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Sales, Inventories, and Real Interest
Rates: A Century of Stylized Facts

WP 12-02

Luca Benati
University of Bern
Thomas A. Lubik
Federal Reserve Bank of Richmond

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

Sales, Inventories, and Real Interest Rates:
A Century of Stylized Facts∗
Luca Benati
University of Bern†

Thomas A. Lubik
Federal Reserve Bank of Richmond‡

April 2012
Working Paper No. 12-02

Abstract
We use Bayesian time-varying parameters structural VARs with stochastic
volatility to investigate changes in both the reduced-form and the structural
correlations between business inventories and either sales growth or the real
interest rate in the United States during both the interwar and the post-WWII
periods. We identify four structural shocks by combining a single long-run
restriction to identify a permanent output shock as in Blanchard and Quah
(1989), with three sign restrictions to identify demand- and supply-side transitory shocks. We produce several new stylized facts which should inform the
development of new models of inventories. In particular, we show that (i) during both the interwar and the post-WWII periods, the structural correlation
between inventories and the real interest rate conditional on identified interest
rate shocks is systematically positive; (ii) the reduced-form correlation between
the two series is positive during the post-WWII period, but in line with the
predictions of theory it is robustly negative during the interwar era; and (iii)
during the interwar era, the correlations between inventories and either of the
two other series exhibits a remarkably strong co-movement with output at the
business-cycle frequencies.
Keywords: Bayesian VARs; stochastic volatility; time-varying parameters;
structural VARs; long-run restrictions; sign restrictions; inventories; monetary policy; monetary regimes.
JEL codes: C11; C32; E32.
∗

The views expressed in this paper are those of the authors and should not be interpreted as
those of the Federal Reserve Bank of Richmond or the Federal Reserve System.
†
Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001 Bern, Switzerland. Email: luca.benati@vwi.unibe.ch
‡
Research Department, Federal Reserve Bank of Richmond, 701 E Byrd Street, Richmond, VA
23218, USA. E-mail: Thomas.Lubik@rich.frb.org

Introduction

Inventories have been at the center of modern economics ever since its inception as
a quantitative discipline. Starting with accelerator models designed to capture the
inventory cycle, inventory behavior has attracted an enormous amount of attention
in macroeconomics. The chief reason is that while inventory investment makes up
only a small fraction of GDP, roughly one-half of a percent, it can contribute up to
90 percent of its cyclical variation.1 At the same time, the inventory literature is rife
with puzzles in the sense that theoretical inventory models cannot reproduce salient
empirical facts in the data. Maccini, Moore, and Shaller (2010) classify two sets
of puzzles. The traditional puzzles tend to describe the unconditional properties of
inventories, such as the relative volatilities of inventories, production and sales, their
persistence, and their comovement relationships with potential determinants. The
second set of puzzles concerns the relationship between interest rates and inventories.
While theory derives clear predictions - in particular, increases in interest rates lower
inventory investment as the cost of holding inventories rises - this relationship is much
more diﬃcult to establish empirically.
In this paper we discuss one explanation of why these puzzles exist, namely the
surprising diﬃculty of establishing invariant stylized facts over almost a century of
U.S. data. We focus on the relationship between final sales, inventories and interest rates and utilize data from 1919 on. Using Bayesian time-varying parameter
VARs with stochastic volatility we study changes in both the reduced-form and the
structural correlations between these variables in the United States during both the
interwar and the post-WWII periods. Our estimates are based on the identification
of four structural shocks, whereby we combine a single long-run restriction to identify
a permanent output shock with three sign restrictions to identify transitory demandand supply-side shocks.
Our main findings are the following. We first show that the absence of a negative
correlation between inventories and interest rates only pertains to the post-WWII
period. During the interwar era, the correlation is, in fact, strongly negative. Second, we find that the behavior of this relationship is asymmetric over the business
cycle and over subperiods. During the interwar period, the interest rate-inventories
correlation exhibits positive co-movement with real output at business-cycle frequencies. Although the correlation is overall systematically negative, it is relatively less
negative in business-cycle upswings, and even more negative during downturns. During the post-WWII period, on the other hand, the correlation does not exhibit any
clear-cut systematic pattern of co-movement with the cyclical component of real output. Finally, estimates from the structural VAR specification show that the negative
reduced-form correlation of the interwar era is induced by demand- and supply-side
transitory shocks, and, especially at longer horizons, by the permanent output shock.
The contribution of interest rate shocks, on the other hand, is uniformly positive. As
1

See Blinder and Maccini (1991a) and Blinder and Maccini (1991b).

for the post-WWII era, the positive reduced-form correlation between inventories and
the interest rate can be traced back to the positive structural correlations induced
by interest rate shocks, and, to a lesser extent, by demand-side transitory shocks and
the permanent output shock.
We also look at the relationship between inventories and sales. During the interwar era, the correlation is mostly positive. At business-cycle frequencies it exhibits
strong negative co-movement with the cyclical component of real output. During
the post-WWII period, and in line with the evidence reported in Wen (2005), the
correlation is mostly negative at very short horizons, and uniformly positive at longer
horizons. Evidence from the structural VAR suggests that the positive correlation of
the interwar era is due to the demand shock, and in later periods of the sample, due
to the permanent output shock, whereas the correlation conditional on the other two
shocks is negative. As for the post-WWII period, the evolution of the reduced-form
correlation closely mirrors the evolution of the structural correlation conditional on
the permanent output shock, being strongly positive both in the earlier and in the
most recent parts of the sample at all horizons, and, at shorter horizons, being instead
comparatively smaller, and sometimes negative, during the Great Inflation years.
Our results suggest that identified interest rate shocks have systematically and
robustly induced a positive correlation between business inventories and the real
interest rates during both the interwar and the post-WWII periods. The puzzle of the
absence of a negative correlation between the two series over the post-WWII period
identified in the previous literature is therefore deeper, and more intriguing, than
previously thought. Even conditional on a structural shock for which both economic
theory, and simple intuition, suggest that the correlation should be negative, our
results highlight that, in fact, it has been systematically positive. It follows that
the reduced-form relationship that masks this conditional behavior is therefore due
to the changing nature of the comovement with output and the importance of the
underlying shocks. In that sense, the interwar period was, in fact, markedly diﬀerent
from the post-WWII era.
We see our paper as making the following contributions to the literature. First, we
establish that the reduced form relationships between inventories and salient variables
has varied quite substantially since the Great Depression. While it is possible to
identify periods of stable reduced-form relationships, the changes are often suﬃciently
large that we find it diﬃcult to treat the stylized facts as time-invariant. We argue
that this presents a distinct challenge to theoretical models, which is not easy to
resolve.
We conjecture that a solution of this issue may be found by recognizing that
a positive structural innovation to the interest rate has two eﬀects. First, by raising the real interest rate, it causes, ceteris paribus, optimal inventories holdings to
be reduced, which turns inventory investment negative. Second, by engineering a
decline in output growth, it causes an unanticipated fall in sales, which translates
into unplanned inventories accumulation. Ultimately, whether an interest rate shock
3

will cause an increase or a decrease in inventories will crucially hinge upon which of
these two eﬀects dominates. If the latter eﬀect turns out to be stronger, an interest
rate shock will be associated with an increase in the real interest rate, and with an
accumulation of inventories. Crucially, diﬀerent sample and historical periods may
have been associated with the dominance of either the first or the second eﬀect, and
this might explain why diﬀerent periods have been associated with correlations with
a diﬀerent sign.
Our second contribution is to the broader empirical inventory literature, in that
we introduce Bayesian time-varying VAR techniques, and the use of long time series.
To the very best of our knowledge, this is the first study of inventory relationships
which allows for time-variation in both the coeﬃcients and the covariance matrix
of innovations, which is based on structural VAR methods, and which utilizes long
time series, in particular including the interwar period. Our paper is thus part of a
recent literature which, following Cogley and Sargent (2002)’s seminal contribution,
estimates Bayesian VARs with time-varying parameters.2
It has been noted before that the behavior of inventories is diﬀerent across sample
periods. Perhaps most notably, changes in inventory management are often associated
with changes in the behavior of other aggregate time series, as, for instance, in the
Great Moderation.3 Much of the discussion thus focuses on selected episodes, but
does not take into account whether the identified stylized facts are stable over other
periods. One approach to the issue of sample selection is to identify time periods of
interest, such as the shift from the Great Inflation to the Great Moderation. This
approach rests on the ability of the researcher to identify such episodes in the data.
This may be plausible in the case of the Great Moderation, but is less convincing
otherwise. An alternative is to instead compute statistics of interest using a rolling
window approach. In this paper, we take the latter to its logical conclusion and
estimate a Bayesian VAR with time-variation in both the lag-coeﬃcients and the
covariance matrix of reduced-form innovations, for sales, inventories, inflation and
nominal interest rates.
The remainder of the paper is organized as follows. The next section describes the
Bayesian methodology we use to estimate the time-varying parameters VARs with
stochastic volatility. We detail our identification strategy in the case of the structural
VAR, and we discuss the methodology to compute the VAR’s impact matrix. Section
3 discusses estimation results and evidence from the reduced-form specification, while
Section 4 examines the structural evidence. We particularly focus on the eﬀects of
the identified monetary policy shocks and use the structural evidence to understand
time variation in the reduced-form correlations. Section 5 concludes.
2
3

See Cogley and Sargent (2005), Primiceri (2005), and Benati and Goodhart (2011).
See Kahn, McConnell, and Perez-Quiros (2002).

Methodology

2.1

A Bayesian Time-Varying Parameter VAR with Stochastic Volatility

We specify the following time-varying parameter VAR(p) model:
0

 = 0 + 1 −1 +  +  − +  ≡   +  

(1)

The notation is standard. The vector  ≡ [∆ , ∆ ,   ,  ]0 collects the data series of
interest. ∆ is the log-diﬀerence of real sales, while   is inflation, computed as the
log-diﬀerence of the relevant price index. We use the GNP deflator for the interwar
period, and the GDP deflator for the post-WWII years.  is the short-term rate,
specifically, the three-month commercial paper rate and the three-month Treasury
bill rate, respectively, for the two sample periods.4 Finally, ∆ is the change in real
inventories normalized by potential output5 . For the interwar period we use Balke
and Gordon’s (1986) estimate of potential GNP, whereas for the post-WWII years
we use the Congressional Budget Oﬃce’s estimate of potential GDP. For a complete
description of the data and of their sources, see Appendix A.
The overall sample periods are 1919Q1-1941Q4 for the interwar period and 1949Q12011Q1 for the post-WWII era. For both sample periods we use the first 10 years
of data to compute the Bayesian priors. Consequently, the eﬀective sample periods
are 1929Q1-1941Q4 and 1959Q1-2011Q1, respectively. As it is customary in the literature on Bayesian time-varying parameter VARs6 , we set the lag order to  = 2.
We collect the VAR’s time-varying coeﬃcients–that is, the elements of the matrices
0 , 1 , ...,  –in the vector  , and we postulate that they evolve according to:
( | −1 , ) = ( )  ( | −1 , )

(2)

with ( ) being an indicator function that rejects unstable draws, thus enforcing a
stationarity constraint on the VAR.  ( | −1 , ) is given by:
 = −1 +   

(3)

with   ∼ (0 ).
The VAR’s reduced-form innovations in (1) are assumed to be zero-mean, normally
distributed. We factor the time-varying covariance matrix Ω as:
−1 0
Var( ) ≡ Ω = −1
  ( ) 
4

(4)

We do not annualize the short rate in order to make its scale comparable to that of inflation.
Specifically, if  is the relevant short-term rate, reported as, say, 10%,  is computed as  =
(1 +  100)14 − 1.
5
We discuss in detail the motivation behind this scaling assumption in the data Appendix A.
6
See e.g. Cogley and Sargent (2002), Cogley and Sargent (2005), Primiceri (2005), Benati (2008),
and Benati and Goodhart (2011).

where the matrices  and  are defined as:
⎡
⎡
⎤
1 0
1
0
0
⎢ 0 2 0
⎢
⎥
21
0 ⎥
 ≡ ⎢
 ≡ ⎢
⎣ 0
⎣
⎦
0 3 0
31
0
0
0 4
41

0
1
32
42

0
0
1
43

The  ’s are assumed to evolve as geometric random walks:

⎤
0
0 ⎥
⎥
0 ⎦
1

(5)

(6)

ln  = ln −1 +   

For future reference, we define  ≡ [1 , 2  3  4 ]0 . As in Primiceri (2005), we
postulate that the non-zero and non-unity elements of the matrix  , which we collect
in the vector  ≡ [21 , 31 , ..., 43 ]0 , evolve as driftless random walks:
 = −1 +   .
Finally, we assume the vector [0 ,
⎡
⎤
⎡

⎢
⎢  ⎥
⎢
⎥
⎢
⎣   ⎦ ∼  (0  ) , with  = ⎣

1

 0 ,  0 ,  0 ]0 to be distributed as:
⎡ 2
⎤
1 0
4 0 0 0
⎢ 0 22
0  0 0 ⎥
⎥ and  = ⎢
⎣ 0 0
0 0  0 ⎦
0 0 0 
0 0

(7)

0
0
 23
0

⎤
0
0 ⎥
⎥
0 ⎦
 24
(8)

2
where  is such that  ≡ −1
   .
We impose a block-diagonal structure on  for parsimony, since the model is already quite heavily parameterized. Moreover, allowing for a completely generic correlation structure among diﬀerent sources of uncertainty would preclude any structural
interpretation of the innovations, as discussed in Primiceri (2005). Finally, following
Primiceri (2005) we adopt the additional simplifying assumption of a block-diagonal
structure for :
⎤
⎡
1 01×2 01×3
 ≡   (  ) =   (  ) = ⎣ 02×1 2 02×3 ⎦ 
(9)
03×1 03×2 3

with 1 ≡ Var( 21 ), 2 ≡ Var([ 31   32 ]0 ), and 3 ≡ Var([ 41   42   43 ]0 ). This
implies that the non-zero and non-unity elements of  which belong to diﬀerent rows
evolve independently. As discussed in Primiceri (2005, Appendix A.2), this assumption drastically simplifies inference, since it allows us to perform Gibbs sampling on
the non-zero and non-unity elements of  equation by equation.

2.2

Estimation and Identification

We estimate (1)-(9) via standard Bayesian methods. Appendix B discusses our choices
for the priors, and the Markov-Chain Monte Carlo algorithm we use to simulate the
6

posterior distribution of the hyperparameters and the states conditional on the data.
We identify four structural shocks by combining a single long-run restriction, in order
to identify a permanent output shock as in Blanchard and Quah (1989), with three
sign restrictions to identify demand- and supply-side transitory shocks.
As a preliminary assessment of the plausibility of our long-run assumption, we
perform standard unit root tests on the logarithms of real sales and real output.7 We
find that we cannot reject the null of a unit root for either variable at conventional
significance levels. We also assess whether the two series are cointegrated. We can
reject the null of a unit root in the diﬀerence of these variables. The rejection is very
strong for the post-WWII period, whereas it is weaker for the interwar period. We
therefore proceed under the assumption that, for either period, real sales and real
output share a common stochastic trend. We thus identify permanent output shocks
based on the restriction that they are the only shocks exerting a permanent impact
on log sales.
The remaining three shocks are identified based on a standard set of sign restrictions reported in Table 1. We identify the interest rate shock based on the restriction
that it has a non-negative impact on the interest rate, and a non-positive impact on
both inflation and real sales growth. A demand shock is postulated to have a nonnegative impact on either the interest rate, inflation, or real sales growth. Finally,
a transitory supply shock is diﬀerentiated from the other two shocks because it is
assumed to induce negative co-movement between inflation and real sales, whereas
its impact on the interest rate is left unconstrained. These restrictions are the same
used by Benati (2008) and Benati and Goodhart (2011), the only diﬀerence being that
both papers featured output growth instead of real sales growth. These identifying
assumptions are compatible with a wide range of macroeconomic models.
An important aspect of our identification strategy is that, although we constrain
the sign of the response of sales growth to the three transitory shocks, we instead leave
the response of the change in inventories unconstrained. As for the permanent output
shock, the responses of all variables to it are left unconstrained by assumption. This
implies that the structural correlations between inventories and sales, and between
inventories and the real interest rate–that is, the correlations conditional on either
of the four structural shocks–are left entirely unconstrained. These correlations will
be one of the key objects of our investigation.

2.3

Computing the Structural Impact Matrix

For each quarter, and for each draw from the ergodic distribution, we compute the
time-varying structural impact matrix, 0 , by combining the methodology proposed
by Rubio-Ramirez, Waggoner, and Zha (2005) for imposing sign restrictions, and the
procedure proposed by Gali and Gambetti (2009) for imposing long-run restrictions
within a time-varying parameters VAR context. Specifically, let Ω =   0 be the
7

All of these results are available from the authors upon request.

eigenvalue-eigenvector decomposition of the VAR’s time-varying covariance matrix
1
Ω , and let ̃0 ≡  2 . We draw an  ×  matrix, , from a standard-normal
distribution and compute the  decomposition of , that is, we compute matrices
 and  such that  =  · . The “intermediate estimate” of the time-varying
structural impact matrix can then be computed as ̄0 = ̃0 · 0 .
Following Gali and Gambetti (2009, Section II), we then compute a local approximation to the matrix of the cumulative impulse response functions (IRFs) to the
VAR’s structural shocks as:8
̄∞ = [ − 1 −  −  ]−1 ̄0 
|
{z
}

(10)

∞ = ̄∞  = 0 ̄0  = 0 0 

(11)

0

where  is the  ×  identity matrix. We then rotate the matrix of the cumulativeIRFs via an appropriate Householder matrix9  in order to introduce zeros in all
of the second row of ̄∞ , that is, in the row corresponding to log real sales. The
second row of the resulting local approximation to the matrix of the cumulative IRFs,

2·
= [ 00(−1)×1 ], with 0( −1)×1 being a vector of ( − 1) zeros, and 
is given by ∞
being a non-zero entry.
This implies that the first shock is the only one exerting a long-run impact on the
level of log sales, and therefore on the level of log output. If the resulting structural
impact matrix 0 = ̄0  satisfies the sign restrictions we store it. Otherwise
we discard it, and we repeat the procedure until we obtain an impact matrix which
satisfies both the sign restrictions and the long-run restriction at the same time.

Reduced-Form Correlations

In discussing our estimation results we proceed in several steps. We first present
estimates for the time-varying reduced-form correlations of the forecast errors. This
should give us an idea of how the variables have co-moved over the sample period, and
in particular of how such co-movements may have changed over time. This establishes
our set of stylized facts since the late 1920s. In the subsequent Section we discuss the
same type of evidence, but this time conditional on the identified structural shocks.
Finally, we use the structural estimation evidence in order to shed light on the sources
of time-variation in the reduced-from correlations. We find it convenient throughout
to discuss the two sample periods separately.
8

The only diﬀerence with respect to Gali and Gambetti (2009) is that they compute the local
approximation to the matrix of the cumulative IRFs based on the companion form of the VAR,
whereas we compute it directly based on the VAR itself.
9
We compute the Householder matrix via Algorithm 5.5.1 of Golub and VanLoan (1996).

3.1

The Inter-War Period

Conceptually in line with Cogley, Primiceri, and Sargent (2010), Figures 1 and 2
show, for the interwar period, evidence of changes in the correlation between the
reduced-form forecast errors for the change in real inventories and either the ex post
real interest rate, or real sales growth, at various horizons. Specifically, the top rows in
the two figures show the median and the one- and two-standard deviation percentiles
of the posterior distribution of the correlations between the forecast errors for the
relevant series generated by the time-varying VAR at each point in time; the middle
rows show the fractions of the draws from the posterior distribution for which the
correlation is positive; and the bottom rows show the business-cycle components of
the series shown in the middle rows, together with the business-cycle component of
the logarithm of real GNP.10 For each quarter, the correlations between the forecast
errors for the relevant series at the various horizons generated by the time-varying
VAR were computed based on the estimated covariance matrix of the forecast errors,
following Cogley, Primiceri, and Sargent (2010).11
Starting from the relationship between the change in real inventories and the ex
post real interest rate, the evidence reported in the top row of Figure 1 clearly suggests
that the correlation is consistently negative over the entire sample period and for the
diﬀerent forecasting horizons. This impression is confirmed by the results reported in
the middle row, although the fraction of draws for which the correlation is positive
exhibits significant fluctuations over the sample period, both on impact and at all
horizons. In particular, the fraction of draws reaches a minimum in April 1933, when
the U.S. went oﬀ the Gold Standard, and it achieves local maxima in 1932 and 1936.
To gain further insight into the driving forces behind these patterns, we decompose
the fraction of draws into cyclical and trend components. When we look at the business cycle component, that is, at the component associated with frequencies between
6 quarters and 8 years, we see that, at all horizons, the business-cycle component of
the fraction of draws for which the correlation is positive exhibits a striking positive
correlation with the business-cycle component of the logarithm of real GNP. On the
other hand, casual inspection does not suggest any systematic lead or lag patterns
between the series. Although the correlation is systematically negative during the en10

Business-cycle components have been extracted via the band-pass filter proposed by Christiano
and Fitzgerald (2003). Following established conventions in business-scycle analysis–see e.g. Baxter
and King (1999) and Stock and Watson (1999)–business-cycle frequencies have been defined as those
pertaining to fluctuations with frequencies of obscillation between 6 quarters and 8 years.
11
See expressions (9) to (11) of Cogley, Primiceri, and Sargent (2010). An important point to
stress here is that, given the two-sided nature of the estimates of the time-varying VAR produced
by the Gibbs sampler, the VAR-generated forecast errors we are working with (and therefore their
estimated covariance matrices) should only be regarded as approximations to the authentic objects
we would obtain if we had estimated the VAR based on recursive samples. Such an approximation is
routinely used in the literature–beyond Cogley, Primiceri, and Sargent (2010), see e.g. Benati and
Surico (2008)–because of the significant computational burden associated with recursive estimation
of time-varying parameters VARs.

tire period, it also tends, equally systematically, to be less negative during economic
upswings, and even more negative during downswings. This suggests a fundamental
asymmetry in the way economic upswings and downturns evolve, and thus a challenge
for theoretical business cycle models. Furthermore, as we now discuss with respect
to the relationship between the change in real inventories and real sales growth, this
systematic pattern of variation at the business-cycle frequencies does not appear to
be an isolated result.
The evidence reported in the top row of Figure 2 on the relationship between
the change in real inventories and real sales growth suggests that the correlation is
mostly positive over the sample period, and for the diﬀerent forecasting horizons.
The exceptions are the very early quarters of the sample and the very last quarter.
We note, however, that in most cases a zero correlation is contained within the
one-standard deviation bands. In the middle panel, the fraction of draws shows
significant variation over the sample period, although, consistent with the evidence
we just discussed, most of the draws fall on the positive side. When we decompose
the fraction of draws into a trend and a business cycle component, we see that, at the
business-cycle frequencies, the fraction of draws for which the correlation is positive
exhibits a remarkably strong negative correlation with the logarithm of real GNP.
We can summarize our findings as follows. In the inter-war period, the real rate
and inventories strongly comove negatively, as theory would suggest. Increases in real
rates raise the cost of maintaining inventory holdings. What underlies this reducedform correlation is a stable low-frequency relationship, whereas time-variation is generated by strongly pro-cyclical business cycle components. In economic expansions,
the business cycle component reduces the eﬀect of a negative long-run relationship.
This finding is related to Wen’s (2005) argument that the inventory facts are conditional on the sampling frequency. The correlation between sales and inventory growth
rates is positive, although not as strongly as for the other variables. This pattern
is largely driven by the trend component. The business cycle component is strongly
countercyclical, to the eﬀect that trend and business-cycle movements can both reinforce and dampen each other in diﬀerent periods. This points towards a theory that
oﬀers predictions for both short-run and medium-run business cycle movements.12
Finally, the negative correlation between the business cycle component of GDP and
the sales-inventories correlation suggests a more dominant role for demand factors.13
We will delve deeper into this based on our structural identification scheme.

3.2

The Post-WWII Period

We now turn to the post-WWII period, which has been the main focus of previous
work in the literature. We report the same kind of evidence as for the first sample
period in Figures 3 and 4. We note that the correlation between the reduced-form
12
13

See Comin and Gertler (2006).
See e.g. West (1995).

forecast errors for the real ex post interest rate and the change in real inventories
at various horizons is almost uniformly positive over the entire sample period and
for the various horizons. The relationship is stronger at longer horizons, and much
less so for the contemporaneus correlation (that is, on impact). The correlation is
weakest (and negative) in the very first quarters of the sample, for which a minority
of the draws from the posterior distribution is positive. Similarly, the correlation on
impact is mildly negative during the middle of the 1980s, when a slight majority of
the draws is associated with a negative correlation.
Our finding of a positive correlation between interest rates and inventories is consistent with the previous empirical literature on this topic–see Ramey and West
(1999)–and constitutes one of the puzzles in the inventory literature.14 What the
overall positive correlation over this time period masks, however, is the substantial
time-variation in this pattern, which has not been documented before. These movements are especially apparent on impact, when the correlation turns from negative
to strongly positive during the period up to the Volcker disinflation. It then changes
sign during the first half of the 1980s, after which it exhibits a strong hump-shaped
pattern, with a peak of one around the turn of the century. The contemporaneous
correlation shows a dramatic decrease during subsequent years, with the fraction of
draws for which the correlation is positive falling below 50 percent in the last few
quarters of the sample. Evidence for other horizons is weaker, but it replicates the
very broad features of the evidence on impact.
As in Figures 1 and 2, in the bottom panel of Figure 3 we plot the business-cycle
component of the fraction of draws for which the correlation has been positive, together with the business-cycle component of log real GDP. In sharp contrast with
the interwar period, at the business-cycle frequencies the fractions of draws for which
the correlation has been positive does not exhibit any clear-cut, stable relationship
with the logarithm of real GDP, comoving with it sometimes positively, and sometimes negatively, but without any systematic discernible pattern. This again presents
a challenge for theoretical models, in terms of explaining both a positive long-run relationship and time variation in the short-run together with an apparent disconnect
with movements in GDP.
We now turn to the relationship between inventories and sales growth (see Figure
4). We find that, first, the correlation between the two series’ forecast errors is
predominantly negative on impact, but it becomes more and more positive at longer
horizons. This is conceptually in line with the evidence reported in Wen (2005).
Second, there is a significant extent of time-variation in the correlations between the
forecast errors at the various horizons. This is especially apparent from the middle row
of Figure 4. Specifically, both on impact and at all subsequent horizons, the fraction
of draws for which the correlation is positive has exhibits a broadly V-shaped pattern
over the sample period, although with significant short-run fluctuations around such
a secular movement.
14

See e.g. Maccini, Moore, and Shaller (2010).

Some of these fluctuations coincide with well-known events in U.S. post-WWII
monetary history. The fraction of draws for which the correlation is positive exhibits
significant decreases during the collapse of Bretton Woods and the Volcker disinflation
of the early 1980s, which is especially apparent for horizons up to one year ahead.
This provides evidence that a non-negligible portion of the time-variation in the
correlation between the forecast errors documented in Figure 4 originates from some
of the same fundamental macroeconomic forces which have shaped U.S. post-WWII
macroeconomic dynamics, specifically the evolution of the U.S. monetary regime.
Contrary to the previous results, the fraction of draws on impact shows a much
diﬀerent pattern than those at longer horizons, which suggests underlying structural
diﬀerences.
Finally, as shown in the bottom row, the frequency-domain-based decomposition
of the evolution of the fraction of draws points towards a positive co-variation with
the logarithm of real GDP at business-cycle frequencies since the early 1980s, whereas
evidence for the former period is not clear-cut. At best, there is some mild negative
co-variation. This suggests a possible element of continuity between the earliest part
of the post-WWII period and the interwar era. As we discussed in the previous subsection, the business-cycle component of the fraction of draws exhibits a remarkably
strong negative correlation with the cyclical component of log real GDP. This indicates that the most recent period represents a discontinuity with the pattern that
prevailed during previous decades.
In summary, we find that the real interest rate-inventory growth correlation is
positive for the post-WWII period - which is in line with the previous literature -,
but markedly diﬀerent from the inter-war period. The sales-inventories correlation is
mildly positive, but there are substantial diﬀerences over the various time horizons,
and substantial time-variation in the business cycle component. In particular, the
impact period exhibits diﬀerent patterns along these dimensions than the other time
horizons. We also note that the Volcker disinflation marks a turning point for many
of these statistics. We return to this issue, and how it relates to the causes of the
Great Moderation below.

Structural Evidence

The correlation patterns in the data that we discussed in the previous section represent a set of stylized facts, some of which are well known in the literature. However,
our approach of using a time-varying parameters model reveals a substantial extent
of variation in these facts since the end of the 1920s. We now impose our structural
identification scheme on the reduced-form VAR, and we interpret the reduced-form
findings in light of our structural identification. To recap, we identify a permanent output shock based on a long-run restriction, and three temporary shocks - an
interest-rate shock, a demand shock, and a supply shock - based on sign restrictions.
We first discuss the dynamic behavior of sales, inventories and the real interest rate
12

by analyzing the IRFs to these shocks. We then compute structural forecast error
correlation and use these to dig deeper into the previously discussed stylized facts.

4.1
4.1.1

Impulse Response Functions
The Inter-War Period

Figures 5 to 8 show the normalized IRFs of the change in inventories, real sales growth,
and the ex post real interest rate to each of the four identified structural shocks over
the inter-war period for the first and the last quarter of the sub-sample (1929Q4
and 1941Q4, respectively), and for 1933Q3, in the middle of the Great Depression.15
A permanent output shock leads to a temporary fall in the ex post real rate, and
a temporary increase in sales growth. The former reponses are reasonably tightly
estimated, while the error bands of the sales growth responses include zero both at
the beginning and at the end of the sample. However, the median estimates show
an increase on impact, while there is some variation in inventory growth behavior in
subsequent periods.
In response to a positive interest rate shock, the ex post real rate rises (recall that
its median impact response has been normalized to unity), but swiftly returns to zero
within a year. The estimated median response of sales growth is negative (which
we have imposed in identification), while that of inventories is positive, although the
extent of uncertainty with respect to the latter is large enough that it is not possible to
make statements with any confidence. While the response of sales is consistent with a
priori theorizing - a monetary contraction reduces sales -, the response of inventories
is less obvious. One possibility is that the economic contraction leads to a build-up
in unsold final goods and thus inventory accumulation. We note, however, that the
reduced-form correlation between inventories and the ex post real rate is negative
during the inter-war period (see Figure 1). This suggests that another underlying
driving process explains the co-variation of these two variables.
A positive demand shock (see Figure 7) causes a temporary increase in the change
in inventories, and a temporary decrease with a hump-shaped adjustment pattern in
the ex post real rate. Similarly, a transitory supply shock leads to temporary increases
in both sales growth and the ex post real rate, whereas for inventories uncertainty is
once again so large that we cannot rule out a zero impact at any horizon. Median
estimates, however, show a temporary decrease in the change in inventories.
15

The IRFs to a permanent output shock are normalized so that the median of the distribution of
the cumulative IRFs of sales growth is equal to one (which implies that the median long-run impact
of the shock on the logarithm of real GNP is, likewise, equal to one). The IRFs to an interest rate
shock are normalized so that the median impact at zero on the ex post real rate is equal to one.
Finally, the IRFs to the transitory supply and demand shocks are normalized so that the median
impact at zero on sales growth at zero is equal to one. We compute the IRFs using the Monte Carlo
integration procedure described in Appendix C, which allows us to tackle the uncertainty originating
from future time-variation in the VAR’s structure.

We thus find that an unconditional negative comovement between the interest rate
and inventories can be traced back to either temporary demand or supply shocks.
These shocks are separately identifiable through their impact on sales relative to
inventories. We found that the reduced-form correlation between these two series in
the inter-war period is largely positive, which would point to a more dominant role
for demand shocks (see Figure 11). Finally, none of the IRFs for either of the three
series exhibits any discernible time-variation over the sample period.
4.1.2

The Post-WWII Period

We report the normalized IRFs of the change in inventories, real sales growth, and
the ex post real interest rate to each of the four identified structural shocks in Figures
9-12 for selected quarters. On impact, a permanent output shock raises both sales
growth and the ex post real rate, which then reverts back to zero. Whereas the
responses of sales growth are quite tightly estimated and do not exhibit a significant
extent of variation across time periods, this is often not the case for the ex post
real rate. In particular, as the bottom row of Figure 9 shows, around the time
of the Great Inflation the extent of uncertainty is significantly larger than either
before or after. The response of the change in inventories, on the other hand, exhibits
some time-variation. It is consistently negative on impact over the entire post-WWII
period, albeit with varying degrees of uncertainty. In the latter half of the sample
period, a pronounced hump-shaped response emerges, whereas in the earlier part the
adjustment is more monotonic. What is striking is the diﬀerence with respect to the
inter-war period, when the response on impact is positive. Similarly, the behavior of
the ex post real rate is diﬀerent between the two sub-samples, whereas the response
of sales does not diﬀer much.
Turning to the interest rate shock in Figure 10, none of the variables’ impulse
responses exhibit significant time-variation. The ex post real rate rises on impact,
and then reverts monotonically to zero. The speed of mean-reversion appears tobe
smaller in the first decades of the sample, but evidence on this is weak, given the large
extent of uncertainty. Mean-reversion is significantly faster for sales growth, which
jumps down on impact, then oscillates around zero for a few quarters. Finally, the
response for inventories is positive on impact, which then revert to zero monotonically.
Overall, the post-war impulse responses exhibit the same pattern as those for the prewar period, thereby reflecting the positive reduced-form correlation between the two
series.
A positive demand shock (see Figure 11) is estimated to lead to a small and
statistically insignificant decrease in the ex post real rate, the magnitude of which
remains essentially unchanged over the sample period. The evolution of the posterior
distributions of the impulse responses suggests that the dynamics of the real rate are
not significantly diﬀerent from zero in the latter part of the sample, and comparatively
much stronger in the earlier part. As for sales growth, responses on impact tend to be

positive, but insignificant, with a more drawn-out behavior at the end of the sample.
The responses of inventories show a sizeable extent of time-variation, with the impact
being negative in the early part of the sample, and then turning positive around
the time of the Great Inflation. However, the extent of uncertainty is substantial.
Once again, the corresponding patterns for the inter-war period are broadly similar,
although the behavior of inventories and sales resemble more closely the latter part
of the post-war sample.
Finally, the responses to the transitory supply shock reported in Figure 12 point
towards no time-variation for any of the three variables. Sales rise, inventories fall,
and the real rate increases significantly and persistently. Sales revert back to zero after
one quarter, while the response of inventories is more drawn-out, with a pronounced
hump-shaped pattern in the latter part of the sample. This mirrors the behavior
of inventories in response to a permanent supply shock. Comparison to the pre-war
responses do not reveal substantial diﬀerences.
The main finding in this section is the comovement pattern between inventories
and the real interest rate. Their responses to an identified interest rate shock suggest
that the puzzle of a positive correlation between the two series over the post-WWII
period does not uniquely pertain to their reduced-form relationship. Arguably, this
is perhaps more surprising than the positive reduced-form correlation. In principle, a
positive reduced-form correlation is compatible with a negative correlation conditional
on interest rate shocks, which is, in fact, what we would expect to find based on
theoretical considerations. A positive reduced-form correlation can result from the
fact that the negative correlation generated by interest rate shocks is dominated by
the positive correlation induced by at least one of the other shocks and that the
fractions of variance of inventories and the real interest rate explained by the shocks
generating positive conditional correlations are suﬃciently large. Our results suggest
that even interest rate shocks generate a positive co-movement between the two series,
which deepens the inventory-interest rate puzzle.

4.2

Structural Correlations

We now turn to an analysis of the structural correlations between the series, that is,
to the correlations conditional on the identified structural shocks. The evidence on
the evolution of the reduced-form correlations between inventories and either sales
growth or the real rate naturally raises two questions. First, which structural shocks
generate such correlations in the data? Second, what are the causes of time variation
in the correlations? More specifically, is the time-variation due to changes in the
relative importance of the diﬀerent shocks hitting the economy, or due to changes in
the way their impact is propagated through the system? In this section we address
the first question, whereas we answer the latter in the next .

4.2.1

The Inter-War Period

Figures 13 and 14 show the structural correlations between inventories and either
the ex post real interest rate or sales growth, for both the long run and one quarter
ahead. We also report the fraction of draws for which the structural correlations
are positive. We only consider structural correlations at horizons greater than zero
because, on impact, the correlation between two series conditional on a single shock
is by construction always equal to one. At horizons greater than zero, on the other
hand, the dynamics of the system do matter, so that even correlations conditional on
a single shock are no longer necessarily equal to one.
As documented above, the reduced-form correlation between the change in inventories and the ex post real rate is systematically negative during the entire inter-war
period. Figure 13 shows that the negative correlation is due to transitory demand
and supply shocks, which induce a uniformly and substantially negative correlation
over the entire period. This is re-inforced by the permanent output shock, for which
the conditional correlation turns substantially negative after the early 1930s, and especially at longer horizons. In line with our discussion of the IRFs, the correlation
conditional on interest rate shocks is systematically positive over the entire sample
period. This highlights once again the puzzling nature of the relationship between
interest rates and business inventories. It is also apparent from the figures that the
years leading up to the abandonment of the Gold Standard exhibit a significantly different relationship between real rates and inventories under permanent supply shocks,
whereas the eﬀect on the other identified shocks is much more muted.
Figure 14 shows a similar set of graphs for the correlation between inventories
and sales. The mostly positive correlation of the inter-war era is largely due to the
permanent output shock, especially after April 1933. As for the contribution of the
demand shock, it is close to borderline. In particular, it is slightly positive at very
short horizons, but essentially neutral at longer horizons. Finally, the interest rate
shock and the transitory supply shock systematically induce a negative correlation,
which is not strong enough to overturn the contributions of the other two shocks.
4.2.2

The Post-WWII Period

We now look at the structural correlations for the variables of interest in post-war
data, which are reported in Figures 15 and 16. Although the reduced-form correlation
between the real rate and inventories is almost uniformly positive during this period,
Figure 15 depicts a complex picture for its structural sources. A positive correlation is
induced over the entire sample period, and over all horizons, by the interest rate shock,
which particularly at shorter horizons generates a conditional correlation of almost
one. The permanent output shock plays a supporting role at short horizons, but less
so at longer horizons. The correlation it induces also exhibits a significant extent of
time-variation. The demand shock contributes to an overall positive correlation only
during the period of the Great Moderation and, to a lesser extent, before the collapse
16

of Bretton Woods. During the Great Inflation the contribution of this shock to the
overall correlation is in fact negative. This is consistent with the idea, espoused in
Maccini, Moore, and Schaller (2004) that the monetary regime may play a crucial
role in determining the comovement between inventories and real rates.
We previously found that the reduced-form correlation between inventories and
sales is strongly positive at most horizons, with the exception of the very short ones.
In Figure 16, we note that both the interest rate and the transitory supply shock
generate a uniformly negative correlation at all horizons over the sample period. The
demand shock contributes positively to the correlation before the beginning of the
Volcker stabilization, and especially during the Great Inflation years. At the same
time, the evolution of the reduced-form correlation closely mirrors the evolution of the
correlation conditional on the permanent output shock. It is very strongly positive
at the beginning of the sample, then it oscillates around zero around the time of the
Great Inflation, before turning positive during the most recent part of the sample. In
the long run, this large degree of time variation in the structural correlation disappears
in favor of a robustly positive correlation.
A comparison between these patterns across our two main sample periods shows
that the role of the identified interest rate shock in generating a negative corrrelation
between sales and inventories and a positive correlation between real rates and inventories is the same. Similarly, transitory supply shocks imply the same reduced-form
correlation in the pre-war and post-war samples. The diﬀerences in the reduced-form
behavior across periods are thus largely explained by the permanent supply shocks
and, to a lesser extent, by the demand shocks. The permanent shock induces a positive correlation between sales and inventories in both sample periods, but switches
sign in the conditional correlation between interest rate and inventories between the
two periods, as do demand shocks. Moreover, we find that during times of extreme
economic turbulence, such as the abandonment of both the Gold Standard and the
Bretton Woods system of fixed exchange rates, the structural correlations shift, in the
extreme case changing their signs. The most prominent example is the one conditional
on the permanent productivity shock. Inventory models that attempt to capture this
feature of the data might therefore have to incorporate regime shifts.

4.3

Understanding Time-Variation in the Correlations

Our final exercise attempts to delve deeper into the underlying causes of the changes in
the reduced-form and structural correlations over the two sample periods. Specifically,
we attempt to disentangle the contribution of the time-varying VAR-coeﬃcients and
the time-varying components of the variance-covariance matrix. These results are
reported in Figure 17. For both sample periods we show the medians of the posterior
distributions of the VAR’s time-varying coeﬃcients (the  ’s), of the logarithms of
the diagonal elements of the matrix  (the  ’s), and of the oﬀ-diagonal elements
of the matrix  (the  ’s).
17

As for the inter-war period, the first column points towards essentially no timevariation in the VAR coeﬃcients. Moreover, there is no time-variation for three
volatilities out of four. The main source of variation pertains instead to the oﬀdiagonal elements of the covariance matrix. This implies that the VAR coeﬃcients
cannot have played a significant role in fostering changes in the two correlations. On
the other hand, it is not possible to further disentangle the separate roles played by
changes in the structural shocks’ volatilities, and changes in the way the shocks have
impacted the economy.16 What we can conclude is that the VAR coeﬃcients played
essentially no role in causing the variation of the reduced-form correlations. Turning
to the post-WWII era, the bottom row of Figure 17 shows time-variation in all of
the VAR elements, although the visual evidence clearly suggests that the extent of
variation for the VAR’s coeﬃcients is comparatively smaller than for elements of the
covariance matrix.
We also conducted analogous experiments for the structural correlations17 . In the
pre-war period, the impact of changes in the VAR’s coeﬃcients is negligible. In the
post-WWII period, on the other hand, for the correlation between inventories and
the real rate the impact is virtually zero conditional on the interest rate shock. It is
minor, and only at longer horizons, for the one conditional on the transitory supply
shock, and significant at longer horizons for the demand non-interest rate shock.
On the other hand, it is quite substantial at all horizons for the permanent output
shock. For the correlation between inventories and sales, the impact is negligible for
all shocks, and at all horizons.
To summarize, evidence for the pre-war period points towards no role of changes in
the VAR coeﬃcients in driving changes in either of the two correlations. Instead, these
originate from changes in both the volatilities of structural shocks, and especially
the way in which such shocks have impacted the economy. For the post-war period,
changes in both the shocks’ volatilities and the way in which they impact the variables
play a dominant role, but it is possible to detect some impact of changes in the VAR’s
coeﬃcients for the correlation between inventories and the real rate at longer horizons.
16

Note that from equation (4), and from the definition of the structural impact matrix 0 , we
−1 0
0
have that under the assumption of unit-variance structural shocks Ω = −1
  ( ) = 0 0 .
However, this is equivalent to

−1 0
0
0 0
Ω = −1
  ( ) = 0 0 = ̂0  ̂0

⎡

 1
⎢ 0
= ̂0 ⎢
⎣ 0
0

0
 2
0
0

0
0
 3
0

⎤
0
0 ⎥
⎥ ̂0 ,
0 ⎦ 0
 4

(12)

where the   ’s are the non-unitary, and possibly time-varying volatilities of the structural shocks.
̂0 is the structural impact matrix associated with these shocks, for any values of the  ’s. Timevariation in the elements of the matrices  and  during the pre-war era cannot be automatically
mapped into time-variation in the  ’s and the elements of ̂0 , due to an identification problem.
17
These results are available from the authors upon request.

Conclusions

We use Bayesian time-varying VAR methods to study the statistical relationship
between inventories, sales and the real interest rate. Reduced-form estimates detail a
substantial degree of time-variation both between the pre-war and post-war sample
periods, but also within them. Our main finding is that the positive correlation
between inventories and interest rates is robust over the entire sample. While this
stylized fact is well-known for the post-WWII period, we further establish that the
reduced-form correlation between these variables is negative in the inter-war period.
However, this pattern is driven demand and supply shocks. Once we condition this
correlation on the identified interest-rate shock, the positive correlation emerges. This
highlights that one of the salient inventory puzzles is more pertinent than the previous
literature let on. Moreover, we also demonstrate the need to establish conditional
stylized facts since the changing patterns of demand and supply shocks render the
notion of fixed relationships meaningless.
As a conjecture for a theoretical mechanism at play we suggest the following.
An unanticipated structural innovation to the interest rate has two eﬀects. First, it
increases both nominal and real interest rates. The latter eﬀect causes, ceteris paribus,
a decrease in firms’ desired level of inventories, and therefore a negative change in
inventories. Second, the interest rate shock will cause an unexpected fall in sales,
and therefore an undesired accumulation of inventories. The notion that an interest
rate shock causes a positive or a negative change in inventories crucially hinges upon
which of these two eﬀects dominates: if the extent of the undesired accumulation of
inventories due to the recessionary eﬀect of the interest rate shock is larger than the
extent of their planned decrease due to the higher real interest rate, an interest rate
shock will be associated with an increase in inventories, rather than a decrease.

The Data

Before describing the data in more detail, we first want to address the issues of trending behavior in the raw series. To the very best of our knowledge, no previous study
has performed any normalisation of the change in real inventories. Within the present
context, however, expressing the change in real inventories in absolute terms rather
than in percentage terms, as is the case for sales, would make the interpretation of
our results especially diﬃcult. Economic growth causes the variance of the change in
real inventories to increase without bounds, thus automatically introducing a systematic element of distortion in any comparison over time and across quarters. Consider
for example the response of the economy to an identified interest rate shock. Since
the change in sales is expressed in percentage terms, once the IRFs have been appropriately normalised on the interest rate, comparing the response of sales across
quarters is appropriate and meaningful. This is however not the case for the change
in real inventories: in this case, what the IRFs are telling us is by how much, in real
dollars, inventories would have changed in each quarter in response to a normalized
interest rate shock. Since the economy in, say, 2009Q1 was significantly larger than
it had been in 1959Q1, comparing the response to a normalized interest rate shock in
the two quarters of the change in inventories expressed in constant dollars does not
provide any meaningful information. This automatically implies that, for our results
to be interpretable, the change in real inventories has to be normalized in such a
way as to eliminate the impact of economic growth. We have chosen to normalize it
by potential, rather than by actual output, in order to avoid distorting the cyclical
properties of the resulting series.

A.1

Interwar period

Quarterly seasonally adjusted series for real GNP (the acronym is RGNP72), real
potential GNP (TRGNP), the GNP deflator (GNPDEF), nominal GNP (GNP), the
change in nominal business inventories (DBUSINOM), and the commercial paper rate
(CPRATE) are all from Balke and Gordon (1986). The sample period is 1919Q11941Q4. Our sample period is dictated by data availability, as the inventories series
from Balke and Gordon (1986) starts in 1919Q1 and is not available for the period
1942Q1-1946Q4.Real sales have been computed as the diﬀerence between nominal
GNP and the change in nominal inventories, deflated by the GNP deflator, whereas
the change in real inventories has been computed as the ratio between the change in
nominal inventories and the GNP deflator (the series thus obtained is near-identical
to Balke and Gordon’s series for the change in real inventories, DBUSI72).

A.2

Post-WWII period

A quarterly seasonally adjusted series for the GDP deflator is from Table 1.1.9 of
the Bureau of Economic Analysis’ National Income and Product Accounts (hence20

forth, NIPA). Quarterly seasonally adjusted series for nominal GDP and the change
in nominal inventories are from Table 1.1.5. of the NIPA. The series for potential
GDP (‘GDPPOT: Real Potential Gross Domestic Product, U.S. Congress: Congressional Budget Oﬃce, Budget and Economic Outlook, Quarterly, Billions of Chained
2005 Dollars’) is from FRED II. The series for real GDP has been computed as the
ratio between nominal GDP and the GDP deflator. The series thus obtained is numerically near-identical to the chain-weighted series for real GDP, GDPC96, which
can be found, e.g., at the St. Louis Fed’s internet data portal, FRED II (‘GDPC96:
Real Gross Domestic Product, 3 Decimal, U.S. Department of Commerce: Bureau
of Economic Analysis, Gross Domestic Product, Seasonally Adjusted Annual Rate,
Quarterly, Billions of Chained 2005 Dollars). This justifies our use of the Congressional Budget Oﬃce’s chain-weighted estimate of potential real GDP in order to
normalize the change in real inventories. The series for the change in real inventories has been computed as the ratio between the change in nominal inventories and
the GDP deflator. Real sales have been computed as the diﬀerence between nominal
GDP and the change in nominal inventories, deflated by the GDP deflator. A monthly
seasonally unadjusted series for the 3-Month Treasury Bill: Secondary Market Rate
(acronym is TB3MS) is from the Board of Governors of the Federal Reserve System,
and it has been converted to the quarterly frequency by taking averages within the
quarter.

Details of the Markov-Chain Monte Carlo Procedure

We estimate (1)-(8) via Bayesian methods. The next two subsections describe our
choices for the priors, and the Markov-Chain Monte Carlo algorithm we use to simulate the posterior distribution of the hyperparameters and the states conditional
on the data, while the third section discusses how we check for convergence of the
Markov chain to the ergodic distribution.

B.1

Priors

For the sake of simplicity, the prior distributions for the initial values of the states–
0 and 0 –which we postulate all to be normal, are assumed to be independent
both from each other, and from the distribution of the hyperparameters. In order to
calibrate the prior distributions for 0 and 0 we estimate a time-invariant version of
(1) based on the first 10 years of data, and we set
h
i
0 ∼  ̂  4 · ̂ (̂ )
(B1)
where ̂ (̂ ) is the estimated asymptotic variance of ̂ . As for 0 , we proceed
as follows. Let Σ̂ be the estimated covariance matrix of  from the time-invariant
21

VAR, and let  be its lower-triangular Cholesky factor, i.e.,  0 = Σ̂ . We set
ln 0 ∼ (ln 0  10 ×  )

(B2)

where 0 is a vector collecting the logarithms of the squared elements on the diagonal
of . As stressed by Cogley and Sargent (2005), “a variance of 10 is huge on a
natural-log scale, making this weakly informative” for 0 .
Turning to the hyperparameters, we postulate independence between the parameters corresponding to the two matrices  and , which is an assumption we adopt
uniquely for reasons of convenience. Further, we make the following, standard assumptions. The matrix  is postulated to follow an inverted Wishart distribution,
¡
¢
 ∼  ̄−1  0
(B3)

with prior degrees of freedom 0 and scale matrix 0 ̄. In order to minimize the
impact of the prior, thus maximizing the influence of sample information, we set 0
equal to the minimum value allowed, the length of  plus one. As for ̄, we calibrate
it as ̄=  × Σ̂ , setting =1.0×10−4 , the same value used in Cogley and Sargent
(2005), and a slightly more “conservative” prior (in the sense of allowing for less
random-walk drift) than the 3.5×10−4 used by Cogley and Sargent (2005). As for ,
we postulate it to be normally distributed with a “large” variance,
 () = (0, 10000·(−1)2 )

(B4)

Finally, as for the variances of the stochastic volatility innovations, we follow Cogley
and Sargent (2002, 2005) and we postulate an inverse-Gamma distribution for  2 ≡
Var(  ):
µ −4 ¶
1
10
2
  ∼ 

(B5)
2 2

B.2

Simulating the posterior distribution

We simulate the posterior distribution of the hyperparameters and the states conditional on the data via the following MCMC algorithm, as found in Cogley and Sargent
(2005). In what follows,  denotes the entire history of the vector  up to time ,
i.e.,  ≡ [01 , 02 , , 0 ]0 –while  is the sample length.
(a) Drawing the elements of  Conditional on   , , and   , the observation
equation (1) is linear, with Gaussian innovations and a known covariance matrix.
Following Carter and Kohn (2004), the density ( |      ) can be factored as
( |      ) = ( |      )


−1
Y
=1

( |+1        )

(B6)

Conditional on  and   , the standard Kalman filter recursions nail down the first
element on the right hand side of (A6), ( |      ) = (   ), with  being
22

the precision matrix of  produced by the Kalman filter. The remaining elements in
the factorization can then be computed via the backward recursion algorithm found,
e.g., in Kim and Nelson (1999), or Cogley and Sargent (2005, appendix B.2.1). Given
the conditional normality of  , we have
−1
(+1 −  ) 
|+1 = | + | +1|

(B7)

−1
| 
|+1 = | − | +1|

(B8)

which provides, for each  from  -1 to 1, the remaining elements in (1), ( |+1 ,   ,
,   ) = (|+1 , |+1 ). Specifically, the backward recursion starts with a draw
from (   ), call it ̃ . Conditional on ̃ , (A7)-(A8) give us  −1| and  −1| ,
thus allowing us to draw ̃ −1 from ( −1|   −1| ), and so on until  = 1.
(b) Drawing the elements of  Conditional on   ,  , and , the orthogonalised
0
innovations  ≡ ( -  ), with  ( ) =  , are observable. Following Cogley
and Sargent (2002), we then sample the  ’s by applying the univariate algorithm
of Jacquier, Polson, and Rossi (1994) element by element.18
(c) Drawing the hyperparameters Conditional on   ,  ,   , and , the innovations to  and to the  ’s are observable, which allows us to draw the hyperparameters, namely the elements of  and the  2 , from their respective distributions.
(d) Drawing the elements of  Finally, conditional on   and  the  ’s are
observable, satisfying
 =  
(B9)
with the  being a vector of orthogonalized residuals with known time-variying variance  . Following Cogley and Sargent (2005), we interpret (B9) as a system of
unrelated regressions. The first equation in the system is given by 1 ≡ 1 , while
the following equations can be expressed as transformed regressions as
³ 1 ´
³ 1 ´ ³ 1
´
−
−
−
22 2 = −21 22 1 + 22 2
(B10)
³ −1 ´
³ −1 ´
³ −1 ´ ³ −1
´
32 3 = −31 32 1 − 32 32 2 + 32 3


³ 1
´
³ 1
´
−2
−2
 ( −1)2 ( −1)2 = −(−1)21  ( −1)2 1 − 
³ 1
´ ³ 1
´
−2
−2
 − (−1)2(−1)2 (−1)2
(−1)2 + (−1)2
(−1)2

where the residuals are independent standard normal. Assuming normal priors for
each equation’s regression coeﬃcients, the posterior is also normal, and can be computed via equations (77) of (78) in Cogley and Sargent (2005, section B.2.4).
18

For details, see Cogley and Sargent (2005, Appendix B.2.5).

Summing up, the MCMC algorithm simulates the posterior distribution of the
states and the hyperparameters, conditional on the data, by iterating on (a)-(d). In
what follows, we use a burn-in period of 50,000 iterations to converge to the ergodic
distribution, and after that we run 10,000 more iterations sampling every 10th draw
in order to reduce the autocorrelation across draws.19

Computing Generalised Impulse Response Functions

Here we describe the Monte Carlo integration procedure we use in Section 4.1 in order
to compute the generalised IRFs to the structural shocks.
Randomly draw the current state of the economy at time  from the Gibbs sampler’s output. Given the current state of the economy, repeat the following procedure
100 times.
• Draw four independent (0, 1) variates (the four structural shocks), and based


 0



on the relationship  = 0  , with  ≡ [
 ,  ,  ,  ] –where  ,  ,  ,
and 
are the permanent output shock, and the interest rate, demand non
interest rate, and transitory supply structural shocks, respectively–compute
the reduced-form shocks  at time .
• Simulate both the VAR’s time-varying parameters and the covariance matrix
of its reduced-form innovations, Ω , 40 quarters into the future. Based on the
simulated Ω , randomly draw reduced-form shocks from +1 to +40. Based
on the simulated  , and on the sequence of reduced-form shocks from  to
+40, compute simulated paths for the four endogenous variables. Call these

simulated paths as ̂+40
, with  = 1, .., 100.
• Repeat the same procedure based on exactly the same simulated paths for the
VAR’s time-varying parameters, the  ; the same reduced-form shocks at times



+1 to +40; and the same structural shocks 
at time ,
 ,  ,  , and 
with the only diﬀerence that, in order to compute the GIRF to shock  , with

 =  , , ,  , you set  = 1. Call these simulated paths as ̃+40
, with
 = 1, .., 100.



≡ ̂+40
− ̃+40
. Finally, compute
For each of the 100 iterations define +40

’s.
each of the 1,000 generalised IRFs as the mean of the distribution of the +40

In this we follow Cogley and Sargent (2005). As stressed by these authors, however, this has
the drawback of “increasing the variance of ensemble averages from the simulation”.

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Table 1: Sign Restrictions
Shock:
Variable:
Interest rate
Inflation
Real sales growth







≥0 ≥0
?
≤0 ≥0 ≤0
≤0 ≥0 ≥0

? = left unconstrained

On impact

One year ahead

Long-run
1

84th percentile

Correlation
between the
two series

0.5

-0.5

95th
percentile

0.5

Median

-1
1930

Wall Street crash

Fraction of
draws for
which the
correlation
is positive

1935

1940

-0.5

-1
1930

5th
percentile
1935

1940

-1
1930

President Roosevelt takes
the dollar off gold

0.8

0.6

0.4

0.2

0
1930

1935

1940

0
1930

1935

1940

16th percentile

1935

1940

0
1930

1935

1940

1935

1940

Business-cycle component of log real GNP
3

-1

-2

Fraction of
draws for
1
which the
correlation 0
is positive:
business-cycle -1
component
-2
-3
1930

1935

1940

-3
1930

1935

1940

-3
1930

Figure 1 United States, interwar period: correlation between the reduced-form
forecast errors for the real ex post interest rate and the change in real
inventories at various horizons, and fractions of draws for which the
correlation is positive

On impact

One year ahead

Long-run
1

84th
percentile

Correlation
between the
two series

0.5

-0.5

95th
percentile

Median

5th
percentile
16th
percentile

-1
1930

Wall Street crash

Fraction of
draws for
which the
correlation
is positive

1935

1940

-1
1930

1935

1940

-1
1930

1940

1935

1940

1935

1940

President Roosevelt takes
the dollar off gold

0.8

0.6

0.4

0.2

0
1930

1935

1940

0
1930

1935

1940

0
1930

Business-cycle component of log real GNP
3

-1

-2

Fraction of
draws for
1
which the
correlation 0
is positive:
business-cycle -1
component -2
-3
1930

1935

1940

-3
1930

1935

1940

-3
1930

Figure 2 United States, interwar period: correlation between the reduced-form
forecast errors for real sales growth and the change in real inventories
at various horizons, and fractions of draws for which the correlation is
positive

On impact

One year ahead

Long-run
1

95th
percentile

84th percentile
0.5

0.5

-0.5

Correlation
between the
two series

-1
1960

Median

1980

2000

Paul Volcker becomes
FED Chairman

Fraction of
draws for
which the
correlation
is positive

16th
percentile

-1
1960

1980

2000

-1
1960

0.8

0.6

0.4

0.2

1980

2000

0
1960

Collapse of
Bretton Woods

1980

2000

0
1960

2000

1980

2000

Business-cycle component of log real GNP

-1

-2

Fraction of
draws for
1
which the
correlation 0
is positive:
business-cycle -1
component -2
-3
1960

1980

Beginning of the
financial crisis

0
1960

5th
percentile

-0.5

1980

2000

-3
1960

1980

2000

-3
1960

1980

2000

Figure 3 United States, post-WWII period: correlation between the reduced-form
forecast errors for the real ex post interest rate and the change in real
inventories at various horizons, and fractions of draws for which the
correlation is positive

On impact

One year ahead

0.5

Correlation
between the
two series

84th
percentile

-0.5

-1
1960

1980

2000

Paul Volcker becomes
FED Chairman

Fraction of
draws for
which the
correlation
is positive

95th
percentile
0.5

0.5

-0.5

16th
percentile

Long-run
1

Median

-1
1960

1980

2000

-0.5

-1
1960

0.8

0.6

0.4

0.2

1980

2000

0
1960

Collapse of
Bretton Woods

1980

2000

0
1960

2000

1980

2000

Business-cycle component of log real GNP

-1

-2

Fraction of
draws for
1
which the
correlation 0
is positive:
business-cycle -1
component -2
-3
1960

1980

Beginning of the
financial crisis

0
1960

5th
percentile

1980

2000

-3
1960

1980

2000

-3
1960

1980

2000

Figure 4 United States, post-WWII period: correlation between the reduced-form
forecast errors for real sales growth and the change in real inventories
at various horizons, and fractions of draws for which the correlation is
positive

95th percentile

Inventories
0.2

84th
percentile

Median
0.1

5th percentile
-0.1

16th percentile
-0.2
0

1929Q4

1933Q4

1941Q4

Sales
0.6

0.4

0.2

-0.2

1929Q4

1933Q4

1941Q4

Ex post real rate
0.1
0
-0.1
-0.2
-0.3
-0.4

1929Q4

1933Q4

1941Q4

Figure 5 United States, interwar period: impulse-response functions
to a permanent output shock in selected quarters

Inventories
1.5

95th percentile
Median

84th
percentile

0.5
0

16th
percentile

-0.5
-1

5th percentile
0

1929Q4

1933Q4

1941Q4

Sales
1
0.5
0
-0.5
-1
-1.5
-2
-2.5

1929Q4

1933Q4

1941Q4

Ex post real rate
2
1.5
1
0.5
0
-0.5

1929Q4

1933Q4

1941Q4

Figure 6 United States, interwar period: impulse-response functions
to an interest rate shock in selected quarters

Inventories

95th percentile
84th
percentile

0.5

Median

16th
percentile

-0.5
0

1929Q4

1933Q4

1941Q4

5th percentile

Sales
2
1.5
1
0.5
0
-0.5
0

1929Q4

1933Q4

1941Q4

Ex post real rate
0.5

-0.5

-1
0

1929Q4

1933Q4

1941Q4

Figure 7 United States, interwar period: impulse-response
functions to a demand non-interest rate shock
in selected quarters

Inventories

95th percentile

1.5

84th
percentile

1
0.5
0

5th
percentile

-0.5
-1

Median

16th percentile

-1.5
-2

1929Q4

1933Q4

1941Q4

Sales
3

-1

1929Q4

1933Q4

1941Q4

Ex post real rate
1.5

1
0.5
0
-0.5
0

1929Q4

1933Q4

1941Q4

Figure 8 United States, interwar period: impulse-response functions
to a transitory supply shock in selected quarters

95th percentile

Inventories

84th
percentile

0.5
0
-0.5

16th
percentile

Median
-1

5th
percentile

-1.5
-2

1960Q1

1970Q1

1980Q1

1990Q1

2010Q1

Sales

2
1.5
1
0.5
0
-0.5
-1

1960Q1

1970Q1

1980Q1

1990Q1

2010Q1

Ex post real rate
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0

1960Q1

1970Q1

1980Q1

1990Q1

2010Q1

Figure 9 United States, post-WWII period: impulse-response functions
to a permanent output shock in selected quarters

Inventories
4

95th percentile
84th
2
percentile
Median
0

16th
percentile

-2

-4

5th percentile
0

1960Q1

1970Q1

1980Q1

1990Q1

2010Q1

Sales
2
1
0
-1
-2
-3
-4

1960Q1

1970Q1

1980Q1

2010Q1

real
rate

Ex
post

1990Q1

1.5
1
0.5
0

1960Q1

1970Q1

1980Q1

1990Q1

2010Q1

Figure 10 United States, post-WWII period: impulse-response functions
to an interest rate shock in selected quarters

Inventories

-2

-4
0

1960Q1

1970Q1

1980Q1

1990Q1

2010Q1

Sales

3
2
1
0
-1
-2
-3

95th
1960Q1
percentile
84th percentile

1970Q1

1980Q1

1990Q1

2010Q1

Ex post real rate

1.5
1
0.5
0

Median
16th
percentile

-0.5
-1

5th percentile
-1.5

1960Q1

1970Q1

1980Q1

1990Q1

2010Q1

Figure 11 United States, post-WWII period: impulse-response
functions to a demand non-interest rate shock
in selected quarters

Inventories

1
0.5
0
-0.5
-1
-1.5
-2

1960Q1

1970Q1

1980Q1

1990Q1

2010Q1

Sales

2
1.5
1
0.5
0
-0.5
-1

1960Q1

1970Q1

1980Q1

1990Q1

2010Q1

Ex post real rate

95th percentile
1

84th
percentile

0.8
0.6

Median

0.4
0.2
0
-0.2

5th
percentile

-0.4
0

16th 1960Q1
percentile

1970Q1

1980Q1

1990Q1

2010Q1

Figure 12 United States, post-WWII period: impulse-response functions
to a transitory supply shock in selected quarters

One quarter ahead

Only
supply
permanent

0.5

-0.5

-1
1930
1

1935

1940

-1
1930
1
0.5

-0.5

-1
1930
1

1935

1940

-1
1930
1

0.5

-0.5

Only
demand
non-interest
rate

One quarter ahead

-0.5

0.5

Only
interest
rate

1935

1940

95th percentile
84th percentile
Median

-1
1930

Only
supply
transitory

Long-run

1935

1940

-1
1930

0.5

-0.5

16th percentile
5th percentile
1935

1940

Wall Street crash

0.8

0.6

0.4

0.2

0
1930
1

1935

1940

-1
1930

1940

0
1930
1
0.8

0.6

0.4

0.2

0
1930
1

1935

1940

0
1930
1

0.8

0.6

0.4

0.2

0
1930

1935

1940

1
0.8

1935

0.8

0.6

-1
1930

Long-run

0
1930

1940

1935

1940

1935

1940

1935

1940

President
Roosevelt
takes
the dollar
off gold

0.8
0.6

0.4

0.2

0
1930

1935

1940

0
1930

II: Fractions of draws for which
the correlation is positive

I: Correlations

Figure 13 United States, interwar period: correlation between real ex post interest rate
and the change in real inventories conditional on individual shocks
40

One quarter ahead

Only
supply
permanent

Long-run

0.5

-0.5

-1
1930
1

-0.5

1935

1940

0.5

Only
interest
rate

-1
1930
1

1935

-0.5

1935

1940

-1
1930
1

1935

0.8

0.6

0.4

0.2

1935

1940

0
1930
1

1935

1940

Wall

0.8

0.8 Street

0.6

0.4

0.2

0
1930
1

1940

Long-run
1

0
1930
1

1940

0.5

-1
1930
1

One quarter ahead
1

crash

1935

1940

0
1930
1

President
Roosevelt
takes
the dollar
off gold

1935

1940

1935

1940

1935

1940

95th percentile
0.5

0.5

Only
demand
non-interest
rate

84th percentile

0.8

0.6

0.4

0.2

16th percentile

-0.5
-1
1930

Only
supply
transitory

Median

0.8

-0.5

1935

1940

-1
1930

0.5

-0.5

-1
1930

1935

1940

-1
1930

5th percentile
1935

1935

0
1930

1940

0
1930

0.8

0.6

0.4

0.2

0
1930

1940

1935

1940

0
1930

II: Fractions of draws for which
the correlation is positive

I: Correlations

Figure 14 United States, interwar period: correlation between real sales growth
and the change in real inventories conditional on individual shocks
41

One quarter ahead

Only
supply
permanent

Long-run

0.5

-0.5

-1
1960
1

-0.5

1980

2000

-1
1960
1

0.5

Only
interest
rate

One quarter ahead

1980

2000

0.8

0.6

0.4

0.2

0
1960
1

1980

-0.5

1980

2000

95th percentile

-1
1960
1

Collapse
of Bretton
Woods

0.2

1980

2000

0
1960
1

1980

2000

0.8

0.4

-1
1960
1

2000

0.8
0.6

-0.5

Long-run

0.6
0.4
0.2

1980

2000

0
1960
1

Paul
Volcker
becomes
FED
Chairman
1980

2000

1980

2000

1980

2000

84th percentile
0.5

Only
demand
non-interest
rate

Median

0.5

-0.5

16th percentile
5th percentile

Only
supply
transitory

-1
1960

1980

2000

-1
1960

0.5

-0.5

-1
1960

1980

2000

-1
1960

1980

2000

0.8

0.6

0.4

0.2

0
1960

1980

2000

0
1960

0.8

0.6

0.4

0.2

0
1960

1980

2000

0
1960

II: Fractions of draws for which
the correlation is positive

I: Correlations

Figure 15 United States, post-WWII period: correlation between real ex post interest rate
and the change in real inventories conditional on individual shocks
42

Beginning
of the
financial
crisis

One quarter ahead

Long-run

0.5

-0.5

Only
supply
permanent

-1
1960
1

1980

2000

-1
1960
1

0.5

-0.5

Only
interest
rate

-1
1960
1980
95th percentile
1

2000

-1
1960
1

1980

One quarter ahead

2000

0.8

0.6

0.4

0.2

0
1960
1

1980

2000

Collapse
of Bretton
Woods

0.8

1980

Long-run

0
1960
1
0.8

0.6

0.4

0.2

0
1960
1

1980

2000

0
1960
1

1980

2000

Paul
Volcker
becomes
FED
Chairman

1980

2000

1980

2000

1980

2000

84th percentile
0.5

0.5

Only
demand
non-interest
rate

Median

-0.5

0.8

0.6

0.4

0.2

16th percentile
5th percentile

Only
supply
transitory

-1
1960

1980

2000

-1
1960

0.5

-0.5

-1
1960

1980

2000

-1
1960

1980

2000

0
1960

1980

2000

0
1960

0.8

0.6

0.4

0.2

0
1960

1980

2000

0
1960

II: Fractions of draws for which
the correlation is positive

I: Correlations

Figure 16 United States, post-WWII period: correlation between real sales growth
and the change in real inventories conditional on individual shocks

Beginning
of the
financial
crisis

Medians of the
VAR coefficients

Medians of the offdiagonal elements of
the covariance matrix

Logarithms of the
medians of the volatilities
0.4

6
-7

0.2

-8

-9
2

Interwar
period

-10

-0.2

-11

-0.4

-12
-2

-0.6

-13
-0.8
-14

-4

-1
-15

-6

1930

1932

1934

1936

1938

1940

1942

-16

-1.2
1930

1932

1934

1936

1938

1940

1942

-9

1932

1934

1936

1938

1940

1942

0.6
0.5

-10

1.5

1930

0.4
-11

0.3

Post-WWII
period

-12

0.2

0.5

0.1

-13

0
-14
-0.5

-15

-1

-16

1960

1970

1980

1990

2000

2010

1960

-0.1
-0.2
-0.3
1970

1980

1990

2000

2010

-0.4
1960

1970

1980

Figure 17 United States: the evolution of the elements of the VAR

1990

2000

2010

Full text of Working Papers (Federal Reserve Bank of Richmond) : Sales, Inventories, and Real Interest Rates : A Century of Stylized Facts, Working Paper 12-02

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