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.