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Working Paper Series

Approximating time varying structural
models with time invariant structures

WP 15-10

Fabio Canova
BI Norwegian Business School and
CEPR
Filippo Ferroni
Banque de France and University of
Surrey
Christian Matthes
Federal Reserve Bank of Richmond

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

Approximating time varying structural models with
time invariant structures
Fabio Canova, BI Norwegian Business School and CEPR
Filippo Ferroni Banque de France and University of Surrey
Christian Matthes Federal Reserve Bank of Richmond
September 8, 2015
Working Paper No. 15-10

Abstract
The paper studies how parameter variation a¤ects the decision rules of a DSGE
model and structural inference. We provide diagnostics to detect parameter variations
and to ascertain whether they are exogenous or endogenous. Identi…cation and inferential distortions when a constant parameter model is incorrectly assumed are examined.
Likelihood and VAR-based estimates of the structural dynamics when parameter variations are neglected are compared. Time variations in the …nancial frictions of
Gertler and Karadi’ (2010) model are studied.
s
Keywords: Structural model, time varying coe¢ cients, endogenous variations,
misspeci…cation.
JEL Classi…
cation: C10, E27, E32.
We thank Michele Lenza, Marco del Negro, Tao Zha, Ferre de Graeve, James Hamilton, Frank
Schorfheide, and the participants to seminars at Goethe University, University of Milan, Bank of England, Carlos III Madrid, Humboldt University Berlin, Federal Reserve Board, and the conferences ESSIM
2015; Identi…cation in Macroeconomics, National Bank of Poland; Econometric Methods for business cycle
analysis, forecasting and policy simulations, Norges Bank; the NBER Summer Institute group on Dynamic
equilibrium models for comments and suggestions. The views presented in this paper do not re‡ those of
ect
the Banque de France, the Federal Reserve Bank of Richmond, or the Federal Reserve system.

1

1 INTRODUCTION

1

Introduction

In macroeconomics it is standard to study models that are structural in the
sense of Hurwicz (1962); that is, models where the parameters characterizing
the preference and the constraints of the agents and the technologies to produce
goods and services are invariant to changes in the parameters describing government
policies. Such a requirement is crucial to distinguish structural from reduced form
models, and to conduct correctly designed policy counterfactuals in dynamic
stochastic general equilibrium (DSGE) models.
Recently, Dueker et al. (2007), Fernandez Villaverde and Rubio Ramirez (2007),
Canova (2009), Rios Rull and Santaeularia Llopis (2010), Liu et al. (2011), Galvao, et
al. (2014), Vavra (2014), Seoane (2014), and Meier and Sprengler (2015) have shown
that DSGE parameters are not time invariant and that variations display small
but persistent patterns. Parameter variations can not be taken as direct evidence
that DSGE models are not structural. For example, Cogley and Yagihashi (2010),
and Chang et al. (2013) showed that parameter variations may result from the
misspeci…cation of a time invariant model, while Schmitt Grohe and Uribe (2003)
indicated that parameter variations may be needed in certain small open economy
models to ensure the existence of a stationary equilibrium.
The approach the DSGE literature has taken to model parameter variations
follows the VAR literature (see Cogley and Sargent, 2005, and Primiceri, 2005):
they are assumed to be exogenously drifting as independent random walks. Many
economic questions, however, hint at the possibility that parameter variations
may instead be endogenous. For example, is it reasonable to assume that a central
bank reacts to in‡
ation in the same way in an expansion or in a contraction? Davig
and Leeper (2006) analyze state-dependent monetary policy rules and describe how
this feature a¤ects structural dynamics. Does the propagation of shocks depend on
the state of private and government debt? Do …scal multipliers depend on inequality,
see e.g. Brinca et al. (2014)? Are households as risk averse or as impatient when
they are wealthy as when they are poor? Questions of this type are potentially
numerous. Clearly, policy analyses conducted assuming time invariant parameters
or an inappropriate form of time variations may be misleading; comparisons of
the welfare costs of business cycles biased; and growth prescriptions invalid.
This paper has three main goals. First, we want to characterize the decision
rules of a DSGE when parameter variations are either exogenous or endogenous,
and in the latter case, when agents internalize or not the e¤ects that their
decisions may have on parameter variations. Second, we wish to provide diagnostics
to detect misspeci…cations due to neglected parameter variations. Third, we want
to study the consequences in terms of identi…cation, estimation, and inference
of using time invariant models when the DGP features parameter variations
and compare likelihood-based and SVAR-based estimates of the structural dynamics
when parameter variations are neglected.
The existing literature is generally silent on these issues. Seoane (2014) uses

2

1 INTRODUCTION
parameter variations as a respeci…cation tool. Kulish and Pagan (2014) characterize
the decision rules of a DSGE model when predictable structural breaks occur.
Magnusson and Mavroedis (2014) and Huang (2014) examine how variations in the
certain parameters may a¤ect the identi…cation of other structural parameters and
the asymptotic theory of maximum likelihood estimators. Fernandez Villaverde et
al. (2013) investigate to what extent variations in shock volatility matter for real
variables. Ireland (2007) assumes that trend in‡
ation in a standard New Keynesian
model is driven by structural shocks; Ascari and Sbordone (2014) highlight that
it may be a function of policy decisions.
The next section characterizes the decision rules in a general setup where
both exogenous and endogenous variations in the parameters regulating preferences,
technologies, and constraints are possible. We consider both …rst order and higher
order perturbed approximations. We present a simple RBC example to provide
intuition for the results we obtain. We show that if parameter variations are
exogenous, structural dynamics are the same as in a model with no parameter
variations. Thus, if one correctly identi…es structural disturbances, she would make
no mistakes in characterizing structural impulse responses, even if she employes a
constant coe¢ cient model. Clearly, variance and historical decompositions exercises
will be distorted, since some sources of disturbances will be omitted. If parameter
variations are instead endogenous, structural dynamics may be di¤erent from those
of a constant coe¢ cient model. Di¤erences exist because the income and substitution
e¤ects present in the constant coe¢ cient model are altered. These conclusions do not
necessarily hold when higher order approximations are used.
Section 3 provides diagnostics to detect misspeci…cation induced by neglecting parameter variations and to distinguish exogenous vs. endogenous parameter variations.
In the context of a Monte Carlo exercise, we show that they are able to detect the
true DGP with high probability
In section 4 we are interested in measuring the identi…cation repercussions that neglected time variations may have for time invariant parameters. Since the likelihood
is constructed using forecast errors, which are generally misspeci…ed when parameter
variations are neglected, one expects the likelihood shape to be both ‡
attened
and distorted. In the context of the RBC example, we show that indeed both
pathologies occur; we also show that weakly identi…ed (time invariant) parameters
do not become better identi…ed when time variations in other parameters exist.
Section 5 considers structural estimation of a time invariant model when the
data is generated by models with time varying parameters. We expect distortions
because the dynamics assumed by the constant coe¢ cient model are generally
incorrect and because shock misaggregation is present. Indeed, important biases in
parameter estimates are present, occur primarily in parameters controlling income and
substitution e¤ects, and do not die away as sample size increase. Estimated impulse
responses di¤er from the true ones both in quantitative and qualitative sense.
Section 6 studies whether a less structural time invariant SVARs model can
capture the dynamics induced by structural shocks. We show that the performance

3

2 THE SETUP

4

is comparable if not superior to the one of structural models. The performance
of SVARs worsens when shocks to the parameters account for a considerable
portion of the variability of the endogenous variables but the deterioration is not
as large as with likelihood -based approaches.
Section 7 estimates the parameters of Gertler and Karadi’ (2010) model of uns
conventional monetary policy, applies the diagnostics to detect parameter variations,
and estimates versions of the model where the bank’ moral hazard parameter is als
lowed to vary over time. We …nd that a …xed coe¢ cient model is misspeci…ed, that
making parameter variations endogenous function of net worth is preferable, and that
the dynamic e¤ects of capital quality shocks on the spread and on bank net worth
can be more persistent than previously thought. Section 8 concludes.

2

The setup

The optimality conditions of a DSGE model can be represented as:
Et [f (Xt+1 ; Xt ; Xt

1 ; Zt+1 ; Zt ;

t+1 ;

t )]

=0

(1)

where Xt is an nx 1 vector of endogenous variables, Zt is an nz 1 vector of strictly
exogenous variables, t = [ 1t ; 2t ]; vector of possibly time varying structural parameters, where 2t is a n 1 nx1 1 vector, nx
nx1 ; appearing in the case agents
internalize the e¤ects that their decisions have on the parameters and 1t is an n 1 1
vector, while f is a continuous function, assumed to be di¤erentiable up to order q,
mapping onto a Rnx space. Since the distinction between variables and parameters is
blurred when we allow for parameter variations, we use the convention that parameters
are those variables that typically assumed to be constant by economists.
The law of motion of the exogenous variables is:
Zt+1 =

(Zt ;

z
t+1 )

(2)

where is a continuous function, assumed to be di¤erentiable up to order q, mapping
onto a Rnz space; z is a ne 1 vector of i.i.d. structural disturbances with mean
t+1
zero and identity covariance matrix; nz
ne ;
0 is an auxiliary scalar;
is a
known ne ne matrix. The law of motion of the structural parameters is:
t+1

=

( ; Xt ; Ut+1 )

(3)

where is a continuous function, assumed to be di¤erentiable up to order q, mapping
onto the Rn space; Ut is a nu 1 vector of exogenous disturbances, n = n 1 (1+nx1 )
nu ; is a vector of constants: The law of motion of Ut+1 is:
Ut+1 =

(Ut ;

u
u t+1 )

(4)

where is continuous and di¤erentiable up to order q, mapping onto the Rnu space;
u is a n
1 vector of i.i.d. disturbances, with mean zero and identity covariance
u
t
matrix, uncorrelated with the z ; and u is a known nu nu matrix.
t+1

2 THE SETUP

5

The decision rule is assumed to be of the form:
Xt = h(Xt

1 ; Wt ;

t+1 ;

)

(5)

where h is a continuous function, assumed to be di¤erentiable up to order q, and
0
mapping onto a Rnx space, t+1 = [ z0 ; u0 ]0 ; = diag[ z ; u ]; Wt = [Zt ; Ut0 ]0 :
t+1 t+1
Few features of the setup need some discussion. First,
t will be serially
correlated if Ut or Xt or both are serially correlated. Second, the vector of
structural disturbances z may be smaller than the vector of exogenous variables
t+1
and the dimension of u may be smaller than the dimension of the structural
t+1
parameters. Thus, there may be common patterns of variations in Zt+1 and
Ut+1 : Third, we allow for time variations in the parameters regulating preferences,
technologies, and constraints, but we do not consider variations in the auxiliary
parameters regulating the law of motion of Zt and Ut ; as we are not interested
in stochastic volatility, GARCH, or rare events phenomena (as in e.g. Andreasan,
2012), nor in time variations driven by evolving persistence of the exogenous
processes. Fourth, (5) makes no distinction between states and controls. Thus, it
has the format of a …nal form (endogenous variables as a function of the exogenous
variables and the parameters) rather than of a state space form (control variables
as a function of the states and of the parameters).

2.1

First order approximate decision rule

We start by studying the implications of structural parameters variation for the optimal
decision rule when a …rst order approximate solution is considered. Taking a linear
expansion of (1) around the steady states leads to
0 = Et [F xt+1 + Gxt + Hxt

1

+ Lzt+1 + M zt + N

t+1

+ O t]

(6)

where F = @f =@Xt+1 , G = @f =@Xt , H = @f =@Xt 1 , L = @f =@Zt+1 , M = @f =@Zt ,
N = @f =@ t+1 O = @f =@ t ; all evaluated at the steady states values of (Xt ; Zt ; t )
and lower case letters indicate deviations from the steady states. Linear expanding
(5) leads to:
xt = P xt 1 + Qzt + Rut
(7)
where P = @h=@Xt

1,

Q = @h=@Zt , R = @h=@Ut ; all evaluated at steady state values.

Proposition 2.1. The matrices P, Q, R satisfy:
P solves F P 2 + (G + N
Given P , Q solves V Q =
Given P , R solves W R =
G + N x)

x )P

+ (H + O

vec(L
vec(N

z +M )

x)

= 0.

and V =

u!u + O u)

0
z

F +Inz

where W =

!0
u

(F P +G+N
F + In

x ).

(F P +

where u = @ =@Ut+1 ; x = @ =@Xt ; z = @ =@Zt ; ! u = @ =@Ut , vec denotes the
columnwise vectorization, and where we assume that all the eigenvalues of z and of
! u are strictly less than one in absolute value.

2 THE SETUP

6

Proof. The proof is straightforward. Substituting (7) into (6), we obtain
0 =[F P 2 + (G + N

x )P

+ [(F P + G + N

+ (H + O

x )R

x )]xt 1

+ F R! u + N

+ [(F P + G + N

u!u

+O

x )Q

+ FQ

z

+L

z

+ M ]zt

u ]ut

Since the solution must hold for every realization of xt
their coe¢ cient to zero and the result obtains.

1,

zt , ut , we need to equate

Corollary 2.2. If x = 0, the dynamics in response to the structural shocks zt are
identical to those obtained when parameters are time invariant. Variations in the j-th
parameter have instantaneous impact on the endogenous variables xt , if and only if the
j th column of N u ! u + O u 6= 0.
Corollary 2.3. If u = 0 and the matrices N x and O
tions have no e¤ ects on the endogenous variables xt .

x

are zero, parameter varia-

Proposition 2.1 indicates that the …rst order approximate decision rule will,
as in a constant coe¢ cient setup, be a VARMA(1,1) but with an additional
set of disturbances. Corollaries 2.2 and 2.3 give conditions under which parameter
variations alter the dynamics induced by structural disturbances. If parameter
variations are purely exogenous, x = 0, the P and Q matrices are identical to
those of a constant coe¢ cient model. Thus, parameter variation adds variability
to the endogenous variables without altering the dynamics produced by structural
disturbances. In other words, suppose an economy is perturbed by technology
shocks. Then, the dynamics induced by these shocks do not depend on whether
the discount factor is constant or time varying, provided technological innovations
are exogenous and unrelated to the innovations in the discount factor.
This result implies that if one is able to identify the structural disturbances z from
t
a time invariant version of the model, she would make no mistakes in characterizing
structural dynamics. Clearly, variance or historical decomposition exercises will be
distorted, since certain sources of variations (the u disturbances) are omitted. One
t
interesting question is whether standard procedures allow a researcher employing a
time invariant model to recover z from the data when the DGP features time varying
t
structural parameters. If not, one would like to know which structural disturbance
absorbs the missing shocks. Sections 5 and 6 study these issues in a practical example.
On the other hand, if parameter variations are purely endogenous, u = 0, the
dynamics in response to structural shocks may be altered. To know if distortions
are present; one needs to check whether the columns of the matrices N x and O x are
equal to zero. If they are not, a researcher employing a time invariant model is likely
to incorrectly characterize both the structural dynamics and the relative importance
of di¤erent sources of disturbances for the variability of the endogenous variables.
The equilibrium dynamics, as encoded in the P matrix, can thus help us to distinguish between models with endogenous time variation featuring di¤erent laws of
motion for the parameters (i.e. the x matrix). Distinguishing between models with

2 THE SETUP

7

exogenous time variation that di¤er in how the parameters respond to exogenous disturbances (i.e. the u matrix) is possible if the cross equation restrictions present in R
are di¤erent across models.

2.2

Higher order approximate decision rule

Are the conclusions maintained when higher order approximations are considered? In
the second order approximation, the …rst order terms are the same as in the linear
approximation. To examine whether quadratic terms will be a¤ected by the presence
of time variations insert (5) in the optimality conditions so that (1) is
0 = Et [F (Xt ; Wt ;

t+1 ;

)]

(8)

The second order approximation of (8) is
Et [(F x xt

1

+ F w wt + F

) + 0:5(F xx (xt

xt

1

F xw (xt

1)

+ F ww (wt

wt ) + F x x t

1

wt ) + F

2

) +

+ F w wt ] = (9)
0

1

Note that F ; F x xt 1 ; F w wt are all zero, see Schmitt Grohe and Uribe (2004).
The second order expansion of (5) is
xt = hx xt

1

+ hw wt + 0:5(hxx (xt

+ hxw (xt

1

wt ) + hx xt

1

xt

1

1)

+ hww (wt

wt ) + h

+ hw wt

2

)
(10)

It is hard to make general statements about the properties of second order solutions
of models with time varying coe¢ cients. As long as Fxx , Fww , Fwx are not a¤ected by
parameter variations, as is the case when variations are exogenous, second order approximations in time varying coe¢ cient and in …xed coe¢ cient models will be the same.
However, when these expressions are a¤ected, the approximations will be di¤erent. As
an example of this latter case, consider the model
Et yt+1 =

0:95
t xt

xt = 0:8 xt
t

= (2

(11)
1

+ 0:2 x + ut

0:5 (exp(

(12)

1 (xt 1

x) + exp(

2 (xt 1

x)) + vt

(13)

where both vt and ut are i.i.d. and x Ext = 1. It is easy to verify that when 1 = 2 ;
the …rst order solution (including only the terms concerning structural dynamics) is
yt = 0:76xt

1

+ 0:95ut

(14)

and it is the same as in the constant coe¢ cient model ( 1 = 2 = 0; vt = 0; 8t), since
N x and O x are both zero. However, the second order solution (including only terms
concerning structural dynamics) is
yt = 0:76xt

1

+ 0:95ut

0:01565x2
t

1

0:2375u2 + 0:038xt
t

1 ut

(15)

2 THE SETUP

8

while the second order solution of the constant coe¢ cient model is
yt = 0:76xt

1

+ 0:95ut

0:01520x2
t

1

0:2375u2
t

0:038xt

1 ut

(16)

The hxx matrix di¤ers in the two cases because, in general, vt may a¤ect yt+1 and
thus alter higher order derivatives.
For higher order approximate solutions, the dynamics induced by structural
shocks in constant coe¢ cient and time varying coe¢ cient models will generally
di¤er, even with exogenous time variations. For example, in a third order approximation, the optimality conditions will feature terms in F x and F w ; which
require a correction of the linear terms to account for uncertainty. Since in
the constant coe¢ cient model some shocks are omitted, one should expect the
correction terms to di¤er in constant and time varying coe¢ cient models.

2.3

Discussion

The results we derived require parameter variations to be continuous and smooth.
This is in line with the evidence produced by Stock and Watson (1996) and with
the standard practice employed in time varying coe¢ cient VAR. Our framework is
‡
exible and can accommodate once-and-for-all breaks (at a known date), as long as
the transition between states is smooth. For example, a smooth threshold exogenously
switching speci…cation can be approximated with t+1 = (1
) + t + a exp(t
T0 )=(b + exp(t T0 )), t = 1; : : : ; T0 1; T0 ; T0 + 1; : : : T , where a and b are vectors,
while t+1 = (1
) + t + a exp( (Xt X))=(b + exp( (Xt X)); where X is the
steady state value of Xt ; can approximate smooth threshold endogenously switching
speci…cations. What the framework does not allow for are Markov switching variations,
occurring at unknown dates, as in Liu, et al. (2011), or abrupt changes, as in Davig
and Leeper (2006), since the smoothness conditions on the f function may be violated.
Note, however, that our model becomes a close approximation to a Markov switching
setup when the number of states is large.
It is important to emphasize that the (linear) solution we derive is a standard VAR
with …xed coe¢ cients and additional shocks. Thus, DSGE models with time varying
coe¢ cients do not generate new issues for aggregation or non-fundamentalness relative to a …xed coe¢ cient DSGE model. More importantly, it is incorrect to consider
time varying coe¢ cient VAR as the reduced form counterpart of continuously varying
coe¢ cient DSGE models. One can show that there exists a state space representation of the solution where the (exogenously) time varying coe¢ cients play the role of
additional states of the model. What Proposition 2.1 shows is that the state space
representation can be solved out to produce a standard VAR representation for the
endogenous variables. Moreover, the proposition indicates that the matrices P
and Q will be time varying only if x is itself time varying. Thus, to match the
time varying coe¢ cient VAR evidence, it is necessary to consider variations in DSGE
auxiliary parameters rather than variations in DSGE structural parameters.
Kulish and Pagan (2014) have developed solution and estimation procedures for
models with abrupt breaks and learning between the states. Their solution for

2 THE SETUP

9

the pre-break and post-break period is a constant coe¢ cient VAR, while for the
learning period is a time varying coe¢ cient VAR. Thus, a few words distinguishing
the two approaches are needed. First, they are interested in characterizing the
solution during the learning period, when the structure is unchanged, while we are
interested in the decision rule when parameters are continuously varying. Second,
their modelling of time variations is abrupt and the solution is designed to deal with
that situation. Third, in our setup expectations are varying with the structure; in
Kulish and Pagan they vary only in anticipation of a (foreseeable) break.
An alternative way of modelling time variations in (3) would be to make
parameters functions of the exogenous rather than the endogenous variables,
( ; Zt ; Ut+1 ) as, for example, in Ireland (2007). While the equations the
t+1 =
coe¢ cients of the decision rule solve are di¤erent, the conclusions we have derived
are unchanged by this modi…cation. For, example, in the …rst order approximation, P
now solves F P 2 +GP +H = 0; given P, Q solves V Q = vec(L z +M +N z z +O z )
and V = Inz (F P + G + F z ); and given P, R solves W R = vec(N u ! u + O u ),
where W = Inz (F P + G + F ! u ).
While there are obvious economic di¤erences between exogenous vs. endogenous
coe¢ cient variations, an alternative (statistical) way to think about the two speci…cations is that in the former each parameter evolves independently and covariations,
if they exist, can be modelled by selecting the matrix u to be of reduced rank.
With endogenous variations, instead a set of observable factors (the X’ drives coms)
mon parameter variations. Thus, u is diagonal and full rank, unless some parameter
variations are purely endogenous.
As (7) makes clear, it is hard to distinguish models with time varying coe¢ cients
from time invariant models with an additional set of shocks. In fact, models with n1
structural shocks and n2 time varying parameters, models with n = n1 + n2 structural shocks and models with n1 structural shocks and n2 measurement errors are
observationally equivalent:
xt = P xt

1

+ Qzt + Rut

(17)

= P xt

1

+ Q zt

(18)

= P xt

1

+ Qzt + vt

(19)

0
where Q = [Q; R]; zt = [zt ; u0 ]0 , vt = Rut . Thus, when designing time variation
t
diagnostics, one must rule out a-priori all these potentially observational equivalent
structures. In applications, procedures like the one described in section 3 or the one of
Seoane (2014) can be used to select the interpretation of the additional shocks.
Finally, it is useful to compare the (linear) solution we derive with the
solution obtained when coe¢ cients are constant but the volatility of the shocks
is stochastic. Neglecting second order terms, the solution in this latter case is
xt = P xt 1 + Qzt + A 2 : Thus, in empirical applications, it is crucial to allow for
t
stochastic volatility to avoid to misrepresent volatility changes for parameter
variations - a point made earlier by Sims (2001).

2 THE SETUP

2.4

10

An example

To convey some intuition into the mechanics of corollaries 2.2-2.3, we use a simple,
closed economy, RBC model. The representative agent maximizes
max E0

1
X

t(

t=1

1
Ct
1

A

Nt1+
)
1+

(20)

subject to the sequence of constraints
Yt (1

gt ) = Ct + Kt

(1

t )Kt 1

1
t Kt 1 Nt

Yt =

where Yt is output, Ct consumption, Kt the stock of capital, Nt is hours worked, and
G
gt = Y t is the share of government expenditure in output. The system is perturbed
t
by two exogenous structural disturbances: one to the technology Zt; and one to the
government spending share, gt , both assumed to follow time invariant AR(1) processes
ln

t

= (1

) ln +

ln gt = (1

g ) ln g

+

ln
g

t 1

ln gt

1

+ et
+ eg
t

(21)

where variables without time subscript denote steady state quantities. There are 12
parameters in the model: 6 structural ones ( is the capital share, the risk aversion
coe¢ cient,
the inverse of the Frisch elasticity of labor supply, A the constant in
front of labor in utility, t the time discount factor, and t the depreciation rate),
and 6 auxiliary ones (the steady state values of the government expenditure share and
of TFP, ( ; g); their autoregressive parameters, ( ; g ); and their standard deviations
( ; g )). We assume that all parameters but t and t are time invariant. Dueker
et al. (2007), Liu et al (2011), and Meier and Sprenger (2015) provide evidence that
these parameters are indeed evolving over time. The …rst order approximation to the
law of motion of ( t , t ) is described below.
The optimality conditions of the problem are:
ACt Nt
t Ct

(1

gt )Yt

= (1

gt )Yt =Nt
(1 gt+1 )Yt+1
= Et
+1
t+1 Ct+1 (
Kt+1
@ t+1
@ t+1
+ Et
u(Ct+1 ; Nt+1 )
Kt )
@Kt
@Kt
= Ct + Kt (1
t )Kt 1

Yt =

)(1

1
t Kt 1 Nt

(22)
t+1

(23)
(24)
(25)

Time variations in t and t a¤ect optimal choices in two ways. There is a direct
e¤ect in the Euler equation and in the resource constraint when t and t are time
varying; and if agents take into account that their decisions may a¤ect parameter

2 THE SETUP

11

variations, there will be a second (endogenous) e¤ect due variations in the derivatives
of t+1 and t+1 with respect to the endogenous states - see equation (23).
Note that varying parameters can not be considered wedges in the sense of
Chari et al. (2007), because there are cross-equation restrictions that need to be
satis…ed. Furthermore, while the rank of the covariance matrix of the wedges is
full, this is not necessarily the case in our setup.
We specialize this setup to consider various possibilities.

2.4.1

Model A: Constant coe¢ cients.

As a benchmark, we let
0

t

t

=

and

t

= . The optimality conditions are

Et [f (Xt+1 ; Xt ; Xt

1 ; Zt+1 ; Zt ;

B C
Et B t
@

)] =

1

+1

ACt Nt
(1
)(1 gt )Yt
Et Ct+1 ( (1 gt+1 )Yt+1 =Kt + 1
(1 gt )Yt Ct + Kt (1
)Kt 1
Yt
Kt 1 Nt1
t

) C
C=0
A

(26)

Xt = (Kt ; Yt ; Ct ; Nt )0 , Zt = ( t ; gt )0 : In the steady state, we have:

K
=
Y

(1 g)
C
;
=1
1 + 1=
Y

2.4.2

"

Model B: Exogenous parameter variations

Set dt =
postulate

t+1 = t .

We let

K
Y

g N
;
=
Y Y

ud;t+1 =
u
Since

t+1

is exogenous, @
0

B 1
Et B
@

1

(dt+1

t+1

;t+1

t+1 =@Kt

=
=@

K
Y

1

(1

d ud;t

u

;t

1

; Y =

) ;

t+1

(1

A
)(1

(1

C
g) Y
(27)

) )0 = Ut+1 and

+ ed;t+1

(28)

+e

(29)

t+1 =@Kt

Et [f (Xt+1 ; Xt ; Xt

;t+1

= 0 and the f function becomes
1 ; Zt+1 ; Zt ;

t+1 ;

+1

ACt Nt
(1
)(1 gt )Yt
dt Ct+1 =Ct ( (1 gt+1 )Yt+1 =Kt + 1
(1 gt )Yt Ct Kt + (1
t )Kt 1
Yt
Kt 1 Nt1
t

t+1 )

1

t )]

C
C=0
A

=

(30)

where Xt = (Kt ; Yt ; Ct ; Nt )0 , Zt = ( t ; gt )0 and t = 1t :
C
With the selected parameterization the steady state values of ( K ; Y ; N ; Y ) coinY
Y
cide with those of the constant coe¢ cient model. In addition, since x = 0, variations

N
Y

1+

#

1
+

:

2 THE SETUP

12

in (dt+1 ; t+1 ) leave the decision rule matrices P and Q as in model A. Thus, as far as
structural dynamics are concerned, models A and B are observationally equivalent.
To examine whether variations in t have an instantaneous impact on Xt , we need
to check the columns of N u ! u + O u .
0
1
0
0
B 1=
= C
C 6= 0
N u u+O u =B
(31)
@ 0
K A
0
0

Note that if dt were a fast moving variable, the impact e¤ect on Xt would depend
on the persistence of shocks to the growth rate of the discount factor. For example,
if d = 0, shocks to the growth rate of the time discount factor have no e¤ects on Xt .
Thus, if only the discount factor is time varying and variations in its growth rate are
i.i.d., models A and B have identical decision rules.

2.4.3

Model C: Endogenous parameter variations, no internalization

Assume that the time variations in the growth rate of the discount factor and in the
depreciation rate are driven by the aggregate capital stock. We specify
t+1

=[

u

(

u

l )e

a (Kt

K)

]+[

u

(

u

l )e

b (Kt

K)

]+U

;t+1

(32)

where a ; b ; u ; l are vectors of parameters and U ;t+1 is a zero mean, i.i.d. vector
of shocks. This speci…cation is ‡
exible and depending on the choice of 0 s;we can
accommodate linear or quadratic relationships, which are symmetric or asymmetric.
To ensure that models C and A have the same steady states, we set l = ( =2, =2).
We assume that agents treat the capital stock appearing in (32) as an aggregate
variable. This assumption is similar to the ’
small k -big k’ situation encountered in
standard rational expectations models or to the distinction between internal and external habit formation. Thus, agents’…rst order conditions do not take into account the
fact that their optimal capital choice changes dt and t so @ t+1 =@Kt = @ t+1 =@Kt = 0
and the equilibrium conditions are then as in (30).Since the f function is the same as
in model B, the matrices N and O are unchanged.
To examine whether parameter variations a¤ect the matrices regulating structural
dynamics note that
0
1
0 0
B 0 1= C (du
=2)( 11
21 ) 0 0 0
C
N x = B
(33)
@ 0 0 A ( u
=2)( 12
22 ) 0 0 0
0 0
0
1
0
0
B 1=
0 C (du
=2)( 11
21 ) 0 0 0
C
O x = B
(34)
@ 0
k A ( u
=2)( 12
22 ) 0 0 0
0
0

2 THE SETUP

13

Endogenous variations in dt ; t leave P and Q una¤ected, unless 1 6= 2 and/or 3 6=
4 , i.e. unless there are asymmetries in the law of motion of (dt ; t ). To verify whether
parameter variations impact on Xt , check the columns of N u ! u + O u . We have:
0
1
0
0
B 1= (du
C
=2)( 1 + 2 )
0
C 6= 0 (35)
N u!u + O u = B
@
0
K( u
=2)( 3 + 4 ) A
0
0
if

1

6=

2,

2.4.4

or

3

6=

4

and regardless of persistence of the shocks to the parameters.

Model D: Endogenous parameter variations, internalization.

We still assume that time variations in the discount factor and in the depreciation rate
are driven by the aggregate capital stock and by an exogenous shock, as in equation
(32). Contrary to case C, we assume that agents internalize the e¤ects their capital
decisions have on parameter variations. The relevant derivatives are
d0
t+1

@dt+1 =@Kt =

0
t+1

@

t+1 =@Kt

=

(
(

u
u

=2)[
=2)[

1e
3e

1 (Kt

K)

3 (Kt K)

+

+

2e
4e

K)

2 (Kt

4 (Kt K)

]

(36)

]

(37)

In order for the steady states of model D to equal to those of model A, we restrict
1 = 2 = 1 , 3 = 4 = 3 . The optimality conditions are:
0 = Et [f (Xt+1 ; Xt ; Xt

0

1 ; Zt+1 ; Zt ;

t+1 ;

t )]

+1

B 1
Et B
@

d0 u(Ct+1 ; Nt+1 )=Ct
t

ACt Nt
(1
)(1 gt )Yt
dt Ct+1 =Ct ( (1 gt+1 )Yt+1 =Kt+1 + 1
(1 gt )Yt Ct Kt + (1
t )Kt 1
Kt 1 Nt1
Yt
t

t+1

+

0
t+1 Kt )

where as before Xt = (Kt ; Yt ; Ct ; Nt )0 , Zt = ( t ; gt )0 but now t = (dt ; t ; d0 ; 0 )0 and
t t
0
1
1
0
1 (Kt K) + e 1 (Kt K) ] + U
2du (du
=2)[e
dt+1
;t+1
B t+1 C
B
=2)[e 3 (Kt K) + e 3 (Kt K) ] + U ;t+1 C
B
C
C = ( ; Kt ; Ut+1 ) = B 2 u ( u
@ d0
A
A
@
(du
=2) [ e 1 (Kt K) + e 1 (Kt K) ]
t+1
0
(Kt K) + e 3 (Kt K) ]
( u
=2) [ e 3
t+1
(39)
The relevant matrices of derivatives evaluated at the steady states are ! u = 02 2 ,

N=

x

@f
@
0

B
=B
@

t+1

0

0 0
B 0 1=
=B
@ 0 0
0 0
0
0

2(
2(

u
u

=2)
=2)

2
1
2
3

0
0
0
0

0
u(C; N )=C
0
0
1
0 0
0 0 C
C;
0 0 A
0 0

u

1
0
0
B
K C
C ; O = @f = B
0 A
@ t @
0
0
0
B
0
=B
@ 2( u
=2) 2
1
0
2(

0
1=
0
0

0
0
K
0

0
0
0
0
1

0
0
0
u

=2)

2
3

C
C:
A

1
0
0 C
C
0 A
0

1

=

C
C (38)
A

2 THE SETUP

14

Clearly, N x 6= 0, and N u ! u + O u = 0. Thus, a shock to the law of motion of
the parameters alters the dynamics produced by structural shocks, even when the
relationship between parameters and states is symmetric:
In sum, parameter variations matter for the structural dynamics either if the
relationship between parameters and the states is asymmetric or if agents internalize
the consequences their decisions have on parameter variations, or both.

2.4.5

Impulse responses

Why are structural dynamics in models C and D di¤erent from those in model A?
To understand what drives economic di¤erences, we compute impulse responses.
For the parameters common to all models, we choose = 0:30, = 0:99, = 0:025,
= 2, = 2, A = 4:50, =1; = 0:90,
= 0:00712, g = 0:18; g = 0:50 and
g = 0:01. For the other parameters, we choose:
Model B:

= 0:985;

Model C : 1 = 0:01;
0:999; u = 0:025.
Model D :
u = 0:999;

1
u

= 0:0001;
= 0:025.

= 0:95 and
= 0:03;

2

2

= 0:002
= 0:2;

1

= 0:016;

1

= 0:2;

= 0:07.
= 0:1,

2

2

= 0:1,

d

=

= 0:5,

d

= 0.0001;

u

=

= 0:1,

Figure 1 reports the responses of hours, capital, consumption, and output to the two
structural shocks in the four models. The …rst column has the responses to technology
shocks; the second has the responses to government expenditure shocks 1 .
Note …rst, that the sign of the responses is unchanged by the presence of
parameter variations. The responses of models C and D di¤er from those of model
A in the shape and the persistence of consumption and capital responses. Di¤erences
occur because income and substitution e¤ects are di¤erent. For instance, in response
to technology shocks, agents work and save less and consume more in models C and D
than in the constant coe¢ cients model, while in response to government expenditure
shocks, consumption falls more and capital falls less relative to the constant coe¢ cients
case. Thus, parameter variations play the same role as uncertainty variations and
make agents desire to smooth less transitory structural shocks.
1

Since the responses of hours and output to government expenditure shocks are di¤erent from what the
conventional wisdom indicates, a few words of explanation are needed. In a standard RBC in response to
government expenditure shocks, hours and output typically increase because of a wealth e¤ect. However,
here the shock a¤ects the share of government expenditure in GDP. Thus, the positive wealth e¤ect on
labor supply is absent because government expenditure increase in exactly the same proportion as output,
thus disincentivizing agents to try to increase private output.

3 CHARACTERIZING TIME VARYING MISSPECIFICATION

Te c h o lo g y s h o c k s

-3

x 10

15

G ex pendit ure s hoc k s

-3

x 10

Hours

0
5
-1
0
-2
-5
5

10

15

20

25

30

35

40

5

Capital
Consumption

20

25

30

35

40

5

10

15

20

25

30

35

40

-3

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

-0.005
-0.01
-0.015
-0.02
-0.025

0.2

5

10

15

20

25

30

35

40

x 10

0.025
0.02
0.015
0.01
0.005

Output

15

-4

0.4

10

-5
-10
-15
5

10

15

20

25

30

35

40

x 10
0

0.1
0.08
0.06
0.04
0.02

-2

-4
5

10

15

20

25

30

35

40

B

C

D

A

Figure 1: Impulse responses, …rst order approximation

3

Characterizing time varying misspeci…cation

Because the decision rules of constant coe¢ cient models are generally misspeci…ed
when the data generating process (DGP) features parameter variations, it is
important to diagnose potential time varying problems. This section describes two
diagnostics useful for the purpose: one based on ”
wedges”and one based on forecast
errors.
Consider the optimality conditions of a constant coe¢ cient model
Et F (Xt

1 ; Wt ;

z
t+1 ;

) =0

(40)

obtained substituting for Xt the decision rule:
Xt = h(Xt

1 ; Wt ;

z
t+1 ;

)

(41)

When Xt 1 has been generated by the constant coe¢ cient model, F is a martingale
di¤erence. When instead Xt 1 has been generated by a time varying coe¢ cient
model
Xt = h (Xt 1 ; Wt ;
(42)
t+1 ; )
E[F (Xt 1 ; Wt ;
F (Xt 1 ; Wt ;

z ;
t+1
z ; )
t+1

z
)] 6= 0; since
t+1 and h6= h : Furthermore,
t+1 6=
will be predictable using past values Xt 1 : To see why,

3 CHARACTERIZING TIME VARYING MISSPECIFICATION

16

consider the …rst order approximate optimality conditions. In this system of
equations, the wedge is
(F (P
(F (Q

Q)

z

+ G(Q

P )2 + G(P

Q) + F (P

(F (P

P )(G

P ))xt

1

+

G))zt +

P )R + GR + F R ! u )ut

(43)

When P = P; Q = Q; as in the exogenously varying model, the wedge reduces to
(GR + F R ! u )ut

(44)

which di¤ers from zero if R 6= 0 and will be predictable using xt j ; j 1; if ! u 6= 0:
When, as in the endogenously varying model, P 6= P; Q 6= Q; the wedge will di¤er
from zero, even when R = 0, and will be predictable using past xt 1 ; even when ! u = 0.
Hence, to detect time varying misspeci…cation, one can compute wedges and
regress them on the lags of the observables. If they are signi…cant, the
martingale di¤erence condition is violated, and there is evidence of time varying
parameters. Note that the diagnostic uses the assumption that the model is
correctly speci…ed up to parameter variations. If the model is incorrect, lags of
the observables may be signi…cant, even without time varying coe¢ cients. Inohue,
Kuo, and Rossi (2015) apply this idea to detect generic model misspeci…cation.
The logic of the forecast error diagnostic is similar. The linearized decision rule
in a constant coe¢ cients model is xt = P xt 1 +Qzt , while in a time varying coe¢ cient
model it is xt = P xt 1 + Q zt + R ut . Let vt be the forecast error in predicting xt
using the decision rules of the constant coe¢ cient model and the data generated from
the time varying coe¢ cient model: The forecast error can be decomposed as
vt = xt j

P xt

1

= Q zt + R ut + (P

P )xt

1

(45)

Thus, forecast errors are functions of the lags of the observables xt 1 when P 6= P:
However, even if P = P , forecasts error linearly depend on the lags of the observables
if ut is serially correlated. Hence, an alternative way to check for parameter variations involves regressing the forecast errors vt on lagged values of the observables and
checking the signi…cance of the regression coe¢ cients.
We apply the two diagnostics to 1,000 samples constructed using the RBC
model previously considered. Table 1 reports the rejection rate of an F-statistic for
the null hypothesis of no time variations at the 0.05 percent con…dence level.
The Euler wedge diagnostic has very good size properties (does not reject the
hypothesis of no time variations) when the model has …xed coe¢ cients; when
it has …xed coe¢ cients but it is locally misspeci…ed - capacity utilization is
neglected; and when the exogenous time variations are i.i.d.. It is somewhat
conservative in detecting time variations when exogenous parameter variations are
persistent and has excellent power properties when variations are endogenous.
The forecast error diagnostic has good size properties when the DGP has no
time variation and no misspeci…cation is present but tends to overreject the null

3 CHARACTERIZING TIME VARYING MISSPECIFICATION

17

DGP

Euler wedge
Forecast errors output
F-test ct 1 ; rt 1 = 0 F-test ct 1 ; nt 1 ; yt 1 = 0
T=1000
T=150
T=1000
T=150
Fixed coe¢ cients
0.00
0.00
0.00
0.00
Fixed coe¤ and capacity utilization
0.001
0.003
1.00
0.98
Exogenous TVC no serial correlation
0.07
0.001
0.91
0.24
Exogenous TVC
0.53
0.40
1.00
0.90
Endogenous TVC
1.00
0.93
1.00
0.99
Table 1: Percentage of rejections at the 0.05 con…dence level of the null of no time variations
in 1000 experiments. The dependent variable is either the Euler wedge or the forecast error
in the output equation. The regressors are lagged consumption and interest rates for the
Euler wedge; lagged output, consumption and hours for the forecast error.
if misspeci…cation is present or exogenous time variations are i.i.d.. On the other
hand, it has good power properties when time variations are present. Because
of the di¤erences they display, it seems wise to use both diagnostics in empirical
applications.

3.1

Exogenous vs. endogenous parameter variations

If the diagnostics of the previous subsection indicate the presence of parameter
variations, one may interested in knowing whether they are of exogenous or
endogenous type. One way to distinguish the two options is to use the DGSE-VAR
methodology of Del Negro and Schorfheide (2004). In a DSGE-VAR, one uses the
DSGE model as a prior for the VAR of the observable data and employs the
marginal likelihood to measure the value of the additional information the DSGE
provides. If the additional observations come from the DGP, the quality of
the estimates improves (standard errors are reduced), and the marginal likelihood
increases. On the other hand, if the additional observations come from a DGP
di¤erent from the one generating the data, biases may be introduced, noise added,
and the precision of the estimates and the …t of the model reduced.
Formally, let L( jy) be the likelihood of the VAR model for data y and let gj ( j j ; Mj )
be the prior induced by the DSGE model Mj using parameters j on the VAR paR
rameters :The marginal likelihood is hj (yj j ; Mj ) = L( jy)gj ( j j ; Mj )d ; which,
for given y; is a function of Mj . Since L( jy) is …xed, hj (yj j ; Mj ) re‡
ects the plausibility of gj ( j j ; Mj ) in the data. Thus, if g1 and g2 are two DSGE-based priors and
h1 (yj 1 ; M1 ) > h2 (yj 2 ; M2 ), there is better support for in the data for g1 .
Thus, for a given data set, a researcher comparing the marginal likelihood
produced by adding data from the exogenous and the endogenous speci…cations
should detect whether the observable sample is more likely to be generated
by one of the two models. We prefer to use the DSGE-VAR device rather

3 CHARACTERIZING TIME VARYING MISSPECIFICATION

18

than comparing the marginal likelihood of di¤erent models directly because small
samples may led to distortions in marginal likelihood comparisons, distortions that will
be reduced in our DSGE-VAR setup.

T1 =150
T1 =750
DGP
Model B Model C Model D Model B Model C Model D
Simulated from B 1.00
0.00
0.00
0.99
0.00
0.00
Simulated from C 0.01
0.99
0.00
0.00
0.98
0.00
Simulated from D 0.00
0.00
1.00
0.00
0.00
0.99
Table 2: Probability that Bayes factor exceeds 3.0 in a sample of 1,000 experiments. Marginal
likelihoods are obtained using T=150 data points produced by the models listed in the …rst
row and T1 simulated data from the model listed in the …rst column. When rows do not sum
to one, the Bayes factor is inconclusive (below 3.0).
Table 2 reports results using this technology in the RBC example. The sample
size is T = 150 and Bayes factors computed when T1 = 150; 750 simulated data from
the DSGE listed in the …rst row are added to the actual data and 1,000 experiments
are run. The statistic is powerful since marginal likelihood di¤erences are quite large,
even when T1 = 150.

3.2

Some practical suggestions

Given that, in practice, we do not know if a model is misspeci…ed or not, we
suggest users the following checklist as a way to approach the diagnostic problem:
i) Take a conventional model that has been used and tested in the literature and
estimate its structural parameters, potentially allowing for time variations in the
variance of the shocks.
ii) Run the time variation diagnostics and, if time variations are found to be
present, check whether endogenous vs. exogenous variations are more appropriate.
When the model is of large scale, running regressions on all potential endogenous
variables leads to overparameterization and muticollinearity. Thus, it is important
to select the relevant variables to make the test powerful. We recommend users to
employ the states of the model, as they determine the endogenous variables. Similarly,
when performing the exogenous vs. endogenous check, having the proper state
variables for the endogenous speci…cation is important to make the comparison
fair. One way do this is to estimate a model with exogenous time variation, take
the smoothed residuals and run auxiliary regressions of the smoothed residuals
on potential determinants of time variations. To avoid overparameterization, we
also suggest users to a-priori shrink the coe¢ cients of the auxiliary diagnostic
regressions toward zero. Rejection of the null of no time variations in this case
provides stronger con…dence that parameter variations are indeed present.

4 PARAMETER IDENTIFICATION
When the diagnostics detect time variations, one needs to specify which parameter
may be time varying for the next stage of the analysis. In theory, one could
specify time variations in all the structural parameters of interest, but this may
lead again to an overparametrized model, which is di¢ cult to estimate. We suggest
two approaches here: either introduce time variations in parameters which have been
documented in the literature to be unstable or in parameters a researcher suspects
variations to be present. Alternatively, one could look at the smoothed residuals of
the time invariant model, equation by equation, and restrict time variations to the
parameters appearing of the equations whose residuals show the largest evidence
of serial correlation.

4

Parameter identi…cation

Since forecast errors are used to construct the likelihood function via the Kalman
…lter, one should expect the misspeci…cation present in the forecast errors
to spread to the likelihood function. In this section we examine whether time
invariant parameters can be identi…ed from a potentially misspeci…ed likelihood
function. Canova and Sala (2009) have shown that standard DSGE models
feature several population identi…cation problems, intrinsic to the models and to
the solution method employed. The issue we are concerned with here is whether
parameters that could be identi…ed if the correct likelihood is employed became
poorly identi…ed when the wrong likelihood is used. In other words, we ask whether
identi…cation problems in time invariant parameters may emerge as a byproduct
of neglecting variations in other parameters. Magnusson and Mavroedis (2014) have
shown that when GMM is used, time variations in certain parameters help the
identi…cation of time invariant parameters. Huang (2014) quali…es the result by
showing that time variations in weakly identi…ed parameters have no e¤ect on the
asymptotic distribution of strongly identi…ed parameters.
Figures 2 and 3 plot the likelihood function of the RBC model in the risk
aversion coe¢ cient and the share parameter ; and in the labor share
and
the autoregressive parameter of the technology ; when the forecast errors of the
correct model (top row) and of the constant coe¢ cient model (bottom row) are used
to construct the likelihood function. The …rst column considers data generated
by the model B, the second and the third data generated by models C and D.
While the likelihood curvature in the correct model is not large, it is easy to
verify that the maximum occurs at = 2; = 2; = 0:30;
= 0:9 for all three
speci…cations. When the decision rules of the constant coe¢ cients model are used
to construct the likelihood function and the true DGP is model B, the likelihood
is ‡
attened and the risk aversion coe¢ cient become very weakly identi…ed. When
the true model features endogenous time variations, distortions are larger. The
likelihood function becomes locally convex in
; and become weakly identi…ed,
and the maximum in the
is shifted away from the true value.

19

4 PARAMETER IDENTIFICATION

20

True R BC B - Estim ated with RBC B
True R BC C - Estim ated with R BC C
True R BC D - Estim ated with R BC D

log likeli

2900

3250

3220

2880

3210
3200

2860

3200

2840
2.5

2.5

2

γ

3150
2.5
2

2
1.5

1.5

2.5

γ

η

3190
2.5

1.5

1.5

2.5

2

2

γ

η

2
1.5

1.5

η

True R BC B - Estim ated with RBC A
True R BC C - Estim ated with R BC A
True R BC D - Estim ated with R BC A

log likeli

1000

2000

2800

1800
500

2600
1600

0
2.5

2.5

2

γ

1400
2.5
2

2
1.5

1.5

2.5

γ

η

2400
2.5

1.5

1.5

2.5

2

2

γ

η

2
1.5

1.5

η

Figure 2: Likelihood surfaces

True R BC B - Estim ated with R BC B
True R BC C - Estim ated with R BC C
True R BC D - Estim ated with R BC D

log likeli

2900

3400

3500

3200
2850

3000
3000

2800
0.35

1

0.3

α

2800
0.35
0.3

0.9
0.25

0.8

ρ

1

α

ζ

2500
0.35

0.25

0.8

ρ

1

0.3

0.9

α

ζ

0.9
0.25

0.8

ρ

ζ

True R BC B - Estim ated with R BC A
True R BC C - Estim ated with R BC A
True R BC D - Estim ated with R BC A

log likeli

2000

2500

3000

2000
0

2500
1500

-2000
0.35

1

0.3

α

0.9
0.25

0.8

ρ

ζ

1000
0.35

1

0.3

α

0.9
0.25

0.8

Figure 3: Likelihood surfaces

ρ

ζ

2000
0.35

1

0.3

α

0.9
0.25

0.8

ρ

ζ

4 PARAMETER IDENTIFICATION

21

These observations are con…rmed by the Koop et al. (2013) statistic, see table 4.
Koop et al. show that asymptotically the precision matrix grows at the rate T for
identi…ed parameters and at rate less than T for underidenti…ed parameters. Thus,
the precision of the estimates, scaled by the sample size, converges to a constant for
identi…ed parameters and to zero for underidenti…ed parameters. Furthermore, the
magnitude of the constant measures identi…cation strength: a large value indicates a
strongly identi…ed parameter; a small value a weakly identi…ed one.
Parameter T=150 T=300 T=500 T=750 T=1000 T=1500 T=2500
DGP Model B, Estimated model A
15.9
17.8
17.2
18.8
18.4
19.3
17.9
28.5
45.7
108.4
81.4
93.6
104.2
90.17
1.8e+4 2.6e+4 4.2e+4 4.2e+4 4.5e+4 4.9e+4 4.37e+4
z
209.2
655.5
2741
2190
2860
3417
2802
g
927.3
973.8 1.7e+4 1.7e+4 2.4e+4 2.3e+4 2.5e+4
140.2
156.2
264.2
215.5
239.1
252.1
229.3
A
28.42
30.67
7.99
10.99
9.15
7.83
9.83
DGP Model C, Estimated model A
822
1033
743
785
759
746
752
2261
3147
2682
2809
2720
2579
2566
3073
2673
2952
2909
2799
2806
2877
z
1.74
2.23
2.44
2.96
3.17
2.82
2.90
g
4.6e+5 4.4e+5 4.3e+5 4.0e+5 3.8e+5 4.4e+5 4.3e+5
1.8e+4 1.1e+4 1.4e+4 1.2e+4 1.1e+4 1.6e+4 1.5e+4
A
351
493
441
505
500
449
444
DGP Model D, Estimated model A
550
575
592
610
545
542
494
3577
2442
2660
2870
2564
2711
2430
1613
1243
1120
1162
1068
1189
1074
z
1.22
1.28
1.44
1.53
1.60
1.62
1.67
g
5.2e+5 6.7e+5 6.5e+5 6.0e+5 5.7e+5 5.8e+5 5.7e+5
1.1e+4 2.5e+4 2.4e+4 1.9e+4 2.1e+4 2.0e+4 2.1e+4
A
488
276
340
382
349
395
334

Table 3: Koop, Pesaran, and Smith diagnostic. Reported are the diagonal elements of the
precision matrix scaled by the sample size
When the DGP is model B and a …xed coe¢ cients model is considered, all parameters are identi…ed, even though A and are only weakly identi…ed. When the DGP
are models C and D, all parameters but g seem identi…able. Interestingly, in models
C and D, g is weakly identi…ed, even when the correct likelihood is used. Thus, time
variations in t and t do not help in the identi…cation of g , in line with Huang (2014).

5 STRUCTURAL ESTIMATION WITH A MISSPECIFIED MODEL

5

Structural estimation with a misspeci…ed model

To study the properties of likelihood -based estimates of a misspeci…ed constant
coe¢ cients model, we conduct a Monte Carlo exercise. We generate 150 or 1,000
data points from versions B, C, D of the RBC model previously considered, estimate
the structural parameters using the likelihood function constructed with the decision
rules of the time invariant model A, and repeat the exercise 150 times using di¤erent
shock realizations. We also estimate the structural parameters using the likelihood
constructed with the correct decision rules (i.e. model B rules if the data is generated
with model B, etc.) for benchmarking estimation distortions.
We consider two setups: one where parameter variations are small (2-5 percent of
the variance of output is explained by shocks to the parameters; henceforth, DGP1) and
one where parameter variations are substantial (around 20 percent of the variance of
output is explained by shocks to the parameters; henceforth, DGP2). Table 4 has the
results for DGP1: it reports the …xed parameters used to generate the data (column
1), the mean posterior estimate (across replications) obtained when the likelihood uses
the correct decision rules (column 2), and the mean posterior estimate, the 5th and
the 95th percentile of the distribution of estimates obtained when the likelihood
function uses the decision rules of the time invariant model, when T=150 (columns
3-5) and when T=1000 (columns 6-8). Table A1 in the appendix has the results for
DGP2. Figures A1 and A2 in the appendix plot the distributions of estimates for
the two DGPs. The vertical line represents the true parameter value; solid black
lines represent distributions obtained with the correct model; solid blue (red) lines
represent the distributions obtained with the incorrect constant coe¢ cient model
when T=150 (T=1000). When the model is correctly speci…ed, the distribution of
estimates should collapse around the true value. Thus, if the mean is away from
the true parameter value and/or the spread of the distribution is large, likelihood
-based methods have di¢ culties in recovering the constant parameters of the data
generating process. Figure 4 presents the impulse responses for DGP1: the …rst two
columns have the responses to technology shocks and government expenditure shocks
in model B, the next two the responses in model C, and the last two the
responses in model D. In each box we report the response obtained using mean
value of the correct distribution of estimates, and the 16th and 84th percentiles of
the distribution of responses obtained using the estimated distribution of parameters
produced by the time invariant model. Figure A3 in the appendix has the
same information for DGP2. Table 5 presents the long run variance decomposition
for DGP1 (table A2 has the information for DGP2) when T=150 and the mean
posterior estimate is used in the computations. In the …rst two columns we have the
contribution of technology and government spending shocks in the correct model; the
last two columns have the contribution when the constant coe¢ cient model is used.
For the two time varying parameters, we set dt = t+1 = t ; and assume that in
model B, t+1
(dt+1 (1
) ; t+1 (1
) )0 = Ut+1 , where
= 0:99; =
0 are independent AR(1) process with
0:025 the components of Ut+1 = (ud;t+1 ; u ;t+1 )

22

5 STRUCTURAL ESTIMATION WITH A MISSPECIFIED MODEL
persistence d = 0:9;
= 0:8; and standard deviations d = 0:002;
= 0:07: For
models C and D, the law of motion of the time varying parameters is t+1 = [ u
(Kt K) ] + [
(Kt K) ] + U
0 = (0:9999; 0:03);
( u
( u
u
t+1 ; where
l )e a
l )e b
u
0
0
a = (0:03; 0:2); b = (0:031; 0:1); Ut+1 is i.i.d. with u =diag(0.03,0.008).
True value Estimated Correct Estimated Time invariant
Estimated Time invariant
Mean
Mean 5 percentile 95 percentile Mean 5 percentile 95 percentile
T=150
T=150
T=1000
DGP Model B
= 2:0
2.00
2.03
1.47
2.88
2.32
1.55
3.37
= 2:0
2.02
1.23
-0.14
2.07
0.96
-0.38
2.04
0.97
0.99
0.97
1.00
0.99
0.96
1.00
z = 0:98
= 0:5
0.47
0.74
0.60
0.96
0.87
0.77
0.98
g
= 0:025
0.03
0.01
0.01
0.02
0.01
0.01
0.05
= 0:3
0.30
0.19
0.11
0.28
0.23
0.15
0.40
A = 4:5
4.55
2.79
1.33
4.12
2.68
1.23
4.06
DGP Model C
= 2:0
2.00
2.42
1.63
3.85
2.85
1.73
6.14
= 2:0
2.00
0.64
-0.26
1.77
0.60
-0.50
1.79
0.98
0.99
0.97
1.00
0.97
0.85
1.00
z = 0:98
= 0:5
0.48
0.43
-0.10
0.96
0.65
0.27
0.98
g
= 0:025
0.03
0.01
0.01
0.02
0.02
0.01
0.09
= 0:3
0.30
0.22
0.13
0.34
0.29
0.18
0.47
A = 4:5
4.49
2.14
1.18
3.47
2.37
1.18
3.66
DGP Model D
= 2:0
2.00
2.58
1.69
3.34
2.40
1.74
3.26
= 2:0
2.01
0.29
-0.28
1.54
1.09
-0.30
1.99
= 0:97
0.96
0.99
0.94
1.00
0.96
0.91
1.00
z
0.48
0.51
-0.26
0.96
0.66
0.39
0.98
g = 0:5
= 0:025
0.02
0.01
0.01
0.03
0.01
0.01
0.02
= 0:3
0.30
0.22
0.14
0.35
0.22
0.15
0.30
A = 4:5
4.52
2.32
1.42
3.68
3.45
1.37
4.51

Table 4: Distributions of estimates, DGP1.
A few features of the results are worth discussing. First, when the correct model
is employed, estimation is successful even when T=150, regardless of the DGP and
of whether time variations are exogenous or endogenous. Thus, numerical distortions
seem minor. Second, with DGP1, a number of distortions occur when a time invariant
model is used in estimation. For example, when exogenous variations are present, the
persistence of government spending shock is poorly estimated (mean persistence is
about 50 percent larger than the true one), while estimates of ; and A are severely

23

5 STRUCTURAL ESTIMATION WITH A MISSPECIFIED MODEL

24

biased downward. The distortions are smaller when the time variations are endogenous
(models C and D). Nevertheless, signi…cant downward biases exist in the inverse of the
Frisch elasticity , in and . Third, the performance of the time invariant model is
roughly independent of whether the data features external or internal endogenous
time variations and does not improve when the sample size increases.
When parameter variations explain a signi…cant portion of output variability, all
features become more striking. For example, when parameter variations are exogenous,
estimating a time invariant model leads to an overestimation of the persistence of the
structural shocks. In fact, the only way a time invariant model can accommodate the
additional dynamics and variability present in the endogenous variables is by increasing
the persistence of both shocks. In models C and D the distortions become considerably
larger and, for example, the mean posterior estimate of inverse of the Frisch elasticity
is now negative. In addition, the distribution of estimates is typically skewed and
multimodal. Thus, neglected parameter variations are more detrimental when they
account for a signi…cant portion of the variability of the endogenous variables.

-3

x 10

Te c h n o lo g y

-3

x 10

Hours

0

G .e x p e n d itu r e

-3

x 10

Te c h n o lo g y

-3

x 10

0

0

-0.5

G .e x p e n d itu r e

-3

x 10

-3

x 10

0
-2

-2
-4

-2

-2
-4

-2

-6

-3

-2

10

20

30

-3

-6

-3

-2.5

G .e x p e n d itu r e

0

-1

-1
-1

-1
-1.5

Te c h n o lo g y

1

0

-8

-4

40

10

20

30

40

10

20

30

40

10

20

30

40

10

20

30

40

10

20

30

40

10

20

30

40

10

20

30

40

30

40

-3

x 10
0.06

0.06

Capital

0.05

-5

0.06

0.05

0.04
0.03

-10

0.04

-0.02

0.03

0.02

0.02
-15

0.01
10

20

30

10

-3

20

30

40

10

-3

x 10

20

30

40

10

-3

x 10

20

30

-0.03

40

10

-3

x 10

-0.02

0.02

-0.03

0.01

40

-0.01

-0.01

0.04

20

30

40

-3

x 10

-3

x 10

x 10

10

Output

0
8

0

8

0

-4

-2

6

10

-2

8

-1

6

-5
6

-6

-3

4

-8

-10

4

4

-10

-4
10

20

30

40

10

-3

20

30

40

10

-4

x 10

20

30

10

-3

x 10

2

40

20

30

40

10

-4

x 10

20

30

40

-3

x 10

-4

x 10

x 10

Consumption

5
-1

4.5

4.5

4

-2

2.5

-6

-2

3

-4

2

4
-2

-4

3.5

3.5

3

-3

3

-6

2.5
2.5

-8
10

20

30

Mo d e l B- A

40

10
A84

20
A16

30

40
B50

10

20

30

Mo d e l C - A

40

10
A84

20

30
A16

40
C50

10

20

30

Mo d e l D - A

40

10

20
A84

Figure 4: Impulse responses, DGP1
Impulse responses are in line with these conclusions. When parameter variations
explain a small fraction of the variability of output, responses to technology shocks are
o¤ in terms of impact magnitude, in particular for output; and the response produced
with estimates of the true model tend to be on the upper limit of the estimated 68
percent band produced with estimates of the incorrect model. Interestingly, output
responses are those more poorly characterized and, consistent with previous …ndings,
the misspeci…cation is larger when the true model features exogenous time variations.

A16

D50

5 STRUCTURAL ESTIMATION WITH A MISSPECIFIED MODEL
The responses to government expenditure shocks obtained with a time invariant model
are di¤erent from those obtained estimating the correct model in terms of magnitude,
shape and persistence. Since the signal that government expenditure shocks produce
is weak, it is not surprising that it is obscured by the presence of parameter variations.
The dynamic distortions obtained when parameter variations matter for the variance of output are generally larger. For example, the persistence of the responses to
technology shocks is poorly estimated. While true responses tend to zero, the bands
obtained estimating a time invariant model do not include zero even after 10 years.
Variable Technology Government Technology Government
DGP: Model B
Estimated: Time invariant
Y
94.100
0.300
0.997
0.004
C
89.500
0.200
0.999
0.001
N
60.200
0.500
0.986
0.014
K
70.200
0.400
0.995
0.006
DGP: Model C
Estimated: Time invariant
Y
97.200
0.300
0.988
0.016
C
88.100
0.300
0.999
0.001
N
44.600
0.600
0.990
0.012
K
84.400
0.200
0.990
0.014
DGP: Model D
Estimated: Time invariant
Y
98.000
0.100
0.993
0.015
C
92.200
0.200
0.998
0.003
N
35.900
0.500
0.973
0.034
K
96.600
0.300
0.992
0.012

Table 5: Long run variance decomposition, DGP1.
What is the contribution of structural shocks to the variability of the endogenous
variables when the forecast errors of the time invariant model are used to construct
the likelihood function? One should expect the structural shocks of the time invariant
model to be a contaminated version of the structural shocks of the time varying DGP
for two reasons. First, the wrong P matrix is used to compute forecast errors. Second,
we are aggregating m (primitive and parameter) shocks into n < m (structural) shocks,
thus generating VARMA decision rules where the n structural shocks are functions of
the leads and lags of the original disturbances (see e.g. Canova and Paustian, 2011).
Thus, even if the P matrix were correctly speci…ed, distortions should occur, unless the
shocks to the parameters are unimportant and feature low persistence.
When parameter variations explain a small portion of output, technology shocks in
the time invariant model absorb the missing variability, regardless of the nature of
parameter variations and the e¤ect seems strong for hours worked. When parameter
variations explain a larger portion of the variance of output, technology shocks still
absorb a large amount of the missing variability but, in some cases, spending shocks

25

6 STRUCTURAL DYNAMICS AND SVAR METHODS

26

also capture the missing variability see, e.g., the case of endogenous variations 2
In sum, for the DGP we consider and the parameterization employed,
estimating a constant parameter model when the DGP features time varying
parameters leads to distortions, regardless of the sample size, of whether variations
are exogenous or endogenous, and of whether parameter variations matter for output
variability or not. The parameters mostly a¤ected are those regulating the estimated
persistence of the shocks and those controlling income and substitution e¤ects.

6

Structural dynamics and SVAR methods

The previous section showed that if time variations are neglected, structural estimates are biased and structural responses distorted. Because of these problems, one
may wonder whether less structural and computationally less demanding methods
can be used if structural dynamics are all that matters to the investigator. Canova
and Paustian (2011) showed that when the model misses features of the DGP,
SVAR methods employing robust sign restrictions can be e¤ective in capturing
qualitative features of structural dynamics. Here we ask if SVARs are good also
when parameter variations are neglected.
The exercise is as follows. Using the illustrative RBC model, we simulate data
from the decision rules of models B, C, and D when parameter variations generate
small output volatility (DGP1). We then estimate a VAR, compute residuals, and
rotate them using an orthonormal matrix. We then keep the resulting impulse
responses if (simultaneously) technology shocks generate a positive response of
hours, capital, output, and consumption on impact and government expenditure
shocks generate a negative response of hours, output, consumption and capital.
These restrictions hold in the four model speci…cations we consider and are robust
to variations of the (constant) structural parameters within a reasonable range.
We repeat the exercise 150 times and collect the distribution of structural
responses when the correct and the time invariant SVAR speci…cations are used.
Figure 5 plots the median response in the correct model (red line) and the 16th and
84th percentiles of the distribution of responses obtained with the time invariant
model.
Overall, SVAR methods are competitive with structural methods when parameter
variations are neglected. When the DGP is model B, the sign and the shape of
the responses are correctly captured. Although the responses to technology shocks
obtained with the correct model are on the upper bound of the band obtained
with the time invariant model and the responses to government spending shocks
2

We have also performed a Monte Carlo exercise allowing the labor share to be time varying. Variations
in the labor share have been documented in, e.g., Rios Rull and Santaeularia Llopis (2010), and there is
evidence that they are strongly countercyclical. This is relevant for our exercise because all four optimatility
conditions are a¤ected by time variations, altering the strength of the income and substitution e¤ect
distortions. Indeed, we do …nd that distortions become quite large and it many cases it becomes di¢ cult to
estimate the time invariant model regardless of the DGP (results available on request).

7 TIME VARYING FINANCIAL FRICTIONS?

27

obtained with the correct model tend to be on the lower bound of the bands
obtained, no major distortions occur. The performance with the other two DGPs
is similar. With model C, it is the magnitude of the response of consumption
that is mainly misrepresented, while with model D it is primarily the persistence
of certain responses that is underestimated.
T4e c h n o lo g y s h o c k s

G - 4e x p e n d itu r e s h o c k s

x 10

x 10

T4e c h n o lo g y s h o c k s

G - 4e x p e n d itu r e s h o c k s

x 10

x 10

T4e c h n o lo g y s h o c k s

Hours

0

0

-5

10

15

-5
-1 0

-5

-1 0

-5

-1 5

-1 0
5

0

0

-5
-5

-1 5

x 10
5

5

0

-1 0

G - 4e x p e n d itu r e s h o c k s

x 10

5
0

5

10

15

-1 0

-1 0

-1 5

20

20

5

10

15

20

5

10

15

20

5

10

15

20

5

10

15

20

5

10

15

20

5

10

15

20

5

10

15

20

0 .0 5
-0 .0 1

-0 .0 1

0 .0 4

-0 .0 2

-0 .0 3

0 .0 3

-0 .0 3

-0 .0 4

0 .0 2

-0 .0 5

0 .0 1

-0 .0 1

0 .0 3

0 .0 5

-0 .0 2

0 .0 4

-0 .0 2

Capital

0 .0 4
0 .0 3
0 .0 2

-0 .0 3

0 .0 1

-0 .0 4

0 .0 2

5

10

15

20

5

-3

10

15

20

5

-3

x 10

10

15

-0 .0 6

20

5

-3

x 10

10

15

20

5

-3

x 10

10

15

20

-3

x 10

-3

x 10

x 10

-2

7

-2

-2
6

6

6

-4

Output

-0 .0 4
-0 .0 5

0 .0 1

-4

5

-4
4

4

4

-6

-6

3

-6
2

-8

2

2
-8

-8
5

10

15

20

5

-3

15

20

5

15

20

5

10

15

20

5

-3

10

15

20

-3

x 10

x 10

-0 .5

3

-3

x 10

x 10
-0 .5

8

4

-2

-1

2 .5

10

-3

x 10

3 .5

Consumption

10

-3

x 10

-1 .5

6

-1
-1 .5

3
-4

2

-2

1 .5

-2 .5

-2

4

2
-2 .5
-6

2

1

-3
5

10

15

M o d e ls B - A

20

5
upper 84

10

15

l o we r 1 6

20
m e a n t ru e

-3

1
5

10

15

M o d e ls C - A

20

5
upper 84

10
l o we r 1 6

15

20
m e a n t ru e

5

10

15

M o d e ls D - A

20

upper 84

l o we r 1 6

m e a n t ru e

Figure 5: Impulse responses, DGP1, SVAR models.
Recall that there are two sources of misspeci…cation in time invariant models:
the P matrix is generally incorrect; aggregation problems are present. Our analysis
indicates that, with the DGP we use, i) distortion in the P matrix are small; ii)
the Q matrix is not very strongly a¤ected by time variations; iii) shock misaggregation
is minor. Because parameter shocks are i.i.d., timing distortions are also small.

7

Time varying …nancial frictions?

We apply the technology we developed to the unconventional monetary policy model
of Gertler and Karadi’ (GK) (2010). Our contribution is three fold. We provide
s
likelihood estimates of the parameters speci…c to the model (the fraction of capital
that can be diverted by banks , the proportional transfer to entering bankers !, and
the survival probability of bankers ), which the authors have informally calibrated
to match a steady state spread, a steady state leverage, and a notional length of
bank activity; we use the diagnostics we developed to gauge the extent of parameter
variations; we estimate a model with time variations in and compare its …t with the

7 TIME VARYING FINANCIAL FRICTIONS?

28

…t of the time invariant model augmented with an extra shock; and examine responses
to capital quality shocks in the …xed coe¢ cient and the time varying coe¢ cient models.
The equations of the GK model are summarized in appendix C. We use U.S. data
from 1985Q2 to 2014Q3 on the growth rate of output, growth rate of consumption,
growth rate of leverage, and growth intermediary demand for assets (credit) and the
spread. The spread is measured by the di¤erence between BAA 10-year corporate
bond yields and a 10-year treasury constant maturity, and it is from the FRED, as
are real personal consumption expenditures and GDP data. Leverage is from Haver
and measures Tier 1 (core) capital as a percent of average total assets. Credit is
measured as total loans (from Haver), scaled by size of US population. While the
data transformation is su¢ cient to eliminate volatility variations, the credit and
the spread variable display a signi…cant structural break in the last few years of the
sample. Thus, we present estimates obtained in the full sample and in the sample
ending at 2007:4.
Equation

T-stat
Creditt 1 Leveraget 1 Spreadt
1
1
Sample 1985:3-2014:3
Y
0.84 2.61
0.24
0.52
10.00
C
-0.85 1.11
0.85
-0.65
0.33
Credit
1.06 2.61
1.65
-0.58
8.49
Leverage -1.11 -2.50 -1.63
0.63
-8.25
Spread -1.26 -3.06 -1.10
0.81
-8.46
Sample 1985:3-2007:4
Y
-1.79 3.87 -2.23
-0.38
6.86
C
-1.37 1.19 -0.26
0.38
1.40
Credit -1.18 3.53 -0.69
-0.08
7.02
Leverage -1.06 -3.46 0.75
0.09
-6.80
Spread 1.16 -3.84 -1.03
0.17
-6.86
Yt

Ct

F-stat
1

4.39
1.26
7.11
7.04
8.16
4.23
0.81
3.60
3.72
4.29

Table 6: Regression diganostic for time variation. The left-hand side of the regression is the
forecast in the equation listed in the …rst column; the right-hand side the variables listed in
the second to the …fth column. Critical levels of the F-stat(5,112)=2.56 and F(5,85)=2.90.
Using a ‡ prior, the posterior mode estimates for the full sample are = 0:245,
at
= 0:464, ! = 0:012; the standard errors are tight (0.0182, 0.0098, 0.0008), making the
estimates highly signi…cant. For the shorter 1985-2007 sample the modal estimates
are = 0:178, = 0:399, ! = 0:008; and the standard errors are 0.0127, 0.0129, 0.0006.
Thus, while estimates of the three crucial parameters are altered when data for
the last …nancial crisis is used, di¤erences are small a-posteriori. For comparison,
GK calibrated these three parameters to = 0:318, = 0:972, ! = 0:002. In the GK
model regulates private leverage: our estimate implies a higher steady state leverage
than the one implied by the authors (our estimate for the full sample is 3.32, GK is

7 TIME VARYING FINANCIAL FRICTIONS?

29

1.38), which is closer to the leverage found in the U.S. in corporate and non-corporate
business sectors over the sample. Our estimates also suggest that bankers’ survival
probability is lower than the one assumed by GK (about 10 years).
With the parameter estimates obtained in the full sample, we perform our diagnostics. Table 6 indicates that the forecast errors of all equations except consumption are
predictable and, typically, lagged consumption and lagged spread matter. The mean
value of the Euler wedge is 0.02 with a standard error of 0.03; but both lagged consumption and lagged investment to output ratios signi…cantly explain its movements
(coe¢ cients are respectively -0.10 and 0.72, with standard errors of 0.01 and 0.13).
When we run our diagnostics on the shorter sample, we reach the same conclusions:
all forecast errors but the one of the consumption equation are predictable, and
lagged consumption and lagged spread matter; lagged consumption and lagged
investment to output ratios predict the wedge (coe¢ cients are respectively -0.13 and
0.96, with standard errors of 0.02 and 0.23). Thus, the time variations we detect are
not due to the crisis and to the potential break it generates.
Parameter Time Invariant Time Invariant Exogenous TVC Endogenous TVC
6 shocks
Function of net worth
h
0.43 (0.006) 0.11 (0.02) 0.19
(0.03)
0.09
(0.02)
0.24 (0.01) 0.97 (0.01) 0.37
(0.03)
0.55
(0.03)
!
0.01 (0.008) 0.02 (0.001) 0.02 (0.002) 0.11
(0.008)
0.46 (0.009) 0.80 (0.01) 0.54
(0.01)
0.52
(0.02)
0.99 (0.004)
0.02 (0.002) 0.03
(0.003)
0.98
(0.008)
u
0.02
(0.007)
1
0.15
(0.009)
2
Log ML -167.97
1098.32
1546.18
1628.69

Table 7: Parameter estimates, Gertler and Karadi model, standard errors in parenthesis.
Armed with this preliminary evidence, we estimate the model allowing to be time
varying. Since regulates leverage and drives movements in the credit and spread
equations, whose smoothed residuals seem to be the most a¤ected by serial correlation,
we opt to make this parameter time varying. We specify
t

= (1

t

= (2

) +
u

(

t 1
u

+ et;

Exogenous variations

) (exp( 1 (Xt
2
Endogenous variations

1

X s )) + exp(

(46)
2

(Xt

1

X s ))) + et;
(47)

where X is net bank wealth. We chose bank net wealth as the relevant state in the
endogenous speci…cation because of its importance for the spread and the credit
variable. Table 7 reports estimates of selected parameters.

7 TIME VARYING FINANCIAL FRICTIONS?

30

In the model with exogenously varying parameters, variations in t very persistent.
Furthermore, estimates of ( ; !; ) are now larger making steady state leverage drop to
about 2.9 and the lifetime of bankers to increase. With the endogenous speci…cation,
estimates of and ! further increase, making steady state leverage fall to 1.9, but
bankers’ survival probability is roughly unchanged. The data seem to require a very
strong asymmetric speci…cation for time variations ( 1 < 2 ), implying a strong negative relationship between the fraction of funds that bankers can divert and their net
worth. Finally, note that, in term of marginal likelihood, the endogenous speci…cation
is superior to both the exogenous one and the …xed coe¢ cient speci…cation augmented
by a shock to bankers’net worth accumulation equation.
To investigate how inference di¤ers in the three estimated models, we plot in …gure
7 the responses of output, in‡
ation, investment, net worth, leverage, and the spread to
a one percent capital quality shock. The constant coe¢ cient speci…cation closely replicates the dynamics presented by GK (see their …gure 3). There is a persistent decline
in output and a temporary but strong decline in in‡
ation. Investment temporarily
falls but it then increases because capital is below its steady state. Bankers’net worth
falls and there is a sharp increase in the spread.

-3

-3

x 10

x 10
1
0

Inflation

Output

-4

-6

-8

-1
-2
-3
-4

-10
5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

-3

x 10

-0.05

4

Net worth

Investments

6

2
0

-0.1
-0.15
-0.2

-2
5

10

15

20

25

30

35

40

-3

x 10
0.2
0.15

Spread

Leverage

3

0.1
0.05

2

1

0
5

10

15

20

25

30

Cons tant

35

40

Ex ogenous

Endogenous

Figure 6: Dynamics in response to a capital quality shock
When we allow to be exogenously varying, responses are qualitatively similar.
Quantitatively, output falls more on impact but less in the short run; net worth falls
less and the spread increases less in the short run. Thus, making exogenously time
varying, reduces the model’ ability to capture recessionary e¤ects on impact.
s

8 CONCLUSIONS
When variations are endogenous, the model possesses an additional mechanism of
propagation of shocks since lower net worth implies higher share of funds diverted by
banks and generally stronger accelerator dynamics. Since the dynamic responses of
net worth are highly persistent, the spread persistently increases making investment
increase less and output fall more and more persistently relative to the other two
cases. Thus, neglecting endogenous variations could impair our ability to correctly
measure the e¤ects of capital quality shocks.
While the economic interpretation of the relationship between
and net worth
is beyond the scope of the paper, attempts to endogenize crucial parameters in the
Gertler and Karadi model exist (see e.g. Bi, Leeper and Leith, 2014; Ferrante, 2014).
Their work could provide the microfundations for the evidence we uncover.

8

Conclusions

This paper is interested in i) characterizing the decision rules of a DSGE when
parameter variations are exogenous or endogenous, and in the latter case, when
agents internalize or not the e¤ects their decisions may have on parameter variations;
ii) providing diagnostics to detect misspeci…cation driven by parameter variations;
and iii) studying the consequences of using time invariant models when the
parameters are time varying in terms of identi…cation, estimation, and inference
We show that if parameter variations are purely exogenous, the contemporaneous
impact and the dynamics induced by structural shocks are the same as in a model
with no parameter variations. However, if parameter variations are endogenous, the
structural dynamics may be di¤erent and the extent of the di¤erence depends on the
detail of the model. We provide diagnostics to detect the misspeci…cation due to
neglected parameter variations and describe a marginal likelihood diagnostics to
recognize whether exogenous or endogenous time variations should be used.
We highlight that certain parameter identi…cation problems noted in the literature
may be the result of misspeci…cation due to neglected time variations. Our Monte
Carlo study indicates that parameter and impulse response distortions may be large
even for modest time variations in the parameters. It also shows that, when parameter variations are neglected, SVAR methods are competitive with more structural
likelihood-based methods, as far as the responses to structural disturbances are concerned.
In the context of the Gertler and Karadi (2010) model, we show that the parameter
regulating the amount of moral hazard is likely to be time varying. When we allow
variations to be linked to the amount of net worth bankers have, the …t of the model
dramatically improves, primarily because there is an additional propagation channel
that makes spread and thus output responses stronger and more persistent.
Overall, our analysis provide researchers with a new set of tools to help them to
assess the quality of their models and respecify certain problematic features.

31

9 REFERENCES

9

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33

9 REFERENCES

34

Appendix A: Additional Monte Carlo …gures and tables

η

ρ

γ

ρ

z

δ

g

α

A

1

Model B

0. 8

0. 8

0. 8

0. 8

0. 6

0. 6

0. 6

0. 6

0. 8

0. 8

0. 6

0. 6

0. 5
0. 4

0. 4

0. 4

0. 4

0. 4

0. 4

0. 2

0. 2

0. 2

0. 2

0. 2

0. 2

0
5

10

-1

0

1

2

3

0. 8

0. 9

1

0. 5

1

1

0. 02

0. 04

0. 2

0. 4

0. 6

0

0

Model C

5

10

1
0. 8

0. 8

0. 8

0. 8

0. 8

0. 6

0. 6

0. 6

0. 6

0. 6

0. 4

0. 4

0. 4

0. 4

0. 4

0. 2

0. 2

0. 2

0. 2

0. 2

0. 5

0. 5

0

0
5

10

-2

0

2

0. 8

0. 9

1

-0. 5 0 0. 5 1 1. 5

1

0. 02

0. 04

0. 2

0. 4

0. 6

1
0. 8

Model D

5

0. 8
0. 6

0. 6

0. 4

0. 4

0. 4

0. 2

0. 2

1

0. 8

0. 6

0. 8

0. 2

0. 6

0. 5

0. 5

0. 5
0. 4
0. 2

0

0
5

10

0

1

2

0. 8

0. 9

1

T rue T = 100

10

-1

0

1

I ncorrect T = 100

0
0. 02

0. 04

0. 2

0. 4

0. 6

I ncorrect T = 1000

Figure A1: Density of estimates; DGP1 (parameter variations explain 2-5 percent of
output variance).

5

10

9 REFERENCES
True

35

Estimated Correct Estimated Time invariant
Estimated Time invariant
Mean
Mean 5 percentile 95 percentile Mean 5 percentile 95 percentile
T=150
T=150
T=1000
DGP Model B
= 2:0
2.00
2.29
1.53
3.87
2.45
1.61
3.09
= 2:0
2.01
1.11
-0.33
2.06
0.25
-0.27
1.95
0.94
0.99
0.96
1.00
0.99
0.97
1.00
z = 0:9
= 0:5
0.47
0.76
0.62
0.96
0.91
0.79
0.98
g
= 0:025
0.03
0.01
0.01
0.03
0.01
0.01
0.01
= 0:3
0.30
0.19
0.11
0.41
0.21
0.10
0.34
A = 4:5
4.53
2.73
1.33
4.14
1.80
1.14
4.16
DGP Model C
= 2:0
2.00
3.40
1.56
7.51
5.19
1.77
22.90
= 2:0
2.00
-0.08
-0.32
0.73
-0.19
-0.35
0.35
0.88
0.99
0.93
1.00
0.99
0.90
1.00
z = 0:9
= 0:5
0.48
0.56
0.08
0.97
0.91
0.59
0.98
g
= 0:025
0.02
0.02
0.01
0.07
0.02
0.01
0.07
= 0:3
0.30
0.26
0.15
0.34
0.26
0.19
0.35
A = 4:5
4.50
1.71
1.25
2.77
2.27
1.24
8.17
DGP Model D
= 2:0
2.00
3.05
1.68
4.59
2.40
1.98
4.81
= 2:0
2.00
-0.06
-0.28
0.54
1.63
-0.27
1.98
= 0:9
0.88
0.98
0.90
1.00
0.92
0.91
1.00
z
0.47
0.42
-0.46
0.96
0.50
0.32
0.97
g = 0:5
= 0:025
0.02
0.01
0.01
0.03
0.01
0.01
0.01
= 0:3
0.30
0.23
0.15
0.32
0.21
0.13
0.27
A = 4:5
4.49
1.91
1.45
3.57
4.10
1.65
4.51
Table A1: Distributions of estimates, DGP2 (parameter variations explain 20 percent of output variance).

9 REFERENCES

36

η

ρ

γ

ρ
z

δ

g

α

A

1
0 .8

0 .8

0 .8

0 .6

0 .6

0 .6

0 .6

0 .4

0 .4

0 .4

0 .4

0 .2

Model B

0 .8

0 .2

0 .2

1
0 .8

0 .2

0 .6
0 .5

0 .5
0 .4
0 .2

0
5

10 15

-1

0

1

2

3

0 .6

0 .8

1

- 0 .5

0

0 .5

1

1

0 .6

0 .6

0 .4

0 .4
0 .2

1

2

1

1

6

8

0 .2

- 0 .5 0

1

4

0 .5

0
0 .8

2

0 .4

0 .2

1

0

0 .6
0 .5

0 .4

0

0 .2 0 .4 0 .6

0 .8

0 .6

0
0 .6

0

1

0 .8

0 .5

5 10152025

0 .0 4

1

0 .8

0 .2

Model C

0 .8

0
0 .0 2

0 .5 1

1

0
0 .0 2

0 .0 4

0 .2

0 .4

0 .6

1

2

4

6

8

2

4

6

8

1
0 .8

Model D

0 .6
0 .5

0 .5

0 .5

0 .5

0 .5

0 .5
0 .4
0 .2

0

0
5

10 15 20

0

1

2

0
0 .6

0
0 .8

1

T rue T = 100

0
-1

0

1

0
0 .0 2

0 .0 4

0 .2

0 .4

0 .6

In c o r r e c t T = 1 0 0In c o r r e c t T = 1 0 0 0

Figure A2: Density of estimates, DGP2 (parameter variations explain 20 percent of
output variance)

9 REFERENCES

37

-3
M o d e l B -A

-3
M o d e l C -A

x 10

-3
M o d e l D -A

x 10

-3
M o d e l B -A

x 10

-3
M o d e l C -A

x 10

-3
M o d e l D -A

x 10

x 10

2
0

Hours

-1

-1
-2

-2

-4
10

20

30

10

20

30

10

20

30

0 .0 5

-5

-6
-8

-3

40

-1 0

40

10

20

30

40

10

20

30

40

10

-0 . 0 1

-0 . 0 3

40

20

30

40

20

30

40

20

30

40

-0 . 0 2

-0 . 0 1 5

30

-0 . 0 1

-0 . 0 2

20

10

-0 . 0 0 5
-0 . 0 1

0 .0 6

0 .0 4

0 .0 4
0 .0 4

0 .0 3
0 .0 2

0 .0 3
0 .0 2

0 .0 2

0 .0 1

0 .0 1
10

20

30

40

10

-3

20

30

40

10

-3

20

30

40

10

-3

x 10

8
6

40

6

4

4

10

20

30

40

20

30

40

30

40

-3

-3

x 10

x 10
0

-5

-5

-1 0

-1 0

-2

-4

2
10

20

0

4

2

-0 . 0 4
10

-3

8

6

30

0

10

10
8

20

x 10

12

-0 . 0 3

-0 . 0 4

-0 . 0 2

x 10

x 10
10

Output

-4

-2

-4

40

0 .0 5

Capital

-2

0

-2

0

0

0

0

10

20

30

40

-1 5

-1 5
10

20

30

40

10

20

30

40

10

-3

-3

-3

-4

-4

-4

x 10

x 10

x 10

x 10

x 10

x 10

4

4

3

Consumption

-2

-2

-2

-4

-4

-4

-6

-6

3

3

2

2
2

1

-6

1
10

20

30

40

A 84

10

A 16

20

30

B 50

40

10

20

30

A 84

40

10

A 16

20

C 50

30

40

10

20

A 84

30

40

A 16

10

D 50

Figure A3: Impulse responses, DGP2 (parameter variations explain 20 percent of
output variance)

9 REFERENCES
small
Variable Technology Government Technology Government
DGP: Model B
Estimated: Time invariant
Y
81.300
0.100
0.998
0.006
C
55.300
0.100
0.998
0.002
N
15.600
0.400
0.978
0.025
K
40.600
0.100
0.994
0.008
DGP: Model C
Estimated: Time invariant
Y
81.900
0.100
0.927
0.082
C
26.500
0.100
0.999
0.001
N
5.400
0.400
0.966
0.039
K
37.400
0.100
0.974
0.030
DGP: Model D
Estimated: Time invariant
Y
82.200
0.100
0.936
0.072
C
32.800
0.100
0.996
0.008
N
10.200
0.500
0.928
0.079
K
60.000
0.400
0.979
0.028
Table A2: Variance decomposition, DGP2 (parameter variations explain 20 percent
of output variance).

38

9 REFERENCES

39

Appendix B : The RBC model with variable capital utilization of section 3
The representative household maximizes the following stream of future utility
max E0

1
X

t

j=0

k s
ct + it = wt nt + rt kt

it = kt
s
kt

(1

= ut kt

n1+
A t
1+

c1
t
1
s
a(ut )kt

)kt

(48)

Tt

(49)
(50)

1

(51)

1

where ct is consumption, it investment, kt the stock of capital, and nt is hours worked.
Household chooses the utilization rate of capital, ut , and the amount of e¤ective capital
s
that she can rent to the …rm, i.e. kt . Household receives earnings from supplying labor
k
and capital services to the …rm, i.e. wt and rt respectively, subject to a cost of changing
s . Finally, T are lump sum taxes levied by the government.
capital utilization, a(ut )kt
t
The production function is
s
yt = zt (kt ) n1
t
A fraction of output is consumed by the government and …nanced with lump-sum taxes.
The government budget constrain is always balanced, i.e.
gt yt = Tt
The optimality conditions of the planner problem are
(1

gt )yt = ct + kt
Ant ct = (1

(1

)(1

a(ut )ut ) kt
gt )yt =nt

1 = Et (ct =ct+1 ) (1
(1

gt )yt =kt

1

1

a(ut+1 )ut+1 + (1

gt+1 )yt+1 =kt )

0

= ut (a (ut )ut + a(ut ))

yt = zt (ut kt
1= +
a(ut ) =

1)

n1
t
1

e

(ut 1)

1

The functional form for the adjustment cost of the capital utilization is in the last
equation and it satis…es a(1) = 0, a0 (1) = 1= +
1, a00 (1) = (1= +
1). If
c
assume that, in the steady state u = 1, the steady states for ( k ; y ; n ; n) are the same
y
y
as in the RBC model without variable capital utilization.

9 REFERENCES

40

Appendix C : The equations of Gertler and Karadi model
exp(%t ) = (exp(Ct )
1 =
exp(

t)

=

h exp(Ct

exp(Rt ) exp(
exp(%t )
exp(%t 1 )

1 ))

h(exp(Ct+1 )

h exp(Ct ))

(52)

t+1 )

(53)
(54)

exp( t ) = (1

exp(Yt )
exp(Lt )
) exp( t+1 )(exp(Rk;t+1 ) exp(Rt )) +

exp( t ) = (1

)+

exp(Lt )' = exp(%t ) exp(Pm;t )(1

exp( t ) =
exp(zt ) =
exp(xt ) =
exp(Kt ) =
exp(Nt ) =
exp(N et ) =

(55)

)

exp(

t+1 )

exp( t+1 ) exp(zt+1 ) exp( t+1 )
exp( t )
1
(1
exp( t )
t)
(exp(Rk;t ) exp(Rt 1 ))(1
t 1 ) exp( t 1 ) + exp(Rt
exp( t )(1
t)
exp(zt )
(exp( t 1 )(1
t 1 ))
exp( t ) exp(Nt )
exp(Qt )
exp(N et ) + exp(N nt )
exp(zt ) exp(Nt

exp(N nt ) = !(1

1 ) exp(

eN e;t ) +

exp(xt+1 ) exp( t+1 )
(56)
(57)
(58)

1)

exp(Ym;t )
exp( ))
+ exp( t ) (exp(Qt )
exp(Kt 1 )
exp(Qt 1 )
exp(Ym;t ) = exp(at ) (exp( t ) exp(Ut ) exp(Kt 1 )) exp(Lt )1
(Int + I s )
(Int + I s )
(Int + I s )
1)2 + i (
1)
exp(Qt ) = 1 + 0:5 i (
s
s
(Int 1 + I
(Int 1 + I
(Int 1 + I s s)
(Int+1 + I s )
(Int+1 + I s ) 2
exp( t+1 ) i (
1)(
)
(Int + I s
(Int + I s )

exp(Gt ) = G

s

1)

(61)
(62)
(64)

exp(Rk;t ) = (exp(Pm;t )

exp(Kt ) = exp( t ) exp(Kt

(60)

(63)

t

t 1 ) exp(Qt ) exp( t ) exp(Kt 1 )

exp( ) = c + b=(1 + ) exp(Ut )1+
exp(Ym )
b exp(Ut ) exp( t ) exp(Kt 1 )
=
exp(Ut )
exp(Pm;t )
Int = exp(It ) exp( ) exp( t ) exp(Kt

(59)

(65)
(66)

(67)
(68)
(69)

1)

(70)

+ Int

(71)

exp(gt )

(Int + I s )
exp(Yt ) = exp(Ct ) + exp(Gt ) + exp(It ) + 0:5 i (
(Int 1 + I s
exp(Ym;t ) = exp(Yt ) exp(Dt )

(72)
1)2 (Int + I s ) +

exp(Kt )
(73)
(74)

9 REFERENCES

41

exp(Dt ) =

exp(Dt

+ (1

1)

)((1

exp(inf lt
exp(inf lt

1)
1)

exp(inf lt )

P

P (1

)

exp(inf lt )

1

)=(1

))

=(1

)

exp(Xt ) = 1= exp(Pm;t )

(75)
(76)

exp(Ft ) = exp(Yt ) exp(Pm;t ) +

exp(

exp(F(77)
t+1 ) exp(inf lt+1 ) (exp(inf lt ))
t+1 )
1
(1 )
exp(inf lt ) P
exp(Zt+1 )
(78)
t+1 ) exp(inf lt+1 )
P

exp(Zt ) = exp(Yt ) +
exp(
exp(Ft )
exp(inf lt ) =
exp(inf lt )
1 exp(Zt )
(exp(inf lt ))1

=

exp(inf lt

1)

P (1

)

(79)
)(exp(inf lt ))1

+ (1

(80)

exp(it ) = exp(Rt ) exp(inf lt+1 )
exp(it ) = exp(it
t

=

at =
t

e

;t

=

t

=

i

(

(Rk;t+1
a

=

gt =

1)

at

1

gt
e

s
Rk

s

(exp(Xt )=( =(

+R )+e

1))) y )

i

exp(ei;t )

;t

(82)
(83)

ea;t

(84)

e

a

1

(85)

;t

eg;t

1
;t 1

t 1

exp(inf lt )

Rt

t 1
g

(81)
1

+e

+e

(86)
;t

;t

Note: t = 0; 8t in the basic estimated model. It appears only in the augmented model
of table 7.

(87)
(88)