HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
The Interest Rate, Learning, and Inventory
Investment
Louis
J.
Maccini
Bartholomew Moore
Huntley Schaller
Working Paper 03-04
January 2003
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E52-251
50 Mennorial Drive
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January 2003.
The
Interest Rate, Learning,
and Inventory Investment
Bartholomew Moore
Department of Economics
Fordham University
441 East Fordham Road
The Bronx, NY 10458
(718)817-3564
Maccini
(Corresponding author)
Department of Economics
Johns Hopkins University
Louis
J.
3400 N. Charles Street
21218
(410)516-7607
maccini@jhu.edu
Baltimore,
MD
bmoore(a)fordham.edu
Huntley Schaller
Department of Economics
Massachusetts Institute of Technology
50 Memorial Drive
02142-1347
Cambridge,
MA
(617)452-4368
schaller@mit.edu
[JEL Classification: E22. Keywords: Inventories, Interest Rates, Learning]
thank
We thank Heidi Portuondo for research assistance. For helpful comments we
our discussant at the 2002 AEA meetings, Serena Ng, and seminar
Ken West,
participants at Brandeis University
for Inventory Research. Schaller
his research program.
and the XII symposium of the International Society
acknowledges the financial support of the
SSHRC
for
ABSTRACT
We
derive parametric tests for the role of the interest rate in specifications based on the
firm's optimization problem.
evidence, finding
little
These Euler equation and decision rule
role for the interest rate.
We
tests
mirror earlier
present a simple and intuitively
appealing explanation, based on regime switching in the real interest rate and learning, of
why
tests
based on the stock adjustment model, the Euler equation, and decision rule -
of which emphasize short-run fluctuations in inventories and the
unUkely
respond
to
uncover a relationship.
much
A
-
Both simple and sophisticated
a highly significant long-run relationship
tests
respond to
confirm our predictions and
between inventories and the
interest rate.
formal model of our explanation yields a previously untested implication that
supported by the data.
are
analysis suggests that inventories will not
to short-run fluctuations in the interest rate, but they should
long-run movements.
show
Our
interest rate
all
is
1
I.
Introduction
In their survey of inventory research, Blinder and Maccini (1991) observe that an
important puzzle with empirical research on inventories
to
have
little
impact on inventory investment.
been conducted over the
survey by
decade.
last
Ramey and West
As an
is
But, very
that the real interest rate
little
seems
research on this issue has
indicator of the lack of work, the recent
(1999) barely mentions the effect of interest rates on
inventories.
The
interest rate
issue
is
a puzzle for
two reasons.
First, the
lack of an effect of the real
on inventory investment severs one of the conventional channels through
which monetary policy influences spending.
changing short-term interest
Monetary policy
implemented by
which standard theory predicts should influence
rates,
Second, the financial press
inventory investment spending.
is
is
replete with statements
by
business people asserting that higher interest rates induce firms to cut inventory holdings.
Although
it is
nominal, there
sometimes not clear whether the
is
under discussion
is
real or
nonetheless the perception of an inverse relationship between inventory
investment and interest
rates.
Earlier empirical
adjustment models.
relationship
interest rate
Yet, almost no evidence exists for such an effect.
work with
Among
the
inventories utilized flexible accelerator or stock
key issues investigated
between inventory investment and
interest rates.
in this
literature
was
the
See Akhtar (1983), Blinder
(1986a), Irvine (1981), Maccini and Rossana (1984), and Rossana (1990) for relatively
recent studies using this approach.' These studies typically assumed that "desired stocks"
depend on the current expected
real interest rate as well as current
expected sales and
Stock adjustment models were first used in empirical work with inventories by Lovell (1961). See Akhtar
(1983) for a survey of the relevant literature prior to the eighties.
Current expected real rates were related to actual rates through
expected input prices.
distributed lag relationships
and empirical work proceeded.
These
however,
studies,
generally failed to establish substantial and systematic evidence of a relationship between
the real interest rate and inventory investment, especially with finished goods inventories
in manufacturing.
Furthermore, the literature was subject to the criticism that
it
lacked a
basis in explicit optimization.
In the eighties, the linear-quadratic
rational expectations
quadratic
began
to
model of inventory behavior combined with
be applied to empirical work on inventories.
model of inventory behavior was developed by
Simon (1960), and was revived
(1982, 1986b)
who used
expectations, the
relationship
model
between the
the
is
in the
model
economics
in analytical
a very fruitful
real interest rate
The
work.
framework
Muth and
Holt, Modigliani,
literature in the eighties
When combined
for empirical work.
by Blinder
with rational
However, the
and inventory movements was not a key issue
under investigation in empirical research with the linear-quadratic model.
research focused on attempts to explain
sales,
why
why
Rather,
production seemed to fluctuate more than
which contradicts the production-smoothing motive
for holding inventories,
and
inventory stocks seemed to exhibit such persistence.
Two
approaches were employed in empirical research.
Euler equations using generalized methods of
this
linear-
moments
One approach
techniques.
estimated
Contributions using
approach include Durlauf and Maccini (1995), Eichenbaum (1989), Kashyap and
Wilcox (1993), Kollintzas (1995), Krane and Braun (1991), Ramey (1991), and West
(1986).
when
In this work,
it
proved
the discount factor,
which
difficult to estimate the structural
is
defined by the real interest
parameters of interest
rate, is
allowed to vary.
Hence, researchers invariably assumed that the discount factor
which of course eliminates by assumption any
is
a given,
known
effect of the real interest rate
value,
on inventory
investment.
Given the
finite
sample problems with generalized methods of moments applied
approach used in empirical work with the linear-quadratic
to Euler equations, another
model was
to estimate
to solve for the optimal choice
it
using
Moore and Schuh
maximum
(1995), and
of inventories, that
likelihood techniques^.
is,
for the decision rule,
and
See Blanchard (1983), Fuhrer,
Humphreys, Maccini and Schuh (2001). However, solving
for the decision rule requires an Euler equation that is linear in
its
variables.
Since Euler
equations are nonlinear in the discount rate, and therefore in the real interest rate,
researchers
again resorted to a constant and
known
discount rate for reasons of
tractability.
A
few
studies departed
interest rates to vary.
from the linear-quadratic framework and allowed
Miron and Zeldes (1988)
utilize
real
an approach with a Cobb-Douglas
production function that emphasizes cost shocks and seasonal fluctuations.
Kahn (1992)
and Bils and Kahn (2000) focus on a more rigorous treatment of the stockout avoidance
motive. In these studies,
when
a check
was made, no
of the model was found when the real interest
An
constant.^
exception
is
Ramey
(1989)
who
rate
effect
on the empirical implications
was allowed
to vary or
developed a model that
was held
treats inventory
stocks as factors of production, and found evidence of interest rate effects through
relevant imputed rental rates.
As
is
model.
well known, the decision rule for optimal inventories can be converted into a stock adjustment
A
key difference between this approach and the earlier stock adjustment principles is that the
now depends on expected future sales, input prices, and real interest rates as well as current
desired stock
expected values.
^
See Miron and Zeldes (1988) and Kahn (1992).
Given the lack of strong evidence of a relationship between the
and inventory movements, a number of authors conjectured
standard model
lend as
much
is
that
as they
it
assumes perfect
want
at
that the
real interest rate
problem with the
capital markets, so that firms
given interest
rates.
may borrow
or
Rather, they argued that capital market
imperfections arising from asymmetric information will impose finance constraints on the
firm's inventory decision. These constraints suggest that the cost of external finance to
the firm
is
inversely related to the firm's internal financial position, as measured
by
liquid
See Kashyap, Lamont and Stein (1994), Gertler and Gilchrist (1994),
assets or cash flow.
and Carpenter, Fazzari and Petersen (1994) for contributions
that such financial variables
to this approach.
They
find
do have an influence on inventory movements of small firms
but not of large firms. This leaves open the relationship between the real interest rate and
inventory
movements
for large firms
The purpose of
real interest rate
empirical
this
paper
and
is to
in the aggregate.
take a fresh look at the relationship between the
and inventory investment.
work on
the
transmission
and
This
We
obviously an important issue for
effectiveness
complements the empirical work underway with
monetary policy.
is
of monetary
policy,
and
interest rate rules as descriptions
of
begin by extending the typical approach taken with the linear-
quadratic model, specifically to obtain a specification in which the interest rate appears in
a separate term with
its
own
coefficient.
Essentially, this involves an appropriate linear
approximation of the Euler equation in the real interest
rate.
This enables us to solve for
the optimal level of inventories as a linear fiinction of the real interest rate and other
variables.
Using the linearized Euler equation as a
starting point,
we
are able to derive
specifications that allow us to parametrically estimate the effect of the interest rate on
We
inventories.
use two approaches
-
the linearized Euler equation and the firm's
decision rule for inventories (which can be derived from the linearized Euler equation).
We
undertake empirical work with monthly data on inventories for the nondurable
aggregate of U.S. manufacturing for the period 1959-1999.
existing puzzle: these specifications reveal
Why
rate
displays
We
suggest that the answer
transitory
variation
around
stability
show an
lies in the
highly
persistently negative real interest rates in the 1970s).
appear to enter regimes that exhibit
results reinforce the
significant effect of the interest rate.
don't the Euler equation and decision rule
on inventories?
which
no
The
effect of the real interest
behavior of the interest
persistent
mean
values
In other words, real interest rates
Regime changes
are infrequent.
In fact, careful econometric study has provided evidence that the real interest rate
If the
Markov regime switching [Garcia and Perron
mean
real interest rate is highly persistent, firms
be a persistent change
in the real interest rate.
may
- may
find
On
little
well
largely ignore short-
Under these
when
there
seems
conditions, econometric
procedures that focus on short-run fluctuations in inventories and the interest rate
as the older stock adjustment or the
is
1996].
run interest rate fluctuations, altering their typical inventory level only
to
(e.g.,
for extended periods with temporary
variation around a persistent level within each regime.
described by
rate,
newer Euler equation and decision
- such
rule specifications
evidence of a relationship.
the other hand, firms will adjust their inventory positions if they believe there
has been a change
in the
underlying interest rate regime. This suggests that estimation of
the long-run relationship between the inventories and the real interest rate
fruitful.
We
use two approaches. The
into interest rate regimes
-
high,
the interest rate
is
low.
simple and intuitive:
medium, and low - and
(detrended) inventories in each regime.
when
first is
We
The second approach
interest rate.
the interest rate are cointegrated.
we
failure to find
calculate the
is
more
mean
level
of
sophisticated.
Using the
derive the cointegrating relationship
Cointegration tests
show
that inventories
and
Estimates of the cointegrating vector uncover a strong
long-run effect of the interest rate on inventories.
view of the
divide our sample
find that inventories are significantly higher
linearized Euler equation as a starting point,
between inventories and the
we
may be
This finding
is
especially striking in
such a relationship using specifications that focus on short-run
fluctuations.
We proceed to formally model the implications of regime switching in the interest
rate.
Of course,
shift to a
new
it is
sometimes
difficult to distinguish
persistent regime.
To
capture this difficulty,
learn the unobservable regime fi^om observable
Under
between a transitory shock and a
we assume
movements
the assumption of regime switching and learning, the
that firms
must
in the real interest rate.
model of optimal inventory
choice yields a distinctive implication: inventories should be based on the firm's
assessment of the probability that the economy
is
currently in a given interest rate regime.
In particular, under the assumption of regime switching and learning, the probabilities of
being in either the high or low interest rate regime should replace the interest rate in the
cointegrating vector for inventories.
significant
evidence that the
probabilities.
We
test
this
implication and find statistically
long-run behavior of inventories
is
linked
to
these
The next section presents
the short-run relationship
new
tests for the role
rule.
the firm's optimization problem.
between inventories and the
Section
III
examines
real interest rate, introducing the
of the interest rate based on the Euler equation and the decision
Section IV analyzes and tests the long-run relationship between inventories and the
Section
real interest rate.
V
introduces the formal
model of regime switching and
learning, derives the distinctive implication of the model, calculates the probabilities,
tests the implication.
II.
VI presents robustness checks, and Section VII concludes.
Section
The Firm's Optimization Problem
We
its
and
begin by assuming a representative firm that minimizes the present value of
expected costs over an infinite horizon.
Real costs per period are assumed to be
quadratic and are defined as
C,
where
=^,Y, +|rr
,/ ,S
,a>
,<^
0.
(Ar,r- +|(iV,.,
+f
X^
,
real sales,
associate with real input prices.
discuss
cointegrating vector.)
exogenously. The
costs on output.
The
first
The
how
and Wt a
,
(We do not
are not directly relevant to the relationship
we
stockout costs, where aX,
is
which we
between inventories and the
is
will
interest rate. In
the modeling of unobservable cost shocks
level of real sales,
term
real cost shock,
include unobservable cost shocks, since they
X, and the
,
two terms capture production
last
(1)
Ct denotes real costs, Yt, real output, Nt, end-of-period real
finished goods inventories,
Section IV.B,
-aX,y
would
affect the
real cost shock, Wt, are given
costs.
The
third term
is
adjustment
inventory holding costs, which balance storage costs and
the target stock of inventories.
Let p, be a variable real discount factor, which
is
given by
/?,
=
,
denotes the real rate of
interest.
The
firm's optimization
problem
is
where
r,
+ r,
1
to
minimize the
present discounted value of expected costs,
^oZ
C,,
(2)
(=0
subject to the inventory accumulation equation,
which gives the change
in inventories as
the excess of production over sales,
7V,-iV,_,=r,-X,
The Euler equation
where from
factor
(3)
}^
=
A^,
that results
- A^,_, + X,
.
from
(3)
this
Observe
optimization problem
is
that (4) involves products
of the discount
and the choice variables and products of the discount factor and the forcing
variables. Linearizing these products around constant values,
which may be
interpreted as
stationary state values or sample means, yields a linearized Euler equation:
E\e[Y,-pY,^,YY[^Y,-ip^Y,^,^P^t.Y,^,)^^{W,-pW,^,)
(5)
where
t]
=J3(0Y + ^W) >
0,
c
= -rj3{9Y + ^W) < 0,
/3=-^,
\
and a bar above a
+r
variable denotes the stationary state value. This linearized Euler equation will serve as a
basic relationship that
III.
we will use
in the empirical
work.
The Short-Run Relationship between Inventories and the Real
Interest Rate
A. Euler Equation Estimation
A common
approach in empirical work on inventories
is
to
apply rational
expectations to eliminate unobservable variables and then use Generalized Methods of
Moments
We first investigate whether a short-
techniques to estimate the Euler equation.
run relationship between inventories and the real interest rate can be found using
this
approach.
Assume
that sales,
X,
general stochastic processes.
from
,
the cost shock, W,
,
and the
real interest rate,
r,
,
obey
Then, use rational expectations to eliminate expectations
(5) to get
e(Y,-'pY,^,) + y(^Y,-2^^Y,^,^-W-^Y^.^] + m,-pW,^,)
where
a:,'
is
a forecast error
Since not
all
and
Y, is
defined by
Y^
(6)
= N, - //,_, + X,
the structural parameters of the Euler equation are identified,
adopt the widely used normalization and
set
S
equal to
1.
We
we
estimate the Euler
10
equation by GMM,'* using a constant,
the variables are linearly detrended.^
GMM
by many
Yt-i, Wt-i,
Nn,
Xm
and tm as instruments.^ All of
Inventory Euler equations have been estimated by
Eichenbaum (1989),
authors, including Durlauf and Maccini (1995),
Kashyap and Wilcox (1993), Kollintzas (1995), Krane and Braun (1991), Ramey (1991),
and West (1986).
A
few papers have allowed
for interest rate variation, for example, Bils
and Kahn (2000), Miron and Zeldes (1988), Kahn (1992), and
of our knowledge, however,
this
paper
is
the
first to
Ramey
To
(1989).
provide parametric
the best
tests in
Euler
equations for possible effects of the interest rate on inventories.
Estimates of the Euler equation under the assumption of a constant interest rate
are presented in the first
equation relevant to these estimates
target inventory-sales ratio,
weeks of sales, which
is
is
A
column of Panel
«,
is
is
of Table
1
.
(The version of the Euler
very precisely estimated and
is
plausible. Further, the estimate of the slope
positive and significant, indicating rising marginal cost.
approximately four
of marginal
E,
,
the parameter associated with observable cost shocks,
theory predicts) but insignificantly different from zero.
parameter, y
but Y
"*
As
is
,
is
cost,
6
These estimates are consistent
with those found by estimating analogous Euler equations in the recent
Interestingly,
The
equation (4) with p^ set equal to a constant.)
is
literature.^
positive (as
The estimated adjustment
positive, a result that is consistent with the existence
of adjustment
cost
costs,
imprecisely estimated and not significantly different from zero.
discussed by West (1995), estimation by
case where they are
1(1), as
long as (in the
on pages 201-202.
'
The interest rate is included because
in the interest rate,
and
it is
it
GMM
is
latter case)
valid both in the case
where
sales are 1(0)
and
in the
they are cointegrated. See particularly the discussion
appears in the Euler equation specification that allows for variation
desirable to use a consistent set of instruments across specifications.
*
The
'
See, for example, Durlauf and Maccini (1995).
results are qualitatively similar if the Euler equation is estimated without detrending.
11
Estimates of the Euler equation under the assumption of a variable interest rate
are presented in the second
simple
test for the effect
of the
negative, but the coefficient
Even
variable
if
7
interest
generally,
is
column of Table
is
interest rate
1
,
Panel A. The
improve the
statistic
on
77
provides a
on inventories. The point estimate of
insignificantly different
from
not significantly different from zero,
rate
t
rj
is
zero.
it is
possible that allowing for a
could improve estimates of the other parameters and, more
fit
of the Euler equation. Informally, a comparison of the
first
and
A suggests that allowing for a variable interest rate makes some
second columns of Panel
quantitative difference in the estimates of the other parameters but
little
qualitative
A formal test procedure, which is based on a comparison of the overidentifying
difference.
restrictions
between the two models,
for the test
is
will tend to
straightforward. If a
be
is
described by
model
is
statistics is distributed
x^
^
(1987).
The
intuition
incorrectly specified, the J statistic for the
large; the difference in J statistics
whether the improvement in specification
Newey and West
between two models provides a
is statistically
significant.
The
model
test
of
difference in J
with degrees of freedom equal to the number of omitted
parameters, here equal to one.^
The Newey- West
test statistic is 2.752, so
it
is
not possible
to reject the constant interest rate restriction.
In the remaining panels of Table
1
,
we check
the robustness of the results to
changes in the specification of the model. In the empirical inventory
mixed evidence on
Panel
B
For the
the importance of adjustment costs
and observable cost shocks.^
presents Euler equation estimates from a specification that includes observable
test,
the
same weighting matrix should be used; we use
interest rate specification, since
'
literature, there is
See, e.g., the surveys
it is
the weighting matrix
from the variable
the "unrestricted" model.
by Blinder and Maccini (1991), Ramey and West (1999), and West (1995).
12
cost shocks but sets y to zero. Estimates of the other parameters {6
dramatically affected.
As
in the Panel
A
coefficient but is insignificantly different
,
^ and
,
results, the interest rate enters
from
zero.
a)
are not
with a negative
Also as in Panel A, the Newey-West
test fails to reject the constant interest rate specification.
C
Panel
sets
^ equal
presents estimates of a specification that allows for adjustment costs but
to zero (so observable cost
shocks do not enter). Panel
D presents
estimates
of a specification that excludes both adjustment costs and observable cost shocks.
neither case
is
In
there statistically significant evidence of a role for the interest rate.
Overall, the Euler equation results presented in this section
statistically significant relationship
consistent with
much
between inventories and the
earlier research,
which has typically found
show no evidence of a
interest rate.
This
is
that inventories are not
significantly related to the interest rate.
B. Decision Rule Estimation
An
alternative approach in empirical
decision rule for optimal inventories that
We
is
work with
inventories
is
to estimate the
implied by the firm's optimization problem.
next explore whether this approach can detect a short-run relationship between
inventories and the real interest rate.
equation, (5),
and X
2
may be
In the appendix,
we show
that the linearized Euler
written as a fourth-order expectational difference equation.
denote the stable roots of the relevant characteristic equation.
We
Let
show
/L
in the
appendix that the firm's decision rule for the optimal level of inventories can be expressed
as a function of current
given by
and expected
fiiture sales, cost
shocks, and the interest rate;
it
is
13
^'''^-
E,N,={X,+ l,)N,_-\^_N,_,+
-zRMr-iAr'
(^-^2)^
(V)
^,^,..
where
'
yP
y5
Assume
Then
'
that the firm carries out its production plans for time
the inventory accumulation equation, (3), implies that (A^,
which means
rp
'
yl5
so that
EX
=
Y,
-E^N^) = -{X^ -E^X^^
Define
in effect that inventories buffer sales shocks.
t,
w;"^
= -{X,- E^X^ ) as
the sales forecast error, then the relationship that defines the firm's actual inventory
position
is
M^
where again
'^,^. is
defined by
Assume now
that
\7"+l
E,^,^.+u:
(9)
(8).
sales,
real
input prices,
and the
real
interest rate
follow
independent AR(1) processes, and that the current information set of the firm includes
lagged values of sales, and current and lagged values of input prices and the interest
We show in the appendix that under these assumptions (9)
N,
with
gives
=T, + {X,+ l,)N,_-X,X,N,_,+T,X,_,+T,,W,+T^r,+
<
(10)
rate.
14
r^-0,
r„,
<0,and
T^ <0.
Note the expected sign of the coefficients
general,
ambiguous, as
it
we
An
we
coefficient
on
sales
in
is,
expect a positive coefficient, which
expect stockout avoidance to dominate.
increase in real input prices,
The
balances production smoothing and stockout avoidance.
Nonetheless, based on prior empirical work,
implies that
in (10).
which increases
It
follows from
F^ <
that
an
costs, should cause a decline in inventories.
increase in the real interest rate should induce the firm to reduce inventory holdings,
as implied
by F^
Under
identified
.
the assumption
the decision rule
specifically
<
is
made above
just identified
assume a
that the stochastic process for sales is
and can be estimated by OLS.
(In later sections,
unit root process, but the result that the decision rule
and can be estimated by
OLS
AR(1),
is
we
just
holds more generally for any AR(1) process.)
Inference can be carried out with standard distributions, regardless of whether sales are
1(0) or 1(1).'°
As noted
above, a number of authors have estimated the decision rule,
including Blanchard (1983), Fuhrer, Moore, and Schuh (1995), and Humphreys, Maccini,
and Schuh (2001)."
Based on previous studies
surprising if
'"
we found
it
would be mildly
was of
the theoretically
that allow for interest rate variation,
that the coefficient
on the
interest rate
See West (1995) for a detailed discussion of estimation and inference issues. Our estimation procedure
is also valid if there are unobservable cost shocks (including serially correlated shocks)
for the decision rule
unless the shocks are
1(1).
We consider the case of 1(0) observable cost shocks in this subsection and 1(1)
observable cost shocks in Section VI.
'
'
These studies have typically estimated
procedures, whereas in this paper
we
structural parameters via nonlinear
maximum likelihood
form parameters. However, since the
and since rj appears only in the
are in effect estimating reduced
relevant structural parameter connected to the real interest rate
is
reduced form parameter ^r the results for the role of the interest
>
estimating structural parameters using nonlinear procedures.
r]
,
rate are unlikely to
be improved by
15
On
predicted sign and statistically significant.
the other hand, to the best of our
knowledge, no one has previously reported estimates of the decision rule for inventories
that allow for a variable interest rate.
Prior studies that have allowed interest rate
variation have estimated Euler equations.
We
begin by considering the most general specification of the decision rule
(allowing for both adjustment costs and observable cost shocks) in Panel
Under
the assumption of a constant interest rate, the coefficients
the decision rule are significantly different
coefficient, indicating that the stockout
have a negative
Under
r),
we
from
zero.
on
all
A
of Table
the variables in
Further, sales has a positive
avoidance motive dominates, and real input prices
coefficient, consistent with the predictions
of the theory.
the assumption of a variable interest rate (specifically an
AR(1) process
see from equation (10) that the interest rate also enters the decision rule.
second column of Panel
positive, a result
which
A
is
shows
that the estimated coefficient
on the
for
The
interest rate is
contrary to the implications of the linear-quadratic
model of
inventories with a variable interest rate, although the coefficient on the interest rate
insignificantly different
essentially
no
effect
B
zero.
on the signs and
real input prices, or the
Panel
from
2.
is
Further, allowing for a variable interest rate has
statistical significance
of the coefficients of
sales,
lagged inventory stocks.
presents decision rule estimates from a specification that includes
observable cost shocks but excludes adjustment costs.
This changes the empirical
specification of the decision rule, leading to the omission of the second lag of inventories.
Panel
C presents
estimates of a specification that allows for adjustment costs but excludes
observable cost shocks. The specification in Panel
D
excludes both adjustment costs and
16
The estimated
observable costs shocks.
positive sign, although
different
from
zero.
on the
Moreover, the signs and
in the decision rule are little affected
is
never significantly
of the other variables
statistical significance
by whether or not
always has a
interest rate
important to note that the coefficient
is
it
coefficient
the interest rate
is
variable or
constant.'
It is
straightforward to summarize the results from the different specifications of
the decision rule; there
rate
is
no evidence of a
on inventories. Again,
between inventories and the
The Long-Run Relationship between Inventories and
Why
is
that
it
is
Figure
1
mean
2%
the figure illustrates, there are long periods
for
much of
real interest rate to a value
1980 and remains high (around
and for much of the 1990s, the
1960s
A
the 1960s.
show no
A possible clue
lies
plots the ex-post real interest rate over the
when
centered around a given mean. For example, the real interest rate
value just below
'"
the Real Interest Rate.
of the real interest rate on inventories?
in the behavior of the interest rate.
As
interest rate.
estimates of the Euler equation and decision rule
statistically significant effect
period 1961-2000.
of the interest
with most earlier research, which has failed
this is consistent
to find a significant relationship
IV.
statistically significant effect
is
the interest rate
centered around a
In the early 1970s, there
is
a shift in the
of about -2%. The real interest rate rises sharply around
5%
on average)
for
much of the
real interest rate returns to a
1980s. In the late 1980s
mean
value that
is
close to
level.
minor exception
is
that the coefficient
of sales
is
a bit
variable. Also, note that excluding observable cost shocks
sales, suggesting that
more
significant
seems
when
the interest rate
is
to reduce the statistical significance
excluding cost shocks creates an omitted variable bias problem.
of
its
17
What
implication does this behavior of the interest rate have for empirical
estimates of the effect of the interest rate on inventories?
highly persistent within an interest rate regime, firms
If the
may
interest rate is
largely ignore short-run
interest rate fluctuations, adapting average inventory levels only
persistent
mean
when
change in the opportunity cost of holding inventories. If
seems
to
be a
this is the case,
then
there
econometric procedures that focus on short-run fluctuations in inventories and the interest
rate
may
find
little
evidence of a relationship.
In this section,
interest rate.
test:
a
We
we
The
consider two tests of the relationship.
are inventories lower
more
focus on the long-run relationship between inventories and the
sophisticated
test,
on average during the high
first is
interest rate
a simple, intuitive
regime? The second
is
based on deriving and estimating the cointegrating vector for
inventories under the assumption of variable interest rates.
A. Tests of Means
Our
first test
of the long-run relationship between inventories and the interest
divides the 1961-2000 period into regimes characterized
Specifically,
we
by
different
mean
rate
interest rates.
classify the period 1972:02-1980:02 in the low-interest-rate regime,
1980:12-1986:04 in the high-interest-rate regime, and the remaining observations in the
medium-interest-rate regime.'^
The
rate
regime and lowest
inventories
^
results are presented in
This
is
is
about
8%
in
the
Table
Inventories are highest in the low-interest-
high-interest-rate
regime.
The
level
of detrended
higher in the low-interest-rate regime than in the high-interest-
an intuitive definition of the regimes;
Section V.
3.
we
introduce a foraml procedure for identifying regimes in
18
The
rate regime.
highly significant.
difference in
To
means between
the high- and low-interest-rate regimes
the best of our knowledge, this
for the long-run effect of the interest rate
is
is
the first time that this simple test
on inventories has been reported.
B. Derivation of the Cointegrating Vector
A more
sophisticated test of the long-run relationship between inventories and the
interest rate is to derive the cointegrating relationship
rate,
between inventories, the
interest
and any other relevant variables. The advantage of the cointegration approach over
the simple test of
means presented
earlier is that
it
accounts for the effect of variables
such as sales and observable cost shocks on the long-run level of inventories.
To
(5), in
derive the cointegrating vector,
such a
way
+ p5
as to put
Note
'"^
Y,
that
is
N- 'a- "<-«' X
H^
again given by
we
helpfiil to rewrite the basic
most of the variables
V
where
it is
in the
form of first differences
+ {\-/3)W, +
rir,+c
=0.
:
(11)
J
[For the derivation of (11) from (5), see the appendix.]
(3).
can express equation (11) as E, [Xt+i] - ^
See Kashyap and Wilcox (1993) and
Ramey and West
^^^ ^^^ appropriate definition of
(1999) for derivations of the cointegrating
relationship for inventories under the assumption that the interest rate
'^
Euler equation,
is
constant.
We are agnostic on the issue of unobserved cost shocks, which are not of primary interest in this paper.
a result, we do not include a term comparable to Ud in Hamilton (2002) or Ramey and West (1999) in
our model and thus do not address the issue of whether unobservable cost shocks are best modeled as 1(0)
As
or
1(1).
Hamilton (2002) argues
that, if
one were to include unobservable cost shocks under his preferred
assumptions, the same variables would appear in the cointegrating vector, but the coefficients would be
altered.
It is
possible to
show
that his
argument can be generalized
to
our model, which includes a time-
varying interest rate and observable cost shocks. Under assumptions similar to Hamilton's, the same
variables appear in our cointegrating vector.
remain the same.
The
coefficients are altered, but signs of the coefficients
19
^^^,
expectations
Rational
X,^2
=
Z,i.2
~ E,
root; in other
;jf,^2will
"will
{Zi+2}
words
b^
serially
X+2 =
that
X,
,
W, and
be 1(1) and the stationarity of Zt+i implies that inventories,
real interest rate will
1
r,
are 1(1).
^,^2
Then
sales, the cost shock,
is 1(0),
TV,
and the
135
H^
ADF
usual sense that the tests
show #,
tests
fail to reject
the above derivation,
p5
135
J
,
X,, W, and
r,
to
be
1(1) variables (in the
the null hypothesis of a unit root).
it
'^
follows that the cointegrating vector
is
the
same
regardless of whether adjustment costs are included in or excluded from the model.
see why, consider the
costs.
Note
will
,
useful to note that
From
Since
^/+2-
be cointegrated, with cointegrating vector
V
It is
error
uncorrected and therefore cannot have a unit
Suppose for the moment
also be 1(0).
expectation
the
that
Since £",{^,^2 1=0'
is 1(0).
^,^,
implies
first
To
term in parentheses in equation (11), which reflects adjustment
that all the elements in this
term enter in the form
AY
.
Thus, if
Y is 1(1), all
the elements in this term will be 1(0).
The parameters a
below equation
form of
(5), that
a regression, X,
their coefficients will
above.
An
,
,
is
S
,
E,
positive.
W^ and
r,^,
are
assumed
When
will
to
be positive, and
the cointegrating vector
have signs opposite to those shown
In other words, in the long run,
we
is
shown,
expressed in the
in the cointegrating vector
expect inventories to be inversely related to
between inventories and the forcing
form of a stock adjustment principle
defined by the "desired" or the "equilibrium" stock of
to express the decision rule for optimal inventories in the
which case the cointegrating relationship is
Such a procedure yields an identical cointegrating vector.
inventories.
is
we have
be on the right hand side of the equation, so
alternative procedure for deriving the cointegrating relationship
variables
in
7
,
20
and the
the cost shock
motive for holding
real interest rate, and, if the accelerator
inventories dominates the production smoothing motive,
we
expect inventories to be
positively related to sales.
C. Cointegration Tests and Estimates of the Cointegrating Vector
Johansen-Juselius tests of cointegration between inventories, sales, observable
cost shocks, and the interest rate are presented in Table 4 for levels, logs, and linear
detrending of the variables. The evidence
is
consistent with the theory: the tests reject the
null hypothesis of no cointegration.
We turn next to
is
DOLS
as described
estimation of the cointegrating vector.
by Stock and Watson (1993).
Our estimation procedure
In contrast to
SOLS
estimation of
cointegrating vectors,
DOLS
assumptions) in
samples. In addition. Stock and Watson (1993) find that
the
minimum
finite
corrects for biases that can arise (except under rather strong
DOLS
has
RMSE among a set of potential estimators of cointegrating vectors.'^
Table 5 presents estimates of the cointegrating vector. The interest rate enters the
cointegrating relationship with a negative sign, and the t-statistic
which
is
is
greater than five,
very strong evidence of a long-run relationship between inventories and the
interest rate.
This
is
a very striking result, especially
when compared with
the results
reported above indicating no evidence of a short-run relationship between inventories and
the interest rate.
'^
DOLS
essentially adds leads
and lags of the
first
differences of the right hand side variables to the
cointegrating regression to ensure that the error term
is
orthogonal to the right hand side variables. (For a
brief description, see, e.g., Hamilton (1994), p. 602-612.) In theory, the
Monte Carlo evidence
number of leads and
lags could be
of fixed investment) that
relatively high numbers of leads and lags are the most effective in reducing bias. Caballero (1994, p. 56)
finds that the bias is smallest when the number of leads and lags is 25 for a sample size of 120. We set the
number of leads and lags to 24. (Recall that we are using monthly data.)
infinite,
but this
is
impractical. There
is
(for the case
21
The point estimate of
detrended variables)
coefficient in Table 5 therefore
economy moves from
interest-rate
impHes a decrease
the high to the
interesting to
comparison of means
on
interest rate (based
in
low
compare these
is
about 6.8%.
The estimated
in inventories of about
results with the findings in
Table 3 shows that inventories are about
and observable cost shocks
in the high-interest-rate
Table
8%
The simple
3.
lower in the high-
~
as the cointegrating regression does
theoretically predicted sign
relationship that
many
The estimated
-0.8
and
11% lower
variables
enter
and a coefficient
the
cointegrating
regression
that is significantly different
with
from zero, a
empirical studies that focus on short-run fluctuations
elasticity
the
fail
of inventories with respect to observable cost shocks
to
is
-1.0.
The only previously
we
~
regime than the low-interest-rate regime.
cost
Interestingly,
of which
as the
interest rate.'^
leads to a qualitatively similar but slightly larger effect. Inventories are about
between
11%
regime than in the low-interest-rate regime. Controlling for other variables,
specifically sales
uncover.
linearly
-0.016 and the difference in the interest rate between the high-
is
regime and the low-interest-rate regime
interest-rate
It is
on the
the coefficient
reported cointegrating vectors for inventories which allow for a variable interest rate
are aware are in Rossana (1993),
which uses
a rather different approach.
Instead of including
the ex post real interest rate in the cointegrating vector, he enters the nominal interest rate and the inflation
rate as separate variables
and (using two-digit industry-level data)
are equal in magnitude and of opposite signs.
results he reports
real interest rate.
It is
tests the restriction that the coefficients
therefore not straightforward to determine from the
whether inventories have an economically or
statistically significant relationship
with the
22
V. Formally Modeling Regime Switching and Learning
In the previous section,
we
suggest an intuitively appealing explanation for the
The behavior of the
lack of a relationship between inventories and the real interest rate.
real interest rate is strongly suggestive
persistent
may
mean
of regime
Because mean
interest rates.
shifts,
with transitory variation around
interest rates are highly persistent, firms
largely ignore short-run variation in the interest rate.
changes in interest rate when there has been a regime
two
explanation,
tests
Consistent with this
shift.
confirm a highly significant long-run relationship between
inventories and interest rate. In this section,
modeling regime
Instead, firms respond to
we go
shifts in the real interest rate
a step further.
and then show how
We begin by formally
this affects inventory
behavior. This leads to a distinctive implication of regime switching and learning for the
long-run behavior of inventories. In subsection D,
we
test this implication.
A. Regime Switching and Learning
Regime switches can be formally modeled by assuming
that the real interest rate
follows
r,
where s,~
i.i.d.
=
N(0,1) and
St
econometricians.
the interest rate regime (with the
In particular, Garcia
in the U.S. is well described
that
is
Regime-switching in the real
"state").
assume
(12)
rs,+^s,-^,
e
that St
when
St
=
1
{1, 2, 3}
by a
interest rate has
follows a
the real interest rate
is
that the real interest rate
Markov switching model.
Markov switching
for
been formally explored by
and Perron (1996) show
three-state
mnemonic S
process'
.
Let
in the low-interest-rate regime,
We
/^
<
when
therefore
Tj
<
St
=
For a comprehensive discussion of Markov switching processes, see Hamilton (1994, Chapter 22).
rj
,
so
2 the
23
real interest rate is in the
moderate
interest rate regime,
rate is in the high-interest-rate regime. St
and
s,
and when
assumed
are
the transition probabilities governing the evolution of St
Collecting these probabilities into a matrix
Interest rate
the
economy has
Pii
P31'
P22
Pn
Pn
Pn
^33.
regimes are not directly observable.
low
=
3 the real interest
be independent. Denote
p.j
=
Prob(iS',
=j
\
= /).
5',_,
we have
Pu
=
P Pn
just entered the
by
to
St
No
interest rate regime.
one announces to firms
Instead, firms
that
must make
inferences about the interest rate regime from their observations of the interest rate.
In
other words, firms learn about the interest rate regime.
To be
precise,
we assume
that the firm
Markov switching process but does not know
therefore infer St
from observed
assessment of the true state by
;r
=
Prob(5, =1|Q,)'
;r.
?Toh(S,=2\n,)
^.
Prob(5,
is
set,
the firm's estimate at date
To understand
t
t-1
denote the firm's current probability
=3|Q,)
Q,
,
includes the current and past values of
of the probability that the real interest rate
the learning process, consider
the current real interest rate to develop
end of period
We
the firm uses
;?z-,_,
The firm must
is.
^„
where the firm's information
TTi,
That
the structure and parameters of the
the true interest rate regime.
interest rates.
Ut.
knows
its
how
the firm uses
probability assessment,
;t,
.
its
is
in
Here,
rt.
regime
i.
observation of
Beginning
together with the transition probabilities in
P
to
at
the
form
24
beliefs about the period
evaluates
;r.,|,^,
interest rate state prior to observing
t
= Prob(S, =
/
1
Q,_|
)
for
i
=
r,
That
.
is
the firm
3 using
1, 2,
71,
\t\i-\
Pk^t-\
n.it\t-\
(13)
n
3/|/-l
Once
the firm enters period
and observes
t
r,
uses the prior probabilities from (13)
it
,
together with the conditional probability densities,
/(r. \S,
-1
=/) =
exp
c..
to update
n.
;r,
for
r(r,-i;-r
r
=
1, 2, 3,
(14)
2o-.
according to Bayes' rule. Specifically,
~—
7Z,,., ,
lt\t—\
-
V 2;r
5',
Jfix,
\ I
•
I
\
I
= i)J
.
f.
-
— *
tori =1,2,3.
(15)
Y.^,^,-^f(J^S,=i)
7=1
Thus, the firm uses Bayes' rule and
its
observations of the real interest rate to learn about
the underlying interest rate regime.
Given
;r,
,
may then be computed
real interest rate,
where
/^
=
r/
p^^x^
the right
= [r,
,
r,
the expected real interest rate,
,
r, ]
,
+ P22h ^ P3ih
+ /j.^r, + p^^x^
Since
;r„ +;;r,, +;r3,
hand side of (16)
E,r„,
as
,r,
y^=
p,
which may be interpreted as an ex ante
y^
,
=
p,^x^
=1 by
+ p,,x, + p,^x^ and
definition,
,
we can
eliminate
7^2,
from
to obtain
= (r, - ^2 ) ^1, + ih - Yi
^3,
+ Yi
(17)
25
Now,
to isolate the expected real interest rate in the linearized Euler equation,
partition (5) so that
E,{e[Y,- pY,^,) + y[^Y,-2/5^Y,^, + ft'-^Y,^,)+^{w,- pw,^,)
(18)
Then, substitute (17) into (18) to get
E,[e{Y,-'pY,^,Yr[^Y,-2p^Y,^, + 'p'^Y,^,)+aw,-'pw,^,)
(19)
+5p{N, - aX,^, )} + r]{y, - y, );z-„ +
To summarize
this sub-section,
we have
r]{y.,
period
t.
In the
i,
().
introduced a formal model of regime
switching and learning. The key variable in the model
probability of being in interest rate regime
- y, )n^, +r]y,+c =
is
tt.,
,
the firm's assessment of the
conditional on the firm's information set in
model of optimal inventory choice under
the assumption of regime
switching and learning, these probabilities replace the interest rate in the Euler equation.
In the next subsection,
we
will
show
that this leads to a distinctive implication
of regime
switching and learning and then use cointegration techniques to test this implication.
B. Derivation of the Distinctive Implication of Regime Switching
We
can
now
and Learning
follow the same approach as in the Section IV.B in order to derive
the long-run implication of regime switching and learning.
(19) so that most of the variables are in
first
differences.
Begin by re-writing equation
26
E,{r(^Y,-l|3^Y,,,+p'^Y,,,) -
+
m^,.^+^,J
oii--p)-
',_
J3S N.
f3S
V
+
Suppose now
X,,W^,
that
^(l-y9)
- a-
,
Again, note
W,
,
that,
,
tt^,
when
and
n^,
will
have signs opposite
We
earlier (just
20
Then
will be 1(1)
TV,
Vih-Yi)
and inventories,
ViYi-Yi)
/3S
fiS
expressed in the form of a
is
be on the right-hand-side of the equation, so their
to those
below equation
shown
in the cointegrating vector above.
(5)) that
complicated fiinctions of the elements of P and
unambiguously for
(20)
+c =
TlYi
the cointegrating vector
coefficients will
have shown
+ (l->^)^^}
X,
1(1).
ps
/55
regression, X^
- /3^AW,,
and the probabilities will be cointegrated with cointegrating vector
sales, the cost shock,
1
and n^ are
n^^
j3SaAX,,,
J
+ 7(^3-72)^3/+
7(ri -72)^1,
-
+ 0AN,
Tv,
77
> 0.
so
it
{y^-Y-y) ^iid
-^-jjare
not possible to sign them
is
mathematically feasible values of P and
all
(7-3
ry.
They
can, however, be
signed for the empirically relevant values. Using empirical estimates of the elements of P
and
rv,^'
we
coefficient
obtain
on
;r„
will
{7\-Y2)<^ and {Y2,-y2)>^^
so the
model
be positive and the coefficient on
tt^ will
be negative.
predicts that the
This accords with our intuition of how the probabilities should affect inventories.
If,
'
for example, there
Since
1(1).
7ti,
and
713,
is
an increase in
;t„
,
the firm believes that the
have a restricted range, one might wonder whether
We note two points.
First, in careful
such as the nominal interest
rate, are
is
better to
is
model them
entering
as 1(0) or
applied econometric research, variables with restricted ranges,
modeled
as 1(1) variables
when
Stock and Watson ( 1993) and Caballero( 1994).) Second, unit root
See the next sub-section.
it
economy
they are highly persistent. (See,
tests indicate that
ti,,
and
113,
e.g..
are 1(1).
27
This will lower the expected opportunity cost of
a persistent low-interest-rate regime.
holding inventories and should therefore lead to an increase in Nt.
cointegrating vector,
we
can see
lead to an increase in Nt (since
regression).
Similarly, since
firm believes the
economy
is
With
this effect.
;t„
//(^^j
will
-;k2)
rjiy^
-;/,)<
,
Looking
an increase in
at the
^r,,
will
be on the right hand side of the cointegrating
>
,
an increase in
ti^^
,
which indicates
that the
entering a persistent high-interest-rate regime, will lead to a
decrease in Nt.
Thus, the distinctive implication of regime
inventories will be cointegrated with the probabilities
on
TT^^
C.
will
be positive and the coefficient on
Calculating the Probabilities
;z"„
and
ti:^,
will
switching and learning
;r„
and
n^^
and
is
that the coefficient
be negative.
k^^
In order to test the distinctive implication of regime switching and learning,
must construct the
probabilities
;r„
and
n^^
.
we
This can be done using the techniques
described in Hamilton (1989 and 1994, Chapter 22).
parameters of a three-state
that
Markov switching process
We
begin by estimating the
for the real interest rate over our
sample period. Our estimates of the elements of the transition probability matrix are
P=
=0.98
;72,
=0.01
^3,
p,2
= 0.02
P22
= 0-98
P32
;7,3=0.00
Our
=0.00'
p,,
estimates of
oi, 02,
;733=0.96
;?23=0-01
ri, X2,
and
= 0.04
xt,
(annualized) are -1.71, 1.61, and 5.15, and our estimates of
and 03 are 1.90, 0.80 and
1.96, respectively.
28
Two
features of the behavior of the real interest rate stand out
First, since
estimates.
highly persistent.
pn,
P22,
and P33 are
regime
regime next period.
this period, there is a
regime next period.
infrequently.
is
a
98%
Similarly, if the
96%
probability that
economy
probability that
it
is
in the
will be in the low-
is
in the high-interest-rate
be
in the high-interest-rate
it
will anticipate that the current regime will persist for
time.
regimes
.7
will
it
economy
Furthermore, once the firm comes to believe that the economy has entered
Second, note that the difference between the
1
that, if the
This suggests that changes in the interest-rate regime will occur
a particular interest rate regime,
some
close to one, the interest rate regimes are
For example, these estimates indicate
low-interest-rate regime this period, there
interest-rate
all
from these
is
large relative to the standard deviations.
mean
of any two
interest rates
For example,
r2
-
ri
=
3.3,
which
is
times as large as the standard deviation of the white noise shock in regime one. This
suggests that, within a given regime, white noise shocks that are sufficiently large to be
mistaken for a regime change will not be common.
Figure 2 plots the behavior of tiu,
njt,
and
7131
as obtained
by applying
equations (13), (14), and (15) to the real interest rate data in our sample,
required for subsequent
illustrate
;r,,
tests,
since the sirai of the three probabilities
in Figure 2 for completeness.) This figure confirms that,
the perspective of the
interest rate consists
Markov switching model, most of the
is
the
filter in
is
not
but
we
(tt^,
1,
when viewed
fi^om
short-run variation in the real
of temporary fluctuations around the mean interest rate for the
current regime. For the most part, the probability of being in a given interest rate regime
29
is
or
close to
the
mean
1
.
Only occasionally does the Markov switching model
identify shifts in
real interest rate.
D. Tests of the Distinctive Implication of Regime Switching and Learning
In Section V.B,
and testable
we show
implication:
probabilities that the
that
regime switching and learning have a distinctive
inventories
economy
in the
is
be
will
with
cointegrated
low and high-interest
nn and
linear detrending of the variables.
7i3t
sales,
observable cost
are presented in Table 6 for levels, logs,
The evidence
is
the
rate regime, respectively.
Johansen-Juselius tests of cointegration between inventories,
shocks, and the probabilities
and n^,
Tin
and
consistent with regime switching and
learning: the tests reject the null hypothesis of no cointegration.
Table 7 presents estimates of the coefficients in the cointegrating vector between
inventories, sales, cost shocks,
and the probabilities
switching and learning, the coefficient on
that
tt^^
is
tt^^
and
tt^^
.
Consistent with regime-
negative and highly significant, implying
an increase in the probability of the high-interest-rate regime reduces inventories in
the long run.
The point estimate of the
rate
from 1.6%
(the
mean
point of reference) to
coefficient on
implies that an increase in the interest
interest rate in the medium-interest-rate regime,
5.1%
(the
reduces inventories by about 7%.
Although
^Tj,
mean
The
interest rate in the high-interest-rate
coefficient
on
7^^,
is
less precisely estimated, the point estimate implies a
moving from
in
Table
5, a
is
the
regime)
positive as predicted.
change
the low-interest-rate to the middle-interest-rate regime, that
change implied by the estimates
which
in inventories,
is
similar to the
decrease in inventories of about 5%. The
30
estimated cumulative effect of a
interest-rate
regime
is
move from
the low-interest-rate regime to the high-
a decrease in inventories of about
1
2%.
VI. Robustness Checks
A. Adjustment Costs and Observable Cost Shocks
In Section III
we examine
the robustness of the Euler equation
results to the inclusion or exclusion
rule
of adjustment costs from the model. As discussed in
Section IV.A, however, adjustment costs
Intuitively, this is
and decision
make no
difference to the cointegrating vector.
because adjustment costs affect dynamics in the short run but do not
affect the long-run relationship.
In the inventory literature,
sometimes that they are
1(1).
it is
sometimes assumed
(See, e.g.,
Rossana (1993, 1998), and West (1995).)
cost shocks have a unit root,
1(0),
we
that cost
shocks are 1(0) and
Hamilton (2002), Ramey and West (1999),
Although
ADF
tests suggest that
observable
consider both possibilities. If observable cost shocks are
we
carry
As Panel
A of
then inventories, sales, and the interest rate will be cointegrated. In Table 8
out cointegration tests for the case where observable cost shocks are
Table 8 reports,
interest
rate.
probabilities
is
we
find evidence of cointegration
The evidence of
In Table 9,
we
between inventories,
cointegration between
even stronger, as shown
in
1(0).
inventories,
sales,
sales,
and the
and the
Panel B.
estimate the cointegrating vector under the assumption that
observable cost shocks are
1(0).
The
predicts, but insignificantly different
coefficient
on the
interest rate is negative, as theory
from zero, as shown
in Panel A.
Panel
B
presents
estimates of the cointegrating vector for the case of regime switching and learning.
The
31
point estimate of the coefficient on
Table
7.
As
We
literature
in
Table
71^,
is
somewhat
greater than the estimate reported in
7, the t-statistics on^Tj, are large.
present the results in Table 9 for completeness, because the inventory
has modeled cost shocks as both 1(0) and 1(1) processes, but the results in Table
ADF
7 are preferred for two reasons. First, as noted above,
cost shocks are 1(1).
very large
observable
Second, the coefficients on observable cost shocks in Table 7 have
t-statistics, raising the potential
of omitted variable bias
shocks are excluded fi-om the cointegrating regression.
likely to
tests suggest that
Thus the
observable cost
if
results in
Table 9 are
be biased due to a misspecification of the cointegrating relationship, specifically
the omission of observable cost shocks.
B. Serial Correlation of the Interest Rate within a
It is
Regime
possible to allow for serial correlation of the interest rate within a regime
through suitable modification of the Markov switching model for the interest
Allowing for
serial
implications. First,
it
correlation of the interest rate within a regime has
yields different time series of
tt^^
resulting probabilities are fairly similar regardless of
correlation).
to a
Second,
it
more complicated
and
two main
k^, (although, in practice, the
whether or not
changes the expectation of the interest rate
set
rate.^^
we
(i.e.,
allow for serial
£',[r,^,]),
leading
of interest-rate-related variables in the Euler equation.
Table 10 presents estimates of the cointegrating vector that allow for
correlation of the interest rate within a regime.
^^
Again,
we
^^
serial
focus on the specification
See Garcia and Perron (1996) for details. Following Garcia and Perron (1996), we focus on the case
where the stochastic component within a regime follows an AR(2) process.
Details of the derivation are available from the authors.
32
including observable cost shocks. (The inclusion or exclusion of adjustment costs
no difference
Table
7,
the point estimate of the coefficient on
magnitude of the estimated coefficient on
Table
7.
As
As
to the cointegrating vector, as noted above.)
in
Table
7, the t-statistics
is
tz^,
is
tTt^^
in the estimates reported in
negative.
somewhat
associated with
tt^^
makes
In fact, the absolute
larger in Table 10 than in
are large (in this case, around
6).
VII. Conclusion
We present a variety of new evidence on the relationship between inventories and
the real interest rate.
effect
of the interest
First,
we
develop a tractable
rate in the Euler equation
findings parallel previous
work based on
way
to parametrically estimate the
and decision
rule for inventories.
the older stock adjustment model: there
is little
evidence of a significant role for the interest rate from the Euler equation or decision
Second,
up
in
we propose
is
own
that the real interest rate
with transitory variation within a regime.
or those of previous researchers.
shows evidence of highly
relationship
Table
3,
regime.
If our explanation
is
Moreover,
interest rate
and find
we
interest rate.
when
the
to
interest rate
in response to short-run variation in
correct, there should
between inventories and the
inventories tend to be lower
much
The key
persistent regimes,
Because of the persistence of
regimes, firms do not adjust their inventories
the interest rate.
rule.
an explanation for the fact that the interest rate does not show
econometric estimates, either our
our explanation
Our
be evidence of a long-run
In fact, there
economy
is
is.
As shown
in
in the high-interest-rate
derive the cointegrating vector between inventories and the
that: 1) inventories
and the
real interest rate (together
with sales and
33
observable cost shocks) are cointegrated; and 2) the estimated coefficient on the real
interest rate in the cointegrating regression is statistically
These
results are consistent
and economically
with our explanation, but
we go
significant.
We
a step further.
formally model optimal inventory choice under the assumption of regime switching and
learning, derive a distinctive implication
from the model, and
Briefly, the implication is that inventories should
implication.
test this
be cointegrated with the probabilities of
being in the high- or low-interest-rate regime. The data confirm this implication.
We
view our
results,
which emphasize regime switching and
learning,
as
complimentary to research that focuses on finance constraints. Finance constraints will
also lead to a relationship
between inventories and the
interest rate that is stronger in the
long run. If finance constraints are important, in the short run inventories
strongly influenced
rate)
and by the
by
financial market conditions (that are not captured
availability
of internal
finance.^"^
may be more
by the
For example, in 2001 and parts of
2002, the interest rate was not high, but financial market conditions
made
it
firms to raise fimds, and the economic downturn squeezed internal funds for
In circumstances like these, the
cost of fiinds
may be
be weak. In the long run, even
affect inventories
in the presence
difficult for
many
firms.
greater than the market
and the relationship between inventories and the observed
interest rate,
still
shadow
interest
of finance constraints, the
interest rate
may
interest rate will
because long-lasting periods of low or high interest rates will span
different short-term financial
market conditions and, when finance constraints do not
bind, firms will adjust inventories to the prevailing interest rate.
^''
As noted
above, Kashyap, Lamont, and Stein (1994), Gertler and Gilchrist (1994), and Carpenter,
Fazzari, and Petersen
least for
some
firms.
( 1
994) find evidence that finance constraints are important for inventory behavior,
at
34
The paper can be extended
in at least
two
One
directions.
is
to test the
the two-digit industries of the nondurables sector of total manufacturing.
to
The
model on
objective
is
see whether certain two-digit industries exhibit a stronger long-run relationship
between inventories and the
interest rate than other two-digit industries.
more
is
substantial extension
relationship
and
to include the durable
is
a long-run
goods sector as well as the nondurable goods
Humphreys, Maccini, and Schuh (2001)
emphasize the importance of such an extension.
inventories are larger and
that durable
whether there
between input inventories (materials and work-in-process inventories) and
the interest rate
sector.
to investigate empirically
The second and
more
establish several stylized facts that
Specifically, they
that input
volatile than output inventories in manufacturing,
goods input inventories are larger and more
input inventories.
show
Such an extension requires a
volatile than
nondurable goods
substantial extension of the
include input inventories as well as output inventories, but
it
and
model
to
would permit an empirical
analysis of the sensitivity of the highly volatile level of input inventories to the interest
rate.
35
REFERENCES
Akhtar, M.A. (1983), "Effects of Interest Rates and Inflation on Aggregate Inventory
Investment in the United States", American Economic Review 73, 319-28.
Bils,
M. and J. Kahn,
"What Inventory Behavior Tells Us about Business
Economic Review, 90, 458-481
(2000),
Cycles", American
Blanchard, O.J. (1983), "The Production and Inventory Behavior of the American
Automobile Industry", Journal of Political Economy, 91, 365-400.
Blinder, A. S. (1982), "Inventories and Sticky Prices:
More on
Macroeconomics", American Economic Review, 72,
Blinder, Alan S. (1986
a),
"More on
3,
the Microfoundations of
334-49.
the Speed of Adjustment in Inventory Models",
Journal of Money, Credit, and Banking,
18, 3, 355, 365.
"Can the Production Smoothing Model of Inventory Behavior
Be Saved?", Quarterly Journal of Economics, 101,3,431,53.
Blinder, Alan S. (1986b),
A
Blinder, A.S. and L.J. Maccini (1991), "Taking Stock:
Critical
Assessment of Recent
Research on Inventories", Journal of Economic Perspectives, 5,73-96.
Caballero R.J., (1994), "Small Sample Bias and Adjustment Costs," Review of
Economics and Statistics 76, 52-58.
,
Carpenter, R. E., S.
M.
Fazzari, and B.C. Petersen (1994) "Inventory Investment,
Internal-finance Fluctuations, and the Business Cycle", Brookings Papers on
Economic Activity,
2,
75-138.
Durlauf, S.N. and L.J. Maccini (1995), "Measuring Noise in Inventory Models", Journal
of Monetary Economics 36, 65-89.
,
Eichenbaum, M.S. (1989), "Some Empirical Evidence on the Production Level and
Production Cost Smoothing Models of Inventory Investment", American
Economic Review, 79, 853-64.
Fuhrer,
J.C.,
G.R.
Inventory
Moore and
Model:
S.
Schuh (1995), "Estimating the Linear-quadratic
Maximum
Likelihood
versus
Moments", Journal of Monetary Economics 35,
Garcia, R. and P. Perron (1996),
Shifts",
Gertler,
"An
1
Generalized
Method
of
15-57.
Analysis of the Real Interest Rate under Regime
Review of Economics and Statistics, 78,
1 1
1-125
M. and C. Gilchrist (1994), "Monetary Policy, Business Cycles, and the Behavior
of Small Manufacturing Firms", Quarterly Journal of Economics 109(2), 309-40.
36
Hamilton,
J.
D. (2002),
"On
the Interpretation of Cointegration in the Linear-Quadratic
Inventory Model", Journal of Economic Dynamics and Control, 26, 2037-49.
Hamilton,
J.
Time
Hamilton,
Holt,
J.
C.C,
D. (1989),
"A New Approach
to the
Economic Analysis of Nonstationary
Series and the Business Cycle", Econometrica, 57, 357-84.
D. (1994), Time Series Analysis, (Princeton University Press).
Muth and H.A. Simon (1960), Planning, Production,
Work
Force,
(Prentice Hall, Englewood Cliffs, NJ).
and
F. Modigliani, J.F.
Inventories
Humphreys, B.
R., L. J. Maccini and S. Schuh (2001), "Input and Output Inventories",
Journal ofMonetary Economics, 47, 347-375.
Owen, (1981) Retail Inventory Investment and
Economic Review, 70, 4, 633-48.
Irvine, F.
the Cost of Capital,
American
Kahn, J.A. (1992), "Why Is Production More Volatile Than Sales? Theory and Evidence
on the Stockout-Avoidance Motive for Inventory-Holding", Quarterly Journal
ofEconomics, 107,481-510.
Kashyap, Anil K. and David W. Wilcox, (1990) "Production and Inventory Control at the
General Motors Corporation in the 1920s and 1930s", American Economic
iJevzm
83, 383-401.
Kashyap, Anil K., O. Lamont, and J. Stein (1994), "Credit Conditions and the Cyclical
Behavior of Inventories", Quarterly Journal ofEconomics, 109(3), 565-92.
KoUintzas, T. (1995), "A Generalized Variance Bounds Test, with an Application to the
Holt et al.Inventory Model", Journal of Economic Dynamics and Control, 19,
59-90.
Krane, Spencer and Steven Braun (1991), "Production Smoothing Evidence from
Physical Product Data", Journal of Political Economy, 99, 558-81.
Lovell, Michael C. (1961), "Manufacturer's Inventories, Sales Expectations, and the
Acceleration Principle", Econometrica, 29,
3,
293-3 14.
Maccini, L.J. and R.J. Rossana (1984), "Joint Production, Quasi-Fixed Factors of
Production, and Investment in Finished Goods Inventories", Journal of Money,
Credit,
and Banking,
16,
218-236.
Miron, J.A. and S.P. Zeldes (1988), "Seasonality, Cost Shocks and the Production
Smoothing Model of Inventories", Econometrica, 56, 877-908.
37
(1987), "A Simple, Positive Definite, Heteroscdasticity and
Autocorrelation Consistent Covariance Matrix", Econometrica, 55, 703-8.
Newey, W. and K. D. West
Ramey, V.A. and K.D. West (1999), "Inventories", in J.B. Taylor and M. Woodford
(eds.), Handbook of Macroeconomics, (Amsterdam: North-Holland).
Ramey, Valerie A.
(1989), "Inventories as Factors of Production and
Fluctuations",
Ramey, Valerie A.
American Economic Review,
(1991),
"Nonconvex Costs and
Economic
79, 3, 338-54.
the Behavior of Inventories", Journal
of Political Economy, 99, 306-34.
Rossana, R.J. (1990), "Interrelated Demands for Buffer Stocks and Productive Inputs:
Estimation for Two-digit Manufacturing Industries," Review of Economics and
Statistics, 72, 19-29.
Rossana, R.J. (1993), "The long-run Implications of the Production Smoothing Model of
Inventories: an Empirical Test", Journal of Applied Econometrics, 8, 295-306.
Rossana, R.J. (1998), "Structural Instability and the Production Smoothing Model of
Inventories", Journal of Business and Economic Statistics, 16, 206-215.
M.W. Watson (1993), "A Simple Estimator of Cointegrating Vectors in
Higher Order Integrated Systems", Econometrica, 61, 783-820.
Stock, J.H., and
West, K.D. (1986), "A Variance Bound Test of the Linear-Quadratic Inventory Model",
Journal of Political Economy, 94, 374-401.
West, K.D. (1995), "Inventory Models", in M. Peseran and M. Wickens, eds.. Handbook
ofApplied Econometrics, Volume I (Basil BlackM^ell, Oxford), 188-220.
38
APPENDIX A
Derivation of Equation
(7).
Using
can be written
(3) in (5) the linearized Euler equation
as the following fourth- order difference equation in Nt:
E\e[N,-N,_,+X,-p{N,^,-N,+X,^,)yy[N,-N,_,-{N,_,-N,_,) + X,-X,_,
-2^(iV,,,-iV,-(iV,-iV,_,) +
Rearranging
X,,-X,) + ^^(7V,,,-7V,,,-(7V,,,-7V,) + X,,,-X,,,)]
we have
E\f{L)N,^2\ = E,^,
(A.l)
where
y/3L
W
V
J
^^.
-^['M^-'P)Y'-jL'
and
—^(W,-pW,^,)-^r,^i —^-
+ ^X,,
Let
X .,\=
1,2,3,4
,
denote the roots of the fourth-order polynomial on the left-hand side
U
of (A.l). Order these roots as
X
— and A
,= ^=
pX,
,=
(8)
^—
PX^
,
with
U
I
<
,1
,
I
,
II'
UJ
U
I
,
I
21
Solve the unstable roots forward to obtain
<
<
U
^
^
.
jl
<
I/I4I.
It
follows that
Suppose
further
vv
that
U
I
,
I
,
11'
UJ<
I
21
1
39
PKK
Note
that
X
j
and
are either real or
/I ,
7T. \V+'
/77,
complex conjugates, so
\^'+'
that
pyv
/^i
+ /I2
(7)
^'^d A^A^ are
real.
Derivation of (10). To resolve the forward sum on the right-hand side of (9) note that
assume
X, =
that sales
ju^
and input prices follow AR(1) processes:
+ p^Xj_^ + £^, where
,
s^,
~
id.d. (0, crl )
W,=/u^ + pJV,_, + s^, where f„, ~
,
We allow for the special case of p^
1 .)
The terms involving
J3A,Z.
where
Note
a,
E,X,^j.^
It
i.i.d. (0,
crl)
and
.
= p^^=l.
X on the right-hand side of (9) can be written as
^+1
/TT^ \>+'
x{Mr-(M)
-X,^2+j
+ '^\^t+]+j ~ '^O^t+j +'^^/-l+y
=-L[^ + 7(2 + /?)]and ao=-L^^0 + r(l + 2j3)-aSj3^.
that, for j
=
0,
1
,
2,
.
.
.
=mA^ + pJ + pIE,
,
E,X,,.
X,_,,,,
= /i, + p^E^
X,,,^.
and E,X,,,,j =
ju^(l
+ p^ +pI) + pIE,
therefore follows that
'^,+2+j +^\^i+\+j
[-(l
~^O^i+j +-^^,-I+;
+ A+^.') + «lO + P.x)-«o]-".x +
-p^+a^p;-a^p^ +
E,X,_^^j
Fand
thus,
we
X,_,
(A.2)
40
\
\7+l
"^r+2+j +«1^,+1+; -^O^t^j
+^X,-Uj
_ fiX[-{\ + p^+ fr) + a,{\ + p^)-a^~\
/"r
\-/SX,
>0
+/3X,
-P.+«l/^.
(A.3)
-«0A+^2 S(y&A,.y^<^<-W
J^')j;=0
For the AR(1) process governing Xt ,the forward sum in (A.3)
E,X,_,,^
Y,(P^)'
/?^,A
=
1
-+-
(l-/?A,)(l-^A,,9j
j=0
is
-X.
(\-/3X,p^)
This in (A.3) gives
f
YiP^T
-X,^,,.+a,X,^,,j-a,X,^^+jX,_,^j
E,
c{p,,X.)^^+p2.. -P.x+«iP;-«oP.+
V
F
(l-y^/l
,/>,)-'
(A.4)
X,_,
where
c{p,,^) =
pX,[-{\ + p^+ pl) +
a,{\
+ p^)-a,]
{l3X;)'[-pl+a,pl-a,p^+/3-')
^
\-px,
Using (A.4)
we
can rewrite the term (A. 2) as
A
m
X(M)' -(y^^)
.'+>
(^ ~ ^
{\-/3X){\-pX,p^)
2 )
1
\-/+l
f
-^,+2+j +«i^,+i+y -«o^/+; '^"«2"^'-i+./
L=0
-
where
c^^.
=
,
_^ J c(/7,
,
A
,)
- c(/3^
,
^ )]
//^
(^A-
+ ^X^,-\
and
1
r^ -;^^l,^(-y9>a,/7^-«0/7,+;5-^)
(l-;9/L,pJ(l-/?/l,/7j
(A.5)
41
2.)
Proceeding as with the terms in X, the terms involving
W on the right-hand side of
(18) can be written as
s \
f
\\
^
r/]
^.
j{A,-z,)
I (Mr
/
^.
\j+i
EXW,^j-m.uj)\ = c^+^^vW,
-(A)
^4-t[cCc„,^i)-cO„,'^2)]a,..
{A,-A,)
-P'A,^_ {j3Af(l-/3pJ
c(/?v.>-^,)
A„> and
=
X-PX,
r^-
(A.6)
[,=oL
%=
where
\/+i
{\-/3A;){\-/3X,pJ
{1-Pp.)
A-^A-y
Note
.
that if
/I ,/l
then
2>
F^ <
{\-/3X,pJ{\-/31,pJ
3.) Similarly,
assuming that
rt
follows
r,
= /j^+ p^
r,_,
+ s^, where
,
e^,
~
i.i.d.(0,a^
j
,
we
obtain that
'
7^
Y/3
A,A,
TT. \>+'
){\-X,)
/
77^
Xi+l
s(Mr-(M)
(A.7)
^/,.i./hc,+r,r,
[,.=0
where
c_
7^
=r
A
,A
r.
.
(l-y5^,){l-y51,)
and r, = f_l] {A,A,)
(/^^^f
(M2f
(\-/3A,){\-fiA,p^)
(l-/3X,)(l-/31,p^)
+ Pr
-,
•Pr
Pr
Note
that, \i
\?^>^ then
T^ <
.
[\-l5X,p;)[\-l3X,p)
4.) Finally,
^-P P\K
kyP-
J
-/+'
/
7T.
\>+l
2:(Ar-(A)
(^-^2)11:
c^ =
-/I ,/t ,
=
rC.
y[\-[5X,)[\-l3X;)
(A.8)
42
5.)
Using the
Eg =
where
from (A.5), (A. 6), (A. 7), and (A. 8)
results
c,^
+Cj^
—
+c^+ —^
^^
^\
-
_^
—
c
in (9)
we have
,
(\-j3A,)(l-fiA,)
and where
r;^.
—0
T^^
<
F^ <
.
<
Equation (10) in the Model witliout Adjustment Costs.
In the
model with y =
that has
equation (3) in (5) yields a second-order difference equation in Nt
one stable and one unstable
root.
Denoting the stable root by
/I
,
,
equation (10)
becomes
N,
= T,+A,N,_,+r,X,_,+r^W,+r^r,+ u%
where T y, =
f-A,
d
(10')
[0p^-[9/3 + a5/3)pl\\-
r.H^|^(i-AAO
PX.P,
\l^x
J
-A,
1
and
l-/^^,A.y
r.
6
1
IPr
\-px,p,j
Derivation of Equation (11).
Use equation
(3) to substitute for Yt
and Yt+i
in equation (5)
and rearrange to get
43
E\y[^Y,-2/3^Y,^,+p'-^Y,^,)-p9{^N,^,+^x,^,)^e^N,-j35a^x,^,
+aW,-/3W,^,)+5J3 N.
a
X.
+
Tir,^,
+
c
=
(A. 11)
/35
Use ^{W, -J3W,J = -J3^AW,^,+{1-J3)^W, and
rjr,.,
= rjAr,^, +W]
in
(A.H)
to get (11).
44
APPENDIX B
Data Description
The
real inventory
and shipments data are produced by the Bureau of
Econoic Analysis and are derived from the Census Bureau's Manufacturers' Shipments,
Inventories, and Orders survey.
1996 chained
shipments
is
dollars,
They
are seasonally adjusted, expressed in millions of
and cover the period 1959:01-1999:02.
An
implicit price index for
obtained from the ratio of nominal shipments to real shipments.
The observable
cost shocks include average hourly earnings of production
nonsupervisory workers for the nominal wage
rate; materials price
and
indexes constructed
from two-digit producer price indexes and input-output relationships (See Humphreys,
Maccini and Schuh (2001) for
details);
Nominal input prices were converted
nominal
interest rate is the
and crude
oil prices as a
measure of energy
prices.
to real values using the shipments deflator.
3-month Treasury
bill rate.
The
Real rates were computed by
deducting the three-month inflation rate calculated by the Consumer Price Index.
45
Table
1
Euler Equation Estimates
Panel A: Including Adjustment Costs and Observable Cost Shocks
Constant Interest
Variable Interest
Rate
Rate
e
y
^
a
6.210
11.113
(2.042)
(2.071)
1.026
1.098
(0.593)
(0.433)
0.814
1.185
(0.469)
(0.484)
1.060
1.317
(7.334)
(4.691)
-.659
7
(-1.407)
Constant
-.043
0.142
(-.347)
(0.599)
Newey-West
2.752
[0.097]
test
Panel B: Including Observable Cost Shoclcs but no Adjustment Costs
Constant Interest
Variable Interest
Rate
Rate
e
6.482
15.072
(2.637)
(2.331)
1.172
1.359
(0.797)
(0.497)
\
a
1.068
1.513
(9.181)
(4.609)
7
-1.003
(-1.780)
Constant
-.045
0.283
(-.441)
(1.045)
Newey-West
3.171
test
[0.075]
Point estimates and
(t
statistics).
The
last
row
reports the
Newey-West
tests the constant interest rate specification against the specification in
value
is
reported in square brackets.
statistic, which
column two; the p-
46
Table
1
Euler Equation Estimates
Panel C: Including Adjustment Costs but no Observable Cost Shocks
Constant Interest
Variable Interest
Rate
Rate
e
Y
7.714
12.401
(2.752)
(2.527)
0.912
1.035
(0.467)
(0.377)
1.094
1.376
(7.450)
(4.921)
a
V
-.687
(-1.377)
Constant
-.023
0.199
(-.183)
(0.856)
Newey-West
2.074
[0.1501
test
Panel D: Excluding Observable Cost Shocks and Adjustment Costs
Constant Interest
Variable Interest
Rate
e
Rate
8.495
17.091
(3.191)
(2.591)
1.131
1.611
(8.702)
(4.611)
a
V
-1.060
(-1.695)
Constant
0.370
-.003
(-.030)
(1.318)
Newey-West
2.991
[0.0841
test
Point estimates and
(t statistics).
The
last
row
reports the
Newey-West
tests the constant interest rate specification against the specification in
value
is
reported in square brackets.
statistic,
which
column two; the p-
47
Table 2
Decision Rule Estimates
Panel A: Including Adjustment Cost and Observable Cost Shocks
Constant Interest
Variable Interest Rate
Rate
Constant
Nm
0.032
0.031
(4.206)
(3.306)
1.072
(23.548)
Nt.2
-.114
(-2.529)
Xt-i
W,
1.072
(23.514)
-.114
(-2.526)
0.045
0.045
(3.736)
(3.733)
-.034
(-4.126)
-.034
(-3.798)
0.001
Tt
(0.081)
Panel B: Including Observable Cost Shocks, no Adjustment Costs
Constant Interest
Variable Interest Rate
Rate
Constant
Nt-i
0.034
0.033
(4.451)
(3.482)
0.959
(101.04)
Xt.,
W,
0.960
(99.052)
0.043
0.043
(3.615)
(3.612)
-.036
(-4.333)
rt
-.036
(-3.979)
0.002
(0.10175)
OLS
Estimate of Decision Rule with
(t-statistic)
48
Table 2
Decision Rule Estimates
Panel C: Including Adjustment Cost, no Observable Cost Shocks
Constant Rate
Constant
N..,
0.007
(2.073)
(0.969)
1.108
(24.377)
N..2
-.130
(-2.844)
Xt-i
Variable Rate
0.013
1.105
(24.330)
-.127
(-2.784)
0.010
0.015
(1.137)
(1.645)
0.024
rt
(1.583)
Panel D: Excluding Observable Cost Shocks and Adjustment Costs
Constant Rate
Constant
Nm
0.014
0.007
(2.227)
(1.045)
0.981
0.980
(118.369)
Xm
Variable Rate
(118.570)
0.006
0.012
(0.729)
(1.313)
rt
0.026
(1.682)
OLS
Estimate of Decision Rule with
(t-statistic)
49
Table
3:
Means of Linearly Detrended
Interest Rate
Mean Log
Inventories in Different Interest Rate
Regime
Regimes
Test Statistic
Low
Medium
High
10.405
10.341
10.327
0.077
Inventories
(5.34)
(Detrended)
[0.000]
The
cells
under Interest Rate Regime report the mean of linearly detrended inventories
column reports
means between the high and low
and the [p-value], where the t-statistic is based on
Newey-West standard errors (allowing for an MA(12) error term) from a regression of
linearly detrended inventories (in logs) on a constant and dummies for high and low
interest rate regimes. Test results are similar for quadratic detresnding and an MA(24).
(in logs).
The
last
the difference in
interest rate regimes, the (t-statistic),
50
Table 4
Cointegration Tests
Inventories, Sales, Observable Cost Shocks and the Interest Rate
Null hypothesis: that the number of cointegration vectors
<1
<2
50.011
27.528
7.854
[0.035]
[0.129]
[0.636]
Logs
68.733
35.984
15.872
[0.000]
[0.015]
[0.094]
Linear detrending
77.879
41.795
19.151
[0.000]
[0.002]
[0.034]
Levels
Johansen-Jeselius test statistics with p-values in square brackets.
is
51
Table 5
Estimated Cointegrating Vector
Inventories, Sales, Observable Cost Shocks, and the Interest Rate
Constant
60901.23
Levels
Logs
0.351
-1070.78
(-5.11)
-401.814
(-5.55)
(2.44)
[0.0001
[0.7201
[0.0151
[0.0001
[0.0001
1.000
-0.015
-0.843
(8.029)
(-5.19)
(-5.27)
10.248
(-1.51)
[0.0101
[0.1321
[0.0001
[0.0001
[0.0001
0.891
-.000
1.069
-0.016
-0.922
(8.76)
(-5.33)
(-5.84)
(5.40)
[0.0001
DOLS
w
r
(0.36)
(2.60)
Detrended
10.248
X
(5.46)
3.407
Linearly
Time
(-0.82)
[0.4151
[0.0001
[0.0001
estimates of the cointegrating vector with (t-statistic) and [p-values1.
[0.0001
52
Table 6
Cointegration Tests
Inventories, Sales, Observable Cost Shocks and the Probabilities
Null hypothesis: that the number of cointegration vectors
<1
<2
Levels
94.11
54.94
[0.0021
[0.0471
[0.0821
Logs
91.68
57.88
35.97
[0.0031
[0.0241
[0.0331
Linear detrending
92.37
58.11
35.95
[0.003]
[0.0231
[0.0331
Johansen-Jeselius test statistics with p-values in square brackets.
32.33
is
53
Table 7
Estimated Cointegrating Vector
Inventories, Sales, Observable Cost Shocks,
Constant
51839.61
Levels
(2.82)
[0.005]
Logs
2.722
(1.02)
[0.308]
Linearly
Detrended
DOLS
and the Probabilities
n\
^3
w
41.767
0.237
3845.719
-5.263
-301.177
(1.03)
(1.06)
(1.73)
(-6.36)
(-4.34)
[0.306]
[0.289]
[0.085]
[0.000]
[0.000]
0.970
0.052
-0.072
-0.647
(4.64)
(1.35)
(-4.23)
(-4.12)
[0.000]
[0.178]
[0.000]
[0.000]
-0.707
(-4.51)
Time
41.767
(-0.36)
[0.722]
X
0.679
0.000
1.015
0.056
-0.073
(2.29)
(1.52)
(4.74)
(1.37)
(-4.32)
[0.023]
[0.131]
[0.000]
[0.171]
estimates of the cointegrating vector
vi^ith (t-statistic)
and
[0.000]
[p-values].
[0.000]
54
Table 8
Cointegration Tests
Inventories, Sales, and Interest-Rate-Related Variables
Panel A: Variable Interest Rate
Null hypothesis: that the number of cointegrating vectors
<=1
Levels
Logs
Linear Detrending
39.93
19.56
[0.012]
[0.031]
38.95
16.23
[0.015]
[0.094]
38.84
15.82
[0.015]
[0.107]
is
<=2
3.17
[0.071]
6.38
[0.010]
6.05
[0.012]
Panel B: Regime-Switching and Learning
Null hypothesis: that the number of cointegrating vectors
<1
<2
34.77
Levels
71.52
[0.001]
[0.047]
[0.258]
Logs
68.46
36.48
20.43
[0.001]
[0.029]
[0.023]
68.13
35.97
19.81
[0.002]
[0.033]
[0.028]
Linear detrending
Johansen-Juselius test statistics with p-values in square brackets.
12.64
is
55
Table 9
Estimated Cointegrating Vector
Inventories, Sales, and Interest-Rate-Related Variables
Panel A: Variable Interest Rate
Levels
Time
15288.81
62.082
0.213
-499.931
(0.56)
(0.81)
(0.52)
(-0.80)
[0.574]
[0.421]
[0.606]
[0.425]
0.584
62.082
0.879
-0.005
(0.16)
(-0.65)
Logs
r
(0.73)
(2.65)
[0.876]
[0.465]
[0.009]
[0.518]
0.099
0.000
0.904
-0.005
(0.27)
(0.09)
(2.61)
(-0.54)
[0.789]
[0.931]
[0.010]
Linearly
Detrended
X
Constant
[0.589]
Panel B: Regime-Switching and Learning
Levels
Logs
Linearly
Detrended
Time
-16692
-32.84
0.733
(-0.97)
n\
n-i
-3686
-Mil
(-1.50)
(-3.07)
(-0.61)
(2.56)
[0.335]
[-0.545]
[0.011]
[0.135]
[0.002]
-5.803
-32.84
1.454
-0.078
-0.094
(-1.94)
(-1.15)
(5.39)
(-1.89)
(-3.24)
[0.053]
[0.250]
-0.476
0.000
(-1.73)
[0.085]
DOLS
X
Constant
[0.000]
[0.072]
1.410
-0.087
-0.094
(0.70)
(5.23)
(-1.84)
(-3.08)
[0.482]
[0.000]
[0.067]
estimates of the cointegrating vector with (t-statistic) and [p-values].
[0.001]
[0.002]
56
Table 10
Estimated Cointegrating Vector
Allowing
Serial Correlation
of the Interest Rate within a Regime
Including Observable Cost Shocks
Constant
Levels
Logs
Linearly
Detrended
DOLS
Time
X
601.618
-13.790
0.602
-2211.669
(0.105)
(-1.098)
(9.528)
(-3.384)
[0.916]
[0.274]
[01
[0.001]
-0.920
0.000
1.141
(-0.899)
(-1.876)
(14.743)
[0.369]
[0.062]
[0]
0.177
0.000
(1.329)
(-1.433)
[0.185]
[0.153]
1.188
(14.963)
[0]
W
K,
TV,
-5556.076
(-6.474)
[01
-63.431
(-1.579)
[0.116]
-0.021
-0.112
-0.279
(-1.224)
(-5.932)
(-2.430)
[0.22203]
[0]
[0.016]
-0.019
-0.117
-0.337
(-1.069)
(-5.989)
(-2.967)
[0.286]
[0]
[0.003]
estimates of the cointegrating vector with (t-statistic) and [p-values].
57
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