2. Overall Outline
• Perturbation and Projection Methods for DSGE
Models: an Overview
• Simple New Keynesian model
– Formulation and log‐linear solution.
– Ramsey‐optimal policy.
– Using Dynare to solve the model by log‐linearization:
• Taylor principle, implications of working capital, News shocks,
monetary policy with the long rate.
• Financial Frictions as in BGG
– Risk shocks and the CKM critique of intertemporal shocks.
– Dynare exercise.
• Ramsey Optimal Policy, Time Consistency, Timeless
Perspective.
4. Outline
• A Simple Example to Illustrate the basic ideas.
– Functional form characterization of model
solution.
– Use of Projections and Perturbations.
• Neoclassical model.
– Projection methods
– Perturbation methods
• Make sense of the proposition, ‘to a first order
approximation, can replace equilibrium conditions with
linear expansion about nonstochastic steady state and
solve the resulting system using certainty equivalence’
5. Simple Example
• Suppose that x is some exogenous variable
and that the following equation implicitly
defines y:
hx, y 0, for all x ∈ X
• Let the solution be defined by the ‘policy rule’,
g:
y gx
‘Error function’
• satisfying
Rx; g ≡ hx, gx 0
• for all x ∈ X
6. The Need to Approximate
• Finding the policy rule, g, is a big problem
outside special cases
– ‘Infinite number of unknowns (i.e., one value of g
for each possible x) in an infinite number of
equations (i.e., one equation for each possible x).’
• Two approaches:
– projection and perturbation
7. Projection
ĝx;
• Find a parametric function, , where is a
vector of parameters chosen so that it imitates
Rx; g 0
the property of the exact solution, i.e.,
for all x ∈ X , as well as possible.
• Choose values for so that
̂
Rx; hx, ĝx;
x∈X
• is close to zero for .
• The method is defined by how ‘close to zero’ is
ĝx;
defined and by the parametric function, ,
that is used.
8. Projection, continued
• Spectral and finite element approximations
ĝx;
– Spectral functions: functions, , in which
ĝx; x∈X
each parameter in influences for all
example:
n 1
ĝx; ∑ i Hi x,
i0
n
H i x x i ~ordinary polynominal (not computationaly efficient)
H i x T i x,
T i z : −1, 1 → −1, 1, i th order Chebyshev polynomial
: X → −1, 1
9. Projection, continued
ĝx;
– Finite element approximations: functions, ,
ĝx;
in which each parameter in influences
over only a subinterval of x ∈ X
ĝx; 1 2 3 4 5 6 7
4
2
X
10. Projection, continued
• ‘Close to zero’: collocation and Galerkin
x : x1 , x2 , . . . , xn ∈ X
• Collocation, for n values of
1 n
choose n elements of so that
̂
Rx i ; hx i , ĝx i ; 0, i 1, . . . , n
– how you choose the grid of x’s matters…
• Galerkin, for m>n values of
x : x1 , x2 , . . . , xm ∈ X
choose the n elements of 1 n
m
∑ wij hxj , ĝxj ; 0, i 1, . . . , n
j1
11. Perturbation
• Projection uses the ‘global’ behavior of the functional
equation to approximate solution.
– Problem: requires finding zeros of non‐linear equations.
Iterative methods for doing this are a pain.
– Advantage: can easily adapt to situations the policy rule is
not continuous or simply non‐differentiable (e.g.,
occasionally binding zero lower bound on interest rate).
• Perturbation method uses local properties of
functional equation and Implicit Function/Taylor’s
theorem to approximate solution.
– Advantage: can implement it using non‐iterative methods.
– Possible disadvantages:
• may require enormously high derivatives to achieve a decent
global approximation.
• Does not work when there are important non‐differentiabilities
(e.g., occasionally binding zero lower bound on interest rate).
12. Perturbation, cnt’d
x∗ ∈ X
• Suppose there is a point, , where we
know the value taken on by the function, g,
that we wish to approximate:
gx ∗ g ∗ , some x ∗
• Use the implicit function theorem to
approximate g in a neighborhood of x ∗
• Note:
Rx; g 0 for all x ∈ X
→
j
R x; g ≡ d j Rx; g 0 for all j, all x ∈ X.
dx j
13. Perturbation, cnt’d
• Differentiate R with respect to and evaluate
x
x∗
the result at :
R 1 x ∗ d hx, gx|xx ∗ h 1 x ∗ , g ∗ h 2 x ∗ , g ∗ g ′ x ∗ 0
dx
′ ∗ h 1 x ∗ , g ∗
→ g x −
h 2 x ∗ , g ∗
• Do it again!
2
R x ∗ d 2 hx, gx| ∗ h x ∗ , g ∗ 2h x ∗ , g ∗ g ′ x ∗
xx 11 12
dx 2
h 22 x ∗ , g ∗ g ′ x ∗ 2 h 2 x ∗ , g ∗ g ′′ x ∗ .
→ Solve this linearly for g ′′ x ∗ .
14. Perturbation, cnt’d
• Preceding calculations deliver (assuming
enough differentiability, appropriate
invertibility, a high tolerance for painful
notation!), recursively:
g ′ x ∗ , g ′′ x ∗ , . . . , g n x ∗
• Then, have the following Taylor’s series
approximation:
gx ≈ ĝx
ĝx g ∗ g ′ x ∗ x − x ∗
1 g ′′ x ∗ x − x ∗ 2 . . . 1 g n x ∗ x − x ∗ n
2 n!
16. Example of Implicit Function Theorem
y
hx, y 1 x 2 y 2 − 8 0
2
4
gx ≃ g∗ x ∗ x − x ∗
− ∗
g
‐4
4
x
‐4
h 1 x ∗ , g ∗ x ∗ h 2 had better not be zero!
g′ ∗
x − − ∗
∗ ∗
h 2 x , g g
17. Neoclassical Growth Model
• Objective:
1−
c −1
E 0 ∑ t uc t , uc t t
1−
t0
• Constraints:
c t expk t1 ≤ fk t , a t , t 0, 1, 2, . . . .
a t a t−1 t .
fk t , a t expk t expa t 1 − expk t .
18. Efficiency Condition
ct1
E t u ′ fk t , a t − expk t1
ct1 period t1 marginal product of capital
− u ′ fk t1 , a t t1 − expk t2 f K k t1 , a t t1 0.
• Here, k t , a t ~given numbers
t1 ~ iid, mean zero variance V
time t choice variable, k t1
• Convenient to suppose the model is the limit
→1
of a sequence of models, , indexed by
t1 ~ 2 V , 1.
19. Solution
• A policy rule,
k t1 gk t , a t , .
• With the property:
ct
Rk t , a t , ; g ≡ E t u ′ fk t , a t − expgk t , a t ,
ct1
k t1 a t1 k t1 a t1
− u ′ f gk t , a t , , a t t1 − exp g gk t , a t , , a t t1 ,
k t1 a t1
fK gk t , a t , , a t t1 0,
• for all a t , k t and 1.
20. Projection Methods
• Let
ĝk t , a t , ;
– be a function with finite parameters (could be either
spectral or finite element, as before).
• Choose parameters, , to make
Rk t , a t , ; ĝ
– as close to zero as possible, over a range of values of
the state.
– use Galerkin or Collocation.
21. Occasionally Binding Constraints
• Suppose we add the non‐negativity constraint on
investment:
expgk t , a t , − 1 − expk t ≥ 0
• Express problem in Lagrangian form and optimum is
characterized in terms of equality conditions with a
multiplier and with a complementary slackness condition
associated with the constraint.
• Conceptually straightforward to apply preceding method.
For details, see Christiano‐Fisher, ‘Algorithms for Solving
Dynamic Models with Occasionally Binding Constraints’,
2000, Journal of Economic Dynamics and Control.
– This paper describes alternative strategies, based on
parameterizing the expectation function, that may be easier,
when constraints are occasionally binding constraints.
22. Perturbation Approach
• Straightforward application of the perturbation approach, as in the simple
example, requires knowing the value taken on by the policy rule at a point.
• The overwhelming majority of models used in macro do have this
property.
– In these models, can compute non‐stochastic steady state without any
knowledge of the policy rule, g.
k∗
– Non‐stochastic steady state is such that
a0 (nonstochastic steady state in no uncertainty case) 0 (no uncertainty)
∗ ∗
k g k , 0 , 0
1
– and 1−
k∗ log .
1 − 1 −
23. Perturbation
• Error function:
ct
Rk t , a t , ; g ≡ E t u ′ fk t , a t − expgk t , a t ,
ct1
− u ′ fgk t , a t , , a t t1 − expggk t , a t , , a t t1 ,
f K gk t , a t , , a t t1 0,
k t , a t , .
– for all values of
• So, all order derivatives of R with respect to its
arguments are zero (assuming they exist!).
24. Four (Easy to Show) Results About
Perturbations
• Taylor series expansion of policy rule:
linear component of policy rule
gk t , a t , ≃ k g k k t − k g a a t g
second and higher order terms
1 g kk k t − k 2 g aa a 2 g 2 g ka k t − ka t g k k t − k g a a t . . .
t
2
– g 0 : to a first order approximation, ‘certainty equivalence’
– All terms found by solving linear equations, except coefficient on past
gk
endogenous variable, ,which requires solving for eigenvalues
– To second order approximation: slope terms certainty equivalent –
g k g a 0
– Quadratic, higher order terms computed recursively.
25. First Order Perturbation
• Working out the following derivatives and
evaluating at k t k ∗ , a t 0
R k k t , a t , ; g R a k t , a t , ; g R k t , a t , ; g 0
‘problematic term’ Source of certainty equivalence
• Implies: In linear approximation
R k u ′′ f k − e g g k − u ′ f Kk g k − u ′′ f k g k − e g g 2 f K 0
k
R a u ′′ f a − e g g a − u ′ f Kk g a f Ka − u ′′ f k g a f a − e g g k g a g a f K 0
R −u ′ e g u ′′ f k − e g g k f K g 0
26. Technical notes for following slide
u ′′ f k − e g g k − u ′ f Kk g k − u ′′ f k g k − e g g 2 f K
k 0
1 f − e g g − u ′ f Kk g − f g − e g g 2 f K 0
k k
u ′′
k k k k
1 f − 1 e g u ′ f Kk f f K g e g g 2 f K 0
k u ′′
k k k
1 fk − 1 u ′ f Kk f k g g 2 0
k
eg fK f K u ′′ e g f K eg k
1 − 1 1 u ′ f Kk gk g2 0
k
u ′′ e g f K
• Simplify this further using:
f K K−1 expa 1 − , K ≡ expk
exp − 1k a 1 −
f k expk a 1 − expk f K e g
f Kk − 1 exp − 1k a
f KK − 1K−2 expa − 1 exp − 2k a f Kk e −g
• to obtain polynomial on next slide.
27. First Order, cont’d
Rk 0
• Rewriting term:
1 − 1 1 u ′ f KK gk g2 0
k
u ′′ f K
0 g k 1, g k 1
• There are two solutions,
– Theory (see Stokey‐Lucas) tells us to pick the smaller
one.
– In general, could be more than one eigenvalue less
than unity: multiple solutions.
gk ga
• Conditional on solution to , solved for
Ra 0
linearly using equation.
• These results all generalize to multidimensional
case
28. Numerical Example
• Parameters taken from Prescott (1986):
2 20, 0. 36, 0. 02, 0. 95, Ve 0. 01 2
• Second order approximation:
3.88 0.98 0.996 0.06 0.07 0
ĝk t , a t−1 , t , k ∗ gk k t − k ∗ ga at g
0.014 0.00017 0.067 0.079 0.000024 0.00068 1
1 g kk k t − k g aa
2
a2
t g 2
2
−0.035 −0.028 0 0
g ka k t − ka t g k k t − k g a a t
29. Conclusion
• For modest US‐sized fluctuations and for
aggregate quantities, it is reasonable to work
with first order perturbations.
• First order perturbation: linearize (or, log‐
linearize) equilibrium conditions around non‐
stochastic steady state and solve the resulting
system.
– This approach assumes ‘certainty equivalence’. Ok, as
a first order approximation.
30. List of endogenous variables determined at t
Solution by Linearization
• (log) Linearized Equilibrium Conditions:
E t 0 z t1 1 z t 2 z t−1 0 s t1 1 s t 0
• Posit Linear Solution:
s t − Ps t−1 − t 0.
z t Az t−1 Bs t Exogenous shocks
• To satisfy equil conditions, A and B must:
0 A2 1 A 2 I 0, F 0 0 BP 1 0 A 1 B 0
• If there is exactly one A with eigenvalues less
than unity in absolute value, that’s the solution.
Otherwise, multiple solutions.
• Conditional on A, solve linear system for B.