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Correction to: Measuring the persistence of high firm growth: choices and consequences

  • Eva Christine ErhardtEmail author
Correction
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Correction to: Small Bus Econ

  https://doi.org/10.1007/s11187-019-00229-7

Unfortunately, the original version of this article was published online with error.

In equations (1) and (2) the variable ‘epsilon’ has disappeared. The corrected equation is shown below.
$$ E\left({g}_{it+\tau}\left|\ {HGF}_{it},{\boldsymbol{x}}_{it-3}\right.\right)={\beta}_0+{\beta}_1{HGF}_{it}+{\boldsymbol{\beta}}_2{\boldsymbol{x}}_{it-3}+{\upvarepsilon}_{it+\tau}\kern0.75em {if\ survival}_{it+\tau } $$
(1)
$$ E\left({g}_{it+\tau}\left|\ {HGF}_{it},{\boldsymbol{x}}_{it-3}\right.\right)={\beta}_0+{\beta}_1{HGF}_{it}+{\boldsymbol{\beta}}_2{\boldsymbol{x}}_{it-3}+{\upvarepsilon}_{it+\tau}\kern0.75em {if\ exit}_{it+\tau } $$
(2)

The data in Tables 1, 3, 4, 5, 6, 13, 14, 15, 16, 17 and 18 were incorrectly displayed and aligned by the Springer proofreaders/or in the proofread stage of Springer.

The corrected Tables 1, 3, 4, 5, 6, 13, 14, 15, 16, 17 and 18 are shown in the next page.

Further, in the section Introdution, third line, the word ‘fact’ has been replaced with the word ‘view’.

The original article has been corrected.

Table 1

Findings and methodological choices of previous studies on the persistence of high-growth

Study

Country

Time

Findingsa

Growth formula

Other choices

Repetition high-growth

Future growth

Overall growth

Absolute

Relative

Composite

Growth indicatorb

Growth period

Growth modec

OECD

Other

Acs (2011)

United States

1998–2006

+

    

x

Employees, sales

4-year

T

Capasso et al. (2013)

Netherlands

1994–2004

  

x

  

Employees

1-year

O/A

Coad (2007)

France

1996–2002

 

  

x

  

Employees, sales

1-year

O

Coad and Hölzl (2009)

Austria

1975–2004

 

  

x

  

Employees

1-year

T

Daunfeldt and Halvarsson (2015)

Sweden

1997–2008

  

x

  

Employees

3-year

T

Daunfeldt et al. (2014)

Sweden

1997–2010

  

x

x

 

x

Employees, sales, productivity, value added

3-year

T

Dillen et al. (2014)

Belgium

2000–2009

   

x

 

Employees, value added

3-year

T

Du and Temouri (2015)

UK

2001–2010

 

+

   

x

 

Sales, productivity

3-year

T

Gabrielsson et al. (2014)

Sweden

2001–2007

    

x

 

Employees, sales

3-year

O

Hölzl (2013)

Austria

1985–2007

+

+

  

x

x

Employees

3-year

T

Moschella et al. (2018)

China

1998–2007

   

x

  

Employees, sales

3-year

T

Satterthwaite and Hamilton (2016)

New Zealand

2005–2014

+

   

x

 

Employees

3-year

T

Senderovitz et al. (2016)

Denmark

2004–2010

 

+

    

x

Employees, sales, profits, equity

4-year

T

a Repetition high-growth: + more than 50%, − less than 50%; Future growth (survivors): + positive, − negative; Overall growth (survivors and exits): + positive, − negative

bIndicators used for either explanatory (high-growth-firms) and dependent variables (persistence of high-growth)

c T total, O organic, A acquired

Table 3

Summary statistics of control variables

Control variables

Absolute HGFs

Relative HGFs

OECD HGFs

Total sample

In mean (S.D.)

  Firm size in levels

114.83

(237.89)

2.71

(6.73)

33.60

(52.91)

11.83

(70.18)

  Age

3.24

(2.25)

1.91

(1.50)

2.81

(1.90)

2.49

(1.79)

  Industry size

31.62

(59.10)

15.55

(34.70)

19.82

(32.39)

9.52

(22.49)

  Industry growth

    Absolute

−6.72

(20.17)

  

−0.08

(0.05)

−2.37

(7.62)

    Relative (percent)

  

−0.09

(0.05)

  

−0.09

(0.05)

In share of total

  Firm size in categories a

    Micro (< 10)

0.22

 

0.96

 

 

0.82

 

    Small (10–49)

0.31

 

0.04

 

0.85

 

0.14

 

    Medium (50–250)

0.36

 

 

0.14

 

0.03

 

    Large (>250)

0.11

 

 

0.01

 

0.01

 

  Industry code

    Primary

0.04

 

0.04

 

0.04

 

0.03

 

    Secondary

0.42

 

0.29

 

0.37

 

0.19

 

    Tertiary

0.55

 

0.66

 

0.60

 

0.78

 

Districts

yes

 

yes

 

yes

 

yes

 

Legal form

yes

 

yes

 

yes

 

yes

 

Observations

3669

 

3655

 

5406

 

369,294

 

Notes. Average values for periods 2001–2004, 2004–2007, and 2007–2010

aAccording to Eurostat

Table 4

Transition probabilities from t to t + 3, absolute HGFs

 

\( {g}_{i,t+3}^{(0)} \) exit

\( {g}_{i,t+3}^{(1)} \) negative growth

\( {g}_{i,t+3}^{(2)} \) zero growth

\( {g}_{i,t+3}^{(3)} \) positive growth

\( {g}_{i,t+3}^{(4)} \) absolute HGF

Total

\( {g}_{i,t}^{(1)} \)

0.352

0.310

0.129

0.205

0.005

1.000

\( {g}_{i,t}^{(2)} \)

0.452

0.086

0.353

0.109

0.001

1.000

\( {g}_{i,t}^{(3)} \)

0.209

0.400

0.063

0.313

0.015

1.000

\( {g}_{i,t}^{(4)} \) = absolute HGF

0.068

0.521

0.002

0.124

0.286

1.000

Table 5

Transition probabilities from t to t + 3, relative HGFs

 

\( {g}_{i,t+3}^{(0)} \) exit

\( {g}_{i,t+3}^{(1)} \) negative growth

\( {g}_{i,t+3}^{(2)} \) zero growth

\( {g}_{i,t+3}^{(3)} \) positive growth

\( {g}_{i,t+3}^{(4)} \) relative HGF

Total

\( {g}_{i,t}^{(1)} \)

0.352

0.310

0.129

0.204

0.005

1.000

\( {g}_{i,t}^{(2)} \)

0.452

0.086

0.353

0.106

0.004

1.000

\( {g}_{i,t}^{(3)} \)

0.203

0.404

0.062

0.323

0.004

1.000

\( {g}_{i,t}^{(4)} \) = relative HGF

0.271

0.376

0.021

0.325

0.007

1.000

Table 6

Transition probabilities from t to t + 3, OECD HGFs

 

\( {g}_{i,t+3}^{(0)} \) exit

\( {g}_{i,t+3}^{(1)} \) negative growth

\( {g}_{i,t+3}^{(2)} \) zero growth

\( {g}_{i,t+3}^{(3)} \) positive growth

\( {g}_{i,t+3}^{(4)} \) OECD HGF

Total

\( {g}_{i,t}^{(1)} \)

0.352

0.310

0.129

0.204

0.006

1.000

\( {g}_{i,t}^{(2)} \)

0.452

0.086

0.353

0.108

0.001

1.000

\( {g}_{i,t}^{(3)} \)

0.215

0.399

0.063

0.293

0.029

1.000

\( {g}_{i,t}^{(4)} \) = OECD HGF

0.078

0.464

0.018

0.326

0.115

1.000

Transition probabilities are calculated using frequencies. Growth brackets are defined based on annualized percentage change in employees as follows: \( {g}_{i,t+\tau}^{(0)} \) = −1 (exit), −1<\( {g}_{i,t+\tau}^{(1)} \)<0 (negative growth), \( {g}_{i,t+\tau}^{(2)} \) = 0 (zero growth), \( {g}_{i,t+\tau}^{(3)} \)>0 (positive growth), and \( {g}_{i,t+\tau}^{(4)} \)= HGF. The definition of growth brackets based on absolute growth or log-differences does not alter results

Table 13

Transition probabilities from t to t + 6, absolute HGFs

 

\( {g}_{i,t+3}^{(0)} \) exit

\( {g}_{i,t+3}^{(1)} \) negative growth

\( {g}_{i,t+3}^{(2)} \) zero growth

\( {g}_{i,t+3}^{(3)} \) positive growth

\( {g}_{i,t+3}^{(4)} \) absolute HGF

\( {g}_{i,t}^{(1)} \)

0.438

0.298

0.115

0.141

0.008

\( {g}_{i,t}^{(2)} \)

0.627

0.116

0.169

0.087

0.002

\( {g}_{i,t}^{(3)} \)

0.315

0.396

0.083

0.189

0.018

\( {g}_{i,t}^{(4)} \) = absolute HGF

0.098

0.697

0.010

0.052

0.142

Table 14

Transition probabilities from t to t + 6, relative HGFs

 

\( {g}_{i,t+3}^{(0)} \) exit

\( {g}_{i,t+3}^{(1)} \) negative growth

\( {g}_{i,t+3}^{(2)} \) zero growth

\( {g}_{i,t+3}^{(3)} \) positive growth

\( {g}_{i,t+3}^{(4)} \) relative HGF

\( {g}_{i,t}^{(1)} \)

0.438

0.298

0.115

0.147

0.003

\( {g}_{i,t}^{(2)} \)

0.627

0.116

0.169

0.087

0.002

\( {g}_{i,t}^{(3)} \)

0.312

0.400

0.082

0.204

0.002

\( {g}_{i,t}^{(4)} \) = relative HGF

0.202

0.540

0.026

0.224

0.008

Table 15

Transition probabilities from t to t + 6, OECD HGFs

 

\( {g}_{i,t+3}^{(0)} \) exit

\( {g}_{i,t+3}^{(1)} \) negative growth

\( {g}_{i,t+3}^{(2)} \) zero growth

\( {g}_{i,t+3}^{(3)} \) positive growth

\( {g}_{i,t+3}^{(4)} \) OECD HGF

\( {g}_{i,t}^{(1)} \)

0.438

0.298

0.115

0.143

0.006

\( {g}_{i,t}^{(2)} \)

0.627

0.116

0.169

0.086

0.003

\( {g}_{i,t}^{(3)} \)

0.324

0.386

0.082

0.192

0.014

\( {g}_{i,t}^{(4)} \) = OECD HGF

0.127

0.622

0.028

0.199

0.024

Transition probabilities are calculated using frequencies. Growth brackets are defined based on annualized percentage change in employees as follows: \( {g}_{i,t+\tau}^{(0)} \) = −1 (exit), −1<\( {g}_{i,t+\tau}^{(1)} \)<0 (negative growth), \( {g}_{i,t+\tau}^{(2)} \) = 0 (zero growth), \( {g}_{i,t+\tau}^{(3)} \)>0 (positive growth), and \( {g}_{i,t+\tau}^{(4)} \)= HGF. The definition of growth brackets based on absolute growth or log-differences does not alter results

Table 16

OLS estimates for growth of surviving firms

Dependent variable: git + τ if survivalit + τ

τ = 3

τ = 6

Absolute HGFs

Relative HGFs

OECD HGFs

Absolute HGFs

Relative HGFs

OECD HGFs

 

(1)

(2)

(3)

(4)

(5)

(6)

Explanatory variable: HGFit

17.077*** (4.624)

−0.069*** (0.010)

0.039*** (0.004)

−52.395*** (8.268)

−0.084*** (0.013)

−0.034*** (0.006)

Control variables

  Firm size dummy (Reference category: 25–49 employees)

    1–4

3.780*** (0.598)

0.106*** (0.003)

0.109*** (0.003)

5.261*** (0.844)

0.078*** (0.006)

0.071*** (0.006)

    5–9

3.995*** (0.521)

0.051*** (0.003)

0.054*** (0.003)

4.585*** (0.833)

0.043*** (0.006)

0.038*** (0.006)

    10–14

4.101*** (0.467)

0.030*** (0.004)

0.028*** (0.004)

3.791*** (0.824)

0.024*** (0.006)

0.025*** (0.006)

    15–19

3.392*** (0.509)

0.022*** (0.004)

0.020*** (0.004)

3.070*** (0.919)

0.008 (0.007)

0.010 (0.007)

    20–24

3.087*** (0.576)

0.019*** (0.005)

0.018*** (0.005)

1.747 (1.133)

0.017* (0.009)

0.017** (0.009)

    50–99

−6.346*** (1.006)

−0.020*** (0.005)

−0.020*** (0.005)

−6.229*** (1.496)

−0.016** (0.007)

−0.016** (0.007)

    100–249

−16.757*** (1.660)

−0.030*** (0.005)

−0.028*** (0.005)

−14.695*** (2.379)

−0.026*** (0.008)

−0.027*** (0.008)

    250–499

−58.492*** (5.299)

−0.060*** (0.009)

−0.056*** (0.009)

−47.508*** (7.745)

−0.041*** (0.015)

−0.042*** (0.015)

    500–1000

−158.630*** (16.392)

−0.050*** (0.012)

−0.045*** (0.011)

−127.839*** (17.243)

−0.032* (0.018)

−0.033* (0.018)

  Age

−1.710*** (0.097)

−0.014*** (0.001)

−0.014*** (0.001)

no

no

no

  Industry size

0.371*** (0.087)

0.000*** (0.000)

0.000*** (0.000)

0.388*** (0.149)

0.000*** (0.000)

0.000*** (0.000)

  Industry growth

1.822*** (0.685)

0.427*** (0.017)

0.421*** (0.017)

2.295* (1.289)

0.340*** (0.032)

0.347*** (0.032)

  Industry code

yes

yes

yes

yes

yes

yes

  District

yes

yes

yes

yes

yes

yes

  Legal

yes

yes

yes

yes

yes

yes

Constant

−1.267 (0.815)

−0.060*** (0.004)

−0.064*** (0.004)

0.930 (3.073)

−0.102*** (0.007)

−0.095*** (0.007)

Observations

95,761

95,761

95,761

28,694

28,694

28,694

R2

0.176

0.061

0.061

0.234

0.052

0.051

Dependent variable: Future growth after high-growth period of surviving firms based on Eq. (1). Growth measure (dependent variable): absolute formula for absolute HGFs = Sit + τ − Sit + τ − 3; relative formula for relative and OECD HGFs = \( {\left(\frac{{\mathrm{S}}_{\mathrm{i}\mathrm{t}+\uptau}}{{\mathrm{S}}_{\mathrm{i}\ \mathrm{t}-3+\uptau}}\right)}^{\frac{1}{3}}-1 \), where S indicates size measured in number of employees. Robust standard errors in parentheses. Coefficients for industry code, district, and legal are available upon request

* p < 0.1; ** p < 0.05; *** p < 0.01

Table 17

OLS estimates for growth of exiting firms

Dependent variable: git + τ if exitit + τ

τ = 3

τ = 6

Absolute HGFs

Absolute HGFs

(1)

(2)

Explanatory variable: HGFit

−187.468*** (13.251)

−41.561* (29.600)

Control variables

  Firm size dummy (Reference category: 25–49 employees)

    1–4

16.729*** (0.689)

8.363*** (0.802)

    5–9

13.559*** (0.684)

7.139*** (0.789)

    10–14

10.557*** (0.697)

5.490*** (0.839)

    15–19

8.326*** (0.788)

4.435*** (0.917)

    20–24

5.347*** (0.910)

3.843*** (1.064)

    50–99

−15.406*** (1.910)

−4.795 (2.964)

    100–249

−41.836*** (4.917)

−29.717*** (5.561)

    250–499

−197.855*** (24.533)

−48.961** (19.715)

    500–1000

−305.331*** (73.982)

−125.530 (98.184)

  Age

−0.037 (0.046)

no

  Industry size

0.040 (0.042)

0.036* (0.022)

  Industry growth

0.205 (0.177)

0.136** (0.067)

  Industry code

yes

yes

  District

yes

yes

  Legal

yes

yes

Constant

−19.306*** (0.716)

−8.855 (0.925)***

Observations

44,514

19,206

R2

0.643

0.163

Dependent variable: Growth after high-growth period of exiting firms based on Eq. (2). Growth measure: absolute formula = Sit + τ − Sit + τ − 3, where S indicates size measured in number of employees. For relative growth, Eq. (2) equals −1 and is not estimated. Robust standard errors in parentheses. Coefficients for industry code, district, and legal are available upon request

* p < 0.1; ** p < 0.05; *** p < 0.01

Table 18

Standard probit estimates for probability of survival

Dependent variable: survivalit + τ

τ = 3

τ = 6

Absolute HGFs

Relative HGFs

OECD HGFs

Absolute HGFs

Relative HGFs

OECD HGFs

(1)

(2)

(3)

(7)

(8)

(9)

Explanatory variable

  HGFit

0.084*** (0.016)

0.058*** (0.011)

0.125*** (0.004)

0.136*** (0.026)

0.230*** (0.016)

0.160*** (0.013)

Control variables

  Firm size dummy (Reference category: 25–49 employees)

    1–4

−0.243*** (0.005)

−0.248*** (0.005)

−0.238*** (0.005)

−0.294*** (0.009)

−0.306*** (0.008)

−0.287*** (0.009)

    5–9

−0.108 *** (0.005)

−0.112*** (0.005)

−0.104*** (0.005)

−0.147*** (0.009)

−0.155*** (0.009)

−0.144*** (0.009)

    10–14

−0.063*** (0.006)

−0.066*** (0.006)

−0.070*** (0.006)

−0.100*** (0.011)

−0.104*** (0.010)

−0.111*** (0.011)

    15–19

−0.029*** (0.007)

−0.030*** (0.007)

−0.032*** (0.007)

−0.049*** (0.012)

−0.051*** (0.012)

−0.056*** (0.012)

    20–24

−0.015* (0.008)

−0.016** (0.008)

−0.017** (0.008)

−0.034** (0.014)

−0.038*** (0.014)

−0.040*** (0.014)

    50–99

0.025*** (0.007)

0.026*** (0.007)

0.027*** (0.007)

0.020 (0.012)

0.021* (0.012)

0.023** (0.013)

    100–249

0.045*** (0.008)

0.048*** (0.007)

0.051*** (0.008)

0.078*** (0.012)

0.080*** (0.012)

0.085*** (0.012)

    250–499

0.055*** (0.013)

0.061*** (0.012)

0.064*** (0.013)

0.071*** (0.023)

0.078*** (0.022)

0.084*** (0.022)

    500–1000

0.083*** (0.015)

0.084*** (0.014)

0.088*** (0.014)

0.095*** (0.029)

0.101*** (0.026)

0.107*** (0.027)

  Age

−0.063*** (0.001)

−0.063*** 0.001

−0.063*** (0.001)

no

no

no

  Industry code

yes

yes

yes

yes

yes

yes

  Industry size

0.001*** (0.000)

0.000*** (0.000)

0.000*** (0.001)

0.002*** (0.000)

0.001*** (0.000)

0.001*** (0.000)

  Industry growth

0.003*** (0.001)

0.716*** (0.026)

0.709*** (0.026)

0.004*** (0.001)

0.954*** (0.044)

0.938*** (0.044)

  District

yes

yes

yes

yes

yes

yes

  Legal

yes

yes

yes

yes

yes

yes

Constant

0.682*** (0.001)

0.682*** (0.001)

0.682*** (0.001)

0.583*** (0.002)

0.583*** (0.002)

0.583*** (0.002)

Observations

140,274

140,274

140,274

49,134

49,134

49,134

Pseudo R2

0.114

0.118

0.119

0.134

0.146

0.146

Dependent variable: Conditional probability of survival based on Eq. (1). It equals 1 for firms which survive until t + τ, and zero in case of firm exit. Average marginal effects presented. HGF and industry growth measured from t − 3 to t, all other variables measured at t − 3. Robust standard errors in parentheses. For growth in t + 6, the indirect measure of age cannot be included as control, because surviving firms are of identical age. Coefficients for industry code, district, and legal can be made available upon request

* p < 0.1; ** p < 0.05; *** p < 0.01

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Chair of Applied Statistics and EconometricsUniversity of MainzMainzGermany

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