Embed Size (px)
Citation preview
Structural VARs, Deterministic and Stochastic Trends:
Does Detrending Matter?
Varang Wiriyawit∗ Benjamin Wong†‡
April 27, 2014
Abstract
We highlight how detrending within a Structural Vector Autoregressions (SVAR) is
directly linked to the identification. Consequences of trend misspecification are investigated
using a prototypical Real Business Cycle model as the Data Generating Process.
Decomposing the different sources of biases in the estimated impulse responses, we find the
biases arising directly from trend misspecifications are not trivial when compared to other
widely studied misspecifications such as lag length truncation and identification. Our
example also illustrates how misspecifying the trend can also distort impulse responses of
even correctly detrended variable within the SVAR system.
JEL Classification: C15, C32, C51, E37
Keywords: Structural VAR, Identification, Detrending, Bias
∗Research School of Economics, The Australian National University. Canberra, ACT 0200, Australia. Email:
[email protected]. Tel: (+61) 2 612 57405.†Reserve Bank of New Zealand & Centre for Applied Macroeconomic Analysis, The Australian National
University. 2, The Terrace, Wellington 6011, New Zealand. Email: [email protected]. Tel: (+64) 4
471 3957.‡The views expressed do not reflect those of the Reserve Bank of New Zealand. We thank Punnoose Jacob,
Timothy Kam, James Morley, Andrea Pescatori, Christie Smith, Rodney Strachan, Wenying Yao and participants
of the Reserve Bank of New Zealand seminar and the 22nd Symposium of the Society of Nonlinear Dynamics and
Econometrics for helpful comments and suggestions. All errors and omissions are our responsibility.
1
1 Introduction
While trends are ubiquitous in macroeconomic time series, dealing with them is often not
straightforward. Within a Structural Vector Autoregression (SVAR) framework, assumptions
regarding the trend has an important role in the identification of structural shocks in a system.
The contribution by Pagan and Pesaran (2008) highlights how choices pertaining to the
handling of trends impact upon the identification of whether shocks in a system are transitory
or permanent. As an example, assuming output evolving according to a stochastic trend
implies at least one structural shock within the system has a permanent impact to the level of
output. The main contribution of the paper focuses the link between the choice treatment of
the trend and how this choice treatment has implications for the identification of shocks in
SVARs. We thereafter quantify different sources of biases induced by trend misspecifications.
To this end, we offer a novel decomposition of the sources of these biases.
A decision whether to difference or deterministically detrend variables cannot be a whimsical
one. It is a prior stand by the researcher on how the variables responding to different identified
shocks must necessarily behave. Unfortunately, our reading of the literature suggests this is
an underappreciated point.1 Some methods of identification in an SVAR framework range
from placing restrictions on contemporaneous relationships, long-run relationships, imposing
restrictions on directional responses of variables to particular shocks or some combination of
those just mentioned. Each has implications for inferences when studying the response of the
economy to different structural shocks. For instance, a common means of orthogonalising the
shocks in an SVAR system is to impose a short-run restriction on the contemporaneous, often
zero, response of variables to particular shocks. Suppose the orthogonalisation happens on a
variable like the difference of real output, with differencing occuring prior to estimation in order
to detrend the variable. Without any other restrictions, all shocks will impose a long-run impact
on the level of real output. We define this type of shock with a long-run impact on the level of
at least one variable in the system as a permanent shock.2 While shocks having a permanent
long-run impact is consistent with some shocks discussed within the macroeconomic literature
(i.e. productivity shocks), they are inconsistent with a large class of shocks (e.g. demand shocks,
monetary shocks etc).
In this paper, we consider standard Real Business Cycle (RBC) models in the spirit of
Hansen (1985) as the Data Generating Process (DGP). Our Monte Carlo experiments are set
up as follows. In our setting, we utilise an illustrate RBC models to draw attention to the role
of trend misspecification. The RBC model under consideration differ only with regard to the
1As anecdotal support to this statement, a plethora of empirical studies routinely state they difference ordetrend using an arbitrary preferred filter without justification (e.g. Peersman and Van Robays, 2012; Han andElekdag, 2012). The inconsistency of the detrending methodology and identifying restrictions can also be seen inPeersman (2005) and subsequently pointed out by Fisher et al. (2013). A more recent development in empiricalmacroeconomics has been the use of large datasets through the FAVAR or GVAR frameworks developed byBernanke et al. (2005) and di Mauro et al. (2007) respectively. Users of such methods often routinely firstdifference all their trending data as a matter of practicality, often with a failure to acknowledge or realise theexplicit link of inducing permanent shocks in all these variables.
2The impact of a shock, ζ, of size 1, at time t to a variable w at time t+ i is∂wt+i
∂ζt. A shock is transitory on w
if limi→∞∂wt+i
∂ζt= 0. Otherwise, it is permanent. If w was first differenced, then limi→∞
∂wt+i
∂ζt=
∑∞j=0
∂4wt+j
∂ζt.
Unless a prior restriction is imposed, this sum will in general not equal to zero, implying a permanent effect onw.
2
trend specification induced by the underlying technology shock process. Output specified in
the DGP will then either evolve according to a stochastic or deterministic trend depending on
the properties of the technology shock in the underlying DGP. Our experiments are to estimate
a bi-variable SVAR using artificial data to identify a generic technology and non-technology
shock. Given a DGP with some unknown trend specification, a trend assumption about output
is made prior to estimation by the econometrician in our study. The trend property of output
is however incorrectly specified. Given the link between identification and trend specification,
this will also cause a misspecification with the identification of the SVAR. Our question is then
how serious are such trend misspecifications for subsequent inference and what are the sources
of these biases.
It is worth offering clarification for readers who may potentially be confused about this
paper’s contribution. The model setup will no doubt be familiar to those acquainted with a
large empirical literature, starting with Galı (1999), questioning the effect of technology shocks
on hours work. Apart from the model features, this is where any similarity ends. The empirical
approach in that body of work almost always assumes a stochastic trend in productivity with
the fiercest debates regarding the specification of hours work (see e.g. Christiano et al., 2003)
and whether VARs are useful mechanisms to study their question of interest (see e.g. Chari et
al., 2008; Linde, 2009). We are neither in a position to add nor have any significant contribution
in that respect. Our interest solely concerns the role of trend misspecification in output or
productivity. In particular, we also study deterministic trends, concepts which are not considered
in that body of work.
Our results show that an incorrect assumption about trend process can cause considerable
biases in the estimated impulse response functions, further highlighting the importance about
the choice of detrending within an SVAR framework. Extending tools used by Ravenna (2007),
Chari et al. (2008) and Poskitt and Yao (2012), we offer a novel decomposition of these biases as
a tool to present a trichotomy between the three sources of the biases. These are biases induced
by truncating the lag length in the estimated VAR, termed truncation bias, biases induced by
identification from an error in orthogonalising the shocks, termed identification bias, and lastly
biases from misspecifying the trending process, termed trend bias. Identification and truncation
bias have received much attention within the profession. Even so, our illustrative example
demostrates that the bias induced directly from trend misspecification compared to the former
two biases are not trivial.
As an SVAR is a system of interdependent equations, the corresponding impulse response
functions of a correctly detrended variable, like hours worked in our study, are also distorted
if the trend in output is misspecified. That is, there can be significant spillover from trend
misspecification of the trending variable to the correctly detrended variable within the system.
While we caution against generalising claims on dealing with trend from our work, the illustrative
example of the simple RBC model offers the empirical plausibility trend misspecification can be
a key source of bias in SVAR studies. This emphasizes the need for researchers using SVARs for
empirical work at least be mindful the treatment of trends and identification are interlinked.
The remainder of the paper is as follows. Section 2 describes the theoretical model and the
identification of the model within an SVAR setting. Section 3 explicitly links the theoretical RBC
3
model with features of the SVAR to motivate the design of the simulation study. The simulation
setup and how we decompose bias to study the consequences of trend misspecification are then
discussed in Section 4. The results are presented in Section 5 before some concluding remarks
in Section 6.
2 Theoretical Model and Identification
We study an RBC model similar to that used by Hansen (1985). The simplicity and
parsimony of the model structure appeals with fewer identifying restrictions for the SVAR.
The following subsections present the theoretical RBC model used as a DGP and a discussion
of the SVAR identification.
2.1 The Theoretical Model
Under this framework, the households’ problem is given by
max{Ct,Ht}
E0
∞∑t=0
βt{
lnCt −(Ht/Bt)
1+η
(1 + η)
}
subject to
Ct + It = RtKt +WtHt (1)
It = Kt+1 − (1− δ)Kt (2)
ln(Bt+1) = ρB ln(Bt) + εB,t+1 (3)
where β ∈ (0, 1) is a discount factor, η > 0 is an inverse short-run (Frisch) labour supply
elasticity, δ ∈ (0, 1) is a depreciation rate of capital, ρB ∈ (0, 1) is a measure of persistence and
εB,t+1 ∼ N (0, σ2B) is a Gaussian shock in the exogenous process.
Households in this economy optimise their expected discounted lifetime utility by choosing
each period consumption (Ct), hours worked (Ht) and next-period capital holdings (Kt+1)
subject to their budget constraint (1), a capital accumulation equation (2) and a stationary
AR(1) exogenous process (Bt). The shock process, εB,t, presented in Equation (3) can be
interpreted as either a shock to labour supply, preference, or a demand of households. In this
paper, we refer to this innovation as a non-technology shock. Furthermore, as the process is
stationary, the shock has a transitory impact upon variables in the system. The sources of
income for households are from supplying capital and labour services to firms. Income is either
consumed or invested. Let It be an investment, Rt be a rental rate of capital and Wt be a wage
rate at period t.
The First Order Conditions for households’ utility maximisation are
1
Ct= βEt
{(1
Ct+1
)(Rt+1 + 1− δ)
}(4)
1
Ct=
Hηt
B1+ηt Wt
(5)
4
These necessary conditions characterise optimal decision rules for households. Equation (4)
is an Euler equation for consumption stating that the marginal rate of substitution between
consumption at period t and consumption at period t + 1 equals the return of capital.
Equation (5) is a labour supply equation stating that the marginal rate of substitution
between consumption and leisure must equal the wage rate.
We can write the problem for firms as
max{Kt,Ht}
{Yt −RtKt −WtHt}
subject to
Yt = Kαt (ZtHt)
1−α (6)
where α ∈ (0, 1) is a capital share and Zt is a technology innovation to productivity.
The firms maximise their profit subject to the labour-augmenting Cobb-Douglas production
function. Here revenue is obtained by selling goods to households, denoted by Yt, while costs
are incurred from renting households’ capital and labour services.
The First Order Conditions for firms’ profit maximisation are then given by
Rt = αKα−1t (ZtHt)
1−α (7)
Wt = (1− α)Z1−αt Kα
t H−αt . (8)
Equations (7) and (8) imply that the rental rate of capital and the wage rate are set equal
to the marginal productivity of an additional unit of capital and labour respectively.
Technology Shock Process
The aim of the theoretical model is to generate the aforementioned RBC model with a
technology shock which can be either a transitory or permanent shock. We consider two
specifications of the technology shock to achieve this. The permanent and transitory
technology shock process will entail a stochastic-trend and a deterministic-trend process
respectively. The former specification can be represented as
Zt+1 =Zt+1
Zt
ln(Zt+1) = (1− ρz) ln(γ) + ρz ln(Zt) + εz,t
where γ is an average growth rate, ρz is persistence in a growth rate of the technology shock
and εz,t ∼ N (0, σ2z) is a Gaussian shock. Under this specification, the technology innovation
has a permanent impact on the level of Yt, Ct,Kt,Wt and Zt causing these variables to inherit
unit roots. Therefore, one can obtain stationary variables with the following transformation;
Yt = YtZt
, Ct = CtZt
, Kt+1 = Kt+1
Ztand Wt = Wt
Zt. Note that Kt+1 is detrended by Zt as it is
determined within period t. Hereinafter, we refer to this model as RBC-rw.
On the other hand, the deterministically-trending process assumes the technology innovation
5
to grow at a constant rate of γ. The process can be represented as
Zt = γtZt
ln(Zt+1) = ρz ln(Zt) + εz,t+1
ln(Zt) = t ln(γ) + ρz ln(Zt−1) + εz,t.
We obtain the stationary equilibrium condition by detrending the variables with the
deterministic trend γ. The transformed variables are then given by Yt = Ytγt , Ct = Ct
γt ,
Kt+1 = Kt+1
γt and Wt = Wtγt . Unlike the RBC-rw specification, the technology shock considered
here only has a transitory impact to all variables in the system. The variables will thus be
trend stationary. We refer this model as RBC-dt.
Competitive Equilibrium Definition
The competitive equilibrium is defined as follows. In a competitive equilibrium, households
will choose allocations of {Ct, Ht,Kt+1}∞t=0 and firms will choose allocations of {Kt, Ht}∞t=0 such
that, given a sequence of prices {Wt, Rt}∞t=0 and exogenous shocks to {Zt, Bt}∞t=0, households and
firms optimise their utility and profit respectively with the market clearing such that Yt = Ct+It.
Data Generating Process
Given this framework, we have Xt = (Kt, Zt, Bt)′ as unobserved state variables,
Qt = (Yt, Ht)′ as observable variables and εt = (εz,t, εB,t)
′ as exogenous shocks where
t = 1, 2, . . . , T . By implementing the first-order approximation, a stable Rational Expectation
Equilibrium solution to the log-linearised system of an RBC model has the following linear
state-space representation,
xt+1 = Rxt + Sεt+1
qt = Mxt (9)
where xt and qt are column vectors of log-deviation state variables and observable variables from
the steady state values, R, S and M are matrices of reduced-form parameters and εtε′t = Σε is
a diagonal covariance-variance matrix.
Given x0, we can simulate data of output and hours worked Q = {Yt, Ht}Tt=0. Hours
worked is always integrated of order zero, Ht ∼ I(0), given underlying trend processes in the
technology shock of interest and the stationary non-technology shock process. The process of
output however depends on the underlying trend process. If the trend process is a
deterministic trend as in the case of RBC-dt, output is trend stationary and integrated of
order zero, Yt ∼ I(0). On the other hand, output is integrated of order one, Yt ∼ I(1), when
the underlying trend process is a stochastic trend as in the case of RBC-rw. The
characteristics of the structural shocks in the system also differ across model specifications.
While the non-technology shock will always have transitory impacts, the technology shock has
transitory impact only under RBC-dt. In the alternative RBC-rw case, technology shock have
a permanent impact upon the level of output.
6
The values of structural parameters used in the DGPs are summarised in Table 1. Most
of these parameter values are standard in the literature, see e.g. Linde (2009). Some of these
choices are set with respect to the objective of our study. In particular, we set the magnitude of
the standard deviation of technology shocks to be twice of that compared to the non-technology
shock. It is well known that the larger the relative magnitude and persistence of any particular
shock, the easier it is to recover properties of the said shock from the SVAR. (see e.g. Erceg et
al., 2005; Paustian, 2007; Chari et al., 2008). Given it will be apparent that the comparison
within our Monte Carlo study will be with respect to technology shocks, this is a natural choice.
At the same time, our study is to examine the estimation distortion of an SVAR induced by
trend misspecification, we set the magnitude of these two shocks largely apart as to isolate this
bias from our analysis.
2.2 Structural VAR Identification
Let Qt = (Yt, Ht)′ be a column vector of (demeaned) observable variables where these “hat”
variables represent time series generated from one of the DGPs described in the previous section
but unknown to an econometrician in the study. Let qt be transformed observable variables,
either by first differencing or linear detrending to achieve stationarity. In general, we can write
down the VAR specification with a transformed variable qt as
Φ(L)qt = ut (10)
= Aνt (11)
where Φ(L) is a lag polynomial, I − Φ1L − Φ2L2 − . . . − ΦpL
p of finite lag order p. A is a
contemporaneous impact matrix whereas ut and νt are the reduced-form and structural
innovations with covariance matrix Σu and Σν respectively. We define the structural shocks in
this system as a technology (νTt ) and a non-technology (νNTt ) shock. Σν is diagonal by
assumption and construction, embedding the idea that the structural shocks in the model are
orthogonal. The econometrician can easily estimate Φ(L) consistently using least squares.
However, the identification issue arises because there is one free parameter in this bivariate
system due to assuming the structural innovations are orthogonal (i.e. Σν is diagonal whereas
Σu is not diagonal).
Therefore, while the reduced-form VAR in Equation (10) can be easily estimated by least
squares, the econometrician can only estimate the SVAR in Equation (11) through invoking
some identifying restrictions. As the following will make clear, identifying strategies are linked
to the trend assumption which the econometrician makes.
Long-Run Restrictions
Suppose the econometrician believes that output exhibits a unit root solely due to a long-
run impact from a technology shock while hours worked is an I(0) process. Blanchard and
Quah (1989) offer an identification strategy to impose this long-run restriction in an SVAR
framework. As output is assumed to be an I(1) process, the series enters an SVAR in first
difference to achieve stationarity. To impose this restriction, first let the vector qt = (∆yt, ht)′,
7
where yt and ht are logged output and hours worked respectively. We can rewrite the VAR in
Vector Moving Average (VMA) form as follows. From Equation (11), we have
qt = Φ−1(L)Aνt
= Ψ(L)νt (12)
where Ψ(L) = Φ−1(L)A. Expanding (12) and substituting in the elements of the vector qt and
νt, we obtain (4ytht
)= Ψ(L)
(νTt
νNTt
)
=
(ψ11(L) ψ12(L)
ψ21(L) ψ22(L)
)(νTt
νNTt
)
=
∞∑i=0
ψi11νTt +
∞∑i=0
ψi12νNTt
∞∑i=0
ψi21νTt +
∞∑i=0
ψi22νNTt
.
By assuming only the technology shock has a long-run impact on output, the required restriction
is then ψ12(1) =∑∞
i=0 ψi12 = 0.
Short-Run Restrictions
If the econometrician believes that both output and hours worked are I(0) processes, all
structural shocks in the system are then assumed to be transitory, with output transitory around
a deterministic trend. Output is linearly detrended before estimation and thus qt = (yt−λt, ht)′
where λ represents the coefficient on the deterministic trend. Regardless of the identification
procedure here, shocks will always be transitory on the variables in qt, namely detrended output
and hours worked. In order to invoke one identifying restriction, we place one zero restriction
directly on the contemporaneous matrix, A. A natural way to identify a technology shock
is to assume that a non-technology shock has no contemporaneous impact on output. The
identification in such a manner will almost by construction allow a large share of the forecast
variance to be explained by the technology shock. This directly appeals to the intellectual
foundations of RBC models, where technology shocks are the dominant drivers of the business
cycle.
This restriction amounts to identification with a Cholesky decomposition of the covariance
matrix, ordering output first. This, by construction, restricts hours worked to not respond
contemporaneously to non-technology shocks. Given this imposes a restriction on the short-run
dynamics in the model, namely the impact response to non-technology shocks, we refer to this
as imposing short-run restrictions. From Equation (11), the proposed Cholesky identification of
this system is then given by(yt − λtht
)= Φ−1(L)
(A11 0
A21 A22
)(νTt
νNTt
).
8
3 Link Between RBC and SVAR
Before discussing the simulation setup, it is worth discussing the link between the
theoretical RBC model, which we use as a DGP, and the SVAR identification. The Blanchard
Quah identification using long-run restrictions allows for permanent shocks given that output
is differenced prior to estimation. One can reconcile features of the long-run restrictions here
with the RBC-rw. The non-technology shock in the theoretical RBC-rw model does not have
any long-run impact on output, which is also used as the identifying restriction to help identify
both shocks.
It is true that long-run restrictions may nest RBC-dt. However, two conditions need to
be satisfied for this to be the case. First, a necessary condition is that the coefficient on the
deterministic-trend term is zero, λ = 0. Otherwise, transformed observable variables qt under
these two schemes would not be identical and the reduced-form coefficient estimates would
not coincide. Second, even if this necessary condition is met, in the long-run restrictions set
up, ψ11(1) =∑∞
i=0 ψi11 = 0 emerges as a solution. This particular solution implies that the
technology shock has no long-run impact as does the non-technology shock. It then becomes
consistent with RBC-dt. While this is a theoretical possibility, the necessary condition on λ = 0
is clearly not plausible when viewing a time series of output.
With the Cholesky decomposition, all shocks are assumed to be transitory, either around a
deterministic trend or around some unconditional mean. Therefore, the features of the
theoretical RBC-dt model where all shocks are transitory are consistent with short-run
restrictions.3 However, there is still a misspecification with the Cholesky decomposition since
output is restricted not to react contemporaneously to a non-technology shock upon impact.
While this is not consistent with the theoretical RBC-dt model, we view this restriction as
providing the linearly detrended SVAR with the “best” shot at matching the theoretical model
as technology shocks are the dominant shocks in the theoretical structure of the model.4 We
stress at this stage that, despite the theoretical misgivings, the Cholesky decomposition is
largely able to recover the properties of the transitory technology shock in our Monte Carlo
simulation. The biases induced by the Cholesky decomposition are thus trivial when the
deterministic trend is properly specified. While the Cholesky decomposition may induce a
distortion of estimated impulse responses to the non-technology shock under this scheme, our
bias decomposition exercise described in the next section will make comparisons possible
between both specifications. Moreover, we view imposing a Cholesky decomposition for
identification as a plausible strategy for the study from an empirical perspective in the SVAR
literature.
Another discrepancy is that the theoretical models under both trend specifications have a
state variable, capital (Kt), which is omitted in the VAR estimation (see e.g Ravenna, 2007;
3Note this does not in anyway imply short-run restrictions are necessarily transitory shocks. The shockshere are transitory because of the detrending, though this impacted on the identification choice to impose arestriction on the short-run dynamics of the model. Another popular identification procedure is to impose impactsign restrictions on variables (e.g. Uhlig, 2005). Such restrictions do not tie down the long-run properties of theshocks. Therefore, even using sign restrictions on differenced data without additional restrictions must entailthese are by construction permanent shocks.
4The consequence of using Cholesky identification when it is not consistent with a theoretical model is studiedby, for example, Carlstrom et al. (2009) in a New Keynesian environment.
9
Chari et al., 2008). This implies that a VARMA or a VAR (∞) process is the correct specification
to map the theoretical model to the SVAR. A finite order VAR will thus be biased from lag
length truncation. We deal with this truncation on three dimensions. First, we estimate a VAR
of a long lag order which obviates this issue to a large extent. Second, we set the magnitude of
the non-technology shock to be small relative to the technology shock in the DGPs as discussed
in Section 2.1. This is with guidance from Chari et al. (2008) who show that, in a two-variable
SVAR system, truncation bias can be reduced as the variance of one of the shocks in the system
gets smaller relative to other shocks. The result here is driven by the existence of a First-Order
Autoregressive Representation for the observed variables and the continuity of impulse response
function in the parameters. That is, when the matrix M in Equation (9) is invertible, the
observable variables qt have an AR(1) representation. This proposition can only be applied to
our RBC framework if the variance of the non-technology shock is zero, which is not the case.
However, as impulse response function is continuous in the parameters, a VAR(p) with a finite
lag order p can be a better approximation of a VAR (∞) when the non-technology shock plays a
smaller role in the system. Third, we linearly decompose the component of the bias induced by
lag length truncation. This allows the comparison of the components of the bias caused directly
by trend misspecification to be compared directly. The decomposition will be described in the
following section.
4 Simulation Setup
As previously discussed, the long-run and short-run restrictions should correctly recover
the correct shock properties for the RBC with the stochastic and deterministic trend in output
respectively. We label these cases as correct specifications. The purpose of this paper is to
determine the magnitude of the distortion in the estimated impulse responses which trend
misspecification in an SVAR induces. We therefore design experiments to answer the question
as summarised in Table 2. Under our first misspecification study, output is generated I(0) by
an RBC model with a deterministic-trend specification (RBC-dt). The econometrician then
wrongly first differences the series and identifies the SVAR using long-run restrictions. In the
second misspecification, output is generated I(1) by an RBC model with a stochastic-trend
specification (RBC-rw). However, the econometrician wrongly detrends the series and
implements short-run restrictions to identify the SVAR. Hours worked is included in levels
under both SVAR restrictions and correctly specified as implied by both DGPs. As mentioned
in earlier, impulse responses to the non-technology shock are misspecified under the short-run
restrictions. The responses to this shock are thus not a fair comparison between the two
misspecification studies of interest. Our analysis is therefore based on the distortion incurred
in the estimated impulse responses to a technology shock.
4.1 Bias Decomposition
In this paper, the distortion due to trend misspecification is measured by a total bias,
defined as an average difference between estimated impulse response functions derived from a
misspecified SVAR and true ones deduced from a corresponding RBC model across all
10
simulations. For example, in Misspecification 1, a total bias in estimated impulse responses
due to trend misspecification is
Total Bias =1
N
N∑i=1
[IRF
(i)(rw,LR)− IRF (RBC-dt)
](13)
where the first term is estimated impulse responses from an SVAR assuming a random-walk
process and imposing long-run restrictions as a shock identifying strategy, the second term is
true responses from an RBC model with a deterministic-trend process and N = 10, 000 is the
number of simulations.
To determine the sources of the error, recall the estimated VMA form
qt = Φ−1(L)Aνt.
The responses of transformed variables qt to structural innovations νt are influenced by both
the estimated reduced-form coefficients Φ(L) and the identifying matrix A. Hence, imprecision
in estimating the reduced-form coefficients and incorrect identifying restrictions will distort the
estimation of impulse responses. Specifically, an incorrect assumption about a trend process
in output leads to trend mistreatment and in turn causes estimation bias in the reduced-form
coefficients, Φ(L). The bias in the coefficient estimates is also exacerbated by the nonlinear
mapping involved with imposing identifying restrictions in the matrix A as the matrix is also a
function of the estimated coefficients. We refer this source of error as trend bias. The incorrect
trend assumption also leads to an error in identifying structural shocks and hence affects the
way an econometrician imposes restrictions on the matrix A (i.e. imposing long-run or short-
run restrictions). We call this error identification bias. Furthermore as mentioned, using a finite
order VAR(p) to approximate the true data generating process induces truncation bias. In sum,
the these three sources constitutes the total bias from recovering the theoretical RBC responses
through the SVAR.
The following describes our main contribution in this paper. To study the consequences
of trend misspecification, our main focus is on the trend and identification biases in estimated
impulse responses of output and hours worked to a technology shock. We quantify the size of
these components by building on some existing tools (see e.g. Ravenna, 2007; Chari et al., 2008;
Poskitt and Yao, 2012) by linearly decomposing the total bias in Misspecification 1 expressed in
Equation (13) as follows.
11
Total Bias =1
N
N∑i=1
[IRF
(i)(rw,LR)− IRF
(i)(dt, LR)
]︸ ︷︷ ︸
Trend Bias
+
1
N
N∑i=1
[IRF
(i)(dt, LR)− IRF
(i)(dt, SR)
]︸ ︷︷ ︸
Identification Bias
+
1
N
N∑i=1
[IRF
(i)(dt, SR)− IRF (RBC-dt)
]︸ ︷︷ ︸
Truncation Bias
.
The first component is the trend bias and is the average difference between estimated impulse
responses with incorrect and correct trend assumptions as implied by a corresponding RBC
model, given identifying restrictions assumed by the econometrician in our study. The bias
due to imposing incorrect identifying restrictions is captured by the second component. This
component is purely generated by identification bias as it is measured by an average difference
between estimated impulse responses with different identifying restrictions, given a consistent
detrending process to a corresponding RBC model. The last component gauges truncation bias
as a result of estimating a VAR of finite lag order. Incidentally, the truncation bias is also the
bias incurred if the trend is correctly specified. That is, the truncation bias in Misspecification
1, which wrongly first differences the data, measures the bias incurred in Correct Specification 2,
if the data was correctly linearly detrended. Similar decomposition of the total bias can also be
done for Misspecification 2. Similarly, the truncation bias in this case measures the bias which
occurs with Correct Specification 1.
5 Results and Discussion
Figure 1 depicts the simulation results under the correct specification studies while Figures 2
and 3 plot the ones under the misspecification studies. In all figures, the true responses derived
from the DGPs are presented by a dashed line with circles. Red area represents estimated
impulse responses of the SVAR from 10,000 data sets of 200 observations.
Given DGPs of interest, the dynamics of the true responses can be described as follows.
Under both trend specifications, output and hours worked increase in response to a positive
technology shock. The behaviour of these impulse responses is however different depending on
the underlying trend process. To recap, a transitory technology shock emerges from a
deterministic-trend specification while a unit-root technology shock has permanent impact on
the level of output. In the former case, the impulse response functions converge back to zero in
the long-run due to the transitory nature of the shock. Contrast against the latter case, the
impulse response increases and reaches a new steady state level instead of converging back to
zero. Hours worked, on the other hand, has a transitory response under both specifications. A
stochastic-trend specification reveals a hump-shape behaviour of hours worked rather than a
sharp increase matching with a response of output depicted under a deterministic-trend
12
specification. This is due to households’ expectation of higher productivity in the future.
Households are thus motivated to substitute some of their time toward current leisure and do
not initially increase their labour supply as much as in the case of a deterministic-trend
specification. Later on, hours worked is adjusted to cope with a new long-run level of output.
Under the correct specification studies, Figures 1a and 1b present impulse response functions
estimated using long-run restrictions and short-run restrictions respectively. As we can see
from Figure 1a, these true responses from RBC-rw can be captured very well by a correctly
specified SVAR using long-run restrictions as the true responses lie neatly within the red area.
However, there is greater sampling uncertainty about the estimated response of hours worked
to a technology shock. This can be seen in the a wide range of red lines which also indicate
a nonzero probability of having a negative response upon impact when the true response is in
fact positive. Specifically, the probability of inferring an incorrect sign upon the initial impact is
0.1793. Note that the estimation error incurred here is also due to an imperfect approximation of
the SVAR to the RBC model, referred to as truncation bias. In the case of short-run restrictions
presented in Figure 1b, the estimated impulse responses can capture the behaviour implied by
RBC-dt. We note that these estimated responses have a tendency to be biased slightly upward
initially and then slightly downwards. On the basis of Figures 1a and 1b, the evidence does
suggest subject to the trend being correctly specified, the SVAR is largely able to recover the
DGP impulse responses. This is a useful starting point because it reveals our parameter and
modelling choices can largely isolated the other sources of biases. This will be useful point of
comparison moving into the trend misspecification study.
Mishandling the trend in output unsurprisingly affects the behaviour of estimated impulse
responses, incurring larger distortion in the estimation than the correct specification cases.
This large distortion is due to an introduction of trend and identification biases in the
estimation. Figure 2a plots the estimated impulse responses from a misspecified SVAR using
long-run restrictions along with the true responses derived from RBC-dt. As the
econometrician incorrectly first differences output and imposes long-run restrictions, the
estimated impulse responses of output to a positive technology shock converges to a new
steady state instead of exhibiting the transitory behaviour implied by the underlying DGP.
Furthermore, the estimated responses are biased upward except for the initial impact. The
bias decomposition is presented in Figure 2b suggesting that the trend bias is non-trivial. This
bias is the main source of the error from the trend mistreatment of output which persistently
contaminates the estimation in all horizons. The identification bias in this study contributes a
relatively small fraction to the total bias and only distorts the estimation of the impulse
response in the short-run dynamic before dying out.
We consider our other misspecification study. Wrongly imposing short-run restrictions will
cause estimated impulse responses of output to die out over time even though the true response
exhibits a unit root as implied by RBC-rw. As zero long-run impact of the output response to
technology shock is imposed by the detrending, and then identification, procedure, we should
expect a downward bias in the estimated impulse response functions. The downward bias should
also get larger at longer horizons because we expect detrending imposes a long-run effect on the
system. Figure 3b validates this is indeed the case. As the trend bias is large at all horizons,
13
particularly noteworthy at longer horizons, the trend bias is the dominant source of error. Our
simulations also show that the reversion of the output level to zero can be slow to take effect,
perhaps reflecting the permanent shock in the underlying DGP. This is however insufficient to
prevent the obvious downward bias due to mishandling the trend in output.
Spillovers to the correctly detrended variable from trend misspecification can also occur.
That is, the trend misspecification however does not only affect the estimated impulse
responses of output to a technology shock, the estimated impulse responses of hours worked to
a technology shock also become distorted as the system is interdependently estimated. In both
misspecifications we consider, the shapes of the impulse response function of hours worked are
fairly well preserved. Even so, they both have a tendency to exhibit upward bias in the initial
periods after the technology shock. The response in Misspecification 2 though, has a tendency
to follow up upward biases with downward biases after about 10 periods after the shock.
Unlike the estimated impulse responses of output, the dominant source of biases for hours
worked differs and is generally split between trend misspecification and identification bias.
Given the design of the experiment limits the degree of identification bias from a correctly
detrended model, we can interpret much of this identification bias would have been spilled over
from trend mistreatment of the other variable in the system.
A interesting point to note in our analysis is the variance of the simulations is notably
larger with the RBC-rw specification, namely the DGP specification with a stochastic trend.
Time series with deterministic trends, on the other hand, will have less noise by construction.
Therefore, it will almost be no surprise if the underlying data generating process has a
deterministic trend, there is less uncertainty in the estimated impulse response functions. This
feature carries over even in the presence of trend misspecification. Particularly, this means less
uncertainty with the estimators with trend misspecification when the econometrician assuming
a stochastic trend, and thus use long-run restrictions, compared to a situation where there is
indeed a stochastic trend and the econometrician correctly models the system as such.
Nonetheless, given the DGP is beyond the control of the econometrician, there is little solution
reducing uncertainty apart from the availability of more data.
The results from our Monte Carlo study should be sufficient to raise warning flags.
Identification of SVARs receive much attention given their role in allowing the researcher to
interpret the data within the context of their chosen model. Our Monte Carlo simulations
reveal that trend misspecification are non-trivial, or could even potentially be a greater source
of bias compared to widely studied issues like identification and truncation biases. While it
would be tempting to conclude on the basis from Figures 2b and 3b that a misspecified SVAR
with short-run restrictions provides larger distortion than a misspecified SVAR with long-run
restrictions, we caution against such interpretations as premature. The illustrative example we
consider is a very stylised and simple model. Empirical reality dictates richer and larger model
structures. It is very much an empirical question whether the patterns from our Monte Carlo
study carry over. One needs to keep in mind that a large share of the bias in richer systems
are inherent plagued by identification issues which can be challenging to isolate even before
considering the role of trend misspecification (see e.g. Carlstrom et al., 2009; Castelnuovo,
2012, for examples of models with typical identifying strategies). Our experiment nevertheless
14
stresses the role of trend misspecification impacting on identification. It also highlights the role
of trend misspecification can be large, even relative to identification bias, the latter issue which
receives much attention by the profession.
Empirically, beginning with Galı (1999), there is a strong tradition in using long-run
restrictions to identify technology shocks (see e.g. Christiano et al., 2003; Francis and Ramey,
2005; Linde, 2009).5. The main empirical question within that area of the literature pertains
to the response of hours worked to technology shocks. We note at least within the context of
our experiment that the strategy of using long-run restrictions, by assuming technology are
permanent for the empirical question of interest, is sound. The truncation bias is the total bias
using a Cholesky decomposition after linear detrending if technology indeed has a linear trend.
The additional biases induced by trend misspecification, assuming a stochastic trend and
imposing long-run restrictions in that situation, are not large. We note, of course, we have not
consider a multitude of other issues within that literature, a key component focusing on the
data generating process of hours worked.
To sum up the results from our Monte Carlo experiments, trend misspecification is a source
of bias which is comparable to identification. Spillovers to biases to correctly detrended variables
within the system are not trivial. These reiterate a reminder that researchers should careful link
their detrending procedures with the identification of the structural model they have in mind.
5.1 Alternative Experimental Set-up
In this section, we look into alternative experimental set-ups to explore possible scenarios an
econometrician can face. The setting varies from different parameter values used in the DGPs
to alternative detrending strategies the econometrician might employ.
Degree of Persistence in the Permanent Technology Shock
In contrast to the theoretical responses we have in the previous section, some empirical
studies suggest that a positive permanent technology shock should lead to a fall in hours worked
instead of an increase (see e.g. Galı, 2004; Francis and Ramey, 2005; Kimball et al., 2006). In
this section, we consider a theoretical framework in which hours worked falls after a positive
technology shock. Linde (2009) demostrates that an RBC model can account for both positive
and negative responses to hours worked. The latter can be generated by allowing for a highly
persistent growth rate of technology shock in RBC-rw. We therefore repeat the simulation
exercises with RBC-rw as a DGP and increase the persistence in the permanent technology shock
from 0.25 to 0.5. Other parameter values remain the same as used previously. In this section,
we label the SVAR with long-run restrictions as a correctly-specified framework whereas the
SVAR with short-run restrictions is a misspecified one. Recall that, in the following figures, the
true responses are presented by a dashed line with circles whereas red area represents estimated
impulse responses of the SVAR from 10,000 data sets of 200 observations.
5We acknowledge a constructed productivity series is often used instead of output in such exercises. Resultsusing a simulated productivity produce similar conclusions. The choice of output is largely for comparativepurposes. Larger scale studies, often with New Keynesian roots and considering only transitory shocks, oftenconsider detrended output instead of detrended productivity due to its theoretical setup (see e.g. Boivin andGiannoni, 2006; Carlstrom et al., 2009).
15
In our exercise, we misspecify the trend as a deterministic trend and use the Cholesky
decomposition to identify the technology shock. This is presented in Figure 4. Figure 4a presents
the impulse response functions. In general, output is badly biased, similar to that in Figure 3a.
The results also reveals none of the impulse response functions are able to capture the fall in hours
worked. The decomposition uncovers the predominant sources of biases is mainly identification
bias for hours worked and a mix of identification and trend bias for output. However, absolute
bias is much larger compared to Figure 3a, the specification with less persistence in the growth
rate. Note that, there is a large degree of truncation bias. This reveals that even if the the
trend was properly specified, the bias would still be large. This is not entirely surprising. Given
a more persistent growth rate in the DGP, it is going to be the case that a higher lag order
is needed to fully model the dynamics. The truncation of the lag length is thus going to be
significant. Even so, truncation bias is still not as dominant as trend and identification biases.
Degree of Persistence in the Transitory Technology Shock
We investigate the sensitivity of a more persistent technology shock under RBC-dt to total
bias. Instead of 0.25 which is considered to be relatively low persistent given this framework, we
therefore increase the shock persistence to 0.95. As a consequence, the technology shock to both
output and hours worked takes at least 40 periods to revert to the steady state. We then conduct
an experiment as in Misspecification 1. That is, output series is incorrectly first differenced and
long-run restrictions are imposed as a shock identifying strategy by an econometrician. The
total bias decomposition is presented in Figure 5 with a dash line represented a total bias in
Misspecification 1 given a low persistence in the transitory technology shock for a comparison.
We can see that, at longer horizons, the trend bias is the predominant source of bias. This is
similar to a case of lower persistence in the transitory technology shock except for one difference;
the horizon which this trend bias dominates the source of biases kicks in much later, the more
persistent the technology shock is. The total bias on average unexpectedly shrinks at shorter
horizons. The reason is that, once the degree of persistence approaches 1, the deterministic trend
process behaves more like a unit-root process. Assuming a stochastic trend specification in this
case can therefore still capture features of the underlying process, but only in the short-run. The
trend bias due to incorrectly first differencing eventually and rapidly expands, resulting in larger
total bias towards the end. In essence, this reveals the fact that assuming a permanent shock,
and thus first differencing, results in an inability to mimic the transitory long-run behaviour of
the underlying DGP. Even though the persistence of the technology shock has increased, the
general conclusion is still similar to that shown previously. That is, trend bias is still the main
source of error when wrongly assuming a stochastic trend.
Alternative Detrending Strategies
The HP filter is a widely-used tool in empirical study to extract the cyclical component
of the series (see e.g. Cesaroni, 2008; Leu, 2011; Dias and Dias, 2013). It is therefore worth
investigating the performance of the filter in estimating the true impulse responses when the
econometrician does not have adequate information regarding the underlying trend process in
output. We consider a case where an econometrician employs the HP filter on output series as a
16
detrending methodology, and then implements short-run restrictions to identify structural shocks
in the system.6 Figure 6 plots total biases given the DGPs of interest with different degrees of
persistence in the technology process.7 We find that the performance of the HP filter depends on
the degree of persistence in the technology shock. This is not surprising. As discussed in Canova
and Ferroni (2011), the relatively persistent shock process in an RBC model would produce the
variability of the series at longer horizons. The HP filter however attributes this low frequencies
cycle as the non-cyclical component, and thus measures the true cyclical component with error.
Specifically, the higher persistence in the technology shock, the larger the trend bias induced by
the HP filter. Another way of thinking about this is if there is a persistence component in the
cycle, the HP filter will mistake part of this persistent cycle as a trend and thus systematically
filter it out. Given one expects the persistence of technology shocks to be large empirically, we
should not expect the performance of the filter to be satisfactory.
At this stage, one might suspect that estimating all series in differences may be flexible
enough to consider both transitory and permanent shocks without the econometrician taking a
stand. While conceptually, differencing does produce permanent shocks, it is possible empirical
exercises may reveal impulse responses which appear transitory. We therefore consider a common
empirical strategy of just first differencing and impose short-run a Cholesky decomposition to
identify the technology shock. Note such an approach has no theoretical support as both shocks
are allowed to be permanent. We therefore generate the RBC model with two transitory shocks,
RBC-dt, and then first difference output and take a Cholesky decomposition to explore whether
the original transitory technology shock can be recovered. Figure 7 suggests any confidence in
recovering underlying transitory shocks despite first differencing may be misplaced. In particular,
none of the impulse response functions are able to recover the underlying transitory response of
output. Moreover, the impulse response functions are so badly biased, missing both the short
and long-run properties of the underlying DGP. This reveals an uncomfortable reality of SVAR
practitioners. Taking a stand on the underlying shock properties and model them as such is
mandatory. As this is an identification exercise, the data cannot speak without imposing any
structure. First differencing as an empirical strategy, with the intention of being agnostic about
the transitory or permanent impacts of a shock, is unfortunately misguided.
6 Conclusion
In this paper, we highlight how the choice of detrending directly feeds into SVAR
identification. While identification in SVARs receive much attention given its role for sensible
empirical analysis, trend misspecification remain possible blindspots for researchers using
SVARs. In an illustrative example using a Monte Carlo study, we demonstrate that the biases
directly attributable to trend misspecifications can be non-trivial. While our example can be
construed as being model, or even parameter, specific, this is at least raises the empirical
6As we simulate quarterly data for our experiments, the smoothness parameter for the HP filter is 1600 asrecommended by Hodrick and Prescott (1997).
7In this exercise, the degrees of persistence used under RBC-dt are 0.25 and 0.9 for low and high casesrespectively. Under RBC-rw, on the other hand, 0.25 and 0.5 are set for low and high persistence in the growthrate of a technology shock.
17
possibility of significant biases emanating from trend misspecification feeding into and
exacerbating biases from identification.
We are deliberately minimalist with the SVAR structure. More work is needed before
general prescriptive advice on detrending in SVARs is possible. Our approach is largely
designed for the purpose of isolating the different sources of biases by keep any potential
identification misspecification to a minimum. This serves in keeping the analysis tractable
while drawing attention to the role of trend misspecification. A natural question is how
important are then these sources of biases within richer and larger model environments. To
this end, the decomposition of the biases which we put forward in this paper is a useful tool for
addressing these questions in future research.
References
Bernanke, Ben, Jean Boivin, and Piotr S. Eliasz, “Measuring the Effects of Monetary Policy: A Factor-augmented Vector Autoregressive (FAVAR) Approach,” The Quarterly Journal of Economics, January 2005,120 (1), 387–422.
Blanchard, Olivier Jean and Danny Quah, “The Dynamic Effects of Aggregate Demand and SupplyDisturbances,” American Economic Review, September 1989, 79 (4), 655–73.
Boivin, Jean and Marc P. Giannoni, “Has Monetary Policy Become More Effective?,” The Review ofEconomics and Statistics, August 2006, 88 (3), 445–462.
Canova, Fabio and Filippo Ferroni, “Multiple Filtering Devices for the Estimation of Cyclical DSGE Models,”Quantitative Economics, 03 2011, 2 (1), 73–98.
Carlstrom, Charles T., Timothy S. Fuerst, and Matthias Paustian, “Monetary Policy Shocks, CholeskiIdentification, and DNK Models,” Journal of Monetary Economics, October 2009, 56 (7), 1014–1021.
Castelnuovo, Efrem, “Monetary Policy Neutrality: Sign Restrictions Go to Monte Carlo,” Marco FannoWorking Papers 0151, Dipartimento di Scienze Economiche Marco Fanno October 2012.
Cesaroni, Tatiana, “Estimating Potential Output using Business Survey Data in a SVAR Framework,” MPRAPaper 16324, University Library of Munich, Germany February 2008.
Chari, V.V., Patrick J. Kehoe, and Ellen R. McGrattan, “Are Structural VARs with Long-run RestrictionsUseful in Developing Business Cycle Theory?,” Journal of Monetary Economics, 2008, 55 (8), 1337 – 1352.
Christiano, Lawrence J., Martin Eichenbaum, and Robert Vigfusson, “What Happens After aTechnology Shock?,” NBER Working Papers, July 2003, (9819).
di Mauro, Filippo, L. Vanessa Smith, Stephane Dees, and M. Hashem Pesaran, “Exploring theinternational linkages of the euro area: a global VAR analysis,” Journal of Applied Econometrics, 2007, 22 (1),1–38.
Dias, Maria Helena Ambrosio and Joilson Dias, “Macroeconomic Policy Transmission and InternationalInterdependence: A SVAR Application to Brazil and US,” EconomiA, 2013, 14 (2), 27 – 45.
Erceg, Christopher J., Luca Guerrieri, and Christopher Gust, “Can Long-Run Restrictions IdentifyTechnology Shocks?,” Journal of the European Economic Association, December 2005, 3 (6), 1237–1278.
Fisher, Lance A, Syeon-seung Huh, and Adrian Pagan, “Econometric Issues when Modelling with aMixture of I(1) and I(0) Variables,” NCER Working Paper Series 97, National Centre for Econometric ResearchOctober 2013.
Francis, Neville and Valerie A. Ramey, “Is the Technology-driven Real Business Cycle Hypothesis Dead?Shocks and Aggregate Fluctuations Revisited,” Journal of Monetary Economics, 2005, 52 (8), 1379 – 1399.
Galı, Jordi, “Technology, Employment, and the Business Cycle: Do Technology Shocks Explain AggregateFluctuations?,” American Economic Review, March 1999, 89 (1), 249–271.
18
Galı, Jordi, “On The Role of Technology Shocks as a Source of Business Cycles: Some New Evidence,” Journalof the European Economic Association, 04/05 2004, 2 (2-3), 372–380.
Han, Fei and Selim Elekdag, “What Drives Credit Growth in Emerging Asia?,” IMF Working Papers 12/43,International Monetary Fund February 2012.
Hansen, Gary D., “Indivisible Labor and the Business Cycle,” Journal of Monetary Economics, 1985, 16 (3),309–327.
Hodrick, Robert J and Edward C Prescott, “Postwar U.S. Business Cycles: An Empirical Investigation,”Journal of Money, Credit and Banking, February 1997, 29 (1), 1–16.
Kimball, Miles S., John G. Fernald, and Susanto Basu, “Are Technology Improvements Contractionary?,”American Economic Review, December 2006, 96 (5), 1418–1448.
Leu, Shawn Chen-Yu, “A New Keynesian SVAR Model of the Australian Economy,” Economic Modelling,2011, 28 (12), 157 – 168.
Linde, Jesper, “The Effects of Permanent Technology Shocks on Hours: Can the RBC-model Fit the VAREvidence?,” Journal of Economic Dynamics and Control, 2009, 33 (3), 597–613.
Pagan, A.R. and M. Hashem Pesaran, “Econometric Analysis of Structural Systems with Permanent andTransitory Shocks,” Journal of Economic Dynamics and Control, 2008, 32 (10), 3376–3395.
Paustian, Matthias, “Assessing Sign Restrictions,” The B.E. Journal of Macroeconomics, 2007, 7 (1), 23.
Peersman, Gert, “What Caused the Early Millennium Slowdown? Evidence Based on Vector Autoregressions,”Journal of Applied Econometrics, 2005, 20 (2), 185–207.
and Ine Van Robays, “Cross-country Differences in the Effects of Oil Shocks,” Energy Economics, 2012, 34(5), 1532–1547.
Poskitt, Donald Stephen and Wenying Yao, “VAR Modeling and Business Cycle Analysis: A Taxonomy ofErrors,” Monash Econometrics and Business Statistics Working Papers 11/12, Monash University, Departmentof Econometrics and Business Statistics April 2012.
Ravenna, Federico, “Vector Autoregressions and Reduced Form Representations of DSGE Models,” Journal ofMonetary Economics, 2007, 54 (7), 2048–2064.
Uhlig, Harald, “What are the Effects of Monetary Policy on Output? Results From an Agnostic IdentificationProcedure,” Journal of Monetary Economics, 2005, 52 (2), 381–419.
19
Table 1: List of Parameters Specified in the Model
Structural parameters True Value
β Discount factor 0.99α Capital share 0.33δ Depreciation rate of capital 0.025η Inverse short-run labor elasticity 0γ Average growth rate of technology shock 1.0074ρz Persistence in technology shock 0.25ρB Persistence in non-technology shock 0.8σz Standard deviation of technology shock 0.007σB Standard deviation of non-technology shock 0.003
Table 2: Monte Carlo Simulation
SVAR DGP
RBC-rw: Yt ∼ I(1) RBC-dt: Yt ∼ I(0)
Long-Run Restrictions Correct Specification 1 Misspecification 1
Short-Run Restrictions Misspecification 2 Correct Specification 2
20
Figure 1: Impulse Responses to a Positive Technology Shock under Correct Specifications
(a) Long-Run Restrictions (Correct Specification 1)
(b) Short-Run Restrictions (Correct Specification 2)
21
Figure 2: Monte Carlo Simulations Implementing Long-Run Restrictions (Misspecifications 1)
(a) Impulse Responses to a Positive Technology Shock
(b) Total Bias Decomposition
22
Figure 3: Monte Carlo Simulations Implementing Short-Run Restrictions (Misspecifications 2)
(a) Impulse Responses to a Positive Technology Shock
(b) Total Bias Decomposition
23
Figure 4: Monte Carlo Simulations Implementing Short-Run Restrictions (Misspecifications 2)in a Case where Hours Worked Negatively Responds to a Positive Technology Shock
(a) Impulse Responses to a Positive Technology Shock
(b) Total Bias Decomposition
24
Figure 5: Total Bias Decomposition Given a High Persistence in a Technology Shock(Misspecification 1)
Figure 6: Total Bias in the case of Using the HP Filter and Implementing Short-Run Restrictions
25
Figure 7: Monte Carlo Simulations Using First Difference and Implementing Short-RunRestrictions
26
