A Study of Production Functions Taking Account of Endogeneity and Selectivity with Copula Methods

行政院國家科學委員會

獎勵人文與社會科學領域博士候選人撰寫博士論文

成果報告

A Study of Production Functions Taking Account of

Endogeneity and Selectivity with Copula Methods

核 定 編 號 ： NSC 99-2420-H-004-018-DR 獎 勵 期 間 ： 99 年 08 月 01 日至 100 年 07 月 31 日 執 行 單 位 ： 國立政治大學金融系 指 導 教 授 ： 黃台心 博 士 生 ： 謝子雄 公 開 資 訊 ： 本計畫可公開查詢

中 華 民 國 102 年 08 月 01 日

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國立政治大學商學院金融學系

博士論文

Department of Money and Banking

College of Commerce

National Chengchi University

Doctoral Dissertation

考慮內生性與樣本選擇之生產邊界估計方法—

關聯結構法與共同邊界法之應用

An Estimation of Production Frontiers Taking Account of

Endogeneity and Selection under the Framework of

Copula Methods and Metafrontier Models

指導教授：黃台心 博士

研究生：謝子雄 撰

中華民國一○二年七月

July, 2013

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An Estimation of Production Frontiers Taking Account of

Endogeneity and Selection under the Framework of Copula

Methods and Metafrontier Models

Department of Money and Banking

National Chengchi University

By

Zixiong Xie

July 1, 2013

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謝辭

為了等寫謝辭的這一天，我花了整整七年的時間，這箇中滋味，唯有自知。 在這過程中，受許多師長幫忙。其中，我最要感謝黃台心老師的論文指導，引領 我進入學術研究的大門。感謝口試委員 (王泓仁老師、胡均立老師、陳坤銘老師 及林建秀老師) 的意見讓我的論文更加完善。沈中華老師的提攜與陳仕偉老師的 鼓勵是讓我前行的動力。另外，我要特別感謝徐士勛老師對我的照顧與經濟上的 支持，讓我在求學的過程中無後顧之憂。班上同學的陪伴，也讓漫長且苦悶的日 子添加一些快樂。 為了求取博士學位，我花了七年時間。而我未來的老婆—莊佳樺，則花了整 整十五年只為了等我完成學業。家人則等了我三十三年，我才能開始為家裡付出 一些心力。家人的陪伴與支持，是我最大的後盾與力量。最後，我將此論文獻給 我最為思念的父親。

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Abstract

Plants in Taiwan’s manufacturing are characterized as small- and medium-size with frequent exit and entry and the scale of survivors varies considerably with business cycles. Plants’ choices on whether to exit or to stay and continuing plants’ options on input quantities count on both technical efficiency and productivity. This entails a selection and a simultaneity problems in the estimation of production frontiers.

This dissertation proposes a new approach to solve both issues under the framework of the stochastic frontier approach. More specific, we extend Olley and Pakes’ (1996) and Levinsohn and Petrin’s (2003) approaches to a stochastic production frontier and use copula methods to deal with simultaneity and selection at the same time. Based on the proposed method, we further conduct a metafrontier analysis to compare the technical efficiency and technology gap ratio between exit and continuing firms, which are operating under different technologies and subject to simultaneity and selection. The data of Taiwan’s electronic and food products industries are arbitrarily chosen to illustrate our empirics. Some results are obtained in this dissertation: first, the proposed model solves the problems of simultaneity and selectivity in the production function that exists in ordinary least square estimation; second, there is a serious downward bias in technical efficiency when the conventional stochastic frontier approach ignores simultaneity or sample selection problem; third, the results of metafrontier analysis find that, there is little difference in technology gap ratio between exit and continuing firms. The primary determinant on whether a firm can keep operating in the industry is its managerial ability, rather than its adoption of technology.

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Table of Contents

1. Introduction ... 1

2. Simultaneity and Selectivity in a Production Frontier ... 5

2.1 OP/LP Approach ... 7

3. Stochastic Frontier Model with Simultaneity and Selectivity ... 11

3.1 Controlling for Simultaneity and Selectivity ... 12

3.2 Estimation Procedure Using Copula Methods... 14

4. Switching Production Function and Metafrontier Analysis ... 17

4.1 Switching Production Frontiers ... 18

4.2 Metafrontier Analysis ... 18

5. Data Description ... 23

6. Empirical Results of the SFSS Model ... 27

6.1 Productivity and Technical Efficiency ... 31

7. Empirical Results of the Metafrontier Models ... 33

8. Conclusion ... 37

Appendix A. Deriving the Likelihood Function of the SFSS Model ... 38

Appendix B. The Derivation of

F Q

_

( )

... 41

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List of Tables

Table 1: Descriptive Statistics, Electronics Industry ... 51

Table 2: Descriptive Statistics, Food Products Industry ... 52

Table 3: Correlation Coefficient of Variables, Electronics Industry ... 53

Table 4: Correlation Coefficient of Variables, Food Products Industry ... 54

Table 5: Parameter Estimates of the Electronics Industry ... 55

Table 6: Parameter Estimates of the Food Products Industry ... 56

Table 7: Average Firm-specific Productivity Growth, Electronics Industry ... 57

Table 8: Average Firm-specific Productivity Growth, Food Products Industry ... 57

Table 9: Descriptive Statistics of Technical Efficiency, Electronics Industry ... 58

Table 10: Descriptive Statistics of Technical Efficiency, Food Products Industry ... 58

Table 11: The Group-specific Stochastic Frontier Estimates ... 59

Table 12: The Estimates of the Industry’s Metafrontier ... 60

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List of Figure

Figure 1: Technical Efficiency Measures, Electronics Industry ... 46

Figure 2: Technical Efficiency Measures, Food Products Industry ... 46

Figure 3: TGR of Continuing and Exit Firms, Electronics Industry ... 47

Figure 4: TGR of Continuing and Exit Firms, Food Products Industry ... 48 Figure 5: TE Score of Continuing and Exit Firms, Electronics Industry ... 49 M

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1. Introduction

Production function is a function that specifies the output of a firm for all combinations of inputs. Rather than just representing the result of economic choices, the link between output and inputs implicitly show firm’s technology, decision making, managerial ability, external shocks, and so forth. More precise estimates of production function become more relevant when we are going to investigate the firm’s characteristics.

The conventional ordinary least squares (OLS) is a useful tool to estimate the parameters of a production function. Two shortcomings of the OLS are worth mentioning. The first is that it overlooks possible contemporaneous correlation between input choices and the unobserved firm-specific productivity (Marschak and Andrews (1944) and Olley and Pakes (1996) (henceforth, OP)). This simultaneity problem causes the OLS estimates to be upwardly biased when inputs are positively correlated with productivity that has serial correlation.1 The second is that it ignores the selection problem induced by endogenous exit decision. This selectivity bias entails a downward bias in capital coefficient when a firm’s decision of exit in the next year is affected by current productivity level. This arises from the fact that the conditional expectation of productivity on current inputs and available information is decreasing in capital, as claimed by OP. Overall, the OLS procedure would overestimate labor coefficient and underestimate capital coefficient, leading to a misled measure of returns to scale. Levinsohn and Petrin (2000) note that the productivity changes are often over-predicted when adopting OLS method and the direction of productivity movement cannot be accurately captured.

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OP proposes a semi-parametric approach to solve both simultaneity and selection problems. They suggest the use of investment, an observed variable, to proxy unobserved productivity and the use of estimated survival probabilities to correct the selectivity bias. However, many of the sample firms report zero investment due possibly to pronounced adjustment costs, which introduces the serious truncation problem. Levinsohn and Petrin (2003) (henceforth, LP) recommend the use of intermediate inputs (such as material, fuel, and electricity expenses) as the proxy to avoid the zero investment problem and thus ensure monotonicity condition of OP estimators to be held.

This dissertation aims to solve both problems of simultaneity and selectivity, on the one hand, and extends OP/LP’s approach to include the technical efficiency of a firm into the production frontier, on the other hand. This requires the production frontier having an extra one-sided error term, representing technical inefficiency, in addition to unobserved productivity and idiosyncratic shocks. The emergence of the composite errors largely complicates the model. We suggest using copula methods, an important tool to deal with the dependence between relevant variables in the area of finance, to derive the likelihood function that simultaneously takes the two problems and composite errors into account. As far as we know, this study appears to be the first time in the literature attempting to disentangle these topics concurrently.

Based on our proposed model that solves simultaneity and selectivity in the production frontier, we attempt to compare the operating technologies of continuing and exit firms and further calculate their comparable technical efficiencies. Since the two groups of firms may operate under different technologies, the metafrontier production function model proposed by Battese et al. (2004) and O’Donnell et al. (2008) and the stochastic metafrontier model, recently proposed by Huang et al. (2012), will be adopted in this dissertation.

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literature review on the simultaneity and sample selection problems. Chapter 3 develops our model that integrates the both problems under the framework of the stochastic frontier analysis (SFA). Chapter 4 introduces the stochastic metafrontier model that is able to modify the both problems. Chapter 5 describes the data source and variable definitions. Chapter 6 and 7 reports empirical results from our proposed model and the metafrontier model, respectively, while Chapter 8 concludes the dissertation.

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2. Simultaneity and Selectivity in a

Production Frontier

Consider a firm’s production function with Cobb-Douglas technology:

0 , it lit k it it y  l  k  (1) , it it Vit    (2)

where y is plant i’s (log)output measured as value-added at time _it t, l the log of _it

its labor input and k the log of its capital input. _it ₀, _l and _k are unknown technology parameters to be estimated. The disturbance term of _it is composed of a firm’s productivity _it and a random shock V . The former is a productivity index _it

that affects the firm’s decision on input hiring and hence is correlated with the input choices and exit behavior. _it is known to the firm and evolves over time according to an exogenous process, but not known to researchers. The latter is an unobservable random disturbance uncontrollable by the firm.

Given equation (1), the OLS estimates for the two inputs are

   

   2

( , ) ( , ) ( , ) (

Cov Cov Cov Cov

ˆ C , ) ( , ) ov Cov( , ) Cov( , ) l l k k l l k k l l k k l k       _ _  ,        2 ( , ) ( , ) ( , ) (

Cov Cov Cov Cov

ˆ C , ) ( , ) ov Cov( , ) Cov( , ) k k l l k l k l l l k k l k       _ _  .

If the two inputs are uncorrelated with the disturbance term (_it), i.e., Cov( , )k  

Cov( , )l 0, the OLS estimates are unbiased. Recently, OP and LP note that the OLS estimates of labor and capital will be overestimated and underestimated, respectively,

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when the unobserved productivity _it is, respectively, positively and negatively correlated with labor and capital. The first bias comes from simultaneity problem, arising from the fact that input choice of labor is responsible to the firm’s beliefs about _it. More labor is expected to be hired in response to a higher value of current productivity _it , i.e., Cov( , )l 0 , leading to an upward bias in the labor coefficient. The second bias stems from self-selection when a firm’s decision on exit is affected by its productivity. Specifically, firms accumulating larger capital stocks can expect to earn more future profits for any given level of current productivity. These firms will keep running at lower  realizations. Thus, the self-selection induced by exit decision implies that the conditional expectation of  on current inputs, survival, and all available information at t1 will be decreasing in capital, i.e, Cov( , )k  0, which results in a downward bias in the capital coefficient.

There are three conventional approaches to resolve the simultaneity problem, namely, instrumental variables (IV), generalized method of moments (GMM), and fixed effects model. The first approach of the IV requires researchers to collect extra variables that are (highly) correlated with the explanatory (endogenous) variables and uncorrelated with the disturbance term. IV estimators can be shown to be consistent, provided the instruments satisfy the above requirements. The economics of production theory suggests that input prices may be valid instruments because they are market-determined and directly influence the choices of inputs, but not directly enter the production function. However, input prices are often not reported by firms or cannot be recovered from their accounting data. Even though input prices are sometimes available, they may not be determined by the demand and the supply in a perfectly competitive market. It is well-known that the product price in an imperfectly competitive market is itself a function of the quantities sold. This invalidates the use of input prices as valid instruments. Even worse, the use of the IV approach is unable to solve the selectivity problem, arising from a firm’s non-random behavior of exit.

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GMM technique, developed by Blundell and Bond (1998). They extend the standard first difference GMM estimation of Arellano and Bond (1991) to include additional moment conditions formed by the lagged difference of the explanatory variables that are treated as extra instruments. This approach shares the same drawback as the IV approach, i.e., it leaves the self-selection problem intact.

As for the last approach, when panel data are available, the use of fixed effects model is able to solve the simultaneity problem (Hoch, 1962; Mundlak, 1961; Harrison, 1994). To correct for the simultaneity problem, the fixed effects estimation assumes that the unobserved firm-specific productivity is time-invariant. Thus, consistent parameter estimates can be obtained by using either the within-group or first difference estimation. An additional advantage of this approach lies in its capability of at least partially solving the selectivity, if exit decisions are determined by the time invariant unobserved firm-specific productivity. Unfortunately, the time-invariant assumption is strong and invalid particularly for long panel data and for data containing major environmental changes, e.g., deregulation, trade liberalization, technical change, etc.

2.1 OP/LP Approach

OP propose a novel technique to handle both problems of simultaneity and self-selection in the context of production function. They recommend using investment to control for the correlation between input variables and _it and estimating the probit model to account for a firm’s exit decision. Later, LP argue for the use of intermediate inputs, such as materials, fuel, and electricity, to control for simultaneity, instead of investment, since in some datasets the number of observations with zero values of investment is large, which reduces the sample size considerably. We now briefly describe the estimation algorithm of OP/LP.

Differing from OP, LP specifies electricity input e as a function of _it kit and

,

it

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condition ensures the following inverse function of the unobserved productivity it

to be held, i.e., 1 ( _it, _it) ( , ). it eit k e h k eit it it  _  _ (3) Substituting equations (2) and (3) into (1) yields a partially linear regression equation:

( , ) , it l it it kit it it y l  e V (4) where 0 ( ) ( , ) , it k eit it kkit hit k eit it     . (5)

The estimation procedure is divided into two steps. In the first step, one approximates _it( ) with a fourth order polynomial series in capital and electricity and then estimate equation (4) by OLS procedure or approximates _it( ) using kernel estimation method of Robinson (1988). The coefficient estimate of labor, ˆl, is

consistent since the series expansions or kernel estimator of _it( ) controls for the unobserved productivity and the error term of V_it is uncorrelated with inputs by assumption.

In the second step, one attempts to identify _k and correct selectivity bias. The identification of _k requires separating the effect of capital on electricity function

) (·

h from the effect of capital on output. Moreover, to consider a firm’s exit decision in its production function, we need additional information on a firm’s decision of liquidation. OP/LP use estimates of hˆ_it1(ˆ_it1_kk_it1) and predicted survival probabilities Pˆ_it to respectively control for simultaneity and selectivity biases. The bias term, g P h( _it, _it1) say, arising from both simultaneity and selectivity, can thus be approximated by the fourth order polynomial series in (P hˆ_it,ˆ_it1). The proposed regression equation is expressed as:

4 1 1 4 0 0 ˆ ˆ ( , ) ˆ _, ˆ ˆ it it k it it it m j m k it mj it it it it j l it it m y l g P h P h k V k V       _             





(6)

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where the innovation it(it E[ it | it1,Iit 1]) and V are uncorrelated with it

.

it

k 2 We can estimate (6) using nonlinear least square method to obtain the consistent estimate of _k.

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3. Stochastic Frontier Model with

Simultaneity and Selectivity

This chapter intends to develop a new approach that solves simultaneity and self-selection problems under the framework of stochastic frontier approach, in which the error term of _it in (2) contains an extra terms of a one-sided error, uit, that

reflects technical inefficiency of the firm. We believe that this is the first time term

it

u entering the OP/LP type of model. Differing from it that captures the

unobserved evolution of firm-level productivity, technical inefficiency uit is

exploited to represent managerial inabilities of a firm in the production process, which is assumed to be independent of V_it and it, and uncorrelated with inputs. This

allows us to generalize the OP/LP model to a stochastic frontier setting, dated back to Aigner et al. (1977) and Meeusen and van den Broeck (1977). We therefore refer our model as the stochastic frontier model with simultaneity and selection (SFSS).

The SFSS is specified as:

0 , if 1; 0, otherwise, it it it it it it l k it V u l k I y          (7)



1 1, 1



1 ( ) 0 . it it it kit it it I  γ z  _{ } e _   (8)

Equation (7) is a two-input production frontier that incorporates a productivity index

it

 , a technical inefficiency term u_it, and a random shock V_it. u_it is assumed to be a half normal random variable with a mean of zero and a constant variance, i.e.,

2

~ (0, )

it iid N u

u  ;3 V_it is assumed to be a two-sided error with a mean of zero and a

3 _Term

it

u can also be assumed to be distributed as truncated normal, exponential, or gamma. Since all of the four distributions result in similar efficiency scores, Ritter and Simar (1997) argue for the use

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constant variance, i.e., V_it ~iid N(0,_V2). Variables it, uit and Vit are assumed to

be mutually independent. We define the indicator function of I_it 1 for the ith firm at time t if the firm continues to operate and I_it 0 if the firm exits the market. We specify the survival probability to be dependent of some threshold value of productivity, it1, which is a function of kit1 and eit1,

4

and z , a vector of it

covariates that influences the exit decision of a firm. The disturbance term of  is it

conventionally assumed to be a standard normal random variable, making equation (8) a probit model. Following OP/LP, equations (7) and (8) lead to the estimation of the production frontier of continuing firms after solving for the problems of simultaneity and selection. The production frontier of exit firms will be considered in the next chapter, where the metafrontier model is introduced.

3.1 Controlling for Simultaneity and Selectivity

Equations (7) and (8) involve the stochastic frontier, unobserved productivity, and limited-dependent variable concepts. To estimate the SFSS, conventional stochastic frontier algorithm is not directly applicable since it does not allow for the presence of unobserved productivity and exit decision. Also, Heckman’s (1979) two-step estimation procedure cannot be employed due to the existence of it and uit. Both

(7) and (8) are better to be estimated jointly to yield efficient and consistent estimators. Recently, Lai et al. (2009) propose a new approach that considers the sample selection problem in the context of a stochastic frontier model using copula functions. Copula methods is useful to capture the dependence between (7) and (8), provided it has

been appropriately dealt with.

Following the idea of OP/LP, we specify electricity e_it as a function of _it and

it

k , i.e., e_it e_it(_it,k_it). Under the condition of monotonicity the function can be inverted to be _it h k e_it( _it, _it) as shown in (3). The production frontier of (7)

distribution, such as truncated normal or gamma.

4 _{OP argues that a firm will choose to stay in the market if its productivity is greater than the threshold}

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becomes ( , ) , it l it it it it it it y l  k e V u (9) where 0 ) ). ( , _k ( , it k eit it kit h k eit it it     (10)

Similar to Fan et al. (1996), equation (9) is a semi-parametric stochastic frontier model, where the functional form of h_it( ) is unknown. Rearranging (9), we get





, ( ) , it llit it kit eit Vit uit y      (11) where ( , ) ( , ) , it k eit it it k eit it    (12) and

 

2 . E u_{it u}     

Equations (11) and (12) are alike to equation (17) of Fan et al. (1996). Pseudo-maximum likelihood algorithm listed in equations (10)-(15) of Fan et al. (1996) can be applied to obtain consistent estimates of labor, ˆl, and predicted values

of ˆit(ˆitˆ). Then, following the second step of OP/LP (shown in (6)), the

simultaneity bias can be approximated by powers of hˆit1(ˆit1kkit1):

1 4 0 * ˆ , m it k it m it it it m v y  k  h _ u     



(13)

where y_it* y_itˆ_{l it}l and v_it(_itV_it) is assumed to be distributed as N(0,_v2). The error term of v_itu_it in (13) is no longer correlated with capital and hence solving the simultaneity bias problem, while the selection problem is yet to be solved.

Equation (8) is used to account for a firm’s decision on exit or stay. Following OP, we use the fourth order polynomial expansion in (k_it1,e_it1) to approximate

(k ,e )

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4 0 1 * ˆ , if 1; 0, otherwise, m k it m it i m it it t h k I y       _ _   



₍₁₄₎



0 ,



1 it it it I  α w   (15) where _it v_itu_it and 4 1 1 4 0 0 . m j m it it mj i j m t it k e          



α w γ z

The composed error _it in (14) and _it in (15) are here assumed to be correlated. The difficulty one is now facing is how to jointly estimate (14) and (15), in which the core issue is how to derive the joint distribution of the composed error _it and _it.

3.2 Estimation Procedure Using Copula Methods

Let F_( ) and F_( ) be the marginal cumulative distribution functions (cdf) of _it and  and _it, f_( ) and f_( ) are corresponding probability density functions (pdf). Define β( _k, ₀,,₄),  (_u2 _v2 1/ 2) , and    _u / _v. Following Lai et al. (2009) who utilize the Gaussian copula function to model the dependency between the production frontier and selection equation, the log likelihood function of (14) and (15) using the entire sample can be expressed as







*



| 1 1 1 ˆ (1 ) ln Pr( 0 | ; ) ln ( , , , , ) l , , 1; , , ln Pr( 1| ; n , , , , ) N it it it it it T it it it it it i i t t I I L k I I I f_ y     _             _{ }          _{ }      



α β α β α w w w α  (16) where





 

1 2 * | 1 1 ˆ , , , , ( ( )) 1 , , 1; , , Pr( 1| ; ) it it it it it it it it it it f F f k y I I                             _{ }           w α w w β α α  (17) 2 ( ) it it , it f_           _           (18)

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



Pr(I_it 0 |w_it; )α    α w_it ,





Pr(I_it 1|w_it; )α   α w_it , ( )

  and  ( ) denote standard normal pdf and cdf, respectively, and  is the coefficient dependence between F_( ) and F_( ) . Note that the conditional density function of f_|( ) in (17) is derived based on the Gaussian copula function that

describes the dependence between F_( ) and F_( ) . Appendix A gives detailed derivation of (16) and (17).

Unfortunately, the objective function given in (16) is rather complicated and difficult to be estimated when the number of unknown parameters is large. Lai et al. (2009) alternatively propose a simpler two-step estimation procedure. The first step aims to estimate (15) by the standard probit model, using the entire sample, to get the estimates of α. The objective function is expressed as













1 1 1 max ln ( ) (1 ) ln Pr( 0 | ; ) ln Pr( 1| ; ) . T it t it it it it N i t L I I I I   



    α α w α w α (19)

Given the estimates of α, ˆα say, one can estimate the remaining parameters ( , , , )β    using the subsample that satisfy I_it 1. The objective function in the second step is formulated as













, , , | 1 1 1 2 ˆ max ln , , 1; l , , , | 1 and ˆ _{ˆ ˆ} ln , , , , , n Pr 1| ; . T i it N it i t t it it it it L k f I I I         _         _     



β β α α w w β α  (20)

A final problem remains to be resolved when maximizing either (16) or (20). That is, there does not exist a closed form of F_( ) in (17), i.e.,

) ( ) ( , it it F f_ d        



(21)

since f_( ) has no close form as shown by (18). Tsay et al. (2013) recently develop a method of approximation to (21), which relies on the use of the error function and

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neither relies on simulation methods like Greene (2009) nor on Gaussian quadrature like Tsionas and Papadogonas (2006). This study chooses to apply the approximation approach of Tsay et al. (2013) to deduce F(it). Appendix B gives the detailed

derivation.

After getting the parameter estimates of the SFSS model, we follow Jondrow et al. (1982) to calculate technical efficiency score:













2 1 / . / | it it it it i it t u TE E e               _        _   (22)

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4. Switching Production Function

and Metafrontier Analysis

This chapter is devoted to estimate and compare the technical efficiency of different groups, i.e., exit and continuing firms, that are potentially operating under different technologies. One could first estimate a common frontier by pooling all the data of the various groups and compute and compare technical efficiencies (TEs) for the groups of firms. However, the so-derived common frontier may not necessarily envelop the group-specific frontiers. It would also lack good reason if one simply estimates the individual group-specific frontiers and compares the TEs among groups, because these TE scores are measured relative to distinct production frontiers. A metafrontier production function model, proposed by Battese et al. (2004) and O’Donnell et al. (2008), is able to envelop group frontiers by imposing appropriate restrictions. They suggest a two-step procedure for estimating the metafrontier, which utilizes the SFA in the first step to estimate the group-specific frontier and the mathematical programming techniques in the second-step to estimate the deterministic metafrontier. However, their second-step estimates from mathematical programming suffer from no statistical properties and the estimation results tend to be confounded with idiosyncratic shocks. We instead apply the stochastic metafrontier approach, developed by Huang et al. (2012), to obtain the parameter estimates of the metafrontier, which preserves statistical properties and free from the impacts of shocks.

Recall that the production frontiers of exit and stay firms should be described by two distinct regression equations. Before conducting a metafrontier analysis, we use a switching regression approach that allows for correcting endogenous liquidation decision and simultaneity in such a way as to estimate group-specific production

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frontiers consistently.

4.1 Switching Production Frontiers

Based on the SFSS model, we are able to estimate the group-specific frontiers where both endogenous liquidation decision and simultaneity are considered. For continuing firms, the model is the same as equations (7) and (8), and repeated as follows:

10 1 1 1 1 1 1 1 1 , if 1; 0, otherwise, it it it it it i t l k i t V l k u I y         



1 1, 1



1 ( ) 0 . it it it kit it it I  γ z  _{ } e _  

As for exit firms, the model is similar to continuing firms’, but the indicator function has to be changed into I_it 0, i.e.,

00 0 0 0 0 0 0 0 0 , if 0; 0, otherwise, it it it it it i t l k i t V l k u I y          (23)



1 1, 1



1 ( ) 0 . it it it kit it it I  γ z  _{ } e _   (24)

The above two cases explicitly consider the endogenous decision on whether exiting or staying at the market through the indicator function. Their production frontiers can be jointly estimated, which leads to consistent and asymptotically efficient estimates.

Following the estimation algorithm described in Chapter 3, the simultaneity-corrected production frontiers are specified as:

4 0 * 1 ˆm _,

jit jk jit jm jit it jit m j k y   h v u      



(25)

where hˆ_jit1(ˆ_jit1_jkk_jit1), j0 corresponds to exit firms, and j1 stay firms.

4.2 Metafrontier Analysis

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et al. (2004). For simplicity, the production frontier with unobserved productivity and is re-written as: 4 0 1 ˆ ln ( , ) m

jit jit jk jit jm jit jit jit j j j it j jit l m k v u y l h f             



x b (26)

Here _jit v_jitu_jit, ln f x_j( _jit,b_j) is the logarithm of a production function, x_jit a vector of explanatory variables, and b_j (  _jl, _jk, _j0,,_j4) the unknown parameters corresponding to x_jit. The deterministic metafrontier production function

, ( _jit M)

M

f x b that envelops all individual group’s frontiers f_j is expressed as:

( , ) ( , ) , , , M jit M jit j jit u j M j i t f x b  f x b e  or, equivalently, , ln f_j(x_jit b_j)ln f_M(x_jit,bM)uM_jit, j i, ,t (27)

where bM _{signifies the parameters of the metafrontier function and} uM_jit 0 implies that

( , M) ( , ), 0,1

M it fj jit j j

f x b  x b 

Following Battese et al. (2004), the technology gap ratio (TGR) is defined as:

, ) 1 ( ) ( exp( ) , jit j M M i j M j t t it ji f T f GR  x b  u  x b (28)

and the technical efficiency relative to the meta-frontier TEM_jit is formulated as:

.

M

jit jit jit

TE TE TGR (29)

Battese et al. (2004) suggest estimating bM by mathematical programming techniques, i.e., linear or quadratic programming. Since equation (26) is a nonlinear regression model, a nonlinear programming (NP) technique is required to solve the following problem:

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0 1 1 0 1 1 1 1 ˆ ln ( , ) ln ˆ , ) ˆ ln ( , ) ln , min ( ˆ s.t ( ). M M M N T N T M jit it j jit j j i t j i t M j t j M it i j f L u f f f         







b x b x b x b x b (30)

We thus use the GAUSS module of Constrained Optimization to solve nonlinear optimization problem of (30) and obtain M ( M, M, ₀M, , ₄M)

l k

   

  

b .

As mentioned above, estimates from mathematical programming are lack of statistical properties and apt to be contaminated with statistical noises. Huang et al. (2012) recently propose a new two-step stochastic approach to estimate the stochastic metafrontier. Both Huang et al. (2012) and Battese et al. (2004) share the same first step. Their main difference comes from the second step, i.e., the metafrontier of Huang et al. (2012) is constructed on the basis of the stochastic frontier, rather than in the deterministic setting. We now turn to the second step of the new approach.

Given the first step SFSS estimates of the group-specific frontiers f xˆ_j( _jit,bˆ )_j , 0, 1

j , for (26), the estimation error of the group-specific frontier is calculated as:

ˆ

ln ( ,ˆ ) ln ( , ) ˆ M

jit j jit j jit

j j jit jit

f x b  f x b   v . (31)

The metafrontier frontier of (27) can be re-formulated by replacing the unobserved group-specific frontiers f xj( jit,bj) on the left-hand side with its estimated

counterpart, f xˆj( jit,bˆ )j , from (31), i.e.,

ˆ

ln fj(xjit,bˆj)ln fM(xjit,bM)vMjit uMjit. (32)

Equation (32) turns out to be a standard SFA with vM_jituM_jit being the composed error term. Parameter vector b can be estimated by the ML. Since this procedure M

resembles the conventional SFA, it is referred to as the stochastic metafronteir (SMF) regression. Since vM_jit involves residual ˆ_jit, the variance of the disturbance term in (32) is heteroskedastic. This leads the estimated covariance matrix of the parameters to be inconsistent. This problem can be solved by relying on the use of the “sandwich” estimator for the covariance matrix. See, for example, White (1982).

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The SMF allows for the estimated group-specific frontier in excess of the metafrontier, i.e., ˆ ( ,ˆ ) ( , M)

jit j ji

j M t

f x b  f x b , due to the existence of the estimation error of vMjit. However, the metafrontier is always higher than the group-specific

frontier, i.e., ( , ) ( , M)

jit j M jit j

f x b  f x b . The TGR must always be less than or equal to unity and is defined as:





* 1 | M jit u j M jit it TGR E e   , (33)

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5. Data Description

The primary source of data is the plant-level annually surveyed data of Taiwanese manufacturing industry, spanning 1997 to 2000 and 2002 to 2005.5 The survey is called the Industry, Commerce, and Service Census (ICSC) conducted by the Directorate General of Budget, Accounting and Statistics (DGBAS) of Taiwan. Among the 23 two-digit industries, we select two of them, i.e., electrical machinery and electronics industry, and food products industry as our targets. The former is known as a high-tech and highly capital-intensive industry with swift technological advance, while the latter is characterized as a traditional industry and carries the opposite traits of the former.

We define the output variable (y) in the production function as the value-added that is equal to the sales revenue minus the sum of expenses on raw materials and electricity. The capital input ( k ) is defined as the net amounts of operating fixed assets. The labor input ( l ) is measured as the number of employee. The electricity expenses (e) is identified as the intermediate input. Note that all of the dollar-valued variables are deflated by Taiwan’s consumer price index (CPI) with the base year of 2006 and all variables are further transformed by taking the natural logarithm.

The dummy variable I_it 1 if firm i stays in the market; I_it 0, otherwise. The determinants of exit used in this study are classified into three parts. The first is the threshold of unobserved productivity _it1 that is a function of k_it1 and e_it1. Following OP, _it1 is substituted by the fourth order polynomial series expansion in

1 1

(k_it,e_it ). The second consists of a set of macroeconomic variables such as EXCHANGE_t (annual percentage change in the exchange rate of New Taiwan’s

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Dollar versus the US dollar) and GDPGrowth_t (GDP growth rate at time t). The former intends to examine whether a depreciation or appreciation in domestic currency influence the probability of exit, while the latter wants to examine whether the macroeconomic condition affect firms’ current decision on exit. Variable EXCHANGE_t is taken from Taiwan Economic Journal and GDPGrowth_t from

Quarterly National Economic Trends published by the DGBAS. The third type of

determinants are firm-/industry-specific variables. Variable CPTL2OUTPUTit is

calculated as the ratio of capital to output, representing a firm’s sunk cost; AGEit

signifies the age of a firm; PCMt is computed as sales revenue minus costs and

divided by the sales revenue, which evaluates the profitability of the industry as a whole.

[Table 1 and Table 2 here]

Tables 1 and 2 respectively report descriptive statistics of the variables for the both industries. Each table is divided into three panels, i.e., the entire sample, continuing plants, and exit plants. After removing missing data and extreme values, the sample of electrical machinery and electronics industry consists of 5,512 plants. Among them, 3,404 plants are classified as continuing plants with a total of 11,891 plant-year observations and the rest of 2,108 plants belong to exit firms with a total of 4,453 plant-year observations. Food products industry consists of 1,976 continuing plants with a total of 7,104 plant-year observations and 559 exit plants with a total of 1,199 plant-year observations.

As far as the whole sample is concerned, the mean values of y, l , k , and e

in both industries are much larger than their medians, reflecting the distributions of these variables are skewed to the right. This implies that most of Taiwanese electronic and food products plants are small and medium enterprises (SMEs). The exit rate of electronics industry is 38.24% that is much higher than that of food product industry (22.05%). This is because the electronics industry is facing a highly uncertain

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atmosphere and keen competition relative to the food products industry. According to the indices of PCM and CPT2OUTPUT, the electronics industry is more profitable than the food products industry, but the former incurs higher initial sunk costs than the latter.

The continuing plants are inclined to be larger than the exit ones, as the former has higher average values of output and inputs than the latter. This indicates that smaller plants tend to have higher probability of leaving the market. In addition, plants with greater sunk costs (CPT2OUTPUT) have higher probability of exiting the market in both industries.

[Table 3 and Table 4 here]

Tables 3 and 4 summarize the correlation coefficient matrices of all variables for the two industries. Panel (A) reports correlation coefficients for variables in the production function and Panel (B) for variables in the selection equation. These tables reveal that all of the variables in the production function are significantly and positively correlated with each other. The magnitudes of the correlation coefficients in the selection equation are relatively smaller and their signs vary substantially. It is noteworthy that the correlation coefficient between EXCHANGE and PCM is positive in the electronics industry, while the reverse is true in the food products industry, implying that the electronics (food products) firms can earn higher profit from the depreciation (appreciation) of domestic currency. This may be attributed to the fact that most of the electronics firms in Taiwan devote themselves to export their products and the depreciation of New Taiwan’s dollar stimulates their sales revenue.

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6. Empirical Results of

the SFSS Model

Tables 5 and 6 present parameter estimates of the production and selection equations for the two chosen industries. Columns 1-4 of the upper panel, respectively, list estimates of OLS, Heckman’s sample selection model, the conventional SFA, and the SFA with sample selection (SFAS) of Lai et al. (2009).6 These four models do not take the simultaneity problem into account, since the unobserved productivity  is precluded from the production function. Columns 5-6 of the upper panel summarize estimates of OP/LP and our proposed SFSS model, respectively. The latter two models consider both simultaneity and selection problems, but the SFSS further generalizes the OP/LP model to a stochastic frontier framework.

[Table 5 and Table 6 here]

The OLS estimates of labor and capital are 1.000 (1.081) and 0.153 (0.170) for the case of electronics (food products) industry. The sum of the two coefficients is equal to 1.153 (1.251), suggesting that these plants in both industries are operating under technology of increasing returns to scale. The coefficient of labor is found to be more than six times as large as that of capital in both industries. One may suspect that labor coefficient is inclined to be overestimated, while capital coefficient

6

Lai et al. (2009) combine the conventional SFA with the sample selection. Their model can be specified as: 0 , if 1; 0, otherwise, l it k it it it it it k v u l I y        



0 .



1 it it it I  α w  

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underestimated. This is ascribable to the failure of the OLS to modify both simultaneity and selectivity problems, where the former problem entails an upward bias in the labor coefficient and a downward bias in the capital coefficient and the latter problem incurs a downward bias in the capital coefficient, as pointed out by OP.

Column 2 gives the corrected estimates from the Heckman’s (1979) sample selection model. Estimated  that describes the correlation between unobserved determinants of propensity to exit the market _it and unobserved determinants of output v_it for electronics (food products) industry is statistically significant, suggesting that the unobservables (i.e., I_it 0) is correlated with one another (i.e.,

1

it

I  ). The correlation further implies the firm’s liquidation decision affects its production non-randomly and hence affects its production process. Compared to the OLS estimates, the coefficient of labor is slightly decreased, but the coefficient of capital is slightly increased in both industries.

Columns 3 and 4 include an extra non-negative random variable u,

representing technical inefficiency, in the production function, while the unobserved productivity it is still excluded from the two models. Column 3 corresponds to the

conventional SFA and Column 4 is the SFAS model that considers a firm’s liquidation decision. Differing from the OLS and the Heckman’s approach, attributing all deviations from the production frontier to noise (e.g., measurement error, random shocks, etc.), the stochastic frontier framework assigns all deviations to both noise and inefficiency. For the case of electronics (food products) industry, the estimated labor coefficient of the SFA is slightly decreased, but the estimated capital coefficient is somewhat increased as compared to the OLS results. Similar outcomes can be detected from the SFAS estimates. In sum, the coefficient estimates of labor and capital are, respectively, decreased and increased when technical inefficiency is included in the production function, irrespective of the liquidation decision.

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estimates of the OP/LP method and the SFSS method, respectively. Recall that both methods deal with the problems of simultaneity and selectivity, but the SFSS further deliberates technical inefficiency. To eliminate the positive bias of the labor coefficient, we reformulate the SFSS model to a semi-parametric stochastic frontier model of (11) to control for the unobserved productivity and re-estimate the production function. This leads to a consistent parameter estimate of labor. For the case of electronics (food products) industry, the labor coefficients in columns 5 and 6 are 0.927 and 0.900 (0.819 and 0.809), respectively, which are 7.30% and 10.00% (24.93% and 25.07%) lower than those of OLS.

The above results confirm that the production function considering the unobserved productivity is indeed able to fix the upward bias in labor coefficient at least to some extent. In addition, the upward bias of labor coefficient in food products industry is much larger than in the electronics industry. This uncovers that firms’ decision on labor hiring in food products industry is positively and more closely related to their current productivity _t than in the electronics industry. One possible explanation is that labor input plays a more important role in food products plants than in electronics plants that undertake a highly capital-intensive technology.

We next attempt to estimate k after correcting for selection in the production

function. OP/LP recommend the use of polynomial series expansion in

1 1 1

ˆ ₍ ˆ ₎

it it kkit

h_  _  _ and predicted survival probabilities Pˆit to control for both

simultaneity and selectivity. Note that our SFSS model uses powers of hˆit1 to

control for simultaneity in production frontier and uses selection equation to simultaneously account for firm’s decision of exit, in addition to consider potential production inefficiency. Our estimation results of the electronics industry show that the capital coefficient increases from 0.153 (OLS) to 0.191 for the OP/LP and to 0.250 for the SFSS model, indicating that the capital coefficients of the OP/LP and SFSS are, respectively, raised by 24.84% and 63.40% relative to that of OLS. This is congruent with the expectation that the presence of simultaneity and selection

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problems is likely to predict a downward bias in coefficient of capital. Moreover, the capital coefficient of SFSS is about 30.89% higher than the OP/LP. This validates the inclusion of technical inefficiency in the production function. Likewise, both capital coefficients of OP/LP and SFSS in food products industry are greater than that of OLS and the capital coefficient of SFSS exceeds that of OP/LP.

Except for the consideration of technical inefficiency, our SFSS has another exclusive feature. That is, it explicitly models production frontier and selection equation as simultaneous equations. Such a model of structural equations is better to be jointly estimated like our SFSS. Conversely, OP/LP implicitly assume production function and selection equation to be uncorrelated and thus suggest using the two-step method to remove the selection bias. Since the SFSS results in the estimated values of the dependence being equal to 0.981 and 0.740 for the two industries, the uncorrelation assumption may not be desirable.

When both simultaneity and selectivity biases are eliminated and technical inefficiency is incorporated, the ratio of labor share to capital share is reduced from 6.535( 1.000 / 0.153) to 3.600



0.900 / 0.250



for electronics industry and from 6.359( 1.081/ 0.170) to 3.275



0.809 / 0.247



for food products industry. The measure of returns to scale in electronics (food products industries) lowers from 1.152 to 1.150 (from 1.251 to 1.056). The finding of increasing returns to scale seems to be reasonable, since most of the sample plants are small- and medium-sized as pointed out in the Chapter 5 of data description. These plants are anticipated to be operating at the decreasing portion of the long-run average costs.

The determinants of a plant’s liquidation are also important. The less sunk costs and the older the plants are, the more likely they choose to stay in the market. However, the former results contradict most of empirical studies, e.g., Dunne and Roberts (1991) and Fotopoulos and Spence (1998), that assert that capital requirements are barriers to exit. Moreover, a depreciation in domestic currency has