | Generalized transfer functions 73 | 87
Thank you very much for your attention!
Perturbation in Macroeconomics
Detailed Derivations | 74 | 87
Part V : The detailed derivations
I State space solution
I Nonlinear MA solution
Perturbation in Macroeconomics
Detailed Derivations | 75 | 87
State Space Solution
Perturbation in Macroeconomics
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
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
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
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
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
Detailed Derivations | 81 | 87
Nonlinear MA Solution
Perturbation in Macroeconomics
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
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, . . .
Perturbation in Macroeconomics
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).
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)
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
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)