INTRODUCTION
Introduction
The increasing demand for metal products, particularly iron and steel, in everyday life and the construction sector highlights the significance of the metal manufacturing industry As reported by the World Steel Association, Vietnam's steel market ranked as the seventh largest in Asia by the end of 2011, reflecting a growth rate aligned with the nation's economic expansion This industry holds substantial potential for further growth, driven by rising incomes and a continuing trend in construction activities.
According to the Viet Nam Chamber of Commerce and Industry (VCCI), small and medium-sized enterprises (SMEs) constitute 97% of all businesses in Vietnam, employing over half of the domestic workforce and contributing more than 40% of the nation's GDP This vibrant sector plays a crucial role in driving economic growth in Vietnam However, SMEs in the metal manufacturing industry currently face significant challenges, including outdated technology and a reliance on imported materials Therefore, analyzing the technical inefficiency levels of Vietnamese SMEs is essential for sustaining and enhancing the benefits derived from this vital industry.
Technical efficiency measures how effectively a firm utilizes its inputs to generate outputs, with the production frontier representing the maximum output achievable from a specific input level Firms operating on this frontier are deemed technically efficient, while those significantly below it are considered technically inefficient To assess technical efficiency, analysts create a production-possibility boundary and evaluate the distance of firms from this boundary to determine their inefficiency levels.
Technical efficiency can be measured using two primary approaches: deterministic and stochastic The deterministic method, known as Data Envelopment Analysis (DEA), was introduced by Charnes, Cooper, and Rhodes in 1978 and utilizes linear programming to create a frontier based on input and output data, without needing to specify a production function However, its deterministic nature assumes that data is free from statistical noise In contrast, the stochastic approach, or Stochastic Frontier Analysis (SFA), was first discussed by Aigner, Lovell, and Schmidt in 1977, as well as by Meeusen and Broeck in the same year SFA requires a specific functional form for the production function and acknowledges the presence of noise in the data, making it more commonly used in practice, as the assumption of noiseless data is often unrealistic.
Since its introduction by Aigner et al (1977) and Meeusen and Broeck (1977), the concept of technical efficiency has evolved significantly, with key contributions from researchers such as Pitt and Lee (1981), Schmidt and Sickles (1984), and Battese and Coelli (1988, 1992, 1995) This methodology has become a vital tool for assessing the performance of various production units, including firms, regions, and countries, as demonstrated in studies by Battese and Corra (1977), Bravo-Ureta and Rieger (1991), and Cullinane et al (2006) For a comprehensive overview of this literature, refer to Greene (2008).
Despite extensive literature on estimating technical efficiency, researchers often struggle to select the most suitable model for analysis Early models focused on cross-sectional data, requiring assumptions about the distribution of technical inefficiency and its independence from other model components Critics like Pitt and Lee (1981) and Schmidt and Sickles (1984) argued that these models could not consistently estimate technical inefficiency, leading to the development of panel data models Initially, these models assumed time-invariant technical inefficiency, as seen in the works of Battese & Coelli (1988) and others However, researchers later challenged this assumption, advocating for models that account for time-variation in inefficiency, such as those proposed by Cornwell et al (1990) and Kumbhakar (1990) Battese and Coelli (1995) introduced a model that allows technical inefficiency to vary over time and other factors, while Greene (2005) presented "true" fixed and random models that enable unrestricted time variation of inefficiency, distinguishing it from firm-specific factors.
This thesis evaluates the technical efficiency of Vietnamese metal manufacturing firms using panel-data stochastic frontier models It also reviews various panel data models for analyzing technical inefficiency and provides insights on model selection in this area The research utilizes an unbalanced panel dataset from the metal manufacturing industry for the year 2005.
2007 and 2009 which is withdrawn from Vietnamese SMEs survey The result shows different technical efficiency levels among those stochastic frontier models.
Research objectives
- To give a review of panel-data stochastic frontier models;
- To apply those models to investigate the technical efficiency of SME firms in metal manufacturing industry in Viet Nam.
LITERATURE REVIEW
Efficiency measurement
The primary economic function of a business involves transforming inputs into outputs, showcasing its production capability The productivity of a firm is measured by the output-to-input ratio, which indicates how effectively the production unit operates In economic terms, improvements in productivity are widely regarded as a key indicator of a firm's performance and efficiency.
The terms productivity and efficiency need to be discriminated in the context of firm production
Productivity encompasses all factors determining the efficiency of output derived from specific input levels, often referred to as Total Factor Productivity (TFP) In contrast, efficiency pertains to the production frontier, which represents the maximum achievable output for a given input level A firm is deemed technically efficient when it operates on this frontier; any production below this threshold indicates inefficiency, with greater distances from the frontier signifying higher inefficiency Variations in productivity may arise from changes in efficiency, adjustments in input amounts and proportions, advancements in technology, or a combination of these elements (Coelli et al., 2005).
Efficiency measurement can be evaluated through two primary approaches: input-oriented and output-oriented measures Input-oriented measures focus on minimizing the amount of inputs required to achieve a specific output, while output-oriented measures emphasize maximizing output from a given set of inputs For instance, an isoquant curve illustrates the minimum input combinations necessary for a certain output level, indicating technical efficiency when a firm operates on this frontier The iso-cost line helps determine the optimal input ratio for cost minimization Technical efficiency (TE) is calculated as the ratio of actual output to potential output, while allocative efficiency (AE) is determined by the ratio of optimal output to actual output The product of AE and TE yields the firm's overall economic efficiency (EE) Conversely, when analyzing a single input-output scenario, the f(X) curve represents the maximum output achievable with varying input levels, with technical efficiency also being assessed on this frontier.
Numerous studies have measured and analyzed technical efficiency (TE) using two primary approaches: Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA) The following section provides a concise overview of these two methodologies.
Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA)
Data Envelopment Analysis (DEA) is a non-parametric approach for assessing firm efficiency, initially proposed by Charnes, Cooper, and Rhodes in 1978, focusing on constant returns to scale This methodology was later enhanced by Banker, Charnes, and Cooper in 1984 to accommodate decreasing and variable returns to scale For detailed guidance on DEA, refer to the works of Banker et al (1984), Charnes et al (1978), and Fare, Grosskopf, and Lovell.
(1994), Fọre, Grosskopf, and Lovell (1985) and Ray (2004)
In a scenario involving n firms, known as Decision Making Units (DMUs), each firm utilizes m types of inputs to generate s types of outputs The Data Envelopment Analysis (DEA) model, focused on an output-oriented approach, is defined by the objective function: max ℎ 0 = ∑ ∑ 𝑠 𝑟=1 𝑢 𝑟 𝑦 𝑟0.
In this study, we analyze the efficiency of decision-making units (DMUs) by utilizing a mathematical framework where \( j = 1,2, \ldots, n \), \( r = 1,2, \ldots, s \), and \( i = 1,2, \ldots, m \) The variables \( x_{ij} \) and \( y_{rj} \) represent the ith input and rth output of the jth DMU, while \( u_r \) and \( v_i \) denote the weights assigned to outputs and inputs, derived from a maximization problem as outlined by Charnes et al (1978) By applying a piece-wise frontier approach from Farrell (1957) alongside linear programming techniques, we construct a production frontier to evaluate efficiency This methodology enables us to calculate the efficiency level of each firm by comparing the ratio of outputs to inputs against the established frontier.
Since its recognition in 1978, Data Envelopment Analysis (DEA) has emerged as a significant method for efficiency analysis across various industries in both the private and public sectors Wei (2001) outlined five key developments in DEA research, highlighting the increasing number and quality of numerical methods and computer programs available Over the years, new DEA models have been introduced, including the additive model, log-type DEA model, and stochastic DEA model Additionally, the economic and management foundations of DEA have been examined more thoroughly, enhancing its practical applications The contributions of mathematicians to the theoretical aspects of DEA have further propelled its advancement, leading to both theoretical improvements and empirical applications of this non-parametric approach.
Aigner et al (1977) and Meeusen and Broeck (1977) suggested the method of production stochastic frontier to measure firms’ efficiency The model can be described mathematically as below:
The output of firm \( i \) is represented by the equation \( Y_i = f(X_i, \beta)e^{v_i} - u_i \), where \( X_i \) denotes the vector of inputs and \( \beta \) represents the parameters to be estimated In this model, \( v_i \) and \( u_i \) are two critical error terms; \( v_i \) accounts for random statistical noise, assumed to follow a normal distribution with a zero mean, while \( u_i \) reflects inefficiency, indicating how far a firm is from optimal production Various distributions have been proposed for \( u_i \), including half-normal, exponential, gamma, and a non-negative truncation of the normal distribution This highlights the trade-off between Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA) in evaluating firm performance.
DEA and SFA, while both methods for efficiency analysis, differ significantly in their approaches, each offering unique advantages and disadvantages DEA, a non-parametric method, operates deterministically without specifying a production function, assuming data is free from noise, which can be a limitation in real-world scenarios where measurement errors and random factors exist In contrast, SFA employs a stochastic approach using parametric tools, accommodating statistical noise and allowing for greater flexibility with imperfect data However, SFA requires careful model specification, including assumptions about functional forms and error distributions Additionally, DEA considers all factors contributing to a firm's inefficiency, while SFA differentiates between noise and inefficiency, leading to generally higher efficiency estimates Researchers must carefully weigh these trade-offs when selecting between DEA and SFA for their analyses.
Data Envelopment Analysis (DEA) offers significant advantages in complex production environments, as it does not require a specific production function, thereby simplifying the relationship between inputs and outputs However, DEA lacks statistical properties, making it impossible to assess its goodness of fit In contrast, Stochastic Frontier Analysis (SFA) provides econometric tools to evaluate model suitability and effectively addresses statistical noise DEA is preferable in tightly controlled industries with stable production processes, where output is consistently determined by input Conversely, SFA is more appropriate for industries like metal manufacturing, where firms face inevitable random fluctuations from both domestic and international markets, as well as policy changes Given these industry characteristics, SFA emerges as the superior model for analysis, which will be further explored in the subsequent section on SFA methodologies, including both cross-sectional and panel data models.
The cross-sectional Stochastic Frontier Model
The cross-sectional stochastic frontier model in Aigner et al (1977) can be describe as:
In the equation \( y_i = f(x_i, \beta) + v_i - u_i \), \( v_i \) represents random noise, while \( u_i \) indicates technical inefficiency, constrained to be non-negative (\( u_i \geq 0 \)) To effectively differentiate between these two residual components, specific assumptions regarding their distributions are required Notably, \( v_i \) is assumed to follow a symmetric normal distribution, whereas \( u_i \) reflects the distance from the production frontier, necessitating a non-negative value Suggested distributions for \( u_i \) include half-normal (Aigner et al., 1977), exponential (Meeusen & Broeck, 1977), gamma (Greene, 1990), or a non-negative truncation of the normal distribution \( N(m_{it}, \sigma^2) \) (Battese & Coelli, 1988, 1992, 1995).
Both Ordinary Least Squares (OLS) and Maximum Likelihood (ML) methods are employed to calculate technical inefficiency in econometrics However, the error term 𝜀, comprising an asymmetric component 𝑢 and a symmetric component 𝑣, leads to a non-normal distribution of 𝜀, resulting in a downward bias in the OLS intercept In a simple regression model, the estimator for 𝜀, represented as 𝑦̅, does not equal the true intercept 𝛼 To address this bias, Winsten (1957) proposed Corrected Ordinary Least Squares (COLS), while Afriat (1972) and Richmond (1974) introduced Modified Ordinary Least Squares (MOLS), both of which adjust the intercept upward by incorporating the maximum or average values of OLS residuals Despite their utility, COLS and MOLS face challenges, including non-statistical significance in estimates In contrast, ML is favored for its asymptotic properties and capability to handle asymmetrically distributed residuals, making it a more reliable choice than OLS.
With technical inefficiency (𝑢) following a half-normal distribution i.e 𝑢~𝑁 + (0, 𝜎 𝑢 2 ) (Aigner et al., 1977) the log-likelihood function is:
The equation \(2\sigma^2 \sum_{i=1}^{I} \epsilon_i^2\) defines the relationship between efficiency and inefficiency in a firm, where \(\sigma^2 = \sigma_v^2 + \sigma_u^2\) and \(\lambda = \frac{\sigma_u^2}{\sigma_v^2}\) A value of \(\lambda = 0\) indicates full efficiency, while \(\lambda = 1\) signifies total inefficiency In this context, \(y\) represents a vector of logarithmic outputs, and \(\epsilon_i\) is calculated as \(v_i - u_i = \ln q_i - x_i' \beta\) Additionally, \(\Phi(x)\) denotes the cumulative distribution function of a random variable following a normal distribution \(N(0,1)\) at \(x\) (Coelli et al., 2005) The iterative optimization procedure outlined by Judge et al (1982) is utilized for solving this function Furthermore, the log-likelihood function with an exponential distribution is expressed as \( \ln L(Y|\alpha, \beta, \lambda, \sigma^2) = -N(\ln \sigma_u + \sigma_v^2)\).
The case of truncated normal distribution i.e 𝑢~𝑁 + (𝜇 𝑢, 𝜎 𝑢 2 ) has the log-likelihood function: ln 𝐿(𝑌|𝛼, 𝛽, 𝜆, 𝜎 2 ) = −𝑁
Log likelihood function for Stochastic Frontier Model with gamma distribution of u can be found in Greene (1990): ln 𝐿 = ∑ (𝑃𝑙𝑛Θ − 𝑙𝑛Γ(𝑃) + Θ 2 𝜎 2
𝑖 ] + ln ℎ[𝑃 − 1, 𝜀 𝑖 ]) (2.3.4) with Θ and P are two parameters of a gamma distribution
Figure 2 – 3: Distribution of technical inefficiency
Figure (2 - 3) displays the probability density functions for four types of distributions of 𝑢 The gamma, exponential, and half-normal distributions exhibit certain limitations, as they tend to cluster most observations in the low-value area of 𝑢, indicating a high level of firm efficiency and low technical inefficiency However, this assumption may not hold true across various industries where high efficiency is not feasible In contrast, the truncated normal distribution offers greater flexibility by allowing inefficiency to be allocated across a broader range of positive values, making it a more accurate representation of 𝑢.
Consider a Cobb-Douglas production function as: ln 𝑌 𝑖 = 𝛽 ln 𝑋 𝑖 + 𝑣 𝑖 − 𝑢 𝑖 (2.3.5)
The technical inefficiency of a firm is determined by the ratio of its observed output (𝑌 𝑖) to the maximum feasible output (𝑌 ∗), which represents the output achievable under full efficiency, where the value of 𝑢 𝑖 is zero.
With 𝑢 𝑖 follows the 𝑁(𝑚 𝑖, 𝜎 2 ) distribution in Battese and Coelli (1995), the parameter for inefficiency (or, in other words, also efficiency) can be analyzed with some determinants with the regression equation follow:
The Technical Inefficiency Model, as proposed by Battese & Coelli (1995), involves estimating the vector of parameters 𝛿 alongside the determinants vector 𝑍 𝑖 The distribution of 𝜔 𝑖 is characterized as a truncation of the normal distribution 𝑁(0, 𝜎 2), allowing for simultaneous estimation with the Stochastic Frontier.
Stochastic frontier model with panel data
When utilizing the stochastic frontier model with cross-sectional data, three key issues emerge, as highlighted by Schmidt and Sickles (1984) The primary concern is the inconsistency in estimating technical inefficiency Many studies rely on the methodology established by Jondrow, Lovell, Materov, and Schmidt (1982) to assess the technical inefficiency levels of individual firms within the sample.
The estimation of technical inefficiency involves using the unknown parameters 𝜇 ∗ and 𝜎 ∗, represented by their estimators 𝜇̂ ∗ and 𝜎̂ ∗, which introduces sampling bias that is complex to account for but diminishes with larger samples Despite this, the estimation of technical inefficiency remains inconsistent regardless of sample size, as noted by Schmidt and Sickles (1984) Additionally, the ambiguity in the distribution of the error term 𝑢 complicates the separation of technical inefficiency from statistical noise 𝑣, making it difficult to validate assumptions with cross-sectional data Furthermore, the assumption that 𝑢 is uncorrelated with other regressors leads to endogeneity issues, resulting in biases within the model; Schmidt and Sickles (1984) argue that this endogeneity is inevitable, as firms eventually recognize their inefficiency and adjust input usage accordingly.
Panel data models, which utilize data from multiple firms over several time periods, effectively address key limitations in traditional analyses (Greene, 2008) Firstly, the increased observations over time enhance the consistency of technical inefficiency estimates, particularly when time series data is extensive Secondly, by treating technical inefficiency as a fixed effect, these models become distribution-free, making distribution assumptions optional (Greene, 2008) Lastly, the assumption of uncorrelatedness is relaxed, allowing certain panel models to incorporate correlation effects The following section will delve into the development of panel data stochastic frontier models, tracing their evolution since their inception.
4.1 Time-invariant models a Within estimation with fixed effects and GLS estimation with random effects from
Schmidt and Sickles (1984) propose utilizing panel data to assess time-invariant technical inefficiency through both fixed and random effects models The mathematical representation of this model is given by the equation: ln 𝑦 𝑖𝑡 = 𝛼 + 𝑋 𝑖𝑡 ′ 𝛽 + 𝑣 𝑖𝑡 − 𝑢 𝑖.
Schmidt and Sickles (1984) employ a log-linear function where the error term 𝑣 𝑖𝑡 is uncorrelated with 𝑋 𝑖𝑡 ′ 𝛽 and 𝑢 𝑖 Their within estimator utilizes dummy variables to derive separate intercepts for each firm, reflecting its unique technical inefficiency This approach offers advantages by eliminating the need for assumptions regarding the uncorrelatedness of 𝑢 with other variables and the distribution of 𝑢 After computation, each firm's effect is compared to the highest in the sample, estimating inefficiency as 𝑢̂ 𝑖 = max(𝛼̂ 𝑖 ) − 𝛼̂ 𝑖 The authors advocate for a large sample of firms to accurately identify the most efficient firm, ideally with a substantial number of firms across multiple time periods However, since this method functions as a fixed effects estimation using panel data, it inadvertently incorporates time-invariant but firm-specific effects, such as capital stock, which may not accurately represent inefficiency.
The authors address the limitations of the within estimator by proposing assumptions regarding the uncorrelatedness of \( u \) and \( X_{it}' \beta \), enabling the application of Generalized Least Squares (GLS) estimation for improved \( u \) estimation This method's advantage lies in its ability to differentiate time-invariant regressors, a capability the within estimator lacks However, the validity of the uncorrelatedness assumption warrants testing To tackle the issues of uncorrelatedness and distribution assumptions, the authors recommend two alternative methods: the Hausman and Taylor (1981) estimation, which relaxes the uncorrelatedness assumption, and maximum likelihood estimation, which is more sophisticated and tailored to specific distributions of \( u \).
The two models discussed represent some of the simplest approaches to understanding technical efficiency They highlight the challenges of accurately estimating technical inefficiency levels and recommend using fixed and random effects models for more consistent estimates, particularly when the time period (T) is large relative to the number of firms (N) However, the absence of a specific distribution complicates the estimation of true inefficiency, as it intertwines with firm-specific factors Pitt and Lee (1981) introduced a half-normal distribution with maximum likelihood estimation, which is further elaborated in the following section, focusing on a model with time-invariant efficiency.
This study analyzes panel data from the Indonesian weaving industry to assess the levels and sources of technical inefficiency The research tests the hypothesis of whether technical inefficiency remains constant over time or varies, utilizing three distinct models The authors propose three scenarios, with the first case indicating that inefficiency is time-invariant, fluctuating only among individuals and represented by the index 𝑢 𝑖.
*Note: Pitt and Lee (1981) use a linear function
In the second case, technical inefficiency is independent of time and among individuals, which leads back to the cross-sectional model as in Aigner et al (1977) That is:
With 𝐸(𝑢 𝑖𝑡 𝑢 𝑖𝑡 ′ ) = 0 and 𝐸(𝑢 𝑖𝑡 𝑢 𝑗𝑡 ′ ) = 0 for all 𝑖 ≠ 𝑗 and 𝑡 ≠ 𝑡′
The final case is the intermediate of these two when the technical inefficiency is assumed to be correlated with time That is:
With 𝐸(𝑢 𝑖𝑡 𝑢 𝑖𝑡 ′ ) ≠ 0 and 𝐸(𝑢 𝑖𝑡 𝑢 𝑗𝑡 ′ ) = 0 for all 𝑖 ≠ 𝑗 and 𝑡 ≠ 𝑡′
The first and second models are estimated using the maximum likelihood method, while the intermediate model employs generalized least squares due to the intractability of the maximum likelihood procedure in this case A comparison between the first two models and the third model is conducted using a χ² test, as detailed by Joreskog and Goldberger (1972), which indicates that the third model is appropriate, suggesting that technical inefficiency varies over time Although the paper does not specify the measure of technical inefficiency for each firm, it can be determined using the method proposed by Jondrow et al (1982), which infers the value of each uᵢ from the corresponding εᵢ values.
The latest model, while demonstrating greater precision, overlooks the distribution of technical inefficiency and fails to provide a measure of inefficiency Pitt and Lee (1981) proposed a model characterized by time-invariant inefficiency following a half-normal distribution, advocating for further exploration into time-varying inefficiency However, this half-normal distribution may not always be reasonable To address this, Battese and Coelli (1988) introduced a more flexible approach using a truncated normal distribution The details of this model will be elaborated in the following section.
Battese and Coelli (1988) introduce a model where technical inefficiency is represented by a truncated normal distribution, building on Stevenson’s (1980) work in stochastic production frontier estimation Given the three-year data availability, they assume time-invariant inefficiency Their new distribution, denoted as 𝑁 + (𝜇, 𝜎 𝑢 2), is more comprehensive than previous models, such as the half-normal distribution proposed by Pitt and Lee (1981) and Schmidt and Sickles (1984).
𝜇 = 0, the distribution becomes half normal With development in calculating the likelihood function from Stevenson (1980), the model is estimated with maximum likelihood method The model can be described as: ln 𝑦 𝑖𝑡 = 𝛼 + 𝛽𝑙𝑛𝑥 𝑖𝑡 + 𝑣 𝑖𝑡 − 𝑢 𝑖 (2.4.6)
*Note: Battese and Coelli (1988) use a Cobb-Douglas function
This paper significantly contributes to the field of stochastic frontier analysis by estimating technical efficiency at both the industry and firm levels using the logarithmic form, specifically the Cobb-Douglas functional form Rather than relying on the mean of technical inefficiency, E(u), and calculating efficiency as 1 - E(u) as done by Jondrow et al (1982), the authors propose that the technical efficiency level should be expressed as exp(-u) in the logarithmic case Additionally, the paper clarifies the formula for technical efficiency by discussing the properties of the truncated distribution of u.
Research into the time-varying characteristics of technical inefficiency is a critical area for further exploration, as highlighted by various studies Pitt and Lee (1981) advocate for examining time-varying technical inefficiency based on their empirical findings, while Schmidt and Sickles (1984) assert that firms will ultimately recognize and adapt to their inefficiency levels over the long term Despite Battese and Coelli's (1988) assumption of fixed inefficiency due to limited long-term data, they propose alternative models that accommodate variations in inefficiency over time This shift towards models incorporating time-varying inefficiency addresses the limitations of the traditional assumption of time-invariant inefficiency, paving the way for more dynamic approaches in future research.
4.2 The time varying models a The model of Cornwell et al (1990)
Research on technical inefficiency often varies in how it addresses the distribution of inefficiencies and their correlation with inputs When assumptions about uncorrelatedness and distribution are established, they can become rigid and limit the study's effectiveness While panel data can alleviate these assumptions, it also requires treating technical inefficiency as constant over time, which is a significant limitation (Cornwell et al., 1990) To overcome the constraint of time-invariant efficiency, Cornwell et al propose a model that allows firm effects to vary over time, thereby introducing flexibility into the analysis This model is represented by the equation: ln 𝑦 𝑖𝑡 = 𝛼 𝑖𝑡 + 𝛽 ln 𝑋 𝑖𝑡 + 𝑣 𝑖𝑡 (2.4.7).
Cornwell et al (1990) utilize a Cobb-Douglas function that incorporates time-varying firm effects, represented by 𝛼 𝑖𝑡, which evolve over time The authors propose a quadratic function of time, where the parameters differ among firms, highlighting the dynamic nature of these effects in their analysis.
METHODOLOGY
Overview of Vietnamese metal manufacturing industry
This study categorizes firms into two types: basic metal manufacturing and fabricated metal manufacturing (excluding machinery and equipment), as defined by the International Standard Industrial Classification (ISIC), Revision 4 The first group focuses on smelting and refining ferrous and non-ferrous metals, utilizing metallurgic techniques with materials sourced from the mining industry, such as metal ore, pig iron, or scrap This sector requires significant investments in physical assets, representing only 9% of the small and medium enterprises in the sample In contrast, the second group, which comprises 91% of the firms, produces structural metal products, metal containers, and steam generators, catering to more widely demanded products.
The metal manufacturing industry in Vietnam holds significant potential due to the high demand for metal products in daily use, production, and construction Despite being a young and developing economy, the industry remains immature, with approximately 80% of iron and steel materials utilized for construction, as reported by the World Steel Association Additionally, the growing domestic demand for metal materials in machinery, automotive, and consumer goods manufacturing presents opportunities for growth However, the industry faces challenges due to the recent economic downturn, which has led to reduced public construction investments and a subsequent decline in demand for metal materials The Vietnamese Steel Association reported a nine percent drop in steel consumption in 2012, further exacerbated by rising input costs such as electricity, water, and labor.
This study aims to analyze the technical efficiency of firms in the crucial metal manufacturing industry Notably, a significant portion of the sample consists of micro and household firms, accounting for 74.5%, while medium-sized firms represent only 4% The analysis is based on data collected during a specific period, highlighting the industry's composition and the need for efficiency evaluation.
Between 2005 and 2009, the economic landscape differed significantly from today's conditions, which raises concerns about potential sample bias in this study's conclusions regarding the industry Therefore, it is essential for readers to approach the findings with caution, particularly when considering their implications for policy recommendations.
Analytical framework
In this study, we apply panel-data stochastic frontier models to assess the technical inefficiency levels of Vietnamese SMEs in the metal manufacturing sector The production function is estimated using inputs and outputs data, employing two distinct functional forms: Cobb-Douglas and Translog This analysis is grounded in the technical inefficiency effects model developed by Battese and Coelli.
In 1995, a set of firm-specific variables was incorporated into the model to serve as sources or determinants of technical inefficiency The findings were subsequently analyzed across different models to assess how each assumption and model specification influenced the determination of technical efficiency.
Research method
Firm efficiency will be assessed using the Stochastic Frontier Model, which requires careful selection of an appropriate functional form to establish the production frontier Various production functions can be utilized, including Linear, Cobb-Douglas, Quadratic, Normalized Quadratic, Translog, Generalized Leontief, and Constant Elasticity of Substitution (CES) forms, as reviewed by Griffin, Montgomery, and Rister (1987) According to Coelli et al (2005), an effective functional form should be flexible, linear in parameters, regular, and parsimonious A flexible functional form of ith-order possesses sufficient parameters to provide an accurate differential approximation.
The Linear and Cobb-Douglas production functions are classified as first-order flexible, while other forms are considered second-order flexible Many of the production functions discussed are linear in their parameters Notably, both the Cobb-Douglas and Translog functions can be transformed into linear forms by applying logarithms to both sides of their equations.
A regular functional form adheres to the economic regularity properties of production functions, either inherently or through basic constraints In contrast, a parsimonious function is defined as the simplest form that effectively addresses the problem at hand Flexible functions tend to impose fewer assumptions or restrictions on the production function's properties, while parsimonious functions preserve degrees of freedom Researchers must navigate the trade-off between flexibility and parsimony when selecting the appropriate functional form for their analysis.
Researchers often prefer the Cobb-Douglas function for its simplicity and ease of use, while the Translog function is favored for its greater flexibility The Cobb-Douglas model features constant production elasticity and a fixed elasticity of factor substitution, which limits its adaptability compared to the Translog model, which allows for testable properties of the production function This flexibility makes the Translog model more realistic and less restrictive; however, it has its drawbacks The inclusion of cross and squared terms in the Translog model increases the number of parameters, which can lead to potential correlation issues among them Additionally, a limited number of observations can reduce the degree of freedom due to the increased parameters For comparison, both the Cobb-Douglas and Translog functions can be specified with three inputs: capital (K), labor (L), material (M), and indirect cost (I).
Cobb-Douglas functional form: ln 𝑌 𝑖𝑡 = 𝛽 0 + 𝛽 1 ln 𝐾 𝑖𝑡 + 𝛽 2 ln 𝐿 𝑖𝑡 + 𝛽 3 ln 𝑀 𝑖𝑡 + 𝛽 4 𝑙𝑛𝐼 𝑖𝑡 + 𝑉 𝑖𝑡 − 𝑈 𝑖𝑡 (3.1) Translog functional form: ln 𝑌 𝑖𝑡 = 𝛽 0 + 𝛽 1 ln 𝐾 𝑖𝑡 + 𝛽 2 ln 𝐿 𝑖𝑡 + 𝛽 3 ln 𝑀 𝑖𝑡 + 𝛽 4 𝐼 𝑖𝑡 + 1
𝛽 6 ln 𝐾 𝑖𝑡 ln 𝑀 𝑖𝑡 + 𝛽 7 ln 𝐾 𝑖𝑡 ln 𝐼 𝑖𝑡 + 𝛽 8 ln 𝐿 𝑖𝑡 ln 𝑀 𝑖𝑡 + 𝛽 9 ln 𝐿 𝑖𝑡 ln 𝐼 𝑖𝑡 +
𝛽 10 ln 𝑀 𝑖𝑡 ln 𝐼 𝑖𝑡 + 𝛽 11 (ln 𝐾 𝑖𝑡 ) 2 + 𝛽 12 (ln 𝐿 𝑖𝑡 ) 2 + 𝛽 13 (ln 𝑀 𝑖𝑡 ) 2 + 𝛽 14 (ln 𝐼 𝑖𝑡 ) 2 ] +
In this analysis, we denote firms with the index \( i \) and time periods with \( t \) The output of firm \( i \) at time \( t \) is represented by \( Y_{it} \), while capital input is denoted as \( K_{it} \) and labor input as \( L_{it} \) Additionally, materials are represented by \( M_{it} \) and indirect costs by \( I_{it} \) The term \( V_{it} \) accounts for statistical noise, which is modeled as \( N(0, \sigma_v^2) \) Finally, \( U_{it} \) represents technical efficiency (TE), following a specific non-negative distribution.
The Translog function simplifies to the Cobb-Douglas function in the absence of squared and interaction terms The Cobb-Douglas form is characterized by constant proportionate returns to scale, a consistent elasticity of factor substitution, and the assumption that all input pairs are complementary However, these assumptions render the Cobb-Douglas function more restrictive.
A likelihood ratio test (LR) can be used to test for goodness of fit between these two functional forms with:
𝐻 1 : otherwise where 𝐿(𝐻 0 ) is the log-likelihood value for null model (𝐻 0 ) and 𝐿(𝐻 1 ) is the log-likelihood value for alternative model (𝐻 1 ), the test is given by:
The test statistic is approximately followed a chi-squared distribution with the degree of freedom (df) is the difference between df of null model and df of alternative model
This thesis utilizes two computer programs, STATA and FRONTIER 4.1, to estimate Stochastic Frontier Models Since the introduction of the model, STATA has significantly advanced its commands for estimating these models, particularly with the "frontier" command designed for cross-sectional data.
“xtfrontier” for panel data are those popular STATA commands to estimate technical efficiency
The "frontier" command in STATA can handle models with half normal, truncated normal, exponential, or gamma distributions, while "xtfrontier" addresses both time-invariant and time-varying models as proposed by Battese and Coelli in 1988 and 1992, respectively However, users often encounter difficulties when testing additional models with just these commands Fortunately, the recent work of Belotti, Daidone, Ilardi, and Atella (2012) has introduced the "sfcross" and "sfpanel" commands, expanding STATA's capabilities for utilizing these models with straightforward syntax Additionally, FRONTIER 4.1 from Coelli (1996) supports models from Battese and Coelli's 1992 and 1995 studies.
In the analysis of logarithmic functional forms, specifically Cobb-Douglas and Translog, technical efficiency (TE) is determined using two distinct methods The first method, proposed by Jondrow et al (1982), calculates TE with the formula TE = exp[−E(u|ε)] The second method, suggested by Battese & Coelli (1988), utilizes the formula TE = E[exp(−u)|ε] The resulting TE value will always be a positive number less than one, representing the ratio of actual output to the maximum potential output achievable without inefficiency.
It means TE is a comparison between the output of a real firm and the output of an efficient firm
The estimation process for time-invariant fixed and random effects models, as outlined by Schmidt and Sickles (1984), parallels the approach used in panel regressions with fixed and random effects These models are estimated using least squares methods, specifically the within estimator and generalized least squares (GLS) After estimation, each firm's effect is assessed against the highest effect in the sample, allowing for the calculation of inefficiency as 𝑢̂ 𝑖 = max(𝛼̂ 𝑖 ) − 𝛼̂ 𝑖 This model operates under the premise that there is always one efficient firm in the sample, which is assigned a technical efficiency level of 100% (𝑢 = 0) This methodology is consistent with other models, such as those developed by Cornwell et al., which are also estimated using least squares methods.
The fixed effects models discussed by Cornwell et al (1990) and Lee and Schmidt (1993) involve numerous parameters, leading to biases due to coincident parameter situations Specifically, Cornwell et al.'s model requires \(N \times 3\) parameters for its time function of technical inefficiency, making it unsuitable for datasets with only three time periods Additionally, the command developed by Belotti et al (2012) for Lee and Schmidt's model inaccurately assesses firms' technical efficiency levels, as it compares these levels to the highest from all years instead of the highest for each individual year Consequently, the technical efficiency levels derived from this model will not be included in our results.
The models developed by Pitt and Lee (1981) and Battese and Coelli (1988) are both estimated using the maximum likelihood method, differing primarily in their distribution assumptions Pitt and Lee utilize a half-normal distribution for the error term 𝑢, whereas Battese and Coelli adopt a truncated normal distribution, introducing an additional parameter, 𝜇, which represents the mean of the normal distribution for 𝑢 The likelihood functions for both models are detailed in their original publications, with the Pitt and Lee model effectively serving as a special case of the Battese and Coelli model when 𝜇 is set to zero.
The models developed by Kumbhakar (1990) and Battese & Coelli (1992) exhibit similar characteristics, as both utilize the maximum likelihood method for estimation and consider technical inefficiency as a time-dependent variable Kumbhakar's model defines the time function as \( u_{it} = \gamma(t)u_i \), with \( \gamma(t) = (1 + \exp(bt + ct^2))^{-1} \), where \( u_i \) remains constant over time but varies among firms and follows a half-normal distribution In contrast, Battese & Coelli's approach employs the function \( u_{it} = \eta_{it}u_i \), where \( \eta_{it} = \exp[-\eta(t - T)] \).
The truncated normal distribution at zero, represented as \( u \sim N(\mu, \sigma_u^2) \), allows the data to determine the temporal behavior of \( u \) While the model proposed by Battese and Coelli (1992) offers simpler calculations, Kumbhakar's (1990) approach provides greater flexibility in illustrating the dynamics of technical inefficiency.
The technical inefficiency effects model proposed by Battese and Coelli (1995) incorporates variables such as firm age, size, ownership type, and location, highlighting the significant impact of age on technical inefficiency while also considering the influence of time, a focus of many prior studies To avoid biases associated with two-step estimations, the Stochastic Frontier Model and the technical inefficiency effects model are estimated concurrently Greene (2005) presents "true" random and fixed effects models with an exponential distribution of technical inefficiency The authors advocate for a "brute force" approach, applying the maximum likelihood method to simultaneously estimate all parameters, including constant terms, in the "true" fixed effects model, with the likelihood function detailed in the original research.