GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.13

The Sigmoidal Investment Function

Yuzo Honda and Kazuyuki Suzuki

November 2008

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

THE SIGMOIDAL INVESTMENT FUNCTION*

Yuzo Honda(Osaka University)
andKazuyuki Suzuki(Meiji University)
November, 2008

Abstract

Based on the investment theory of Abel and Eberly (1994), we develop an analyticalmodel of adjustment costs, which produces a sigmoidal investment function. We also
estimate the piecewise linear investment function, which includes as special cases
linear models, models with one threshold, the original model of Abel and Eberly, which
has two thresholds, and sigmoidal models. Empirical evidence clearly supports the
sigmoidal model. The threshold estimate of Tobin’s q is 0.91. The investment ratio does
not respond at value of Tobin’s q below 0.91, but begins to react sensitively as Tobin’s qpasses 0.91.
Keywords: Tobin’s q, financial constraints, irreversibility of investment, unlisted
Japanese firms, piecewise linear function
JEL classification: E22, G31
Correspondence: Yuzo Honda, School of Economics, Osaka University, Toyonaka,
Osaka 560-0043, JAPAN1

THE SIGMOIDAL INVESTMENT FUNCTION*

Yuzo Honda(Osaka University)
andKazuyuki Suzuki(Meiji University)

Investment is one of the most influential factors for explaining macroeconomic booms
and slumps because of its volatile nature. However the models derived fromneoclassical theory have not performed well when compared with ad hoc accelerator
models of investment (see, e.g., Bernanke et al., 1988).
There are at least five reasons for the poor empirical performance of neoclassicalmodels. First, firms might face the financial constraints, which neoclassical theory
ignores. According to neoclassical theory, Tobin’s q is a sufficient statistic for theinvestment ratio. However, many empirical studies have found the cash flow variable
has a statistically significant effect on investment. Firms facing financial constraints
may be unable to invest even if they wish to. Fazzari et.al (1998), having divided theirsample into groups of firms with high and low dividends, find that severely financiallyconstrained firms react more sensitively to cash flow. They also demonstrate that firms
are not indifferent between using internal and external funds for investment. Having
divided their sample into firms with main banks and those without, Hoshi et al. (1991)
estimate separate investment equations for these groups. Having found that the2

estimated coefficient of the cash flow variable is significantly smaller for firms withmain banks, Hoshi et al. (1991) argue that main banks mitigate the asymmetric
information problem and reduce the agency cost of lending. From this viewpoint, cash
flow is an important component of internal funds, and the agency cost of internal funds
is lower than that of external funds. Thus, the investment behaviour of firms issensitive to the volume of internal funds.
Second, the fundamentals that drive investment might be mismeasured. In particular,
most studies use average q as a measure of fundamentals. Average q in turn is definedas the ratio of the firm’s intrinsic value to the replacement cost of its assets. Because
the firm’s intrinsic value is unobservable, most authors have used its market value asa proxy. However, people might make mistakes in evaluating the firm’s intrinsic value
and/or may speculate on the stock market. Thus, the stock market value may be
misleading, and might be a poor proxy for the intrinsic value. Instead of using stock
market values, Cummins et al. (2006) propose using analysts’ forecasts to measure
average q.
Third, in practice, fixed investment is often infrequent and lumpy. Among others, Abeland Eberly (1994, 2002) formalise this idea in constructing neoclassical non-linear
investment functions. The derived investment functions incorporate an inactive regionin which investment does not respond to Tobin’s q. In these models, the adjustment
costs of investment incorporate not only standard convex costs, but also lumpy fixed
costs and/or those related to the irreversibility of investment. Some authors assert that
adjustment costs are asymmetric because firms might incur additional costs above acertain threshold (e.g., when their investments exceed replacement investment). The
asymmetric adjustment costs with a certain threshold could also produce an inactive
3

region in the investment function.
Fourth, some researchers argue that because the lumpy nature of investment
originates at the plant level, aggregation over plants distorts the shape of theinvestment function. To understand the relationship between adjustment costs and thenon-linearity in investment, among others, Caballero et al. (1995, 1997), Goolsbee and
Gross (2000) and Cooper and Haltiwanger (2006) use plant-level data.
Fifth, Cooper and Ejarque (2001) argue that allowing the profit function at the firmlevel to be strictly concave to reflect its market power is sufficient to replicate
regression results based on q-theory when profits significantly affect investment.
Empirical findings largely support the non-linear investment function instead of thelinear one, but there is no consensus on the shape of the non-linearity. For example,
the investment theory of Abel and Eberly (1994) suggests the non-linear investment
function illustrated in Figure 1. The investment function in Figure 1 incorporates apart in which investment is insensitive to Tobin’s q. However Barnett and Sakellaris(1998) suggest a different shape based on their empirical work. Making use of grid
methods, they suggest that the investment ratio is a convex function of average q forsmall values of q, and is a concave function for larger values of q. The purpose of thispaper is to provide estimates of a non-linear investment function, taking into accountthe five problems mentioned above.
[Figure 1 around here]
Our empirical findings indicate that the firm’s investment function is non-linear andhas three parts and two thresholds. According to our estimated investment function,
investment is insensitive to Tobin’s q for small values of q, is sensitive above the firstthreshold value of q, and is relatively insensitive above the second threshold value of q.
4

Although we used piecewise linear functions, incorporating the assumption ofsmoothness gives the estimated function a sigmoidal shape.
The outline of the paper is as follows. In Section 1, we explain the relationship between
various kinds of adjustment costs and the corresponding types of non-linearity ininvestment functions. In Section 2, we describe our approach to estimation. In Section
3, we explain how to estimate a piecewise linear model with two thresholds, choose thebest model among these models with two thresholds and compare this model withlinear models and non-linear models that have one threshold. In Section 4, we report
our empirical results and compare them with those from other studies. In Section 5, we
discuss the implications of our findings and directions future research.

1. Adjustment Costs and Types of Non-linear Investment Functions
In this section, we review the relationship between adjustment costs and the type of
non-linearity in investment functions.
1.1. Standard Convex Adjustment Costs
The standard neoclassical investment literature assumes convex adjustment costs.
The simplest form is represented by the quadratic adjustment cost function. In thiscase, one can derive the investment ratio as a linear function of Tobin’s q (as shown inAppendix 1).
1.2. The Irreversibility of Investment in Abel and Eberly (1994)
Arrow (1968) argued that gross investment cannot take a negative value because oflarge disposal costs. Instead of following Arrow (1968) in assuming that theadjustment cost becomes infinite, Abel and Eberly (1994) quantify the irreversibility ofinvestment as the difference between the purchase price ( p it+ ) and the resale price ( p it− )5

of investment goods at time t, assuming that p it+ > p it− . The subscript i denotes the priceof investment goods.
Following Barnett and Sakellaris (1998), we briefly explain how incorporating thistype of investment irreversibility into the adjustment cost generates an investment
function that has an inactive component.
The augmented adjustment cost function G (I t , K t −1 ) takes the form of

G (I t , K t −1 ) = C(I t , K t −1 ) + p it+ I t D(I t > 0) + p it− I t D(I t < 0) if I t ≠ 0 ,= C(I t , K t −1 )if I t = 0 ,(1-1)

where I t , C(I t , K t −1 ) and D denote gross investment, the standard convex adjustment
cost function, and an indicator function, respectively. The indicator function D takes avalue of unity when the condition in parentheses is satisfied and is zero otherwise.
Note that the augmented adjustment cost G is discontinuous at the value of I t = 0 .
The first-order condition for investment is given by:
p it / p t + ∂G (I t , K t −1 ) / ∂I t = q t / p t for I t ≠ 0 ,(1-2)

where p t and q t denote the output price and the shadow price of capital at time t,
respectively (see Appendix 1 for details). Denoting the right-hand neighbourhood andthe left-hand neighbourhood of zero by 0 + and 0 − , respectively, we have:

(

)

(

)

p it+ / p t + ∂C 0 + , K t −1 / ∂I t = q t / p t at I t = 0 + ,andp it− / p t + ∂C 0 − , K t −1 / ∂I t = q t / p t at I t = 0 − .

(1-3)

Dividing (1-3) by p it+ / p t throughout yields()

1 + C +I / p it+ / p t = q t / p it+

at I t = 0 + ,

and

(

)

p it− / p it+ + C I− / p it+ / p t = q t / p it+ at I t = 0 − ,
6

(1-4)

(

)

(

)

where C +I = ∂C 0 + , K t −1 / ∂I t and C I− = ∂C 0 − , K t −1 / ∂I t , respectively. Equations (1-4)
show that investment is a discontinuous function of q; this is illustrated in Figure 1.
1.3. Linear Adjustment Cost in Barnett and Sakellaris (1998)
Instead of assuming that the purchase price exceeds the resale price of investment
goods, Barnett and Sakellaris (1998) postulate that the firm incurs an incremental
linear adjustment cost θ when its investment exceeds replacement investment. They
assume an asymmetric adjustment cost,
G (I t , K t −1 ) = C(I t , K t −1 )if (I t / K t −1 ) ≦δ

= C(I t , K t −1 ) + θ(I t − δK t −1 ) if (I t / K t −1 ) > δ .

(1-5)

The investment function based on the augmented adjustment cost (1-5) is illustrated in
Figure 2. Under this assumption, investment is inactive at the value of the investment
ratio I t / K t −1 = δ in Figure 2.
1.4. An Alternative Specification of Irreversibility
Alternatively, taking into account the irreversibility of investment, one could specify
the augmented adjustment cost as:
2

⎞
α ⎛ I
G (I t , K t −1 ) = 1 ⎜⎜ t − δ ⎟⎟ K t −1
2 ⎝ K t −1⎠

if (I t / K t −1 ) ≦δ

2

⎞
α ⎛ I
= 2 ⎜⎜ t − δ ⎟⎟ K t −1 + θ(I t − δK t −1 )2 ⎝ K t −1⎠

if (I t / K t −1 ) > δ ,(1-6)

where α1 > α 2 > 0 . The augmented adjustment cost is composed of two quadratic
functions. It is asymmetric and thus not differentiable at (I t / K t −1 ) = δ . Equation (1-6)
is illustrated in Figure 3.
[Figure 3 around here]
As in Barnett and Sakellaris (1998), equation (1-6) includes the incremental linear cost7θ(I t − δK t −1 ) , which implies that the firm incurs an additional linear cost as well as thestandard convex cost when its investment exceeds replacement investment. However,given the assumption that α1 > α 2 > 0 , the firm also incurs increasingly large costs
when making negative net investments (whenI t − δK t −1 < 0 ) because of theirreversibility of investment. As argued by Arrow (1968), the firm incurs prohibitive
costs as α1 goes to infinity with the value of α 2 held constant.
The specification of the augmented adjustment cost in (1-6) together with thefirst-order condition for investment (1-2) yields the non-linear investment function
illustrated in Figure 4.
[Figure 4 around here]
The slope of the third part of the non-linear investment function in Figure 4 is muchsteeper than that of the first part. This is because the slopes of the first and third partsof the investment function in Figure 4 depend on ( 1 / α ) (see Appendix 1), and becauseof the assumption that α1 > α 2 > 0 . The larger the value of α , the less steep is theslope of the investment function in Figure 4. Indeed, as α1 tends to infinity, the slopeof the first part approaches the horizontal line at the height of δ , and the investment
function looks like the one illustrated in Figure 5.
[Figure 5 around here]
1.5. Potentially Prohibitive Adjustment Costs for Extremely Large Investments
The assumption of standard convex adjustment costs implies that the second
derivative of the adjustment cost function is positive. There are usually two reasons forthis. First, as the firm expands its investment, the costs of retraining labour, andmanagement and other associated costs become increasingly high. Second, investment
goods prices might rise as the firm purchases more investment goods.
8In this sub-section, we postulate that the cost becomes prohibitive as the firm’s
investment ratio exceeds a certain threshold. We believe that there is some limitation
to expanding investment, partly because, as the firm significantly expands its
investment, management cannot consider an infinite number of investment projects atthe same time and partly because supply shortages of investment goods eventually
make investment goods prices prohibitively high.
Suppose that the augmented adjustment cost is given by:
2

⎞
α ⎛ I
G (I t , K t −1 ) = G 1 = 1 ⎜⎜ t − δ ⎟⎟ K t −1
2 ⎝ K t −1
⎠

if

2

= G2 =⎞
α2 ⎛ It⎜⎜− δ ⎟⎟ K t −1 + θ(I t − δK t −1 )2 ⎝ K t −1⎠

(I t / K t −1 ) ≦δ

if δ < (I t / K t −1 ) ≦λ

2

⎧
⎫⎤
α ⎡ I
α
= G 3 = 3 ⎢ t − ⎨λ − 2 (λ − δ )⎬⎥ K t −1α32 ⎢⎣ K t −1 ⎩⎭⎥⎦+ θ(I t − δK t −1 ) +α2 ⎛ α2 ⎞⎜1 −⎟(λ − δ )2 K t −1
2 ⎜⎝ α 3 ⎟⎠if

(I t / K t −1 ) > λ ,(1-7)

where α 3 and λ are positive constant parameters such that α 3 > α 2 and λ > δ . The
augmented adjustment cost function (1-7) is illustrated in Figure 6.
[Figure 6 around here]
The augmented adjustment cost function in (1-7), together with the first-ordercondition for investment (1-2) and the assumption of α1 being infinite, yields the
non-linear investment function illustrated in Figure 7.
[Figure 7 around here]
The non-linear investment function now has two kinks. Above the first threshold, thefirm increases its investment ratio dramatically, as q increases. However, there is somesaturation point for the investment ratio, λ , beyond which the investment ratio

9increases only moderately as q increases. The investment function in Figure 7 is
S-shaped or ‘sigmoidal’. If we assume a ‘smooth’ investment function, then a logistic
curve would be a good approximation.
Empirical studies such as those of Eberly (1997) and Barnett and Sakellaris (1998)
suggest this shape. Abel and Eberly (2002) argue that aggregation over heterogeneous
capital goods might yield an S-shaped investment function. To choose between a linear
model, a logistic model and a hybrid of the two, Honda and Suzuki (2000) use the
Akaike Information Criterion and the Schwartz Information Criterion. They alsosuggest that their investment function takes the logistic form for a sample of large
Japanese manufacturers.

2. The Basic Model
Taking into account the relationship between adjustment costs and the type of
non-linearity of investment functions, and following Tachibana’s (2007) analysis ofpiecewise linear models, we estimate a piecewise linear investment function for thei th firm,It= β10 + β11q t −1 + γ1L t −1 + γ 2 L t −1 ⋅ D t if q t −1 ≦q1
K t −1= β 02 + β12 q t −1 + γ1L t −1 + γ 2 L t −1 ⋅ D t if q1 < q t −1 ≦q 2
= β 30 + β13 q t −1 + γ1L t −1 + γ 2 L t −1 ⋅ D t if q t −1 > q 2 ,(2-1)

where L t −1 and D t denote the log of total fixed assets, and a dummy variable that is
unity if the i th firm has never issued corporate bonds and zero otherwise, respectively.
The parameters β and γ are to be estimated. The first and second threshold valuesof Tobin’s q are denoted by q1 and q 2 , respectively, and q t denotes Tobin’s q at time t.
10

To avoid simultaneity bias, we use one-period lagged values for q t in equation (2-1).
Hansen (1999), Chida (2003), and Bo et al. (2006) carefully analyse thresholdregression models. However, none of them uses Tobin’s q as a right-hand-side thresholdvariable, and their focus differs from ours. Because we are interested in testing thevalidity of Tobin’s q theory, the right-hand-side threshold variable should be Tobin’s q.
We have explicitly clarified the relationship between the types of adjustment costs andthe shape of non-linearity of the investment function in the previous section to extend
Tobin’s q theory in such a way that the investment function has an inactive part.
2.1. Financial Constraints
Based on a similar sample to ours, Honda and Suzuki (2006) find that financial
constraints significantly affect the investment ratio. To incorporate financial
constraints, equation (2-1) includes the terms L t −1 and L t −1 ⋅ D t as right-hand-side
variables. When a firm has a large amount of capital stock at time t − 1 , L t −1 becomeslarge. Given a sufficient stock of capital, such a firm is expected to make lessinvestment at time t. Hence, the expected sign of γ1 is negative. On the other hand,
we expect γ 2 to be positive. Firms often use fixed capital as collateral when theyborrow money, and fixed capital is a rough indicator of ‘creditworthiness’, which isparticularly important for smaller firms that have never issued corporate bonds. The
larger the value of collateral, the less likely is the financial constraint to be binding,and thus the larger is the investment ratio. Note that we include the interaction
dummy L t −1 ⋅ D t rather than the simple dummy variable D t in equation (2-1). This isbecause including the simple dummy variable D t together with individual dummy
variables in a panel data regression causes perfect multi-collinearity.

11

Among others, Abel and Eberly (1994, 2002), Eberly (1997) and Barnett and Sakellaris(1998) exclude financial constraints from their non-linear estimated equations.
However, Fazzari et al. (1998), Hoshi et al. (1991) and Honda and Suzuki (2006),
among others, argue that financial constraints are important determinants ofinvestment ratios. Thus, omitting these financial constraint variables from thenon-linear regressions might seriously bias the estimates. Thus, we include L t andL t ⋅ D t on the right-hand side of equation (2-1).

2.2. Sample Selection
We use data on unlisted Japanese auto parts suppliers for a number of reasons. First,
data on listed large corporations might distort the true investment behaviour of firms
because large corporations tend to produce multiple products in different fields. When
there are multiple products, the relationship between the investment ratio and Tobin’s
q is less clear.
At the other extreme, some authors use plant-level data. Doms and Dunne (1998) findthat most fixed investments are infrequent but lumpy at the plant level. Incorporating
the finding of Doms and Dunne (1998), Caballero et al. (1995) develop a model in which
fixed investments are not made gradually but are undertaken only when the difference
between the desired and the existing capital stock exceeds a certain threshold.
We do not use plant-level data because it is difficult to measure Tobin’s q at the plantlevel. In addition, firms examine all potential projects when making decisions on fixed
investments. They do not investment in one plant without considering the situations inother plants.
There is another advantage using data on unlisted Japanese auto parts suppliers.
Cooper and Ejarque (2003), among others, point out that the q-theory of investment
12

does not apply to firms that have market power. Unlisted Japanese auto parts
suppliers supply auto parts to giant auto manufacturers such as Toyota, and are lesslikely to have market power. Thus, sampling unlisted smaller firms that have limited
market power is appropriate for our purposes.
2.3. Data on Tobin’s q
Because we use data on unlisted firms, stock price data are not available. Therefore,
we have constructed our measure of Tobin’s q based on the balance sheet and the profit
and loss account of each firm. (See Appendix 2 for details.)

3. Piecewise Linear Models
As was explained in Section 1, the investment theory on Tobin’s q suggests that there
are at most two thresholds in a piecewise linear model. We first explain how weestimate a piecewise linear model with two thresholds, and then discuss the procedure
for choosing between models that contain zero, one, or two thresholds.
3.1. Estimation of a Model with Two Thresholds
In this sub-section, we explain how to estimate a piecewise linear model with two
thresholds. Rewriting equation (2-1) as an econometric model yields:()+ (β + β q ) I (q < q ≦q )+ (β + β q ) I (q > q )y it = β 10 + β 11 q it −1 I (q it −1 ≦q 1 )
2
0

2
1

it −1

1

it −1

3
0

3
1

it −1

it −1

2

2

+ γ1L it −1 + γ 2 L it −1 ⋅ D it + φWi + u it ,(3-1)

where we define y it = I it / K it −1 . Subscript i denotes the i th firm. I( ⋅ ) and u it denotethe indicator function and the disturbance term, respectively. Wi denotes theindividual dummy variable, which is unity for the i th firm and zero otherwise, and φ13

is the corresponding coefficient parameter. The disturbance terms are assumed to be
independently and normally distributed. Given the values of L it −1 , D it , Wi , and u it ,equation (3-1) is illustrated in Figure 8. Our estimation method is based on grid
methods. We first provide a combination of respective numbers for the values of q1
and q 2 , and then estimate equation (3-1) by using the fixed-effects model.
[Figure 8 around here]
To estimate equation (3-1), given the values of q1 and q 2 , we transform equation(3-1). The conditions for our piecewise linear model to be continuous at the respective
points q1 and q 2 are given by:
β10 + β11q1 = β 02 + β12 q1

and
β 02 + β12 q 2 = β 30 + β13q 2 .

(3-2)

Solving for β10 and β 02 in equation (3-2), expressing these two parameters in terms ofβ 30 , β11 , β12 , and β13 and then substituting the resulting equations into equation (3-1)

yields:y it = β 30 + β11G it + β12 J it + β13 M it + γ1L it −1 + γ 2 L it −1 ⋅ D it + u it ,where
G it = (q it −1 − q1 )I(q it −1 ≦q1 ) ,J it = (q1 − q 2 )I(q it −1 ≦q1 ) + (q it −1 − q 2 )I(q1 ≦q it -1 < q 2 ) ,and
M it = q it −1 [1 − I(q it −1 ≦q1 ) − I(q1 < q it -1 ≦q 2 )] + q 2 [I(q it −1 ≦q1 ) + I(q1 < q it -1 ≦q 2 )] .

(3-3)

One can estimate equation (3-3) using standard computer software.
Our sample is an unbalanced panel. There are 1,553 observations on 104 firms for the
sample period from 1977 to 2006. However, after excluding outliers, our final number
14

of observations is 1,527. (See A2-1 in Appendix 2.) Table 1 provides summary statistics
for our sample of 1,527 observations. Figure 9 plots these observations with theinvestment ratio on the vertical axis and the lagged Tobin’s q on the horizontal axis.
This shows that the ratio of investment to capital stock jumps upward near the valueof one of the lagged Tobin’s q values.
[Table 1 around here][Figure 9 around here]
3.2. The Best Model with Two Thresholds
We first choose the best model among those models with two thresholds. Our procedurecomprises two steps. First, we select our benchmark model by using ‘likelihood’ as ourcriterion. Then, in the second step, we compare this benchmark model with all othermodels, and choose the best model by using as our criterion the ‘encompassing
principle’ (based on non-nested J-tests).
3.2.1. Likelihood
We first provide a combination of respective numbers for the values of q1 and q 2 ,and estimate equation (3-3). Similarly, we repeat the same procedure for all
combinations of the respective values of q1 and q 2 , and choose the best model, whichis the one that produces the largest likelihood. We define this model as our benchmark
model.
More concretely, we vary the values of q 2 from 1.0 to 2.5 by 0.01 and vary the valuesof q 2 − q1 from 0.0 to 2.5 by 0.01 respectively, and calculate the log likelihood for eachcase. It turns out that the pair q1 = 0.91 and q 2 = 1.20 yields the largest log
likelihood of 384.21. Table 2 reports the estimation results of this benchmark model.
[Table 2 around here]15

3.2.2. J-Tests
To select the best estimated model with two thresholds, we use Davidson andMackinnon’s (1981) non-nested testing procedure based on J-Tests. The model selection
procedure proceeds as follows. Suppose that we wish to test the above benchmark
model B, with q1 = 0.91 and q 2 = 1.20 , against an alternative model A with q1 = q1A
and q 2 = q 2A , where q1A and q 2A are exogenously given values. We first estimate thealternative model A with q1 = q1A and q 2 = q 2A in equation (3-3), and obtain thepredicted values of y it , which we denote by ŷ itA . Superscript A indicates model A. In
the second step, we add these predicted values ŷ itA to the right-hand side of thebenchmark model B, in which q1 = 0.91 and q 2 = 1.20 , and apply an artificial
regression to the resulting equation. We are interested in the significance of thecoefficient estimate of the predicted values ŷ itA . If it is statistically significant, theimplication is that the alternative model A has some additional information that ourbenchmark model B does not have. Otherwise, model A adds no information to ourbenchmark model B. Table 3 reports the results of these tests. Because none of thet-values indicates significance, all alternative models have no additional information.
(All t-statistics reported in the paper are based on heteroskedasticity-consistent
standard errors.)[Table 3 around here]
We then switch the respective roles of models A and B, and test the significance of thecoefficient estimate of the predicted values ŷ Bit in another artificial regression modelwith q1 = q1A and q 2 = q 2A . If it is significant, our benchmark model has some
additional information that model A does not have. Otherwise, the benchmark model
adds no information to model A. Table 4 reports the results of these tests. We reject the16

null hypothesis that our benchmark model has no additional information at least at
the 10% significance level in 77 out of 83 cases. We cannot reject the null hypothesis in
only six cases. The respective values of q1 and q 2 in these six alternative models
cluster around 0.91 and 1.2 in our benchmark model. This suggests that these six
alternative models are so similar to our benchmark model that the tests are notsufficiently powerful to discriminate between them.
[Table 4 around here]
In summary, we cannot reject the null hypothesis that an alternative model has no
additional information in all the 83 cases reported in Table 3. In Table 4, we reject thenull hypothesis that our benchmark model has no additional information at least at
the 10% significance level in 77 cases. These results imply that our benchmark model
dominates these 77 alternative models. We cannot reject the null hypothesis that ourbenchmark model has no additional information in six cases in Table 4. However, therespective values of q1 and q 2 in these six alternative models are so close to those in
our benchmark model that our tests do not have sufficient power to discriminate
between them. These results indicate that our benchmark model is the best model
among those non-linear models with two thresholds.
3.3. Linear Models and Models with One ThresholdIn the previous sub-section we found that our benchmark model is the best model
among those models with two thresholds. In this sub-section we compare ourbenchmark model with linear models and with non-linear models with one threshold.
For this, we use both standard F-tests and J-tests.
3.3.1. F-Tests

17

We have found that model (3-3) with q1 = 0.91 and q 2 = 1.20 is the best model amongmodels with two thresholds. To compare this model with the equivalent model with one
threshold, we use a standard F-test to test the null hypothesis that β11 = β12 in model(3-3) with q1 = 0.91 and q 2 = 1.20 .
Substituting β11 = β12 into equation (3-3) yields the restricted model:

y it = β 30 + β11 (G it + J it ) + β13 M it + γ1L it −1 + γ 2 L it −1 ⋅ D it + u it .

(3-4)

We estimate (3-3) with q1 = 0.91 and q 2 = 1.20 , estimate (3-4) with q 2 = 1.20 , andthen calculate the F-value, which turns out to be 6.24. Under the null hypothesis, thisstatistic has an F-distribution with (1, 1418) degrees of freedom. We reject the null
hypothesis that β11 = β12 at the 5% significance level. Therefore, we reject the modelwith one threshold.
Similarly, we compare our best model with two thresholds with a linear model, and test
the null hypothesis that β11 = β12 = β13 . Substituting these two restrictions into equation(3-3) yields the restricted equation:y it = β 30 + β13 (G it + J it + M it ) + γ1L it −1 + γ 2 L it −1 ⋅ D it + u it .

(3-5)

Having estimated (3-3) with q1 = 0.91 and q 2 = 1.20 , and (3-5), we test the null
hypothesis that β11 = β12 = β13 . The F-value turns out to be 7.48. This statistic has
F-distribution with (2, 1418) degrees of freedom under the null. We reject the null
hypothesis at the 1% significance level. Therefore, we reject the linear model.
3.3.2. Non-nested J-Tests
In sub-section 3.3.1, we confined our attention to model (3-3) with q1 = 0.91 andq 2 = 1.20 , and tested restrictions on the coefficients in this model. In this sub-section,
we first search for the best model among those with one threshold. Then, we compare18this model with the best model with two thresholds, which we validated in sub-section
3.2.
The procedure used to search for the best model with one threshold is the same as thatused to search for the best model with two thresholds. In the first step, we find thebenchmark model based on the likelihood; i.e., we find the one-threshold mode with thelargest likelihood. In the second step, we use grid methods to compare the best modelwith two thresholds with all alternative models with one threshold, including thebenchmark model with one threshold.
In the first step, we vary the value of the threshold q1 from 0.0 to 2.5 by 0.01, and findthat the model with q1 = 1.34 gives the largest log likelihood of 381.8763. Ourestimated benchmark model with q1 = 1.34 is reported in Table 5.
[Table 5 around here]
In the second step, we vary the value of threshold q1 from 0.5 to 2.4 by 0.1, which
produces 20 alternative models with one threshold. We also include the benchmark
model with one threshold (with q1 = 1.34 ) as another alternative model. Therefore, we
compare the best model with two thresholds ( q1 = 0.91 and q 2 = 1.20 ) with these 21
alternative models with one threshold. Tables 6 and 7 report the results of the
non-nested J-tests.
[Tables 6 and 7 around here]
The number in each cell in Table 6 is the t-value of the coefficient of the fitted variableŷ itA in the corresponding alternative model with one threshold. Because none of these
indicates statistical significance, the alternative models with one threshold have no
additional information.

19

On the other hand, all the t-values in Table 7 indicate significance at least at the 5%
level. This implies that the best model with two thresholds ( q1 = 0.91 and q 2 = 1.20 )has some additional information that is not incorporated in any of the alternativemodels.
The exceptional case in Table 6 is that of q1 = 1.20 . In this case, the estimatedcoefficient of the fitted variable ŷ itA is unusually large. We suggest that this is theresult of multi-collinearity. Recall that our best model with two thresholds has
q1 = 0.91 and q 2 = 1.20 .
The results of this section indicate that our best model with two thresholds ( q1 = 0.91
and q 2 = 1.20 ) is the best model among linear models, models with one threshold, andmodels with two thresholds.

4. Results and Discussion
In this section, we summarise our findings and discuss their relationship to those ofexisting studies.
4.1. Non-linearity
We apply the neoclassical theory of investment that specifies investment as a functionof Tobin’s q. With the introduction of non-convex adjustment costs into the model, Abeland Eberly (1994) show theoretically that the investment function incorporates aninactive portion in which investment does not respond to Tobin’s q. Although theconsensus seems to be that the empirical literature supports this theory and the notion
at investment is a non-linear function of Tobin’s q, there is no consensus about the typeof non-linearity exhibited by the investment function.

20

For example, although Bo et al. (2006) rely on real-options theory and adopt a
threshold variable that differs from Tobin’s q, they essentially estimate an investment
function with one threshold. Based on our evidence the investment function is
non-linear and sigmoidal with two thresholds. This empirical finding confirms those of
Eberly (1997), Barnett and Sakellaris (1998), and Honda and Suzuki (2000).
4.2. Irreversibility
Our preferred estimated investment function is reported in Table 2. The estimate of
the slope coefficient in the first regime, β11 , is 0.0134 and its t-value is 0.2362. Thisshows that the null hypothesis that the slope in the first regime is zero cannot berejected. This suggests the existence of an inactive portion in which investment isinsensitive to Tobin’s q, because of the irreversibility of investment. This finding
confirms the theory of Abel and Eberly (1994), and is also consistent with the recent
empirical findings of Chirinko and Schaller (2008), who estimate a Euler equation andobtain empirical evidence of an irreversibility premium. Our estimates also indicatethat this inactive part holds for the domain q≦0.91. When Tobin’s q exceeds thisthreshold value of 0.91, the investment ratio starts to respond sharply to changes in
Tobin’s q.
What is the best estimate of the height of the investment ratio I / K for the domainq≦0.91? The floor of the investment ratio in the first regime is given by:
β10 + γ1L it −1 + γ 2 L it −1 ⋅ D it .

(4-1)

(See equation (2-1).)
Substituting our preferred estimates from Table 2 with q1 = 0.91 and q 2 = 1.20 into
equation (3-2) yields our preferred estimate for β10 . Making use of this estimate of β10
together with the estimates of γ1 and γ 2 from Table 2 enables us to estimate
21

equation (4-1) for each firm. We obtain such estimates for 104 firms. The simple
average of equation (4-1) over these 104 firms turns out to be 0.1992. This represents
the estimated depreciation rate δ in Figure 7.
4.3. Estimates of the Thresholds
Our preferred estimate of the first threshold q1 is 0.91. This implies that a firm starts

to increase its investment ratio when Tobin’s q exceeds 0.91. This estimate 0.91 issmaller than those from existing studies. The corresponding benchmark estimatesobtained by Barnett and Sakellaris (1998) are 1.95 for their first model and 1.13 fortheir second model. Honda and Suzuki (2000) report a corresponding benchmark
estimate of 1.62.
There are too few empirical studies to draw definitive conclusions on why theseestimates differ. However, our estimates may differ from theirs for several reasons.
First, unlike Barnett and Sakellaris (1998) and Honda and Suzuki (2000), we control
for the influence of financial constraints by adding the variables L and L ⋅ D to theright-hand side of equation (2-1).
Second, our estimate of the first threshold applies to smaller unlisted firms whereas
those of Barnett and Sakellaris (1998) and Honda and Suzuki (2000) apply to larger
listed firms.
Third, our estimates are for Japanese firms whereas those of Barnett and Sakellaris(1998) are for US firms. More empirical evidence must be accumulated to explain
estimation difference across different studies.
4.4. Financial Constraints
To control for the potential effects of financial constraints, we added γ1L it −1 andγ 2 L it −1 ⋅ D t to the right-hand side of equation (2-1). Our estimate of γ1 is -0.1622 in
22Table 2. The negative estimate implies that a firm with sufficient capital stock at the
beginning of a period undertakes less investment in that period, as expected. The
t-value of -10.2899 indicates that our estimate is highly significant.
D t in equation (2-1) is a dummy variable that is unity for firms that have never issued

corporate bonds. Our estimate of γ 2 is 0.0588 in Table 2. This implies that the level ofthe capital stock is a rough indicator of ‘creditworthiness’, and that relatively large
firms can borrow and invest more easily. The result is as expected. The t-value of
2.3592 indicates that the parameter is highly significant. Our results support thenotion that financial constraints matter and confirm those of in Honda and Suzuki(2006).
4.5. Market Power
Cooper and Ejarque (2001) argue that allowing the firm’s profit function to be strictly
concave to reflect its market power is sufficient to replicate regression results based on
q-theory in which profits significantly affect investment. Partially because of thisargument, we have carefully selected our samples. More specifically, we have chosen touse data on auto parts suppliers. The sampled firms are small and unlisted. They are
price takers and do not have the power to control their product prices. Therefore, thearguments of Cooper and Ejarque (2001) do not apply to our study.

5. Concluding Remarks
Based on the investment theory developed by Abel and Eberly (1994), we haveprovided a simple analytical model of adjustment costs, which produces a sigmoidal
investment function. We have also estimated the piecewise linear investment function,
which includes the linear model, a model with one threshold, the original model of Abel23and Eberly (1994) with two thresholds, and the sigmoidal model as special cases. Our
empirical evidence clearly supports the sigmoidal investment function.
The sigmoidal investment function casts serious doubt on the credibility of existinglarge macro-econometric models that specify a linear investment function.
Is it possible to aggregate sigmoidal investment functions over firms? If so, under what
conditions? Although answering these questions is important for understanding thevolatile nature of investment in practice, they are beyond the scope of our research.
What is the threshold value of Tobin’s q above which a firm undertakes newinvestment? Our estimate is 0.91. Are these threshold values the same acrossindustries and countries? Do they change over time? Answers to these questions
require estimated thresholds for different samples, which is a task for future research.

24

Appendix 1.Derivation of the Standard Investment Function
Following Abel (1980, 1990), we briefly derive the standard investment function in thisappendix. A firm maximises its value V, which is the sum of future discounted net cash
flows, with respect to capital investment I and other variable inputs, such as labour
input, X. To simplify the exposition, we ignore taxes, which can easily be incorporated
if necessary.

Vt =

∞

∫ [p {F(K , X ) − C(I , K )} − w X − p I ] R (t, s )ds ,
t

s −1

s

s

s

s −1

s

s

is s

(A-1)

s
where R (t , s ) = exp⎛⎜ − rv dv ⎞⎟ .
⎝ t
⎠

∫

K s , X s , I s , p s , w s , p is , F(K s−1 , X s ) , R (t , s ) , and rv denote the capital stock at theend of period s, variable input in period s, investment input in period s, the productprice in period s, the variable input price in period s, the investment goods price inperiod s, a homogeneous production function of degree one, the discount factor that
discounts the net cash flow of period s to their value at time t, and the discount rate attime v, respectively. C(I s , K s −1 ) is the adjustment cost incurred when a firm changes
its capital stock K. C is assumed to be strictly convex with respect to I, andhomogeneous of degree one with respect to I and K.
C represents the standard convex adjustment cost function. A firm maximises its
value Vt , given by (A-1), subject to the dynamic constraint,& = I − δK ,
K
t
t
t −1

(A-2)

where ⋅ denotes the derivate with respect to time.
The current value Hamiltonian of this maximisation problem is given by:
H t = p t {F(K t −1 , X t ) − C(I t , K t −1 )} − w t X t − p it I t + q t (I t − δK t −1 ) ,

25

(A-3)

where q t is the shadow price of capital. The above maximisation problem yields thefirst-order condition:
p it / p t + ∂C(I t , K t −1 ) / ∂I t = q t / p t .

(A-4)

The shadow price q t can be shown to be the sum of the discounted values of marginal
profits that accrue from an additional unit of capital installed at time t; see Abel (1980,
1990) for details.
When we assume a quadratic adjustment cost function:C(I t , K t −1 ) =2⎞α ⎛ It⎜⎜− δ ⎟⎟ K t −1 ,2 ⎝ K t −1
⎠

(A-5)

we have:⎞⎛ I∂C t / ∂I t = α⎜⎜ t − δ ⎟⎟ .
K⎠
⎝ t −1(A-6)

By substituting (A-6) into (A-4), we can derive the investment ratio I / K as a linear
function of q:⎞It
1 p it ⎛ q t⎜⎜=δ+− 1⎟⎟ .
K t −1
α p t ⎝ p it
⎠

(A-7)

This is the standard investment equation on which empirical studies are based.

26

Appendix 2.Data
We describe data in this appendix. Because our sample comprises unlisted firms, stock
price data are not available. Hence, we obtain our data from the financial statementsof unlisted firms.
A2.1. Data Sources
The sample data are selected from the financial statements of 104 unlisted
corporations that produce automobile parts. These data are complied by Tokyo Shoko
Research (TSR). The Kaisha-Sokan (Corporation List) is used to identify automobile
parts suppliers. Although we attempted to include all suppliers, the TSR database does
not cover all firms. Our final sample comprises information on 104 corporations.
The potential sample period is from 1977 to 2006, or 30 years. However, because few
firms are represented for the entire sample period, we use an unbalanced panel. The
total number of observations is 1,553. We discard as outliers cases with negativevalues of q t −1 . We also discard those samples for which any value of the gross profit
rate, q t −1 , or I / K is more than five standard deviations from the corresponding mean(see, e.g., Abel and Eberly (2002) for similar treatment of the data). Therefore, our final
number of observations is 1,527.
The fact that most of our sample consists of data with missing observations hampers
use of the perpetual inventory method as used by Hoshi and Kashyap (1990) andHayashi and Inoue (1991), and poses difficulties in measuring the replacement cost ofcapital, which is needed to measure marginal q. Hence, we follow Kaplan and Zingales(1997), Polk and Sapienza (2008), Almeida and Campello (2007) and others, and use
the book value of the nominal capital stock or total assets.
A2.2.Tobin’s q27

We define the gross profit rate as:(1 − τ )π + δK −1 ,

(A-8)

K −1

where τ , π , δ , and K −1 denote the corporate tax rate, operating profit, thedepreciation rate, and capital stock at the beginning of the period, respectively. When
we discount this value (A-8) by the cost of capital (1 − τ )r + δ with the assumption of

static expectations, we have Tobin’s q, q = V / K −1 , where V =(1 − τ )π + δK −1 and r(1 − τ )r + δdenotes the interest rate. We use the average loan rate of each firm for r.
A2.3.Corporate Investment
We use the same definition of corporate investment as in Honda and Suzuki (2006),
which improves measuring investment used in existing studies, including that ofHayashi and Inoue (1991). Anyone can download this working paper, Honda andSuzuki (2006), from http://www2.econ.osaka-u.ac.jp/library/local/e_HP/e_g_shiryo.html,
School of Economics, Osaka University.

Footnote* The authors thank Kazuo Ogawa for his advice on computational techniques. The
first author also gratefully acknowledges financial support from Grant-in-Aid No.
17203016 and the Global Center of Excellence Program at Osaka University, bothsponsored by the Ministry of Education, Culture, Sports, Science, and Technology.

28

Tables
Table 1: Summary Statistics from 1,527 Observations

Mean

Std Dev

Minimum

Maximum

I/K

0.31920.2327-0.16621.5964q

1.39140.46510.03284.0472

L

14.5747

1.126610.746217.7081

Table 2: Benchmark Estimates of the Model with Two Thresholds( q 1 = 0.91 and q 2 = 1.20 )Estimates
2.5645β 03 (intercept of the third regime)(11.0512)
0.0134β11 (slope of the first regime)(0.2362)0.3598β12 (slope of the second regime)(4.9787)
0.0354β13 (slope of the third regime)(1.8413)-0.1622γ 1 (coefficient of L)(-10.2899)
0.0588γ 2 (coefficient of L･D)

(2.3592)

The log likelihood of the estimated equation is 384.2105. Numbers in parentheses are
t-values.

Throughoutthepaper,

allheteroskedasticity-consistent standard errors.
29

reported

t-values

are

based

on

Table 3: Testing the Significance of ŷ itA in the Benchmark Model

q2

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0.2

-0.01

-0.66

0.26

0.60

1.09

0.83

0.91

1.10

0.3

-0.04

-0.43

0.25

0.60

1.08

0.83

0.90

1.09

0.4

0.47

0.36

0.46

0.70

1.15

0.87

0.93

1.12

0.5

0.85

0.85

0.69

0.83

1.24

0.94

1.00

1.17

0.6

0.93

0.96

0.73

0.85

1.25

0.95

1.00

1.18

0.7

0.38

0.27

0.43

0.68

1.13

0.85

0.92

1.10

0.8

0.47

0.36

0.42

0.67

1.13

0.85

0.91

1.09

0.91

0.49

0.35

0.64

1.11

0.84

0.91

1.09

q1

1.0

-0.49

0.30

0.63

1.11

0.86

0.93

1.12

1.1

-0.41

0.32

0.67

1.15

0.90

0.98

1.16

0.35

0.64

1.11

0.84

0.91

1.09

1.2

Note: The number in each cell is the t-value for the coefficient of the fitted variable ŷ itA .
Because none of these indicates statistical significance, the alternative models convey
no additional information.

30

Table 4: Testing the Significance of ŷ itB in the Alternative Model

q2

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0.2

3.87***

2.90***

2.31**

2.44**

2.43**

2.85***

3.11***

3.24***

0.3

3.82***

2.68***

2.19**

2.40**

2.41**

2.84***

3.10***

3.24***

0.4

3.67***

2.31**

1.93**

2.28**

2.35**

2.81***

3.08***

3.23***

0.5

3.51***

1.88*

1.70*

2.19**

2.31**

2.79***

3.08***

3.23***

0.6

3.40***

1.51

1.57

2.16**

2.30**

2.80***

3.09***

3.25***

0.7

3.36***

1.29

1.62*

2.23**

2.36**

2.83***

3.11***

3.26***

0.8

3.12***

0.58

1.60

2.28**

2.41**

2.88***

3.15***

3.29***

0.91

2.73***

1.73*

2.37**

2.47**

2.91***

3.16***

3.29***

q1

1.0

1.39

2.02**

2.49**

2.54**

2.93***

3.15***

3.25***

1.1

2.18**

2.40**

2.60***

2.57***

2.85***

2.99***

3.05***

2.64***

2.70***

2.64***

2.80***

2.86***

2.89***

1.2

Note: The number in each cell is the t-value for the coefficient of the fitted variable ŷ itB .
***”, **, and * indicate that the fitted variableŷ itB is significant at the 1% level, 5% level,and 10% level, respectively, which implies that the benchmark model conveys more
information than does each alternative model.

31

Table 5: Benchmark Estimates of the Model with One Threshold( q1 = 1.34 )Estimates
2.5797β 02 (intercept of the third regime)(11.0218)
0.1742β11 (slope of the first regime)(5.6493)0.0313β12 (slope of the second regime)(1.5338)-0.1627

γ 1 (coefficient of L)(-10.1861)
0.0577γ 2 (coefficient of L･D)

(2.3046)

The log likelihood of the estimated equation is 381.8763. Numbers in parentheses are
t-values.

Table 6: Testing the Significance of ŷ itA (the Fitted Values from the Model with One
Threshold) in the Best Model with Two Thresholds

q1

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.34

1.4

t-value

0.85

0.96

-0.27

-0.36

0.22

0.49

0.41

-

0.35

0.39

0.35

q1

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

t-value

0.49

0.64

1.03

1.11

0.94

0.84

0.78

0.91

1.01

1.09

Note: The number in each cell is the t-value for the coefficient of the fitted variable ŷ itA
in an alternative model with one threshold. Because none indicates statisticalsignificance, alternative models with one threshold convey no additional information.

32

Table 7: Testing the Significance of ŷ Bit (the Fitted Values from the Model with Two
Thresholds) in Alternative Models with One Threshold

q1

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.34

1.4

t-value

4.12***

4.13***

4.10***

4.09***

4.05***

3.91***

3.52***

3.14***

2.67***

2.53**

2.49**

q1

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

t-value

2.45**

2.51**

2.40**

2.47**

2.67***

2.86***

3.04***

3.11***

3.19***

3.24***

Note: Number in each cell is the t-value for the coefficient of the fitted variable ŷ itB . ***,**, and * indicate that the fitted variableŷ itB is significant at the 1% level, 5% level,and 10% level, respectively, which implies that the best model with two thresholds( q1 = 0.91 and q 2 = 1.20 ) conveys additional information not conveyed by eachalternative model.

33

FiguresFig. 1.TheIrreversibility of Investment in Abel and Eberly (1994)
I

K

0

q

Fig. 2.Linear Adjustment Costs in Barnett and Sakellaris (1998)
I

K

δ

0

q

34

Fig. 3.Asymmetric Adjustment Costs

G

0

δ

I

K

Fig. 4.An Alternative Specification of Irreversibility
I

K

δ

0

q

35

Fig. 5.A Horizontal First Part
I

K

δ

0

q

Fig. 6.Prohibitive Adjustment Costs for Extremely Large Investments

G

0

δ

36

λ

I

K

Fig. 7.A Non-linear Investment Function with Two Kinks
I

K

λ

δ

q

0

Fig. 8.A Model with Two Thresholds

y = β 30 + β13qyy = β 02 + β12 qy = β10 + β11q

0

q2

q1

37

q

Fig. 9.Sample Plot of Investment Ratio against Lagged Tobin’s q (N=1,527)
I

K

1.50

1.25

1.00

0.75

0.50

0.25

0.00

lagged q

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

38

References
Abel, A. B (1980). ‘Empirical investment equations: an integrative framework’, in (K.
Brunner and A. Meltzer, eds.), On the State of Macroeconomics, vol. 12 of theCarnegie-Rochester Conference Series on Public Policy, a Supplementary series to
the Journal of Monetary Economics, pp. 39-91.
Abel, A. B (1990). ‘Consumption and investment’, in (B. M. Friedman and F. H. Hahn,eds.), Handbook of Monetary Economics, vol. II, pp. 725-78, Amsterdam:
North-Holland,Abel, A. B. and Eberly, J. C. (1994). ‘A unified model of investment under uncertainty’,
American Economic Review, vo1. 84(5), pp. 1369–84.
Abel, A. B. and Eberly, J. C. (2002). ‘Investment and q with fixed costs: an empiricalanalysis’, mimeo, Northwestern University.
Almeida, H. and Campello, M. (2007). ‘Financial constraints, asset tangibility, andcorporate investment’, The Review of Financial Studies, vol. 20(5), pp. 1429-60.
Arrow, K. J (1968). ‘Optimal capacity policy with irreversible investment’, in (J. N.
Wolfe, ed.), Value, Capital and Growth. Papers in Honour of Sir John Hicks, pp.
1-19, Edinburgh: Edinburgh University Press.
Barnett, Steven A., and Plutarchos Sakellaris (1998). ‘Nonlinear Response of Firm
Investment to Q: Testing a Model of Convex and Non-Convex Adjustment Costs’,
Journal of Monetary Economics, vol. 42 (2), pp. 261–88.
Bernanke, Ben S., Henning Bohn, and Peter C. Reiss. (1988). ‘Alternative Non-Nested
Specification Tests of Time-Series Investment Models’, Journal of Econometrics, vol.
37(3), pp. 293-326.
Bo, Hong, Jan Jacobs, and Elmer Sterken (2006). ‘A Threshold Uncertainty Investment
39

Models for the Netherlands’, Applied Financial Economics, vol. 16, pp. 665-73.
Caballero, Ricardo J., Engel, Eduardo M.R.A. and John C. Haltiwagner (1995). ‘PlantLevel Adjustment and Aggregate Investment Dynamics’, Brookings Papers on
Economic Activity, 2, pp. 1-39.
Caballero, Ricardo J., and Eduardo M.R.A. Engel (1999). ‘Explaining Investment
Dynamics in U.S. Manufacturing: A Generalized (S, s) Approach’, Econometrica, vol.
67(4), pp. 783-826.
Chirinko, Robert S. and Huntley Schaller (2008). ‘The Irreversibility Premium’,(CESIFO Working Paper 2265).
Chida, Ryokichi (2003). ‘Uncertainty and Investment: Empirical Study Based on theUser Cost of Capital Approach’, (in Japanese), (Mimeo, Meiji University).
Cooper, Russell and João Ejarque (2001). ‘Exhuming Q: Market Power vs.
Capital Market Imperfections’, (NBER Working Paper 8182).
Cooper, Russell and João Ejarque (2003). ‘Financial Frictions and Investment:
Requiem in Q’, Review of Economic Dynamics, vol. 6(4), pp. 710-28.
Cooper, Russell and John C. Haltiwanger (2006). ‘On the Nature of Capital Adjustment
Costs’, The Review of Economic Studies, vol. 73(3), pp. 611-33.
Cummins, Jason G., Kevin A. Hassett and Stephen D. Oliner (2006).
‘Investment Behavior, Observable Expectations, and Internal Funds’,
American Economic Review, vol. 96(3), June, pp. 796-810.
Davidson, R. and J.G. Mackinnon (1981). ‘Several tests for model specificationin the presence of alternative hypotheses’, Econometrica, vol. 49(3), pp.
781-93.
Dixit, Avinash K. and Robert S. Pindyck (1994). Investment Under Uncertainty,
40

Princeton, N.J.: Princeton University Press.
Doms, Mark and Timothy Dunne (1998). ‘Capital Adjustment Patterns in
Manufacturing Plants’, Review of Economic Dynamics, vol. 1(2), pp. 409-29.
Eberly, Janice C. (1997). ‘International Evidence on Investment and Fundamentals’,
European Economic Review, vol. 41(6), pp. 1055-78.
Fazzari, Steven, Glenn Hubbard, and Bruce Petersen (1988). ‘Financing
Constraints and Corporate Investment’, Brooking Papers on Economic
Activity 1, pp. 141–95.
Goolsbee, Austan and David B Gross (2000). ‘Estimating the Form of
Adjustment Costs with Data on Heterogeneous Capital Goods’, (Mimeo,
University of Chicago).
Hansen, Bruce E (1999). ‘Threshold Effects in Non-Dynamic Panels: Estimation,
Testing, and Inference’, Journal of Econometrics, vol. 93(2), pp. 345-68.
Hayashi, Fumio (1982). ‘Tobin’s Marginal q and Average q: A NeoclassicalInterpretation’, Econometrica, vol. 50(1), pp. 213–24.
Hayashi, Fumio, and Tohru Inoue (1991). ‘The Relation between Firm Growthand Q with Multiple Capital Goods: Theory and Evidence from Panel Dataon Japanese Firms’, Econometrica, vol. 59(3), pp. 731-53.
Honda, Yuzo, and Kazuyuki Suzuki (2000). ‘Estimation of the Investment Thresholdsof Large Japanese Manufacturers’, Japanese Economic Review, vol. 51(4), pp.
473–91.
Honda, Yuzo, and Kazuyuki Suzuki (2006). ‘Is Cash Flow a Proxy for Financing
Constraints in the Investment Equation? The Case of Unlisted Japanese Firms’,
Discussion Paper 06-24, Osaka University.
41

Hoshi, Takeo, and Anil Kashyap (1990). ‘Evidence on q and investment for JapaneseFirms’, Journal of the Japanese and International Economies, vol. 4(4), pp. 371-400.
Hoshi, Takeo, Anil Kashyap, and David Scharfstein (1991). ‘Corporate
Structure, Liquidity, and Investment: Evidence from Japanese Panel Data’,
Quarterly Journal of Economics, vol. 106, February, pp. 33–66.
Kaplan, Steven N., and Luigi Zingales (1997). ‘Do Investment – Cash Flow
Sensitivities Provide Useful Measures of Financing Constraints?’, Quarterly
Journal of Economics, vol. 112(1), pp. 169–215.
Polk, Christopher, and Paola Sapienza (forthcoming, 2008). ‘The Stock Market andCorporate Investment: A Test of Catering Theory’, The Review of Financial Studies.
Tachibana, Minoru (2008). ‘Inflation Zone Targeting and the Federal Reserve’, Journalof the Japanese and International Economies, vol. 22(1), pp. 68-84.

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