Evolution of Firm Dynamics:
With Minimal Assumptions∗
Kim P. Huynh†

David T. Jacho-Chávez‡
Marcel Voia¶

Robert J. Petrunia§

August 24, 2009

Abstract
This paper investigates the evolution of firm distributions for entrant manufacturingfirms in Canada using nonparametric methods. These nonparametric methods allow
a flexible method based on functional principal components or dynamic densities tocharacterize how these densities evolve over time. This method is applied to a novel
administrative firm-level database from Statistics Canada to investigate the evolutionof the 1985 and 1989 cohorts of new entrants. We find that firm leverage (debt-to-asset
ratio) distributions have persistent deviations from the initial distributions and bootstrap test statistics suggest that the distributions are different across all time periods.
Firm size and labour productivity have transitory deviations and some of the distributions are the same across all time periods. Univariate finite mixture and stochastic
dominance tests are used to conduct pairwise comparisons as robustness measures.
We find that these static pairwise comparisons confirm the dynamic evolution of these
densities. This method illustrates the efficacy of functional principal components toanalyze firm distributions.

Keywords: Firm Distributions, Functional Principal Components, Nonparametric Methods.
∗

The assistance and hospitality of Statistics Canada Business and Labour Market Analysis and theEconomics Analysis department is gratefully acknowledged. We are especially grateful to Garnett Picot forhis support and encouragement in this project. We thank Victor Aguirregabiria and the participants of2009 Canadian Economics Association. We are indebted to Steve Romer for his help with computing issues.
The contents of this paper have been subject to vetting and pass the Disclosure Rules & Regulations set
forth by Statistics Canada. We thank Susan Brunet and Bob Gibson for vetting and sending our results
expeditiously. All remaining errors are the responsibility of the authors.
†
Department of Economics, Indiana University, 105 Wylie Hall, 100 S Woodlawn, Bloomington, IN 47405,
USA. Phone: +1 (812) 855 2288. Fax: +1 (812) 855 3736. E-mail: kphuynh@indiana.edu.
‡
Department of Economics, Indiana University, 105 Wylie Hall, 100 S Woodlawn, Bloomington, IN 47405,
USA. Phone: +1 (812) 855 728. Fax: +1 (812) 855 3736. E-mail: djachoch@indiana.edu.
§
Department of Economics, Lakehead University, 955 Oliver Road, Thunder Bay, ON P7B 5E1,
CANADA. Phone: +1 (807) 343 8384. Fax: +1 (807) 346-7936. Email: rpetruni@lakeheadu.ca¶
Department of Economics, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, CANADA.
Phone: +1 (613) 520-2600 x3546. Fax: +1 (613) 520-3906. Email: mvoia@connect.carleton.ca

1

1Introduction

Understanding the evolution of firm distributions has been a major concern among economists,businesses, and policymakers. This paper investigates the evolution of firm debt-to-asset ratio (leverage), labour productivity, and size distributions of manufacturing firms in Canada.
Previous literature has focused mostly on productivity and firm size distributions. The novelty of this analysis is the inclusion of financial information (debt-to-asset ratio) of small,young, and privately-owned firms. However, little is known about the evolution of financial
variables due to the lack of detailed financial data on small, private firms. This paper uses
the T2LEAP, a unique administrative firm-level database combining information from the
Corporate Tax Statistical Universe File (T2SUF) with Longitudinal Employment Analysis
Program (LEAP) database in Canada. Information in the database allows us to analyze
the evolution of debt-to-asset ratio distributions, in addition to size and productivity distributions, for two entry cohorts (1985 and 1989) in the manufacturing sector. The analysis
focuses on nonparametric methods to examine variable distributions from both a static anddynamic perspective.
Beginning with Gibrat (1931), firm size distributions (FSD) have been a concern forstudies on industry dynamics. Typically, FSD for an industry are found to be skewed andlognormal in shape. Work on firm dynamics suggests financial frictions are relevant in theevolution of firm size distributions. Cabral and Mata (2003) document that the FSDs movetowards a lognormal distribution with age. One possible explanation for the changes in FSD
associated with age is a learning and selection process as suggested in Jovanovic (1982).
Cabral and Mata (2003) provide evidence against selection and suggest a simple model where

2the relaxing of financial constraints with firm age generate the dynamics observed for FSD.
Angelini and Generale (2008) provide further evidence of the impact of financial constraints
on the evolution of FSD. Recent work using the T2LEAP by Huynh and Petrunia (2008)
documents empirically that a firm’s growth depends nonlinearly on its leverage (debt-to-asset
ratio) even when controlling for firm size and age. Huynh, Petrunia, and Voia (forthcoming)
find a similar nonlinear conditional relationship between a firm’s survival prospects and its
initial leverage; survival rates increase with leverage for firms in the low to mid range of theleverage distribution and fall for firms at the upper end.
We perform the following procedures to capture both the static and dynamic statisticalproperties of variable distributions:
1. Functional principal component analysis (FPCA) is used to perform dynamic analysis of variable distributions for each entry cohort. Debt-to-asset ratio sees persistentchanges to a cohort’s distribution over time. Movements of the size and productivity
distributions are mostly transitory.
2. The equality of densities of each variable’s distribution across years is tested via abootstrap procedure. The bootstrap procedure analyzes distributions over time forentry cohorts between 1985 and 1993. Across time debt-to-asset ratio distributions arefound to be not jointly equal statistically for all cohorts. Joint equality of an entry
cohort’s across time productivity distributions is rejected for every cohort except the1990 entry cohort. We do not reject equality in across time size distributions for the
1985, 1991, 1992, and 1993 entry cohorts, while joint equality is rejected for all otherentry cohorts.
33. Pairwise comparisons of distributions are made using stochastic dominance tests andfinite mixture analysis. Finite mixtures are more stable for size and productivity
distribution, while continual movements occur for the mixtures of the debt-to-asset
ratio distribution. Stochastic dominance tests indicate the debt-to-asset ratio andsize distributions change with an entry cohort’s age, while no movement occurs for acohort’s productivity distribution.
The rest of the paper is organized as follows: Section 2 provides a description of thedata used in the paper. Section 3 discusses the methodology and the results of this dynamic distributional analysis. Pairwise comparison of the cross-sectional aspects of variable
distributions are examined in Section 4. Finally, section 5 concludes.

2

T2LEAP: Firm-Level Data

2.1

Data Description

The firm-level data used in this study comes from the T2LEAP database maintained by
Statistics Canada. This database was created through the merging of two administrative
databases; employment information from the Longitudinal Employment Analysis Program(LEAP) is linked to financial records from the Corporate Tax Statistical Universe File
(T2SUF). A firm is incorporated if it files a corporate (T2) tax return. T2LEAP uses abusiness registry number (BSNUM) to track all incorporated firms operating in a given
year. The data effectively covers the universe of incorporated Canadian firms hiring workers. The T2LEAP database contains firm details from 1984 until 1997. A firm’s entry into
the database allows us to identify birth year and place firms into entry cohorts. Birth year
is measured as the first year in which a firm both hires employees and files a corporate tax
4return. We do not observe a birth year for firms existing in the database in 1984. A firm isconsidered to exit when it fails to hire employees and/or file a tax return. The removal of thefirm occurs in all years subsequent to exit. Each firm receives an unique identify, BSNUM,
which ensures that exit and re-entry does not occur.
The paper uses the following firm level information available in the T2LEAP database:
Annual measures of a firm’s employment, sales, assets, and debt. Each firm is classified
by a three-digit Standard Industry Classification (SIC) code. The paper only considers
manufacturing firms. The following generated variables are calculated using the firm information: Leverage or the debt-to-asset ratio (Debtit /Assetit ), and labour productivity or(Salesit /ALUit ), where the Average Labour Units (ALU) measure a firm’s yearly employment. Analysis is performed on leverage, the logarithm of sales and labour productivity.

2.2

Univariate Densities

We perform the analysis on two entry cohorts: (i) 1985 and (ii) 1989. Focusing on new
entrants allows us to capture the dynamic process related to the post-entry evolution of new
firms. The 1985 entry cohort is the first identifiable birth cohort, and thus, is the entry
cohort with the longest time-series. The 1989 entry cohort provides a comparison group,
but also allows for us to separate age effects from time effects. The 1989 cohort is chosen
since 1989 is the first year after the Canada-US free trade agreement (CUSFTA) and lastyear prior to the recession of early 1990s.
Figures 1 illustrates the densities for leverage, logarithm of sales, and labour productivity
for the two entry (1985 and 1989) cohorts. In each figure there is a density for ages one,four and seven. In the first row, the leverage distribution shows a clear pattern of moving

5

leftwards left with age. As firms age (survive) they reduce their debt relative the asset holdings. For the 1989 entry cohort, there is clear decrease in leverage with age as the leverage
distribution shifts right with age. Alternatively, the decrease is not uniform as a crossing
occurs between the age four and age seven leverage distributions for the 1985 cohort. The
results are not surprising since at age seven the 1985 cohort is in the midst of a deep recession
in Canada while age seven for the 1989 would have seen a recovery.
The second row contains the logarithm of sales, which is our measure of firm size. The
distinguishing feature of firm size is that the densities shift to the right as the entrant cohort
ages. This stylized fact is known as the ‘trend to bigness’, see Lucas (1978). The difference
between the two cohorts is that the 1989 has more mass in terms of larger size firms while1985 has smaller firms.
Finally, row three contains the densities for labour productivity. In contrast to the twoother variables, the labour productivity distribution does not show as much movement muchwith age. The 1989 cohort has higher productivity and has less dispersion in labour productivity than the 1985 cohort. Overall, the labour productivity for the most part increases
with age for both cohorts.
Based on these observations, we note that as firms age they reduce their leverage, getlarger, and more productive. Between cohort comparison shows that members of the 1989
cohort reduce their leverage faster, are larger in size, and are more productive. As shownin Huynh, Petrunia, and Voia (forthcoming), the over time changes to sales and leverage
distributions likely result from two factors: (i) selection due to exit of firms with age and(ii) changes from initial conditions for surviving firms. The next section describes a general
method to describe the evolution of these densities in a formal statistical fashion.
6

3

Dynamic Distribution Analysis

We make use of functional principal component analysis (hereafter FPCA), as suggested
by Kneip and Utikal (2001), to describe the underlying population densities of each cohort
and age group jointly in labour productivity, leverage, and firm size distributions. To thebest of our knowledge, the usage of this tool is novel in understanding firm distributions
and industrial dynamics. It is a powerful device that is able to decompose the changes in
distributions into dynamic factors. In the time-series and panel data literature Bai and Ng(2002) and Pesaran (2006) have proposed to use factor models. The difference lies in how
the factors are estimated. We focus on nonparametric kernel methods since we want to place
minimal structure on how the dynamic factors are computed. The next section presents
these procedures.

3.1

Functional Principal Component Analysis

The approach of Kneip and Utikal (2001) analyzes jointly the underlying population densities{ft }Tt=1 . In particular, to characterize differences and similarities of {ft }Tt=1 , we assume their
expansions into the first L principal components, g1 , g2 , . . . , gL , and represent each ft interms of the model
ft = fµ +LXθtj gj ,(1)

j=1

where fµ =PT

t=1 ft /T is the common mean and L ≤ T corresponds to the number of nonzero

eigenvalues of the empirical covariance operator.1 Model (1) implies that each ft can be
1R
Consider the space of square-integrable functions ξ such that ξ 2 (x) w (x) dx < ∞, where w (x) > 0 issome continuous, uniformly boundedR weighting function ∀x ∈ D ⊂ R that lies in the support of ft , ∀t; and2define the scalar product hξ1 , ξ2 i = ξ1 (x) ξ2 (x) w (x) dx and kξk = hξ, ξi. Then the empirical covariancePT
operator of {ft }Tt=τ is given by Vξ = (1/T ) t=τ hft − fµ , ξi (ft − fµ ), where τ is the start year and T is
the last observed year. In this paper we use the uniform weight function, i.e. w (x) = 1, ∀x ∈ D.

7

obtained by adding to fµ a transformation of compromising common components g1 , g2 , . . . ,gL , with varying strengths encapsulated in the coefficients θtj . Since ft represent densities
obtained for each time period t = 1, . . . ,T , then the time evolution of their respective
coefficients θt1 , θt2 , . . . , θtL provides information about the evolution of the main differences
and similarities between the underlying distributions.
The unknown g1 , g2 , . . . , gL and θt1 , θt2 , . . . , θtL , can be obtained from the T × T
matrix M, whose elements are defined by Mts = hft − fµ , fs − fµ i, ∀t, s = 1, . . . ,T . In
particular, the unknown components gr , and parameters θtr relate to the T eigenvectors,
pr = (p1;r , . . . , pT ;r )> , r = 1, . . . , T , with corresponding eigenvalues λ1 ≥ λ2 ≥ . . . , ≥ λT of
M through:−1/2θtr = λ1/2
r pt;r , gr = λr

T
X

PT

pt;r , and ft = Pt=1
T

t=1

θtr ft2
t=1 θt.

(2)

Kneip and Utikal (2001) provide consistent estimators of θtr ,2 and gr . The next section
describes the results from these FPCA estimates.

3.2

Test of Densities via Bootstrap

The data contains the universe of all firms for the period of 1984-1997. In essence there
are 13 cohorts, 12 of which have a known entry date while those in the 1984 have censoredentry date. The first step is to construct a test of whether the densities of these variables
come from the same distribution, i.e. H0 : L = 0 as suggested in (Kneip and Utikal 2001).
They proposed using ρb =PT

r=1 λr as a plausible test statistic, and the following bootstrapbprocedure for approximating its distribution: In each of 399 replications T subsamples ofsizes n∗1 , . . . , n∗T are drawn with replacement from the pooled empirical cdf of the entire
2

However, no asymptotic distribution was derived.

8

sample, i.e.

PT

b∗ =
t=1 nt , and calculated ρPTb∗r=1 λr . The null distribution of the test statistic

is then approximated by the empirical distribution of ρb∗ . The results are shown in Table 1.
The bootstrap test statistic for the leverage distribution rejects the null hypothesis of equal
distributions across all cohorts at the 10% significance level. The result for size and labour
productivity is different as there are some cohorts that does not reject the null hypothesisof equal distributions. Namely, the 1975, 1991, 1992, and 1993 cohorts for size while 1990
cohort for labour productivity.
Table 1: Bootstrap Test Statistics and P-values

Cohort
84
85
86
87
88
89
90
91
92
93

Leverage
476.422205.937
193.368167.574
146.996123.089
101.14586.00071.18055.195p-value
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.020
0.093
0.090

Size
p-value
13.440 0.028
11.610 0.318
11.391 0.015
9.376
0.003
7.973
0.000
5.777
0.005
4.528
0.010
3.139
0.667
2.621
0.233
2.070
0.419

LabourProductivity
147.856
97.76286.35843.95547.09243.15239.21428.96614.42010.975p-value
0.000
0.000
0.020
0.005
0.045
0.045
0.278
0.098
0.050
0.015

Note: P-values calculated based on 399 bootstrap replications in each cohort.

3.3

Dynamic Scree Plotsb r / PT λbb
Figure 2 displays λr=1 r , where λr represent the estimated eigenvalues of the FPCA
decompositions of the 1985 and 1989 cohorts across the three variables. The estimatedeigenvalues are plotted across time and borrowing from the principal components literature
we call these Dynamic Scree Plots to emphasize the time-series dimension of these objects.
For leverage the dynamic scree plots show that the eigenvalues do not decline until the final
9

time period at which it falls to zero by definition. For firm size there is a marked decline in
the eigenvalues. Labour productivity shows a decline and then plateau in the eigenvalues.
Figure 2 shows that the first 3 components for size and labour productivity explain at least
60% of the total density time variation, therefore we proceed to analyze their time pattern
next.

3.4

Estimated Dynamic Strength Components

Figure 3 contains the plots of the three principal components when analyzing the dynamic
properties of the variable distributions. The time evolution of θbtr is compared against θb1r ,so that the estimated θbtr − θb1r for r = 1, . . . , 3 are plotted. For each variable, we highlightthe movements of the first three principal components, as suggested in Figure 2.
The first row contains the leverage θbtr − θb1r . All the components of the leverage distribution
exhibit a high degree of persistence. The movement of θbtr − θb1r tracks the movement of θbtr
as θb1r is constant. For the 1985 cohort, the first component has a steep drop, slight increase,and then flattens with a permanent negative difference between θbt,1 and θb1,1 . The second andthird components show some variability at the beginning of the time sample but then have asimilar time pattern of falling below, at a constant rate, the corresponding component value
at age one. The 1989 cohort shows the same pattern in the first component. However, thesecond and third components, although displaying some variability, do not show a remarked
divergence from θb1r as θbtr − θb1r is close to zero. For both cohorts, there are persistent changes
occurring to their respective leverage distribution. Both cohorts experience variability incomponent values over the first five years, while after age five (1990 for the 1985 cohort and1994 for the 1989 cohort) the value of θbtr − θb1r remains constant for all three components.

10

The results suggest a cohort’s leverage distribution initially experiences turbulent movements
but stabilizes relatively shortly after entry.
The second row displays these changes in estimated principal components when analyzing
the logarithm of sales distribution. For 1985 cohort, each component is transitory with atleast one incidence of a short-lived deviations as indicated by the spikes. The 1989 cohort isdifferent in that the second component difference has a permanent and persistent decrease.
For the 1985 cohort, the size distribution does appear to be somewhat stable after age 3(1988), but spikes in θbtr − θb1r are observed at age eight (1993) for the second component
and at age ten (1995) for the third component. For the 1989 cohort, the size distribution
appears to stabilize after age five (1994) as the value of θbtr − θb1r remains constant for all threecomponents. As will leverage, the size distribution appears to move towards an equilibrium
distribution with most of the movement occurring in the first five years of an entry cohort’s
life.
Finally, for labour productivity the deviations in the principal components for labour
productivity distribution are for the most part transitory. Unlike the other variables, stability
occurs at early ages followed by periods of instability. The 1985 cohort shows more transitory
and volatility in the differenced components. The volatility happens around 1989 and in the
1990s. The 1989 cohorts are for the most part stable until the end of the time period.
Both the 1985 and 1989 cohort suffer a spike in 1994. After 1994, the values of θbtr − θb1r
remain stable at close to zero for the 1985 cohort, while the component values continue tomove around for the 1989 cohort. Age effects are a possible explanation for the observed
movements. Aggregate effects provide alternative explanations. A recession occurred in
Canada between 1990 and 1992. Further, the Canada-US free trade agreement (CUSFTA)
11

was enacted in 1989, while the North American free trade agreement (NAFTA) came into
effect in 1994. These events probably had an impact, at least temporarily, on individual
firm’s productivity.

4

Empirical Evidence from Static Distributions

Having analyzed the joint dynamic behaviour of the leverage, size and labour productivity
distribution, and in view of results in Table 1 we proceed to look at the the source of theobserved time changes for each distribution.

4.1

Finite Mixture Decompositions

The finite mixture models are used to understand the changes in heterogeneity over timeand between cohorts of the outcome variables of interest. Our analysis of the 1985 and1989 entry cohorts moves to checking for the presence of mixtures in the individual variable distributions at cross-sectional level. We estimate univariate finite mixtures using themethodology described in McLachlan and Peel (2000). There are 2 × tj decompositions of
the distributions of each variable of interest requiring 2 × tj estimates of the K-class probability density functions, with tj being the length of time for cohort j. In this case, if fk (y)is the k th class probability density function and pk denotes the proportion of the k th class,then the finite mixture model with K components is defined as f (y) =PK

k=1 pk fk (y).

Theproportion pk can be interpreted as the prior probability of observing a sample from classk, with the property that is greater or equal to zero and that they sum to one (pk ≥ 0 andPK

k=1 pk = 1). The objective is to estimate the parameters of the class probability densities

and the proportions pk of each class, while fixing an upper bound on the number of possible

12

classes. The class of probabilities densities are assumed to have parametric components thatcan be estimated. We tested different parametric specifications and chose the ones that fit
better the true distributions. BIC measures and tests of equality of distributions are used
to evaluate which parametric specification fits better the true data.
The parameters of such mixtures can be estimated by maximum likelihood. For labour
productivity and logarithm of sales, a mixture of normal distributions is estimated with
2k)
. A mixture of
the k th probability density function given by fk (y|θ) = σ √1 2π exp −(y−µ
2
2σ
k

k

lognormal distributions is used for the leverage with the k th probability density function given2k), where y is the variable of interest and the parameters
by fk (y|θ) = yσ 1√2π exp −(log2σy−µ2kk

of interest are θ = {K, pk , µk , σk }> with k = 1, ..., K and K is the number of mixtures. The
parameter pk gives the weight on each mixture,PK

k=1 pk = 1, and µk , σk provide the meanand standard deviation for each mixture respectively. All the parameters of interest with the
exception of the number of types are estimated by the maximum likelihood. The number oftypes is selected using the AIC criteria, i.e. AICk = −2 log l (θ|y) + 2dk is minimized withrespect to k, where dk is equal to the dimension of the model and acts as a correction term
without which one would choose the model that maximizes the unconditional log-likelihood.
Tables 3 and 4 present the results of finite mixtures analysis for the distribution of logsales for the 1985 and 1989 entry cohort, respectively. There are various similarities as well as
differences between the two cohorts. At age one, the log-sales distribution for both cohorts iscomposed of three mixtures, while at age four and seven two mixtures are identified for bothcohorts. At age one both cohorts have the highest share of sales associated to medium size
firms, while for ages four and seven both cohorts have the highest share of sales associatedwith the smaller firms. The 1985 cohort presents an interesting feature at the age of four,13when the 1989 cohort enters. It shows a concentration of the sales for the smaller firms (72percent). In terms of mean sales by type, cohort 1985 has higher means than cohort 1989 atages one and four. At age seven, when compared with 1989 cohort, cohort 1985 has a higher
mean for the highest shares firms, but lower means for the other firms. The variances of thesales types are showing some variation between the cohorts at different ages of the firms.
The difference is that cohort 1989 shows higher variation than cohort 1985 at ages one andseven while the reverse is true at age four.
Table 5 and 6 examine the presence of finite mixtures for the leverage distribution of
the 1985 and 1989 entry cohorts, respectively. For this outcome variable, more differences
are observed between the two cohorts than for the previous outcome variable. The same
number of mixtures (three) are identified for ages one and four for both cohorts, with cohort
1989 keeping the same number of mixtures at age seven while cohort 1985 showing only
two mixtures at that age. For cohort 1985 the low leverage and medium leverage firms are
dominating the distribution at ages one and four, while for the cohort 1989 only at age seven
the low and medium leverage are dominating the distribution, for the other ages the low andhigh leverage firms are dominating the distribution. In the same tone, cohort 1985 presents
less concentration of types of firms than cohort 1989, which shows a higher concentration offirms with medium leverage. For cohort 1985 there is steady decrease in leverage over time,with very small changes between age four and seven. In contrast, there is a steady decrease
in mean leverage for the 1989 cohort only for ages one and four and an increase in leverageduring year seven. Finally, there is a higher variation in leverage for cohort 1989 than forcohort 1985 especially at age seven.
Finite mixture analysis for labour productivity is presented in tables 7 and 8 for the14

1985 and 1989 cohorts, respectively. For both cohorts, there is a reduction in the numberof mixture types. By age seven, there are only two types for both cohorts: low and highproductivity. The difference in mean productivity between the high and low types is small
between 0.4 and 0.7 with even smaller differences for cohort 1989. While the mean estimatedlabour productivity values for these two groups are almost equal at all ages, the estimatedstandard deviations are very different and provide the distinguishing factor between theestimated types.
4.1.1

Coefficient of Variation of Mixture Components

Figure 4 illustrates the results of the finite mixture components of all time periods for the1985 and 1989 cohort. The information is collapsed into a measure of a weighted coefficientof variation: CVk = pk µk /σk , where k = 1, . . . , K, pk is the probability of the mixture k,µk is the mean of the mixture k, and σk is the standard deviation of the mixture k. The
figures have discontinuous plots since for some cases the mixture component is not present
in certain years. The first row of the figure shows a downward trend in the first and second
mixture component for leverage in both the 1985 and 1989 component. This confirms thedensity pictures which illustrates that the leverage distribution is shifting leftwards. The
third component does not show a downward trend but rather volatility.
The firm size finite mixture components is in the second row. The first stark patternis that the first component is always larger than the second component for both cohorts.
There is a trend of the first component increasing while the second component is decreasing.
This trend is indicative of the first component is getting larger while the second component
is getting smaller. The difference between the two components is larger in the 1985 cohort

15

relative to the 1989 cohort. Once again, this confirms the density estimation that shows that1985 cohort is more heterogeneous.
The third row contains the labour productivity finite mixture components. The firstcomponent is larger than the second component for the most part in both cohorts. In the1985 cohort there is marked decrease in the first component and increase in the second
component in 1993. Part of it maybe due to an emergence of a third component which
may decrease the weight in the first component and increase it for the second component.
A similar case is shown for the 1989 cohort in year 1991 where the emergence of anothercomponent causes a large change in the first and second component.

4.2

Stochastic Dominance in the Presence of Finite Mixtures

To complete the static analysis we need to integrate the information obtained from finite
mixture decompositions into a unified measure of comparison, which can be done using
stochastic dominance testing. These tests are used to compare whether pair of distributions
are equal and if not what it is the level of dominance, i.e. equality of two distributions (EoD),
first-order stochastic dominance (FOSD) and second-order stochastic dominance (SOSD), as
explained in Table 2.
Let Yi,t;l represent either leverage, size or debt-to-asset ratio of firm i = 1, . . . , nt , atage t, in cohort l, and let its distribution be Ft;l (y) ≡ Pr[Yi,t;l ≤ y]. Analogous, let Yi,s;jrepresent the outcome variable of interest of firm i = 1, . . . , ms , at age s, in cohort j with[t;l]its corresponding distribution Fs;j (y) ≡ Pr[Yi,s;j ≤ y]. Furthermore, let D1 (y) = Ft;l (y),[t;l]and define higher orders for D(∗) as Do (y) =

Ry
0

[t;l]

Do−1 (x)dx. Note that we can also write

Ry
[t;l]
Do (y) = 1/(o − 1)! 0 (y − x)o−1 dFt;l (x). Table 2 provides the null hypothesis and the test

16

Table 2: Test of Distributions
H0
EoD:
Ft;l (y) = Fs;j (y)FOSD: Ft;l (y) ≤ Fs;j (y)
[t;l]
[s;j]
SOSD: D2 (y) ≥ D2 (y)abTest Statisticb = p nt ms supx |Fbt;l (y) − Fbs;j (y)|Knt +msb = p nt ms supx [Fbt;l (y) − Fbs;j (y)]Dnt +msb 2[t;l] (y) ≥ D
b 2[s;j] (y)]b = p nt ms supx [DH
nt +ms

Linton, Maasoumi, and Whang (2005) is used to test EoD and FOSD.
Andrews (2000) and Dufour (2006) are used to test SOSD.

statistics for each case.
Linton, Maasoumi, and Whang (2005) is used to test EoD and FOSD, while Andrews(2000) and Dufour (2006) are used to test SOSD. However, since the above test statistics
depend on nuisance parameters (the number of finite mixtures in each variable’s distribution),
their critical values are tabulated by the following parametric bootstrap procedure:∗∗
1. Sample nt and ns values, (Y1,t;l, . . . , Yn∗t ,t;l ) and (Y1,s;j, . . . , Yn∗s ,s;j ), from the estimateddistributionsRy0

fb(u)du =R y PK0

bk fk (u) du.
k=1 pb

2. Using the bootstrap samples in step 1, calculate the empirical distribution functions∗∗b ∗, Db ∗ , and Hb ∗ accordingly.
Fbt;l(y) and Fbs;j(y), and calculate K

3. Repeat steps 1-2 B many times and define the critical value as the smallest value of yb ∗, Db ∗ , and Hb ∗ are at or
subject to at least 100(1 − α)% of the obtained B values of K
below y.
b ∗, Db ∗ , or Hb ∗ are greater than the critical value found in step 3.
4. Reject if K

17

4.2.1Results of Stochastic Dominance Tests

The presentation of the tests refers to a) within a cohort comparisons. Analogous tests areused for b) between cohorts comparisons. Table 9 presents distribution comparisons overtime within each cohort, i.e. t 6= s and l = j. For the 1985 log-sales distribution, age one
is SOSD dominated by ages four and seven, with ages four and seven stochastically equal.
For the 1989 cohort, the age one is equal to age four and both are SOSD by age seven. The
finding may suggest that cohort 1985 grew faster than cohort 1989.
The leverage story is very similar with the log-sales story but in opposite direction andwith a higher magnitude. Therefore, for the leverage distribution, age one SOSD age four andseven for both cohorts, while the age four distribution does not look stochastically different
than the age seven distribution. This finding indicates that for cohort 1985 we may have aconvergence for the leverage distribution after age four. The convergence to an equilibrium
leverage distribution appears to be longer for 1989 cohort as we find this cohort’s age four
leverage distribution SOSD its age seven distribution. Labour productivity does not show
any changes in its distribution as we cannot reject equality of distributions across all ages
for both cohorts.
Table 10 presents the comparison of the distributions between the entry cohorts, i.e.
t = s and l 6= j. For log-sales variable, cohort 1985 SOSD cohort 1989 at age four, while thetwo cohorts distributions at age one and seven are stochastically equal. In the leverage casethe 1989 cohort SOSD leverage distribution for the 1985 cohort at all ages. The finding may
suggest that pre free trade entry firms needed less leverage to survive than post free trade
entry firms as they were more protected by differential tariffs. In contrast, for the labour18

productivity distribution we do not find any statistical difference between the cohorts at any
ages.

5

Concluding Remarks

Our analysis looks at the evolution of debt-to-asset ratio, size and productivity distributions
for two entry cohorts (1985 and 1989) of manufacturing firms. Functional principal component analysis provides a concise method to visualize how distributions evolve over time.
Functional principal component analysis provides a method to capture dynamics and breaks
a variable’s distribution into dynamic principal components. The results show deviations ineach entry cohort’s debt-to-asset ratio distribution are persistent, while deviations are transitory for productivity and size distributions. A persistent component allows a variable’s
distribution to move around a steady state distribution. Transitory components capturechanges in a variable’s distribution over time, such as a movement towards an equilibrium
distribution. The transitory components can account for deviations in a distribution occurring over time.
Static analysis makes pairwise comparisons of distributions using stochastic dominancetests and finite mixture analysis. The pairwise static distributional analysis confirms thefindings of the dynamic functional principal component analysis. The distribution for the
debt-to-asset ratio moves across time, while the distributions for productivity and size tend
to be more stable over time. The functional principal component analysis and the static pairwise analysis provide effective methods to describe the evolution of a variable’s distribution
over time with minimal assumptions.

19

References
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Boundary of the Parameter Space,” Econometrica, 68(2), 399–406.
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Models,” Econometrica, 70(1), 191–221.
Cabral, L. M., and J. Mata (2003): “On the evolution of the firm size distribution:
Facts and theory,” American Economic Review, 93(4), 1075–1090.
Dufour, J.-M. (2006): “Monte Carlo tests with nuisance parameters: A general approach
to finite-sample inference and nonstandard asymptotics,” Journal of Econometrics, 133(2),443–477.
Gibrat, R. (1931): Les Inegalités Economiqués. Librairie du Recueil Sirey. Paris.
Huynh, K. P., and R. J. Petrunia (2008): “Age Effects, Leverage, and Firm Growth,”
mimeo.
Huynh, K. P., R. J. Petrunia, and M. C. Voia (forthcoming): “The Impact of Initial
Financial State on Firm Duration Across Entry Cohorts,” Journal of Industrial Economics, forthcoming.
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670.
Kneip, A., and K. J. Utikal (2001): “Inference for Density Families Using Functional
Principal Component Analysis,” Journal of the American Statistical Association, 96(454),
519–542.
Linton, O., E. Maasoumi, and Y.-J. Whang (2005): “Consistent Testing for Stochastic
Dominance under General Sampling Schemes,” Review of Economic Studies, 72(3), 735–
765.
Lucas, R. E. (1978): “On the Size Distribution of Business Firms,” Bell Journal of Economics, 9(2), 508–523.
McLachlan, G., and D. Peel (2000): Finite Mixture Models. Wiley, New York.
Pesaran, M. H. (2006): “Estimation and Inference in Large Heterogeneous Panels with a
Multifactor Error Structure,” Econometrica, 74(4), 967–1012.

20

Year
1986
1986
1986
1989
1989
1992
1992

Table 3: Log of Size (Sales) Distribution: Cohort 1985
Type Share
µ
s.e.
σ
s.e.
logL
N
I
0.610 5.861 0.079 1.085 0.131 -4371.73 2578
II
0.247 6.415 0.151 2.048 0.135
III
0.143 5.373 0.113 0.501 0.216
I
0.717 6.206 0.052 1.017 0.067 -3377.69 1993
II
0.283 7.048 0.155 1.828 0.110
I
0.697 6.242 0.087 1.094 0.116 -2804.305 1615
II
0.303 7.005 0.205 1.813 0.159
Table 4: Log of Size (Sales) Distribution: Cohort 1989

Year
1990
1990
1990
1993
1993
1996
1996

Type
I
II
III
I
II
I
II

Share
µ
0.506 5.93
0.291 5.232
0.203 6.175
0.536 5.771
0.464 6.522
0.838 6.138
0.162 8.098

s.e.
0.217
0.129
0.161
0.086
0.133
0.121
1.179

σ
1.07
0.681
2.222
0.978
1.808
1.276
1.657

s.e.
logL
N
0.159 -3793.08 2266
0.193
0.162
0.143 -3049.33 1710
0.106
0.077 -2558.86 1402
0.349

Note: The parameters of the finite mixtures: number of types (k), mean of type k (µk ),and standard deviation of type k (σk ) were obtained by maximizing the log-likelihood
associated to the following density function:flog-sales (y|θ) =

K
X
k=1

pk

1
√

σk 2π


exp


−(y − µk )2.
2σk2

The number of types were chosen according to the number for which the likelihood is thehighest and the other parameters of interest are stable.

21

Table 5: Leverage Distribution: Cohort 1985
Year
1986
1986
1986
1989
1989
1989
1992
1992

Type
I
II
III
I
II
III
I
II

Share
µ
0.514 0.822
0.354 0.931
0.132 1.018
0.391 0.830
0.378 0.697
0.231 0.716
0.581 0.696
0.419 0.812

s.e.
0.025
0.013
0.080
0.024
0.061
0.115
0.051
0.017

σ
0.365
0.149
1.584
0.181
0.375
0.996
0.685
0.212

s.e.
logL
0.062 -1596.73
0.051
0.103
0.068 -1687.58
0.177
0.101
0.065 -1606.77
0.061

N
2578

1993

1615

Table 6: Leverage Distribution: Cohort 1989
Year
1990
1990
1990
1993
1993
1993
1996
1996
1996

Type
I
II
III
I
II
III
I
II
III

Share
µ
0.718 0.966
0.224 1.167
0.058 0.514
0.618 0.911
0.250 1.191
0.132 0.459
0.519 0.819
0.389 0.699
0.092 1.736

s.e.
0.013
0.057
0.141
0.028
0.065
0.176
0.019
0.086
0.762

σ
0.239
1.746
0.131
0.239
1.886
0.149
0.240
0.597
4.959

s.e.
logL
0.042
0.076 -1508.82
0.105
0.060
0.085 -1441.67
0.125
0.069
0.164 -860.71
0.301

N
2266

1710

1402

Note: The parameters of the finite mixtures: number of types (k), mean of type k (µk ),and standard deviation of type k (σk ) were obtained by maximizing the log-likelihood
associated to the following density function:
fleverage (y|θ) =

K
X
k=1

pk

1
√

yσk 2πexp

−(log y − µk )2.
2σk2

The number of types were chosen according to the number for which the likelihood is thehighest and the other parameters of interest are stable.

22

Table 7: Labour Productivity Distribution: Cohort 1985
Year
1986
1986
1986
1989
1989
1992
1992

Type
I
II
III
I
II
I
II

Share
µ
0.702 4.342
0.224 5.052
0.074 4.299
0.755 4.428
0.245 4.852
0.814 4.492
0.186
4.9

s.e.
0.035
0.418
0.221
0.018
0.071
0.018
0.093

σ
0.543
0.843
2.111
0.473
1.125
0.493
1.286

s.e.
logL
0.071 -3084.8
0.204
0.178
0.043 -2021.03
0.084
0.039 -1643.81
0.111

N
2578

1993
1615

Table 8: Labour Productivity Distribution: Cohort 1989
Year
1990
1990
1990
1993
1993
1996
1996

Type
I
II
III
I
II
I
II

Share
µ
0.531 4.432
0.376 4.887
0.093 4.569
0.780 4.630
0.220 4.675
0.715 4.594
0.285 4.976

s.e.
0.037
0.105
0.201
0.020
0.088
0.024
0.074

σ
0.455
0.857
2.081
0.545
1.491
0.481
1.091

s.e.
logL
N
0.077 -2744.68 2266
0.137
0.201
0.043 -2001.57 1710
0.110
0.05 -1472.24 1402
0.091

Note: The parameters of the finite mixtures: number of types (k), mean of type k (µk ),and standard deviation of type k (σk ) were obtained by maximizing the log-likelihood
associated to the following density function:
flabour productivity (y|θ) =

K
X

pk

k=1

1
√

σk 2π


exp


−(y − µk )2.
2σk2

The number of types were chosen according to the number for which the likelihood is thehighest and the other parameters of interest are stable.

23

Table 9: Stochastic Dominance: Within Cohorts
1985
Age 1
Age 4
Age 7

1989
Age 1
Age 4
Age 7

Age 1
eq
.
.

Sales
Age 4
-2
eq
.

Age 1
eq
.
.

Sales
Age 4
eq
eq
.

Age 7
-2
eq
eq

Age 1
eq
.
.

LeverageAge 4 Age 7
2
2
eq
eq
.
eq

Labour Productivity
Age 1 Age 4 Age 7
eq
eq
eq
.
eq
eq
.
.
eq

Age 7
-2
-2
eq

LeverageAge 1 Age 4 Age 7
eq
2
2
.
eq
2
.
.
eq

Labour Productivity
Age 1 Age 4 Age 7
eq
eq
eq
.
eq
eq
.
.
eq

Note: The results are read in terms of row versus columns. (eq) stands for equality of
distributions, (2) stands for second order stochastic dominance and a negative number
indicates that the column dominates the row, here (-2) stands for second order stochastic
dominated distribution. For example, for the 1985 cohort, line one (columns 2, 3 and 4 for leverage) reads that Age 1 SOSD both Age 4 and Age 7, while line two (columns 3 and
4 - for leverage) reads that Age 4 is stochastically equal to Age 7 distribution. All results
are significant at 5%.

Table 10: Stochastic Dominance: Between Cohort 1985 & 1989

Age 1
Age 4
Age 7

Sales

Leverage1985 vs. 1989eq
2eq

1985 vs. 1989
-2
-2
-2

LabourProductivity
1985 vs. 1989
eq
eq
eq

Note: The results are read in terms of row versus columns. For example column 2 (forleverage) reads that at each age cohort 1989 SOSD cohort 1985. All results are significant
at 5%.

24

Figure 1: Distributions
85 Cohort

89 Cohort

Leverage

1.5

1.0

0.5

0.0
0.0

0.5

1.0

1.5

2.0

0.0

14

0

0.5

1.0

1.5

2.0

Size

0.3

0.2

0.1

0.0
0

2

4

6

8

10

12

2

4

6

8

10

12

0.8
Age 1
Age 4
Age 7

Labour Productivity

0.6

0.4

0.2

0.0
2

3

4

5

6

7

8

2

a

3

4

5

6

7

8

Cross-validated bandwidths are 0.2598, 0.3214, and 0.3585 for firms aged 1, 4 and 7-years
old respectively in the 1985 cohort (First column).
b
Cross-validated bandwidths are 0.2642, 0.2707, and 0.3114 for firms aged 1, 4 and 7-years
old respectively in the 1989 cohort (Second column).
cGaussian kernels are used throughout.
25

Figure 2: Dynamic Scree Plot
Leverage
0.30

0.25

0.20

0.15

0.10

0.05

0.00

Size
0.30

0.25

0.20

0.15

0.10

0.05

0.00

Labour Productivity
0.30

0.25

0.20

0.15

0.10

0.05

85 Cohort
89 Cohort

0.00
2

4

br /
Note: Each plot displays λ

6

PT

8

10

r=1 λr for the 1985 and 1989 cohorts.

b

26

12

Figure 3: Estimated Dynamic Strength Components
85 Cohort

89 Cohort

2

Leverage

0

−2

−4

−6
1986

1988

1990

1992

1994

1996

1986

1988

1990

1992

1994

1996

1986

1988

1990

1992

1994

1996

1986

1988

1990

1992

1994

1996

1990

1992

1994

1996

1.5

1.0

Size

0.5

0.0

−0.5

−1.0

−1.5

1st Comp.
2nd Comp.
3rd Comp.

Labour Productivity

4

2

0

−2

−4
1986

1988

1990

1992

1994

1996

1986

Note: Each figure plots θbtr − θbt0 r for r = 1, 2, 3.

27

1988

Figure 4: Coefficient of Variation
85 Cohort

89 Cohort−0.1

Leverage

−0.2

−0.3

−0.4

−0.5

1986

1988

1990

1992

1994

1996

1986

1988

1990

1992

1994

1996

1986

1988

1990

1992

1994

1996

1986

1988

1990

1992

1994

1996

1990

1992

1994

1996

4

Size

3

2

1

8

1st C.V.
2nd C.V.
3rd C.V.
4th C.V.

Labour Productivity

6

4

2

0
1986

1988

1990

1992

1994

1996

1986

1988

Note: Each plot displays CVk = pk µk /σk where k = 1, . . . , K, pk is the probability ofthe mixture k, µk is the mean of the mixture k, and σk is the standard deviation of the
mixture k.

28