News Shocks and the Slope of the
Term Structure of Interest Rates
ONLINE APPENDIX
André Kurmann
Christopher Otrok
Federal Reserve Board
University of Missouri
Federal Reserve Bank of St. Louis
April 5, 2012
This appendix reports details on
VAR identi…cation methodology
Baseline VAR results reported in the paper
Contemporaneous TFP shock identi…cation
Robustness checks for baseline VAR
– with respect to OLS estimation
– with respect to alternative variable de…nitions
Larger VAR results reported in the paper
The …t of interest rate rule with VAR results
This appendix accompanies the paper with the same title and is not intended for publication. The results in
this appendix do not necessarily represent the views of the Federal Reserve System or the Federal Open Market
Committee.
1
A
Identifying Structural Shocks: Two VAR Approaches
Here we present details on the two approaches to VAR identi…cation in the paper. The …rst approach, proposed by Uhlig (2003), is purely statistical and extracts the largest 1 or 2 (or 3 or 4)
shocks that explain the maximal amount of the forecast error variance (FEV) in a target variable,
which in our case is the slope of the term structure. The second identi…cation approach is motivated by a key result from the …rst identi…cation: news about future TFP play an important
role in explaining movements in the slope. To assess this interpretation formally, we follow Barsky
and Sims (2010) and extend the FEV maximization approach of Uhlig (2003) by using TFP as
the target variable and imposing the extra restriction that the identi…ed shock is orthogonal to
contemporaneous TFP.
A.1
Review of VAR basics
Consider a reduced-form VAR of the form
+ ut ;
(1)
1 vector of variables observed at time t; and ut is a m
1 vector of one-step-ahead
Yt = B1 Yt
where Yt is a m
1
+ B2 Yt
2
+ ::: + Bp Yt
p
prediction errors with variance-covariance matrix E[ut u0t ] = . Constant terms are dropped to save
on notation. The vector moving average representation of this reduced-form VAR is
Yt = [B(L)] 1 ut = C(L)ut ,
where B(L)
I
B1 L
:::
Bp Lp , and C(L)
(2)
I + C1 L + C2 L2 + :::
Identi…cation of the structural shocks amounts to …nding a mapping A between the prediction
errors ut and a vector of mutually orthogonal shocks "t ; i.e. ut = A"t . The key restriction on A is
that it needs to satisfy
= E[A"t "0t A0 ] = AA0 . This restriction is, however, not su¢ cient to identify
~ = A, where Q is
A because for any matrix A, there exists some alternative matrix A~ such that AQ
an orthonormal matrix, that also satis…es
= A~A~0 .1 This alternative matrix maps ut into another
~"t . For some arbitrary matrix A~ satisfying
vector of mutually orthogonal shocks ~"t ; i.e. ut = A~
~ = A such that Q"t = ~"t . Then,
To see this, consider an orthonormal matrix Q (i.e. QQ0 = I) and de…ne AQ
0 ~0
0
0 0
0 0 ~0
0
~
~
~
~
= E[A"t "t A ] = E[AQ"t "t Q A ] = E[A~"t ~"t A ] = AA because E[~"t ~"t ] = QE["t "0t ]Q0 = QQ0 = I.
1
2
= A~A~0 (e.g. the Cholesky decomposition of
), identi…cation therefore reduces to choosing an
orthonormal matrix Q.
A.2
Extracting the most important shocks
Uhlig’s (2003) statistical approach consists of …nding the n < m columns of Q de…ning the n
mutually orthogonal shocks that explain most of the FEV of some variable in Yt over forecast
horizon k to k. Formally, denote the k-step ahead forecast error of the i-th variable yi;t in Yt by
yi;t+k
Et yi;t+k = e0i
"k 1
X
~ t+k
Cl AQ"
l=0
l
#
,
(3)
where ei is a column vector with 1 in the i-th position and zeros elsewhere. Then Uhlig’s (2003)
approach solves
3
2
k 1
k X
X
~ n Q0 A~0 C 0 5 ei
Cl AQ
Qn = arg max e0i 4
n
l
Qn
(4)
k=k l=0
subject to Q0n Qn = I, where Qn contains the columns of Q de…ning the n most important shocks.
To implement this problem, consider …rst …nding the shock – i.e. the column q1 of Q – that
explains most of the FEV of variable yi 2
2
3
k X
k 1
X
~ 1 q10 A~0 Cl0 5 ei
q1 = arg max e0i 4
Cl Aq
q1
k=k l=0
subject to q10 q1 = 1. The objective to be maximized can be expressed as
3
2
k 1
k X
k
k X
i
h
X
X
0 ~0 0
0 ~0 0 5
0
04
~
~
Cl Aq1 q1 A Cl ei =
trace (ei ei )(Cl Aq1 )(q1 A Cl )
ei
k=k l=0
k=k l=0
=
k X
k
X
k=k l=0
h
i
~ 1)
trace (q10 A~0 Cl0 )(ei e0i )(Cl Aq
2
3
k
k
XX
= q10 4
A~0 Cl0 (ei e0i )Cl A~5 q1
k=k l=0
=
2
q10 Sq1 ,
Without loss of generality, we order this vector …rst in Q.
3
(5)
with
S
k X
k
X
~
A~0 Cl0 (ei e0i )Cl A,
k=k l=0
where the formulation in the second line takes advantage of the fact that for any three square
matrices D, E, F of same dimension trace(DEF ) = trace(F DE). The maximization problem in
(5) can therefore be expressed as a Lagrangian
L = q10 Sq1
(q10 q1
(6)
1)
with …rst-order condition
Sq1 = q1 .
Inspection of this solution reveals that this is simply the de…nition of an eigenvalue decomposition,
with q1 being the eigenvector of S that corresponds to eigenvalue . Furthermore, since q10 q1 = 1, we
can rewrite the …rst-order condition as
= q10 Sq1 ; i.e. eigenvalue
is the objective to be maximized.
The partition q1 that maximizes the variance is therefore the eigenvector associated with the largest
eigenvalue : Likewise, q2 is the second principal component and so forth for all the n components
of Qn that we want to extract.3
A.3
Identifying TFP News Shocks
In the second part of our empirical analysis, we identify a news shock about future innovations to
TFP. As in Barsky and Sims (2010), we assume that TFP is an exogenous process driven by two
orthogonal innovations: a traditional technology shock that a¤ects TFP contemporaneously; and
a news shock that is revealed today but impacts TFP only in the future. More formally, let the
logarithm of TFP be de…ned by a moving average process of the form
,
log T F Pt = v(L)"current
+ d(L)"news
t
t
3
(7)
As explained by Uhlig (2003), there are two other pairs (or rotations) of eigenvectors that explain together an
equal fraction of the total FEV of the slope as the two eigenvectors associated with the two largest eigenvalues of
S. We compared all of our results with these two other rotations and found that for each rotation, there exists one
shock that explains over 50% of slope movements. This shock has very similar properties than the …rst shock we
consider in the text.
4
where "current
and "news
are uncorrelated innovations; and v(L) and d(L) are lag polynomials with
t
t
as the contemporaneous TFP
the only restriction that d(0) = 0. This restriction de…nes "current
t
can a¤ect TFP immediately in period t,
as the TFP news shock; i.e. while "current
shock and "news
t
t
"news
can a¤ect TFP only in period t + 1 and later.
t
In a VAR with (an exogenous measure of) TFP ordered …rst, speci…cation (7) with d(0) = 0
is identi…ed as the shock associated with the
implies that the contemporanous TFP shock "current
t
…rst column of the matrix A~ obtained from the Cholesky decomposition.4 The news shock "news
t
then necessarily covers all movements in TFP that are not related to "current
. This restriction cannot
t
be imposed to hold at all horizons. Instead, Barsky and Sims (2010) propose to identify "news
as
t
the innovation that explains most of future movements in TFP but nothing of current TFP. This
is a restricted version of Uhlig’s (2003) approach presented above but now applied to TFP rather
than the slope. It amounts to solving the Lagrangian problem in (6) subject to the side-constraint
that the identi…ed shock has no contemporaneous e¤ect; i.e. that the …rst element of the extracted
eigenvector q is zero.
A.4
Comparison of Uhlig’s approach applied to spread and long bond
rate
An important question is whether Uhlig’s identi…cation yields similar results independent of the
target variable on which the approach is applied. To show that this is not the case, we applied
Uhlig’s approach on other variables in the baseline VAR. Generally, the extracted shocks from these
alternative identi…cation is very weakly correlated with our slope shock (and therefore with the
TFP news shock). As an illustration, we show here results for the case where we applied Uhlig’s
approach to the long bond rate. This is an especially interesting alternative target variable because
it is part of the de…nition of the spread. As can be seen from the below …gures, the results are very
To see this, recall that A~ obtained from the Cholesky decomposition is a lower-triangular matrix. Hence, the
only shock in ~"t associated with A~ that can have an immediate e¤ect on the …rst variable in Yt (i.e. TFP) is the …rst
element in ~"t :
4
5
di¤erent from the ones obtained for the slope.
5-year bond yield
Total factor productivity
1
fr a ctio n o f FEV e xp lain e d
fr a ctio n o f FEV e xp lain e d
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.8
0.6
0.4
0.2
0
40
0
5
10
15
q u a r te r s
Consumption
fr a ctio n o f FEV e xp lain e d
fr a ctio n o f FEV e xp lain e d
0.6
35
40
0.4
0.2
0
5
10
15
20
25
30
35
25
30
35
40
30
35
40
0.8
0.6
0.4
0.2
0
40
0
5
10
15
q u a r te r s
20
q u a r te r s
Federal Funds rate
Spread (5-year - Fedfunds)
1
fr a ctio n o f FEV e xp lain e d
1
fr a ctio n o f FEV e xp lain e d
30
1
0.8
0.8
0.6
0.4
0.2
0
25
Inflation
1
0
20
q u a r te r s
0
5
10
15
20
25
30
35
0.8
0.6
0.4
0.2
0
40
q u a r te r s
0
5
10
15
20
25
q u a r te r s
Fig. A.1: Fraction of FEV explained by level shock
The …rst …gure shows that the extracted ’long-bond shock’accounts for 75% or more of the FEV
of the long bond yield but only very little of TFP and the spread. Furthermore, the correlation
between the extracted long-bond shock and the slope shock of the paper is 0.237 for the full sample;
0.49 fro the pre-84 sample; and -0.44 for the post-84 sample (note the negative sign!).In other words,
the shock that explains most of the FEV of the long-bond yield is very di¤erent from the slope shock
(and therefore the TFP news shock). Likewise the IRFs to the level shock (shown below in solid
6
black lines) are very di¤erent from the IRFs to our slope shock (shown in dashed red lines).
Total factor productivity
1
0.5
0.5
percent
percent
Spread (5-year - Fedfunds)
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
qu ar ter s
20
25
30
35
40
25
30
35
40
30
35
40
qu ar ter s
Consumption
Inflation
0.5
1.2
percent
percent
1
0.8
0.6
0.4
0
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
qu ar ter s
5-year bond yield
0.5
0
0
percent
percent
Federal Funds rate
0.5
-0.5
-1
20
qu ar ter s
-0.5
0
5
10
15
20
25
30
35
-1
40
qu ar ter s
0
5
10
15
20
25
qu ar ter s
Fig. B.1.2: Impulse responses to 1% level shock (solid black lines and grey
con…dence intervals) and impulse responses to 1% slope shock (red dashed
lines)
In contrast to the reaction to the slope shock, the spread hardly moves on impact of the long-bond
shock. Conversely, the long-bond yield jumps down by almost 50 basis points whereas the long-bond
yield barely reacts in response to the slope shock. The results suggest that the long-bond shock is
akin to what the …nance literature has identi…ed as a ’level shock’; i.e. a shock that moves both
the long and the short end of the term structure roughly equally. The long-bond shock also has a
sizable impact on in‡ation but moves TFP and consumption relatively little. In particular, boht
TFP and consumption return back towards their initial level suggesting that the extracted longbond shock does not relate to a permanent technology improvement. All of these results con…rm
that the extracted level shock looks like a very di¤erent shock than the slope shock (and therefore
the TFP news shock) identi…ed in the paper.
7
B
Details on baseline VAR results reported in the paper
Here we report details for the baseline VAR of the paper. For both the slope identi…cation and
the TFP news identi…cation, we show the full set of variance decompositions and impulse responses
with corresponding con…dence intervals. We also show a decomposition of the spread’s impulse
reponse into an expectations hypothesis part and a term premia part. This decomposition takes
the spread between the nominal yield iTt on a T -period zero-coupon bond (in our case the 5-year
treasury bond) and nominal yield it on a 1-period bill (in our case, the Federal Funds rate) and
expresses is as the sum of two parts
iTt
T 1
1X
it =
T i=0
1
i
T
E [ it+i jYt ] + tpt ,
where the E[ it+i jYt ] denote expectations of future short rates as implied by the VAR based on
information Yt ; and tpt denotes term premia. This type of decomposition is standard in the term
structure literature. Notable examples are the seminal paper by Cambpell and Shiller (1987) or
more recently Diebold, Rudebusch and Aruoba (2006) and Evans and Marshall (2007).
8
Baseline VAR results for slope identi…cation
Total factor productivity
1
1
0.8
0.8
fraction explained
fraction explained
Spread (5-year - Fedfunds)
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.6
0.4
0.2
0
40
0
5
10
15
quarters
0.8
0.8
fraction explained
fraction explained
1
0.6
0.4
0.2
0
5
10
15
20
25
30
35
0
40
0
5
10
15
fraction explained
0.8
0.6
0.4
0.2
20
25
30
35
40
30
35
40
5-year bond yield
0.8
15
20
quarters
1
10
40
0.2
Federal F unds rate
5
35
0.4
1
0
30
0.6
quarters
0
25
Inflation
1
0
20
quarters
Consumption
fraction explained
B.1
25
30
35
0.6
0.4
0.2
0
40
quarters
0
5
10
15
20
25
quarters
Fig. B.1.1: Fraction of FEV explained by slope shock
9
Spread (5-year - Fedfunds )
Total factor productivity
0.5
0.5
percent
1
percent
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
quarters
20
25
30
35
40
25
30
35
40
30
35
40
quarters
Consumption
Inflation
0.5
1.2
1
percent
percent
0
0.8
0.6
0.4
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
quarters
20
quarters
Federal Funds rate
5-year bond y ield
0
0
percent
0.5
percent
0.5
-0.5
-1
-0.5
0
5
10
15
20
25
30
35
-1
40
quarters
0
5
10
15
20
25
quarters
Fig. B.1.2: Impulse responses to 1% slope shock
10
0.8
S pread
Term premium
S pread under E H
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0
5
10
15
20
25
30
35
40
0.4
Long rate
Term premium
Long rate under E H
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
5
10
15
20
25
30
35
40
Fig. B.1.3: Decomposition of spread and long-rate response to slope shock
into Expectations Hypothesis part and term premia part
11
Baseline VAR results for TFP news shock identi…cation
T otal factor productivity
1
1
0.8
0.8
fraction explained
fraction explained
Spread (5-year - F edfunds)
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.6
0.4
0.2
0
40
0
5
10
15
quarters
0.8
0.8
fraction explained
fraction explained
1
0.6
0.4
0.2
0
5
10
15
20
25
30
35
0
40
0
5
10
15
fraction explained
0.8
0.6
0.4
0.2
20
25
30
35
40
30
35
40
5-year bond yield
0.8
15
20
quarters
1
10
40
0.2
F ederal F unds rate
5
35
0.4
1
0
30
0.6
quarters
0
25
Inflation
1
0
20
quarters
Consumption
fraction explained
B.2
25
30
35
0.6
0.4
0.2
0
40
quarters
0
5
10
15
20
25
quarters
Fig. B.2.1: Fraction of FEV explained by TFP news shock
12
Spread (5-year - F edfunds)
T otal factor productivity
0.7
0.4
0.6
percent
percent
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
5
10
15
20
25
30
35
0
40
5
10
15
quarters
Consumption
30
35
40
30
35
40
30
35
40
Inflation
0
1
0.8
percent
percent
25
0.1
1.2
0.6
0.4
0.2
-0.1
-0.2
-0.3
-0.4
0
-0.5
5
10
15
20
25
30
35
40
5
10
15
quarters
20
25
quarters
Federal F unds rate
5-year bond yield
0
0
-0.2
-0.1
percent
percent
20
quarters
-0.4
-0.6
-0.2
-0.3
-0.8
-0.4
5
10
15
20
25
30
35
40
5
quarters
10
15
20
25
quarters
Fig. B.2.2: Impulse responses to 1% TFP news shock
13
0.6
Spread
Term premium
Spread under EH
0.5
0.4
0.3
0.2
0.1
0
-0.1
0
5
10
15
20
25
30
35
40
0.3
Long rate
Term premium
Long rate under E H
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
5
10
15
20
25
30
35
40
Fig. B.2.3: Decomposition of spread and long-rate response to TFP news shock
into Expectations Hypothesis part and term premia part
14
B.3
Correspondence of extracted shocks from slope identi…cation and
TFP news identi…cation
news shock
slope shock
3
2
1
percent
0
-1
-2
-3
-4
-5
-6
1965
1970
1975
1980
1985
1990
1995
2000
2005
date
Fig. B.3.1: TFP news shock (solid black line) and Slope shock (dashed red line)
extracted from baseline VAR
The correlation between the two shocks is 0.86 over the full sample; 0.91 over the pre-84 sample;
and 0.58 over the post-84 sample.
15
Results for contemporaneous TFP shock identi…cation
T otal factor productivity
1
1
0.8
0.8
fraction explained
fraction explained
Spread (5-year - F edfunds)
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.6
0.4
0.2
0
40
0
5
10
15
quarters
1
0.8
fraction explained
fraction explained
1
0.6
0.4
0.2
0
5
10
15
20
25
30
35
0
40
0
5
10
15
fraction explained
0.8
0.6
0.4
0.2
20
25
30
35
40
30
35
40
5-year bond yield
0.8
15
20
quarters
1
10
40
0.2
F ederal F unds rate
5
35
0.4
1
0
30
0.6
quarters
0
25
Inflation
0.8
0
20
quarters
Consumption
fraction explained
C
25
30
35
0.6
0.4
0.2
0
40
quarters
0
5
10
15
20
25
quarters
Fig. C.1: Fraction of FEV explained by contemporaneous TFP shock
16
Total factor productivity
1
0.5
0.5
percent
percent
Spread (5-year - Fedfunds)
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
quarters
20
25
30
35
40
25
30
35
40
30
35
40
quarters
Consumption
Inflation
0.5
1.2
percent
percent
1
0.8
0.6
0.4
0
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
quarters
5-year bond yield
0.5
0
0
percent
percent
Federal Funds rate
0.5
-0.5
-1
20
quarters
-0.5
0
5
10
15
20
25
30
35
-1
40
quarters
0
5
10
15
20
25
quarters
Fig. C.2: Impulse responses to 1% contemporaneous TFP shock
D
Robustness checks
Here we report a variety of robustness checks of our baseline results. First, for the slope identi…cation, we show result when the baseline VAR is estimated via OLS and con…dence bands are
obtained via bootstrapping. Second, we report robustness checks with respect to various alternative
de…nitions of di¤erent variables in the VAR. In all cases, corresponding robustness checks for the
news shock identi…cation are very similar. We therefore do not report separate robustness checks
for the news shock identi…cation (results are, however, available from the authors upon request).
17
Baseline VAR results for OLS estimation
Total factor productivity
1
1
0.8
0.8
fraction explained
fraction explained
Spread (5-year - Fedfunds)
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.6
0.4
0.2
0
40
0
5
10
15
quarters
0.8
0.8
fraction explained
fraction explained
1
0.6
0.4
0.2
0
5
10
15
20
25
30
35
0
40
0
5
10
15
fraction explained
0.8
0.6
0.4
0.2
20
25
30
35
40
30
35
40
5-year bond yield
0.8
15
20
quarters
1
10
40
0.2
Federal Funds rate
5
35
0.4
1
0
30
0.6
quarters
0
25
Inflation
1
0
20
quarters
Consumption
fraction explained
D.1
25
30
35
0.6
0.4
0.2
0
40
quarters
0
5
10
15
20
25
quarters
Fig. D.1.1: Fraction of FEV explained by slope shock (OLS estimates)
18
Total factor productivity
1
0.5
0.5
percent
percent
Spread (5-year - Fedfunds)
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
quarters
20
25
30
35
40
25
30
35
40
30
35
40
quarters
Consumption
Inflation
0.5
1.2
percent
percent
1
0.8
0.6
0.4
0
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
quarters
5-year bond yield
0.5
0
0
percent
percent
Federal Funds rate
0.5
-0.5
-1
20
quarters
-0.5
0
5
10
15
20
25
30
35
-1
40
quarters
0
5
10
15
20
25
quarters
Fig. D.1.2: Impulse responses to 1% slope shock (OLS estimates)
19
D.2
Baseline VAR results for alternative variable de…nitions
The following robustness checks each replace one variable of the VAR with an alternative variable,
leaving all other variables as in the baseline speci…ciation.
Replacing total consumption by consumption of non-durables and services (nd&s)
Total factor productivity
1
0.8
0.8
fraction explained
fraction explained
Spread (5-year - Fedfunds)
1
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.6
0.4
0.2
0
40
0
5
10
15
quarters
1
0.8
fraction explained
fraction explained
1
0.6
0.4
0.2
0
5
10
15
20
25
30
35
0
5
10
15
fraction explained
fraction explained
0.8
0.6
0.4
0.2
20
25
30
35
40
30
35
40
5-year bond yield
0.8
15
20
quarters
1
10
40
0.2
0
40
Federal Funds rate
5
35
0.4
1
0
30
0.6
quarters
0
25
Inflation
0.8
0
20
quarters
Consumption (nd&s)
25
30
35
40
quarters
0.6
0.4
0.2
0
0
5
10
15
20
25
quarters
Fig. D.2.1: Fraction of FEV explained by slope shock
20
Spread (5-year - Fedfunds)
Total factor productivity
0.5
0.5
per cent
1
per cent
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
quarters
20
25
30
35
40
25
30
35
40
30
35
40
quarters
Consumption (nd&s)
Inflation
0.5
1.2
1
per cent
per cent
0
0.8
0.6
0.4
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
quarters
20
quarters
Federal Funds rate
5-year bond yield
0
0
per cent
0.5
per cent
0.5
-0.5
-1
-0.5
0
5
10
15
20
25
30
35
40
quarters
-1
0
5
10
15
20
25
quarters
Fig. D.2.2: Impulse responses to 1% slope shock
21
Replacing the GDP de‡ator as the measure for in‡ation with the PCE de‡ator
Spread (5-year - Fedfunds)
Total factor productivity
1
fr a ctio n e xp la in e d
fr a ctio n e xp la in e d
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.8
0.6
0.4
0.2
0
40
0
5
10
15
quarters
35
40
0.6
0.4
0.2
0
5
10
15
20
25
30
35
25
30
35
40
30
35
40
0.8
0.6
0.4
0.2
0
40
0
5
10
15
quarters
20
quarters
Federal Funds rate
5-year bond yield
1
fr a ctio n e xp la in e d
1
fr a ctio n e xp la in e d
30
1
fr a ctio n e xp la in e d
fr a ctio n e xp la in e d
1
0.8
0.6
0.4
0.2
0
25
PCE inflation
0.8
0
20
quarters
Consumption
0
5
10
15
20
25
30
35
40
quarters
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
quarters
Fig. D.2.3: Fraction of FEV explained by slope shock
22
Total factor productivity
1
0.5
0.5
p e r ce n t
p e r ce n t
Spread (5-year - Fedfunds)
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
q u a r te r s
20
25
30
35
40
25
30
35
40
30
35
40
q u a r te r s
Consumption
PCE inflation
0.5
1.2
p e r ce n t
p e r ce n t
1
0.8
0.6
0.4
0
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
q u a r te r s
5-year bond yield
0.5
0
0
p e r ce n t
p e r ce n t
Federal Funds rate
0.5
-0.5
-1
20
q u a r te r s
-0.5
0
5
10
15
20
25
30
35
40
q u a r te r s
-1
0
5
10
15
20
25
q u a r te r s
Fig. D.2.4: Impulse responses to 1% slope shock
23
Replacing the GDP de‡ator as the measure for in‡ation with the CPI de‡ator5
Total factor productiv ity
1
0.8
0.8
fraction explained
fraction explained
Spread (5-year - Fedfunds )
1
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.6
0.4
0.2
0
40
0
5
10
15
quarters
fraction explained
fraction explained
1
0.8
0.6
0.4
0.2
5
10
15
20
25
30
35
0
40
0
5
10
15
20
30
35
40
30
35
40
quarters
Federal Funds rate
5-y ear bond yield
1
0.6
0.4
0.2
10
25
0.2
0.8
5
40
0.4
1
0
35
0.6
0.8
0
30
quarters
fraction explained
fraction explained
1
0
25
CPI inflation
0.8
0
20
quarters
Cons umption
15
20
25
30
35
40
quarters
0.6
0.4
0.2
0
0
5
10
15
20
25
quarters
Fig. D.2.5: Fraction of FEV explained by slope shock
5
The only exception is when we replace the GDP de‡ator with the CPI to compute in‡ation. As noted in the
text, in‡ation in that case responds to a TFP news shock with a larger drop than the Fed Funds rate, implying an
increase in the real short rate. We prefer either the GDP de‡ator or the PCE de‡ator over the CPI as a measure
of in‡ation because the CPI su¤ers from substitution bias and is a¤ected by large swings in food and energy prices
that make it excessively volatile.
24
Spread (5-year - Fedfunds)
Total factor produc tivity
0.5
0.5
percent
1
percent
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
quarters
20
25
30
35
40
25
30
35
40
30
35
40
quarters
Cons umption
CPI inflation
0.5
1.2
1
percent
percent
0
0.8
0.6
0.4
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
quarters
20
quarters
Federal Funds rate
5-year bond yield
0
0
percent
0.5
percent
0.5
-0.5
-1
-0.5
0
5
10
15
20
25
30
35
40
quarters
-1
0
5
10
15
20
25
quarters
Fig. D.2.6: Impulse responses to 1% slope shock
25
Replacing the Federal funds rate in the computation of the spread with the 3-month treasury
bill rate
Total factor productivity
1
0.8
0.8
fraction explained
fraction explained
Spread (5-year - 3-month)
1
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.6
0.4
0.2
0
40
0
5
10
15
quarters
fraction explained
fraction explained
1
0.8
0.6
0.4
0.2
5
10
15
20
25
30
35
0
40
0
5
10
15
20
30
35
40
30
35
40
quarters
Federal Funds rate
5-year bond yield
1
0.6
0.4
0.2
10
25
0.2
0.8
5
40
0.4
1
0
35
0.6
0.8
0
30
quarters
fraction explained
fraction explained
1
0
25
Inflation
0.8
0
20
quarters
Consumption
15
20
25
30
35
40
quarters
0.6
0.4
0.2
0
0
5
10
15
20
25
quarters
Fig. D.2.7: Fraction of FEV explained by slope shock
26
Total factor productivity
1
0.5
0.5
p e r ce n t
p e r ce n t
Spread (5-year - 3-month)
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
quarters
20
25
30
35
40
25
30
35
40
30
35
40
quarters
Consumption
Inflation
0.5
1.2
p e r ce n t
p e r ce n t
1
0.8
0.6
0.4
0
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
quarters
5-year bond yield
0.5
0
0
p e r ce n t
p e r ce n t
Federal Funds rate
0.5
-0.5
-1
20
quarters
-0.5
0
5
10
15
20
25
30
35
40
quarters
-1
0
5
10
15
20
25
quarters
Fig. D.2.8: Impulse responses to 1% slope shock
27
Replacing the 5-year treasury bond yield in the computation of the spread with the 10-year
treasury bond yield6
Spread (10-year - Fedfunds)
Total factor productivity
1
fra ction explained
fra ction explained
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.8
0.6
0.4
0.2
0
40
0
5
10
15
qu arters
Consumption
fra ction explained
fra ction explained
0.6
35
40
0.4
0.2
0
5
10
15
20
25
30
35
25
30
35
40
30
35
40
0.8
0.6
0.4
0.2
0
40
0
5
10
15
qu arters
20
qu arters
Federal Funds rate
10-year bond yield
1
fra ction explained
1
fra ction explained
30
1
0.8
0.8
0.6
0.4
0.2
0
25
Inflation
1
0
20
qu arters
0
5
10
15
20
25
30
35
40
qu arters
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
qu arters
Fig. D.2.9: Fraction of FEV explained by slope shock
6
The 10-year zero coupon bond yield series is taken from Gurkanyak, Sack and Wright (2007). Since this series
is available only starting in 1971:4, the sample is shortened to 1971:4-2005:2.
28
T otal factor productivity
1
0.5
0.5
percent
percent
Spread (10-year - F edfunds)
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
quar ters
20
25
30
35
40
25
30
35
40
30
35
40
quar ters
Consumption
Inflation
0.5
1.2
percent
percent
1
0.8
0.6
0.4
0
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
quar ters
10-year bond yield
0.5
0
0
percent
percent
F ederal F unds rate
0.5
-0.5
-1
20
quar ters
-0.5
0
5
10
15
20
25
30
35
40
quar ters
-1
0
5
10
15
20
25
quar ters
Fig. D.2.10: Impulse responses to 1% slope shock
Replacing the 5-year bond – Federal funds rate spread with a term structure slope factor
29
computed as in Diebold and Li (2006)
Total factor productivity
1
0.8
0.8
fraction explained
fraction explained
Slope
1
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
0.6
0.4
0.2
0
40
0
5
10
15
quarters
0.8
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
35
0
40
0
5
10
15
fraction explained
fraction explained
0.8
0.6
0.4
0.2
20
25
30
35
40
25
30
35
40
Level
0.8
15
20
quarters
1
10
40
0.2
Federal Funds rate
5
35
0.4
1
0
30
0.6
quarters
0
25
Inflation
1
fraction explained
fraction explained
Consumption
1
0
20
quarters
25
30
35
40
quarters
0.6
0.4
0.2
0
0
5
10
15
20
quarters
Fig. D.2.11: Fraction of FEV explained by slope shock
30
Total factor productivity
1
0.5
0.5
percent
percent
Slope
1
0
-0.5
0
0
5
10
15
20
25
30
35
-0.5
40
0
5
10
15
quarters
20
25
30
35
40
25
30
35
40
25
30
35
40
quarters
Consumption
Inflation
0.5
1.2
percent
percent
1
0.8
0.6
0.4
0
-0.5
0.2
0
0
5
10
15
20
25
30
35
-1
40
0
5
10
15
quarters
Level
0.5
0
0
percent
percent
Federal Funds rate
0.5
-0.5
-1
20
quarters
-0.5
0
5
10
15
20
25
30
35
40
quarters
-1
0
5
10
15
20
quarters
Fig. D.2.12: Impulse responses to 1% slope shock
31
E
Robustness checks with larger VARs
Here we report details for the robustness checks with larger VARs in Section 5 of the paper. For
both the slope identi…cation and the TFP news identi…cation, we show the full set of variance
decompositions and impulse responses with corresponding con…dence intervals.
Results for VAR with investment and equipment&software price
de‡ators
40
fraction of FEV explained
40
fraction of FEV explained
Spre ad (5 -y ear - Fedfun ds )
40
fraction of FEV explained
fraction of FEV explained
Slope identi…cation
1
0.5
0
0
5
10
15
20
25
30
35
Total fac tor p rodu c tiv ity
1
0.5
0
0
5
10
15
fraction of FEV explained
quarters
1
0.5
0
0
5
10
15
20
25
30
35
fraction of FEV explained
1
0.5
0
5
10
15
20
25
30
35
10
15
20
25
30
35
40
quarters
fraction of FEV explained
0.5
5
35
40
0
0
5
10
15
20
25
30
35
40
25
30
35
40
30
35
40
In fla tio n
1
0.5
0
0
5
10
15
20
quarters
1
0
30
0.5
quarters
Fed eral Fu nds ra te
0
25
Equip&So ft pric e inde x
1
quarters
C o ns u mp tio n
0
20
quarters
inv es tment p ric e inde x
quarters
fraction of FEV explained
E.1
5 - y e ar b ond y ie ld
1
0.5
0
0
5
10
15
20
25
quarters
Fig. E.1.1: Fraction of FEV explained by slope shock
32
Total fa c to r pr oduc tiv ity
1
0.5
0.5
percent
percent
Spread (5-y ear - Fedfunds )
1
0
-0.5
0
5
10
15
20
25
30
35
0
-0.5
40
0
5
10
15
quarters
30
35
40
25
30
35
40
25
30
35
40
30
35
40
0.5
0
percent
percent
25
Equip&Soft pric e index
0.5
-0.5
-1
20
quarters
inv es tment pric e index
0
5
10
15
20
25
30
35
0
-0.5
-1
40
0
5
10
15
quarters
20
quarters
C ons umption
In flation
0.5
percent
percent
1
0.5
0
0
5
10
15
20
25
30
35
0
-0.5
-1
40
0
5
10
15
quarters
5- y ear bond y ield
0.5
0
percent
percent
0.5
-0.5
-1
20
quarters
Federal Funds rate
0
5
10
15
20
25
30
35
40
quarters
0
-0.5
-1
0
5
10
15
20
25
quarters
Fig. E.1.2: Impulse responses to 1% slope shock
33
TFP news identi…cation
Total fac tor produc tivity
0.5
0
0
5
10
15
20
25
30
35
40
fraction of FEV explained
fraction of FEV explained
Spread (5-year - Fedfunds)
1
1
0.5
0
0
5
10
15
quarters
5
10
15
20
25
30
35
40
fraction of FEV explained
fraction of FEV explained
0.5
0
0
0
5
10
15
25
30
35
40
5
10
35
40
25
30
35
40
30
35
40
0.5
0
0
5
10
15
20
quarters
5-y ear bond yield
0.5
0
30
quarters
1
0
fraction of FEV explained
20
25
1
Federal Funds rate
15
20
25
30
35
40
quarters
fraction of FEV explained
fraction of FEV explained
fraction of FEV explained
15
20
quarters
0.5
10
40
Inflation
1
5
35
0.5
quarters
0
30
1
Cons umption
0
25
Equip&Soft price index
1
0
20
quarters
inv estment price index
1
0.5
0
0
5
10
15
20
25
quarters
Fig. E.1.3: Fraction of FEV explained by TFP news shock
34
Spread (5-year - Fedfunds)
Total factor productivity
1
p e r ce n t
p e r ce n t
1
0.5
0
-0.5
0
5
10
15
20
25
30
35
0.5
0
-0.5
40
0
5
10
15
q u a r te r s
30
35
40
25
30
35
40
25
30
35
40
30
35
40
0.5
p e r ce n t
p e r ce n t
25
Equip&Soft price index
0.5
0
-0.5
-1
20
q u a r te r s
investment price index
0
5
10
15
20
25
30
35
0
-0.5
-1
40
0
5
10
15
q u a r te r s
20
q u a r te r s
Consumption
Inflation
p e r ce n t
p e r ce n t
0.5
1
0.5
0
0
5
10
15
20
25
30
35
0
-0.5
-1
40
0
5
10
15
q u a r te r s
5-year bond yield
0.5
p e r ce n t
p e r ce n t
0.5
0
-0.5
-1
20
q u a r te r s
Federal Funds rate
0
5
10
15
20
25
30
35
40
q u a r te r s
0
-0.5
-1
0
5
10
15
20
25
q u a r te r s
Fig. E.1.4: Impulse responses to 1% TFP news shock
35
Results for VAR with GDP, investment and real S&P500 price
Spread (5-year - Fedfunds)
fr action explain ed
fr action explain ed
Slope identi…cation
1
0.5
0
0
5
10
15
20
25
30
35
40
Total factor productivity
1
0.5
0
0
5
10
15
1
0.5
0
0
5
10
15
20
25
30
35
40
fr action explain ed
fr action explain ed
0.5
5
10
15
20
25
30
35
40
fr action explain ed
fr action explain ed
1
0.5
5
10
15
20
40
30
35
40
30
35
40
30
35
40
1
0
0
5
10
15
20
25
Real S&P500 index
1
0.5
0
0
5
10
15
20
25
q ua r ter s
Inflation
0
35
0.5
q ua r ter s
0
30
q ua r ter s
1
0
25
Gross domestic product
q ua r ter s
Investment
0
20
q ua r ter s
Consumption
fr action explain ed
fr action explain ed
q ua r ter s
25
30
35
40
q ua r ter s
fr action explain ed
E.2
Federal Funds rate
1
0.5
0
0
5
10
15
20
25
q ua r ter s
5-year bond yield
1
0.5
0
0
5
10
15
20
25
30
35
40
q ua r ter s
Fig. E.2.1: Fraction of FEV explained by slope shock
36
Spread (5-year - F edfunds)
T otal factor productivity
percent
percent
0.4
0.6
0.4
0.2
0
5
10
15
20
25
30
35
0.2
0
40
5
10
15
quarters
percent
percent
0.5
0
10
15
20
25
30
35
40
5
10
40
15
20
25
30
35
40
30
35
40
30
35
40
Real S&P500 index
percent
2
percent
35
quarters
Investment
1
0
-1
1.5
1
0.5
0
5
10
15
20
25
30
35
40
5
10
15
quarters
20
25
quarters
Inflation
F ederal Funds rate
0
percent
0
-0.2
-0.4
5
10
15
20
25
30
35
40
-0.2
-0.4
-0.6
5
quarters
10
15
20
25
quarters
5-year bond yield
0
percent
30
0.8
0.6
0.4
0.2
0
-0.2
quarters
percent
25
G ross domestic product
1
5
20
quarters
Consumption
-0.1
-0.2
-0.3
5
10
15
20
25
30
35
40
quarters
Fig. E.2.2: Impulse responses to 1% slope shock
37
Spread (5-year - Fedfunds)
fr a ctio n e xp la in e d
fr a ctio n e xp la in e d
TFP news identi…cation
1
0.5
0
0
5
10
15
20
25
30
35
40
Total factor productivity
1
0.5
0
0
5
10
15
1
0.5
0
0
5
10
15
20
25
30
35
40
fr a ctio n e xp la in e d
fr a ctio n e xp la in e d
1
0.5
5
10
15
20
25
30
35
40
fr a ctio n e xp la in e d
fr a ctio n e xp la in e d
1
0.5
5
10
15
20
25
30
35
40
q u a r te r s
fr a ctio n e xp la in e d
40
30
35
40
30
35
40
30
35
40
1
0
0
5
10
15
20
25
Real S&P500 index
1
0.5
0
0
5
10
15
20
25
q u a r te r s
Inflation
0
35
0.5
q u a r te r s
0
30
q u a r te r s
Investment
0
25
Gross domestic product
q u a r te r s
0
20
q u a r te r s
fr a ctio n e xp la in e d
fr a ctio n e xp la in e d
q u a r te r s
Consumption
Federal Funds rate
1
0.5
0
0
5
10
15
20
25
q u a r te r s
5-year bond yield
1
0.5
0
0
5
10
15
20
25
30
35
40
q u a r te r s
Fig. E.2.3: Fraction of FEV explained by TFP news shock
38
Total factor productivity
perc ent
perc ent
Spread (5-year - Fedfunds)
0.4
0.2
0.4
0.2
0
5
10
15
20
25
30
35
0
40
5
10
15
quarters
25
30
35
40
30
35
40
30
35
40
30
35
40
Gross domestic product
perc ent
perc ent
20
quarters
Consumption
1
0.5
1
0.5
0
0
5
10
15
20
25
30
35
40
5
10
15
quarters
2
perc ent
perc ent
25
Real S&P500 index
2
1
0
-1
5
10
15
20
25
30
35
1
0
40
5
10
15
quarters
20
25
quarters
Federal Funds rate
perc ent
Inflation
perc ent
20
quarters
Investment
0
-0.2
-0.4
0
-0.2
-0.4
-0.6
5
10
15
20
25
30
35
40
5
quarters
10
15
20
25
quarters
perc ent
5-year bond yield
0
-0.1
-0.2
-0.3
5
10
15
20
25
30
35
40
quarters
Fig. E.2.4: Impulse responses to 1% TFP news shock
39
Correspondence between the two extracted shocks
news shock
slope shock
3
2
1
0
percent
-1
-2
-3
-4
-5
-6
1965
1970
1975
1980
1985
1990
1995
2000
2005
date
Fig. E.2.5: TFP news shock (solid black line) and Slope shock (dashed red line) extracted
from larger VAR with output, investment and real S&P500 price
The correlation between the two shocks is 0.84 over the full sample; 0.88 over the pre-84
sample; and 0.68 over the post-84 sample.
F
Fit of interest rate rule with VAR results
As discussed in the …nal section of the main text, an interesting question is whether the joint
response of the Federal Funds rate, in‡ation and output to a TFP news shock as implied by our
VAR estimates is consistent with an interest rate rule for monetary policy.
40
The interest rate rule we consider is
it = i t
1
where it denotes the Federal Funds rate;
+ (1
t
)[
t
+
y
yt ],
denotes in‡ation; and
yt denotes output growth.7
The black solid line shows the impulse response of the Federal Funds rate to a TFP news shock as
estimated from the large VAR above (Figure E.2.4), with the 68% con…dence bands in grey. The
blue dotted line shows the impulse response of the Federal Funds rate as implied by the above rule
for
= 0:9,
= 1:5,
y
= 0:5. The red solid line with dots shows the impulse response of the
7
Results are very similar if we replace output growth by a measure of the output gap, speci…ed as yt
potential output yt is de…ned as
yt = yt 1 + (1
)trendt
and trendt is proportional to the path of TFP in response to the news shock.
41
yt , where
Federal Funds rate as implied by the same rule but for
= 0:5,
= 2,
y
= 0:5.
Federal Funds rate
0.1
0
-0.1
percent
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
5
10
15
20
25
30
35
40
quarters
Fig. F.1. Impulse response of Federal Funds rate to TFP news shock as estimated
from VAR (black solid line) and as implied by two calibrations of the interest rate
rule (blue dotted line and red line with dots).
As can be seen from this …gure, the …rst calibration (blue dotted line) …ts the impulse response from
the VAR very closely except for the impact response. This is because of the high degree of interest
rate smoothing
= 0:9. The second calibration (red line with dots) …ts the impulse response from
the VAR somewhat less closely after 10 quarters out but provides a better match of the impact
response.
42