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A nonparametric methodology for evaluating convergence in a multi-input multi-output setting
Accepted Manuscript
A nonparametric methodology for evaluating convergence in a multi-input multi-output setting I.M. Horta, A.S. Camanho PII: DOI: Reference:
S0377-2217(15)00387-2 10.1016/j.ejor.2015.05.015 EOR 12939
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
25 May 2014 12 December 2014 4 May 2015
Please cite this article as: I.M. Horta, A.S. Camanho, A nonparametric methodology for evaluating convergence in a multi-input multi-output setting, European Journal of Operational Research (2015), doi: 10.1016/j.ejor.2015.05.015
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ACCEPTED MANUSCRIPT Highlights The paper presents a novel nonparametric methodology to evaluate convergence We develop two new indexes to evaluate α-convergence and β-convergence The indexes developed allow evaluations using multiple inputs and outputs The methodology complements productivity assessments based on the Malmquist index The methodology is applied to Portuguese construction companies operating in 2008-2010
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• • • • •
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A nonparametric methodology for evaluating convergence in a multi-input multi-output setting I.M. Horta a,∗ , A.S. Camanho a de Engenharia, Universidade do Porto. Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
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a Faculdade
Abstract
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This paper presents a novel nonparametric methodology to evaluate convergence in an industry, considering a multi-input multi-output setting for the assessment of total factor productivity. In particular, we develop two new indexes to evaluate σ-convergence and β-convergence that can be computed using nonparametric techniques such as Data Envelopment Analysis. The methodology developed is particularly useful to enhance productivity assessments based on the Malmquist index. The methodology is applied to a real world context, consisting of a sample of Portuguese construction companies that operated in the sector between 2008 and 2010. The empirical results show that Portuguese companies tended to converge, both in the sense of σ and β, in all construction activity segments in the aftermath of the financial crisis.
Introduction
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Key words: Convergence, Productivity, Malmquist index, Data Envelopment Analysis, Construction Industry
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Convergence has been extensively discussed in the economic growth literature over the past decades (see Temple (1999) and Islam (2003) for a literature review). The first studies devoted their attention to whether poor or low productive countries would catch up with their rich or highly productive counterparts. This is an issue of utmost importance for researchers and policymakers interested in worldwide welfare. Two main concepts of convergence appear in the classical literature (see Barro and Sala-i Martin (1992)). β-convergence analyzes if poor countries tend to grow faster than rich countries, whereas σ-convergence examines if the dispersion of the productivity for a group of countries tends to decrease over time. To measure these concepts, most studies use a single productivity measure, such as income per capita or Gross Domestic Product (GDP) per capita, or a measure of total factor productivity estimated using econometric methods. The evaluation of convergence in a multi-input multi-output setting has not been addressed in the literature. ∗ Corresponding author. Tel: +351 225081639. E-mail address: [email protected]
ACCEPTED MANUSCRIPT This research contributes to the literature by developing a nonparametric methodology for the evaluation of convergence in a multi-input multi-output setting, to enhance productivity assessments based on the Malmquist index. In particular, we explore the use of Shephard distance functions, estimated through Data Envelopment Analysis (DEA), to calculate β-convergence and σ-convergence. In addition, this paper applies the two new indexes to evaluate convergence in a sample of Portuguese construction companies that operated in the sector between 2008 and 2010.
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The remainder of this paper is organized as follows. Section 2 describes the new nonparametric indexes to evaluate σ-convergence and β-convergence, and explains their computation using DEA. Section 3 presents the empirical application, including the motivation, the description of the data set and the discussion of the results. The last section concludes and points topics for future research.
Evaluation of convergence using nonparametric techniques
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2.1 A new nonparametric σ-convergence index
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The concept of σ-convergence can be explained as follows: “a group of economies are converging in the sense of σ if the dispersion of the their real per capita GDP levels tends to decrease over time” (Sala-i Martin, 1996). The notion of σ-convergence can be expressed, in mathematical terms, as shown in expression (1). (1)
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σt+1 < σt
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σt is the standard deviation of the logarithm of the real per capita GDP levels across all economies in period t, whereas σt+1 is a similar measure in a subsequent period. This definition can be extended to a more general setting, consisting of the assessment of Decision Making Units (DMUs) (e.g., representing countries or organizations), where σ-convergence explores whether the dispersion of values of a productivity indicator, measured as an output to input ratio, tends to decrease (or increase) over time.
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Our aim is to generalize the measurement of convergence to a multi-input multi-output setting, and thus we explore σ-convergence using the Shephard distance function. The Shepard distance function is a generalization of the production function that allows us to consider simultaneously multiple inputs and outputs. As this paper is devoted to output oriented assessments, all indexes and measures described have an output orientation. m s Consider n DMUs in time period t that use inputs xt ∈ R+ to produce outputs y t ∈ R+ , and in t+1 m t+1 s a subsequent time period t + 1 use inputs x ∈ R+ to produce outputs y ∈ R+ . In period t, the production technology T t consists of the set of all feasible input/output combinations for a certain production process, as shown in (2).
T t = {(xt , y t ) : xt can produce y t } 3
(2)
ACCEPTED MANUSCRIPT Following Shepard (1970), the output distance function for DMU jo in relation to the technology T t is defined as shown in (3). (
yt Dt (xt , y t ) = min θ : xt , θ
∈ Tt
)
(3)
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This function is defined as the reciprocal to the maximum proportional expansion of the output vector y t , given inputs xt , i.e., Dt (xt , y t ) ≤ 1. This means that it corresponds to the efficiency score of DMU jo , in the sense of Farrell (1957). Thus, in order to estimate σ-convergence we can use efficiency measures. In other words, the spread of productivity levels can be estimated using the spread of efficiency measures. Efficiency is a relative measure that compares the productivity of a DMU jo with the best productivity levels of the sample. Therefore, a sample with larger dispersion of productivity levels will also have larger dispersion of efficiency. Fare et al. (1992) were the first to note that input and output distance functions could be estimated using DEA models (Charnes et al., 1978).
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For a given DMU jo , the ratio between the efficiency score in period t + 1 and in t, as presented in (4) is a measure of convergence towards the best practice frontier (see Fare et al. (1994)). This implies that there is convergence to the frontier if efficiency increases from period t to t + 1. Note that this ratio corresponds to the Efficiency Change (EC t,t+1 ) component of the Malmquist index proposed by Fare et al. (1992).
(4)
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Dt+1 (xt+1 , y t+1 ) = EC t,t+1 Dt (xt , y t )
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To determine σ-convergence in an industry, we propose calculating the geometric mean of expression (4) for all DMUs in the sample, as shown in (5). To distinguish our measure of σ-convergence, defined using Shepard distance functions and calculated using DEA models, from the traditional σ-convergence measure (see expression 1), we refer to it as σ ˆ -convergence hereafter.
σ ˆ -convergence =
n Y
j=1
1/n
ECjt,t+1
(5)
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The σ ˆ -convergence index may be greater, equal or smaller than one. A σ ˆ -convergence index greater than one indicates convergence (i.e. the DMUs moved closer to the best practice frontier from period t to t + 1), whereas a score less than one means divergence (i.e. the DMUs moved away from the best practice frontier between t to t + 1). A score equal to one indicates that, on average, the DMUs are located at a similar distance to the frontier in period t and t + 1. The advantage of using σ ˆ -convergence over the traditional measure of σ-convergence is that it allows accounting for multiple inputs and multiple outputs, as it can be estimated using nonparametric techniques such as DEA. The basic ideas behind the calculation of the σ ˆ -convergence can be illustrated in Figure 1. This figure presents 10 DMUs (e.g., countries), whose activity is represented by an output (e.g., GDP) and an input (e.g., population) in two time periods. The ratio y/x represents a 4
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productivity measure. The best practice frontiers (BF) at time periods t + 1 and t are also plotted in Figure 1.
Fig. 1. Illustrative example one input-output
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Consider DMU H, represented by point H t in time period t and point H t+1 in time period t + 1. In the example, the efficiency score of DMU H in period t measured in relation to the best practice in t, corresponds to the ratio oH t /oa. The efficiency score of DMU H in period t + 1, measured in relation to the best practice frontier in period t + 1, corresponds to the ratio o0 H t+1 /o0 d. Hence, the change in efficiency between period t and t + 1 is measured as shown in expression (6).
EC t,t+1 =
o0 H t+1 /o0 d oH t /oa
(6)
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The value of EC t,t+1 for DMU H is greater than one (i.e., 1.09), as DMU H is closer to the best practice frontier in t + 1 than in t. This indicates convergence towards the best practice frontier.
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Table 1 presents the original data set used in Figure 1, the output distance function and the productivity for each DMU in each time period. Recall that in a single input-output setting, the efficiency measure is the ratio between the observed productivity of DMU jo and the maximum productivity observed in other DMUs in the same time period. Thus, if maximum productivity is equal to one (i.e, the maximum ratio of output to input is one), the two measures would coincide. In the example shown in Table 1, the maximum productivity in period t was observed in DMU C, whose output to input ratio is equal to one, and thus productivity and efficiency coincide (see fourth and fifth columns of Table 1). Furthermore, if the maximum productivity remains equal to one in periods t and t + 1 (i.e., the technology does not change between period t and t + 1), the output distance functions (or efficiency scores) and the productivity ratios would be identical in the two periods. 5
ACCEPTED MANUSCRIPT Table 1 Data and σ ˆ -convergence for the DMUs considered in the illustrative example DMU
yt
xt
Productivity
Dt (xt , yt ) (∗
(yt /xt ) At
0.5
2
0.25
yt+1
DMU
xt+1
Productivity (yt+1 /xt+1 )
)
0.25
At+1
3 3.5 4
2
Dt+1 (xt+1 , yt+1 ) ( ∗∗
ECt,t+1
)
1.50
1.00
4.00
4
0.88
0.58
0.93
3
1.33
0.89
0.89
Bt
2.5
4
0.63
0.63
Bt+1
Ct
3
3
1.00
1.00
Ct+1
6
5
1.20
0.80
1.00
5.5
4
1.38
0.92
1.10
4
5
0.80
0.80
Dt+1
Et
5
6
0.83
0.83
Et+1
Ft
2
3
0.67
0.67
Ft+1
5
4.5
1.11
0.74
1.11
Gt
2
4.5
0.44
0.44
Gt+1
1.8
1.5
1.20
0.80
1.80
1.14
0.76
1.09
1.13
0.75
1.38
0.91
0.61
0.64
Ht
1.75
2.5
0.70
0.70
Ht+1
4
3.5
It
3
5.5
0.55
0.55
It+1
4.5
4
Jt
4.25
4.5
0.94
0.94
Jt+1
4.55
5
∗
(yt /xt )/max(yt /xt ) ;
∗∗
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Dt
σ ˆ -conv.=1.22
(yt+1 /xt+1 )/max(yt+1 /xt+1 )
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In this illustrative example, using expression (5), the value of σ ˆ -convergence is 1.22, which indicates convergence between period t and t + 1. Using expression (1) the standard deviation of the log of productivity in period t + 1 is equal to 0.07, and the standard deviation of the log of the productivity in period t is equal to 0.18. So, both measures indicate the occurrence of convergence between period t and t + 1.
4 3 Density 2 1 0
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Note that the existence of σ ˆ -convergence can be confirmed with a visual inspection of Figure 1, as our illustrative example consists of a single input and a single output. For a multi-input multi-output setting, a possible way to visualize the existence of convergence is to use kernel densities that represent the values of the distance functions (or efficiency scores) of the sample (see Fare et al. (2006) and Jamasb et al. (2008)). The narrowing of these curves over time or their skewness towards the maximum efficiency score (i.e., one) signals convergence. Figure 2 exhibits the kernel density estimates of the distance functions for time period t and t + 1. Following Simar and Zelenyuk (2006), we use Gaussian kernel function and Silverman (1986) rule of thumb to determine the bandwidth. From Figure 2, we can observe that the distribution of the distance function narrowed and the mode of the distribution moved towards higher levels in time period t + 1, which provides visual evidence of convergence.
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.4
.6 Distance function
Distance function t
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1
Distance function t+1
Fig. 2. Kernel density estimates of the efficiency scores for t and t + 1
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ACCEPTED MANUSCRIPT 2.2 A new nonparametric β-convergence index The concept of β-convergence can be explained as follows: “there is absolute β convergence if poor economies tend to grow faster than rich ones” (Sala-i Martin, 1996). Suppose we have data on real per capita GDP for a sample of economies. Let log(yt /xt ) be the logarithm of an economy’s GDP per capita at time t, and γt,t+1 be the economy’s growth rate of GDP between t and t+1, measured as log((yt+1 /xt+1 )/(yt /xt )). β-convergence can be estimated using regression as shown in expression (7). (7)
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γt,t+1 = α + βlog(yt /xt ) + ε
If β < 0, we say that the data set exhibits β-convergence, meaning that countries with high levels of per capita GDP in t exhibit smaller growth rates of per capita GDP.
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To evaluate β-convergence, we need to analyze the movements of the DMUs located on the best practice frontier (rich countries) and also the movements of the DMUs located on the worst practice frontier (poor countries) between periods t and t + 1. In this sense, there is β-convergence if the distance between the frontiers of richer countries and poor countries is smaller in t + 1 than in t.
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The process of convergence between period t and t + 1 may occur due to the decline of the best practice frontier whilst the worst practice frontier improves, remains unchanged or declines at a smaller rate. It may also be associated to the improvement of the worst practice frontier whilst the best practice frontier declines, remains unchanged or improves at a smaller rate. The process of divergence between period t and t+1 follows a similar rationale, i.e., the best practice frontier improves whilst the worst practice frontier improves less than the best practice frontier, remains unchanged, or regresses, or the worst practice frontier declines whilst the best practice frontier declines less than the worst practice frontier, remains unchanged or improves. There is no evidence of convergence or divergence when the distance between the frontiers remains unchanged over time, i.e., the frontiers improve or decline at the same rate, or keep unchanged.
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In a single input-output setting, the distance between the best and worst frontiers (DBW ) in periods t and t+1 can be measured, for a given DMU jo using expression (8) or (9). Expressions (8) and (9) measure the change in distance between the two frontiers, evaluated for the inputoutput mix of DMU jo , in period t and in t + 1, respectively.
t DBW
Dt (xt , y t ) W t (xt , y t ) = t+1 t t D (x , y ) W t+1 (xt , y t )
t+1 DBW
(8)
Dt (xt+1 , y t+1 ) W t (xt+1 , y t+1 ) = t+1 t+1 t+1 D (x , y ) W t+1 (xt+1 , y t+1 )
(9)
W t (xt , y t ) corresponds to the reciprocal to the maximum proportional reduction of the output vector y t , given inputs xt , required to reach the minimum productivity level observed in the sample in period t, whilst Dt (xt , y t ) is the reciprocal to the maximum proportional expansion 7
ACCEPTED MANUSCRIPT of the output vector y t , given inputs xt , to reach the maximum productivity level. Note that W t (xt , y t ) ≥ 1, with 1 meaning that the DMU is located on the worst practice frontier, whilst Dt (xt , y t ) ≤ 1, with 1 meaning that the DMU is located on the best practice frontier. In a single input-output setting, the values of expressions (8) and (9) are identical for all DMUs. However, in a multi input-output setting, the distance between the best and worst frontiers in periods t and t + 1 can be different. Thus, for a given DMU jo , the change in distance between t,t+1 the best and worst frontiers (DBW ) can be measured as the geometric mean of the change in distance between the two frontiers evaluated at t (xt , y t ) and at t + 1 (xt+1 , y t+1 ) as follows:
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t,t+1 DBW =
Dt (xt+1 , y t+1 ) Dt (xt , y t ) × W t (xt , y t ) W t (xt+1 , y t+1 )
Dt+1 (xt , y t ) Dt+1 (xt+1 , y t+1 ) × W t+1 (xt , y t ) W t+1 (xt+1 , y t+1 )
!1/2
(10)
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Note that expression (10) can be rearranged as shown in (11), such that the numerator becomes the Technological Change (T C t,t+1 ) component of the Malmquist index proposed by Fare et al. (1992). Thus, the numerator measures the distance between the best practice frontiers in t and t + 1. Similarly, the denominator measures the distance between the worst practice frontiers (or the lowest productivity levels observed in the sample) in t and t + 1. Hereafter, we call this measure Worst Practice Change (W P C t,t+1 ), as it is computed in relation to the worst practice frontier.
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!1/2
W t (xt+1 , y t+1 ) W t (xt , y t ) × W t+1 (xt , y t ) W t+1 (xt+1 , y t+1 )
!1/2
=
T C t,t+1 W P C t,t+1
(11)
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t,t+1 = DBW
Dt (xt+1 , y t+1 ) Dt (xt , y t ) × Dt+1 (xt , y t ) Dt+1 (xt+1 , y t+1 )
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To determine the β-convergence index in an industry, we propose calculating the geometric mean of expression (11) for all DMUs in the sample, as shown in (12). To distinguish our measure ˆ of β-convergence from the traditional β-convergence measure, we refer it as β-convergence hereafter.
1/n
T Cjt,t+1 ˆ β-convergence = t,t+1 W P C j=1 j n Y
(12)
ˆ ˆ The β-convergence index may be greater, equal or smaller than one. A β-convergence index greater than one indicates that the worst and the best practice frontiers are more distant in time period t + 1 than in period t (i.e., divergence occurs between periods t and t + 1). A ˆ β-convergence index smaller than one indicates that both frontiers are closer in period t + 1 8
ACCEPTED MANUSCRIPT than in period t (i.e., convergence occurs between periods t and t + 1). A score equal to one means that both frontiers are located at a similar distance in periods t + 1 and t.
ˆ β-convergence =
n Y
j=1
n Y
j=1
1/n
T Cjt,t+1
1/n
W P Cjt,t+1
=
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ˆ The β-convergence extends the traditional method for the estimation of β-convergence in two ways. First, it is applicable to multi-input multi-output settings. Second, it can be decomposed into two components as presented in expression (13). The Average Technological Change (ATCt,t+1 ) and the Average Worst Practice Change (AWPCt,t+1 ), corresponding to the numerator and denominator of expression (13), which characterize the average movements of the best practice frontiers and worst practice frontiers, respectively, between period t and t + 1.
ATCt,t+1 AWPCt,t+1
(13)
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ˆ The basic ideas behind the calculation of the β-convergence are illustrated in Figure 3. This figure represents the 10 DMUs presented in Table 1, and illustrates the best practice frontier (BF) and the worst practice frontier (WF) at time periods t and t + 1.
Fig. 3. Illustrative example one input-output
In the case of DMU H, considering its position in t (H t ), the distance between the best practice frontier and the worst practice frontier in period t + 1 is measured by ob/obW , whereas the distance between frontiers in period t is oa/oaW . Thus, the change in distance between the two frontiers in periods t and t + 1 is given by the ratio of ob/obW to oa/oaW . Similarly, considering the position of DMU H in period t + 1 (H t+1 ), the change in distance between the two frontiers in periods t and t + 1 is given by the ratio of o0 d/o0 dW to o0 c/o0 cW . The geometric mean of these two ratios, corresponding to expression (10), defines the change in distance between the two frontiers in periods t and t + 1 for DMU H as follows. 9
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t,t+1 DBW
ob/obW o0 d/o0 dW × oa/oaW o0 c/o0 cW
=
!1/2
(14)
t,t+1 The value of DBW for DMU H is less than one (i.e., 0.43), as the distance between the best and worst practice frontiers diminished from t to t + 1. As the illustrative example consists of t,t+1 a single input-output, the value of DBW is equal for all DMUs in the sample, and so the value t,t+1 ˆ of β-convergence is also equal to DBW .
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t,t+1 Table 2 shows the values of DBW , T C t,t+1 , and W P C t,t+1 for each DMU in the sample. In ˆ this example, β-convergence is equal to 0.43, which means convergence occurred between t and t + 1 (i.e., the distance between the best and worst frontiers reduced from t to t + 1). Using the traditional method for the estimation of β-convergence, shown in expression (7), we obtain a β-convergence value equal to -1.15, which also indicates convergence between t and t + 1. The values of ATCt,t+1 and AWPCt,t+1 are 1.5 and 3.5, respectively. This means that both the best and worst practice frontiers improved productivity between t and t + 1 (i.e., both components ˆ of β-convergence are greater than one). However, the worst practice frontier improved more than the best practice frontier, as the value of AWPCt,t+1 is greater than the value of ATCt,t+1 .
Table 2 ˆ Value of β-convergence and its components for the DMUs considered in the illustrative example DMU
TCt,t+1
A
1.5
B
1.5 1.5
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C D
1.5
Dt,t+1 BW
3.5
0.43
3.5
0.43
3.5
0.43
3.5
0.43
1.5
3.5
0.43
F
1.5
3.5
0.43
G
1.5
3.5
0.43
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E
WPCt,t+1
1.5
3.5
0.43
I
1.5
3.5
0.43
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Y
J
1.5
3.5
0.43
ATCt,t+1 = 1.5
AWPCt,t+1 = 3.5
ˆ β-conv.=0.43
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To illustrate further the information that can be obtained by analyzing the components of ˆ β-convergence, we use three cases of divergence, as illustrated in Figure 4. The insights obˆ tained exploring the components of β-convergence would not be available using the traditional regression approach to estimate β-convergence. ˆ In the three cases presented in Figure 4, β-convergence is greater than one, meaning that divergence occurred between the best and worst observed productivity levels in t and t + 1. In particular, in case a, the best practice frontier improved between t and t + 1 (ATCt,t+1 > 1), whereas the worst practice frontier declined between t and t+1 (AWPCt,t+1 < 1). In case b, both the best practice and worst practice frontiers improved between t and t+1, but the worst practice frontier improved less than the best practice frontier (ATCt,t+1 > 1 and AWPCt,t+1 > 1, with AWPCt,t+1 < ATCt,t+1 ). In case c, both the best practice and worst practice frontiers declined between t and t+1, but the worst practice frontier declined more than the best practice frontier (ATCt,t+1 < 1 and AWPCt,t+1 < 1, with AWPCt,t+1 > ATCt,t+1 ). 10
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Fig. 4. Three cases of divergence between period t and t + 1
ˆ The computation of σ ˆ -convergence and β-convergence using DEA
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2.3
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The standard DEA model, proposed by Charnes et al. (1978), can be used to estimate the distances in relation to the best practice frontier. The distances measured in relation to the ˆ worst practice frontier embodied in the β-convergence index can be obtained using the inverted DEA model, first proposed by Yamada et al. (1994). This model consists of swapping the inputs and the outputs of the standard DEA model. The inverted DEA model was previously used in the literature in other contexts different from the purpose of this paper. For instance, it was proposed to estimate intervals for efficiency scores (Entani et al., 2002), to detect outliers (Johnson and McGinnis, 2008), to evaluate credit risk (Paradi et al., 2004), to support the design of financial investment strategies (Kadoya et al., 2008), or to construct a lower frontier to delimitate a contract zone (Hadley and Ruggiero, 2006).
t
t
−1
[D (x , y )]
m X
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t
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The standard DEA model and the inverted DEA model (as presented in Entani et al. (2002)) ˆ used to compute σ ˆ -convergence and β-convergence are shown in (15) and (16), respectively. Both are specified with an output orientation, seeking adjustments to output levels, and assume constant returns to scale.
= min
m X
[W t (xt , y t )]−1 = max
vi xijo
i=1
s X
r=1
s X
ur yrjo = 1
m X
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r=1 s X
subject to
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subject to
ur yrj −
i=1
r=1 m X
vi xij ≤ 0, j = 1, . . . , n
vi ≥ 0, i = 1, . . . , m ur ≥ 0, r = 1, . . . , s
vi xijo
i=1
i=1
ur yrjo = 1 vi xij −
s X
r=1
ur yrj ≤ 0, j = 1, . . . , n
vi ≥ 0, i = 1, . . . , m ur ≥ 0, r = 1, . . . , s
(15)
(16)
The variables of models (15) and (16) are ur and vi , corresponding to the weights associated to the outputs yrj (r = 1, . . . , s) and inputs xij (i = 1, . . . , m), respectively. The objective function of model (15) is greater or equal to one, where one indicates that the DMU is located on the 11
ACCEPTED MANUSCRIPT frontier. The objective function of model (16) returns a value smaller or equal to one, such that a value equal to one indicates that the DMU is located on the worst practice frontier.
3
Empirical study
3.1 Construction industry context
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The construction industry (CI) is a major sector in Europe. According to the European Construction Industry Federation (FIEC, 2010), the construction sector accounted for 10.4% of European Gross Domestic Product and 7.2% of Europe’s total employment in 2008. The CI plays a crucial role in the economy and society, as it provides the infrastructure and housing that supports other economic activities.
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The European CI has been witnessing considerable changes in the last years. Since the 1990’s, European construction activity has experienced a positive development. This was mainly boosted by the introduction of the euro currency that implied low interest rates, and by the large scale subsidies for infrastructure projects. However, the growth trend in the European CI started to invert in 2008. Europe was hit by a financial crisis that severely impacted the CI activity. The crisis affected most European countries and subsectors of the CI activity. In particular, the residential segment that constituted the largest market for the CI was strongly hit by the credit constraints and the lack of consumer confidence, which postponed investment in property. The civil engineering segment was less affected by the crisis due to national economic recovery plans that prioritized infrastructure projects. Concerning the effects of the crisis in the European countries, Eastern countries (e.g. Poland, Romania, Bulgaria) were the least affected by the economic downturn, whereas the construction activity in Western countries (e.g. Portugal, Spain, France) has dramatically dropped.
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To overcome the financial crisis in the European CI, the European Construction Industry Federation and the European Federation of Building and Woodworkers (FIEC and EFBWW, 2009) proposed the following measures: i) to increase the public investment in infrastructure projects and sustainable construction, ii) to provide incentives for buying and renovating houses, iii) to facilitate the access to mortgages, iv) to provide incentives for “social housing”, v) to ensure the efficient use of investment resources, vi) to apply temporary unemployment schemes, vii) to create a more sustainable and transparent financial system. The urgency of implementing such measures at European and national levels was recently recalled (FIEC and EFBWW, 2010). To survive and prosper in this adverse context, construction companies need robust methods to evaluate performance for a timely realignment of strategies and procedures. The development of robust methods to provide enhanced knowledge of the construction sector is even more relevant in periods of economic recession, as companies are more prone to bankruptcy. These arguments, in conjunction with the scarce literature on CI performance assessment, have motivated an empirical application devoted to the characterization of the evolution of the CI in the aftermath of the financial crisis in Portugal. This study is the first to evaluate convergence in the context of the construction industry. Previous research on the Portuguese CI primarily focused on the evaluation of construction companies’ performance using data from internet benchmarking 12
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3.2 Data and variables
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The data set used to illustrate the methodology developed consists of a balanced panel data of 8308 Portuguese construction companies that operated in the sector between 2008 and 2010. This corresponds to a total of 16616 observations in the two years analyzed. In order to create groups of similar construction companies, the analysis was done by splitting companies into the four major CI activity segments: buildings (5667 companies), road networks and related infrastructure (807 companies), hydraulic works (153 companies), and mechanical & electrical facilities (1681 companies).
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Before defining the data set, we applied an outlier detection procedure. In particular, an observation was considered an outlier when the ratio output to any of the three inputs was outside the interval of the median plus and minus two standard deviations (e.g., see Kapelko et al. (2014)). The data set concerns companies that comply with the minimum obligatory requirements to operate in the sector in the period analyzed (i.e., at least 100% on liquidity and 5% on financial autonomy). The data used came from the regulatory board of Portuguese Construction and Real Estate (INCI), through the icBench project (for further details on this project, see Costa et al. (2007)).
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To evaluate the convergence of the Portuguese construction companies, the inputs and outputs of the production process need to be specified. According to previous studies carried out on CI performance (Pilateris and McCabe, 2003; You and Zi, 2007; Horta et al., 2013), as well as in other industries (Johnes and Johnes, 2013; Delen et al., 2013), we selected one output and three inputs to represent key financial measures of companies: liquidity, leverage, profitability and cost accounting. The output concerns the value of sales, which represents a profitability measure. The inputs are: total current liabilities (that measure the amount due to creditors and suppliers) to proxy liquidity, shareholders’ funds (that measure the shareholders’ participation in the company) to proxy leverage, and operating costs (which include the fixed and the variable costs of production) to proxy cost accounting. The selection of variables was constrained by data availability. As the inputs and outputs are measured in monetary terms, our measure of productivity reflects economic aspects of companies activity. Table 3 reports the mean and the standard deviation (SD) of the variables used in the assessment of the CI activity sectors for the period 2008 and 2010.
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Table 3 Descriptive statistics of data (Ke) Activity
Operating costs
Shareholders’ funds
Total current liabilities
Value of sales
segment
Mean
SD
Mean
SD
Mean
SD
Mean
SD
1310
5330
434
1509
710
3263
1437
5597
Buildings Road networks
3078
10349
1328
4807
1843
6705
3343
11013
Hydraulic works
7785
15342
2883
5877
4654
10334
8304
16099
Mechanical & electrical
3150
12418
809
3572
1494
6926
3404
13403
From Table 3, it is possible to observe that the companies analyzed within each activity sector 13
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3.3 Results
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The kernel density estimates of the efficiency scores for each activity segment in 2008 and 2010 are presented in Figure 5. From Figure 5, we can observe that the mode of the distribution moved towards higher levels of efficiency in 2010 in all activity sectors, but particularly in hydraulic works and mechanical & electrical sectors. This provides evidence of convergence in the sense of σ.
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Fig. 5. Kernel density estimates of the efficiency scores for 2008 and 2010 by activity profile
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ˆ Table 4 provides the quantification of σ ˆ -convergence values and presents the values of βconvergence and its components (Average Technological Change and Average Worst Practice Change) for each CI activity segment in the period analyzed. σ ˆ -convergence indicates whether the industry average moved closer or further away from the best practice frontier between peˆ riod t and t + 1, whereas β-convergence reflects the average movements of the firms that define the best and the worst frontiers between period t and t + 1. These two convergence indexes (ˆ σ ˆ and β) should be used as complementary measures in convergence assessments.
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Table 4 ˆ Values of σ ˆ -convergence and β-convergence by activity profile Activity
σ ˆ -convergence
ˆ β-convergence
ˆ Components of β-convergence
segment
index
index
ATCt,t+1
AWPCt,t+1
Buildings
1.36
0.61
0.76
1.25
Road networks
1.06
0.60
0.90
1.51
Hydraulic works
3.05
0.29
0.32
1.11
Mechanical & Electrical
3.30
0.34
0.30
0.88
The results of σ ˆ -convergence results presented in Table 4 confirm that companies moved closer to the best practice standards in all activity segments from 2008 to 2010 (ˆ σ -convergence greater 14
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than 1). Thus can be interpreted as an indication that construction companies became more homogenous in terms of efficiency. Hydraulic works and mechanical & electrical companies exhibited the greatest values of σ ˆ -convergence, whereas buildings and road networks exhibited lower values of σ ˆ -convergence. Buildings and road networks segments include large general contractors and also subcontractors. The structural differences between these companies make it difficult for most companies to follow closely the best practices in financial terms implemented by the benchmark companies of their segments. Conversely, hydraulic works and mechanical & electrical segments are mainly characterized by specialized subcontractors. In Portugal, the current stagnation of the construction activity implies that subcontractors have to face a fierce competitive environment to survive in the sector. Hence, it is likely that they follow the practices of their peers (for instance, the control of financial indicators or the adoption of cost reduction strategies) to ensure viability in the market.
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ˆ Concerning the β-convergence results, we can conclude that the gap between the best and the ˆ worst companies narrowed in all activity sectors from 2008 to 2010 (β-convergence less than 1). ˆ The β-convergence between companies was even more exacerbated in the hydraulic works and ˆ mechanical & electrical facilities (β-convergence index equal to 0.29 and 0.34, respectively). Despite the improvement of the worst practice companies (AWPCt,t+1 greater than 1) in most activity sectors, the process of convergence was mainly motivated by the considerable technological decline of the best practice companies (ATCt,t+1 less than 1). These results indicate that the best practice companies were not able to maintain the cutting edge standards of financial achievements observed in previous periods due to the slowing down of the Portuguese CI activity. This reveals that the financial crisis caused an inward shift of the best practice frontier. In turn, the worst companies could acquire technological, financial and managerial know-how from the best practice companies, and probably have worked hard to survive in the stagnant CI market. The success of their strategies is revealed by the values of AWPCt,t+1 greater than one.
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These results are in line with previous research, which concluded that convergence in productivity is likely to be observed during economic recession, both in terms of σ and β-convergence. For instance, Escribano and Stucchi (2014) analyzed a sample of Spanish manufacturing firms in the period 1991 and 2005, and concluded that productivity tended to converge in recession periods.
Conclusions
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This paper presents a novel nonparametric methodology to evaluate convergence considering a multi-input multi-output context for the evaluation of total factor productivity. This methodology is particularly relevant to complement productivity assessments using the Malmquist index. In particular, we develop two new indexes to evaluate σ-convergence and β-convergence that can be estimated using nonparametric techniques such as DEA. The major advantage of using ˆ our σ ˆ -convergence and β-convergence indexes over traditional approaches is that they allow evaluations using multi-inputs and multi-outputs, which typically characterize the activity of ˆ DMUs in a real world context. In addition, our β-convergence index can be decomposed into ˆ two components that provide insights on the sources of β-convergence. The methodology developed was applied to a real world context by evaluating panel data 15
ACCEPTED MANUSCRIPT consisting of 8308 Portuguese construction companies between 2008 and 2010. The analysis was done separating construction companies into four main activity profiles (buildings, road networks and related infrastructure, hydraulic works, and mechanical & electrical facilities). The empirical results showed that Portuguese construction companies tended to converge in the sense of σ ˆ and βˆ in all activity sectors in the aftermath of the financial crisis. The information concerning the status of the CI sector at a national level is important for company managers and also administrative authorities. The detailed knowledge of the market and its evolution over time is essential for defining industry strategies and policies, as they can support decisions concerning investments, subcontracting practices, mergers and acquisitions, and partnerships.
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For future research, it would be of interest to develop other measures of convergence, such as a nonparametric conditional β-convergence. This measure would enable the evaluation of convergence in a sample of DMUs with different characteristics. It would also be interesting to develop enhanced convergence indexes that can account for slacks, or to complement the indexes using the bootstrapping technique to test the significance of the results obtained. In addition, the methodology developed could be applied to other activity sectors (e.g. manufacturing or mining) to enable cross industry comparisons, or to other countries or worldwide regions to undertake international benchmarking exercises.
Acknowledgments
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The authors are grateful to Professor Jorge Moreira da Costa and to the regulatory board of Portuguese Construction and Real Estate (INCI), for enabling access to the data used in this study. The funding of this research through the scholarship SFRH/BPD/86294/2012 from the Portuguese Foundation of Science and Technology (FCT) is also gratefully acknowledged.
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References
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Barro, R. J. and Sala-i Martin, X. (1992). Convergence. Journal of Political Economy, 100(2):223. Charnes, A., Cooper, W. W., and Rhodes, E. (1978). Measuring efficiency of decision making units. European Journal of Operational Research, 2(6):429–444. Costa, J. M., Horta, I. M., Guimaraes, N., Novoa, H., Cunha, J. F., and Sousa, R. (2007). icBench - a benchmarking tool for portuguese construction industry companies. International Journal for Housing Science and Its Applications, 31(1):33–41. Delen, D., Kuzey, C., and Uyar, A. (2013). Measuring firm performance using financial ratios: A decision tree approach. Expert Systems with Applications, 40(10):3970–3983. Entani, T., Maeda, Y., and Tanaka, H. (2002). Dual models of interval DEA and its extension to interval data. European Journal of Operational Research, 136(1):32–45. Escribano, . and Stucchi, R. (2014). Does recession drive convergence in firms productivity? evidence from spanish manufacturing firms. Journal of Productivity Analysis, 41(3):339–349. Fare, R., Grosskopf, S., Lindgren, B., and Roos, P. (1992). Productivity changes in swedish pharamacies 19801989: A non-parametric malmquist approach. Journal of Productivity Analysis, 3(1-2):85–101. 16
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Fare, R., Grosskopf, S., and Margaritis, D. (2006). Productivity growth and convergence in the european union. Journal of Productivity Analysis, 25(1-2):111–141. Fare, R., Grosskopf, S., Norris, M., and Zhang, Z. (1994). Productivity growth, technical progress, and efficiency change in industrialized countries. The American Economic Review, 84(1):66–83. Farrell, M. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A, 120(3):253–290. FIEC (2010). Construction in europe: Key figures. Technical report, European Construction Industry Federation, Brussels, Belgium. FIEC and EFBWW (2009). The global economic crisis and its consequences for the european construction industry: Positive measures and concerns of the european social partners EFBWW and FIEC. Technical report, European Construction Industry Federation and the European Federation of Building and Woodworkers, Brussels, Belgium. FIEC and EFBWW (2010). Emerging from the crisis: fostering growth and jobs for a sustainable construction industry. Technical report, European Construction Industry Federation and the European Federation of Building and Woodworkers, Brussels, Belgium. Hadley, L. and Ruggiero, J. (2006). Final-offer arbitration in major league baseball: A nonparametric analysis. Annals of Operations Research, 145(1):201–209. Horta, I. M., Camanho, A. S., and Costa, J. M. (2010). Performance assessment of construction companies integrating key performance indicators and data envelopment analysis. Journal of Construction Engineering and Management, 136(5):581–594. Horta, I. M., Camanho, A. S., Johnes, J., and Johnes, G. (2013). Performance trends in the construction industry worldwide: an overview of the turn of the century. Journal of Productivity Analysis, 39(1):89–99. Horta, I. M., Camanho, A. S., and Moreira da Costa, J. (2012). Performance assessment of construction companies: A study of factors promoting financial soundness and innovation in the industry. International Journal of Production Economics, 137(1):84–93. Islam, N. (2003). What have we learnt from the convergence debate? Journal of Economic Surveys, 17(3):309–362. Jamasb, T., Pollitt, M. G., and Triebs, T. (2008). Productivity and efficiency of US gas transmission companies: A european regulatory perspective. Energy Policy, 36(9):3398–3412. Johnes, G. and Johnes, J. (2013). The persistence of efficiency. Expert Systems with Applications, 40(14):5661–5672. Johnson, A. L. and McGinnis, L. F. (2008). Outlier detection in two-stage semiparametric DEA models. European Journal of Operational Research, 187(2):629–635. Kadoya, S., Kuroko, T., and Namatame, T. (2008). Contrarian investment strategy with data envelopment analysis concept. European Journal of Operational Research, 189(1):120–131. Kapelko, M., Oude Lansink, A., and Stefanou, S. E. (2014). Assessing dynamic inefficiency of the spanish construction sector pre- and post-financial crisis. European Journal of Operational Research, 237(1):349–357. Paradi, J. C., Asmild, M., and Simak, P. C. (2004). Using DEA and worst practice DEA in credit risk evaluation. Journal of Productivity Analysis, 21(2):153–165. Pilateris, P. and McCabe, B. (2003). Contractor financial evaluation model (CFEM). Canadian Journal of Civil Engineering, 30(3):487–499. Sala-i Martin, X. X. (1996). The classical approach to convergence analysis. The Economic Journal, 106(437):1019–1036. Shepard, R. (1970). Theory of cost and production functions. Princeton, princeton university press edition. Silverman, B. (1986). Density Estimation for Statistics and Data Analysis. Monographs on 17
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A nonparametric methodology for evaluating convergence in a multi-input multi-output setting I.M. Horta, A.S. Camanho PII: DOI: Reference:
S0377-2217(15)00387-2 10.1016/j.ejor.2015.05.015 EOR 12939
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
25 May 2014 12 December 2014 4 May 2015
Please cite this article as: I.M. Horta, A.S. Camanho, A nonparametric methodology for evaluating convergence in a multi-input multi-output setting, European Journal of Operational Research (2015), doi: 10.1016/j.ejor.2015.05.015
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ACCEPTED MANUSCRIPT Highlights The paper presents a novel nonparametric methodology to evaluate convergence We develop two new indexes to evaluate α-convergence and β-convergence The indexes developed allow evaluations using multiple inputs and outputs The methodology complements productivity assessments based on the Malmquist index The methodology is applied to Portuguese construction companies operating in 2008-2010
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• • • • •
1
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A nonparametric methodology for evaluating convergence in a multi-input multi-output setting I.M. Horta a,∗ , A.S. Camanho a de Engenharia, Universidade do Porto. Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
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a Faculdade
Abstract
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This paper presents a novel nonparametric methodology to evaluate convergence in an industry, considering a multi-input multi-output setting for the assessment of total factor productivity. In particular, we develop two new indexes to evaluate σ-convergence and β-convergence that can be computed using nonparametric techniques such as Data Envelopment Analysis. The methodology developed is particularly useful to enhance productivity assessments based on the Malmquist index. The methodology is applied to a real world context, consisting of a sample of Portuguese construction companies that operated in the sector between 2008 and 2010. The empirical results show that Portuguese companies tended to converge, both in the sense of σ and β, in all construction activity segments in the aftermath of the financial crisis.
Introduction
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Key words: Convergence, Productivity, Malmquist index, Data Envelopment Analysis, Construction Industry
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Convergence has been extensively discussed in the economic growth literature over the past decades (see Temple (1999) and Islam (2003) for a literature review). The first studies devoted their attention to whether poor or low productive countries would catch up with their rich or highly productive counterparts. This is an issue of utmost importance for researchers and policymakers interested in worldwide welfare. Two main concepts of convergence appear in the classical literature (see Barro and Sala-i Martin (1992)). β-convergence analyzes if poor countries tend to grow faster than rich countries, whereas σ-convergence examines if the dispersion of the productivity for a group of countries tends to decrease over time. To measure these concepts, most studies use a single productivity measure, such as income per capita or Gross Domestic Product (GDP) per capita, or a measure of total factor productivity estimated using econometric methods. The evaluation of convergence in a multi-input multi-output setting has not been addressed in the literature. ∗ Corresponding author. Tel: +351 225081639. E-mail address: [email protected]
ACCEPTED MANUSCRIPT This research contributes to the literature by developing a nonparametric methodology for the evaluation of convergence in a multi-input multi-output setting, to enhance productivity assessments based on the Malmquist index. In particular, we explore the use of Shephard distance functions, estimated through Data Envelopment Analysis (DEA), to calculate β-convergence and σ-convergence. In addition, this paper applies the two new indexes to evaluate convergence in a sample of Portuguese construction companies that operated in the sector between 2008 and 2010.
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The remainder of this paper is organized as follows. Section 2 describes the new nonparametric indexes to evaluate σ-convergence and β-convergence, and explains their computation using DEA. Section 3 presents the empirical application, including the motivation, the description of the data set and the discussion of the results. The last section concludes and points topics for future research.
Evaluation of convergence using nonparametric techniques
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2.1 A new nonparametric σ-convergence index
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The concept of σ-convergence can be explained as follows: “a group of economies are converging in the sense of σ if the dispersion of the their real per capita GDP levels tends to decrease over time” (Sala-i Martin, 1996). The notion of σ-convergence can be expressed, in mathematical terms, as shown in expression (1). (1)
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σt+1 < σt
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σt is the standard deviation of the logarithm of the real per capita GDP levels across all economies in period t, whereas σt+1 is a similar measure in a subsequent period. This definition can be extended to a more general setting, consisting of the assessment of Decision Making Units (DMUs) (e.g., representing countries or organizations), where σ-convergence explores whether the dispersion of values of a productivity indicator, measured as an output to input ratio, tends to decrease (or increase) over time.
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Our aim is to generalize the measurement of convergence to a multi-input multi-output setting, and thus we explore σ-convergence using the Shephard distance function. The Shepard distance function is a generalization of the production function that allows us to consider simultaneously multiple inputs and outputs. As this paper is devoted to output oriented assessments, all indexes and measures described have an output orientation. m s Consider n DMUs in time period t that use inputs xt ∈ R+ to produce outputs y t ∈ R+ , and in t+1 m t+1 s a subsequent time period t + 1 use inputs x ∈ R+ to produce outputs y ∈ R+ . In period t, the production technology T t consists of the set of all feasible input/output combinations for a certain production process, as shown in (2).
T t = {(xt , y t ) : xt can produce y t } 3
(2)
ACCEPTED MANUSCRIPT Following Shepard (1970), the output distance function for DMU jo in relation to the technology T t is defined as shown in (3). (
yt Dt (xt , y t ) = min θ : xt , θ
∈ Tt
)
(3)
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This function is defined as the reciprocal to the maximum proportional expansion of the output vector y t , given inputs xt , i.e., Dt (xt , y t ) ≤ 1. This means that it corresponds to the efficiency score of DMU jo , in the sense of Farrell (1957). Thus, in order to estimate σ-convergence we can use efficiency measures. In other words, the spread of productivity levels can be estimated using the spread of efficiency measures. Efficiency is a relative measure that compares the productivity of a DMU jo with the best productivity levels of the sample. Therefore, a sample with larger dispersion of productivity levels will also have larger dispersion of efficiency. Fare et al. (1992) were the first to note that input and output distance functions could be estimated using DEA models (Charnes et al., 1978).
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For a given DMU jo , the ratio between the efficiency score in period t + 1 and in t, as presented in (4) is a measure of convergence towards the best practice frontier (see Fare et al. (1994)). This implies that there is convergence to the frontier if efficiency increases from period t to t + 1. Note that this ratio corresponds to the Efficiency Change (EC t,t+1 ) component of the Malmquist index proposed by Fare et al. (1992).
(4)
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Dt+1 (xt+1 , y t+1 ) = EC t,t+1 Dt (xt , y t )
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To determine σ-convergence in an industry, we propose calculating the geometric mean of expression (4) for all DMUs in the sample, as shown in (5). To distinguish our measure of σ-convergence, defined using Shepard distance functions and calculated using DEA models, from the traditional σ-convergence measure (see expression 1), we refer to it as σ ˆ -convergence hereafter.
σ ˆ -convergence =
n Y
j=1
1/n
ECjt,t+1
(5)
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The σ ˆ -convergence index may be greater, equal or smaller than one. A σ ˆ -convergence index greater than one indicates convergence (i.e. the DMUs moved closer to the best practice frontier from period t to t + 1), whereas a score less than one means divergence (i.e. the DMUs moved away from the best practice frontier between t to t + 1). A score equal to one indicates that, on average, the DMUs are located at a similar distance to the frontier in period t and t + 1. The advantage of using σ ˆ -convergence over the traditional measure of σ-convergence is that it allows accounting for multiple inputs and multiple outputs, as it can be estimated using nonparametric techniques such as DEA. The basic ideas behind the calculation of the σ ˆ -convergence can be illustrated in Figure 1. This figure presents 10 DMUs (e.g., countries), whose activity is represented by an output (e.g., GDP) and an input (e.g., population) in two time periods. The ratio y/x represents a 4
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productivity measure. The best practice frontiers (BF) at time periods t + 1 and t are also plotted in Figure 1.
Fig. 1. Illustrative example one input-output
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Consider DMU H, represented by point H t in time period t and point H t+1 in time period t + 1. In the example, the efficiency score of DMU H in period t measured in relation to the best practice in t, corresponds to the ratio oH t /oa. The efficiency score of DMU H in period t + 1, measured in relation to the best practice frontier in period t + 1, corresponds to the ratio o0 H t+1 /o0 d. Hence, the change in efficiency between period t and t + 1 is measured as shown in expression (6).
EC t,t+1 =
o0 H t+1 /o0 d oH t /oa
(6)
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The value of EC t,t+1 for DMU H is greater than one (i.e., 1.09), as DMU H is closer to the best practice frontier in t + 1 than in t. This indicates convergence towards the best practice frontier.
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Table 1 presents the original data set used in Figure 1, the output distance function and the productivity for each DMU in each time period. Recall that in a single input-output setting, the efficiency measure is the ratio between the observed productivity of DMU jo and the maximum productivity observed in other DMUs in the same time period. Thus, if maximum productivity is equal to one (i.e, the maximum ratio of output to input is one), the two measures would coincide. In the example shown in Table 1, the maximum productivity in period t was observed in DMU C, whose output to input ratio is equal to one, and thus productivity and efficiency coincide (see fourth and fifth columns of Table 1). Furthermore, if the maximum productivity remains equal to one in periods t and t + 1 (i.e., the technology does not change between period t and t + 1), the output distance functions (or efficiency scores) and the productivity ratios would be identical in the two periods. 5
ACCEPTED MANUSCRIPT Table 1 Data and σ ˆ -convergence for the DMUs considered in the illustrative example DMU
yt
xt
Productivity
Dt (xt , yt ) (∗
(yt /xt ) At
0.5
2
0.25
yt+1
DMU
xt+1
Productivity (yt+1 /xt+1 )
)
0.25
At+1
3 3.5 4
2
Dt+1 (xt+1 , yt+1 ) ( ∗∗
ECt,t+1
)
1.50
1.00
4.00
4
0.88
0.58
0.93
3
1.33
0.89
0.89
Bt
2.5
4
0.63
0.63
Bt+1
Ct
3
3
1.00
1.00
Ct+1
6
5
1.20
0.80
1.00
5.5
4
1.38
0.92
1.10
4
5
0.80
0.80
Dt+1
Et
5
6
0.83
0.83
Et+1
Ft
2
3
0.67
0.67
Ft+1
5
4.5
1.11
0.74
1.11
Gt
2
4.5
0.44
0.44
Gt+1
1.8
1.5
1.20
0.80
1.80
1.14
0.76
1.09
1.13
0.75
1.38
0.91
0.61
0.64
Ht
1.75
2.5
0.70
0.70
Ht+1
4
3.5
It
3
5.5
0.55
0.55
It+1
4.5
4
Jt
4.25
4.5
0.94
0.94
Jt+1
4.55
5
∗
(yt /xt )/max(yt /xt ) ;
∗∗
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Dt
σ ˆ -conv.=1.22
(yt+1 /xt+1 )/max(yt+1 /xt+1 )
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In this illustrative example, using expression (5), the value of σ ˆ -convergence is 1.22, which indicates convergence between period t and t + 1. Using expression (1) the standard deviation of the log of productivity in period t + 1 is equal to 0.07, and the standard deviation of the log of the productivity in period t is equal to 0.18. So, both measures indicate the occurrence of convergence between period t and t + 1.
4 3 Density 2 1 0
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Note that the existence of σ ˆ -convergence can be confirmed with a visual inspection of Figure 1, as our illustrative example consists of a single input and a single output. For a multi-input multi-output setting, a possible way to visualize the existence of convergence is to use kernel densities that represent the values of the distance functions (or efficiency scores) of the sample (see Fare et al. (2006) and Jamasb et al. (2008)). The narrowing of these curves over time or their skewness towards the maximum efficiency score (i.e., one) signals convergence. Figure 2 exhibits the kernel density estimates of the distance functions for time period t and t + 1. Following Simar and Zelenyuk (2006), we use Gaussian kernel function and Silverman (1986) rule of thumb to determine the bandwidth. From Figure 2, we can observe that the distribution of the distance function narrowed and the mode of the distribution moved towards higher levels in time period t + 1, which provides visual evidence of convergence.
.2
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.6 Distance function
Distance function t
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1
Distance function t+1
Fig. 2. Kernel density estimates of the efficiency scores for t and t + 1
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ACCEPTED MANUSCRIPT 2.2 A new nonparametric β-convergence index The concept of β-convergence can be explained as follows: “there is absolute β convergence if poor economies tend to grow faster than rich ones” (Sala-i Martin, 1996). Suppose we have data on real per capita GDP for a sample of economies. Let log(yt /xt ) be the logarithm of an economy’s GDP per capita at time t, and γt,t+1 be the economy’s growth rate of GDP between t and t+1, measured as log((yt+1 /xt+1 )/(yt /xt )). β-convergence can be estimated using regression as shown in expression (7). (7)
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γt,t+1 = α + βlog(yt /xt ) + ε
If β < 0, we say that the data set exhibits β-convergence, meaning that countries with high levels of per capita GDP in t exhibit smaller growth rates of per capita GDP.
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To evaluate β-convergence, we need to analyze the movements of the DMUs located on the best practice frontier (rich countries) and also the movements of the DMUs located on the worst practice frontier (poor countries) between periods t and t + 1. In this sense, there is β-convergence if the distance between the frontiers of richer countries and poor countries is smaller in t + 1 than in t.
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The process of convergence between period t and t + 1 may occur due to the decline of the best practice frontier whilst the worst practice frontier improves, remains unchanged or declines at a smaller rate. It may also be associated to the improvement of the worst practice frontier whilst the best practice frontier declines, remains unchanged or improves at a smaller rate. The process of divergence between period t and t+1 follows a similar rationale, i.e., the best practice frontier improves whilst the worst practice frontier improves less than the best practice frontier, remains unchanged, or regresses, or the worst practice frontier declines whilst the best practice frontier declines less than the worst practice frontier, remains unchanged or improves. There is no evidence of convergence or divergence when the distance between the frontiers remains unchanged over time, i.e., the frontiers improve or decline at the same rate, or keep unchanged.
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In a single input-output setting, the distance between the best and worst frontiers (DBW ) in periods t and t+1 can be measured, for a given DMU jo using expression (8) or (9). Expressions (8) and (9) measure the change in distance between the two frontiers, evaluated for the inputoutput mix of DMU jo , in period t and in t + 1, respectively.
t DBW
Dt (xt , y t ) W t (xt , y t ) = t+1 t t D (x , y ) W t+1 (xt , y t )
t+1 DBW
(8)
Dt (xt+1 , y t+1 ) W t (xt+1 , y t+1 ) = t+1 t+1 t+1 D (x , y ) W t+1 (xt+1 , y t+1 )
(9)
W t (xt , y t ) corresponds to the reciprocal to the maximum proportional reduction of the output vector y t , given inputs xt , required to reach the minimum productivity level observed in the sample in period t, whilst Dt (xt , y t ) is the reciprocal to the maximum proportional expansion 7
ACCEPTED MANUSCRIPT of the output vector y t , given inputs xt , to reach the maximum productivity level. Note that W t (xt , y t ) ≥ 1, with 1 meaning that the DMU is located on the worst practice frontier, whilst Dt (xt , y t ) ≤ 1, with 1 meaning that the DMU is located on the best practice frontier. In a single input-output setting, the values of expressions (8) and (9) are identical for all DMUs. However, in a multi input-output setting, the distance between the best and worst frontiers in periods t and t + 1 can be different. Thus, for a given DMU jo , the change in distance between t,t+1 the best and worst frontiers (DBW ) can be measured as the geometric mean of the change in distance between the two frontiers evaluated at t (xt , y t ) and at t + 1 (xt+1 , y t+1 ) as follows:
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t,t+1 DBW =
Dt (xt+1 , y t+1 ) Dt (xt , y t ) × W t (xt , y t ) W t (xt+1 , y t+1 )
Dt+1 (xt , y t ) Dt+1 (xt+1 , y t+1 ) × W t+1 (xt , y t ) W t+1 (xt+1 , y t+1 )
!1/2
(10)
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Note that expression (10) can be rearranged as shown in (11), such that the numerator becomes the Technological Change (T C t,t+1 ) component of the Malmquist index proposed by Fare et al. (1992). Thus, the numerator measures the distance between the best practice frontiers in t and t + 1. Similarly, the denominator measures the distance between the worst practice frontiers (or the lowest productivity levels observed in the sample) in t and t + 1. Hereafter, we call this measure Worst Practice Change (W P C t,t+1 ), as it is computed in relation to the worst practice frontier.
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!1/2
W t (xt+1 , y t+1 ) W t (xt , y t ) × W t+1 (xt , y t ) W t+1 (xt+1 , y t+1 )
!1/2
=
T C t,t+1 W P C t,t+1
(11)
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t,t+1 = DBW
Dt (xt+1 , y t+1 ) Dt (xt , y t ) × Dt+1 (xt , y t ) Dt+1 (xt+1 , y t+1 )
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To determine the β-convergence index in an industry, we propose calculating the geometric mean of expression (11) for all DMUs in the sample, as shown in (12). To distinguish our measure ˆ of β-convergence from the traditional β-convergence measure, we refer it as β-convergence hereafter.
1/n
T Cjt,t+1 ˆ β-convergence = t,t+1 W P C j=1 j n Y
(12)
ˆ ˆ The β-convergence index may be greater, equal or smaller than one. A β-convergence index greater than one indicates that the worst and the best practice frontiers are more distant in time period t + 1 than in period t (i.e., divergence occurs between periods t and t + 1). A ˆ β-convergence index smaller than one indicates that both frontiers are closer in period t + 1 8
ACCEPTED MANUSCRIPT than in period t (i.e., convergence occurs between periods t and t + 1). A score equal to one means that both frontiers are located at a similar distance in periods t + 1 and t.
ˆ β-convergence =
n Y
j=1
n Y
j=1
1/n
T Cjt,t+1
1/n
W P Cjt,t+1
=
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ˆ The β-convergence extends the traditional method for the estimation of β-convergence in two ways. First, it is applicable to multi-input multi-output settings. Second, it can be decomposed into two components as presented in expression (13). The Average Technological Change (ATCt,t+1 ) and the Average Worst Practice Change (AWPCt,t+1 ), corresponding to the numerator and denominator of expression (13), which characterize the average movements of the best practice frontiers and worst practice frontiers, respectively, between period t and t + 1.
ATCt,t+1 AWPCt,t+1
(13)
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ˆ The basic ideas behind the calculation of the β-convergence are illustrated in Figure 3. This figure represents the 10 DMUs presented in Table 1, and illustrates the best practice frontier (BF) and the worst practice frontier (WF) at time periods t and t + 1.
Fig. 3. Illustrative example one input-output
In the case of DMU H, considering its position in t (H t ), the distance between the best practice frontier and the worst practice frontier in period t + 1 is measured by ob/obW , whereas the distance between frontiers in period t is oa/oaW . Thus, the change in distance between the two frontiers in periods t and t + 1 is given by the ratio of ob/obW to oa/oaW . Similarly, considering the position of DMU H in period t + 1 (H t+1 ), the change in distance between the two frontiers in periods t and t + 1 is given by the ratio of o0 d/o0 dW to o0 c/o0 cW . The geometric mean of these two ratios, corresponding to expression (10), defines the change in distance between the two frontiers in periods t and t + 1 for DMU H as follows. 9
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t,t+1 DBW
ob/obW o0 d/o0 dW × oa/oaW o0 c/o0 cW
=
!1/2
(14)
t,t+1 The value of DBW for DMU H is less than one (i.e., 0.43), as the distance between the best and worst practice frontiers diminished from t to t + 1. As the illustrative example consists of t,t+1 a single input-output, the value of DBW is equal for all DMUs in the sample, and so the value t,t+1 ˆ of β-convergence is also equal to DBW .
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t,t+1 Table 2 shows the values of DBW , T C t,t+1 , and W P C t,t+1 for each DMU in the sample. In ˆ this example, β-convergence is equal to 0.43, which means convergence occurred between t and t + 1 (i.e., the distance between the best and worst frontiers reduced from t to t + 1). Using the traditional method for the estimation of β-convergence, shown in expression (7), we obtain a β-convergence value equal to -1.15, which also indicates convergence between t and t + 1. The values of ATCt,t+1 and AWPCt,t+1 are 1.5 and 3.5, respectively. This means that both the best and worst practice frontiers improved productivity between t and t + 1 (i.e., both components ˆ of β-convergence are greater than one). However, the worst practice frontier improved more than the best practice frontier, as the value of AWPCt,t+1 is greater than the value of ATCt,t+1 .
Table 2 ˆ Value of β-convergence and its components for the DMUs considered in the illustrative example DMU
TCt,t+1
A
1.5
B
1.5 1.5
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C D
1.5
Dt,t+1 BW
3.5
0.43
3.5
0.43
3.5
0.43
3.5
0.43
1.5
3.5
0.43
F
1.5
3.5
0.43
G
1.5
3.5
0.43
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WPCt,t+1
1.5
3.5
0.43
I
1.5
3.5
0.43
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1.5
3.5
0.43
ATCt,t+1 = 1.5
AWPCt,t+1 = 3.5
ˆ β-conv.=0.43
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To illustrate further the information that can be obtained by analyzing the components of ˆ β-convergence, we use three cases of divergence, as illustrated in Figure 4. The insights obˆ tained exploring the components of β-convergence would not be available using the traditional regression approach to estimate β-convergence. ˆ In the three cases presented in Figure 4, β-convergence is greater than one, meaning that divergence occurred between the best and worst observed productivity levels in t and t + 1. In particular, in case a, the best practice frontier improved between t and t + 1 (ATCt,t+1 > 1), whereas the worst practice frontier declined between t and t+1 (AWPCt,t+1 < 1). In case b, both the best practice and worst practice frontiers improved between t and t+1, but the worst practice frontier improved less than the best practice frontier (ATCt,t+1 > 1 and AWPCt,t+1 > 1, with AWPCt,t+1 < ATCt,t+1 ). In case c, both the best practice and worst practice frontiers declined between t and t+1, but the worst practice frontier declined more than the best practice frontier (ATCt,t+1 < 1 and AWPCt,t+1 < 1, with AWPCt,t+1 > ATCt,t+1 ). 10
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Fig. 4. Three cases of divergence between period t and t + 1
ˆ The computation of σ ˆ -convergence and β-convergence using DEA
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2.3
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The standard DEA model, proposed by Charnes et al. (1978), can be used to estimate the distances in relation to the best practice frontier. The distances measured in relation to the ˆ worst practice frontier embodied in the β-convergence index can be obtained using the inverted DEA model, first proposed by Yamada et al. (1994). This model consists of swapping the inputs and the outputs of the standard DEA model. The inverted DEA model was previously used in the literature in other contexts different from the purpose of this paper. For instance, it was proposed to estimate intervals for efficiency scores (Entani et al., 2002), to detect outliers (Johnson and McGinnis, 2008), to evaluate credit risk (Paradi et al., 2004), to support the design of financial investment strategies (Kadoya et al., 2008), or to construct a lower frontier to delimitate a contract zone (Hadley and Ruggiero, 2006).
t
t
−1
[D (x , y )]
m X
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The standard DEA model and the inverted DEA model (as presented in Entani et al. (2002)) ˆ used to compute σ ˆ -convergence and β-convergence are shown in (15) and (16), respectively. Both are specified with an output orientation, seeking adjustments to output levels, and assume constant returns to scale.
= min
m X
[W t (xt , y t )]−1 = max
vi xijo
i=1
s X
r=1
s X
ur yrjo = 1
m X
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r=1 s X
subject to
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subject to
ur yrj −
i=1
r=1 m X
vi xij ≤ 0, j = 1, . . . , n
vi ≥ 0, i = 1, . . . , m ur ≥ 0, r = 1, . . . , s
vi xijo
i=1
i=1
ur yrjo = 1 vi xij −
s X
r=1
ur yrj ≤ 0, j = 1, . . . , n
vi ≥ 0, i = 1, . . . , m ur ≥ 0, r = 1, . . . , s
(15)
(16)
The variables of models (15) and (16) are ur and vi , corresponding to the weights associated to the outputs yrj (r = 1, . . . , s) and inputs xij (i = 1, . . . , m), respectively. The objective function of model (15) is greater or equal to one, where one indicates that the DMU is located on the 11
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Empirical study
3.1 Construction industry context
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The construction industry (CI) is a major sector in Europe. According to the European Construction Industry Federation (FIEC, 2010), the construction sector accounted for 10.4% of European Gross Domestic Product and 7.2% of Europe’s total employment in 2008. The CI plays a crucial role in the economy and society, as it provides the infrastructure and housing that supports other economic activities.
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The European CI has been witnessing considerable changes in the last years. Since the 1990’s, European construction activity has experienced a positive development. This was mainly boosted by the introduction of the euro currency that implied low interest rates, and by the large scale subsidies for infrastructure projects. However, the growth trend in the European CI started to invert in 2008. Europe was hit by a financial crisis that severely impacted the CI activity. The crisis affected most European countries and subsectors of the CI activity. In particular, the residential segment that constituted the largest market for the CI was strongly hit by the credit constraints and the lack of consumer confidence, which postponed investment in property. The civil engineering segment was less affected by the crisis due to national economic recovery plans that prioritized infrastructure projects. Concerning the effects of the crisis in the European countries, Eastern countries (e.g. Poland, Romania, Bulgaria) were the least affected by the economic downturn, whereas the construction activity in Western countries (e.g. Portugal, Spain, France) has dramatically dropped.
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To overcome the financial crisis in the European CI, the European Construction Industry Federation and the European Federation of Building and Woodworkers (FIEC and EFBWW, 2009) proposed the following measures: i) to increase the public investment in infrastructure projects and sustainable construction, ii) to provide incentives for buying and renovating houses, iii) to facilitate the access to mortgages, iv) to provide incentives for “social housing”, v) to ensure the efficient use of investment resources, vi) to apply temporary unemployment schemes, vii) to create a more sustainable and transparent financial system. The urgency of implementing such measures at European and national levels was recently recalled (FIEC and EFBWW, 2010). To survive and prosper in this adverse context, construction companies need robust methods to evaluate performance for a timely realignment of strategies and procedures. The development of robust methods to provide enhanced knowledge of the construction sector is even more relevant in periods of economic recession, as companies are more prone to bankruptcy. These arguments, in conjunction with the scarce literature on CI performance assessment, have motivated an empirical application devoted to the characterization of the evolution of the CI in the aftermath of the financial crisis in Portugal. This study is the first to evaluate convergence in the context of the construction industry. Previous research on the Portuguese CI primarily focused on the evaluation of construction companies’ performance using data from internet benchmarking 12
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3.2 Data and variables
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The data set used to illustrate the methodology developed consists of a balanced panel data of 8308 Portuguese construction companies that operated in the sector between 2008 and 2010. This corresponds to a total of 16616 observations in the two years analyzed. In order to create groups of similar construction companies, the analysis was done by splitting companies into the four major CI activity segments: buildings (5667 companies), road networks and related infrastructure (807 companies), hydraulic works (153 companies), and mechanical & electrical facilities (1681 companies).
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Before defining the data set, we applied an outlier detection procedure. In particular, an observation was considered an outlier when the ratio output to any of the three inputs was outside the interval of the median plus and minus two standard deviations (e.g., see Kapelko et al. (2014)). The data set concerns companies that comply with the minimum obligatory requirements to operate in the sector in the period analyzed (i.e., at least 100% on liquidity and 5% on financial autonomy). The data used came from the regulatory board of Portuguese Construction and Real Estate (INCI), through the icBench project (for further details on this project, see Costa et al. (2007)).
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To evaluate the convergence of the Portuguese construction companies, the inputs and outputs of the production process need to be specified. According to previous studies carried out on CI performance (Pilateris and McCabe, 2003; You and Zi, 2007; Horta et al., 2013), as well as in other industries (Johnes and Johnes, 2013; Delen et al., 2013), we selected one output and three inputs to represent key financial measures of companies: liquidity, leverage, profitability and cost accounting. The output concerns the value of sales, which represents a profitability measure. The inputs are: total current liabilities (that measure the amount due to creditors and suppliers) to proxy liquidity, shareholders’ funds (that measure the shareholders’ participation in the company) to proxy leverage, and operating costs (which include the fixed and the variable costs of production) to proxy cost accounting. The selection of variables was constrained by data availability. As the inputs and outputs are measured in monetary terms, our measure of productivity reflects economic aspects of companies activity. Table 3 reports the mean and the standard deviation (SD) of the variables used in the assessment of the CI activity sectors for the period 2008 and 2010.
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Table 3 Descriptive statistics of data (Ke) Activity
Operating costs
Shareholders’ funds
Total current liabilities
Value of sales
segment
Mean
SD
Mean
SD
Mean
SD
Mean
SD
1310
5330
434
1509
710
3263
1437
5597
Buildings Road networks
3078
10349
1328
4807
1843
6705
3343
11013
Hydraulic works
7785
15342
2883
5877
4654
10334
8304
16099
Mechanical & electrical
3150
12418
809
3572
1494
6926
3404
13403
From Table 3, it is possible to observe that the companies analyzed within each activity sector 13
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3.3 Results
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The kernel density estimates of the efficiency scores for each activity segment in 2008 and 2010 are presented in Figure 5. From Figure 5, we can observe that the mode of the distribution moved towards higher levels of efficiency in 2010 in all activity sectors, but particularly in hydraulic works and mechanical & electrical sectors. This provides evidence of convergence in the sense of σ.
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Fig. 5. Kernel density estimates of the efficiency scores for 2008 and 2010 by activity profile
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ˆ Table 4 provides the quantification of σ ˆ -convergence values and presents the values of βconvergence and its components (Average Technological Change and Average Worst Practice Change) for each CI activity segment in the period analyzed. σ ˆ -convergence indicates whether the industry average moved closer or further away from the best practice frontier between peˆ riod t and t + 1, whereas β-convergence reflects the average movements of the firms that define the best and the worst frontiers between period t and t + 1. These two convergence indexes (ˆ σ ˆ and β) should be used as complementary measures in convergence assessments.
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Table 4 ˆ Values of σ ˆ -convergence and β-convergence by activity profile Activity
σ ˆ -convergence
ˆ β-convergence
ˆ Components of β-convergence
segment
index
index
ATCt,t+1
AWPCt,t+1
Buildings
1.36
0.61
0.76
1.25
Road networks
1.06
0.60
0.90
1.51
Hydraulic works
3.05
0.29
0.32
1.11
Mechanical & Electrical
3.30
0.34
0.30
0.88
The results of σ ˆ -convergence results presented in Table 4 confirm that companies moved closer to the best practice standards in all activity segments from 2008 to 2010 (ˆ σ -convergence greater 14
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than 1). Thus can be interpreted as an indication that construction companies became more homogenous in terms of efficiency. Hydraulic works and mechanical & electrical companies exhibited the greatest values of σ ˆ -convergence, whereas buildings and road networks exhibited lower values of σ ˆ -convergence. Buildings and road networks segments include large general contractors and also subcontractors. The structural differences between these companies make it difficult for most companies to follow closely the best practices in financial terms implemented by the benchmark companies of their segments. Conversely, hydraulic works and mechanical & electrical segments are mainly characterized by specialized subcontractors. In Portugal, the current stagnation of the construction activity implies that subcontractors have to face a fierce competitive environment to survive in the sector. Hence, it is likely that they follow the practices of their peers (for instance, the control of financial indicators or the adoption of cost reduction strategies) to ensure viability in the market.
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ˆ Concerning the β-convergence results, we can conclude that the gap between the best and the ˆ worst companies narrowed in all activity sectors from 2008 to 2010 (β-convergence less than 1). ˆ The β-convergence between companies was even more exacerbated in the hydraulic works and ˆ mechanical & electrical facilities (β-convergence index equal to 0.29 and 0.34, respectively). Despite the improvement of the worst practice companies (AWPCt,t+1 greater than 1) in most activity sectors, the process of convergence was mainly motivated by the considerable technological decline of the best practice companies (ATCt,t+1 less than 1). These results indicate that the best practice companies were not able to maintain the cutting edge standards of financial achievements observed in previous periods due to the slowing down of the Portuguese CI activity. This reveals that the financial crisis caused an inward shift of the best practice frontier. In turn, the worst companies could acquire technological, financial and managerial know-how from the best practice companies, and probably have worked hard to survive in the stagnant CI market. The success of their strategies is revealed by the values of AWPCt,t+1 greater than one.
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These results are in line with previous research, which concluded that convergence in productivity is likely to be observed during economic recession, both in terms of σ and β-convergence. For instance, Escribano and Stucchi (2014) analyzed a sample of Spanish manufacturing firms in the period 1991 and 2005, and concluded that productivity tended to converge in recession periods.
Conclusions
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This paper presents a novel nonparametric methodology to evaluate convergence considering a multi-input multi-output context for the evaluation of total factor productivity. This methodology is particularly relevant to complement productivity assessments using the Malmquist index. In particular, we develop two new indexes to evaluate σ-convergence and β-convergence that can be estimated using nonparametric techniques such as DEA. The major advantage of using ˆ our σ ˆ -convergence and β-convergence indexes over traditional approaches is that they allow evaluations using multi-inputs and multi-outputs, which typically characterize the activity of ˆ DMUs in a real world context. In addition, our β-convergence index can be decomposed into ˆ two components that provide insights on the sources of β-convergence. The methodology developed was applied to a real world context by evaluating panel data 15
ACCEPTED MANUSCRIPT consisting of 8308 Portuguese construction companies between 2008 and 2010. The analysis was done separating construction companies into four main activity profiles (buildings, road networks and related infrastructure, hydraulic works, and mechanical & electrical facilities). The empirical results showed that Portuguese construction companies tended to converge in the sense of σ ˆ and βˆ in all activity sectors in the aftermath of the financial crisis. The information concerning the status of the CI sector at a national level is important for company managers and also administrative authorities. The detailed knowledge of the market and its evolution over time is essential for defining industry strategies and policies, as they can support decisions concerning investments, subcontracting practices, mergers and acquisitions, and partnerships.
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For future research, it would be of interest to develop other measures of convergence, such as a nonparametric conditional β-convergence. This measure would enable the evaluation of convergence in a sample of DMUs with different characteristics. It would also be interesting to develop enhanced convergence indexes that can account for slacks, or to complement the indexes using the bootstrapping technique to test the significance of the results obtained. In addition, the methodology developed could be applied to other activity sectors (e.g. manufacturing or mining) to enable cross industry comparisons, or to other countries or worldwide regions to undertake international benchmarking exercises.
Acknowledgments
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The authors are grateful to Professor Jorge Moreira da Costa and to the regulatory board of Portuguese Construction and Real Estate (INCI), for enabling access to the data used in this study. The funding of this research through the scholarship SFRH/BPD/86294/2012 from the Portuguese Foundation of Science and Technology (FCT) is also gratefully acknowledged.
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References
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