Hitotsubashi University
Institute of Innovation Research

Detecting endogenous effects
by aggregate distributions:
A case of lumpy investments

Chaoqun Lai
Makoto Nirei
IIR Working Paper WP#09-03

May 2009

Institute of Innovation Research
Hitotsubashi University
Tokyo, Japan
http://www.iir.hit-u.ac.jp

Detecting endogenous effects by aggregate distributions:
A case of lumpy investments
Chaoqun Lai

Makoto Nirei∗

Utah State University

Institute of Innovation Research
Hitotsubashi University
May 25, 2009

Abstract
This paper studies the effect of strategic complementarity among firms’ lumpy investments on
the fluctuations of aggregate investments. We investigate an extensive panel data set on Italianmanufacturing firms. We first show that the fluctuations of fraction of firms that experience large
investment rates in a region-year follow a double-exponential distribution. We then estimate
the degree of the strategic complementarity within a region directly by estimating the firm’s
decision on lumpy investments. We propose a simple sectoral model which is capable of generating
the double-exponential distribution for the aggregate fluctuations that arise from the strategic
complementarity among firms’ lumpy investments. We argue that the shape and magnitudeof the aggregate fluctuations observed in the data are consistent with the degree of strategic
complementarity estimated at the micro-level in the same data.
Keywords: Strategic complementarity, endogenous effect, non-Gaussian fluctuations
JEL codes: L16, E22∗ Corresponding author, E-mail: nirei@iir.hit-u.ac.jp, Phone: +81-42-580-8417, Fax: +81-42-580-8410. We are grate-

ful to Luigi Guiso and Fabiano Schivardi for generously sharing the data. This research is initiated during Nirei’s visit
at Ente Einaudi. We are benefitted by comments from Marc Henry, Ryo Kambayashi, Young-Gak Kim, Andrea Tiseno,and Hajime Tomura.

1

1

Introduction

This paper empirically studies the aggregate fluctuations that arise from the complementarity of firms’
decisions. Our strategy is to focus on the distribution shape of the fluctuations in order to differentiate
the aggregate consequence of the endogenous or exogenous mechanism of the fluctuations.
Models of interacting individuals have studied the possibility that the interaction gives rise toaggregate shifts endogenously (Brock and Durlauf [4], Glaeser, Sacerdote, and Scheinkman [15]). It
has been recognized that the models of endogenous effects are often unidentified econometrically, as
Manski [21] formulated as a reflection problem. In the context of sectoral comovements, it has been
proposed to utilize the heterogeneous input-output relations to instrument the endogeneity by Shea
[23, 24] and Bartelsman, Caballero, and Lyons [2]. In the context of information spillovers, Guiso
and Schivardi [16] recently tackles the problem by utilizing additional information on the reference
network of firms.
In this paper, we analyze the same data on Italian firms as Guiso and Schivardi, with a newconcentration on large investment episodes. First, we establish that the aggregate fluctuation of the
lumpy investments follows a non-normal, exponential distribution empirically. We show that thenormality hypothesis is rejected statistically, and is also dominated statistically by an alternative
double-exponential hypothesis. Secondly, we present a simple sectoral model with lumpy investmentsthat generates the exponential distribution, while the model generates a normal distribution if it lacks
the lumpy behaviors. The model predicts that the parameter of the distribution of the aggregate
fluctuations is determined by the degree of strategic complementarity among firms’ lumpy investments.
Thirdly, we estimate the strategic complementarity from the firm-level data. We use the inputoutput matrix to identify the endogenous effects for the firm’s decision on lumpy investments. In
accordance with Bartelsman et al., we observe the externality effect through the output-weighted
sectoral activities in the short run. Finally, we show that the complementarity estimated at the microlevel is consistent with the fluctuation distribution estimated at the aggregate level. While Guiso and
Schivardi investigate information spillovers of continuous investment decisions within an industry, we
focus on interactions of lumpy investments within a region. By finding the exponential distribution,
we argue that the lumpiness of the investment decision has an important consequence on aggregate
fluctuations. Also, we use the input-output relations to identify the endogenous effects, while Guiso
and Schivardi utilize the interview data that describe which firms observe which upon their investment

2

decisions.
Aggregate consequences of micro-level lumpy investments have been a topic of extensive discussion.
On the one hand, researchers traditionally emphasize the “law of large numbers” effect by which
individual lumpy investments cancel out with each other in aggregation. For example, Long andPlosser’s [19] sectoral business cycles model has met such arguments by Lucas [20] and Dupor [13]. On
the other hand, researchers have investigated the effects of the density of firms around the threshold for
lumpy adjustments on aggregate investments. If the capital level of many firms are positioned near to
the threshold level for adjustments and far from the desired level of capital, then aggregate investment
is likely to increase in near future. This mechanism has been studied empirically by Caballero andEngel [6] and Cooper, Haltiwanger, and Power [10]. However, theoretical investigations have shownthat the distribution of firms’ gaps between desired and actual capital has an invariant distribution
and its convergence is fast (Caplin and Spulber [8] and Caballero and Engel [5]), implying that someaggregate shocks need to be present in order for the extensive margin to fluctuate, such as in themodel of Caplin and Leahy [7]. In this line of research, the lumpy investments provide a mechanismof amplifying the exogenous aggregate shocks and of deferring the timing of the impacts.
There is a possibility that the interaction of firms’ lumpy investments causes the aggregate fluctuations without aggregate shocks (Nirei [22]). Consider a one-sided (S,s) policy in which firms tend
to be distributed uniformly over an inaction band. If there are a continuum of firms, we obtain the“neutrality” result in which lumpy investments do not cause aggregate fluctuations (Caplin and Spulber [8], Caballero and Engel [5]). Now, suppose that there are a large, finite number of firms which
are distributed evenly in the inaction band with slight disturbances. Also suppose that the lumpy
investments are strategic complements. Then, when a firm that is the closest to the threshold decides
to invest, it can cause another lumpy investment of the firm that is positioned at the second closest
to the threshold. The investment of the second closest can further cause an investment of the thirdclosest, and so on. This chain reaction stops at the point where there is no firm that is positioned
close enough to the threshold. The mechanism is similar to the domino game: the line of falling tiles
stops at where two adjacent tiles are standing apart too far. Since the stopping point is altered greatly
by a slight change in standing points, the domino effect generates a varying degree of amplification ofindividual shocks.
We investigate such domino effects in the interaction of lumpy investments in the Long-Plosser
type model. The Long-Plosser model exhibits a multiplier effect of idiosyncratic shocks. If the3

individual behavior is lumpy, then the multiplier itself behaves stochastically, because the it depends
on the initial random state just like the standing points of domino tiles. In the model, we showthat the aggregate fluctuations generated by the domino effect follow an exponential tail, whereas thestandard Long-Plosser model generates a normal distribution. We argue that the particular shape ofthe distribution of aggregate fluctuations may serve as a symptom that differentiates the endogenous
shock model from the exogenous shock model. Often the sources of the exogenous common shock aredriven by many factors. In those cases, the central limit theorem predicts that the common shockshould follow a normal distribution even when the factors that comprise the common shock follow
non-normal distributions. Conversely, when we find a normal distribution in the aggregate variable,
it is reasonable to include an unspecified set of exogenous shocks in the model.
The central limit theorem does not characterize the aggregate distribution if there are endogenous
effects, since the firms’ actions are correlated in that case. Let us consider the simplest case ofendogenous effects where the probability for agent i to act (ai,r = 1) depends on the realized action of
its neighbor i − 1. Suppose that Pr(ai,r = 1) = βai−1,r . In this case, an action by i = 1 may cause aPN
domino effect on the successive agents. The distribution of the number of agents who act, i=1 ai,r ,follows an exponential distribution. The distribution shape depends on the precise structure of theinteractions. In the case of the herd behavior model (Banerjee [1]), for example, the agent i’s action
is affected by the actions taken by any agent j who acted before i, i.e., j = 1, 2, . . . , i − 1. In this case,there is a probability mass on the event that all agents act. The exponential distribution generated
by domino effects appears to be robust as long as the domino effect does not degenerate to such a
deterministic cascade. We will show that this is in fact the case in our sectoral model.
The rest of the paper is organized as follows. Section 2 describes the data and presents an evidence that the fraction of firms that engage in large investments follows the exponential distribution
rather than the normal distribution. Section 3 presents a model of lumpy investment that generates
the exponential distribution. The model provides a testable prediction that relates the macro-levelproperty, namely the distribution parameter of aggregate fluctuations, to the micro-level property,
namely the degree of strategic complementarity. In Section 4, we numerically simulate the calibrated
model, and show that the empirical distribution is fitted well by the simulated distribution. In Section
5, we estimate the firm’s decision on lumpy investment directly, and match the estimated strategic
complementarity with the distributional estimate. Section 6 concludes.

4

0

5

Percent
10

15

20

Firm−Level Investment Distribution

0 .05.1.15.2.25.3.35.4.45.5.55.6.65.7.75.8.85.9.95 1
Investment Rate

Figure 1: Histogram of firms’ investment rates IP K(i, t)

2Fluctuations of aggregate lumpy investments

2.1

Data and variables

We use longitudinal data of Italian firms drawn from the Company Accounts Data Service (CADS).
The data set we use was compiled by Guiso and Schivardi [16]. The annual data set covers over 30,000
firms from 20 regions, 25 industries, and 15 years from 1982 to 1996.
We focus on the number of firms that experience large investment episodes in a given referencegroup in each year. First, we define an investment-capital ratio, IP K(i, t), for each firm i and year t.
Figure 1 depicts a histogram of IP K(i, t). We can see that the distribution is highly skewed to theright. 24% of the samples have investment rates less than 2%. The tail is long, and it implies that a
relatively small fraction of firms has large impacts on aggregate investments.
Such a long tail in investment rates is observed commonly as in Doms and Dunne [12] and Cooper
et al. [10]. Doms and Dunne also found the salient spiking of investment behaviors, in which firms
adjust capital actively in a short period and show little adjustments in other periods. Along withthe literature following their findings, we are most concerned with the aggregate consequence of suchspiking investments. Thus, we divide the heterogeneous behaviors of IP K into two polar groups,

5lumpy investments and inactions. That is, we convert the investment-capital ratio into a binary
variable, d(i, t), which takes 1 if IP K(i, t) > d¯ and zero otherwise. We take the threshold d¯ at
20% for our estimation, in order to be comparable with the literature on lumpy investments ([10] forexample). The sum of the lumpy investments defined as such accounts for about 20% of the aggregate
investments in our data. The results obtained in this paper are not affected very much when wechange the threshold to 10% or 30%.
This paper is concerned with the fraction of investing firms, X(G· , t) =Pi∈G· d(i, t)/#G· , where

G· is a reference group and #G· is the number of firms in G· .1 We drop the groups that have firms
less than 10, because the behavior of the fraction X can be overly volatile for the groups that have asmall number of firms.2 We then define a centered fraction variable X̃(G· , t) ≡ X(G· , t)−hX(G· , t)iG· .
Namely, yearly common effects across reference groups is subtracted from X(G· , t). We work withthe region reference group Gr which is a set of firms that operate in the same region r, as well as theindustry-region reference group Gl,r where l is an industry. The variables of interest are summarizedin Table 1.
Figure 2 plots the histogram of X(Gl,r , t) for each year. We observe a general pattern of the
distributions: they are fairly centered and their center location fluctuates over years. In this paper,
we regard the annual common shock as exogenous, and concentrate on the remaining fluctuations
observed across reference groups. The left panel of Figure 3 shows the histogram of X̃(Gl,r ) in which
the yearly effect on X is subtracted. The histogram of X̃(Gl,r ) is fairly symmetric. In the right panel
is the centered histogram for region reference groups X̃(Gr ), which shows the similar pattern to theindustry-region histogram.
Our goal is to characterize and explain the distribution of the fraction of investing firms X̃. Figure
4 shows semi-log plots of the cumulative distributions of X̃(Gl,r , t) and X̃(Gr , t) for positive andnegative sides. To produce the plot, we first divide X̃ into two groups depending on the sign of X̃,and for each positive or negative group we rank |X̃| in a descending order. Then, the log of the rank
divided by the total number of observations is plotted against |X̃| for positive and negative groups.
1 The number of observations in the reference G may change over time. The time subscript t is suppressed here only·

for simplicity.
2 Original data file contains 306363 observations. We drop observations with missing industry or region codes, andone outlier of the variable “ioverk”. Then we are left with 283210 observations. When we work with the fraction
variables X, we also exclude small reference groups that contain less than 10 firms. In the regression analysis thatinvolves a lag variable, we exclude observations in the initial year 1982.

6

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

0
10 15
0

5

Density

5

10 15

0

5

10 15

1983

0

1

0

.5

1

1996

0

5

10 15

1995

.5

0

.5

1

0

.5

1

X_IRY
Graphs by annorif

Figure 2: Histograms of X(Gl,r , t) for each year

800

55
50

700
45
600

40
35
Frequency

Frequency

500

400

300

30
25
20
15

200

10
100
5
0

−0.4

−0.2

0
X(I,r,t) tilde

0.2

0.4

0

0.6

−0.5 −0.4 −0.3 −0.2 −0.1
00.1
X(r,t) tilde

Figure 3: Histograms of X̃(Gl,r ) (left) and X̃(Gr ) (right)

7

0.2

0.3

0.4

0.5

Obs

Mean

Std. Dev.

Min

Max

Mediand(i, t)283210

0.224

0.417

0

1

0

X(Gl,r , t)

3234

0.213

0.141

0

0.830

0.185

#Gl,r

3234

84

154

10

1465

33

X(Gr , t)

298

0.208

0.119

0

0.538

0.179

#Gr

298

950

1396

10

7919

381

X(Gl , t)

362

0.210

0.116

0

0.571

0.183

#Gl

362

782

936

10

4825

392

Table 1: Description of variables. X is the number of firms that engage in lumpy investments divided
by the total number of firms in a reference group. Reference groups with less than 10 firms are dropped
from the observation.

Thus, the vertical line shows the percentile in the descending order. We observe a steeper slope forthe regional distribution compared to the industry-regional distribution.
We now fit parametric distributions to the empirical distribution. Our null hypothesis is the normal
distribution. An alternative hypothesis is the double-exponential distribution, in which the distribution of X̃(Gl,r , t) follows exponential distributions for the positive and negative sides with possibly
different means λ+ and λ− . Namely, the likelihood function is Pr(X̃ = x̃|x̃ > 0) = (1/λ+ )e−(1/λ+ )x̃
and Pr(X̃ = x̃|x̃ < 0) = (1/λ− )e−(1/λ− )(−x̃) . The other alternative hypothesis is the Laplace distribution in which both positive and negative sides of X̃ follow an exponential distribution with the samemean λ: Pr(X̃ = x̃) = (1/(2λ))e−(1/λ)|x̃| .
Table 2 shows the results of the maximum likelihood estimation for each parametrization and foreach reference group. We start from the industry-region reference group X̃(Gl,r , t). The estimatedmean of the exponential distribution is 0.071 (standard error 0.002) for the positive side and 0.062(standard error 0.002) for the negative side of X̃. The estimated λ for the Laplacian is 0.066, whichis the middle value for the exponential slopes for the positive and negative sides. The exponential
hypothesis has a larger log likelihood value than the Laplacian hypothesis, because the Laplacian
distribution is equivalent to the double-exponential distribution with restriction λ+ = λ− .
We test the normality hypothesis by a likelihood-based test. Let L(i; H) denote the likelihood of

8

0

10

G(r),positiveG(r),negativeG(l,r),positiveG(l,r),negative−1

1−F(X)

10

−2

10

−3

10

0

0.1

0.2

X

0.3

0.4

0.5

Figure 4: Semi-log plots of cumulative distributions of |X̃(Gl,r , t)| and |X̃(Gr , t)| for positive andnegative values. The distributions are cumulated from above.

9

Industry-region X̃(Gl,r , t)Exponential
ML estimates

λ+

0.071
(0.002)λ−Laplaceλ0.066

Normalµ(0.001)

0.062

3337

Vuong’s statistic

4.98

σ

3185

-0.50
10.70

0.000

Exponential
λ+(0.002)(0.002)
Log Likelihood

Region X̃(Gr , t)

0.090

0.031
(0.003)

λ−

λ

0.031

(0.000)(0.003)

3200

531.9

Null

1.30

Normalµ(0.002)

0.031

Null

Null

Laplace(0.003)σ

0.042(0.000)

518.4

519.7
Null

-0.12
3.06

0.000

Null

Null

Table 2: Parameter estimates for various distributions. Standard errors are in parentheses.
sample point i under the hypothesis H. Define the log likelihood ratio for each i as li = log L(i; H1 ) −√log L(i; H0 ). We use Vuong’s statistic V ≡ N ¯li /Std(li ) which follows a standard normal distribution
if the hypotheses H0 and H1 are “equivalent” in the sense of Kullback-Leibler information. Thus, if
V computed for H1 against the null H0 is greater than 1.96, then the null is rejected in favor of thealternative at the 5% significance level. Vuong’s statistics are reported in Table 2 for the exponential
and the Laplacian hypotheses against the Gaussian null hypothesis and for the exponential against the
Laplacian. The exponential is favored against the Gaussian and the Laplacian at the 1% significance,
while the Laplacian is not significantly different from the Gaussian. We repeat the same test forX̃(Gr , t) on the right side of Table 2. Vuong’s statistics suggest that the exponential hypothesis isfavored against the Laplacian hypothesis at 1% significance level, but it is not significantly different
from the Gaussian. Note that we have fewer observations for the region group compared to theindustry-region group. Thus, we lose the testing power somewhat due to the limited number ofobservations for the case of regional distribution.
We now turn to another normality test which is based on higher moments. Table 3 shows the higher
moments of X̃. The large kurtosis indicates that X̃ is leptokurtic. Standard normality tests such as
skewness-kurtosis test, Shapiro-Wilk test, and Shapiro-Francia test reject the normality hypothesis
overwhelmingly for both X̃(Gl,r , t) and X̃(Gr,t ).

10

Variable

Observations

Mean

Std Deviation

Skewness

KurtosisX̃(Gl,r , t)

3234

0

0.090

0.537

5.274

X̃(Gr , t)

298

0

0.042

0.692

6.703

Table 3: Moments of X̃

We also conduct a test that is only based on kurtosis but not on skewness. The concentration of the
4th moment is useful in order to test among a class of symmetric distributions, in our case the Laplacianand the Gaussian. The true kurtosis is equal to 3 under the Gaussian hypothesis and 6 under the
PN
PN
Laplacian hypothesis. Consider a method of moment estimator ( i=1 x4i /N )/( i=1 x2 /N )2 . Then,its asymptotic variance is V (x4i )/σ 2 N . Under the Gaussian hypothesis, the variance is equal to
96/N . Thus, the sample kurtosis in Table 3 rejects the Gaussian hypothesis at 1% significance, eventhough the estimator is not efficient. In contrast, the asymptotic variance of the MOM estimatoris (8! − 4!2 )/28 N = 155.25/N under the Laplacian hypothesis. Thus, the MOM test permits the
Laplacian hypothesis at 5% significance for X̃(Gr , t).
The good fit of the exponential parametrization can be seen in Figure 4. In the semi-log scale,an exponential distribution function would show as a linear line. Our plot demonstrates that thepositive side of the distribution of X̃(Gl,r , t) is well fitted by a linear line. The negative side has akink beyond which the distribution shows a faster decline than the positive side. The faster decline
in the negative side can be caused by the boundary effect of X(Gl,r , t) which takes only non-negativevalues. Since X(Gl,r , t) has mean 0.21 and standard deviation 0.14 as in Table 1, it is natural thatthe distribution of X̃ shows a boundary effect that is seen in the empirical distribution at around−0.17. This explanation is also consistent with the patterns of yearly distributions shown in Figure
2. The distribution is skewed in the years when the distributions are close to zero, such as in 1984
and 1992, whereas the distribution is not skewed in the boom years when the distributions are far
from zero, such as in 1994, 1995, and 1996. The downward kink may be the reason why Vuong’s test
cannot reject the Gaussian hypothesis for X̃(Gl,r , t). The right panel of Figure 4 shows a distributionof X̃(Gr , t). The distribution does not exhibit the kink for the negative side, since the zero-bound is
not binding for X̃(Gr , t) due to its small standard deviation.
In sum, we show that the distribution of the fraction of firms that engage in lumpy investments

11exhibits an exponential decay rather than a Gaussian decay at the industry-region level or the region
level. Thus, the empirical distribution favors the model that generates an exponential pattern thanthe models that generate normal distributions for the aggregate fluctuation of investments. The
fluctuation we investigated captures the extent of aggregate fluctuations that are not explained by
the common annual shocks. For the case of the regional level, the variation of X̃(Gr , t) accounts for athird of the total variation in X(Gr , t). In the following sections, we try to understand the mechanismthat generates the exponential pattern of the aggregate fluctuation.

3

A model of endogenous investment fluctuations

This section presents a simple model of endogenous investment fluctuations that generates an exponential tail for the distribution of X as we observed previously. Consider that there are N firms that
produce differentiated goods with a production function:y i = Ai n α
i .

(1)

Ai is an exogenous productivity, and ni is input goods. The returns-to-scale of production is α < 1.
We assume that the firm is constrained on the operation scale ni to a discrete set:±2
ni ∈ {1, λ±1i , λi , . . .}(2)

where λi > 1 is the exogenous parameter for lumpiness. The constraint represents the situation wherefirms must choose an integer for the number of plants, although we restrict growth rates rather thanlevels for the sake of tractability.
Input ni is the composite good produced by using all the goods in a CES manner:σ/(σ−1)NX
1/σ (σ−1)/σ 
ni = 
χi,j zi,j
,

(3)

j=1

where σ > 1. Note that the input weight χi,j is different across i. By χi,j we incorporate theheterogeneous inter-industrial effects in the input-output relations. Define the aggregate index of
input prices for industry i as a weighted sum:
1/(1−σ)NXPi ≡ 
χi,j p1−σ
.
j
j=1

12

(4)

∗
Then, the derived demand for good j is given in an isoelastic form as: zi,j= (pj /Pi )−σ χi,j ni . Thus,∗
changes proportionally to the weight χi,j when the input ni varies. The
the derived demand zi,jP∗minimized cost satisfies j pj zi,j
= Pi n i .

We assume that each firm i is owned by household i as in a backyard production economy. Household i’s income is the profit πi from producing good i. Households have a common utility function
P
σ/(σ−1)
1/σ iN(σ−1)/σU (Ci ) over a composite consumption good: Ci =. Aggregate varij=1 χC,j (zC,j )P i
PP
ables across households are defined as zC,j = i zC,j , C = i Ci , and π = i πi . Also define theaggregate index of consumer price:1/(1−σ)NX,χC,j p1−σ
PC ≡ 
j

(5)

j=1

P i∗
P
−σ
i
is:
χC,j i Ci . The
and normalize it to 1. Then the derived demand for zC,j
i zC,j = (pj /PC )Pi∗optimal expenditure satisfies j pj zC,j
= PC Ci = πi .
P
The equilibrium condition for good i is yi = j zj,i + zC,i . Thus, the total demand function is
isoelastic:yi = p−σi Yi ,(6)

where Yi is the aggregate demand factor for good i:
Yi ≡X

Pjσ nj χj,i + CχC,i .

(7)jAggregate output is defined as Y ≡1−1/σ 1/σYi .
i=1 yiPN

Each firm is a monopolistic supplier of a differentiated good. Its profit is defined as πi = pi yi −P

j pj zi,j .

The second term, the cost of inputs, is equal to Pi ni at optimum by the result of cost

minimization. The monopolist’s problem is thus maxni pi yi − Pi ni subject to (1), (2), and (6). We
define an “inaction band” of ni by the lower and upper thresholds [n∗i , λi n∗i ). We can find such
thresholds by noting that the profits must be equalized at the two boundaries of the inaction band(see [22]). Then the bound is solved as:α(1−1/σ)

n∗i =

λi

λi − 1−11! 1−α(1−1/σ)

1−1/σ

Ai

1/σ

Yi

Pi−1

(8)

Consider the situation where the productivity Ai is a random variable independent across i. For
each realization of the productivity profile, an equilibrium profile of ni is determined by the threshold13

rule n∗i ≤ ni < λi n∗i and the discreteness constraint (2). We are interested in the probability distribution of the fraction of investing firms (the fraction takes a negative value if firms adjust downward),
when the exogenous productivities Ai ’s are slightly perturbed.
Suppose that we are at an equilibrium, and now a small perturbation on Ai affects the equilibrium.
An increase in Ai raises n∗i , and then induces i to increase ni to λi ni if i was near the lower thresholdn∗i . The increase in ni raises the demand for other goods Yj ’s. The shift in demand increases n∗j ’s,
and thus the firms near the lower threshold n∗j may then choose to increase nj by a factor of λj . This
propagation process continues until the economy reaches a new equilibrium. We regard this process
as a fictitious tatonnement, and an equilibrium selection algorithm as in Vives [25] and Cooper [9].
The fictitious tatonnement is defined in Appendix A.1 precisely. The selected equilibrium has theproperty that the equilibrium is the closest to the initial equilibrium in the direction of the search(namely, the direction of the initial adjustment).
In what follows we analytically show that the model generates the exponential distribution for
the fluctuations of the fraction of investing firms. The analytical characterization is carried out byembedding the fictitious tatonnement in a stochastic process ex ante the random variables realize,
following the method presented in a separate paper [22]. Here we consider the case when the input
weight χi,j is the same across i: χj = χi,j for all i. The numerical simulations of the heterogeneouscase under calibrated parameters are presented in the next section. The aggregate output is simplified
P 1/σ 1−1/σ σ/(σ−1)
as Y = ( i χi yi). Then we obtain the optimal threshold as:
1−1/σ

n∗i

1

= ai Ai1−α(1−1/σ) Y σ(1−α(1−1/σ))1 (σ−1)(1−α(1−1/σ))
1−1/σ
X
1
1
1−
α(1−1/σ) = ai Ai1−α(1−1/σ) χjσ Aj σ nj

(9)
(10)

j

where

α(1−1/σ)ai ≡λiλi − 1

−1

1

χiσ .

(11)

By the Taylor series expansion as in Appendix A.2, we obtain:d log n∗i =NXβ(σ − 1)d log njd log Ai + βbj+ O(N −2 )
α
log
λ
j
j=1

(12)

where
β

≡

1
σ(1/α − 1) + 1
14

(13)

1

bj

≡

1− 1

α(1−1/σ)α(1−1/σ)λj−1
χjσ Aj σ nj,0P σ1 1− σ1 α(1−1/σ) α(1 − 1/σ) .
nl,0l χ l Al(14)

Now define the initial gap between the actual input and the lower threshold as follows:si ≡log ni − log n∗ilog λi(15)Here ni is the initial level of input that is determined by the productivity and the aggregate demand before the perturbation. Thus si is the state associated to the initial equilibrium before theperturbation. Then, the optimal rule is to invest upon the perturbation if:


N
∗
X
1  β(σ − 1)d log nj 
d log ni=−d log Ai + β
bj
si < −log λilog λiαlog λjj=1(16)

for large N . From (16), we see that the investment decision depends on the exogenous shock on Ai
and the endogenous shock on Y . An investment by firm j increases Y , which reduces the gap andincreases the likelihood of firm i to invest. The magnitude of this impact of log λj on the gap of ithrough the strategic complementarity is:−βbj d log nj
log λi log λj(17)Note that for an upward lumpy adjustment, d log nj / log λj = 1. Also, the first fraction in (14) is 1/N
if firms are homogeneous, and the second fraction is approximately log λj when the lumpiness is small(λj is close to 1). Thus, the impact on the gap variable is roughly equal to β/N in the homogeneouscase. Consider a random variable of si unconditional on λi and Ai that is distributed on the unit
circumference of a circle. We suppose, as a first-order approximation of the distribution, that thedensity around si = 0 is constant q. Then the probability that j’s adjustment induces i to adjustis equal to qβbj / log λi . The probability is reduced to qβ/N for the homogeneous case with small
lumpiness.
Consider the fictitious tatonnement process that starts from the initial state (si ). Suppose that
there is one firm j that adjusts capital at the first round of the tatonnement upon the perturbation.
Then, the number of firms that are induced to adjust in the second step of the tatonnement conditionalon the adjustment of firm j follows a convolution of a Bernoulli trial with probability qβbi / log λj acrossi. Thus the number of firms that are induced to adjust in the second step unconditionally follows
a mixture across j of the convolution above, which is an integer random variable with mean φ ≡15

PNqβE( i=1 bi )E(1/ log λj ). Then, the number of firms mu that adjust in each step u of the tatonnement
conditional to mu−1 follows a distribution that is represented as a mu−1 -times convolution of a certain
integer random variable with mean φ. The integer random variable is identically distributed acrossPu
u as long as N is so large that v=1 mv /N is small and thus the density of si of the affected firms
is constant at q (see [22] for further details and generalizations). Then we can utilize the following
theorem.
Theorem 1 (Otter) Consider a branching process mu , u = 1, 2, . . . , that starts from m1 = 1. Letφ denote the mean number of children in u + 1 born by each parent in u. Define the total numberPT
M = u=1 mu . Then:
Pr(|M | = m|m1 = 1) = C0 (eφ−1 /φ)−m m−1.5(18)

for a large integer m, where C0 is a constant.
When N tends to infinity, mu approximately follows a Poisson distribution with mean φmu−1 . By
using this, the analytical result can be sharpened when the lumpiness λi , the productivity Ai , andthe input weight χi,j are homogeneous across i, j.
Proposition 1 When λi , Ai and χi,j are common across i, j, M follows a symmetric probability
distribution function:
Pr(|M | = m | m1 ) = m1 e−φ(m+m1 ) φm (m + m1 )m−1 /m!.

(19)

The tail of the distribution function is approximated as:√
Pr(|M | = w | m1 ) ∼ (m1 e(1−φ)m1 / 2π)(eφ−1 /φ)−m m−1.5 .

(20)

Proof: This result draws on Nirei [22]. See Appendix A.3.
Proposition 1 shows that M follows a distribution that is a mixture of power and exponentialfunctions as seen in (20). Since the exponential function declines faster than the power function, thetail of the distribution is dominated by the exponential part. We will argue that this corresponds toour empirical findings that X follows an exponential distribution. Proposition 1 further shows thatthe slope of the exponential distribution is determined by φ. By Equation (18), the exponential slope
is φ − 1 − log φ, and the mean and standard deviation of the exponential part is determined by the16

inverse of the slope. Since the slope is decreasing in φ in the region φ < 1, a large φ implies a large
fluctuation of M . This is intuitive, because φ is the mean number of firms that are induced to invest
by a firm’s lumpy investment in the fictitious tatonnement.
In the model, φ is directly affected by β, which summarizes the impact of a lumpy investment
on the gap variable si of other firms in the micro-level investment decision (16). In this sense, φis the parameter of strategic complementarity among firms’ lumpy investments. From (13), a small
returns-to-scale (small α) or a low mark-up (large σ) implies a small β and φ, and thus a steep slope.
This prediction will be confirmed in a calibrated simulation in the next section.
Since β appears in the single firm’s decision rule (16), we can estimate it from the panel data
independently from the estimation of the distribution of the aggregate fluctuations. Thus, Proposition
1 provides a testable implication that connects the two estimates at the micro level and the macrolevel. In Section 5, we estimate the firm’s decision at the micro level, and use this connection to test
the validity of the model.

4

Numerical simulations

In this section, we simulate the model under calibrated parameters. We calibrate the distribution of
λi to an exponential distribution with lower bound 1.1 and mean 1.2, which mimics the histogram ofinvestment rate in our data (shown in Figure 1). We set χi,j as a transpose of the Italian input-output
matrix normalized to 1 column-wise.3 We also set Ai to mimic the distribution of total input demandacross industry in the input-output matrix. The returns to scale is set at α = 0.8 and 0.9. The σ isset at 6 and 11, which corresponds to the mark-up rate µ at 20% and 10%, respectively (the mark-up
rate satisfies µ = 1/(σ − 1)).
We compute the distribution of the number of investing firms numerically under the calibration.
We first draw the profiles of λi and Ai , and compute the initial equilibrium for the realization. Then
we add a small shock to Ai (the variance of the shock is set so that the mean number of firms that
adjust to the shock initially is 1), and compute a new equilibrium. The left panel of Figure 5 shows thehistogram of the fraction of firms that adjust their capital upward in the transition from the old to the
3 We use the 1985 version of the Italian transaction table to identify the industrial relation. We convert the original

4-digit industry code of the Italian firms in our data set to 2-digit industry code that is consistent with the OECDclassification. With the data available we end up with 25 industries.

17

5

0

10

10

!=0.9,µ=0.2!=0.9,µ=0.1!=0.8,µ=0.2!=0.8,µ=0.1

410G(r),positiveG(r),negative
Simulation, positive
Simulation, negative−1

1−F(X)

Frequency

3

10

10

2

10

1

10

−2

10
0

10
−0.05

−0.04

−0.03

−0.02 −0.01
00.01
Fraction of investing firms

0.02

0.03

0.04

0

0.05

0.1
X

0.15

0.2

Figure 5: Left: simulated histogram of the fraction of investing firms when the sizes of lumpiness,
industry, and input weights are heterogeneous. Right: the cumulative distribution of |X̃(Gr , t)| andthe simulated distribution.

new equilibrium (a negative number means that those firms adjust capital downward). The verticalaxis is log-scaled, and thus a linear line implies an exponential decay. The simulated histogram clearly
conforms to the exponential pattern.
Next, we try to match with the empirical exponential distribution of X̃(Gr , t) shown in Figure 4.
The variance of X̃(Gr , t) is much larger than the variances of the simulated distributions shown inthe cases α = 0.8, 0.9. Thus, we repeat the simulation with a higher returns to scale at α = 0.95, andalso computed the sum of the fraction of firms across 60 periods. We compare the resulting simulated
distribution with the distribution of X̃(Gr , t) in the right panel of Figure 5. We observe a goodfitting. The summation across periods has a similar effect to increasing the perturbation size. The
time-accumulation does round the shape of exponential distribution toward the normal distribution
near the mean 0, as predicted by the central limit theorem, but it still preserves the exponential tail far
from the mean. Moreover, it improves the fitting around 0, because the empirical distribution showsa slight deviation from the exponential distribution that is well captured by the rounding effect.
P
The left panel of Figure 6 shows the distribution of the log-deviation in aggregate output i pi yi .
The plots show that the aggregate output largely inherits the exponential pattern in the distribution

18

7

6

10

10

!=0.9,µ=0.2!=0.9,µ=0.1!=0.8,µ=0.2!=0.8,µ=0.1

5

10

6

10

4

5

10
Frequency

Frequency

10

3

10

2

4

10

3

10

10

1

2

10

10

0

10
−0.015!=0.9,µ=0.2!=0.9,µ=0.1!=0.8,µ=0.2!=0.8,µ=0.1

1

−0.01
−0.00500.005
0.01
Log deviation of aggregate output in lumpy economy

10
−6

0.015

−4
−2
0
2
4
Log deviation of aggregate output in smooth economy

6
−4

x 10

Figure 6: Simulated histograms of the log-deviations of aggregate output when adjustments are lumpy(left) and when adjustments are smooth (right)of the fraction of investing firms, although we observe a fatter tail than the exponential for the caseα = 0.9 and µ = 0.2. We can analytically show that the inheritance of the exponential decay fromthe fraction of investing firms to the aggregate output is exact if the input weights and the lumpiness
are homogeneous.
In contrast to the exponential pattern we observe so far, the aggregate fluctuation follows a normal
distribution if there is no discreteness constraint (2). The right panel of Figure 6 shows that the log
deviation of aggregate output Y when the same realizations of the random variables occur as in
the lumpy economy we have observed so far. In the semi-log plot, a normal distribution looks as aparabola. The plot clearly shows the parabola for the case of continuous adjustments.
We can derive the normal distribution in the smooth economy analytically when λi and χi,j arecommon across i. By taking the first order condition with respect to ni and combining with theequilibrium conditions, we obtain the equilibrium aggregate output as:σ(1−α(1−1/σ))Y =! (σ−1)(1−α(1−1/σ))−αα (1− σ1 ) 1−α(1−1/σ)1
X  1− 1 1  1−α(1−1/σ)
1
α 1−
Ai σ χiσ
.
σ
i

(21)

As we can see above, the equilibrium output Y is a weighted sum of idiosyncratic shocks Ai , andfollows a normal distribution as N tends to infinity.

19

Another salient difference between the distributions in Figure 6 is the magnitude of fluctuations.
The standard deviation in the smooth economy is one-digit smaller than that in the lumpy economy
even though they incur the same realization of random shocks. Thus, the micro-level lumpy adjustments have a strong effect of magnifying exogenous independent shocks. It is not that the smootheconomy is completely lacking the magnifying mechanism. Consider a simple case where the microPNdecision is positively dependent on the aggregate such as xi = φ j=1 xj /N + . Then the exogenous
shock will be magnified by the multiplier 1/(1 − φ). In the case of lumpy adjustments, however, themultiplier effect at the micro-level is non-linear because of the threshold rule. Thus, the aggregation
mechanism works as if the multiplier itself is stochastic, because the multiplier depends on the profile
of underlying heterogeneous productivity before the perturbation. This is just like the small deviations in the standing points of domino tiles greatly affect the resulting number of falling tiles. The
exponential tail we have observed in the aggregates captures the nature of this stochastic multipliereffect, and thus it is a symptom of the endogenous fluctuation mechanism.

5

Consistency with micro-level estimate

In this section, we test the model in the previous section by examining whether the strategic complementarity observed in the micro data is strong enough to generate the exponential distribution
observed in the aggregate data. In the model, the magnitude of fluctuations is largely determined byβ, the degree of strategic complementarity, which is determined by σ and α. The σ was calibrated toa reasonable markup rate 10%, but α was taken as a free parameter. In the simulations, α was set sothat β was at around 0.67. In this section, we provide an empirical estimate that justifies this choiceof β.
We start by directly estimating the individual firm’s decision rule on lumpy investment. Each firm
faces a binary choice d(i, t) ∈ {0, 1} whether or not to engage in a lumpy investment. We would liketo estimate the binary decision rule in the model (16) by a probit model. According to the optimal
decision rule, the lumpy investment is more likely to occur when the perturbation in productivity∆ log Ai and/or the perturbed derived demand ∆Y is large. (Note that, in (16), ni is the equilibrium
input before the perturbation, and thus predetermined by the levels of Ai and Y .) By using (17),P∆Y is rewritten as β l bl Xl Nl where l is an industry, bl is bj for any firm j in industry l (by abuseof notation), Xl is the fraction of firms in industry l that engage in lumpy investments after the20

perturbation, and Nl is the number of firms in industry l. As seen in the definition (14), bl is theweight that is determined by the importance of industry l as an input good and the relative size ofthe industry l.
Our goal is to estimate the strategic complementarity in lumpy investments within a region, namely,
the impact of a lumpy investment on the likelihood of lumpy investment of other firms within the same
region. Thus, we are most concerned with β, the impact of the fraction of firms that engage in lumpy
investment in Gri on i’s likelihood for investment spiking. A simple regression of di,t on Xri ,t would
pick up a common shock effect that affects all the firms in a region-year. Thus, we include a region-year
dummy to control for the common shock. Because Xri ,t is constructed by the left-hand-side variable
d(i, t) in the same region, however, a direct estimation would still cause an endogeneity bias in theestimate of β. We avoid this issue by the structure of the model in which the impact derived demandis heterogeneous across industries. Namely, we will identify the endogenous effects within region byusing the industrial dimension as an instrument. This method follows Shea [23] and Bartelsman et al.
[2]. Our estimator satisfies the conditions in Brock and Durlauf [4] for the identification of this typeof estimator.
We follow a two-step procedure to construct a proxy variable that captures the portion of thederived demand in a region-year that is affected by industrial exogenous shocks. First, we regress
Xli ,ri ,t on a constant and year-dummies, and construct X̃li ,ri ,t by subtracting the year effect (capturedby the year-dummy) from Xli ,ri ,t . Repeat the same procedure for the industry-year fraction Xli ,t andconstruct X̃li ,t that is the industry-year fraction less common year effects. Thus, X̃li ,t captures theindustry-wide common shock on lumpy investments, which we deem exogenous to the firm in industries
other than li . We regress X̃li ,ri ,t on X̃li ,t and obtain linear predictors. The predictors X̂li ,ri ,t are thenused as proxies for the exogenous portion of the variation in X̃li ,ri ,t .
Next, we normalize the input-output matrix Π so that the sum for each row is equal to 1. We callit ΠOW which stands for “output-weighted” following Bartelsman et al. [2]. ΠOW (j, k) represents a“demand-pull” or “customer” impact from k-th column to j-th row. This is seen as follows. Supposethat there is one percent increase in the production in industry k. Now ΠOW (j, k) represents thefraction of industry j’s output demanded by k. Thus the demand for j is increased by ΠOW (j, k), otherP
OW
OW
things being equal. Then we define Xi,t
≡ k6=li ΠOW (li , k)X̂k,ri ,t #Gk,ri /#Gri . Xi,trepresents
the total exogenous demand shock coming from all the firms in the same region but in the different
industries. Likewise, we construct an input-weighted matrix ΠIW , which represents a “supply-push”
21Output-weighted

Input-weighted

HybriddΦ/dx(s.e.)

dΦ/dx(s.e.)

dΦ/dx(s.e.)∆TFP

0.0021***(0.0006)

0.0021***(0.0006)

0.0021***(0.0006)

Lagged Cash Flow

-3.3e-07(2.1e-06)-3.3e-07(2.1e-06)-3.3e-07(2.1e-06)

Investment in (li , ri )0.0023*(0.0012)0.0020*(0.0012)0.0022*(0.0012)X OW

1.42***(0.37)

1.38***(0.38)

0.41(0.64)

X IW

0.79

Loglikelihood-87376.19-87382.72(0.63)-87375.99

Table 4: Probit estimation of firm’s lumpy investment

or “supplier” impact from j-th row to k-th column, by normalizing Π column-wise. Using ΠIW as aIWweight, we obtain Xi,t.

Then, we obtain a probit estimation equation as follows:
OW
IW
Pr(d(i, t) = 1) = Φ(Di,t γD + Zi,t γZ + β OW Xi,t+ β IW Xi,t
).

(22)

D’s are dummy variables for industry, region, year, and region-year pair. Zi,t includes the measureof productivity growth for firm i, the cash flow (normalized by capital) in the previous period, andthe aggregate investment in the industry-region which firm i belongs to, Ili ,ri ,t . The construction ofthese variables in Zi,t is explained in Appendix B. The region-year dummy is included to address
OW
IW
the region-year common shocks. Xi,t
and/or Xi,t
are output- or input- weighted averages of the
OWIW
exogenous industrial shocks. Xi,tand Xi,t
are the exogenous portion of the impact of the derived

demand that arises from lumpy investment of other firms in the same region.
The estimation result is summarized in Table 4. The effect of total factor productivity is significantly positive, which is consistent with the decision rule (16). We observe a positive and significant
estimate of β OW , whereas the estimate of β IW is nonsignificant. This result is consistent with the“within” estimate of Bartelsman et al. [2]. In the short-run, the inter-industrial effect seems to flowfrom the downstream to upstream. The aggregate investment in the industry-region has a weakly significant mild effect. The lagged cash flow does not show a significant effect on the lumpy investments.
Now we examine if the strategic complementarity estimated above can explain the exponential
distribution estimated in Section 2. The mean of the exponential distribution of |X̃| is estimated at
22

0.03 for the region-wide fluctuations in Table 2. The median number of firms in Gr is 381 by Table
1. Hence, the median number of affected firms is about 12. By solving 1/12 = φ − 1 − log φ, weindirectly obtain the estimate of φ as φ̂ = 0.65. This number, estimated from aggregated data, falls
within the 90% confidence interval of φ, (0.64, 2.12), which is directly estimated from the micro-level
investment decisions in Table 4. Thus, our model of endogenous fluctuations have passed a test ofinternal consistency with the data we investigated.
This is not to say that our estimation of the strategic complementarity has explained all of the
exponential fluctuation we observed. First, the connection between the complementarity and theexponential slope was analytically established only for the case when the input weight χi,j are common
across i. The actual input-output matrix shows a great deal of heterogeneity across rows, and hence it
may alter the relationship between φ and the slope in the model. The importance of the heterogeneityof the input weights in propagation mechanism has been emphasized by Horvath [17]. Secondly,
the confidence interval of φ is rather wide, and it will pass any exponential distribution that has asufficiently flat slope. The meaningful content of the internal test we passed lies on the lower boundof the estimated φ: the observed exponential distribution has a standard deviation that is larger thanwhat the minimum estimate of the strategic complementarity at the micro-level implies.

6

Conclusion

This paper argues that the distribution of the fraction of firms that engage in large investments
provides a useful test for the existence of endogenous effects among the firms’ investment decisions.
Testing of the endogenous effects is often a challenge because it is observationally equivalent to themodel with exogenous common shocks. We argue that, in a binary choice model, the common shockmodel results in the normal distribution of the number of investing firms when the exogenous common
shock consists of many independent factors, whereas the endogenous effects leads to a non-normal
distribution that is better characterized by an exponential distribution.
We investigate the panel of investment data for Italian firms. We construct the fraction of firms thatengage in the investment with which the investment-capital ratio of the firm exceeds certain threshold. The fraction samples are constructed for region-year groups and industry-region-year groups.
Those samples constitute a distribution of the fraction of investing firms for each definition of thereference group. The results show that the normal distribution hypothesis is rejected by the empir23

ical distributions, whereas the exponential distribution hypothesis is consistent with the empirical
distributions.
We present a simple model of lumpy investments when the firms’ investment decisions are interrelated by strategic complementarity. The complementarity stems from the fact that firms are linked by
input-output relations. When a firm increases production, it generates an increased factor demand,and thus provides an incentive for the firms in the upstream of the input-output relations to produce more. This model generates a non-normal distribution with exponential tail for the fraction of
investing firms, while it generates a normal distribution if there is no lumpy investments in the model.
Finally we estimate the degree of strategic complementarity by a micro-level estimation of the firm’s
decision of the lumpy investment. We utilize the input-output relations to instrument the endogeneityin estimation. Then, we compute the degree of strategic complementarity that is required by the modelso that the model explains the empirical magnitude of fluctuations in the fractions of investing firms.
We confirmed that the result is consistent with the model prediction.

AppendixA

ProofsA.1

Fictitious tatonnement

The tatonnement that converges to an equilibrium is a time series of the input profile nu = (n1,u , n2,u , . . . , nN,u )for u = 1, 2, · · ·. It starts from the initial equilibrium that is defined for a realization of (Ai,0 , λi ). The
initial point is denoted by n0 . There is a corresponding threshold rule at the initial equilibrium. Its
lower bound profile is denoted by n∗0 . The gap variable is accordingly defined as (si,0 ).
Now a perturbation i is drawn independently across i. The productivity is perturbed such thatlog Ai,1 = log Ai,0 + . Then the threshold is altered even if no firms change their input level at n0 .
The new lower threshold under (A1 , n0 ) is denoted by n∗1 . The gap variable changes to s1 . We apply
the threshold rule, and some firms find it optimal to adjust their input. The best reply is denoted byn1 .
Then the second round starts. Under (A1 , n1 ), we compute a new profile of lower threshold n∗2 , a
gap variable s2 , and a best reply n2 . We iterate this process until nu converges. The stopping time is24denoted by T . Thus, ni,T = ni,T −1 for all i.

A.2

Impact of a lumpy investment on total demand

We start from:σ (σ−1)X 1 1− 1 α(1−1/σ)Y0 = 
χjσ Aj σ nj,0
.

(23)

j

Suppose that firm i chooses to invest. Then log ni,1 − log ni,0 = log λi . We calculate the new
aggregate output by the Taylor series expansion around Y0 as follows:1

1

log Y1 − log Y01−α(1−1/σ)∞X
χiσ Ai σ ni,0(α(1 − 1/σ))k−1 (log λi )k= α P 1 1− 1
+ O(N −2 )α(1−1/σ)
k!
σ
σ
nj,0
k=1
j χ j Aj
1

1− 1

(24)

α(1−1/σ)α(1−1/σ)χiσ Ai σ ni,0λi
−1
+ O(N −2 ).
= α P 1 1− 1α(1−1/σ) α(1 − 1/σ)
σ
σ
χ
A
n
j
j,0
j j

(25)

Note that the first term is of order N −1 , and the higher order term results from the derivative of thesummation in the denominator of the first term with respect to ni . From this we obtain the expression(17) and (12).

A.3

Proof of Proposition 1

In the homogeneous set-up, the probability of the Bernoulli trial qβbi / log λj becomes qβ/N . Thus,
the number of firms mu that are induced to adjust upward in u by mu−1 firms who adjusted in theprevious step of the tatonnement u−1 follows a Poisson distribution with mean qβmu−1 asymptoticallyas N → ∞.
Since a Poisson distribution is infinitely divisible, the Poisson variable with mean φmu−1 is equivalent to a mu−1 -times convolution of a Poisson variable with mean φ. Thus the process mu for u ≥ 2
conditional to mu−1 is a branching process with a step random variable being a Poisson with mean φ
for u ≥ 3, and m2 follows a Poisson with mean φm1 . Since φ ≤ 1, the process mu reaches 0 by a finite
stopping time with probability one (see [14]). Thus the best response dynamics is a valid algorithmof equilibrium selection in a sense that the convergence is achieved by a finite stopping time T . By
using the property of the Poisson branching process [18], we obtain an infinitely divisible distribution
called Borel-Tanner distribution for the accumulated sum M conditional to m2 as:Pr(M = m | m2 ) = (m2 /m)e−φm (φm)m−m2 /(m − m2 )!
25

(26)

for m = m2 , m2 + 1, . . .. Using that m2 follows the Poisson distribution with mean φm1 (note that
m1 is not necessarily an integer), we obtain (19) in the Proposition as follows:
Pr(M = m | m1 )=mX((m2 /m)e−φm (φm)m−m2 /(m − m2 )!)e−φm1 (φm1 )m2 /m2 !

m2 =0
m
X

=

(φm1 e−φ(m+m1 ) /m)(φm)m−m2 (φm1 )m2 −1 /((m − m2 )!(m2 − 1)!)=

(φm1 e−φ(m+m1 ) /m)(φm + φm1 )m−1 /(m − 1)!

=

m1 e−φ(m+m1 ) φm (m + m1 )m−1 /m!.

m2 =1

Approximation (20) is obtained by applying the Stirling’s formula m! ∼(27)√

2πe−m mm+0.5 :m1 e−φ(m+m1 ) φm (m + m1 )m−1 /m!
√
m1 e−φm1 (e−φ φ)m (m + m1 )m−1 /( 2πe−m mm+0.5 )√= (m1 e−φm1 / 2π)(e1−φ φ)m m−1.5 (1 + m1 /m)m (1 + m1 /m)−1√∼ (m1 e(1−φ)m1 / 2π)(eφ−1 /φ)−m m−1.5 .
∼2This completes the proof.

B

(28)

Construction of the productivity measure and aggregate
investments

To estimate the total factor productivity Ai , we follow the procedure similar to Cooper and Haltiwanger [11] and Bayer [3]. We start with the static optimization problem of a firm under a CobbDouglas production function:
Y = ALαL K αK .

(29)

The first order condition for the wage-taking firm implies wL = αL Y . Then we obtain:
1
h α
i 1−α
αK
L
L
Y = A( )αLK 1−αL .
w(30)

Using this, we estimate the productivity of capital as:log(Y ) −αKlog(K),
1 − αL26(31)

where Y is the value added.
The expenditure shares for both labor and capital are heterogeneous across firms, and it is impossible to directly estimate αL and αK due to the dynamic structure of the panel data as Bayer [3]pointed out. We instead calculate the average expenditure shares of firms for each industry and use
them for αL and αK above.
We also use an aggregate investment variable in order to capture its direct effect on d(i, t) in the
regression analysis. The aggregate investment is defined as:P
I(Gli ,ri , t) = P

i∈Gli ,ri Ii,ti∈Gli ,riPT

t=1 Ii,t /(#Gli ,ri T )

,

(32)

where Ii,t represents the real investment of firm i in year t and T is the total years in observation.
We take logarithm of I(Gli ,ri , t) and then subtract its yearly trend. We use the residual, i.e. thelog-deviation of the aggregate investment in the industry-region, as a regressor in Zi,t .

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