The document discusses methods for solving dynamic stochastic general equilibrium (DSGE) models. It outlines perturbation and projection methods for approximating the solution to DSGE models. Perturbation methods use Taylor series approximations around a steady state to derive linear approximations of the model. Projection methods find parametric functions that best satisfy the model equations. The document also provides an example of applying the implicit function theorem to derive a Taylor series approximation of a policy rule for a neoclassical growth model.

1. Solution Methods for DSGE Models
and Applications using Linearization
Lawrence J. Christiano

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.

3. Perturbation and Projection
Methods for Solving DSGE Models
Lawrence J. Christiano

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:
hx, y  0, for all x ∈ X
• Let the solution be defined by the ‘policy rule’,
g:
y  gx
‘Error function’
• satisfying
Rx; g ≡ hx, gx  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
ĝx;  
• Find a parametric function, , where is a
vector of parameters chosen so that it imitates
Rx; g  0
the property of the exact solution, i.e.,
for all x ∈ X , as well as possible.

• Choose values for so that
̂
Rx;   hx, ĝx; 
x∈X
• is close to zero for .
• The method is defined by how ‘close to zero’ is
ĝx; 
defined and by the parametric function, ,
that is used.

8. Projection, continued
• Spectral and finite element approximations
ĝx; 
– Spectral functions: functions, , in which
 ĝx;  x∈X
each parameter in influences for all
example:
n 1
ĝx;   ∑  i Hi x,   
i0
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
ĝx; 
– Finite element approximations: functions, ,
 ĝx; 
in which each parameter in influences
over only a subinterval of x ∈ X
ĝx;   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
̂
Rx i ;   hx i , ĝx 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 hxj , ĝxj ;   0, i  1, . . . , n
j1

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:
gx ∗   g ∗ , some x ∗
• Use the implicit function theorem to
approximate g in a neighborhood of x ∗
• Note:
Rx; g  0 for all x ∈ X
→
j
R x; g ≡ d j Rx; 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 hx, gx|xx ∗  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 hx, gx| ∗  h x ∗ , g ∗   2h x ∗ , g ∗ g ′ x ∗ 
xx 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:
gx ≈ ĝx
ĝx  g ∗  g ′ x ∗   x − x ∗ 
 1 g ′′ x ∗   x − x ∗  2 . . .  1 g n x ∗   x − x ∗  n
2 n!

15. Perturbation, cnt’d
• Check….
• Study the graph of
Rx; ĝ
x∈X
– over to verify that it is everywhere close
to zero (or, at least in the region of interest).

16. Example of Implicit Function Theorem
y
hx, y  1 x 2  y 2  − 8  0
2
4
gx ≃ 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 uc t , uc t   t
1−
t0
• Constraints:
c t  expk t1  ≤ fk t , a t , t  0, 1, 2, . . . .
a t  a t−1   t .
fk t , a t   expk t  expa t   1 −  expk t .

18. Efficiency Condition
ct1
E t u ′ fk t , a t  − expk t1 
ct1 period t1 marginal product of capital
− u ′ fk t1 , a t   t1  − expk t2  f K k t1 , a t   t1    0.
• Here, k t , a t ~given numbers
 t1 ~ iid, mean zero variance V 
time t choice variable, k t1
• Convenient to suppose the model is the limit
→1 
of a sequence of models, , indexed by
 t1 ~ 2 V  ,   1.

19. Solution
• A policy rule,
k t1  gk t , a t , .
• With the property:
ct
Rk t , a t , ; g ≡ E t u ′ fk t , a t  − expgk t , a t , 
ct1
k t1 a t1 k t1 a t1
− u ′ f gk t , a t , , a t   t1 − exp g gk t , a t , , a t   t1 , 
k t1 a t1
 fK gk t , a t , , a t   t1   0,
• for all a t , k t and   1.

20. Projection Methods
• Let
ĝk t , a t , ; 
– be a function with finite parameters (could be either
spectral or finite element, as before).

• Choose parameters, , to make
Rk 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:
expgk t , a t ,  − 1 −  expk 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
a0 (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
Rk t , a t , ; g ≡ E t u ′ fk t , a t  − expgk t , a t , 
ct1
− u ′ fgk t , a t , , a t   t1  − expggk t , a t , , a t   t1 , 
 f K gk t , a t , , a t   t1   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
gk 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 − ka 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 expa  1 − , K ≡ expk
  exp − 1k  a  1 − 
f k   expk  a  1 −  expk  f K e g
f Kk   − 1 exp − 1k  a
f KK   − 1K−2 expa   − 1 exp − 2k  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
   
ĝk 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 − ka 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 t1   1 z t   2 z t−1   0 s t1   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 BP   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.