Analysing Two Agnostic Techniques for
Recovering Impulse Response Functions from
Structural Vector Autoregressions
S. Ouliaris∗
A. Pagan
†
J. Restrepo
‡
September 19, 2014
Contents
1 Introduction
2
2 Comparing Two Methods for Generating A Range of Impulse
Responses
5
2.1 The SRC and SRR Methods Applied to a Market Model . . . 6
2.2 Comparing SRC and SRR With a Simulated Market Model . . 9
2.3 Comparing SRC and SRR With a Small Macro Model with
Only Transitory Shocks . . . . . . . . . . . . . . . . . . . . . . 10
3 Combining Parametric and Sign Restrictions
11
3.1 A One Permanent and Two Transitory Shock Model . . . . . . 13
3.2 A Model with Two Permanent Shocks and One Transitory Shock 15
4 Using Factor Augmented SVARs
17
4.1 The Bernanke et al (2005) Two Step Method . . . . . . . . . . 18
4.2 The Boivin and Giannoni (2009) Two-Step Approach . . . . . 22
5 Conclusion
25
∗
Institute for Capacity Development, International Monetary Fund
School of Economics, University of Sydney, [email protected]
‡
Institute for Capacity Development, International Monetary Fund
†
1
6 References
26
Abstract
Structural VARs are used to compute impulse responses to shocks.
Two problems that have arisen with this process involves the information needed to perform this task. One involves how the shocks
are to be distinguished into technology, monetary effects etc. Increasingly the signs of impulse responses are used for this task. However
it is often desirable to impose some parametric assumption as well
e.g. monetary shocks have no long-run impact on output. Existing
methods for combining sign and parametric restrictions are not well
developed. In this paper we provide a relatively simple way to allow
for these combinations and show how it works in a number of different
contexts. A second issue arising with SVARs is that they need to be
kept relatively small. To get around this it has been suggested that
factors be used to summarize a large set of data and these can then be
used in the SVAR. Some simultaneity issues arise when doing this and
the paper shows that two existing methods to resolve it, and which
have been widely used, fail to do so.
1
Introduction
Structural Vector Autoregressions (SVARs) have become a standard way of
modelling macroeconomic series. A SVAR of order p in N variables yt is
A0 yt = A1 yt−1 + ... + Ap yt−p + Ωεt ,
where Ω is diag{σ 1 , .., σ N ) and A0 has unity on its diagonal. Associated with
the SVAR is the Moving Average representation
yt = C(L)εt = C0 εt + C1 εt−1 + ....
where the Cj are the impulse responses of yt+j to a unit shock in εt . When
some of the variables in yt are differences in I(1) variables the long-run impulse responses in C(1) show the response of y∞ to variations in εt . Because
the SVAR is a set of structural equations there is a limit to the number of
parameters in A0 that can be estimated. A variety of methods have emerged
to deal with this problem. Short-run restrictions set some of the elements of
A0 to zero. When some of the members of yt are differences in I(1) variables
2
long-run restrictions may be imposed on C(1). These translate into restrictions between the elements of all Aj . It is also possible to envisage restrictions
upon the impulse responses Cj themselves. All of these methods are examples of how parametric restrictions can be used to distinguish between the
shocks inet .
Many extensions have been made to the SVAR modelling framework described above. One is to allow p to be infinite - see Lutkepohl and Saikkonen
(1997). Another, motivated by the fact that estimateion of A0 requires instruments, finds these by using information about structural change in the
parameters of the SVAR - see Herwartz and Lütkepohl (2011) and Lanne and
Lütkepohl (2008). A third has been the suggestion that the shocks εt can
be differentiated by the signs of the impulse responses in Cj , rather than by
whether they have precise numerical values - see Faust(1998), Canova and
De Nicoló (2002), Uhlig (2005) and Peersman (2005). Finally, because even
a moderate p means that N must be kept relatively small in order to handle
the fact that the number of RHS regressors in any equation of the SVAR is
N × p can easily exceed the number of observations T. Yet there are often
many variables available to measure macroeconomic outcomes and so some
parsimonious way of introducing these variables into a SVAR is needed. One
solution has been the Factor Augmented Vector Autoregression (FAVAR) of
Bernanke et al (2005). A survey of this last approach is available in Lütkepohl
(2014).
As mentioned above one of the popular methods for estimating impulse
responses in SVARs is that involving sign restrictions (SR). In Uhlig’s (2005)
terminology the aim of the exercise is to be "agnostic", in the sense that
no precise parametric restrictions are imposed upon the SVAR coefficients
when finding the impulse responses to the shocks present in the SVAR system.
Rather, much weaker conditions coming from the need for impulse responses
to have certain signs are applied. This method involves two steps. In the
first step a large number of impulse responses for the uncorrelated shocks εt
are generated. Then, in the second stage, these are judged by whether they
have the expected sign restrictions. Those that pass this test are retained,
after which the resulting set of such responses needs to be summarized in
some way. Because the impulse responses found in the first step involves recombining an initial set of impulse responses, we will designate this approach
as SRR, where the R stands for re-combination. Section 2 first discusses this
strategy in terms of the simple market model set out in Fry and Pagan (2011).
Then it proposes a new approach that involves finding impulse responses by
3
varying the unknown coefficients of the SVAR equations i.e. it aims to find a
range of values for any unknown elements in A0 . We designate this method
as SRC, where C stands for coefficients.
In SRC, after finding the impulse responses for a range of possible values
of the unknown coefficients, the responses are referred to the postulated
signs to determine if they should be retained. Hence SRR and SRC are
differentiated solely by how they proceed in the first stage i.e. in the way by
which they produce a range of impulse responses whose signs can be judged.
Section 2.2 simulates some data from the market model and we find that
both of the sets of impulse responses generated by SRR and SRC contain a
close approximation to the true impulse responses. Morover, the frequency
of rejection of the impulse responses arising from their not matching the
postulated signs is very close for both methods. Section 2.3 then applies
SRC to the three variable empirical macro model which was analysed by Fry
and Pagan using SRR. Again we find that SRC and SRR produce comparable
results in terms of rejection frequency, but SRC seems to give a wider range
of impulse responses that are compatible with the sign restrictions.
The macro model used in section 2.3 has only transitory shocks and there
are no parametric restrictions on A0 or C(1). Therefore section 3 therefore
looks at the same model when there is a combination of parametric and
sign restrictions. It turns out that SRC is extremely useful in this context,
particularly when long-run restrictions on C(1) are involved. This is because
the parametric restrictions upon the SVAR can be imposed first, after which
impulse responses can be generated and tested for whether they match the
prescribed sign restrictions. The simple three variable model is examined
in two ways, depending on the number of permanent and transitory shocks
present in the system. In one case there is a close correspondence between
SRC and SRR while, in the other, there are differences. Again SRC produces
a wider range of impulse responses.
Sign restrictions are not the only "agnostic" procedure that is popular in
SVAR work. Another is the FAVAR model of Bernanke et al (2005) mentioned above. Agnosticism arises here since one does not specify a tightly
structured SVAR with a small number of variables. Instead the SVAR is
constructed so that it incorporates a very large number of variables. Specifically, this is done by assuming that the large number of variables have a factor
structure. Factors enable one to model items such as an interest rate rule
without insisting that this depend solely upon inflation and output, thereby
responding to the knowledge that monetary authorities look at a very large
4
number of macro-economic series when making their decisions. A difficulty
that arises in this literature is how the factors are to be entered into the
SVAR, given that a contemporaneous relationship between variables such as
an exchange rate and an interest rates can be expected, and the factors are
formed from these variables. This has led to proposals aiming to "purge" the
factors of such interest rate effects before they are included in the SVAR.
The two most popular methods for doing this are Bernanke et al (2005) and
Boivin and Giannoni (2009). We show in section 4 that these methods do not
handle the problem in a satisfactory way and suggest an alternative method
that invokes Bernanke et al’s assumptions, but does produce consistent estimators of the impulse responses. Section 5 then concludes.
2
Comparing Two Methods for Generating
A Range of Impulse Responses
In Fry and Pagan (2011) the following market model was investigated
qt = αpt + σ D ε1t
qt = βpt + σ S ε2t ,
(1)
(2)
where qt is quantity, pt is price, and the shocks εjt are n.i.d.(0, 1) and uncorrelated with one another. The first curve might be a demand curve and
the second a supply curve ( implying that α would be negative and β positive). Because lags are omitted from (1) and (2) this is a structural but not
a SVAR. Nevertheless, it is useful to abstract from lags and this can be done
without loss of generality. Based on this model we could form
σ −1
D qt = (α/σ D )pt + ε1t
σ −1
S qt = (β/σ S )pt + ε2t ,
(3)
(4)
which can be represented as
aqt = bpt + ε1t
cqt = dpt + ε2t .
(5)
(6)
A unit shock to εjt is then equivalent to a one standard deviation shock in
demand (eD,t = σ D ε1t ) and supply (eS,t = σ S ε2t ) shocks. The corresponding
−1
a −b
impulse responses to these shocks will be
.
c −d
5
2.1
The SRC and SRR Methods Applied to a Market
Model
In the standard sign restrictions methodology (SRR) one produces initial
impulse responses by fitting a recursive model
qt = s1 η 1t
pt = φqt + s2 η 2t ,
(7)
(8)
to data on qt and pt , where the η 1t are n.i.d(0, 1) and the sj are the standard
deviations of the errors for the equations. The first stage of SRR is then
implemented by applying some weighting matrix Q to the initial shocks η 1t
and η2t so as to produce new shocks η ∗1t and η ∗2t i.e. η∗t = Qη t . Q is chosen
in such a way as to ensure that QQ = Q Q = I, which means that the new
shocks are also uncorrelated
with unit variances.
One matrix to do this is
cos λ − sin λ
the Givens matrix Q =
, where λ are values drawn from
sin λ cos λ
the range ( 0, π). After adopting this the new shocks η ∗t = Qη t will be
cos λη 1t − sin λη 2t = η ∗1t
sin λη1t + cos λη 2t = η ∗2t .
Using the expressions for η 1t and η 2t in (7)-(8) we would have
(cos λ/s1 )qt − (sin λ/s2 )(pt − φqt ) = η ∗1t
(sin λ/s1 )qt + (cos λ/s2 )(pt − φqt ) = η ∗2t ,
which, after re-arrangement, is
[(cos λ/s1 ) + (sin λ)(φ/s2 )]qt − (sin λ/s2 )pt = η∗1t
[sin λ/s1 ) − (cos λ)(φ/s2 )]qt + (cos λ/s2 )pt = η∗2t .
(9)
(10)
Now this has the same form as (5)-(6) when
a = cos λ/s1 + (sin λ)(φ/s2 ), b = sin λ/s2 ,
c = sin λ/s1 − (cos λ)(φ/s2 ), d = − cos λ/s2
εjt = η ∗jt
6
(11)
(12)
(13)
The latter can hold since both sets of random variables are uncorrelated and
n.i.d.
Now the impulse responses for η∗t are produced by re-combining those
for η t with the matrix Q, and this is generally how the strategy employed
in SRR is described. An alternative view would be that the SRR method
generates many impulse responses by expressing the A0 coefficients of the
SVAR model in terms of λ, and then varying λ over the region (0, π). Once
the impulse responses are found sign restrictions are applied to say which are
to be retained. So we are generating many impulse responses by making all
the market model parameters i.e. A0 depend upon λ and the data (through
φ, s1 and s2 ).
Rather than expressing the model parameters in terms of λ consider the
possibility of going back to (1) and making the coefficient α a function of
some θ, where θ varies over a suitable range. Given a value for θ this fixes
α. However, to compute impulse responses it is necessary to estimate the
remaining parameters in the system. After setting θ to some value θ∗ this
can be done using the following method.
1. Form residuals ê∗1t = qt − α(θ∗ )pt .
2. Estimate σ 1 with σ̂ ∗1 , the standard deviation of these residuals.
3. Using ê∗1t as an instrument for pt estimate β by Instrumental Variables
∗
(IV) to get β̂ .
∗
∗
4. Using β̂ form the residuals ê∗2t = qt − β̂ pt . The standard deviation of
these, σ̂ ∗2 , will estimate the standard deviation of the second shock.
Using earlier results, the contemporaneous impulse responses to one stan
−1 ∗
1 −α(θ∗ )
σ̂ 1 0
∗
dard deviation shocks will be
. Accordingly, just
0 σ̂ ∗2
1 −β̂
as for λ, we can vary θ to generate many impulse responses. These are directly comparable with the impulse responses generated by SRR, except that
they all depend upon θ and the data (via the IV estimation) rather than λ
and the data. Because the technique consists of finding a range of impulse responses by varying the coefficient α (through varying θ) it is the SRC method
mentioned in the introduction.
It is worth looking closer at these two methods. A number of points
emerge.
7
(i) θ will normally be chosen so as to get a range of variation of α that
is (-∞, ∞). This can be done by drawing θ from a uniform (-1,1) density
θ
and then setting α = 1−abs(θ)
. If one insisted that α was the parameter of
a demand curve, then the range could be made (0, −∞), but in most cases
we would not possess such information. By comparison λ is drawn from a
uniform density over (0, π), because of the presence of λ in the harmonic
terms. In both approaches one has to decide upon the number of trial values
of θ and λ to use i.e. how many impulse responses are to be computed
(ii) It might seem as if the standard deviation of shocks can be estimated
with SRC but not with SRR. However, if one compares (3)-(4) with (9)-(10)
it is clear that there is information in the SRR approach that does enable one
to estimate the standard deviations of the shocks, namely that contained in
the parameters a, b, c, d in (5)-(6). The reason why the standard deviations
seem not to be available in SRR is that it focusses on impulse responses to one
standard deviation shocks, and does not estimate the standard deviations per
se. But, as the previous discussion shows, it is clearly present in the approach.
To get standard deviations of the shocks it is necessary to decide on some
normalization of the equations. In the SRC approach a model was estimated
in which the equations were normalized on quantity. Doing the same for SRR
shows that (1/a) and (1/c) will be the estimates of the standard deviations
of the shocks. If a different normalization was chosen then there would be
different estimates of the standard deviations, and this would be true of
both methods. In multivariate systems the standard deviations of the shocks
in the SRR method can be found by working with the contemporaneous
matrix of impulse responses C0 produced by the re-weighting of the initial
contemporaneous responses. Because a structural VAR has a form A0 yt = εt
(ignoring lags), A0 = C0−1 . When a normalization is used, this fixes some
of the elements of A0 , a0ij , to unity, and so the standard deviations for the
shocks can be find by determining what scaling of the C0−1 matrix coming
from SRR is needed to effect this. In the normalization used above, where
a011 and a021 are to be made unity by re-scaling, the standard deviations will
be 1/a011 and 1/a021 .
(iii) In a SVAR with n variables and no parametric restrictions the number
of λj to be generated in the SRR method equals n(n − 1)/2. Thus, for n = 3,
three λ s are needed. This is also true of the number of θ used in SRC. So
problems arising from the dimensions of the system are the same for both
methods. It should be noted however that, when parametric restrictions
8
are also applied along with sign restrictions, the number of θ may be much
smaller, as we will show in a 3 variable case in section 3.
2.2
Comparing SRC and SRR With a Simulated Market Model
To look more closely at these two methods we simulate data from the following market model
qt = −pt + ε1t
(14)
√
qt = 3pt + 2ε2t .
.75 .3536
The true impulse responses are
. Five hundred values for
.25 −.3536
θ and λ were generated from a uniform random number generator. This
enables 500 impulse responses to be produced for each method. Inspecting
these we find that the closest fit to the true impulse responses for each method
was
.7369 .3427
.7648 .3529
SRC =
, SRR =
.
.2484 −.3605
.2472 −.3563
It is clear that both methods have among their range of impulse responses
values a good match to the true impulse responses. Changing the parameter
values for the market model did not change this conclusion. There was some
sensitivity to sample size. In the results above there were 1000 observations.
When it is reduced to 100 observations the equivalent results are
.7421 .3615
.7828 .4119
SRC =
, SRR =
.
.1702 −.3923
.1780 −.3806
so that SRC seems to be a slightly better fit to the true impulse responses,
although they both provide a reasonable match.
Often when sign restrictions are used the retained impulse responses are
summarized by using the medians of the individual responses. Doing this for
the market model produces medians of
.4082 .3119
.6234 .5665
SRC =
, SRR =
.
.5284 −.4432
.3429 −.2655
9
Neither co-incides with the true values nor are they the same. The reason was
mentioned in Fry and Pagan (2011) - although the true impulse responses
lie in the range generated by the methods they do not necessarily lie at the
median. Indeed, while the median response of quantity to a demand shock
is .4082 for SRC, the true response lies at the 89th percentile. Unless one
had some extra information for preferring one set of impulse responses to
another the median has no more appeal than any other percentile. As the
result above shows the percentile at which the true impulse responses lie can
also vary with which method, SRC or SRR, is used. It is worth noting that
SRR generates impulses that are compatible with the sign restrictions 87.8%
of the time and for SRC it is 85.4%. This is a high percentage but, since the
model is correct, this is what would be expected. It is interesting that, as the
sample size is reduced, the percentage of acceptances rises for both methods.
2.3
Comparing SRC and SRR With a Small Macro
Model with Only Transitory Shocks
We will look at the two methods in the context of a three variable SVAR
macro model used in Fry and Pagan (2011). The variables in the system
consist of three variables y1t ,y2t and y3t , where y1t is an output gap, y2t is
quarterly inflation, and y3t is a nominal interest rate. Quarterly data on
these variables used in Fry and Pagan come from Cho and Moreno (2006).
All variables are assumed to be I(0), and so there are three transitory shocks,
labelled productivity, demand and an interest rate. The expected signs of the
contemporaneous impulse responses are given in Fry and Pagan except that
here the productivity shocks are the negative of the cost shocks designated
in Fry and Pagan.
The model fitted is1
y1t = a012 y2t + a013 y3t + a112 y2t−1 + a113 y3t−1 + a111 y1t−1 + ε1t
y2t = a021 y1t + a023 y3t + a122 y2t−1 + a123 y3t−1 + a121 y1t−1 + ε2t
y3t = a031 y1t + a032 y2t + a122 y2t−1 + a123 y3t−1 + a131 y1t−1 + ε2t .
(15)
(16)
(17)
The SRR method begins by setting a012 = 0, a013 = 0 and a023 = 0 to
produce a recursive model, and then recombines the impulse responses found
1
For illustration we assume a SVAR or order one, but in the empirical work it is of
order two.
10
from this model. In contrast, the SRC method proceeds by first fixing a012
and a013 to some values and computing residuals ε̂1t . After this (16) can be
estimated by fixing a023 to some value and using ε̂1t as an instrument for y1t .
The residuals from (16) can then be found and (17) can be estimated using
ε̂1t and ε̂2t as instruments for y1t and y2t . Once all shocks have been found
impulse responses can be computed. Of course three parameters have been
θ1
fixed and so they need to be allowed to vary using a012 = (1−abs(θ
, a013 =
1 ))
θ2
θ3
, a023 = (1−abs(θ
. Note that three different random variables θj are
(1−abs(θ2 ))
3 ))
needed and these correspond to the three λj in the Givens matrices. As for
the market model the methods are computationally equivalent.2
Unlike the market model it is not easy to find impulse responses that
satisfy the sign restrictions. For both methods only around 5% of the impulse
responses are retained. 1000 of these were plotted for SRR in Figure 1 of
Fry and Pagan. Therefore, Figure 1 below gives the same number of impulse
responses from the SRC method (here we convert the productivity shock to
a cost push shock in order to enable the comparison). It seems as if SRC
produces a broader range of impulse responses than SRR, e.g. the maximal
contemporaneous effect of demand on output with SRC is more than twice
what it is for SRR (we note that all impulse responses in the ranges for both
SRC and SRR are valid and observationally equivalent).
3
Combining Parametric and Sign Restrictions
The big advantage of the SRC method is that it is very effective when a combination of parametric and sign restrictions is to be used. If the restrictions
are of the long-run variety, the system must have a set of permanent and
transitory shocks. Our first example in the next sub-section features a model
with one permanent shock and two transitory shocks. We compare SRC and
2
This points to the fact that the impulses found with SRC and SRR do not span the
same space. Thinking of this in the context of the market model it is clear that we could
find a α ( for SRC) that would produce exactly the same α as coming from SRR. But the
estimate of β found by both methods would then differ. The two sets of impulse responses
are connected by a non-singular transformation but this varies from trial to trial. If it did
not then the impulse responses would span the same space.
11
Output Gap Response to a Demand Shock
Inflation Response to a Demand Shock
Interest Rate Response to a Demand Shock
1.4
1.2
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.0
-0.2
-0.2
-0.2
5
10
15
20
25
5
30
Output Gap Response to a Cost-Push Shock
0.2
10
15
20
25
5
30
Inflation Response to a Cost-Push Shock
.8
10
15
20
25
30
Interest Rate Response to a Cost-Push Shock
1.0
0.0
0.8
.6
-0.2
0.6
.4
-0.4
-0.6
0.4
0.2
.2
-0.8
0.0
.0
-1.0
-0.2
-1.2
-.2
5
10
15
20
25
-0.4
5
30
Output Gap Response to an Interest Rate Shock
10
15
20
25
5
30
Inflation Response to an Interest Rate Shock
0.4
1.2
.0
0.0
0.8
-.2
-0.4
0.4
-.4
-0.8
0.0
-.6
-1.2
-0.4
-1.6
5
10
15
20
25
30
15
20
25
30
Interest Rate Response to an Interest Rate Shock
.2
-.8
10
-0.8
5
10
15
20
25
30
5
10
15
Figure 1: 1000 Impulses Responses from SRC Satisying the Sign Restrictions
for the Small Macro Model using the Cho-Moreno Data
12
20
25
30
SRR in this case and find that the methodologies proceed in the same way.
An empirical example is given based on the model used in section 2.3. The
following sub-section changes the model to have two permanent shocks and
one transitory one. Then there are differences in the methodologies.
3.1
A One Permanent and Two Transitory Shock Model
We will look at the two methods in the context of the three variable SVAR
model in section 2.3. Now however we will assume that the log level of
GDP is z1t , and it will be taken to be I(1). In Cho and Moreno the output
gap was formed by regressing z1t against a constant and a time trend and
then using the residuals. Hence ∆z1t = y1t . The SVAR system composed of
y1t (= ∆z1t ), y2t and y3t will then be assumed to have one permanent (supply)
shock in the system plus two transitory shocks associated with demand shock
and an interest rate. By definition these transitory shocks both have a zero
long run effect on output, z1t .
Before imposing any long-run restrictions the system would be
∆z1t = a012 y2t + a013 y3t + a112 y2t−1 + a113 y3t−1 + a111 ∆z1t−1 + ε1t (18)
y2t = a021 ∆z1t + a023 y3t + a122 y2t−1 + a123 y3t−1 + a121 ∆z1t−1 + ε2t (19)
y3t = a031 ∆z1t + a032 y2t + a122 y2t−1 + a123 y3t−1 + a131 ∆z1t−1 + ε2t (20)
Now the two transitory shocks must have a zero long-run effect upon output,
and we take these to be the second and third ones. Following Fisher et al
(2014) this restriction can be imposed on the system (18)-(20) by using the
Shapiro and Watson (1988) approach of replacing (18) with
∆z1t = a012 ∆y2t + a013 ∆y3t + a111 ∆z1t−1 + ε1t ,
(21)
thereby allowing the parameters of (21) to be estimated by using y2t−1 , y3t−1
and ∆z1t−1 as instruments. Once parameter estimates for (21) are obtained
one can get residuals ε̂1t . This is the first step in implementing both the
SRC and SRR approaches. Unlike the market model in which a recursive
model provided the initial impulse responses to be re-combined in SRR, it is
now necessary to use a non-recursive system that incorporates (21) in order
to ensure that there will be one permanent and two transitory shocks in the
system.
Having recovered the permanent shock ε1t by the use of parametric restrictions, it will be the case that the permanent impulse responses are known,
13
and they will not change in different trials i.e. they are not re-combined
in SRR. To understand why this is so, suppose
SVAR is written as
 the 
∆z1t
A0 ζ t = A1 ζ t−1 + εt , where ζ t is the 3×1 vector  y2t  . Then the moving
y3t
average representation is ζ t = C0 εt + C1 εt−1 + ... Because E(εt εt−j ) = 0 for
j > 0, we can recover C0 by regressing ζ t against εt . Looking at that regres0
0
0
sion for the first variable it would have the form ∆z1t = C11
ε1t +C12
ε2t +C13
ε3t
0
. Furthermore, because ε1t is uncorrelated with ε2t and ε3t , C11 can be estimated by regressing ∆zt on ε1t .3 By the same argument all that is needed in
order to recover the impulse responses for the permanent shock are ζ t (data)
and an estimate of ε1t , ε̂1t .
Following on with the SRC methodology, the second equation (19) needs
to be estimated and ε̂1t , y2t−1 , y3t−1 , and ∆z1t−1 are available as the instruments.
This is one fewer than needed. However, when a021 is fixed, this equation
could be estimated, since y2t − a021 ∆z1t would become the dependent variable. Once estimated, residuals ε̂2t would be available, and these can be used
along with ε̂1t , y2t−1 , y3t−1 and ∆z1t−1 to estimate the last equation. Thus
the SRC method replaces a021 with some value. But this is exactly the same
situation as occurred with the market model, i.e. once a021 is replaced by
some function of θ, every θ produces new impulse responses. It is crucial to
note however that, as θ is varied, the long-run restriction is always enforced
by design of the SVAR i.e. by using (21) as part of it. Because these restrictions reduce the number of parameters that have to be estimated, only
one parameter needs to be varied in order to get all the impulse responses.
Lastly, once a range of impulse responses has been found, sign restrictions
can be applied to determine which of the two transitory shocks is demand
and which is monetary policy. Because the permanent shock does not depend
in any way upon the values assigned to a021 , it is invariant to the changing
values of this coefficient.
Now consider SRC. There are two transitory shocks and one permanent
shock. Normally in the SRR methdology one combines all impulse responses
to get a set of new ones. But, as Fry and Pagan (2011) argued, one has to be
careful in doing that. If one combines a permanent shock with a transitory
one it will be permanent, which is satisfactory for producing a new set of
3
The same argument applies to regressing the other two variables in zt to get their
impulse responses to the permanent shock ε1t .
14
permanent shocks, but it is invalid if one wants the resulting shocks to be
transitory. This can be seen by noting that, if the impulse responses to the
initial shocks η t are Cj , then those to η ∗t = Qηt will be Cj∗ = Cj Q . Suppose
one of the shocks ( say the first) has a permanent effect on the first variable.
It then follows that the first column of C(1) will not be all zeros, and so all the
columns of C ∗ (1) will have non -zero elements, unless Q has many zeros, and
this is very unlikely. Therefore if all shocks are combined together every one
of the η ∗t will have permanent effects, which contradicts the requirement that
some of the shocks be transitory. Because of this Fry and Pagan suggested
that the permanent and transitory shocks be combined together in different
blocks with different weighting matrices. Applying this to the example above,
because there is only a single permanent shock this would not be re-combined
with any of the others, and it is only the transitory shocks that would be
re-combined. If a Givens matrix is used as the weighting matrix this would
be done by varying a single λ over the (0, π) range and that is the equivalent
of varying a012 in the SRC approach.
Estimating the SVAR with a permanent shock by the SRC technique
now results in 45% of the responses satisfying all the sign restrictions, as
compared to the 5% with purely transitory shocks. Consequently, it seems
that the data are more compatible with the sign restrictions, provided one
allows for a permanent supply side shock to GDP.
3.2
A Model with Two Permanent Shocks and One
Transitory Shock
We now consider the model of the preceding sub-section but with two permanent shocks and one transitory shock. The two permanent shocks will be
taken to be supply and demand and the transitory shock will be that to interest rates. This is a case where the I(0) variable y2t has a structural shock
associated with it which has a permanent effect upon output (z1t ). Fisher et
al (2014) deal with such a case, noting that re-writing (21) as
∆z1t = a012 y2t + a013 ∆y3t + a111 ∆z1t−1 + a112 y2t−1 + ε1t ,
(22)
will allow for two permanent shocks.
Because y2t appears as a level on the RHS of (22), rather than as a
difference ∆y2t , there are not enough instruments to estimate it using IV.
Accordingly, in the SRC approach, we would need to form ∆z1t − a012 y2t , and
15
θ1
use this as the dependent variable. This is done by setting a012 = 1−abs(θ
.
1)
After estimation ε̂1t can be computed and then the second equation will
θ2
be estimated with a023 = 1−abs(θ
.4 A range of impulse responses will be
2)
produced by generating many θ1 and θ2 values. In this case, unlike that of
the preceding sub-section, the impulse responses of all shocks depend on the
values used for a012 and a023 . Notice that, if there was no transitory shock
assumption, then it would be necessary to vary a013 as well, so the parametric
restriction of a transitory shock reduces the number of parameters that need
to be estimated by one.
So the above describes SRC. How would this be done with SRR? First,
it needs to be recognized that there has to be one transitory shock, and
that is imposed by working with (22) in any system used to generate initial
shocks. To overcome the fact that an extra instrument is needed, SRR would
generally work with a recursive structure, i.e. put a012 = 0 when generating
initial impulse responses. In the same way a021 would be set to zero when
estimating the second equation, i.e. the initial impulses would be found by
imposing the transitory shock, a012 = 0 and a023 = 0. If one followed the
principle outlined above of separately combining permanent and transitory
shocks, then there would be one value of λ (when Givens is the weighting
matrix) used to combine the two initial permanent shocks. Since there is only
one transitory shock it cannot be combined together with the permanent
shocks. If they were combined, all shocks would have permanent effects.
Thus it is clear that, in this instance, SRC and SRR will be different. It
would seem that SRR would not provide as complete a range of impulse
responses because there is only one free parameter that is varied, λ, and
there is only one impulse response for the transitory shock — that from the
initial model. If one wanted to combine all three shocks together (as done in
SRC) it would be necessary to find a Q matrix with the requisite zeros to
enforce the long-run zero impact on output of the transitory shock. It seems
much simpler just to proceed with the SRC technique.
After applying SRC to this model it is found that about 26 percent of the
trials satisfy the desired sign restrictions, as compared with the 45 percent
when there was just a single permanent shock. One could interpret this as
4
There are other parametric restrictions that might be applied apart from long-run ones
and these would also generate instruments. Suppose that the second shock is not transitory
but is taken to have a zero contemporaneous impact on output. Then the VAR (reduced
form) residuals for ∆z1t can be used as an instrument in the second structural equation.
Consequently, zero restrictions upon C0 are easily handled in the SRC methodology.
16
the data being rather negative about the idea that there are both permanent
demand and supply shocks, although both models have a superior record of
accepting the sign restrictions than if the assumption is made that all shocks
are transitory.
4
Using Factor Augmented SVARs
Often many variables may be available which are expected to influence macroeconomic outcomes. Thus financial factors and confidence might be important to decisions. Because there is rarely a single measure of these there is a
tendency to utilize many approximate measures, particularly involving data
surveying the attitudes of financial officers, households or businesses. There
are far too many of these measures to put them all into a SVAR, so some
aggregation needs to be performed. For a small system involving macroeconomic variables such as the unemployment rate, industrial production and
employment growth, Sargent and Sims (1977) found that two dynamic factors could explain 80 percent or more of the variance of those variables. One
of the factors was primarily associated with real variables and the other with
inflation. Bernanke et al (2005) extended this approach and they proposed
augmenting a SVAR with a small number of factors.
There are two difficulties in implementing a factor oriented approach.
One is how to measure the factors and the other is how to enter these into a
SVAR, particularly in deciding on how to estimate the contemporaneous part
of the SVAR. Bernake et al suggested two estimation methods for this model.
One involved using a Gibbs sampling technique that essentially integrated
the factors out of the likelihood, while the other was a two stage strategy
that was less computationally demanding and required fewer distributional
assumptions. Many applications of the model have been made and mostly
these have been done with the two-stage strategy which we examine below.
A primary concern in the two-stage approach is how to account for the simultaneous determination of some of the variables from which factors are formed
and interest rates. In section 4.1 we show that the method Bernanke et al
use to account for this simultaneity does not deliver consistent estimators of
the impulse responses of interest. Nevertheless we show that it is possible to
utilize their framework to find a method that will provide consistent estimators. The following sub-section studies a modification of what was done by
Bernanke et al suggested by Boivin and Giannoni (2009). We find that this
17
approach would be expected to yields inconsistent estimators of the impulse
responses as it fails to account for the simultaneity
4.1
The Bernanke et al (2005) Two Step Method
The system Bernanke et al seem to have in mind consists of
Xt = ΛFt + Λr Rt + et
Ft = Φ11 Ft−1 + Φ12 Rt−1 + ε1t
Rt = Φ21 Ft−1 + Φ22 Rt−1 + BFt + ε2t ,
(23)
(24)
(25)
where Xt is an N × 1 vector of "informational variables", Ft is a K × 1 vector
of factors and Rt is a nominal interest rate. N is much greater than K. (23) is
their equation (2), while (24) − (25) correspond to their equation (1), except
it is written as an SVAR rather than a VAR. The SVAR structure comes
from their examples, in which the factor Ft enters contemporaneously into
the central bank’s decision rule for interest rates along with the statement
(p 401) that "all the factors entering (1) respond with a lag to changes in
the monetary policy instrument". We will focus on the empirical part of the
paper where there is a single observable factor - the interest rate - although
they also suggest that Rt might be replaced by a vector Yt of observables.
In their application Xt in (23) is a large data set of 119 variables, where
Rt is excluded from Xt .5 This data set consists of "fast moving" and "slow
moving" variables, where the difference is that slow moving variables Xts do
not depend contemporaneously on Rt . The fast moving variables will be Xtf .
Given the factor structure the key to estimation is Bernanke et al’s suggestion that the slow moving variables depend contemporaneously on the
factors but not the interest rate, i.e.
Xts = GFt + vt .
Now suppose that K principal components (PCs) are extracted from Xts .
Then Bai and Ng (2006),(2013) show that the asymptotic relation between
the principal components (P Cts ) and the K factors √will be Ft = H(P Cts )+ξ t ,
where ξ t is Op ( √1N ), provided N −→ ∞ such that TN → 0. i.e. the principal
components asymptotically span the space of the factors. Bai and Ng (2013)
5
Here we follow the Matlab program that Boivin supplied that reproduces their results.
.
18
consider what would be needed for H to be the identity matrix, and state
some conditions that would need to be enforced in forming the principal
components, but these methods are unlikely to have been used in the FAVAR
applications. Nevertheless, as we show below, H is not needed if we are not
interested in the impact of shocks on the factors.
Given the relation between factors and PCs (24) and (25) can be expressed
as
s
) + Φ12 Rt−1 + ε1t + {Φ11 ξ t−1 − ξ t )
H(P Cts ) = Φ11 H(P Ct−1
s
Rt = Φ21 H(P Ct−1 ) + Φ22 Rt−1 + BH(P Cts ) +
ε2t + {Φ21 ξ t−1 + Bξ t }.
(26)
(27)
Now the terms in curly brackets will not affect the asymptotic distribution
of the OLS estimators of Φij provided N is large, so we will ignore them in
the analysis which follows.6 This means that (26)-(27) will be written as
s
P Cts = H −1 Φ11 H(P Ct−1
) + H −1 Φ12 Rt−1 + H −1 ε1t
s
Rt = Φ21 H(P Ct−1
) + Φ22 Rt−1 + BH(P Cts ) + ε2t .
(28)
(29)
This is an SVAR in Rt and P Cts except that, unlike a regular SVAR, the
shocks in the P Cts equations are contemporaneously correlated.7 Once the
SVAR in P Cts and Rt is estimated the impulse responses to the monetary
shock ε2t can be found.
To determine the impact upon members of Xtf and Xts it is necessary
to express these in terms of the SVAR variables. Thus, using the mapping
between factors and principal components,
Xts = GH(P Cts ) + vt ,
and the regression of Xts on P Cts consistently estimates GH, because Ft is
assumed uncorrelated with vt . Consequently, the impulse responses of Xts to
the monetary shocks can be computed. In general
Xt = (ΛH)P Cts + Λr Rt + et ,
6
(30)
Bai and Ng (2006) observe that the conditions on N are sufficient to ensure that
the estimation of the factor by the principal components does not affect the asymptotic
distribution of regressions using the principal components - see Bai and Ng (2006, p 1137).
7
Because the same regressors appear in (28) the OLS estimator is efficient. But the
non-diagonal covariance matrix for the errors coming from Hε1t means that this needs to
be allowed for when getting standard errors of H −1 Φ11 H and H −1 Φ12 .
19
and a regression will give the weights ΛH and Λr .
Now, the K principal components of X ( P Ctx ) can be written as
P Ctx = w Xt ,
where w are the matrix of weights found from the PC analysis. Then, using
(30),
P Ctx = w [(ΛH)P Cts + Λr Rt + et ]
= (w ΛH)P Cts + w Λr Rt + w et ,
and a regression of P Ctx against P Cts and Rt consistently estimates (w ΛH)
and w Λr . Adding this to the system of SVAR equations results in the system
s
P Cts = A111 P Ct−1
+ A112 Rt−1 + H −1 ε1t
s
Rt = A121 P Ct−1
+ A122 Rt−1 + A021 P Cts + ε2t
P Ctx = A031 P Cts + A032 Rt + w et ,
(31)
(32)
(33)
where Akij is the coefficient of the j th variable in the i’th equation for the
k th lag. By assumption all shocks are contemporaneously uncorrelated, but
the presence of an unknown H in H −1 εt will mean that it is not possible
to calculate impulse responses with respect to the shocks ε1t . It is also not
possible to estimate H from the covariance matrix of H −1 ε1t , because the
former has K 2 elements and the latter K × (K + 1)/2. As Bai and Ng note,
making H triangular would identify it.
The above analysis describes a SVAR that can be used to find impulse
responses of variables in Xt to interest rate shocks. However, this is not the
system that Bernanke et al work with. Rather they first regress P Ctx against
P Cts and Rt to estimate A031 and A032 , and then use F̃t = P Ctx − Â032 Rt in a
block recursive SVAR with the form
F̃t = Ψ11 F̃t−1 + Ψ12 Rt−1 + η1t
Rt = Ψ21 F̃t−1 + Ψ22 Rt−1 + DF̃t + η 2t ,
(34)
(35)
where η jt are assumed to be white noise and uncorrelated. Can one recover
such a SVAR from the one involving P Cts and Rt ? To investigate this we will
assume that Â032 = A032 , Â031 = A031 , so that inverting (33) to get P Cts and
20
then using the definition of F̃t gives P Cts = (A031 )−1 (F̃t −w et ). Accordingly, it
is clear that the SVAR involving F̃t will deviate from that with P Cts owing to
the presence of terms involving w et in the errors of the former. Substituting
P Cts = (A031 )−1 (F̃t − w et ) into (31) and (32) we would get n SVAR like
(34)-(35) that had the specific form
F̃t = A031 A111 (A031 )−1 F̃t−1 + A031 A112 Rt−1 + {w et − A031 A111 (A031 )−1 w et−1
+A031 H −1 ε1t }
(36)
1
0 −1
1
0
0 −1
Rt = A21 (A31 ) F̃t−1 + A22 Rt−1 + A21 (A31 ) F̃t
+{ε2t − A121 (A031 )−1 w et−1 − A021 (A031 )−1 w et },
(37)
with the errors of the SVAR equations in curly brackets. Due to the presence
of w et and w et−1 in the errors this clearly produces a VARMA structure.
Moreover the errors of these equations are contemporaneously correlated due
to the joint presence of w et in each of the errors. One might account for
the first phenomenon by allowing for a very high order VAR. However, the
implication of the second is that F̃t will be correlated with the error term for
the Rt equation. Consequently, if one regresses Rt on F̃t−1 , Rt−1 and F̃t one
will get inconsistent estimators for the parameters of the Rt equation, and
therefore the same will be true of the impulse responses.
To investigate this in a relatively simple context suppose that there is a
single factor and we are looking at the response of the first variable X1t to
the monetary shock. To compute the zero lag impulse response we would use
the relation
X1t = λ1 Ft + λ1r Rt + e1t ,
but with Ft replaced by F̃t . This is because it has been assumed that the
SVAR involving F̃t is recursive and so the monetary shock ( assumed to be
the error of the Rt equation) should have no contemporaneous impact upon
F̃t . Now there is
a small correlation between F̃t and e1t because F̃t involves
N
a term w et =
i=1 wi eit . If we make the assumption that E(eit ejt ) =
0 for all i = j and E(e2it ) = σ 2 then the covariance will depend upon
2
E[ N
weights wj from a principal components
i=1 wi eit e1t ] = σ w1 . Now the
N
analysis have the property that i=1 wi2 = 1, meaning that wi is 0(N −1/2 ),
making σ 2 w1 small for large N. This would mean that any contemporaneous
impulse responses found using F̃t rather than P Cts should be much the same.
21
However differences come once the SVAR begins to be used to compute
impulse responses, owing to the fact that F̃t is correlated with the error of
the Rt equation and this leads to inconsistent estimators of the coefficients
of the Rt equation in the SVAR involving F̃t . To see this we observe that
w et is in F̃t and also in the error term of
the Rt equation i.e. the covariance
2
between F̃t and the error will involve E( N
i=1 wi eit ) . Under the simple case
we used above this will be σ 2 , which does not change with N.8
Figure 2 presents some impulse responses for a range of variables to a
shock in the Federal Funds Rate. This example is taken from Bernanke et
al. The two impulses presented in each graph are for the FAVAR based on
just the slow moving variables and also the results from "purging" method
used by Bernanke et al. The size of the shock is the same as used by those
authors. The variables presented are LEHCC (Average Hourly Earnings
of Construction Workers), CMCDQ (Real Personal Consumption Expernditures), PUNEW (the CPI, all items), EXRJAN (Yen/ Dollar Exchange
Rate), IP (Industrial Production Index) and HSFR (Total Housing Starts).
For all variables except housing starts the impulses are cumulated, since those
variables were measured as the change in the logarithms. Consequently, the
responses measure the impact of interest rate shocks upon the level of the
CPI, industrial production etc. As Fisher at al (2014) point out this specification means that the level of industrial production and consumption will be
permanently affected by a one period interest rate shock, and this is apparent from the graphs. There are differences between the two sets of responses,
notably for industrial production, the exchange rate and the CPI. The inconsistent estimates found from using the Bernanke et al approach are much
larger than those found using the SVAR in slow moving variables.
4.2
The Boivin and Giannoni (2009) Two-Step Approach
Boivin and Giannoni propose an iterative procedure to remove the effect of
the observable factor Rt . This starts with the N series in the
matrix
Xt .
Ft
It assumes there are K factors Ct among the Xt , where Ct =
. Here
Rt
8
Of course if A021 = 0 then F̃t would be uncorrelated with the w et in the error, but
this would be a case where one could order the system so that Rt is first, and there would
be no issue over "purging" P Ctx of Rt so that the former can be ordered first.
22
Response of LEHCC to an interest rate shock
Response of GMCDQ to an interest rate shock
.08
.20
.04
.15
.00
.10
-.04
.05
-.08
-.12
.00
-.16
-.05
-.20
-.10
-.24
-.28
-.15
1
5
10
15
20
25
OPR
30
35
40
45
48
1
5
10
15
20
BBE
25
OPR
Response of PUNEW to an interest rate shock
30
35
40
45
48
BBE
Response of EXRJAN to an interest rate shock
.2
.10
.1
.08
.0
-.1
.06
-.2
-.3
.04
-.4
-.5
.02
-.6
-.7
.00
1
5
10
15
20
25
OPR
30
35
40
45
48
1
5
10
15
20
BBE
25
OPR
Response of IP to an interest rate shock
30
35
40
45
48
45
48
BBE
Response of HSFR to an interest rate shock
.0
.01
.00
-.1
-.01
-.02
-.2
-.03
-.3
-.04
-.05
-.4
-.06
-.5
-.07
1
5
10
15
20
25
OPR
30
35
40
45
48
1
BBE
5
10
15
20
25
OPR
30
35
40
BBE
Figure 2: Comparison of Variable Responses to Interest Rate Shocks for OPR
and Bernanke et a SVAR Approaches
23
Ft represents K − 1 unobservable factors and Rt is an observed factor. It is
unclear whether Xt contains Rt in their paper. In the analysis that follows
we assume it does but relax the assumption at the end.
As before the model for Xt is
Xt = Λf Ft + Λr Rt + et .
(38)
(0)
In their first step they extract K − 1 principal
P Ct from Xt . It
components
H11 H12
Ft
(0)
must be that P Ct = HCt =
so that the latent factors
H21 H22
Rt
(0)
−1
relate to the PCs as Ft = H11
(P Ct − H12 Rt ), assuming that H11 has an
inverse given it is square. Substituting into (38) we have
(0)
−1
Xt = Λf H11
(P Ct − H12 Rt ) + Λr Rt + et
(0)
−1
−1
= Λf H11
P Ct + (Λr − Λf H11
H12 )Rt + et
(0)
= A(0) P Ct + B (0) Rt + et
(0)
They then regress Xt on P Ct and Rt to get estimates Â(0) and B̂ (0) .
(0)
After this they form Φt = Xt −B̂ (0) Rt and proceed to take K−1 principal
(1)
(0)
components of it, calling these P Ct . Because Φt = Xt − B̂ (0) Rt = Λf Ft +
(1)
(1)
(1)
(ΛR − B̂ (0) )Rt +et this would mean that P Ct = H (1) Ct = H11 Ft + H12 Rt .
Accordingly,
(1)
(1)
(1)
Xt = Λf (H11 )−1 (P Ct − H12 Rt ) + Λr Rt + et
(1)
(1)
(1)
(1)
= Λf (H11 )−1 P Ct + (Λr − Λf (H11 )−1 H12 )Rt + et
(1)
= A(1) P Ct + B (1) Rt + et
(1)
They then regress Xt against P Ct and Rt , followed by repeated application of the loop described above until convergence. At this point (iteration
L) they identify the factor Ft with P Ct(L) . In order for this approach to be
(j)
correct it is clear that the influence of Rt upon P Ct must tend to zero.
(0)
Take the special case where N = 2, K = 2. Then we have P Ct =
w1 X1t + w2 Rt . The regression of X1t against w1 X1t + w2 Rt and Rt will be of
the form
24
(0)
X1t = a1 P Ct + a2 Rt
= a1 (w1 X1t + w2 Rt ) + a2 Rt
(1)
2
This regression will give a1 = 1/w1 , a2 = − w
so that Φ1t = X1t +
w1
w2
R.
w1 t
(1)
Since there is only one series here the principal component of Φ1t is just
(1)
2
itself, leading to P C1t = X1t + w
R . Then consider the regression of X1t on
w1 t
(1)
P C1t and Rt . This has the form
(1)
X1t = b1 P Ct + b2 Rt
(2)
(1)
2
and we will find b1 = 1, b2 = − w
. Thus Φ1t = Φ1t and the iteration will
w1
2
stop. The resulting estimate of Ft is X1t + w
R and hence the influence of Rt
w1
has not been removed from the principal component, and so it cannot span
just the space of the factors.
This result might be a consequence of not working with enough series in
Xt . To assess this we constructed a small Monte Carlo study which had
Rt
F1t
F2t
Xjt
=
=
=
=
.5F1t + .5F2t + ε1t
ε2t
ε3t
aj1 F1t + aj2 F2t + ε4t ,
where aj1 and aj2 were found by simulation from a uniform density over
(0,1). We allowed for N =100 series in Xt and set the sample size to 50000
observations. After the process converged we regressed the K − 1 = 2 estimated principal components P Ct against F1t , F2t and Rt . For the first PC
the coefficient of Rt is 5.5 and for the second one it is .84. With such a large
sample size one would expect these coefficients to be zero if the principal
components had been purged of Rt . If Xt did not include Rt the equivalent
regressions would give coefficients of 5.2 and -.36.
5
Conclusion
The paper has looked at two agnostic approaches to finding impulse responses
from SVARs. It was argued that the sign restrictions literature involves
25
two stages. In the first stage many impulse responses are generated for
uncorrelated shocks. The second stage then determines what these shocks
should be named based on sign restrictions. When all shocks are transitory,
and there are no parametric restrictions, the second stage just involves recombination of the impulse responses. However, when there are parametric
restrictions of either a short-run or long run nature then re-combination is
less straightforward, as one has to respect the parametric restrictions. A
new method of handling these cases is given which involves searching over
alternative structural equations coefficients. It was shown to be simple to
apply and to perform as well as impulse response combination when all shocks
were transitory, but much easier to apply under a range of specifications
incorporating both permanent and transitory shocks.
The second agnostic procedure involves summarizing a large range of
variables as factors to be used for empirical work with small systems. Because
there are some simultaneity issues arising from the fact that a number of the
variables from which factors are formed depend contemporaneously upon
interest rates, proposals have been made to purge the factors of these effects
before using them into SVARs. We show that the two methods currently
suggested to correct for this simultaneity do not do so. Rather a restriction
that certain variables do not depend contemporaneously upon interest rates
is needed for identification. Factors constructed from these variables can
be used in a SVAR, and then the results can be translated into impulse
responses for all variables. An empirical example is given to show that there
are significant differences in the computed impulse responses from the last
method compared to those currently in use.
6
References
Bai, J. and S. Ng (2006), " Confidence Intervals for Diffusion Index Forecasts
and Inference for Factor-Augmented Regressions", Econometrica, 74, 1133—
1150
Bai, J. and S. Ng (2013), "Principal Components Estimation and Identification of Static Factors", Journal of Econometrics, 176, 18-29.
Bernanke B, J. Boivin and P. Eliasz (2005), "Measuring the effects of
monetary policy: A factor-augmented vector autoregressive (FAVAR) approach", Quarterly Journal of Economics ,120, 387-422
Boivin, J. and M.P. Giannoni (2009), "Global Forces and Monetary Policy
26
Effectiveness", in J. Gali and M. Gertler (eds), International Dimensions of
Monetary Policy, University of Chicago Press, Chicago, 429-478.
Canova, F. and G. De Nicoló (2002), “Monetary Disturbances Matter
for Business Fluctuations in the G—7” Journal of Monetary Economics 49,
1131—59
Cho, S. and A. Moreno (2006), "A Small-Sample Study of the NewKeynesian Macro Model", Journal of Money, Credit and Banking, 38, 146181
Faust, J. (1998). “The Robustness of Identified VAR Conclusions about
Money.” Carnegie—Rochester Conference Series on Public Policy , 49, 207—
44.
Fisher, L.A., H-S Huh and A.R. Pagan (2014), "Econometric Methods
for Modelling Systems with a Mixture of I(1) and I(0) Variables"
Fry R and A. Pagan (2011), Sign restrictions in structural vector autoregressions: A critical review", Journal of Economic Literature, 49, 938-960.
Lütkepohl, H. (2014), "Structural Vector Autoregressive Analysis in a
Data Rich Environment: A Survey," DIW Discussion Paper 1351 , German
Institute for Economic Research, Berlin
Herwartz.H. and H. Luetkepohl, 2011. "Structural Vector Autoregressions with Markov Switching: Combining Conventional with Statistical Identification of Shocks," Economics Working Papers ECO2011/11, European
University Institute.
Lanne. M. and H. Lütkepohl, 2008. "Identifying Monetary Policy Shocks
via Changes in Volatility," Journal of Money, Credit and Banking, 40, 11311149.
Lütkepohl, H. P. Saikkonen( 1997), "Impulse response analysis in infinite
order cointegrated vector autoregressive processes," Journal of Econometrics,
81, 127-157
Peersman, G. (2005), "What Caused the Early Millennium Slowdown?
Evidence Based on Autoregressions", Journal of Applied Econometrics, 20,
185-207.
Sargent, T and C. Sims (1977), "Business Cycle Modeling without Pretending to Have too Much a-priori Theory", in C. Sims et al (eds) New
Methods in Business Cycle Research ( Minneapolis, Federal Reserve Bank of
Minneapolis)
Shapiro M and Watson M. (1988), "Sources of business cycle fluctuation",
in NBER Macroeconomics Annual 3: 111-148.
Uhlig, H. (2005), “What Are the Effects of Monetary Policy on Output?
27
Results from an Agnostic Identification Procedure.” Journal of Monetary
Economics, 52, 381—419
28