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Corporate Savings, the Marginal Value of Cash, and the Discount Rate:
Can Persistent Financial Constraints Be Detected in Aggregate Data?
March 19, 2013
Huntley Schaller
Department of Economics
Carleton University
1125 Colonel By Drive
Ottawa ON K1S 5B6
Canada
Tel: (613) 520-3751
Fax: (613) 520-3906
Email: [email protected]
Keywords: corporate savings, new dynamic structural corporate finance models,
evaluation of investment projects, calculation of NPV, cointegration estimation of
corporate finance models
JEL codes: G31, G32, E22
I would like to thank Jim Poterba and seminar participants at the Econometric Society,
MIT, and the National University of Singapore for helpful comments, Heidi Portuondo
for research assistance, and the SSHRC for financial support.
2
Abstract
In the Modigliani-Miller world, there is no reason for firms to save. In the real
world, the corporate sector accounted for about 80% of total savings in the US in 2005.
Since then, corporate cash holdings have exploded, fuelling a lively public debate. New
dynamic structural models [Riddick and Whited (2009) and Bolton, Chen, and Wang
(2011)] provide a potential framework for understanding corporate savings and cash
holdings. This paper derives an important new implication of these models: Firms that
face persistent finance constraints will use a discount rate that is higher than the required
equity return to evaluate investment projects. This is a dramatic departure from the
standard understanding of how firms evaluate investment projects in the fields of
corporate finance and macroeconomics. In addition to deriving this new result, this paper
uses a highly innovative empirical strategy for testing this implication of the new
dynamic structural models of corporate savings. Using econometric theory, we show that
persistent financial constraints will push the estimated elasticity of the capital stock with
respect to the interest rate component of user cost toward 0. Using econometric
techniques that are appropriate for nonstationary variables – specifically the cointegrating
regression between the capital stock and the components of user cost – we test whether
the effects of persistent financial constraints can be detected in aggregate data. The main
empirical result is that the estimated elasticity of the capital stock with respect to the
interest rate component of user cost is approximately 0 (and, in most specifications,
precisely estimated). Based on our theoretical and econometric analysis, this means that
the effects of persistent financial constraints can be detected in aggregate data.
3
I. Introduction
In the Modigliani-Miller world, there is no motivation for firms to save because
they can always get additional cash by accessing external financial markets. In the real
world, the situation is radically different. As of 2005, corporate savings accounted for
80% of total savings in the US. Corporate savings also account for a substantial
proportion of total savings in other countries – 45% for China, 55% for Germany, and
80% for Japan.1
Since 2005, corporate savings has exploded in many countries, fuelling a heated
debate, in which public figures who are typically circumspect with respect for corporate
policy have staked out positions that are strongly critical of the corporate sector. For
example, Mark Carney, currently Governor of the Bank of Canada and soon to take over
as Governor of the Bank of England, has sharply criticized companies for the “almost
$600-billion in unused cash being held by non-financial companies in Canada. … This is
dead money. … If companies can’t figure out what to do with it, then they should give it
to shareholders.” [“Stop sitting on your cash piles, Carney tells Corporate Canada”,
Financial Post, August 22, 2012]
In the Modigliani-Miller world, firms do not face finance constraints. Before the
recent financial crisis, many macroeconomic models ignored financial frictions. If firms
do not face financial constraints in the real world, then it makes sense to build
macroeconomic models that do not include financial frictions. One possible argument for
this approach is that financial frictions might only affect relatively small businesses, and
a large share of firm activity is accounted for by large firms. On the other hand, financial
constraints might be important in the real world, as suggested by many papers in the
corporate finance literature. This paper is based on recent theoretical advances and
provides an entirely new type of evidence on the importance of finance constraints at the
aggregate level.
If the firm can readily raise outside financing, as is assumed in the ModiglianiMiller theorem and many macroeconomic models, then intuition suggests that there is no
1
The facts on corporate savings as a proportion of total savings are documented by Karabarbounis and
Neiman (2012).
4
reason for the firm to save. An earlier generation of models of finance constraints (and
their effects) were static, however, so it was difficult to clearly work out the analysis of
corporate savings, since saving is an inherently dynamic process. In fact, it is hard to get
a clear understanding of finance constraints, let alone corporate savings, in a static model.
To see this, suppose that external financing is unavailable (or simply costly) today, so a
firm faces a finance constraint. To avoid this situation in the future, the firm can build up
cash holdings. If these cash holdings eventually become large enough, the firm will face
little likelihood of a binding financial constraint.2
A new generation of structural models in corporate finance are designed to
address the inherently dynamic nature of financial constraints and corporate savings. The
most prominent are Riddick and Whited (2009) and Bolton, Chen, and Wang (2011). (We
refer to these as RW and BCW, respectively.) Because they are designed to work out the
firm’s optimal policy in a dynamic setting, they use relatively sophisticated mathematics.
For example, BCW use stochastic calculus to solve a two-boundary optimal stopping
problem. Both the RW and BCW models contain essentially the same elements and have
similar implications. Here, we focus on the BCW model. Despite its mathematical
sophistication, the BCW model provides rich economic intuition. We draw heavily on
this intuition to make the important contributions of the RW and BCW models accessible
to a wider audience.
The first contribution of this paper is to derive a new result from the BCW model:
When the firm faces persistent financial constraints, it employs a higher discount rate.
The paper is organized as follows. Section II describes the BCW model. Section
III draws out the new implication from the BCW model. Section IV outlines the relevant
econometric theory. Section V briefly describes the data. Section VI reports the empirical
results. Section VII concludes.
II. BCW Model
2
This can be clearly seen in the behavior of some emerging countries. After witnessing the costs of
external financing (e.g., in the form of stringent austerity imposed by outsiders) during the Asian financial
crisis in the late 1990s, a number of emerging economies appear to have made a conscious decision to build
up large cash holdings (in the form of foreign-exchange reserves).
5
The standard q model does not explain why firms hold cash. The key idea of the
BCW (2011) model is that, if firms face potential finance constraints, it makes sense for
them to build up cash reserves. In the BCW (2011) model, a firm faces liquidation if it
runs out of cash. Liquidation is costly because the liquidation value of the firm’s assets is
only a fraction l of their continuation value. The alternative to liquidation is to issue
equity, but there are fixed (and marginal) costs of issuance. This leads to a two-boundary
(S, s) model, where w is the state variable. At the lower boundary ( w = 0 ), the firm runs
out of cash and is either liquidated or issues equity. At the upper boundary ( w ), the firm
avoids building up too large a reserve of cash by paying out dividends.
Let P be the value of the firm and p ( w) = P ( K,W ) K be the scaled value of the
firm (scaled by the replacement cost of its assets K ). The shadow value of cash is
p¢ ( w) , which BCW refer to as the “marginal value of cash”. In the standard q model,
p¢(w) is 1. Intuitively, this is because the marginal value of cash is equal to 1; i.e., an
additional dollar of cash is simply worth one dollar, because the firm can frictionlessly
obtain external financing. In the BCW model, if a firm is financially constrained, the
marginal value of cash is greater than 1.
BCW show that p  w  is a convex function of w. The slope of p  w  is the
marginal value of cash. Intuitively, the marginal value of cash is highest when the firm is
close to running out of cash. As the firm builds up its cash holdings, the marginal value
of cash initially declines rapidly, then more slowly. With a big enough cash holding, the
marginal value of cash approaches 1. At the upper boundary w , the marginal value of
cash is equal to 1 and the firm is no longer financially constrained. Because the firm faces
a cash carry cost, it does not want to overaccumulate cash, so, when its cash holdings hit
the upper boundary, it pays out dividends.
Using an innovative empirical approach, based on robust nonparametric
econometrics, Bolton, Schaller, and Wang (2012) estimate the function p  w  . Consistent
with the BCW model, the estimated function is convex. Since the slope of p  w  is the
marginal value of cash, Bolton, Schaller, and Wang (2012) are able to use a discrete
approximation to the slope of the estimated p  w  function to obtain an estimate of the
6
marginal value of cash. In order to give the reader a feel for the BCW model, based on an
empirical estimate of the two key functions, we reproduce the estimated p  w  and
p  w  functions in Figure 1.
III. New Implication
In this section, we start with the equations of the BCW model. The objective is to
prove a new implication that flows from the model: Firms that face persistent financial
constraints use a higher discount rate, when evaluating investment projects, than the
standard Weighted Average Cost of Capital. As noted earlier, the BCW model uses
relatively sophisticated mathematical techniques – specifically stochastic calculus,
applied to a two-boundary optimal stopping problem. Here, we make the conscious
decision to use simpler mathematics in order to make the main result transparent to a
wider audience. Our impression is that the standard toolkit for financial economists – and
economists more generally – includes algebra and calculus, but not stochastic calculus
and the mathematics of optimal stopping problems. (Within financial economics, option
pricing is an exception, when it comes to the standard toolkit. If an individual with this
toolkit feels it would be worthwhile to add a technical appendix that works out the proof
using stochastic calculus and wishes to provide this proof, we would be happy to include
it in a subsequent version of the paper. This is an open invitation – and a serious one.) By
making a few simple assumptions that capture the key aspects – specifically, persistent
financial constraints – we are able to greatly reduce the mathematical complexity of the
proof, relying mainly on standard algebra and calculus, rather than stochastic calculus.
In standard q theory, beautifully formalized by Hayashi (1982), investment is a
linear function of marginal q:
1
i = a + q,
q
(1)
where i = I K , I is investment, K is the capital stock, a is a constant, q is the
adjustment cost parameter, and q is marginal q. Hayashi (1982) assumes perfect
7
financial markets so that the Modigliani-Miller theorem holds. In the presence of costly
external finance, where the Modigliani-Miller theorem no longer holds, Bolton, Chen,
and Wang (2011) show that investment also depends on the marginal value of cash:
i=a +
q
,
q p¢ ( w )
1
(2)
where p¢ ( w) is the marginal value of cash, W is the firm’s cash (i.e., reserves of liquid
assets), and a = 1 q . (This can be derived from equation (10) in Bolton, Chen, and
Wang (2011) using the facts that: 1) p¢ ( w) = PW ( K,W ) ; and 2) q = PK ( K,W ) . See the
lines of text just above equation (13) in BCW (2011) for the statement of these facts.)
Standard q theory, as in Hayashi (1982), is the special case of the BCW model where the
firm is not financially constrained, so p¢ ( w) = 1.
Marginal q is the present discounted value of future marginal products of capital
e
 rs
 ds ,
(3)
where r is the real interest rate and  is the marginal product of capital. Equation (2)
says that investment is a function of marginal q divided by the marginal value of cash
e
 rs
 ds
p  ws 
.
(4)
To capture the idea of persistent financial constraints in a tractable way, assume
that
ws = w ,
(5)
8
where w is a constant such that p  w  1 . This assumption implies that the firm is
persistently financially constrained.3 Using (5), rewrite (4) as
e
 rs
 ds
p  w 
  e B e rs  ds   e B  rs   ds ,
(6)
where e- B º ( p¢ ( w)) . Since p  w  1 ,  p  w   1 , so B  0 . This implies that
-1
r(s) º B + rs > rs .
1
(7)
Equation (7) says that, if the firm faces persistent financial constraints, the effective
discount rate is higher than it would be in the standard q model with perfect financial
markets.
IV. Econometric Theory
Caballero (1994, 1999) argues that stationary econometric techniques do a poor
job of capturing the relationship between the capital stock and user cost. Intuitively, the
problem is that the short-run relationship between the capital stock and user cost is
dominated by adjustment frictions. Unfortunately, there is little agreement about the
exact nature of these adjustment frictions. In contrast, there is widespread agreement in
the literature that, in the long run, the capital stock depends on user cost. Caballero (1994,
1999) therefore recommends estimating the following cointegrating regression between
the capital stock and user cost:
kt   0   R Rt  zt
Rt  ut
3
(8)
If we relax the assumption that ws is a constant, the math becomes much less transparent because we
cannot take 1 p¢ ( ws ) within the integral in equation (4), so it is necessary to use the stochastic differential
equation for p  ws  , which, in turn, depends on more than a dozen equations in the BCW model that fully
specify the two-boundary optimal stopping problem .
9
where k is the log capital/output ratio, R is the log of user cost, and z and u are
stationary.4 Intuitively, the cointegrating regression captures the long-run relationship
between the capital stock and user cost. It therefore abstracts from all of the issues about
adjustment frictions that muddy this relationship in the short run. Caballero (1994) and
Schaller (2006) find that the relationship between the capital stock and user cost emerges
more clearly by estimating the cointegrating regression.
Above, we have shown that, when the firm faces persistent financial constraints,
the interest rate component of user cost no longer provides an accurate description of the
discount rate used by the firm in evaluating investment projects. Instead, the discount rate
is higher because it incorporates the effect of persistent financial constraints on the
marginal value of cash. In terms of econometric theory, we can think of this as a
measurement error problem. In other words,
Rt* = Rt + et
(9)
where Rt is based on r , which incorporates the effect of persistent financial constraints,
and Rt* is based on r , the equity required rate of return, which fails to incorporate the
effect of persistent financial constraints.
Consider the classic errors-in-variables problem with a single independent
variable, where the true model is:
yt = xt b + ut , (10)
but we can only observe xt* , which is measured with error
xt* = xt + et , (11)
where xt is orthogonal to both ut and et and all the variables are stationary. It is
straightforward to show that the OLS estimator ˆ is inconsistent:
æ
ö
ç 1 ÷
. (12)
plim b̂ = b ç
2 ÷
ç 1+ s e ÷
çè s 2 ÷ø
x
4
This relationship can be obtained by solving the firm’s problem (under the consumption of Cobb-Douglas
technology) for the frictionless capital stock and relaxing the unit user cost elasticity constraint. See, e.g.,
Caballero (1999, p. 816-821).
10
Two important implications can be drawn from equation (12).5
First, measurement error leads to "attrition bias." In other words, the OLS
estimate ˆ will be closer to zero than the true parameter  . This is sometimes known as
the "Iron Law of Econometrics." To see this result, use equation (11) to show that
s x2 = s x2 + s e2 ,
*
(13)
since xt* is orthogonal to et . Thus  x2   x2* . Second, the degree of bias depends on the
signal/noise ratio: the greater is the ratio of the variance of et (noise) to xt (signal), the
greater the bias – and thus, the closer b̂ will be to 0.
Because we estimate a cointegrating regression, measurement error would have
no effect on the coefficient estimate if the measurement error were stationary. The
intuition for this result is straightforward. Using equation (11), rewrite equation (10) as
yt  xt   ut  et  ,
(14)
so that the RHS variable is observable and measurement error enters separately in the
regression. If the measurement error is I(0), it will drop out of the cointegrating
relationship, because the cointegrating relationship only includes the I(1) variables. If
financial constraints are persistent, however, the measurement error will also tend to be
persistent (i.e., I(1)). Retaining the other assumptions of the classical econometric
analysis of measurement error, we can obtain a result very much like equation (11) {in
progress}. Thus, if financial constraints are persistent, the coefficient on the RHS variable
will be biased towards 0. The greater the degree of variation in the discount rate that is
due to persistent financial constraints, the smaller will be the signal to noise ratio. Based
on the empirical evidence from papers such as Whited (1992) and Chirinko and Schaller
(2004), the variation in the discount rate due to financial constraints is large relative to
the variation in the interest rate. Thus, persistent financial constraints will tend to lead the
coefficient on the interest rate component of user cost to be close to 0. This implication of
persistent financial constraints is the centerpiece of our empirical approach.
For those not familiar with the intricacies of nonstationary econometrics, there is
a tendency to believe that the superconsistency property of cointegrating regressions
5
For a more detailed statement of the classical errors-in-variables econometric analysis, see, e.g., Maddala
(1977), p.292-293.
11
implies that coefficient bias will be minimal (in the absence of I(1) measurement error).
Stock and Watson (1993) provide the relevant econometric theory, simulation results, and
intuition. They show that SOLS (i.e., OLS estimation of cointegrating regressions) can
lead to serious coefficient bias in samples of the length available to macroeconomists.
A researcher with an orientation to theoretical econometrics can read Stock and
Watson (1993) for a detailed mathematical analysis of SOLS coefficient bias and how
DOLS estimation can solve this problem. Here we present an intuitive explanation,
tailored to the cointegrating relationship between the capital stock and the cost of capital,
that follows Caballero (1994, 1999).
Asymptotically, SOLS yields consistent estimates of the coefficients in the
cointegrating regression. In the presence of adjustment frictions, though, SOLS will tend
to produce biased estimates in samples of the size normally available for aggregate time
series estimation. Analytical results in Caballero (1994) show that SOLS could be
downward biased (i.e., biased towards 0) by 50 to 60% for a sample of 120 observations
(e.g., 30 years of quarterly data) and 70 to 80% for a sample of 50 observations, if
adjustment frictions are large.
To explain the intuition for the SOLS bias, it will be helpful to ignore the constant
term. Let k* be the frictionless capital stock (measured in logs and normalized by the log
of output) and let it be a linear function of user cost:
kt*   R Rt
(15)
Adjustment frictions (broadly defined) will cause a gap zt between the actual capital stock
kt and the frictionless capital stock. Thus the actual capital stock will be equal to the
frictionless capital stock plus zt:
kt   R Rt  zt
(16)
In the presence of adjustment frictions, k* will typically fluctuate more than k, since k
will respond only slowly and partially to shocks. Since k is a sum of the random
variables k* and z.
var(k )  var(k * )  var( z)  2cov(k * , z)
(17)
so the variance of k can be smaller than the variance of k* only if cov(k*,z) is negative.
However, the OLS estimates of k* and z (i.e., kˆ*  ˆ R R and zˆ  k  ˆ R R ) are orthogonal
12
by construction, which implies var (kˆ* ) is less than var(k). In order to achieve this, OLS
will tend to bias the estimate of  R toward 0.6 Monte Carlo simulations by Caballero
(1994) show that if the actual capital stock responds sluggishly to shocks, the OLS
estimate can be biased downward by a factor of two or more.
The necessary condition for unbiased SOLS estimation of  0 and  R is that zt be
uncorrelated with us for all s and t. 7 This strong condition arises because it is only under
this condition that R will be uncorrelated with the error term z since:
cov( Rt , zt )  cov( R0  R1  R2  ...  Rt , zt )
 cov(u21  u22  ...  u2t , zt )
(18)
One solution to the problem of small sample bias in SOLS is the DOLS estimator
proposed by Stock and Watson (1993). Dynamic OLS (DOLS) addresses the problem of
finite sample bias by replacing the original error term z by a new error term v, which is
constructed to be orthogonal to R.8 The intuition is straightforward. OLS projects the
dependent variable onto the space spanned by the right hand side variables. The
remaining variation in the dependent variable is orthogonal to the right hand side
variables. Suppose z were projected onto the space spanned by all leads and lags of ΔR
(which is equivalent to the space spanned by u ). The error term vt from this regression
will be orthogonal to Rs since:
cov( Rs , vt )  cov( R0  R1  ...  Rs , vt )  0
(19)
The last equality follows from the fact that vt is orthogonal to all leads and lags of ΔRt by
construction.
As noted above, the assumptions required for SOLS estimation are unlikely to be
satisfied in estimating the user cost elasticity, primarily because of the slow adjustment of
the capital stock to shocks and the resulting correlation between shocks to user cost ( u )
and the gap (z) between the frictionless capital stock and the actual capital stock. The
6
This argument follows Caballero (1994, 1999).
Early statements of SOLS bias, in a more general context, can be found in Bannerjee et al (1986) and
Stock (1987).
8
In addition to addressing the SOLS bias, DOLS has other attractive properties. In the most general case
considered by Stock and Watson (1993), DOLS is asymptotically efficient (when interpreted
semiparametrically). Perhaps more important, in Monte Carlo simulations of the most general case (Case
C) considered by Stock and Watson (1993), DOLS has the lowest RMSE among a set of estimators of
cointegrating regressions.
7
13
preferred estimation procedure is therefore DOLS (rather than SOLS). Caballero (1994)
and Schaller (2006) estimate the following empirical specification:
kt   0   R Rt 
p
  R
s  p
s
t s
 t .
(20)
Persistent financial constraints lead to measurement error in the interest rate
component of user cost, not the other components. This motivates our innovation on the
earlier work of Caballero (1994) and Schaller (2006). To isolate the effect of persistent
financial constraints, we decompose user cost into three components. The log of user cost
is the sum of the logs of the interest rate, tax, and capital goods price components9:
Rt º Rtr + Rtt + Rtp .
(21)
Both Caballero (1994) and Schaller (2006) find that large values of p (as shown in
the summation in equation (20 )) are necessary to deal with coefficient bias. (In the
related case of inventories, where the cointegrating regression includes more right-handside variables, Maccini, Moore, and Schaller (2012) also find that a large value of p is
required to overcome coefficient bias.) In this paper, we face a tradeoff. Ideally, we
would consider very large values of p, but this quickly eats up degrees of freedom when
there are three components of the cost of capital. We therefore consider three
specifications. In the first specification, we include only one component at a time; e.g.,
k t   o   r Rtr 
p

s  p
r
s
Rtr s   t
(22)
We refer to this as the Single Components specification. The advantage of this
specification is that it only has one right-hand-side variable, so we can consider large
values of p. The disadvantage is that it might suffer from omitted variable bias, since the
other components are not included. In the second specification, we include all three
components, with each entering separately:
k t   o   r Rtr    Rt   p Rtp 
9
p

s  p
 sr Rtr s 
p

s  p
 s Rt s 
p

s  p
p
s
Rtp s   t
(23)
See Clark (1993, p.319) and Tevlin and Whelan (2003) for other decompositions of user cost. Clark does
not, however, include more than one term in any of his regressions.
14
We refer to this as the Full Decomposition specification. In the third specification, isolate
one component at a time (e.g., the interest rate component) and combine the other two
components (e.g., the tax and capital goods components):
k t   o   r Rtr    & p Rt & p 
p

s  p
 sr Rtr s 
p

s  p
&p
s
Rt&s p   t , (24)
where
Rt & p  Rt  Rtp
We refer to this as the Partial Decomposition specification. As discussed above, we use
DOLS estimation, so each of these specifications includes p leads and lags of the first
differences of each right-hand-side variable.
In the data, the tradeoff between a larger value of p and the inclusion of separate
terms for each component of user cost turns out not to matter that much. With one or two
exceptions, we are able to obtain precisely estimated values of the elasticity of the capital
stock with respect to the interest rate component of the cost of capital. These estimates
are essentially the same, regardless of whether we include only a single component of
user cost, separate out one component and combine the other two, or include each of the
components separately.
V. Data
This paper uses Canadian aggregate data for the period 1962:1 to 1999:4. (We
would like to extend the sample to 2011 or 2012, but there are technical problems due to
the fact that Statistics Canada is in the process of changing over to a new system of
national accounts.) In particular, the investment data is non-residential, gross, real, fixed
capital formation (seasonally adjusted) for the business sector from the National Income
and Expenditure Accounts. In the Canadian data, investment is divided into equipment
and non-residential structures. To form a series for total investment, we sum equipment
and structures.
The capital stock is calculated by the perpetual inventory method using a
depreciation rate of 0.13 for equipment and 0.06 for structures. The total capital stock is
the sum of the capital stocks for equipment and structures.
15
Output is matched to the investment data, which are for the business sector, by
subtracting government expenditures from GDP.
A more detailed description of the data is contained in the Data Appendix, which
provides Statistics Canada series numbers and discusses details such as the calculation of
the present value of depreciation allowances.
VI. Empirical results
A. Equipment
The first three rows of Table 1 present unit root tests for the three components of user
cost. In each case, the existence of unit root is a good description of the data: Augmented
Dickey-Fuller test fail to reject the null hypothesis of a unit root for any of the three
components of user cost. The same is true for k.
Table 2 reports Johansen-Juselius tests of the null hypothesis that there is no
cointegrating vector. In six of the seven specifications we consider, the null hypothesis of
no cointegration is rejected.
Table 3 reports the estimated elasticities of the capital stock with respect to the
three components of user cost. The estimated elasticity of the capital stock with respect to
the interest rate component is close to 0 in two of the three specifications and
insignificantly different from 0 in all three specifications. This is consistent with the
presence of persistent financial constraints. In contrast, the elasticity of the capital stock
with respect to capital goods prices, for example, is negative, substantial, and precisely
estimated. In each specification, the point estimate is around -0.9.
B. Structures
As Table 4 shows, the data are consistent with a unit root in each individual
component of user cost, each of the combined components, and k.
Table 5 reports Johansen-Juselius tests of the null hypothesis that there is no
cointegrating vector. In all of the Single Components specifications, the null hypothesis
of no cointegration is rejected. For the other two specifications, the Johansen-Juselius
tests are more mixed.
Table 6 shows that the estimated elasticity of the capital stock with respect to the
interest rate component is approximately 0 in all three specifications. In each case, the
16
standard errors are small, so we can say with considerable statistical confidence that the
elasticity is quite close to 0. As with the equipment capital stock, the evidence for the
structures capital stock is consistent with persistent financial constraints.
C. Total Capital Stock
As Table 7 shows, the data are consistent with a unit root in each individual
component of user cost, each of the combined components, and k.
Table 8 reports Johansen-Juselius tests of the null hypothesis that there is no
cointegrating vector. In all of the specifications, the null hypothesis of no cointegration is
rejected.
Table 9 shows that the estimated elasticity of the capital stock with respect to the
interest rate component is close to 0 in all three specifications. As with the equipment
capital stock and the structures capital stock, the evidence for the total capital stock is
consistent with persistent financial constraints.
VI. Conclusion
This paper builds on recent theoretical advances in our understanding of corporate
savings and corporate cash holdings. The key step forward has been to develop dynamic
structural models of corporate savings that start from the idea that corporate cash
holdings are a way for the firm to avoid costly liquidation and to mitigate the effects of
costly external finance; in effect, to reduce the impact of financial constraints by building
up a cash reserve that allows the firm both to take advantage of promising investment
opportunities and to cope with adverse cash flow shocks. Two of these pioneering
theoretical papers are Riddick and Whited (2009) and Bolton, Chen, and Wang (2011).
BCW derive a new model of corporate investment. The intuition for their model
flows from the idea that cash has a shadow value greater than 1 when external equity
finance is costly. The intuition for this result is very similar to the idea that one dollar’s
worth of capital inside the firm can have a shadow value greater than 1 when there are
costs of adjusting the capital stock. Similarly, if there are external financing costs, a
dollar inside the firm can have a value greater than 1 to the firm.
17
In the standard q theory, investment depends on the present value of future
marginal products of capital. When cash has a shadow value greater than 1, investment
depends on marginal q (the present value of future marginal products of capital) divided
by the shadow value of cash. The intuition is straightforward. If a firm is thinking about
spending an additional dollar on an investment project, it must take into account the fact
that the marginal value of cash is greater than 1. The standard q model of investment is
the special case of the BCW model for a firm that is not financially constrained.
The main theoretical contribution of our paper is to show that the BCW model of
investment can be recast as a model in which the firm bases its investment on the present
value of future marginal products of capital, but in which the firm uses a higher discount
rate because it faces persistent financial constraints. This version of the new dynamic
structural models of corporate savings and investment looks very similar to the standard q
theory, except that the firm uses a higher discount rate.
Previous empirical work has tried to estimate the effect of financial constraints on
the firm’s discount rate. Papers such as Whited (1992) and Chirinko and Schaller (2004)
find that the variability in the discount rate is due to financial constraints is large relative
to the variation in the interest rate. These papers use firm-level panel data to detect the
effect of financial constraints. Our paper takes a different tack. Our objective is to see if
we can detect evidence of persistent financial constraints in the aggregate data.
Our approach is based on the pioneering insight of Caballero (1994), who pointed
out that there is much uncertainty about the dynamic path of investment. Theoretically,
this arises because we are not very certain about the nature of adjustment frictions. In the
standard q model, the adjustment frictions are represented by a convex adjustment cost
function, but there is a large literature that suggests that nonconvexities in capital
adjustment costs are probably quite important at the micro level and may influence
investment at the aggregate level. {References} Caballero emphasizes that a wide variety
of theoretical models all imply the same long-run relationship between the capital stock
and the variable that determines the capital stock in the long run – user cost. From an
econometric perspective, the substantial costs of adjusting the capital stock imply that the
response of the firm can often be spread over many years. In the early empirical
investment literature, researchers tried to address this through creativity in the
18
specification of distributed lags, but this approach suffers from a deep flaw. The lagged
coefficients are not structural parameters; in fact, in many investment models, they are
functions of the variables that determine investment. Caballero cuts through all of these
problems by estimating the long-run relationship between the capital stock and user cost,
using the econometrics of nonstationary variables. Using both econometric theory and
Monte Carlo simulations, Caballero finds that alternative approaches can lead to severe
bias.
Our main theoretical contribution is to show that the firm effectively uses a higher
discount rate when it faces persistent financial constraints. Our main methodological
contribution is to show that, from an econometric perspective, the divergence between the
measured interest rate and the firm’s actual discount rate is a form of measurement error.
The classical econometric analysis of errors-in-variables teaches us that the coefficient
estimate of a variable that is measured with error will be biased towards 0. The larger the
variance of the measurement error, relative to the variance of the true variable, the closer
the coefficient estimate will be to 0. In view of the evidence from firm-level panel data
studies on the variability of the discount rate due to financial constraints, relative to the
variability of the interest rate, the bias due to measurement error could take the estimated
effect of the interest rate close to 0, if the firm is subject to persistent financial
constraints.
The classical econometric analysis of errors-in-variables assumes that the
variables are stationary. Caballero (1994) shows that the variables are I(1). Although we
cannot directly observe the measurement error, it is likely to be I(1) if the financial
constraints are persistent. We therefore derive the counterpart to the classical econometric
analysis for stationary variables in the case where both the variables and the measurement
error are I(1). By the standards of proofs in the field of nonstationary econometrics, this
derivation is fairly straightforward, but we have not been able to find it anywhere in the
published literature, so this may be another contribution of our paper. {In progress}
User cost can be decomposed into three components – the interest rate, tax, and
capital goods price components. Our theoretical result involves only the interest rate
component. This suggests that the bias towards 0 will affect the estimated elasticity of the
capital stock with respect to the interest rate component of user cost. This motivates the
19
innovation in our empirical specification, relative to Caballero (1994). We estimate the
relationship between the capital stock and the three components of user cost, rather than
assuming that the coefficient on each of the three components will be the same.
Our main empirical result is that the estimated user cost elasticity of the capital
stock with respect to the interest rate component of user cost is close to 0. This is
consistent with persistent financial constraints. To the best of our knowledge, this is the
first time the new dynamic structural models of corporate savings and investment have
been used in an effort to detect the effect of financial constraints on aggregate data.
We examine the robustness of our main empirical result in two dimensions. First,
we consider three different types of capital – equipment, structures, and the total capital
stock. Our main empirical result holds for all three. Second, to address technical
econometric issues, we consider three different specifications (Single Components, Full
Decomposition, and Partial Decomposition). Our main empirical result holds for all three.
Our conclusion can be stated simply. Using recent developments in the economic
theory of corporate savings and investment – and the most advanced econometric
techniques that are currently available – we detect the effect of persistent financial
constraints in aggregate data.
The presence of persistent financial constraints has several important
implications. First, it provides empirical evidence that supports the emphasis on costly
external finance in the recent dynamic structural models of corporate savings and
investment, such as Riddick and Whited (2009) and Bolton, Chen, and Wang (2011).
Second, before the recent financial crisis, DSGE macro models tended to incorporate few
financial frictions. The entire process of financial intermediation, including the possibility
of costly external financing and the role of corporate savings in mitigating financial
constraints, played little role in standard, pre-crisis macro models.10 Our main empirical
result is that the effects of persistent financial constraints are detectable in aggregate data.
This provides further impetus for macroeconomists to incorporate the possibility of costly
external finance into macro models. More specifically, it suggests that part of this
10
Bernanke, Gertler, and Gilchrist incorporates some financial frictions, so it is closer to the RW and BCW
models, but it is still some distance from these models, because it does not allow for corporate savings.
20
research agenda should be to integrate corporate savings into the modeling of firm
behavior in macro models.
21
Data Appendix
Capital stock is calculated by the perpetual inventory method, specifically using
the following formula
K t 1  1   K t  I t 1
The annual depreciation rates are set at .13 for equipment and .06 for structures.
(Note that since the investment data are quarterly, the depreciation rate used in the
formula above is a quarterly rate.)
To construct the initial values of the capital stock, we use the current dollar
measures of the capital stock (D818267 for equipment, D818265 for building
construction, and D818266 for engineering construction11) for 1960 and deflate using the
appropriate price index for investment goods produced by the System of National
Accounts Division of Statistics Canada (D15605 for equipment and D15604 for
structures). For example, for equipment we multiply the current dollar capital stock for
1960 by the ratio (equipment price index for 1992)/(equipment price index for 1960). The
initial capital stock for structures is the sum of the building construction and engineering
construction capital stocks, constructed as just described.
The investment data are business sector, non-residential, gross, real, fixed capital
formation (seasonally adjusted) from the National Income and Expenditure Accounts,
which is divided into non-residential structures (D14854) and equipment (D14855). Since
the quarterly investment data are reported at annual rates, we divide by 4 to obtain
investment in a given quarter.
Our measure of total capital stock is simply the sum of our measures of equipment
and structures capital stock.
The cost of capital is calculated as follows
 1   t  u t  Pt K
~
 Y
Rt  it       tK 
1


t

 Pt


where i is the nominal interest rate,  is set at .13 for equipment, .06 for structures, and
.08 for total,  is a fixed risk premium (set at 6%),  K is the rate of inflation for
investment goods,  is the present value of depreciation allowances, u is the investment
tax credit rate,  is the corporate tax rate, P K is the price of investment goods, and P Y
~
is the price of output. R is expressed as a annual rate, so it ,  ,  , and  tK are all
expressed as annual rates. The corporate tax rate is the combined federal and Ontario
(provincial) tax rate on income other than small business or manufacturing income.
The nominal interest is a three month T-bill rate (B14060). The interest rate data
is monthly and starts in 1962. In order to transform it into quarterly data we took the
corresponding three-month average for each of the quarters. For example, 1962:1=
(Jan/62 + Feb/62 +Mar/62)/3, where the dates refer to the interest rate for that date. The
following points were missing form the original series: February 1970, November 1970,
February 1971, April 1973. In these cases we constructed the quarterly data by obtaining
the average of the interest rates for the two months that were available for each of those
quarters. For example, the interest rate for 1970:1 = (Jan/70 + Mar/70)/2.
11
Statistics Canada divides structures into “building construction” and “engineering construction”.
22
The corporate tax and investment tax credit rates are drawn from Finances of the
Nation (previously The National Finances), published by the Canadian Tax Foundation,
various issues, supplemented for the period since 1996 by personal communication with
the Department of Finance.12 Because the investment tax credit applies only to
equipment, u  0 for structures. When we examine the total business sector capital stock,
we multiply the statutory ITC rate for each quarter by the ratio of equipment investment
to the sum of structures and equipment investment for that quarter.
The present value of depreciation allowances (per dollar of investment) is calculated
as follows
[Hayashi 1982, p. 221-222]:
T
 t   t  Dn, t 1  it n
t
n 1
where Dn, t  is the depreciation allowance at time t for an asset at age n and T is the
asset life for tax purposes. In general, depreciation allowances in Canada are based on the
declining balance method.13 Essentially, the Canadian declining-balance method sets
Dn, t  equal to the depreciation rate for that class of assets divided by two times the
purchase cost in the first year (with the idea that, on average, assets are purchased halfway through the year) and the depreciation rate times the remaining undepreciated value
of the asset in subsequent years. Thus,
D1, t   .5 tT


Dn, t   1  .5 tT 1   tT

n2
 tT , n  2
where  T is the depreciation rate for tax purposes. The present value of depreciation
allowances will therefore be:
T
 t  tT
1
T
T
T
T 1 t
1  it 1  ...
t 
 1  .5 t  t  t 1  it   1  .5 t  t  t
2
1  it



 t  tT
2
 t  tT




 1  .5  t  1  it 

T
t
T
t
1  .5  
T
t
t
1
 1   tT

s 0 1  i
t






s
T
t
2
it   tT
For asset class 29 (the primary category for equipment between May 8, 1972 and 1987,
inclusive), three-year straight-line depreciation at rates 25%/50%/25% was applied, so
the present value of depreciation allowances was:
.5 t
.25 t
 t  .25 t 

1  it 1  it 2
12
The ITC was in place from 1969 to 1985 with two exceptions (October 10, 1966 to March 9, 1967 and
April 19, 1969 to August 15, 1971). We assign the ITC rate based on the rate that prevailed for the majority
of a given quarter (e.g., setting the ITC rate to zero for the last quarter of 1966 and the first quarter of
1967).
13
A detailed discussion of the basic structure and historical information on rates is available in Buckwold
(1990), p. 93-105. The rates were verified by direct personal contact with the Department of Finance.
23
For structures, the rate was 5% before 1988 and has been 4% since then. The standard
rate for equipment was 20% before 1972, 40% from 1988 to 1989 inclusive, 30% for
1990, 25% from 1991 until February 26, 1992, and 30% from February 26, 1992 on.
For total investment,  is a weighted average of the corresponding  ' s for equipment
and structures, with the weights corresponding to the proportions of equipment and
structures investment in a given quarter.
The price indexes for business sector structures and equipment and software
investment are D15604 and D15605, respectively, and (D15603) for the total. The price
index for output is the GDP deflator at market prices (D15612).
We attempt to match output as closely as possible to the investment data, which
are for the business sector. To do this, we subtract government expenditures – net
government current expenditure on goods and services (D14848), government gross fixed
capital formation (D14849), and government inventories (D14850) – from GD(D14872).
24
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Erickson, Timothy and Toni Whited, 2000, “Measurement Error and the relation between
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26
Figure 1
27
Figure 2
28
Table 1
Unit Root Tests
Equipment
Variable
Rr
Coefficient
0.91
t-statistic
-2.42
p-value
0.370
m
8
R
0.97
-1.63
0.780
3
0.96
-2.54
0.307
5
0.90
-2.63
0.266
7
0.91
-2.45
0.355
8
0.91
-3.05
0.120
3
0.97
-2.67
0.247
5
R
R
R
R
k
p
r &
r& p
&p
The table represents Augmented Dickey-Fuller tests. Rr is the interest rate component of
the user cost. R  is the tax component of the user cost. Rp is the price component of the
user cost. R r & is the combined interest rate and tax component of the user cost. Rr&p and
R & p are defined comparably. All of the components of user cost are expressed in logs.
k is the log capital/output ratio. See Sections II and III for detailed definitions. The
variable m is the number of augmenting lags.
29
Table 2
Cointegration Tests
Equipment
Specification
Single Components
Full Decomposition
Partial Decomposition
Variables
k, R r
Test Statistic
20.53
p-value
0.022
k, R
20.48
0.023
k, R p
32.00
0.001
k , R r , R , R p
41.57
0.399
k , R p , R r &
37.08
0.024
k , R , R r & P
39.68
0.012
k , R r , R & p
41.22
0.008
The column “Test Statistic” reports Johansen-Juselius tests of the null hypothesis that the
number of cointegrating vectors is zero. See Table 1 for definitions of the variables. The
Single Components specifications include k and a single component of user cost. The
Full Decomposition includes k and all three components of user cost. The Partial
Decomposition specifications include a single component of user cost and the
combination of the two remaining components.
30
Table 3
Estimated Elasticities
Equipment
Specification
Single Components
Full Decomposition
Partial Decomposition
Rr
.92
(.61)
.04
(.06)
.02
(.40)
R
-.27
(1.45)
.13
(.10)
-.85
(.51)
Rp
-.88
(.04)
-.87
(.03)
-.87
(.04)
The column labelled R r reports the long-run interest elasticity of the capital stock, the
column labelled R  the long-run tax elasticity, and the column labelled R p the long-run
elasticity with respect to the price of capital goods. Standard errors are in parentheses.
Estimation is by DOLS. See equation (XX) for the precise specification of the Single
Components estimates, equation (XX) for the Full Decomposition estimates, and
equation (XX) for the Partial Decomposition estimates. We set p to minimize the
Bayesian Information Criterion (BIC).
31
Table 4
Unit Root Tests
Structures
Variable
Rr
R

R
R
R
R
k
p
r &
r& p
&p
Coefficient
0.89
t-statistic
-2.52
p-value
0.316
m
10
0.94
-2.20
0.490
3
0.95
-2.60
0.282
3
0.90
-2.35
0.406
10
0.89
-2.53
0.317
10
0.95
-2.32
0.426
3
0.95
-2.58
0.291
5
The table represents Augmented Dickey-Fuller tests. Rr is the interest rate component of
the user cost. R  is the tax component of the user cost. Rp is the price component of the
user cost. R r & is the combined interest rate and tax component of the user cost. Rr&p and
R & p are defined comparably. All of the components of user cost are expressed in logs.
k is the log capital/output ratio. See Sections II and III for detailed definitions. The
variable m is the number of augmenting lags.
32
Table 5
Cointegration Tests
Structures
Specification
Single Components
Full Decomposition
Partial Decomposition
Variables
k, R r
Test Statistic
22.87
p-value
0.012
k, R
27.90
0.003
k, R p
36.02
0.000
k , R r , R , R p
45.66
0.232
k , R p , R r &
33.62
0.062
k , R , R r & p
30.73
0.116
k , R r , R & p
29.43
0.154
The column “Test Statistic” reports Johansen-Juselius tests of the null hypothesis that the
number of cointegrating vectors is zero. See Table 1 for definitions of the variables. The
Single Components specifications include k and a single component of user cost. The
Full Decomposition includes k and all three components of user cost. The Partial
Decomposition specifications include a single component of user cost and the
combination of the two remaining components.
33
Table 6
Estimated Elasticities
Structures
Specification
Single Components
Full Decomposition
Partial Decomposition
Rr
.03
(.05)
.00
(.07)
.01
(.06)
R
.14
(.43)
.06
(.46)
-.01
(.41)
Rp
-.42
(.68)
-.22
(.84)
-.18
(.73)
The column labelled R r reports the long-run interest elasticity of the capital stock, the
column labelled R  the long-run tax elasticity, and the column labelled R p the long-run
elasticity with respect to the price of capital goods. Standard errors are in parentheses.
Estimation is by DOLS. See equation (4) for the precise specification of the Single
Components estimates, equation (5) for the Full Decomposition estimates, and equation
(6) for the Partial Decomposition estimates.
34
Table 7
Unit Root Tests
Total
Variable
Rr
Coefficient
0.92
t-statistic
-2.22
p-value
0.476
m
8
R
0.86
-3.34
0.060
7
Rp
0.97
-2.32
0.422
4
R r &
0.92
-2.25
0.462
7
R r& p
0.92
-2.20
0.487
8
R & p
0.94
-2.54
0.310
3
k
0.96
-2.79
0.201
5
The table represents Augmented Dickey-Fuller tests. Rr is the interest rate component of
the user cost. R  is the tax component of the user cost. Rp is the price component of the
user cost. R r & is the combined interest rate and tax component of the user cost. Rr&p and
R & p are defined comparably. All of the components of user cost are expressed in logs.
k is the log capital/output ratio. See Sections II and III for detailed definitions. The
variable m is the number of augmenting lags.
35
Table 8
Cointegration Tests
Total
Specification
Single Components
Full Decomposition
Partial Decomposition
Variables
k, R r
Test Statistic
23.84
p-value
0.009
k, R
37.51
0.000
k, R p
26.37
0.004
k , R r , R , R p
91.02
0.000
k , R p , R r &
36.22
0.031
k , R , R r & p
54.38
0.000
k , R , R & p
37.89
0.020
The column “Test Statistic” reports Johansen-Juselius tests of the null hypothesis that the
number of cointegrating vectors is zero. See Table 1 for definitions of the variables. The
Single Components specifications include k and a single component of user cost. The
Full Decomposition includes k and all three components of user cost. The Partial
Decomposition specifications include a single component of user cost and the
combination of the two remaining components.
36
Table 9
Estimated Elasticities
Total
Specification
Single Components
Full Decomposition
Partial Decomposition
Rr
.17
(.06)
.03
(.10)
.06
(.08)
R
-1.58
(.32)
.25
(.49)
-.80
(.86)
Rp
-.31
(.02)
-.33
(.15)
-.28
(.09)
The column labelled R r reports the long-run interest elasticity of the capital stock, the
column labelled R  the long-run tax elasticity, and the column labelled R p the long-run
elasticity with respect to the price of capital goods. Standard errors are in parentheses.
Estimation is by DOLS. See equation (4) for the precise specification of the Single
Components estimates, equation (5) for the Full Decomposition estimates, and equation
(6) for the Partial Decomposition estimates.