Firm exit and technical inefficiency

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DOI 10.1007/s00181-005-0045-2

ORIGINAL PAPER

Efthymios G. Tsionas . Theodore A. Papadogonas

Firm exit and technical inefficiency

Received: 9 November 2004 / Accepted: 24 May 2005 / Published online: 30 March 2006© Springer-Verlag 2006

Abstract The purpose of the paper is to examine formally the fundamentalimplication that technical inefficiency (TI) is related to firm exit. Traditionalstochastic frontier models allow for the measurement of TI but do not allow for adirect effect of TI on exit. We propose a model which allows for such effects andconsists of a stochastic frontier model, and an additional equation that describes theprobability of exit as a function of covariates and TI. Since TI is unobserved,econometric complications arise, and obtaining consistent estimates is non-trivialdue to the presence of integrals in the likelihood function. We propose and im-plement maximum likelihood estimation one step, employing data for 3,404manufacturing firms in Greece. We find significant positive effects from TI on theprobability of exit. We also propose and provide measures of TI that respect the factthat unobserved TI affects the probability of exit and compare them to TI measuresfrom the traditional stochastic frontier model.

Keywords Exit . Technical inefficiency . Stochastic frontier models

JEL Classification C21 . C23

1 Introduction

Stochastic frontier models are part of the arsenal of applied researchers working inmany different fields. The measurement of technical inefficiency (TI) has alwaysbeen thought to be extremely important for applied policy analysis and industrial

E. G. Tsionas (*)Department of Economics Athens, University of Economics and Business, 76 Patission Street,Athens 104 34, GreeceE-mail: [email protected]

T. A. PapadogonasDepartment of Accounting and Finance, Athens University of Economics and Business,76 Patission Street, Athens 104 34, GreeceE-mail: [email protected]

Empirical Economics 31:535–548 (2006)

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studies because inefficient producers cannot survive in the long-run provided theforces of competition in the sector are reasonably strong. There is also a significantamount of work on exit and survival of firms originating from the influential papersof Audretsch (1991) and Audretsch and Mahmood (1994, 1995). The twoliteratures are, surprisingly, separated contrary perhaps to one’s prior expectationssince technically inefficient firms should be prime candidates for failure. If this isnot the case there are two problems. First of all, the systematic study of efficiencythrough advanced models and techniques as the ones presented in Kumbhakar andLovell (2000) would be limited in scope. Secondly, research in industrial or-ganization would have to explain why inefficient firms—that are under thecontinual pressure of competition—do eventually manage to survive in theindustry.

Although there exist numerous studies of applications of stochastic frontiermodels, only two (Dimara et al. 2003; Wheelock and Wilson 1995) have beendevoted to the central issue in efficiency analysis, namely whether TI indeed affectsthe exit of firms in a significant way and, if so, whether the effect is quantitativelyimportant. These studies overlooked some important econometric problems whichare likely to lead one to misguided conclusions about the impact of TI on exit. Fromthe econometric point of view the issue may seem to be trivial. Several methods ofestimation are available to obtain TI measures which, in turn, can be related tosome survival variables in a second stage involving the application of OLS orrelated techniques. A typical two-step procedure would be to estimate the sto-chastic frontier using maximum likelihood, and obtain the technical inefficiencymeasures, say bui; using the Jondrow et al. (1982) method.1 Then, the secondequation can be estimated using bui instead of ui.

Estimates in the second stage will be inconsistent because they ignore the error-in-variables problem associated with the fact that bui is estimated. The problem islikely to be severe because, in addition, the bui estimates from the first stage will notbe efficient since they ignore information about ui contained in the secondequation, so the discrepancy between ui and bui should be expected to be large,exacerbating the errors-in-variables problem in the second stage. Therefore, there isclear need for one-step estimators of the stochastic frontier jointly with the equa-tion that determines exit of firms from the industry.

In this paper, we propose a model that allows TI to be a determinant of exit. Themodel involves two equations. The first equation is a stochastic production frontier,and the second equation provides the probability of exit from the industry, which isparameterized in terms of covariates as well as unobserved TI. Three alternativesare proposed for the probability function, namely the probit, logit, and extremevalue specifications. Since the likelihood function of the model cannot be obtainedin closed form due to the presence of certain integrals associated with the fact thatTI is latent, we use numerical integration to obtain the likelihood function, and MLtechniques to obtain parameter estimates that satisfy the usual asymptotic prop-erties. We also obtain TI measures based on the conditional distribution of one-sided error given the data: Such TI measures generalize the Jondrow et al. (1982)concept and respect the fact that TI is a determinant of exit.

1 An alternative is to obtain inefficiency measures using the data envelopment analysis (DEA)technique as in Wheelock and Wilson (1995). Dimara et al. (2003) have used inefficiencyestimates derived from a Battese and Coelli (1995) model.

536 E. G. Tsionas, T. A. Papadogonas

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We apply the new techniques using data for 3,404 manufacturing firms inGreece. The remainder of the paper is organized as follows. A brief review of theliterature on the determinants of exit is presented in Section 2. The econometricmodel is presented in Section 3. Technical efficiency measurement is discussed inSection 4. The data and empirical results are provided in Section 5. The finalsection provides some concluding remarks.

2 Determinants of exit

Recently, a growing body of literature examines the main factors which determinethe likelihood of business dissolution. Typically, these studies model some measureof firm exit as a function of several variables, designed to reflect structural in-centives and barriers to exit and firm characteristics. The most commonly usedexplanatory variables that have been used in the literature are described below.

2.1 Economies of scale

The minimum efficient scale (MES) level of output is defined as the smallest levelof output needed to exploit the scale economies of an industry. Firms operating at alevel of output below MES are characterized as suboptimal, since they are facedwith a cost disadvantage compared to larger firms. As a consequence, the like-lihood of exit is expected to be positively associated to the MES level of theindustry concerned. Evidence supporting this argument has been found in Dunneand Hughes (1994), Wagner (1994), Audretch and Mahmood (1995), Audretchet al. (2000) and Mahmood (2000).

2.2 Industry growth

Slowly growing or declining markets are expected to induce larger exit rates thanrapidly growing ones. Most existing studies provide empirical support for a neg-ative relation between market growth and exit rates (Caves and Porter 1976; Dunneand Roberts 1991; Austin and Rosenbaum 1990; Doi 1999; Segarra andCallejón 2002). However, there is evidence to the contrary. Shapiro (1983),MacDonald (1986), and Evans and Siegfried (1993) found no relationship,while Audretch et al. (2000) report a positive relationship, which they explainby the greater uncertainty that characterizes high growth industries.

2.3 Profitability

Many studies suggest that high profits have a negative impact on exit. Althoughsuch a relation would seem self-evident to an economist, the empirical results aremixed. Dunne and Roberts (1991), Mayer and Chappel (1992), Doi (1999) andAudretch et al. (2000) found that the higher the price-cost margin in an industry, thelower the gross exit rate. However, other studies (Austin and Rosenbaum 1990;

Firm exit and technical inefficiency 537

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Evans and Siegfried 1993) report no relationship between profitability and exitrates.

2.4 Market concentration

The successful collusive oligopolistic behavior of firms in highly concentratedmarkets is assumed to reduce the likelihood of firm closure. As a consequence,market concentration is expected to have a negative impact on exit rates. Flynn(1991) and Doi (1999) provide evidence that is consistent with this hypothesis.

2.5 Capital requirements

In industries with high capital requirements firms are more committed to theirresources, and thus the capital intensity of an industry is expected to be a barrier toexit. Evidence supporting such a negative relationship between capital intensityand exit rates can be found in MacDonald (1986), Austin and Rosenbaum (1990),Dunne and Roberts (1991) and Audretch et al. (2000).

2.6 Sunk costs

Industries which are characterized by a large proportion of sunk costs in total fixedassets are expected to have lower exit rates, since the durability and specificity ofassets in these industries makes exit more costly. Fotopoulos and Louri (2000)provide evidence for the existence of a negative relationship between high sunkcost and the likelihood of firm exit.

2.7 R&D

R&D is a special investment which is regarded as endogenous sunk cost and, assuch, is considered as a barrier to exit. On the other hand, Audretch et al. (2000)argued that R&D intensive industries are associated with greater uncertainty, whichleads to higher exit rates. Thus, the impact of R&D on exit should be regarded apriori as indeterminate. Doi (1999) reports a negative relationship, while Audretchet al. (2000) and Segarra and Callejón (2002) provide evidence for a positiverelationship. Also, Mahmood (2000) finds that the sign of the coefficient of theR&D variable varies across industries.

2.8 Firm size

There is broad agreement that the likelihood of exit declines with firm size. Largefirms are equipped with large amounts of financial, physical, human and otherresources which improve the possibility of exploiting scale economies and copingwith external shocks, and protect them from failure. A negative relationshipbetween firm size and the likelihood of exit has been found by Hall (1987), Dunne


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and Hughes (1994), Wagner (1994), Audretch and Mahmood (1995), Baldwin(1995), Audretch et al. (2000), Mahmood (2000) and Segarra and Callejón (2002).

2.9 Age of firm

Age is considered as one of the key determinants of firm exit. The generalconsensus of the empirical studies examining the impact of age on exit is that (afterinfancy) the likelihood of exit declines with age (Dunne et al. 1988; Audretch1991, 1994, 1995; Mahmood 1992; Mata and Portugal 1994; Wagner 1994;Agarwal and Gort 1996; Agarwal 1997).

2.10 Leverage

A high level of debt requires high interest payments, thus increasing company riskand reducing the likelihood of survival. A positive effect of the leverage variable(the ratio of debt to total assets) on the likelihood of exit has been found byFotopoulos and Louri (2000).

3 The econometric model

We consider the stochastic production frontier 2

yi ¼ x 0i�þ vi � ui; i ¼ 1; . . . ; n; (1)

where yi is the log of production, xi is a k×1 vector of log of inputs, β is a k×1vector of parameters, vi is a two-sided error term representing measurement error,and ui ≥ 0 is TI, see Aigner et al. (1977), Meeusen and van den Broeck (1977), andKumbhakar and Lovell (2000) for a survey. We observe the indicator Ii=1 if thefirm exited the industry, and zero otherwise. We consider the probability that weobserve an exit, namely

P Ii ¼ 1 xi; zi; uijð Þ ¼ F z 0i� þ �ui

� �; i ¼ 1; . . . ; n; (2)

where zi is an m×1 vector of covariates that influence the exit process, γ and δrepresent m×1 and scalar parameters, respectively, and F is a cumulativedistribution function (cdf ) in standard form. This model merges the stochasticfrontier and limited-dependent variable concepts. Traditional stochastic frontiersdo not allow for the presence of an additional equation, like Eq. (2), that allows TIto be an explanatory variable,3 and limited-dependent variable models do not allow

2Cost frontiers can be considered by using +ui instead of −ui in Eq. (1).3 It seems that more emphasis has been placed on frontier models where technical inefficiency is adependent variable and can be explained using covariates, see Kumbhakar et al. (1991) andBattese and Coelli (1995).


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for the presence of unobserved explanatory variables. Two-step procedures will notwork in general as we have explained in the introductory section. Therefore,Eqs. (1) and (2) must be estimated jointly. To estimate the model we make thefollowing assumptions and we state the model in compact form:

yi xi; zij ; ui � N x0i�� ui; �2v

� �;

P Ii ¼ 1 xi; zi; uijð Þ ¼ F z0i� þ �ui� �

;ui � N 0; �2

u

� �� ;yi and Ii are independent, conditionally on xi, zi, ui.

Denote by � ¼ �0; � 0; �; �v; �u½ �0 the parameter vector. The joint distribution ofendogenous variables yi and Ii has density

f yi; Ii xi; zi; �jð Þ ¼ 2��2v

� ��1=2��2

u

�2

� ��1=2Z10

exp � yi þ ui � x0i�� 2

2�2v

� u2i2�2

u

" #

� F z 0i� þ �ui� �Ii 1� F z 0i� þ �ui

� �� 1�Iidui

This expression can be easily derived if we notice that f yi; Ii xi; zi; �jð Þ ¼R10f yi ui; xi; zi; �jð Þ � f Ii ui; xi; zi; �jð Þp ui �jð Þdui; meaning that conditional on the

latent ui we have a relatively simple likelihood owing to the distributionalassumptions and in order to obtain the density of observed variables we have tointegrate against the distribution of ui For purposes of estimation technicalinefficiency is not known so we cannot proceed conditional on ui, which the reasonwhy the integral appears in Eq. (3) to indicate that we need unconditional (marginal)inferences. For typical choices of the cdf, namely probit, logit or extreme value, theintegral that appears in the density function f yi; Ii xi; zi; �jð Þ cannot be computedin closed form. If F is the logistic cdf, we have F wð Þ ¼ 1þ exp �wð Þ½ ��1: If Fis the extreme value cdf, we obtain the cdf F wð Þ ¼ exp �exp �wð Þ½ �: For the

probit specification, the cdf is � wð Þ ¼ Rw�1

2�ð Þ�1=2 exp �t2�2

� �dt; i.e. the

standard normal cdf. For all these choices the fact that the density functioncannot be computed in closed form implies that the likelihood function must becomputed by resorting to numerical techniques.

The log-likelihood function is

log L �; y; I; x; zð Þ ¼Xni¼1

log f yi; Ii xi; zi; �jð Þ: (4)

ML estimates can be obtained by maximizing the log-likelihood function inEq. (4) with respect to the parameters θ.

From the density function in Eq. (3) two things can be noted. First, efficientestimation of β (as well as σv and σu) cannot be based on estimating Eq. (1) alone,

(3)


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since ui appears in both the production function and the exit equation, implying that(unless δ=0) it is not enough to maximize a half-normal stochastic frontierlikelihood. Second, proper measures of TI have to take into account the exitequation. To compute the log-likelihood in Eq. (4) we write the expression inEq. (3) as

f yi; Ii xi; zi; �jð Þ ¼ 2��2v

� ��1=2��2

u

�2

� ��1=2Z10

g ui; �ð Þdui; (5)

where

g ui; �ð Þ ¼ exp � yi þ ui � x 0i�� 2

2�2v

� u2i2�2

u

" #F z 0i � þ �ui� �Ii 1� F z 0i� þ �ui

� �� 1�Ii :

(6)

Using Gaussian quadrature to approximate the integral,4 we have

qi �ð Þ ¼Z10

g ui; �ð Þdui � qi �ð Þ ¼XMm�1

wmF z 0i � þ �um� �Ii 1� F z 0i� þ �um

� �� 1�Ii ;

i ¼ 1; . . . ; n;

where wm and um denote weights and base points, respectively for M-pointGaussian quadrature. This operation can be carried out easily as it requiresexclusively computation of the cdf at a number of points for a total of nMevaluations for a single computation of the log-likelihood function. If the cdfcorresponds to the extreme value or logit specifications, it is available in closedform. For the probit specification, the cdf is computed, like always, using numericalapproximation techniques so estimation of the probit model is relatively moredifficult. To avoid the infinite end-point of integration we can change the variable

from inefficiency u to efficiency r=exp(−u) to obtain qi �ð Þ¼R10g �log ri; �ð Þr�1

i dri;

and revise appropriately the Gaussian integration formula. Finally, the log-likelihood function can be approximated by

log L �; y; I; x; z;ð Þ ¼ � n=2ð Þln 2��2v

� �� n=2ð Þln ��2u

�2

� �þXni�1

logbqi �ð Þ;

4 An alternative technique to compute the integral would be by simulation resulting in maximumsimulated likelihood (MSL) estimation. Since the integrals involved in the likelihood function areunivariate, Gaussian quadrature is significantly more precise and computationally far lessexpensive compared to MSL. In this study we have used a ten-point rule to implement Gaussianquadrature.


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which can be maximized using standard iterative techniques5 to yield themaximum likelihood estimator.

4 Technical efficiency measurement

To provide technical inefficiency measures on a firm-specific basis we consider theconditional distribution of ui given the parameters and the data. From expressionsEq. (3) or (6) it is clear that the kernel density of this conditional distribution is

given by g ui; �ð Þ / fTN ui ��i ; �

2�

�� F z 0i� þ �ui� �Ii 1� F z 0i� þ �ui

� �� 1�Ii , also pre-sented in Eq. (6). The mean of the distribution is given by

E ui yi; Ii; xi; zijð Þ ¼

R10uig ui; �ð ÞduiR1

0g ui; �ð Þdui

; i ¼ 1 . . . ; n; (7)

where the denominator is the normalizing constant. This expression can be derived

after noticing that what we need is E (ui|data). By definition E ui datajð Þ ¼R10up u datajð Þdu; where p(u|data) is the density of u given the data. From Eq. (3) it

is clear that the joint distribution of endogenous variables, yi and Ii, along with uiconditional on xi and zi is

f yi; Ii; ui xij ; zi; �ð Þ

� / exp � yi þ ui � x 0i�� 2

2�2v

� u2i2�2

u

" #F z 0i � þ �ui� �Ii 1� F z 0i� þ �ui

� �� 1�Ii

Given Eq. (6), this is simply f yi; Ii; ui xi; zi; �jð Þ / g ui; �ð Þ: Therefore,

E uijð Þ ¼ R10u p u datajð Þ=K½ �du; where K ¼ R1

0p u datajð Þdu is the normalizing

constant of the distribution. Since K is independent of u, we finally obtain theexpression in Eq. (7) as the ratio of two integrals.

These integrals can be computed using Gaussian quadrature. A change ofvariables from ui to ri=exp(–ui) yields average firm-specific technical efficiency,E(ri|yi, Ii, xi, zi). Higher order moments of the conditional distribution ri |,Ii,xi,zican be easily computed using the same principles. This technical efficiencymeasure explicitly allows for the fact that TI is a determinant of exit and thusaccounts for all the information provided by Eqs. (1) and (2).

5We have used the BFGS algorithm with numerical derivatives as implemented in the GAUSSsoftware package. The programs are available on request from the first author. We have firstestimated the logit model. The parameter estimates were supplied as starting values for theextreme value model, and parameter estimates from that model were supplied as starting valuesfor the probit specification.


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5 Data and empirical results

All data used in this paper refer to manufacturing. The firm level data have beenobtained through the ICAP directory. ICAP Hellas S.A. is a Greek private researchcompany which collects balance sheet data for all private firms of corporate forms(societés anonymes and limited liability companies) in Greece, together with theiremployment, establishment date, location and ownership status. Firms thatunderwent a merger or were acquired by other firms or switched to another industrywere excluded from the sample in order to avoid biased results. The industry leveldata are derived from the Annual Industrial Surveys of the National StatisticalService of Greece, with the exception of R&D data which are provided by theGeneral Secretariat for Research and Technology. The data refer to the years 1995and 1999.

We use labor (number of persons employed, in logarithmic form, 1995) andcapital (proxied by the firm’s fixed assets in logarithmic form, 1995) as inputs andsales (in logarithmic form, 1995) as output in the stochastic production frontier(which is assumed to be a Cobb–Douglas function) along with regional andsectoral dummy variables to capture heterogeneity. We have three regions (Athens,Thessaloniki and other) and 13 two-digit sectoral dummies so we have 14 dummyvariables.

Firms that operated in year 1995 but do not continue in 1999 are defined asexiting firms. The data originate from 3,404 firms in 1995, out of which 351 havebeen identified as exit firms during the period under examination. For the exitprobability equation we include as independent variables the following (allindustry level variables refer to the corresponding four-digit industry of primaryactivity of a firm):

1. Minimum efficient scale (MES) of the industry (measured by the average salesof the largest firms producing more than half of the industry sales, divided byindustry sales, 1995, as suggested by Comanor and Wilson 1967)

2. Industry growth (deflated sales growth, 1995–1999)3. Price–cost margin (PCM) of the industry (defined as sales minus costs divided

by sales)4. The Herfindahl index of concentration (1995) calculated directly by use of the

individual firm data5. Capital–labor ratio of the industry (fixed assets divided by the number of

persons employed, 1995)6. Industry inverse sunk cost variable (the ratio of second-hand bought

machinery and equipment over the total investments in machinery andequipment, 1995)

7. Industry R&D intensity (R&D divided by sales, 1995)8. Firm size (the logarithm of the firm’s total assets, 1995)9. Age of firm (1995)

10. Leverage (the ratio of the firm’s debt to total assets, 1995)11. Multinational dummy (a dummy variable that indicates whether the firm is

multinational or not)12. Relative size (the ratio of a firm’s sales divided by the sales of the minimum

efficient scale firm of the industry)13. The technical efficiency measure as described above.


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The empirical results are reported in Table 1. The coefficients of labor andcapital in the stochastic frontier are 0.186 and 0.746, respectively, suggestingdecreasing results to scale (0.932). From the estimates we see that the industry-level variables that affect significantly the probability of exit are MinimumEfficient Scale (MES), concentration (Herfindahl index), capital intensity (industrycapital–labor ratio) and sunk costs. All signs are consistent with the a prioriexpectations (see Section 2).

Table 1 Parameter estimates obtained from maximum likelihood estimation

Logit Probit Extreme value

Coeff. t-value Coeff. t-value Coeff. t-value

Constant 9.141 132.48 9.137 637.96 9.137 679.56Dum17 −0.383 13.29 −0.382 25.39 −0.380 16.97Dum19 −0.318 7.83 −0.311 21.67 −0.311 23.44Dum20 −0.424 12.39 −0.414 28.89 −0.414 31.21Dum2122 −0.334 10.31 −0.332 17.23 −0.327 16.06Dum23 0.246 12.34 0.237 16.51 0.237 17.84Dum24 −0.003 0.10 −0.001 0.08 −0.000 0.01Dum25 −0.308 10.76 −0.310 21.56 −0.309 21.09Dum26 −0.475 13.01 −0.467 24.12 −0.472 19.40Dum2728 −0.310 12.40 −0.313 19.30 −0.316 10.32Dum29 −0.453 13.66 −0.451 31.27 −0.451 33.36Dum3033 −0.346 6.91 −0.345 24.03 −0.345 25.94Dum3435 −0.700 16.33 −0.697 48.57 −0.697 52.54Dum36 −0.747 20.30 −0.744 51.63 −0.743 52.98DumAth 0.211 8.43 0.209 14.14 0.210 14.59DumThes 0.211 6.21 0.205 10.82 0.211 12.56Capital 0.186 23.38 0.186 42.16 0.187 42.13Labor 0.746 58.77 0.746 53.82 0.745 58.01Constant 0.193 7.66 0.080 4.27 0.050 2.71MES −0.337 6.80 −0.233 8.17 −0.199 5.97Industry growth 0.085 1.19 0.008 0.27 0.017 0.38PCM −0.025 1.44 −0.013 0.91 −0.013 0.98Herfindahl −0.200 7.65 −0.020 0.91 −0.006 0.21Industry K/L −0.141 5.56 −0.093 5.98 −0.086 3.03Inverse sunk 0.090 4.79 0.080 4.81 0.066 4.47Industry R&D −0.005 0.45 −0.007 0.45 −0.006 0.40Firm size −0.592 15.08 −0.301 13.84 −0.211 9.78Age −0.001 0.11 −0.000 0.03 0.000 0.01Leverage 1.925 10.09 0.976 13.38 0.814 8.60Multinational 1.106 4.71 0.509 4.73 0.362 2.82Relative size −0.360 3.63 −0.133 3.03 −0.094 1.81δ 0.936 9.56 0.446 8.28 0.318 5.53σv 0.595 47.18 0.597 59.57 0.595 50.70σu 0.683 23.72 0.678 40.33 0.682 37.41


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Industry growth, industry price–cost margins (PCM) and industry R&D in-tensities turned out to be statistically insignificant. However, these results are notsurprising, given that there is no consensus in the empirical literature concerningthe impact of these factors on exit.

The firm-level statistically significant variables are firm size, relative size andleverage, all with the expected signs. The age variable is not significantly related toexit.

Being a multinational increases the probability of exit in Greece. This result isconsistent with the recent fact that several labor-intensive multinational companiesin Greece relocate to neighboring countries where the labor costs are lower.

Technical inefficiency is also highly statistically significant at conventionallevels of significance, and its impact on exit risk is positive. Summary statistics fortechnical efficiency are reported in Table 2 for all three specifications of the exitprobability, and a simple normal–half-normal stochastic frontier model that ignoresthe exit equation and its dependence on technical efficiency. Differences of averagetechnical efficiency between continuing and exiting firms are close to three per-centage points (63% and 60%, respectively). This difference is statistically sig-nificant at the 5% level as the t-statistics for equality of means are 3.73 for the logitand extreme value, and 3.63 for the probit model. It should be noted that thiscomparison is unconditional whereas the comparison of coefficient estimates forthe effect of technical inefficiency reported in Table 1 are ceteris paribus. Theceterisparibus condition is important here because the argument is that technicalinefficiency is statistically significant for exit given a vector of variables thatdetermine the environment of the firm, the sectoral conditions, the financial distressetc.

The technical efficiency distributions are provided in Fig. 1, from which6 it isclear that all three specifications produce similar efficiency distributions which are,at the same time, somewhat different compared to the simple normal–half-normalstochastic frontier model. The fact that efficiency distributions are similar is, ofcourse, reassuring in the sense that the results do not depend dramatically on thechoice of a distribution F for the exit event.

Table 2 Summary statistics for efficiency measures

Mean Median SD

Logit (all firms) 0.627 0.642 0.118Continuing firms 0.630 0.644 0.115Exit firms 0.602 0.624 0.135

Extreme value (all firms) 0.627 0.642 0.118Continuing firms 0.630 0.644 0.115Exit firms 0.602 0.623 0.135

Probit (all firms) 0.628 0.643 0.117Continuing firms 0.631 0.645 0.114Exit firms 0.604 0.625 0.134

Traditional half-normal frontier 0.591 0.605 0.125

6 The logit and extreme value specifications give nearly the same results for technical inefficiencymeasures so in Fig. 1 it is not possible to distinguish between the two kernel densities.


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The normal–half-normal stochastic frontier model gives average efficiency59.1% while according to the new model proposed in this paper average efficiencyis higher at about 63% across all specifications of the exit probability. Mediantechnical efficiency is 60.5% according to the simple half-normal frontier but64.2% according to the new models.

6 Concluding remarks

In this paper we have proposed a model that allows technical inefficiency to be adeterminant of exit of business firms. Since technical inefficiency is unobservable,two-step estimation of the model does not yield efficient estimators, so we haveproceeded with maximum likelihood estimation of the complete model in a singlestep using numerical integration. The model involves a stochastic productionfrontier and a limited-dependent variable equation that determines the probabilityof exit as a function of covariates and technical inefficiency. Three alternativespecifications have been used for the exit probability, namely probit, logit, andextreme value. Techniques for efficiency measurement have been also proposedand the new methods have been applied to a sample of 3,404 manufacturing firmsin Greece. The technical inefficiency estimates derived from the three specifica-tions of the exit probability are very similar suggesting that, at least in thisapplication, the choice of distribution for the exit probability is immaterial. Thesetechnical inefficiency estimates are, however, different compared to estimatesderived from a stochastic frontier model that ignores the dependence of exit onlatent technical inefficiency. We find significant positive effects from technicalinefficiency on exit suggesting, as expected on a priori grounds, that inefficientfirms are more likely to exit from the industry.

Fig. 1 Technical efficiency measures


Acknowledgements The authors would like to thank an Associate Editor and two anonymousreferees for their numerous helpful comments.

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Firm exit and technical inefficiency