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Federal Reserve Bank of Chicago

A Firm’s First Year
Jaap H. Abbring and Jeffrey R. Campbell

WP 2003-11
Revised November, 2004

A Firm’s First Year
Jaap H. Abbring∗
Jeffrey R. Campbell†
November, 2004

Abstract
This paper determines the structural shocks that shape a firm’s first year by estimating a structural model of firm growth, learning, and survival using monthly sales
histories from 305 Texas bars. We find that heterogeneity in firms’ pre-entry scale
decisions accounts for about 40% of their sales’ variance; persistent post-entry shocks
account for most of the remainder. We find no evidence of entrepreneurial learning.
Variation of the firms’ fixed costs consistent with an annual lease cycle explains their
exit rates. We use the estimated model to price a new bar’s option to exit, which
accounts for 124% of its value.

∗

Department of Economics, Free University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands,
and Tinbergen Institute. Email: jabbring@econ.vu.nl.
†
Federal Reserve Bank of Chicago, 230 South LaSalle Street, Chicago IL 60604, USA and NBER. Email:
jcampbell@frbchi.org
Keywords: firm exit, option value, fixed costs, Bayesian learning
JEL codes: C34, D83, L11, L83
We gratefully acknowledge the research support of the National Science Foundation through grant #013748
to the NBER. Jaap Abbring’s research is supported by a fellowship of the Royal Netherlands Academy of
Arts and Sciences and was in parts carried out at The University of Chicago, University College London, and
the Institute for Fiscal Studies. We would also like to thank Gadi Barlevy and numerous seminar participants
for comments, and the IZA in Bonn for facilitating a joint visit in December 2001. Ronni Pavan and Raghu
Suryanarayanan provided superlative research assistance.

1

Introduction

The risk inherent to a new firm’s life gives value to the option to exit. In this paper, we
quantify this option’s contribution to a new firm’s value using monthly sales histories from
305 new bars in Texas. Doing so requires identifying the fixed costs that exit avoids and the
information that entrepreneurs acquire as their firms age. We accomplish this by estimating a
structural model of firm growth and survival. The estimation distinguishes between persistent
and transitory post-entry shocks and separates these from firms’ different pre-entry scale
decisions. Jovanovic’s (1982) model of entrepreneurial learning and Hopenhayn’s (1992)
model of Markovian firm growth are both special cases nested within it. The model’s point
estimates indicate that firms’ pre-entry scale decisions account for about 40% of their sales’
variance; persistent post-entry shocks to their profitability account for most of the remainder.
All of the growth of surviving firms’ average sales reflects selection. The option to exit
accounts for 124% of a new firm’s value: A firm that must be operated in perpetuity is a
liability. Thus, a policy analysis that fails to examine entrepreneurs’ exit decisions almost
certainly ignores the largest component of a new firm’s value.
Twenty percent of the firms in our sample exit in less than one year. Their monthly exit
rates are low immediately following entry, but they rise quickly after six months. This survival
pattern is consistent with entrepreneurial learning about a firm’s unobservable profitability,
but it can also reflect fixed costs of continuation that vary over time. A reduced-form analysis
of growth and survival cannot distinguish between them, while an identified structural model
that nests them, such as ours, can. We find that entrepreneurial learning plays no role in the
firm’s first year of operation. Instead, fixed costs that are very high in the first month of each
new year and relatively low in the remaining 11 months explain the increase in exit rates.
This pattern is consistent with many of the costs of annual leases for space and equipment
being effectively sunk at the beginning of the year.
The sales of surviving bars in our sample display non-Markovian history dependence:
Their initial sales significantly contribute to forecasts of their sales in subsequent years that
condition on the firm’s recent history. Pakes and Ericson (1998) find a similar result in a long
annual panel of Wisconsin retailers, and interpret this as evidence of Bayesian entrepreneurial
learning. Our analysis builds on theirs, but it accounts for heterogeneity across firms’ preentry scale decisions and transitory shocks observed only by entrepreneurs. We find that these
can fully explain surviving bars’ long history dependence without appealing to entrepreneurial
learning.
The paper’s structural analysis begins with an extension of Jovanovic’s (1982) model of
1

firm selection. A firm’s producer surplus is proportional to its sales; and the logarithm of
each firm’s surplus sums the entrepreneur’s choice of scale before entry, post-entry shocks that
persistently affect profit, and transitory shocks. After an entrepreneur observes her profit, she
must decide either to remain open or to exit and avoid future fixed costs. An entrepreneur
only observes her profit and one of the transitory shocks, and she bases her continuation
decision on her posterior belief regarding the persistent shocks. The scale decision affects the
firm’s sales and costs proportionally, so it has no relevance for the entrepreneur’s continuation
decision. Heterogeneity in the scale decisions enters as a random effect in firms’ sales and
fixed costs, and is thus a potential source of sales’ persistence. We follow Jovanovic and
assume that the persistent and transitory shocks unobserved by the entrepreneur have normal
distributions. Hence, the entrepreneur’s posterior beliefs are also normal with a historydependent mean and an age-dependent variance. The firm exits when the posterior’s mean
falls below an age-specific threshold.
In our model, the entrepreneur observes her scale decision and one of profit’s transitory
shocks. With this information, she can better forecast her firm’s future than can an observer who relies on the public tax records we use. Thus, the model violates Rust’s (1987)
assumption of conditional independence of unobservables. This makes existing identification
strategies for dynamic discrete choice models, such as Hotz and Miller’s (1993), unavailable.
Nevertheless, the firm histories we observe identify the model’s parameters. We show that
differences between all firms’ initial sales and the initial sales of those that subsequently survive reveal the distribution of the entrepreneurs’ observations as well as the rules they use
for their exit decisions. Heuristically, if these samples are very similar we conclude that most
observed variation across firms is irrelevant for entrepreneurs’ exit decisions.
We augment the Kalman filter to account for sample selection, and we use it to calculate maximum likelihood estimates of the entrepreneurs’ optimal exit thresholds and all of
the model’s identified structural parameters except firms’ fixed costs. With these in hand,
we can recover the fixed costs using the profit maximization problem’s Bellman equation.
This two-step approach has both computational and substantial advantages. Because it estimates the entrepreneur’s decision rule directly, our first stage avoids the repeated solution of
the entrepreneur’s dynamic programming problem that is inherent to Rust’s (1987) nested
fixed point algorithm. Moreover, the procedure accounts for the entrepreneur’s superior
understanding of her profit maximization problem. For example, time-varying fixed costs associated with annual lease renewal can manifest themselves in the estimated exit thresholds
even if we fail to appreciate their importance before estimation.
The remainder of the paper proceeds as follows. The next section describes the data,
2

Section 3 presents the structural model, and Section 4 considers the theory of its empirical
content. Section 5 discusses the model’s parameter estimates and their implications for the
value of a new firm’s option to close. The final section offers some concluding remarks.

2

Histories of Firm Growth and Survival

Readily available data sources— such as public tax records, business directories, or economic
census records— allow the construction of data documenting the growth and survival of a
cohort of entering firms. Gort and Klepper (1982), Dunne, Roberts, and Samuelson (1988),
Bahk and Gort (1992), Jovanovic and MacDonald (1994), Holmes and Schmitz (1994), and
Pakes and Ericson (1998) have all examined such data sets and characterized their implications for various aspects of firm and industry growth. Our analysis uses a similar data set
constructed from a panel of Texas alcohol tax returns. The state of Texas collects a 14% tax
on the sale of alcohol for on-premise consumption, and the Texas Alcoholic Beverage Control
Board (TABC) makes these returns publicly available. Returns are filed monthly, and a firm
must file a separate return for each of its establishments. Information included with each
return includes the identity and street address of the establishment’s parent firm, its trade
name, its own street address, the date its alcohol license was issued, the date it was returned
if the establishment no longer operates, and its tax payment for that month.

2.1

Sample Construction

Our observations begin in December, 1993 and end in March, 2001. We divided all sales
observations in our sample by the geometric average of tax returns from the establishment’s
county filed in the same month. This accounts for persistent differences in input prices
across counties, inflation over the sample period, and county-level aggregate fluctuations.
Campbell and Lapham (2004) measure restaurants’ and bars’ aggregate responses to large
county-specific demand shocks. They are an order of magnitude smaller than typical changes
to individual establishments’ sizes. For this reason, we henceforth focus on this idiosyncratic
variation and suppose that the observations of establishments’ survival and scaled sales come
from a stationary environment.
We used alcohol tax identification numbers and the establishments’ street addresses to
group these observations into establishment histories. Our linking process accounts for the
fact that a single establishment may have multiple owners over its lifetime.1 A tax return must
1

In our sample, there are numerous instances of an establishment being transferred from an individual

3

be filed for each establishment in every month, even if the establishment sold no alcohol for
that month. Therefore, our data contain several reports of zero sales. The data also contain
tax returns with a very low tax payment.2 These apparently reflect unobserved shutdown of
the establishment for part of the month or a very small scale of operation. When constructing
establishment histories, we equate any tax payment of less than $750 with zero. We consider
an establishment to be born in the first month that it pays more than $750 in tax, and we
date its exit in the first month that it fails to pay that amount. If an establishment’s tax
payment temporarily falls below $750, we consider that establishment to have temporarily
shut down. We exclude such establishments from our data set altogether.
Alcohol sales is of primary importance for bars, but restaurants substantially profit from
both alcohol and food sales. So that our sample is as homogeneous as possible in this regard,
we focus only on those establishments that present themselves to the public as bars. To
be included in our sample, a firm’s trade name must include the word “bar” or one of 10
other words indicative of a drinking place, and it must not include the word “restaurant” or
one of 20 words indicating the presence of substantial food service.3 Given the limitations
of the data, this minimizes the risk of falsely including restaurants at the expense of falsely
excluding bars.
There are also substantial differences between multiple establishment firms and their
counterparts that only operate one location. The manager of an incumbent firm’s new
establishment can use that firm’s history and experience to plan its operations and judge
its prospects. An entrepreneur starting a single establishment firm has no such information.
Accordingly, we exclude any establishment founded by a firm with two or more establishments
in Texas from our data set. There are 305 single-establishment firms in our data set that
were born in the five years beginning in 1994 and ending in 1999. These single establishment
to a corporate entity with the same address. These appear to be legal reorganizations with few immediate
economic implications.
2
Some of these reflect operation during only part of the first or last month of the establishments’ operation.
When the given dates of license issuance and return indicate that this is the case, we divide the tax payment
by the fraction of the month the establishment operated. Even after this correction, there remain several tax
returns with very small but positive tax payments. The smallest positive tax payment in our data is under
$1.
3
The words that qualify a firm for inclusion in the data set are “bar”, “cantina”, “cocktail”, “drink”,
“lounge”, “pub”, “saloon”, “tap”, and “tavern.” In addition to “restaurant” the words that exclude a
firm from our analysis are “bistro”, “brasserie”, “cafe”, “club”, “diner”, “dining”, “food”, “grill”, “grille”,
“hotel”, ‘oyster”, “restaurante”, “shrimp”, “steak”, “steakhouse”, “sushi”, and “trattoria”. When selecting
this sample, we consider only the trade name listed on the firm’s first tax return that reports a tax payment
greater than $750.

4

firms comprise our sample.

2.2

Summary Statistics

Table 1 reports summary statistics from our sample of firms. For each age we consider, one
to thirteen months old, it reports the number of firms that survived to that age; the mean
and standard deviation of sales’ logarithm among these survivors; and the fraction of them
that did not operate in the following month (the exit rate). Selection during these firms’
first year was extensive. Over the course of the year, 20% of the firms exited. Exit rates
in the months immediately after entry are quite low, and no firm exited following its sixth
month. Thereafter firms’ exit rates increase with age until their twelfth month. The exit
rate then falls again to zero after the thirteenth month. This lack of exits continues into the
(unreported) fourteenth month. The inverted “U” shape of firms’ exit rate as a function of
age apparently reflects an initial delay in exit followed by a cleansing of less-profitable firms
at the end of the first year. In this case, firms that survived their first year were relatively
fit and unlikely to have immediately exited.
Unsurprisingly, survivors’ average size increases quickly with age. Initial average sales of
all bars is slightly greater than the average sales of all license holders. After one year, the
survivors’ average size is approximately 27% greater than this overall average. The standard
deviation of firms’ initial sales is 0.87, and this dispersion changes little over the first year.

2.3

Firm Growth

To better understand the importance of entrepreneurs’ learning for their exit decisions, Pakes
and Ericson (1998) advocate examining the persistence of surviving firms’ sales. In particular,
they derive nonparametric predictions from two models of learning. In the model of passive
learning, entrepreneurs apply Bayesian updating to learn about a time-invariant and firmspecific parameter, as in Jovanovic (1982). This model implies that a firm’s initial sales will
be useful for forecasting its sales throughout its life. In the active learning model, firms
invest to improve their products and processes. Because the outcome of this investment is
stochastic and very successful firms optimally choose to invest little and allow their knowledge
to depreciate, the observation of a firm’s initial sales becomes progressively less relevant for
forecasting its future. Pakes and Ericson test these models’ contrasting predictions using
panels of Wisconsin retail and manufacturing firms. They find that initial sales improve
forecasts of retailers’ future sales, but manufacturers’ sales appear to be Markovian. They

5

conclude that an approach to firm dynamics based on Bayesian learning is promising for
retail firms.
We have assessed the properties of our sample of bars by conducting a similar empirical
investigation. We estimated density-weighted average derivatives from the regression of a
firm’s sales on its sales in the previous and first months— all in logarithms— using Powell,
Stock, and Stoker’s (1989) nonparametric instrumental-variables estimator.4 These estimates
rely on no distributional assumptions beyond standard regularity conditions, so they are
appropriate for investigating the importance of a firm’s initial sales on its evolution when the
structural parameters relevant for the survival decision are unknown.
Table 2 reports these estimates as well as standard errors based on their asymptotic
distributions.5 All of the derivative estimates are positive and statistically significant at the
5% level. The derivatives with respect to the previous month’s sales are surprisingly similar
across months. They are nearly all between 0.80 and 0.95. The derivatives with respect
to the firm’s sales in its first month are smaller but not negligible. Furthermore, there is
no apparent tendency for the firm’s initial sales to become less relevant for forecasting as
the firm ages. When the dependent variable is the firm’s sales in the third month, the
derivative with respect to the first month’s sales equals 0.154. This is nearly identical to
the analogous coefficient when the dependent variable is the thirteenth month’s sales, 0.168.
The analogous coefficients in the other months vary from a low estimate of 0.021 to a high of
0.190. Apparently, no first-order Markov process can fit surviving firms’ observed sales well.6
Pakes and Ericson (1998) emphasize that the observable differences between the two models they consider only apply to very old firms if sales depend on transitory shocks observed
by only the entrepreneur. Thus, any conclusion about the importance of Bayesian learning
based on the application of their methodology to observations of firms’ first years is necessarily suspect. Non-Markovian dynamics can also arise from permanent and unobservable
differences across entrepreneurs’ choices of their firms’ intended scales. Even with a very
4

If m(x) denotes the expected value of y given x and f (x) is the density of x, then the regression’s densityweighted average derivatives are defined as E[(∂m(x)/∂x)f (x)]/E[f (x)], where the expectation is taken with
respect to f (x). If the regression function is linear, then these equal the linear regression coefficients. If
the regression function depends on an element of x only trivially, then the corresponding density-weighted
average derivative equals zero.
5
To implement this estimation, we follow Powell, Stock, and Stoker’s (1988) recommendation and use the
bias-reducing kernel discussed by Bierens (1987). Before estimation, we scaled both explanatory variables by
their standard deviations. We used a tenth-order kernel with a bandwidth of 2.
6
Expanding the set of regressors to include the past three months’ sales leaves the estimated effect of the
first month’s sales intact.

6

long panel of firm histories, Pakes and Ericson’s procedure cannot distinguish between such
unobserved heterogeneity and true Bayesian learning. The structural analysis we pursue next
overcomes these difficulties.

3

A Structural Model of Firm Growth and Survival

In this section, we present a structural model of firm growth and survival in a monopolistically
competitive industry. A firm’s life begins before entry with the choice of its scale. New firms’
different scales reflect heterogeneity across entrepreneurs’ skills. After entry, a persistent
shock and two transitory shocks affect demand for the firm’s single product. The entrepreneur
observes one of the transitory shocks and the sum of the other one with the persistent shock.
She applies Bayes’ rule to optimally infer the persistent shock from her noisy observations.
Continuation requires payment of fixed costs that are not necessarily constant over time, and
the entrepreneur can irreversibly close the firm to avoid them. She chooses to exit when her
optimal inference of the persistent shock falls below a threshold that depends on the firm’s
age. The pre-entry choice of scale proportionally affects the firm’s producer surplus and the
fixed costs required for continuation, so it has no impact on the exit decision. Instead, it
plays the role of a random effect across firms’ sales. Because the entrepreneur observes both
her scale decision and one of the transitory shocks, we cannot perfectly forecast the firm’s
exit with the observations of sales that we use for estimation.
We assume that firms compete anonymously. That is, the behavior of any single firm only
possibly depends on the behavior of other firms through some aggregate statistics. There is no
direct strategic interaction between any two firms. We next discuss this assumption’s content
and empirical plausibility. We then detail the firm’s stochastic environment and optimization
problem. Following this, we consider the entrepreneur’s procedure for optimally assessing
her firm’s future and deciding upon its survival. With the description of firms’ post-entry
evolution in place, we then consider the entrepreneur’s pre-entry choice of its intended scale.
We finish this section by showing how our framework encompasses two prominent models of
firm dynamics.

3.1

Imperfect Competition and Firm Dynamics

Bars produce heterogeneous goods and compete with each other in local markets. This
compels us to consider imperfect competition as the most likely market structure for our
sample of firms.
7

The theory of competition among a large number of producers offers us two distinct
approaches to consider, monopolistic and oligopolistic competition. In models of monopolistic competition such as Dixit and Stiglitz’s (1977), Hart’s (1985), and Wolinsky’s (1986),
producers compete anonymously, and strategic interaction is absent. Idiosyncratic shocks
to demand and cost have no effect on competitors’ optimal actions, so empirical analysis
can proceed by considering each producer’s choice problem in isolation from those of her rivals. Models of oligopolistic competition, such as Prescott and Visscher’s (1977) and Salop’s
(1979), emphasize strategic interaction. A producer’s actions impact the profits of her neighbors in geographic or product space, so shocks that directly influence only one producer’s
profits can affect her competitors indirectly.7
Campbell and Hopenhayn (2005) suggest distinguishing between these possibilities using
cross-market comparisons of producer sizes. A robust prediction of anonymous monopolistic
competition is that the producer size distribution is invariant to market size if factor prices,
consumer demographics, and technology are held constant. For example, a firm’s sales in
Dixit and Stiglitz’s (1977) model equals the product of the constant demand elasticity with
the sunk cost of entry. Models of oligopolistic competition generally predict that increasing
market size erodes producers’ market power. This is true in Salop’s (1979) model of competition on the circle, where increasing market size— measured with the density of consumers on
the unit circle— decreases the distance between any two firms. This lowers their markups,
so firms must sell more to recover their fixed costs. Hence, with oligopolistic competition we
expect producers’ sizes to be increasing in market size. Campbell and Hopenhayn compare
retailers’ average sales and employment across large and small markets to determine which
of these two approaches is more promising. Their results favor oligopolistic competition.
We have implemented Campbell and Hopenhayn’s procedure for our bars’ parent fourdigit industry, Drinking Places, using exactly the same sample of markets, control variables,
measures of market size, and measures of establishment size that they do. Unlike Campbell and Hopenhayn, we fail to find a statistically significant effect of market size on average
establishment size in Drinking Places. In most of the specifications we have considered, the estimated coefficient on market size is negative and statistically insignificant. This cross-market
comparison of establishment sizes does not refute the assumption that bars are monopolistic
competitors.
We now proceed to present our model of firm dynamics under monopolistic competition.
7

The observation that consumers view similar goods available at different geographic locations as imperfect
substitutes does not immediately imply that competition is oligopolistically competitive.

8

We first discuss the post-entry evolution of a firm of a given scale, and we then detail the
forward-looking scale choice before entry.

3.2

The Stochastic Environment

Consider the life of a single firm, which begins production at time t = 1. The firm is
a monopolistic competitor that produces a single good at a single location. In period t,
consumers demand Qt = eR+Xt +Wt Pt−ε units of the firm’s good, where ε > 1 is the absolute
value of the demand elasticity and Pt is its price. The time-invariant demand-shifter R
reflects a pre-entry choice of the intended scale of operation of the firm. Examples of decisions
affecting R for our sample of bars are the centrality of the bar’s location and the installation
of attractions like a music stage or a mechanical bull. The composite random variable Xt +Wt
shifts the demand curve through time, and we explain the properties of its two components
below. Throughout, we adopt conventional notation and reserve capital letters for random
variables and small letters for their realizations.
An affine cost function, ϑQt + eR κt , describes the firm’s technology, with both ϑ and κt
strictly greater than zero. The fixed costs can vary with the age of the firm and are higher
for firms with larger intended scale. In our application to Texas’ bars, time variation in the
fixed cost κt might reflect periodic renewal of leases for equipment or space. The effect of R
on fixed costs reflects the greater expenditure associated with maintaining a larger firm.
It is straightforward to show that the entrepreneur’s profit-maximizing price choice is
¡ ε ¢
constant, Pt = ε−1
ϑ. The resulting sales and profits are
µ
St

e

= Pt Qt =

ε
ε−1

¶1−ε
eR+Xt +Wt ϑ1−ε

(1)

and
µ
R

(Pt − ϑ) Qt − e κt =

1
ε−1

¶µ

ε
ε−1

¶−ε
eR+Xt +Wt ϑ1−ε − eR κt ,

(2)

We choose the unit of account to set ϑ = (ε−1)/ε, so that log sales are simply St = R+Xt +Wt
and the firm’s profit equals ε−1 eSt − eR κt .
The random variable Xt = At + Ut . Of these, At is the persistent shock to the firm’s
demand and Ut is transitory. The initial value of At is drawn from a normal distribution with
mean µ1 and variance σ12 . Thereafter, a first-order autoregression governs At ’s evolution,
At = µ + ρAt−1 + Zt

with Zt ∼ N (0, σ 2 )
9

(3)

In (3), ρ > 0 and the disturbances Zt are independent over time
The two series of transitory shocks, {Wt } and {Ut }, are independent over time and from
each other and {Zt }. We assume that they are normally distributed with mean zero and
variances γ 2 and η 2 . The entrepreneur cares about these transitory shocks individually
because she observes only one of them, Wt . She also observes Xt , but Ut and At are hidden
from her. The variance of Wt contributes to the model’s econometric error term. Nontrivial
Bayesian learning about At arises from the variance of Ut .
For concreteness, we have assumed that the pre-entry choice of scale and all post-entry
shocks affect the firm’s demand and not its marginal cost. We could alternatively have
assumed that these variables affect marginal cost as well. If we only have data on sales and
survival, these models are observationally equivalent. In this paper, we restrict attention
to the analysis of such data. We do not address the separate identification of idiosyncratic
demand and cost variation, which would require the observation of firms’ prices.

3.3

Bayesian learning and selection

At the end of each period, the entrepreneur must decide whether or not to close the firm
and exit. Exit is an irreversible decision, and its payoff equals zero. The entrepreneur is
risk-neutral and discounts the firm’s future profits with the constant factor δ < 1.
The normality of Zt and Ut imply that the entrepreneur can use the Kalman filter to
calculate an optimal inference of At given the relevant information at hand, (X1 , . . . , Xt ) ≡
X̄t .8 Denote this optimal forecast and its mean squared error with
·³
´2 ¸
Ât ≡ E[At |X̄t ] and Σt ≡ E At − Ât
.
The Kalman filter calculates Ât and Σt recursively using
Ât = µ + ρÂt−1 + λt (Xt − µ − ρÂt−1 ),

(4)

and Σt = η 2 λt , where λ1 ≡ σ12 / (σ12 + η 2 ) and λt ≡ (ρ2 Σt−1 + σ 2 ) / (ρ2 Σt−1 + σ 2 + η 2 ) for
t > 1. The coefficient λt is the Kalman gain, and it measures the informativeness of the
entrepreneur’s observation of Xt . The firm’s sales does not directly reveal Xt , so the entrepreneur’s estimate of At is necessarily more accurate than that of an outside observer.
Although the entire history of Xt is, in principle, relevant for the entrepreneur’s exit
decision, Ât , the firm’s age, and R are sufficient for characterizing the distribution of future
8

Throughout this paper, we denote the vector (a1 , a2 , . . . , at ) with āt .

10

³

´

profits. Define vt Ât , St , R to be the value of a firm of age t and intended scale R to an
entrepreneur who estimates At to be Ât and observes current sales to be St . The Bellman
equation that this value function satisfies is
³
´
vt Ât , St , R = ε−1 eSt − eR κt + δ max{0, Et [vt+1 (Ât+1 , St+1 , R)]}.
(5)
The entrepreneur calculates the expectation Et in (5) using the joint distribution of Ât+1 and
St+1 conditional on the available information at time t, for which (t, Ât , R) is sufficient.
The entrepreneur’s optimal exit policy is simple. The expectation in the right-hand side
of (5) is a function of (t, Ât , R) and it is continuous and increasing in Ât . Therefore, there
exists a threshold value αt such that the entrepreneur chooses to exit if and only if Ât ≤ αt .
The firm’s scale, R, has no impact on the exit decision, because it only scales up the firm’s
surplus and fixed costs. Instead, it plays the role of an unobservable (to us) random effect
on the sales observations.

3.4

Pre-Entry Choice of Intended Scale

With the firm’s exit policy and value function in place, we are now prepared to consider
the entrepreneur’s pre-entry choice of intended scale. For a randomly selected entrepreneur,
the cost of entry with scale R is F eτ R , where τ > 1. The assumption that the entry cost
is convex in eR reflects limits to entrepreneurs’ abilities, as in Lucas (1978). The positive
random variable F embodies heterogeneity across entrepreneurs in those abilities. With this
cost function, doubling the scale more than doubles the costs of entry. The expected value of
a new firm is a linear function of eR , so equating the marginal cost and benefits of increasing
the firm’s scale for an entrepreneur with ability F yields R = (ln (υ/τ ) − ln F ) / (τ − 1),
where υ is the slope of the expected value of new firm with respect to eR . We assume that
ln F has a normal distribution with mean ln (υ/τ ) and variance (τ − 1)2 ν 2 so that R is also
normally distributed, with mean zero and variance ν 2 .

3.5

Existing Models of Firm Dynamics

As we noted in the introduction, the model encompasses versions of both Jovanovic’s (1982)
and Hopenhayn’s (1992) models of firm dynamics. Although both models assume a perfectly
competitive market structure, we see the change to a simple version of monopolistic competition as inconsequential for our purposes. The simplest versions of both models assume
that fixed costs are constant over time; κt = κ for all t. Jovanovic’s model of learning about
11

time-invariant random profitability sets ρ = 1 and σ 2 = 0, while we recover Hopenhayn’s
model with ρ < 1 and η 2 = 0. Our versions of both models add a transitory econometric
error term, Wt , and unobservable heterogeneity, R, to make them suitable for estimation.

4

The Model’s Empirical Content

The TABC panel contains observations of each firm’s sales (St ) and an indicator for its
survival to age t, which we denote with Nt . Because we do not observe St after the firm’s
exit, we set St = 0 if Nt = 0. The sales observations cover the first T = 12 months in the
firm’s life. The TABC panel data set also records NT +1 . That is, the firm’s survival at the
end of the sample period is known. In this section, we describe our sequential procedure for
inferring the structural model’s parameters from the joint distribution of these observations.
In the procedure’s first stage, we treat the exit thresholds (α1 , . . . , αT ) ≡ ᾱT as parameters to be estimated jointly with the parameters that determine the evolution of St .
The procedure’s second stage recovers the sequence of fixed costs κt up to scale using the
first-stage estimates, an assumed value of the discount factor δ, and the Bellman equation.
Unlike Rust’s (1987) nested fixed-point algorithm, our approach to inference separates the
estimation of most of the model’s parameters from the repeated solution of the entrepreneurs’
dynamic programming problem. This separation brings with it a distinct advantage: We estimate many of the model’s structural parameters without assuming that the entrepreneurs’
dynamic program is stationary or otherwise specifying firms’ production possibilities beyond
the sample period.
We use the structural model to estimate the content of entrepreneurs’ information without
actually having that information, so it is particularly important to understand the sources
of the model’s identification. We demonstrate next that a sample of infinitely many firm
histories identifies the exit thresholds and the parameters governing the evolution of St , and
we discuss the features of the data that are particularly influential in this inference. We
then consider the first-stage estimation of these parameters and policies using maximum
likelihood. Finally, we show how to recover the sequence of fixed costs in the procedure’s
second stage. With these in hand, we can calculate the value of the firm’s option to exit and
conduct other policy experiments.

12

4.1

First-Stage Identification

The structural model decomposes the fluctuations of St into four components, the firm’s
scale (R), the transitory shock observed by the entrepreneur (Wt ), and the transitory and
persistent shocks unobserved by the entrepreneur (At and Ut ). In our procedure’s first stage,
we estimate the parameters that characterize these shocks’ distributions— ν 2 , γ 2 , ρ, σ12 , σ 2 ,
η 2 , µ1 , and µ— and the entrepreneurs’ optimal exit thresholds ᾱT . The proof of the following
proposition demonstrates that observations of firm histories identify these parameters and
policies.
¡
¢
Proposition. If T ≥ 3, then the joint distribution of S̄T , N̄T +1 uniquely determines ν 2 ,
γ 2 , ρ, σ12 , σ 2 , η 2 , µ1 , µ, and ᾱT .
The appendix contains the proposition’s proof. Here we describe the three key insights that
it requires.
Blundell and Preston (1998) identify a consumer’s permanent income process using the
covariance of his current income (a noisy proxy) with his consumption (a forward-looking
choice). The proof adapts their approach to the identification of the parameters that govern
Xt . Consider, for example, the identification of V [X1 ] = σ12 + η 2 from the joint distribution
of S1 and N2 . The observed value of S1 is a noisy proxy for the model’s true state variable,
Â1 = µ1 + λ1 (X1 − µ1 ). Its covariance with the forward looking choice N2 is
E[S1 N2 ] − E[S1 ]E[N2 ] =

p
¡
¢
V [X1 ]φ Φ−1 (1 − E[N2 ]) ,

(6)

where Φ(·) and φ(·) are the c.d.f. and p.d.f. of a standard normal random variable. This
immediately identifies V [X1 ]. Intuitively, if S1 is very useful for predicting N2 in a linearin-probabilities model, then much of its variation must arise from the factor that influences
survival, X1 . Otherwise, we conclude that R + W1 dominates its variation. The expression
in (6) translates this intuition into a quantitative measure.9
The model contains two distinct sources of persistence in St , autocorrelation of At and
variation of R across firms. Because the model is nonlinear, standard dynamic panel data
techniques for distinguishing between them, such as Blundell and Bond’s (1998), are inapplicable. We instead exploit the different implications of random effects and autocorrelation
9

Olley and Pakes (1996), Pavcnik (2002), and Levinsohn and Petrin (2003) follow a related identification
strategy. They use nonlinear functions of a continuous decision (investment or materials use) to replace a
firm’s unobserved productivity in the estimation of a production function. We cannot apply this strategy,
because the forward-looking decision we employ is discrete.

13

for the impact of selection on the evolution of St . Consider the regression of S2 on S1 . In
the absence of selection, this regression would be linear. Because X1 and R contribute to its
error, the regression conditional upon survival will also include a Heckman (1979) selection
correction. That is
E[S2 − E[S2 ]|S1 , N2 = 1] = %(S1 − E[S1 ]) + ςm(S1 ).

(7)

In (7), m(S1 ) is the relevant inverse Mills’ ratio, which is known. Hence, we can obtain the
coefficients % and ς from this regression function.
The key to using (7) to separately identify cov[X2 , X1 ] from ν 2 is that these two sources
of persistence have very different implications for the Heckman selection correction. Given
S1 , learning that the firm survived increases the best estimate of X1 and reduces the best
estimate of R. Hence if persistence in St arises mostly from variation in R, then the coefficient
multiplying the Heckman selection correction is negative. If instead the variance of R is small
relative to the variability and persistence of Xt , then selection increases the expectation of S2 .
In addition to this qualitative information, % and ς determine cov[X2 , X1 ] and ν 2 uniquely.
We demonstrate this in the proposition’s proof.
The final key insight required by the proof is that of Pakes and Ericson (1998). As we
noted in Section 2, their procedure applied directly to the original data cannot distinguish
between true entrepreneurial learning, transitory shocks observed by entrepreneurs, and permanent differences in firms’ scales. To reliably apply their procedure to a sample generated
by our model, we must instead consider the joint distribution of X̄T . From this we can
calculate the linear regression of X3 on X2 and X1 ,
X3 − E[X3 ] = β32 (X2 − E[X2 ]) + β31 (X1 − E[X1 ]) + V3 .

(8)

In (8), V3 is a composite error term. If η 2 equals zero, then there is no scope for entrepreneurial
learning, β31 = 0, and β32 = ρ. Otherwise, β31 > 0 and β32 < ρ. It is straightforward to write
β31 and β32 as explicit functions of the model’s structural parameters. These expressions
show that these coefficients directly reveal ρ and η 2 .

4.2

First-Stage Estimation

With the identification of the parameters’ and policies’ established, we now proceed to consider their estimation using maximum likelihood. This poses no conceptual problems, but
computation of the likelihood is nontrivial because the model involves repeated selection
on the basis of a persistent latent state variable. In the following, we denote the vector of
14

parameters and policies estimated in the first-stage with $. Our method of calculating the
likelihood function follows the non-Gaussian state-space approach of Kitagawa (1987), which
utilizes the likelihood’s prediction-error decomposition. If we condition on the realization of
R, then this is
¡

¢

fȲT +1 Y T +1 |R; $ = fY1 (Y1 |R; $)

T
+1
Y

¡
¢
fYt Yt |Y t−1 , R; $ ,

(9)

t=2

where Yt ≡ (St , Nt ) unless t = T + 1, in which case it equals NT +1 .
The first term in (9), the density of Y1 , is known. We calculate the prediction-error decomposition’s remaining terms recursively. We initialize the recursion with fAb1 (b
a1 |Y1 , R; $),
which is normal with known mean and variance. To calculate the likelihood function’s re¡
¢
maining terms, suppose that fAbt−1 ât−1 |Ȳt−1 , R; $ is known. We consider three separate
cases. If the firm exits production following period t − 1, then Nt = 0 and the relevant term
in the likelihood function is the probability of exit conditional on the observed history.
Z αt−1
¡
¢
¡
¢
fYt (0, 0)|Ȳt−1 , R; $ =
fAbt−1 ât−1 |Ȳt−1 , R; $ dât−1
(10)
−∞

Following exit, the evolution of St is trivial and the remaining terms in the prediction error
decomposition identically equal one. In the second case, the firm continues production following period t − 1, but t = T + 1 so the data do not contain the realized value of St . In this
case of right censoring, the final term in the prediction error decomposition is
Z ∞
¡
¢
¡
¢
(11)
fYT +1 1|ȲT , R; $ =
fAbT âT |ȲT , R; $ dâT
αT

In the final case, the firm produces in period t < T + 1, so Nt = 1. For this case, the
term of interest in the likelihood function can be written as
Ã
!
Z ∞
¡
¢
1
St − µ − ρb
at−1 − R
p
fYt (St , 1)|Ȳt−1 , R; $ =
φ p
ρ2 Σt−1 + σ 2 + η 2 + γ 2
ρ2 Σt−1 + σ 2 + η 2 + γ 2 (12)
αt−1
¡
¢
× fAbt−1 ât−1 |Ȳt−1 , R; $ dât−1
bt−1 .
Equation (12) follows from Bayes’ rule and the definition of A
This final case is the only one in which we wish to continue the recursion. To do so, we
¡
¢
at |Ȳt , R; $ , which is
must calculate fAbt b
¡
¢
R∞
f bt ,St (ât , St |ât−1 , R; $) fAbt−1 ât−1 |Ȳt−1 , R; $ dât−1
¡
¢
αt−1 A
¡
¢
fAbt ât |Ȳt , R; $ =
.
(13)
fYt Yt |Ȳt−1 , R; $
15

³

´
bt−1 and R is bivariate normal, so both terms in
b
The distribution of At , St conditional on A
the integrand of (13) are known. Therefore, the recursion can continue.
Iterating on (12) and (13) until the individual either exits or is right censored produces
the likelihood function at any given choice of $ for a fixed value of R. We obtain the
unconditional likelihood function by calculating the expectation of this density with respect
to R. In practice, evaluating the likelihood function requires approximating these integrals.
We do so using Gaussian quadrature procedures. Maximization of the resulting likelihood
function is straightforward; and the corresponding maximum-likelihood estimate, $̂, has the
usual distributional properties.

4.3

Second-Stage Estimation

With $̂ in hand, we can proceed to exploit the entrepreneur’s Bellman equation to estimate
the remaining identified structural parameters. First, we fix the discount factor δ at a
value consistent with a 5% annual rate of interest. Rust’s (1994) nonidentification result
for Markovian dynamic discrete choice models shows that this assumption is unavoidable.
Next, note that the Bellman equation (5) and the linearity of the producer’s surplus, ε−1 eSt ,
and the fixed costs, eR κt , in eR together imply that exit decisions are governed by the scaled
Bellman equation
³
´
gt Ât , Xt + Wt = eXt +Wt − εκt + δ max{0, Et [gt+1 (Ât+1 , Xt+1 + Wt+1 )]}.
(14)
³
´
³
´
In (14), gt Ât , Xt + Wt = εvt Ât , St , R /eR .10 Given an infinite sequence {εκt } and the
parameters estimated in the first stage, we can iterate equation (14) to recover the scaled
value function gt . Obviously, multiplying ε by a positive constant and dividing the sequence
{κt } by that same constant yields an observationally equivalent model. Therefore, we will
focus on estimating {εκt }. This only recovers fixed costs up to scale, but we use this to
express a firm’s fixed costs as a fraction of its expected first-period surplus, εκt /E[eX1 +W1 ].
Note that this ratio does not depend on its intended scale R.11
The Bellman equation (14) suggests a simple procedure for estimating {εκt } from the firststage estimates of the model’s other parameters and of the exit thresholds: Choose the scaled
10

This alternative representation of the exit decision process clarifies the contribution of R to the datagenerating process. It shows that R influences only the firm’s sales and not its exit decisions.
11
Alternatively, we could have used the ratio εκt /E[eXt +Wt |Nt = 1]. This measure of period t’s fixed
costs relative to the expected producer surplus would reflect both the evolution of fixed costs and the typical
growth of surviving firms, so it is not as easily interpreted as the expression we prefer.

16

fixed costs so that the corresponding optimal exit thresholds mimic the estimated thresholds.
A practical problem is that we do not have estimates of αt for t > T . We circumvent this
in our application by assuming that εκt follows an annual cycle. That is, εκt+12 = εκt . This
specification of the scaled fixed costs captures the annual lease cycles mentioned in Section
3. The first T optimal exit thresholds can then be expressed as a known smooth function
of $ and εκ̄12 , ᾱT ($, εκ̄12 ). We estimate the vector of fixed costs with the minimizer of the
ˆT ,
distance between these optimal thresholds and their first-stage estimates, ᾱ
¡
¢
¡
¢
ˆ T )−1 ᾱ
ˆ T − ᾱT ($̂, εκ̄12 ) ,
ˆ T − ᾱT ($̂, εκ̄12 ) 0 V̂(ᾱ
εc
κ̄12 = arg min ᾱ
εκ̄12

ˆ T ) is a consistent estimator for the variance-covariance matrix of ᾱ
ˆ T . This estimawhere V̂(ᾱ
tor of the scaled fixed costs has well-known statistical properties.
Once we have estimated {εκt }, it is straightforward to estimate the expected value of a
new firm relative to its expected first-period surplus. Recall from Subsection 3.4 that υeR
is the expected value of a new firm with intended scale hR. ³The Bellman ´i
equation (14), the
first stage parametes, and {εκt } together yield ευ = E g1 Â1 , X1 + W1 . Using this, we
can estimate the expected value of a new firm relative to its first-period producer’s surplus,
ευ/E[eX1 +W1 ], by simply plugging in our estimates of these parameters. We can also easily
compute the value of a new firm if it must be operated in perpetuity. The difference between
these two gives the value of the option to close the firm.

5

Empirical Results

In this section, we report the results from estimating the model’s parameters by maximum
likelihood using the sample of 305 new bars described in Section 2. We first demonstrate
that the maximum likelihood estimate of η 2 is zero. This implies that entrepreneurs observe
At without error and do no not learn. We then estimate a version of our model in which η 2 is
constrained to equal zero. We find that post-entry idiosyncratic shocks have a half life of 47
months and account for approximately 60% of sales’ variance. The remaining 40% of sales’
variance is due to heterogeneity in pre-entry scale decisions. We conclude with a presentation
of the second-stage estimates of the model’s fixed costs and the policy experiments that yield
the value of the firm’s option to exit.

17

5.1

First-Stage Estimates

We first discuss the estimation of η 2 , which determines the speed and extent of entrepreneurial
learning. Figure 1 summarizes our results. For values of η 2 between zero and 0.45, it plots
the value of the likelihood function after enveloping out the other elements of $. The result
is clear: The likelihood function attains its maximum at η 2 = 0. That is, our maximum likelihood estimates imply that entrepreneurs observe the persistent component of profit without
error. We have conducted the same exercise for several alternative specifications of our model,
and they have all yielded this result. Thus, we find no evidence that entrepreneurial learning
contributes to the dynamics of the firms in this sample.
Table 3 reports the first-stage maximum likelihood estimates of a version of our model in
which η 2 is constrained to equal zero. Below each estimate is its asymptotic standard error,
which we calculated using the outer product estimate of the information matrix. Recall from
Section 2 that no firm in our sample exited following its sixth month. In light of our identification proof’s reliance on selection, this led us to suspect that our data are uninformative
regarding the sixth month’s exit threshold. To determine whether or not this was the case,
we fixed α6 at a variety of large and negative values and chose the other elements of $ to
maximize the likelihood function. The resulting parameter estimates are nearly invariant to
our choice of α6 . Accordingly, we present here estimates based on setting α6 to −2.5.
Consider first the estimates of µ1 , σ1 , γ, and ν; which jointly determine the distribution
of firms’ initial sizes. The estimate of µ1 , 0.069, is virtually identical to the unconditional
mean of firms’ initial size reported in Table 1. However, the standard error attached to this
estimate is relatively large, 0.055. The estimates of σ1 , γ, and ν are 0.714, 0.167, and 0.581.
These imply that the standard deviation of S1 equals 0.94, which is above the estimated
standard deviation of 0.87. At these parameter estimates, the pre-entry choice of location
quality accounts for 38% of the variance of S1 . The persistent post-entry shock accounts for
most of S1 ’s remaining variance. We conclude from this that entrepreneurs gain significant
information about their firm’s prospects immediately after entry. However, firms’ initial sales
also embody substantial heterogeneity of pre-entry scale decisions.
The parameters that govern the evolution of At are ρ, σ, and µ. The estimate of ρ is 0.985,
indicating that post-entry shocks have persistent effects on firms’ sizes. This is estimated
with great precision— its standard error equals 0.0002, so we can reject the hypothesis that
it equals one and Zt permanently changes firm sizes. This implies that the relatively small
firms that compose our sample do not obey Gibrat’s law even after correcting for selection.
An increase in Zt that increases current sales by one percent increases sales twelve and
18

twenty-four months later by 0.84% and 0.70%.
The estimate of the intercept µ implies an estimate of the mean of At ’s ergodic distribution
of 0.093, which is only slightly above our estimate of µ1 . The difference between them is
insignificant at any conventional level. This implies that mean log sales would not grow
much in the absence of exits. Selection explains nearly all the growth in the data. We have
also estimated a version of the model with age-dependent intercepts, but do not reject the
assumption that the intercepts are constant.12 Thus, we do not find evidence of systematic
firm growth, such as that arising from the gradual acquisition of a clientele, as the firm ages.
The estimated standard deviation of Zt is 0.161. This implies that the standard deviation
of At ’s ergodic distribution is 0.94. This is substantially and significantly above the estimate
of σ1 , so the dispersion of an entering cohort’s sizes increases with its age in the absence of
exit. In fact, the dispersion of firm sizes neither increases nor decreases with the cohort’s
age, so selection offsets this underlying increase in variation.
The final estimates to report are those describing the entrepreneurs’ exit policy, ᾱ12 . The
estimate of the initial exit threshold is −1.65. The implied probability that a firm exits
following its first month is 0.008. The estimated exit thresholds tend to rise with the firm’s
age. The estimate of α2 is −1.53, while that for α12 is −0.54. Entrepreneurs apparently
become more selective when deciding on continuation of their operations during the course
of the firms’ first year. There is no Bayesian learning in this specification of the model, and
we have estimated very little drift in average sales. Therefore, accounting for these changes
in the entrepreneurs’ exit policies requires parallel changes in their fixed costs. We measure
these changes below with the procedure’s second stage.
To help gauge how well the estimated model fits the data, Table 4 reports the implied
population values of the summary statistics that we considered in Section 2.13 Like the raw
exit rates in Table 1, the exit rates implied by our model are low in the first half of the year,
and high in the remainder. Substantial increases in mean sales again reflect the importance
of selection effects. Consistently with the data and despite the importance of selection, sales’
standard deviation does not decrease substantially as the cohort ages. Quantitatively, the
exit rates increase faster in the model than in the data. The estimated model’s exit rates in
the ninth and eleventh months are particularly high compared to their empirical counterparts.
Mean sales in the first 9 months are matched well by the model. The large selection effects
in the ninth and eleventh months, however, cause the model to overstate the growth of mean
sales between months 9 and 10 and months 11 and 12, and to generate a small decrease in
12
13

The Appendix shows that such extensions of the model with age-dependent parameters are identified.
We calculated these summary statistics using 100, 000 simulated firm histories from the estimated model.

19

sales’ standard deviation in these months that is absent from the data.14
Similarly, Table 5 provides the population version of the nonparametric regression of surviving firms’ log sales on their log sales in the previous and first months. Recall from Section
2 that its empirical counterpart shows that sales in the first month help predicting current
sales conditional on sales in the previous month. The model generates a similar pattern
of non-Markovian dynamics, even though the estimated model contains no entrepreneurial
learning–the source of non-Markovian dynamics emphasized by Pakes and Ericson (1998).
Instead, unobserved heterogeneity in the scale of operations and transitory shocks observed
by the entrepreneurs are sufficient to generate the observed non-Markovian dynamics in our
short panel of bars.

5.2

Second-Stage Estimates and the Value of the Option to Exit

Table 6 reports estimates of the fixed costs as a fraction of the average first-period producer
surplus, ²κt /E[eX1 +W1 ]. Below each estimated fixed cost is an estimate of its asymptotic
standard error.15 The most remarkable result is that the fixed costs in the first period are
some 10 times larger than the first period’s producer surplus. A claim on a surplus that is on
average above the per-period fixed costs in the remaining 11 months of the year compensates
for this somewhat. Note that the relative fixed costs are estimated to be particularly low
in the seventh month, reflecting the observed low exit rate after six months. We do not
estimate the fixed costs very precisely. The standard error on the first period’s cost is 1.80.
The other estimates’ standard errors are comparable to this when expressed as a fraction
of the corresponding fixed costs. However, all of the estimated fixed costs are statistically
different from zero at conventional sizes. Overall, the sum of the estimated fixed costs equals
14.59 times the expected first-period producer surplus. The standard error for this sum is
2.26.
Our estimates indicate that fixed costs are high at the start of each new year. We
interpret this as the effect of an annual lease cycle. Signing a lease for space or equipment
that is difficult to break effectively sinks its proscribed costs. This explains why exit rates
14

We could pursue a better match of the data by allowing the variances of the structural shocks to vary
over time. The analysis in the Appendix shows that such an extended model is identified. However, we think
the simplicity of our present setup outweighs a slightly better match with the data.
15
Because we set η 2 = 0, the optimal exit thresholds follow an annual cycle. In this special case, we can
b̄ 12 . We calculate
recover εc
κ̄12 as the solution to a set of linear equations based on the assumed optimality of α
these equations’ coefficients and intercepts by simulating a large number of firm histories that begin with
Ât = α̂t . The reported standard errors account for the resulting simulation error.

20

are high near the end of the year, when next year’s lease costs can still be avoided, and low
at the beginning of the year, when the returns on the investment in the lease have not yet
materialized.
With estimates of the model’s structural parameters in hand, we finally compute the value
of the option to close a new firm. In the model, a newly created firm is worth 29.7 times
its expected first-period surplus. The value of the entrepreneur’s exit option is an important
component of this. Consider a hypothetical new firm that faces the same sales process as
the bars in our sample but has no option to exit. Our estimates imply that such a firm is a
liability: Its estimated value is −7.0 times its expected first-period surplus. The fixed costs’
standard errors naturally affect these estimates. The 95% confidence interval for a new firm’s
value is (22.8, 36.7), and the corresponding interval for its value if operated in perpetuity is
(−20.8, 6.7). Together, these estimates imply that the exit option accounts for 124% of the
value of a new firm. The corresponding 95% confidence interval is (0.75, 1.73). The width of
this interval is not small, but it does imply that the option to exit accounts for most of the
value of a new firm.

6

Conclusion

Risk and selection dominate a firm’s first year. Although new bars differ considerably in
their intended scales, persistent post-entry shocks account for the majority of their sales’
cross-sectional variation. Immediately after entry, only very unfavorable shocks induce an
entrepreneur to exit. As the firm’s first anniversary approaches, the entrepreneur raises her
standards for continuation. Firms’ exit rates correspondingly increase over the course of the
first year. In principle, this behavior is consistent with Bayesian learning about persistent
shocks: The entrepreneur delays her exit decision until her posterior beliefs about profitability
are sufficiently tight. However, our estimates indicate that entrepreneurs observe their bars’
persistent shocks without error. Instead, the increased selectivity reflects a substantial fixed
cost incurred at the beginning of each year.
We conclude that a new firm’s value arises mostly from the option to operate it only after
favorable shocks. Hence, an analysis of small business policy that ignores entrepreneurs’
exit decisions almost certainly fails to capture its primary effect on firms’ values. Our estimates directly characterize these decisions as well as the informational differences between
entrepreneurs and policy makers with access only to public information. Hence, they can
support the quantitative evaluation of information-constrained policy interventions. One

21

relevant example is the taxation of entrepreneurship.
The owner of an existing firm that expands by creating a new establishment faces a problem similar to that of an entering firm. In both cases, the new venture carries substantial risk.
However, the owner of an expanding firm has his previous experience as a guide. Presumably,
the new establishment imperfectly copies the firm’s older businesses. For the manufacturing
sector; Dunne, Roberts, and Samuelson (1988) documented that establishments founded by
expanding firms are larger and more likely to survive than other entrants without previous
industry experience. The measurement of how these new establishments’ first years depend
on their parent firms’ histories is a natural application for this paper’s framework. A related
application is the measurement of the sunk costs of exporting, which Roberts and Tybout
(1997) characterized for Columbian manufacturing. Our future research will investigate these
important aspects of firm growth.

22

Appendix
The time-invariance of the model’s parameters plays a limited role in the proposition’s proof. To make this
explicit, we consider a slightly more general model in which the parameters vary over time. In obvious
notation, the parameters of this model are ν 2 , γ̄T2 , ρ̄T , σ̄T2 , η̄T2 , µ̄T , and ᾱT . In this appendix, we prove
¡
¢
Proposition. The distribution of S̄T , N̄T +1 uniquely determines ν 2 ; γ̄T2 ; λ1 ρ2 , ρ3 , . . . , ρT ; σ12 + η12 , ρ22 Σ1 +
σ22 , σ32 , . . . , σT2 ; η22 , . . . , ηT2 ; µ1 , µ2 + ρ2 µ1 , µ3 , . . . , µT ; and (α1 − µ1 ) /σ12 , α2 , . . . , αT .
This proposition implies that all of the model’s parameters except µ1 , α1 , σ12 , η12 , σ2 , and ρ2 are identified.
Complete identification requires one additional restriction, such as ρ2 = ρ3 . For T ≥ 3, the model of Section
3 imposes this and additional restrictions. Hence the proposition stated in Section 4 follows as a corollary.
The proposition’s proof requires some new notation. Recall that St = Nt (Xt + Wt + R). Write the
¡
¢
transition for Xt conditional on X̄t−1 as Xt = βt X̄t−1 + Vt , with βt (X̄t−1 ) ≡ µt + ρt Ât−1 and Vt ≡
ρt (At−1 − Ât−1 ) + Zt + Ut . Note that the regression functions βt can be computed recursively using the
Kalman filter and that β1 = µ1 . The disturbance Vt is normally distributed with mean zero and variance
ωt2 ≡ ρ2t Σt−1 + σt2 + ηt2 and is independent of X̄t−1 ; the processes {Vt } and {Wt } and mutually independent
and independent over time. The recursive specification for the survival process is
(
¡
¢
1 if Nt−1 = 1 and Xt−1 > τt−1 X̄t−2
Nt =
,
0 otherwise
¡
¢
£
¡
¢¤
¡
¢
with τt X̄t−1 ≡ λ−1
αt − βt X̄t−1 + βt X̄t−1 . Note that βt and τt are both affine functions. They
t
are therefore uniquely determined by their restrictions to any set in Rt−1 that is not contained in a linear
subspace of Rt−1 .
The proof proceeds in four steps. We first prove
Lemma 1 (Identification of the distribution of intended scales). ν 2 is identified from the distribution
of (S̄2 , N̄2 ).
Next, define scaled sales to be St∗ ≡ Nt (St − R). We exploit Lemma 1 to prove
Lemma 2 (Identification of the scaled data distribution). The distribution of (S̄T∗ , N̄T +1 ) is identified
from the distribution of (S̄T , N̄T +1 ).
Lemma 2 ensures that we can apply
¢
¡
Lemma 3 (Identification of γ̄T2 , ω̄T2 , β̄T , and τ̄T ). The distribution of S̄T∗ , N̄T +1 uniquely determines
the parameters ω̄T2 and γ̄T2 and the functions β̄T and τ̄T .
With these lemmas in place, Lemma 4 delivers the proposition’s conclusion.
Lemma 4 (Identification of the structural parameters from ω̄T , β̄T , and τ̄T ). Given ω̄T , β̄T , and τ̄T ,
the following parameters of the structural model are uniquely determined: λ1 ρ2 , ρ3 , . . . , ρT ; σ12 + η12 , ρ22 Σ1 +
σ22 , σ32 , . . . , σT2 ; η22 , . . . , ηT2 ; µ1 , µ2 + ρ2 µ1 , µ3 , . . . , µT ; and (α1 − µ1 ) /σ12 , α2 , . . . , αT .
We now present the lemmas’ proofs.

23

¡
¢
Proof of Lemma 1. Write the regression of X1 on S1 as X1 = ι + ζS1 + E1 , where ζ ≡ ω12 / ω12 + γ12 + ν 2 ,
ι ≡ β1 (1 − ζ), and E1 ≡ (1 − ζ) (X1 − β1 ) − ζ (W1 + R). By construction, E1 ⊥⊥S1 , E [E1 ] = 0, and V [E1 ] =
χ2 ≡ ω12 (1 − ζ). Hence, we can write
µ
¶
ι + ζS1 − τ1
Pr (N2 = 1|S1 ) = Φ
.
χ
Because Pr (N2 = 1|S1 ) is data, this identifies χ/ζ = γ12 + ν 2 and (ι − τ1 ) /χ. Because V [S1 ] = ω12 + γ12 + ν 2
and E[S1 ] = β1 are data, we can also identify ω12 , ζ, and χ2 .
Next, consider the autoregression of S2 on S1 without accounting for selection, E [S2 |S1 ] = ξ +ψS1 . Here,
¢ ¡
¢
¡
ψ ≡ β21 ω12 + ν 2 / ω12 + ν 2 + γ12 and ξ ≡ β2 (β1 ) − ψβ1 , with β21 ≡ dβ2 (x)/dx = ρ2 λ1 . The residual from
this regression is E2 ≡ (β21 − ψ) (X1 − β1 ) + (1 − ψ) R + V2 + W2 − ψW1 . We have that
¡
¢
E [E1 E2 ] = (1 − ζ) (β21 − ψ) ω12 + ζψ γ12 + ν 2 − ζν 2 .
Because ζ/χ, (ι − τ1 ) /χ, and χ are identified and (see Heckman, 1979)
³
´
ι+ζS1 −τ1
φ
χ
E [E1 E2 ]
³
´
E [S2 |S1 , N2 = 1] = ξ + ψS1 +
ι+ζS1 −τ1
χ
Φ
χ

both ψ and E [E1 E2 ] are identified. Using that ζ, ω12 , and γ12 + ν 2 are already known, this identifies ν 2 .
Proof of Lemma 2. Denote the densities of S̄t and S̄t∗ on {Nt = 1} by fS̄t (·|Nt = 1) and fS̄t∗ (·|Nt = 1),
respectively. Because R is independent of (S̄t∗ , N̄T +1 ), we have that
Z ∞
¡
¢
¡
¢ ³r´
fS̄t s1 , . . . , st |N̄T +1 =
dr
(15)
fS̄t∗ s1 − r, . . . , st − r|N̄T +1 φ
ν
−∞
on {Nt = 1}. The left-hand side of (15) is data and ν is identified by Lemma 1. Thus, a standard deconvolution argument establishes that fS̄t∗ (·|N̄T +1 ) is identified on {Nt = 1}.16 This immediately identifies
∗
= · · · = ST∗ = 0 on
the distributions of S̄T∗ on {NT +1 = 1} and {NT = 1, NT +1 = 0}. Using that St+1
{Nt+1 = 0}, we can in addition identify the distribution of S̄T∗ on {Nt = 1, Nt+1 = 0} for t = 1, . . . , T − 1.
Because the distribution of N̄T +1 is data, this identifies the distribution of (S̄T∗ , NT +1 ).
~ t+1 to be the random
~t∗ and N
Proof of Lemma 3. The proof proceeds recursively. It is helpful to define S
∗
∗
vectors (St , . . . , ST ) and (Nt+1 , . . . , NT +1 ). Denote their joint density with X̄t−1 by fS~ ∗ ,N~ t+1 ,X̄t−1 (·). We
t
begin with the assumption that this density for a particular age t is known. For t = 1, fS~ ∗ ,N~ 2 ,X̄0 (·) is simply
1
¡
¢
the density of the data. We first show that βt , ωt2 , τt and fWt can be recovered from fS~ ∗ ,N~ t+1 ·|X̄t−1 . We
t
then demonstrate that these parameters in turn identify fS~ ∗ ,N~ t+2 ,X̄t (·) if t < T , allowing the recursion to
t+1
continue.
We begin with the identification of βt , ωt2 and τt . This requires only the knowledge of the expected value
of St∗ conditional on X̄t−1 , the conditional probability of survival, and the expected value of St∗ given this
16

See Feller (1971) for an introduction to deconvolution. Formally, the right-hand side of (15) is a t-variate
convolution of fS̄t∗ (·|Nt = 1) and the singular t-variate density of (R, . . . , R).

24

history and survival to period t + 1. On {Nt = 1}, these are
E[St∗ |X̄t−1 ] = βt (X̄t−1 ),
µ
¶
τt (X̄t−1 ) − βt (X̄t−1 )
, and
E[Nt+1 |X̄t−1 ] = 1 − Φ
ωt
³
´
t (X̄t−1 )
φ τt (X̄t−1 )−β
ωt
³
´,
E[St∗ |X̄t−1 , Nt+1 = 1] = βt (X̄t−1 ) + ωt
τt (X̄t−1 )−βt (X̄t−1 )
1−Φ
ωt
where Φ(·) and φ(·) are the cumulative distribution function and density function of a standard normal
random variable. Because the distribution of (St∗ , Nt+1 ) conditional on X̄t−1 is assumed to be known, these
three equations immediately yield ωt2 and the restrictions of βt and τt to the support of X̄t−1 on {Nt = 1}.
Because the support of X̄t−1 on {Nt = 1} is not contained in any linear subspace of Rt−1 , this identifies βt
and τt .
To obtain γt2 , note that the probability density of St∗ conditional on X̄t−1 can be written as
Ã
¡
¢! µ
¶
Z ∞
¡
¢
x − βt X̄t−1
1
s−x
fSt∗ s|X̄t−1 =
φ
φ
dx
(16)
ωt
γt
−∞ ωt
on {Nt = 1}. Applying a standard deconvolution argument establishes that γt2 is identified.
~ t+1 and X̄T imWe will now show that fS~ ∗ ,N~ t+2 ,X̄t (·) is identified. The independence of Wt from W
t+1
´
³
~ t+1 displays independence.
~∗ , N
plies that the joint distribution of Wt and Xt conditional upon X̄t−1 , S
t+1

Therefore, the probability density of St∗ conditional on these variables is
³
´ Z ∞
³
´ µs − x¶
∗
∗
~
~
~
~
fSt∗ s|X̄t−1 , St+1 , Nt+1 =
fXt x|X̄t−1 , St+1 , Nt+1 φ
dx
γt
−∞

(17)

´
³
~ t+1
~∗ , N
on {Nt = 1}. The left-hand side of (17) and fWt are known, so deconvolution yields fXt ·|X̄t−1 , S
t+1
´
³
~ t+1 on {Nt+1 = 1} ⊂ {Nt = 1}
~∗ , N
on {Nt = 1}. We can immediately recover the distribution of X̄t , S
t+1
by multiplying this conditional distribution by the known joint distribution of the conditioning variables.
~
~∗
Using
³ that both S´t+1 and Nt+1 are identically zero on {Nt+1 = 0}, we can then construct the distribution
~ t+1 . Thus, the recursion may continue.
~∗ , N
of X̄t , S
t+1

Proof of Lemma 4. This proof is straightforward. It is available from the authors upon request.

25

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28

Table 1: Summary Statistics from the First 13 Months in the Lives of New Texas Bars(i)

Age(ii)
1
2
3
4
5
6
7
8
9
10
11
12
13

Firms
305
304
299
297
291
286
286
278
277
268
263
252
244

Sales’ Mean(iii)
0.06
0.08
0.10
0.11
0.12
0.11
0.11
0.17
0.17
0.20
0.20
0.23
0.27

Sales’ Standard Deviation
0.87
0.89
0.86
0.85
0.85
0.87
0.89
0.84
0.86
0.86
0.89
0.89
0.87

Exit Rate(iv)
0.003
0.016
0.007
0.020
0.017
0.000
0.028
0.004
0.032
0.019
0.042
0.032
0.000

Notes: (i) See the text for details regarding the sample’s construction. (ii) Age is measured
in months and equals one for a firm filing its first tax return. (iii) Sales are measured in
logarithms and are relative to the sales of all establishments selling alcoholic beverages for
on-premise consumption, whether or not we classify them as bars. (iv) The exit rate is defined
as the number of firms operating in month t that do not operate in month t + 1 divided by
the number of firms operating in month t.

29

Table 2: Regression Estimates of Sales on Previous and First Months’ Sales(i)

Age
3
4
5
6
7
8
9
10
11
12
13

(ii)

Logarithm of Sales in
Previous Month First Month
0.809
0.154
(0.010)
(0.010)
0.778
0.190
(0.008)
(0.008)
0.942
0.021
(0.008)
(0.008)
0.859
0.088
(0.008)
(0.008)
0.880
0.086
(0.010)
(0.009)
0.851
0.132
(0.009)
(0.008)
0.905
0.073
(0.005)
(0.005)
0.916
0.058
(0.006)
(0.006)
0.894
0.062
(0.006)
(0.006)
0.830
0.122
(0.009)
(0.007)
0.822
0.168
(0.008)
(0.007)

Notes: (i) For the third to thirteenth months, this table reports Powell, Stock, and Stoker’s
(1989) instrumental variable density-weighted average derivative estimates for single-index
regression models of log sales on the logarithms of sales in the previous and first months.
Standard errors are reported in parentheses below each estimate. For each month, the estimation was conducted using the sample of firms that survived to that month. (ii) Age is
measured in months and equals one for a firm filing its first tax return.
30

Table 3: Maximum Likelihood Estimates(i)
Parameter
Estimate
Standard Error(ii)

σ1
σ
ρ
0.714
0.161
0.985
(0.076) (0.005) (0.000)

Parameter
Estimate
Standard Error(ii)

Parameter
Estimate
Standard Error(ii)

α1
α2
-1.65
-1.53
(0.068) (0.038)

α7
α8
-1.22
-1.26
(0.032) (0.063)

γ
0.167
(0.004)

α3
-1.44
(0.049)

α9
-0.83
(0.040)

ν
0.581
(0.000)

α4
-1.38
(0.029)

α10
-0.79
(0.039)

µ1
0.069
(0.055)

µ
0.0014
(0.0030)

α5
α6
-1.34 -2.50
(0.033)
(·)

α11
-0.64
(0.040)

α12
-0.54
(0.025)

Notes: (i) This specification assumes that entrepreneurs face no signal extraction problem
(η 2 = 0). (ii) The standard errors are calculated using an estimate of the information matrix
based on the outer product of the scores. See the text for further details.

31

Table 4: Model Summary Statistics(i)

Age(ii)
1
2
3
4
5
6
7
8
9
10
11
12

Firms(iii)
100.0
99.2
98.5
97.8
97.1
96.5
96.5
95.0
94.5
88.0
85.6
81.1

Sales’ Mean Sales’ Standard Deviation
0.07
0.94
0.08
0.93
0.10
0.93
0.11
0.92
0.12
0.92
0.13
0.92
0.13
0.93
0.15
0.92
0.16
0.92
0.25
0.89
0.28
0.88
0.33
0.86

Exit Rate(iv)
0.008
0.007
0.007
0.007
0.006
0.000
0.016
0.006
0.069
0.026
0.053
0.049

Notes: (i) The reported statistics are calculated from a synthetic sample of 100, 000 firm
histories drawn from the estimated model. (ii) Age is measured in months and equals one
for a newly born firm. (iii) The number of firms is measured in thousands. (iv) The exit rate
is defined as the number of firms operating in month t that do not operate in month t + 1
divided by the number of firms operating in month t.

32

Table 5: Model Population Regression of Sales on Previous and First Months’ Sales(i)

Age
3
4
5
6
7
8
9
10
11
12

(ii)

Logarithm of Sales in
Previous Month First Month
0.663
0.325
0.705
0.238
0.765
0.191
0.797
0.150
0.818
0.142
0.842
0.119
0.867
0.123
0.839
0.114
0.872
0.092
0.875
0.075

Notes: (i) For the third to twelfth months, this table reports the density-weighted average
derivatives from the regression of log sales on the logarithms of sales in the previous and first
months for surviving firms. These are calculated by applying Powell, Stock, and Stoker’s
(1989) estimator to a synthetic sample of 100, 000 firm histories drawn from the estimated
model. (ii) Age is measured in months and equals one for a newly born firm.

33

Table 6: Estimated Fixed Costs as Fractions of First Month’s Producer’s Surplus(i)

Age(ii)
E[eR ]²κ/E[eS1 ]

1
10.23
(1.80)

2
3
4
5
6
0.19
0.24
0.30
0.33
0.44
(0.06) (0.05) (0.06) (0.06) (0.05)

Age(ii)
E[eR ]²κ/E[eS1 ]

7
0.06
(0.00)(iii)

8
9
10
11
12
0.28
0.21
0.68
0.55
1.09
(0.03) (0.02) (0.19) (0.14) (0.40)

Notes: (i) This table reports the minimum-distance estimates of each month’s scaled fixed
costs as described in Subsection 4.3. (ii) Age is measured in months and equals one for a
firm filing its first tax return. (iii) The reported standard error is less than 0.01.

34

Figure 1: Maximum Likelihood Estimation of η 2

−944

Log Likelihood Function

−1200
−1400
−1600
−1800
−2000
−2200
−2400
−2600
−2800
−2988
0

0.05

0.1

0.15

0.2

0.25
2

η

35

0.3

0.35

0.4

0.45

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WP-02-03

Expectation Traps and Monetary Policy
Stefania Albanesi, V. V. Chari and Lawrence J. Christiano

WP-02-04

Monetary Policy in a Financial Crisis
Lawrence J. Christiano, Christopher Gust and Jorge Roldos

WP-02-05

Regulatory Incentives and Consolidation: The Case of Commercial Bank Mergers
and the Community Reinvestment Act
Raphael Bostic, Hamid Mehran, Anna Paulson and Marc Saidenberg
Technological Progress and the Geographic Expansion of the Banking Industry
Allen N. Berger and Robert DeYoung

WP-02-06

WP-02-07

4

Working Paper Series (continued)
Choosing the Right Parents: Changes in the Intergenerational Transmission
of Inequality  Between 1980 and the Early 1990s
David I. Levine and Bhashkar Mazumder

WP-02-08

The Immediacy Implications of Exchange Organization
James T. Moser

WP-02-09

Maternal Employment and Overweight Children
Patricia M. Anderson, Kristin F. Butcher and Phillip B. Levine

WP-02-10

The Costs and Benefits of Moral Suasion: Evidence from the Rescue of
Long-Term Capital Management
Craig Furfine

WP-02-11

On the Cyclical Behavior of Employment, Unemployment and Labor Force Participation
Marcelo Veracierto

WP-02-12

Do Safeguard Tariffs and Antidumping Duties Open or Close Technology Gaps?
Meredith A. Crowley

WP-02-13

Technology Shocks Matter
Jonas D. M. Fisher

WP-02-14

Money as a Mechanism in a Bewley Economy
Edward J. Green and Ruilin Zhou

WP-02-15

Optimal Fiscal and Monetary Policy: Equivalence Results
Isabel Correia, Juan Pablo Nicolini and Pedro Teles

WP-02-16

Real Exchange Rate Fluctuations and the Dynamics of Retail Trade Industries
on the U.S.-Canada Border
Jeffrey R. Campbell and Beverly Lapham

WP-02-17

Bank Procyclicality, Credit Crunches, and Asymmetric Monetary Policy Effects:
A Unifying Model
Robert R. Bliss and George G. Kaufman

WP-02-18

Location of Headquarter Growth During the 90s
Thomas H. Klier

WP-02-19

The Value of Banking Relationships During a Financial Crisis:
Evidence from Failures of Japanese Banks
Elijah Brewer III, Hesna Genay, William Curt Hunter and George G. Kaufman

WP-02-20

On the Distribution and Dynamics of Health Costs
Eric French and John Bailey Jones

WP-02-21

The Effects of Progressive Taxation on Labor Supply when Hours and Wages are
Jointly Determined
Daniel Aaronson and Eric French

WP-02-22

5

Working Paper Series (continued)
Inter-industry Contagion and the Competitive Effects of Financial Distress Announcements:
Evidence from Commercial Banks and Life Insurance Companies
Elijah Brewer III and William E. Jackson III

WP-02-23

State-Contingent Bank Regulation With Unobserved Action and
Unobserved Characteristics
David A. Marshall and Edward Simpson Prescott

WP-02-24

Local Market Consolidation and Bank Productive Efficiency
Douglas D. Evanoff and Evren Örs

WP-02-25

Life-Cycle Dynamics in Industrial Sectors. The Role of Banking Market Structure
Nicola Cetorelli

WP-02-26

Private School Location and Neighborhood Characteristics
Lisa Barrow

WP-02-27

Teachers and Student Achievement in the Chicago Public High Schools
Daniel Aaronson, Lisa Barrow and William Sander

WP-02-28

The Crime of 1873: Back to the Scene
François R. Velde

WP-02-29

Trade Structure, Industrial Structure, and International Business Cycles
Marianne Baxter and Michael A. Kouparitsas

WP-02-30

Estimating the Returns to Community College Schooling for Displaced Workers
Louis Jacobson, Robert LaLonde and Daniel G. Sullivan

WP-02-31

A Proposal for Efficiently Resolving Out-of-the-Money Swap Positions
at Large Insolvent Banks
George G. Kaufman

WP-03-01

Depositor Liquidity and Loss-Sharing in Bank Failure Resolutions
George G. Kaufman

WP-03-02

Subordinated Debt and Prompt Corrective Regulatory Action
Douglas D. Evanoff and Larry D. Wall

WP-03-03

When is Inter-Transaction Time Informative?
Craig Furfine

WP-03-04

Tenure Choice with Location Selection: The Case of Hispanic Neighborhoods
in Chicago
Maude Toussaint-Comeau and Sherrie L.W. Rhine

WP-03-05

Distinguishing Limited Commitment from Moral Hazard in Models of
Growth with Inequality*
Anna L. Paulson and Robert Townsend

WP-03-06

Resolving Large Complex Financial Organizations
Robert R. Bliss

WP-03-07

6

Working Paper Series (continued)
The Case of the Missing Productivity Growth:
Or, Does information technology explain why productivity accelerated in the United States
but not the United Kingdom?
Susanto Basu, John G. Fernald, Nicholas Oulton and Sylaja Srinivasan

WP-03-08

Inside-Outside Money Competition
Ramon Marimon, Juan Pablo Nicolini and Pedro Teles

WP-03-09

The Importance of Check-Cashing Businesses to the Unbanked: Racial/Ethnic Differences
William H. Greene, Sherrie L.W. Rhine and Maude Toussaint-Comeau

WP-03-10

A Firm’s First Year
Jaap H. Abbring and Jeffrey R. Campbell

WP-03-11

7