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Estimation of multi-output production functions in commercial fisheries
Omega 42 (2014) 157–165
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Estimation of multi-output production functions in commercial fisheries$ Trevor C. Collier a, Aaron Mamula b, John Ruggiero c,n a
University of Dayton, USA NOAA Fisheries, Southwest Fisheries Science Center, Santa Cruz, CA 95060, USA c University of Dayton, 509 Miriam Hall, Dayton, OH 45469-2251, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 13 January 2012 Accepted 6 May 2013 Processed by B. Lev Available online 21 May 2013
Measuring the productivity of vessels in a multi-species fishery can be problematic. Typical regression techniques are not capable of handling multiple outputs while Data Envelopment Analysis (DEA) tends to ignore the stochastic nature of production. Applied economists have devoted considerable time to this problem and have developed several methods of dealing with the issue of multiple output technologies in commercial fisheries. Our paper contributes to this literature by providing another method for estimating production functions of vessels operating in multi-species fisheries. We utilize a two-stage model – with data from the West Coast Limited Entry Groundfish Trawl Fishery – using DEA to aggregate output in the first stage. This aggregate index is then used as the dependent variable in a regression framework, allowing for the estimation of the return to different inputs in fisheries production. This provides information that may be particularly important for fisheries managers. & 2013 Elsevier Ltd. All rights reserved.
Keywords: DEA Stochastic frontier Panel data Fisheries
1. Introduction The Magnuson-Stevens Fisheries Conservation and Management Act (MSA) established a legislative mandate for fisheries managers in the U.S. to consider the economic implications of management actions. In response, resource economists have built a large body of research around the issue of developing metrics to assess the economic performance of commercial fishing operations. The frequent omission of reliable input prices from fisheries data has led to the emergence of technical efficiency and related primal productivity measures as important performance metrics for commercial fisheries. The recent move by NOAA to adopt a policy of promoting catch-share management wherever feasible has renewed interest in the topic of performance metrics. In order to fulfill their responsibilities under MSA, and evaluate impacts of newly instituted catch-share programs on producers, it is important for policy makers to have reliable estimates of key performance indicators, such as technical efficiency. Evaluation of technical efficiency for fisheries in which production technologies are characterized by multiple outputs can involve an uncomfortable choice between deterministic methods (such as Data Envelopment Analysis (DEA)), which remain faithful to the multi-output nature of the technology but lack a statistical
☆ The authors want to acknowledge support from NOAA Fisheries and thank Janet Mason for data support and key insights about the fishery. n Corresponding author. Tel.: +1 937 229 2550; fax: +1 937 2293477. E-mail address: [email protected] (J. Ruggiero).
0305-0483/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.omega.2013.05.001
foundation addressing the inherent stochasticity of commercial fishing, and regression-based models (such as Stochastic Frontier Analysis (SFA)), which respects the uncertainty underlying fishing but is ill-equipped to model production of multiple outputs. Our study reconciles this dichotomy by leveraging a newly developed empirical technique [9] to estimate efficiency among harvesters in California's multi-species Groundfish Trawl Fishery. The CJR model has been shown to recover the true inefficiency from simulated data. Moreover, the method permits identification and estimation of key parameters of the production function. This is a potentially valuable development for fisheries managers as estimated marginal input productivities and output elasticities can have important implications for fisheries managers.1 This study provides the first policy-relevant, empirical application of the CJR model.
2. Literature review Measuring vessel performance in a multi-species fishery is complicated by the frequent lack of input price data and the multi-output nature of the production function. Typical regression techniques are not capable of handling multiple outputs. Thus, regression analysis struggles to accurately measure the productivity 1 One way that marginal productivities can be informative for fisheries managers is in the context of input controls (see [11,22,30] for discussion of input controls in fisheries management). If regulated and unregulated inputs can be easily substituted for one another, this may erode the efficacy of input controls.
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of vessels in a multi-species fishery, unless input or output prices are known. The issue of estimating productivity, technical efficiency and/ or harvesting capacity in a multi-species fishery in the absence of input or output prices is a familiar one and the one in which applied economists have invested considerable resources [35,39,19,20]. One approach to handle multiple outputs is to specify an output aggregate that can then be incorporated as the dependent variable in a regression model. One simple measure of aggregate output is the average (or sum) of all relevant species in a fishery (see e.g. [13]). Such an aggregate assumes the marginal rate of technical substitution is constant (and equal to one), implying a constant opportunity cost regardless of the output mix. This measure violates standard microeconomic principles. A popular alternative to the regression based approaches is Data Envelopment Analysis (DEA), a linear programming model that evaluates each production possibility relative to a piecewise linear frontier. DEA, coined by Charnes et al. [7], built on the pioneering work of Farrell [17] by allowing multiple inputs and outputs assuming constant returns to scale. Banker et al. [3] extended DEA to the variable returns to scale technology of Afriat [1]. See [23,24] for recent survey articles. In addition to allowing multiple outputs, the approach is axiomatic and hence does not require a priori specification of the production function. One limitation; however, is that the approach is deterministic and does not allow statistical noise. In this paper, we apply the twostage model of Collier et al. [9]. In the first stage, DEA is applied to obtain an index of aggregate output. In the second stage, regression is used. This approach allows consideration of statistical noise and multiple outputs.2 Finally, approaches based on the directional technology distance function [6,14,8,15] have emerged in recent years as popular methods for representing multi-output production technologies. Using a directional distance function approach, multi-output production functions can be estimated with standard econometric techniques. Applications of the directional distance function ([27,18]); however, have highlighted some important practical challenges, such as the difficulty of dealing with zeros in the output vector with a directional distance function model. This issue can be particularly limiting in the context of commercial fishing, where it can be common for fishermen to target particular species or species aggregates within a multi-species fishery. Our paper contributes to the applied production economics literature by providing another method for estimating production functions of vessels operating in multi-species fisheries. We utilize a recently developed DEA technique that transforms multiple outputs into an aggregate output index without forcing the production frontier to have a constant marginal rate of technical substitution [9]. This aggregate index is used as the dependent variable in a regression framework to estimate the return to effort and technical efficiency in fisheries production. We apply this model to data from the West Coast Limited Entry Groundfish Trawl Fishery. The paper proceeds as follows: in Section 2 we discuss the data and some background on the West Coast Limited Entry Groundfish Trawl Fishery, Section 3 explains the methodology, Section 4 discusses the results and Section 5 concludes.
3. Methodology Assume that each of n vessels employs a vector x ¼ ðx1 ; …; xm Þ of m inputs to produce a vector y ¼ ðy1 ; …; ys Þ of s outputs according to the technology set T ¼ ðx; yÞ : x can produce y: Input 2 An interesting extension beyond the scope of this paper would be to consider imposing the Ultra Passum law [29].
and output vectors for vessel j ðj ¼ 1; …; nÞ are given by ðx1j ; …; xmj Þ and ðy1j ; …; ysj Þ. The output set PðxÞ is defined as PðxÞ ¼ y : ðx; yÞ∈T: An output set consistent with microeconomic theory is presented in Fig. 1, where for convenience we assume only two outputs are produced. We observe a convex output set consistent with increasing opportunity costs. Three possibilities A–C are observed producing maximum output (with different mixes) given input usage. In the case of different types of catch, the increasing costs could reflect costs of relocation and/or that some inputs are better suited for certain species of fish. Past research has aggregated output by adding (or averaging) the various outputs. This implicitly assumes constant costs with an arbitrary weighting scheme. The problem of aggregating with this approach is revealed in Fig. 2, where the production set from Fig. 1 is replicated. As shown, it appears that production possibility A is producing less aggregate output than possibility C, which is itself observed producing less output than production possibility B. This contradicts the assumption that each observation is technically efficient, producing the maximum feasible output given the same input usage. This arises because of the improper linear aggregation. An alternative aggregation consistent with microeconomic theory was recently developed by Collier et al. [9]. Instead of assigning arbitrary weights, aggregate output is estimated nonparametrically using linear programming.3 Aggregate production is measured relative to a fixed isoquant. We illustrate the approach in Fig. 3, where we now assume that 5 vessels A–E are observed producing (catching) two species of fish, y1 and y2. Since the approach is nonparametric, it is not necessary to choose a particular functional form nor assign weights a priori. Similar to our previous figures, vessels A–C are operating efficiently on the isoquant of the output set. Two vessels, D and E are observed producing less output than vessels A–C. Vessel D (E) is producing 100y1D =y1G ð100y1E =y1G Þ percent as much aggregate output. There are two possibly explanations for why D and E produce less output: inefficiency or smaller input usage. Collier et al. [9] measured aggregate output ðAY i Þ for vessel i (i¼1 to n) using the following linear programming model: AY −1 i ¼ Max θ s:t: ∑nj¼ 1 λj ykj ≥θyki
∀ k ¼ 1; …; s
ð1Þ
∑nj¼ 1 λj ¼ 1 λj ≥0
∀ j:
The model seeks the maximum radial expansion (theta) of vessel i's output while maintaining the assumption of convexity. This is achieved via convex combinations, where λJ is the weight placed on observation j's output vector and convexity is defined by maintaining that each λJ is non-negative and they sum to one. The model is equivalent to a data envelopment model without constraints on inputs. One can view this as an additional assumption that all vessels are using the same amount of each input. Aggregate output is the inverse of the maximum radial expansion; for a given output mix, lower levels of output will lead to higher expansions. The resulting index AY i ≤1: Solution of (1) using the data from Fig. 3 results in AY A ¼ AY B ¼ AY C ¼ 1; implying that vessels A–C are achieving the highest aggregate output. Unlike the simple linear aggregation methods, this approach properly evaluates aggregate output. 3 This necessarily assumes that production is output homothetic, which allows us to generate any output isoquant from a base isoquant. The distance between isoquants is only a function of the inputs. See Shephard [36] for a formal proposition and proof.
T.C. Collier et al. / Omega 42 (2014) 157–165
y2
159
Grouping the intercept and the technical inefficiency term, Eq. (2) may be re-written as
A
yit ¼ ðα–ui Þ þ xit β þ εit ¼ αi þ xit β þ εit :
B
ð3Þ
Given the above assumption concerning the error term, ε, Eq. (3) may be estimated using the standard fixed effects (‘within’) estimator6. Estimates of ui that are strictly non-negative are then given by the deviation between each vessel-specific intercept and the maximum intercept:
P(x) C
u^ i ¼ max α^ i −α^ i ≥0:
y1
i
ð4Þ
By construction, the most efficient vessel is deemed completely efficient. The technical efficiency measure is defined as ^ i ¼ expð−u^ i Þ, which is bound by zero and unity. As shown in te ^ i is a consistent estimator as N,T-∞. In the present context [34], te N refers to the number of vessels, and T refers to the number of trips. Two potential drawbacks to the Schmidt and Sickles' [34] approach have been documented in the literature, and are worth mentioning. First, technical inefficiency is assumed to be timeinvariant. In the present context this means that vessels are assumed to have the same level of efficiency across multiple trips in the same year. Second, all heterogeneity across observations is counted as inefficiency. In other words, in excluding the inefficiency term in Eq. (2) there is no other source of individual heterogeneity. We attempt to control for time varying individual variables to minimize this potential issue.
Fig. 1. Standard production set.
y2 A B
C
y1 Fig. 2. Linear aggregation of output.
y2
4. Empirical analysis
A B
D E
y1E y1D y1G
C
y1
Fig. 3. Nonparametric estimation of output aggregate.
Likewise, we observe AY D ¼ y1D =y1G 4 AY E ¼ y1E =y1G : Hence, the method properly evaluates that vessel D produces more than vessel E, but less than A–C. With the aggregate output estimated using the method described above, we then specify a panel production function of the form:4 AY it ¼ α þ xit β–ui þ εit
ð2Þ
where AY it is our estimate of aggregate output for vessel i on trip t, xit is a vector of inputs, ui is the level of technical inefficiency in vessel i, and εit represents purely idiosyncratic shocks which are uncorrelated with the choice of inputs.5 Consistent with the interpretation of u as an inefficiency term, it is assumed that ui ≥0 for all i.
4
The assumption of output homotheticity allows us to specify this two-stage approach. See Primont and Primont [32] for a discussion in terms of distance functions. 5 Although we perform the analysis for one year at a time, we still have a panel of multiple vessels that each take multiple trips within a given year.
This study analyzes harvesting efficiency in the California Limited Entry Groundfish Trawl fishery. The fishery frequently ranks as one of the most significant commercial fisheries in the state in terms of both volume (pounds landed) and value and has been the subject of previous economic inquiry. We provide only a short background; curious readers are directed to Squires and Kirkley [37] and Mason et al. [25] for more depth. The fishery is regulated by species-specific catch limits, gear restrictions, and area closures. Each vessel may land a limited weight of each species over each two-month period throughout the year. Limits may differ by season and be adjusted during the year so that the total allowable catch is not exceeded by the end of the year. The data for the analysis consist of trawl logbook reports collected by California Department of Fish and Game for boats with limited entry groundfish permits from 2004 through 2009. Our study utilizes only those observations on trips returning to 0 California ports and only those vessels fishing south of 40∘ 10 N latitude7. Logbooks contain trawl locations defined by start and end points of each tow, hours towed, the weight and market category of fish landed from the tow, and landing port. Although our data provide an extremely fine breakdown of species harvested, we aggregate into a subset of popular species targets according to convention [38,16,12]. There are over 90 distinct species harvested by West Coast groundfish trawlers. From the landings data we identified several distinct species or species targets comprising at least 1% of the 6 At the suggestion of an anonymous referee, we chose to also estimate a random effects (maximum likelihood) model, assuming a truncated normal distribution for the efficiency term. Since the results are similar, we present the random effects model and the corresponding results in an appendix. 7 Since harvest limits differ significantly north and south of this parallel we confine our analysis to the southern portion of the fishery.
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total landings during the study period. Major species included were Dover sole, sablefish, thornyheads, California halibut, English sole, petrale sole, rex sole, sanddab, and several species of rockfish. In order to reduce the problem to a tractable dimensionality we followed convention and collapsed outputs to a group of four commonly targeted species groupings.8 The four output assemblages used in our model are: (1) a deepwater grouping consisting of Dover sole, sablefish, and thornyheads (DTS), (2) a nearshore mix (NSM), consisting of English, petrale, and rex sole as well as sanddabs, (3) California halibut and (4) a rockfish mix. These four species assemblages accounted for just over 96% of total groundfish harvest among vessels in our sample. This basic level of output aggregation is both practically attractive (a multi-species output model with 90 outputs would almost certainly contain too many zeros to carry out the appropriate matrix inversions) and intuitively reasonable (since trawl gear is necessarily unselective it sometimes makes more sense to think of outputs as co-occuring species complexes – such as DTS – rather than as individual species). Summary statistics for inputs and outputs used in our analysis are provided in Table 1. Table 2 displays the results of our CobbDouglas production function estimates for the years 2004– 2009.9,10 The table shows coefficient estimates and standard errors for production functions with two different dependent variables: (i) natural log of average catch over the four target species and (ii) natural log of the aggregate index calculated as described in linear programming model (1). The coefficient estimates on our inputs are fairly similar across the specifications with different dependent variables within a given year. The coefficient estimates on two hours and days at sea, both of which are measured in natural logs are positive in every instance. The coefficient estimate on “log of two hours” is statistically significant at the one-percent level in every year – except 2009—when using the aggregate index as the dependent variable. However, it is not statistically significant at the one-percent level in any of the years when using average catch as the dependent variable. The coefficient estimate on “log of days at sea” is statistically significant at better than the five-percent level in every year when using the average catch as the dependent variable and when using the aggregate index as the dependent variable. Although the present study focuses primarily on analyzing efficiency scores for vessels in our data sample, it deserves some emphasis that the CJR model provides these estimates while also allowing identification of important, policy-relevant quantities such as the return on fishing effort. The output elasticities, represented by estimated parameters of the production function can aid fisheries managers in a few ways. Notably, the nature of
8 Walden et al. [39] and Färe et al. [16] estimate harvesting capacity and technical efficiency for multi-species fisheries, restricting attention to a subset of total species landed. In addition, it should be noted that our species groupings are consistent with prior biological work on mixed-species modeling [28,33]. In particular Rogers and Pikitch find six assemblages to span the space of harvested species for the Oregon/Washington multi-species groundfish trawl fishery. Our groupings contain one assemblage dominated by a species not harvested outside California (California halibut). The remaining three species groupings are consistent with the Rogers and Pikitch assemblages. 9 We also estimated translog production functions for these same years. The coefficient estimates on the translog model were generally statistically insignificant, so we chose to discuss the Cobb-Douglas model here. The estimates of technical efficiency obtained using the translog model and the Cobb-Douglas model had correlations of greater than.99 for each of the years. These results are available from the authors upon request. 10 Appendix 2 presents the results of a maximum likelihood model where the efficiency term is assumed to have a truncated normal distribution. All technical efficiency estimates from the maximum likelihood model have correlations of greater than or equal to 0.935 with the technical efficiency estimates from the fixed effects model.
technical change in fisheries is of great interest to managers. Technical progress, which increases the effectiveness of effort, can erode the efficacy of input controls. Moreover, changes in the production function resulting from regulatory changes can inform managers regarding the economic consequences of management policy. Kirkley et al. [21] illustrate how estimated parameters of a Cobb-Douglas production function can be used to investigate embodied and disembodied technical change among commercial fishing operations. Although the present analysis does not address technical change (as our data do not include many of the potentially key drivers of such change11), the methodology employed here provides a framework into which drivers of technical change could be easily introduced. Although the coefficient estimates are generally similar across specifications with the two different dependent variables, the corresponding estimates of technical efficiency are not. Table 3 displays summary measures for the technical efficiency estimates using the two different dependent variables across the years 2004–2009, respectively. T-tests of the similarity of the means of technical efficiency were generated using the two different dependent variables. The hypotheses that these means are equal are rejected at the one percent level in every year, 2004–2009. Although there appears to be significant differences in the means of these two distributions of efficiency, KomolgorovSmirnov tests for the equality of distributions fail to reject the null hypothesis that the two distributions of efficiency are equal in each year, 2004–2009. Thus, we have generated two different estimates of efficiency using: (i) a new technique that estimates aggregate output and (ii) using average catch over multiple species. These estimates of technical efficiency have different means, but similar distributions. The question now becomes: does the ranking of vessels in terms of their efficiency score change depending on which measure of output is used? Theoretically speaking, the answer to this question is a resounding yes. Empirically, our models result in correlation coefficients between the efficiency scores estimated using the two different outputs ranging from 0.753 in 2006 to 0.937 in 2005. To illustrate some important differences between our model and the naïve average output, we isolate the subsample of vessels and trips which were estimated to be on the frontier of our measure of aggregate output. Since 2006 shows the lowest correlation, we will focus our discussion on how the estimates of efficiency differ using the two measures of output for specific vessels in 2006. Five trips taken by four different vessels were determined to be on the frontier of our measure of aggregate output in 2006, giving them aggregate output estimates of 100. Table 4 displays the amount of catch for each species group, the average catch, and technical efficiency scores estimated using each of the two measures of output, for each of these trips. You can see that one of the vessels (the one we are calling Vessel #36) was determined to have two trips that were on the frontier of our aggregate output measure. These two trips resulted in average catch of much greater magnitude than the other trips on our frontier and, not surprisingly, this vessel is estimated to be the most efficient vessel when average catch is used as the dependent variable in our production function. Conversely, Vessel #29 has the lowest average catch among the vessels on the frontier, because this vessel chose to target only one species group on this particular trip. In fact, this vessel caught the largest quantity of this species group on any one trip in 2006. 11 Notably, our data do not contain information on harvest limits, which one would suspect influence the production function. Extensions to the current analysis will work to address this data gap and investigate the impact of output constraints on production.
T.C. Collier et al. / Omega 42 (2014) 157–165
161
Table 1 Summary statistics for inputs and outputs. Variable
Min
Mean
Max
St. Dev.
Outputs DTS (lbs) NSM (lbs) CA Halibut (lbs) Rockfish (lbs)
0 0 0 0
6.795 2.135 334 1.102
87.400 50.214 16.852 76.090
11.345 3.927 1.126 3.924
Inputs Tow hours Days at sea
0 0
17.95 1.70
75.23 6.29
14.16 1.08
Table 2 Regression results by year.a Two hours (ln)
2004 Avg. output Aggregate output index 2005 Avg. output Aggregate output index 2006 Avg. output Aggregate output index 2007 Avg. output Aggregate output index 2008 Avg. output Aggregate output index 2009 Avg. output Aggregate output index
a
Days at sea (ln)
Constant
R-square within
between
overall
0.165 (0.165) 0.409nnn (0.117)
0.861nnn (0.137) 0.559nnn (0.120)
6.807nnn (0.394) 1.431nnn (0.280)
0.211
0.791
0.582
0.281
0.767
0.562
0.347n (0.199) 0.593nnn (0.146)
0.822nnn (0.248) 0.415nn (0.168)
5.376nnn (0.451) 0.783nn (0.336)
0.274
0.592
0.626
0.345
0.692
0.678
0.232n (0.135) 0.400nnn (0.068)
1.102nnn (0.300) 0.575nnn (0.117)
5.563nnn (0.295) 1.351nnn (0.162)
0.294
0.784
0.692
0.279
0.757
0.638
0.242 (0.147) 0.590nnn (0.093)
0.915nn (0.355) 0.135 (0.120)
5.801nnn (0.289) 1.194nnn (0.214)
0.190
0.713
0.630
0.248
0.703
0.624
0.479nn (0.179) 0.576nnn (0.155)
0.550n (0.317) 0.312nn (0.144)
5.417nnn (0.367) 1.360nnn (0.366)
0.272
0.740
0.675
0.323
0.873
0.680
0.094 (0.180) 0.245 (0.160)
1.00nnn (0.269) 0.665nnn (0.145)
6.293nnn (0.393) 2.042nnn (0.382)
0.253
0.776
0.670
0.246
0.836
0.634
Asterisks indicate the significance level of the estimate: “nnn” −1% level, “nn” −5% level; “n” −10% level. Standard errors are in parentheses.
There are a potentially large number of reasons why a vessel may target a particular species or species group12 (existing contracts with buyers, unobserved characteristics of a vessel or skipper knowledge of spatial distribution of certain species/species groups for example). Since many of these will generally be unobservable (to the researcher) it is important to have a method capable of accurately assessing the performance of many different types of operators. When estimating a production function with average catch as the dependent variable, one is assuming that all species of fish are perfect substitutes for one another. This type of faulty assumption would lead to an estimate of technical efficiency of
12 It should also be noted that an individual vessel may pursue both generalist and specialist strategies on different trips throughout the year.
0.084 for Vessel #29, whereas using our aggregate output measure leads to a technical efficiency estimate of 0.362. This is one example of how this method of estimating aggregate output is useful when data on market prices is unavailable. An important result here that warrants some emphasis is that specialists – vessels harvesting a small number of the available species – are disproportionately punished by the average catch method (in estimated efficiency terms) relative to generalists. Note the large discrepancies between efficiency scores calculated by the CJR method and the average catch method for vessel #29 in all years. This can have implications in practical settings such as fisheries management where regulators frequently concern themselves with the harvesting efficiency of commercial fishing operations. It is important for managers to be able to evaluate efficiency for a heterogeneous group of producers in a multi-output production setting.
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Table 3 Mean estimated technical efficiency.
Agg. output index Avg. output Difference t-score N
2004
2005
2006
2007
2008
2009
0.358 0.283 0.075 4.752 34
0.384 0.284 0.100 6.699 31
0.476 0.312 0.164 4.699 23
0.414 0.291 0.123 6.473 23
0.441 0.367 0.074 2.908 21
0.436 0.355 0.081 3.080 17
Table 4 Vessels with aggregate output index ¼100 in each year. Year
2004 2004 2004 2004 2004 2005 2005 2005 2005 2005 2005 2006 2006 2006 2006 2006 2007 2007 2007 2007 2007 2007 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2009 2009 2009 2009 2009 2009 2009 2009
Vessel
9 9 29 36 37 9 27 27 36 36 36 9 29 34 36 36 21 26 33 33 33 36 9 21 21 21 21 26 26 30 30 30 36 36 21 30 36 36 36 36 36 36
Trip
1 2 1 1 1 1 1 2 1 2 3 1 1 1 1 2 1 1 1 2 3 1 1 1 2 3 4 1 2 1 2 3 1 2 1 1 1 2 3 4 5 6
y1
10.292 30.188 – 52.748 87.400 689 – 45.152 40.174 12.710 38.973 786 – 9.220 55.574 46.002 – 17.750 41.324 1.166 10.795 25.046 33.932 4.083 124 – 656 20.560 9.620 31.650 30.975 28.477 24.896 480 – 26.333 16.615 31.813 18.785 11.506 15.661 23.816
y2
50.214 46.986 – 3.660 3.708 29.704 10 – 818 1.852 4.555 20.048 – 2.362 1.705 1.878 – 3.568 13.334 23.269 307 1.618 – 23.460 2.139 10 9.916 379 14.714 2.566 4.920 9.352 304 24.302 – 7.614 108 1.824 84 19.588 5.646 2.025
y3
5.132 4.862 – 76.090 6.864 3.408 – 720 19.914 26.595 – 951 – 16.957 6.645 17.148 – 16.284 498 22.888 7.246 13.123 70 828 12.811 – 1.055 17.601 13.021 3.069 2.750 1.791 14.545 10.879 – 933 16.916 7.439 16.508 6.967 14.560 14.251
5. Conclusions Fisheries managers and policy makers are responsible for monitoring performance of commercial fishing fleets and have a legislative mandate to consider the efficiency implications of fisheries management policy.13 A key complication in the measurement of economic performance among fishing vessels is the lack of available data on
13 See NOAA Fisheries' Office of Science and Technology (http://www.st.nmfs. noaa.gov/st5/ExecutiveLegislativeMandates.html) for detail on these responsibilities under the Magnuson-Stevens Fisheries Conservation and Management Act.
y4
– – 16.852 – – – 9.499 – – – – – 8.005 – – – 3.720 – – – 3.044 – – 78 1.820 3.010 2.862 – – – – – – – 4.964 – – – – – – –
Avg. catch
16.410 20.509 4.213 33.125 24.493 8.450 2.377 11.468 15.227 10.289 10.882 5.446 2.001 7.135 15.981 16.257 930 9.401 13.789 11.831 5.348 9.947 8.501 7.112 4.224 755 3.622 9.635 9.339 9.321 9.661 9.905 9.936 8.915 1.241 8.720 8.410 10.269 8.844 9.515 8.967 10.023
Technical efficiency Aggregate output
Avg. catch
0.415 0.415 0.320 1.000 0.582 0.577 0.349 0.349 1.000 1.000 1.000 0.605 0.362 1.000 0.973 0.973 0.470 0.664 0.679 0.679 0.679 1.000 0.460 0.607 0.607 0.607 0.607 0.612 0.612 0.625 0.625 0.625 1.000 1.000 0.624 0.663 1.000 1.000 1.000 1.000 1.000 1.000
0.399 0.399 0.102 1.000 0.511 0.483 0.087 0.087 1.000 1.000 1.000 0.427 0.084 0.705 1.000 1.000 0.131 0.477 0.399 0.399 0.399 1.000 0.437 0.276 0.276 0.276 0.276 0.624 0.624 0.627 0.627 0.627 1.000 1.000 0.229 0.671 1.000 1.000 1.000 1.000 1.000 1.000
input prices. In the absence of such data, technical efficiency has emerged as an important performance metric. We utilize a recently developed DEA technique to transform multiple outputs into an aggregate output index, without forcing the production frontier to have a constant marginal rate of technical substitution [9]. This aggregate index is used as the dependent variable in a regression framework to estimate the return to effort and technical efficiency in fisheries production. We apply this model to data from the West Coast Limited Entry Groundfish Trawl Fishery. Our research addresses an important issue in efficiency measurement among fishing vessels: modeling the multi-output production technology present in multi-species fisheries while
T.C. Collier et al. / Omega 42 (2014) 157–165
163
Table A1 Regression results by year using CJR method and order M method.a Two hours (ln)
2004 Order M index Aggregate output index 2005 Order M index Aggregate output index 2006 Order M index Aggregate output index 2007 Order M index Aggregate output index 2008 Order M index Aggregate output index 2009 Order M index Aggregate output index
a
Days at sea (ln)
Constant
R-square
Correlation between efficiency estimates
within
between
overall
0.422nnn (0.114) 0.409nnn (0.117)
0.522nnn (0.122) 0.559nnn (0.120)
−2.220nnn (0.272) 1.431nnn (0.280)
0.275
0.741
0.54
0.281
0.767
0.562
0.595nnn (0.137) 0.593nnn (0.146)
0.398nn (0.179) 0.415nn (0.168)
−2.855nnn (0.305) 0.783nn (0.336)
0.337
0.710
0.686
0.345
0.692
0.678
0.399nnn (0.068) 0.400nnn (0.068)
0.595nnn (0.125) 0.575nnn (0.117)
−2.425nnn (0.161) 1.351nnn (0.162)
0.279
0.762
0.657
0.279
0.757
0.638
0.513nnn (0.082) 0.590nnn (0.093)
0.273nn (0.128) 0.135 (0.120)
−2.280nnn (0.188) 1.194nnn (0.214)
0.250
0.707
0.643
0.248
0.703
0.624
0.563nnn (0.155) 0.576nnn (0.155)
0.323nn (0.140) 0.312nn (0.144)
−2.402nnn (0.368) 1.360nnn (0.366)
0.314
0.854
0.656
0.323
0.873
0.624
0.212 (0.167) 0.245 (0.160)
0.686nnn (0.151) 0.665nnn (0.145)
−1.614nnn (0.398) 2.042nnn (0.382)
0.228
0.823
0.610
0.246
0.836
0.634
0.984
0.988
0.987
0.99
0.991
0.996
Asterisks indicate the significance level of the estimate: “nnn” −1% level, “nn” −5% level; “n” −10% level. Standard errors are in parentheses.
respecting the inherent stochasticity of commercial capture fisheries. Moreover, our method permits recovery of important, economically meaningful parameters. The estimated coefficients from the translog and Cobb-Douglas production functions represent marginal productivities and output elasticities of the production function. Given the emphasis that has been placed on input controls as a fisheries management tool, these quantities provide potentially valuable information. For instance welfare effects of input controls can be approximated using estimated marginal value product of affected inputs. Additionally, in cases where managers are interested in imposing input controls to limit output, it should be recognized that not all inputs influence fish production identically at the margin. Our method, in contrast to other multi-output methodologies, permits estimation of these important quantities and their associated confidence bounds. Our analysis highlights the potential problems associated with aggregating multiple outputs by taking a simple average. In the absence of price data, this type of aggregation has been used in previous research and can lead to poor estimates of technical efficiency. In particular this is problematic when some outputs are more difficult to produce (species are more difficult to catch). We show that using the two-stage approach developed by Collier et al. [9], we can provide more reasonable estimates of technical efficiency without the need for input or output prices.
the outer frontier to estimate aggregate output. If this outer frontier is influenced by outliers then our estimate of aggregate output will be biased. As a check for robustness, we consider an order-m frontier introduced in [5]. As discussed in [10], the orderm frontier does not envelop all data and hence, is less sensitive to potential outliers.14 Because we are interested in the outermost output isoquant, the procedure involves sampling from all data points with replacement. For each sample, we estimate the aggregate output. This process is repeated a large number of times (2000 in our application) and the average level of aggregate output is used as our measure of aggregate output. In Table A1, we report the stochastic frontier results using the order-m approach (and the approach used in our paper for comparison purposes). The results are similar across approaches. The standard stochastic production frontier model of Aigner et al. [2] and Meeusen and van den Broeck [26] took the following form: lnyi ¼ β0 þ β1 lnxi þ vi −ui
where yi is the output of firm i, xi is an input for firm i, vi is a random error term for firm i from a standard normal distribution and ui is an inefficiency component for firm i from that must be positive. A panel data version of the stochastic frontier model,
Appendix In this appendix, we address some of the concerns raised by two anonymous referees. A potential concern is with our use of
14
Technical details can be found in [10].
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T.C. Collier et al. / Omega 42 (2014) 157–165
Table A2 Results by year using fixed effects and maximum likelihood.
2004 ML truncated normal Fixed effects 2005 ML Truncated Normal Fixed Effects 2006 ML truncated normal Fixed effects 2007 ML truncated normal Fixed effects 2008 ML truncated normal Fixed effects 2009 ML truncated normal Fixed effects
Two Days at hours (ln) sea (ln)
Constant
Correlation between efficiency estimates
0.412nnn (0.112) 0.409nnn (0.117)
0.617nnn (0.146) 0.559nnn (0.120)
2.145nnn (0.406) 1.431nnn (0.280)
0.601nnn (0.176) 0.593nnn (0.146)
0.446nn (0.151) 0.415nn (0.168)
−1.938nnn (0.463) 0.996 0.783nn (0.336)
0.398nnn (0.071) 0.400nnn (0.068)
0.629nnn (0.112) 0.575nnn (0.117)
2.313nnn (0.208) 1.351nnn (0.162)
0.591nnn (0.069) 0.590nnn (0.093)
0.186 (0.115) 0.135 (0.120)
2.059nnn (0.336) 1.194nnn (0.214)
0.600nnn (0.067) 0.576nnn (0.155)
0.350nnn (0.049) 0.312nn (0.144)
2.351nnn (0.335) 1.360nnn (0.366)
0.244nnn (0.088) 0.245 (0.160)
0.716nnn (0.093) 0.665nnn (0.145)
3.108nnn (0.377) 2.042nnn (0.382)
0.935
0.985
0.987
0.99
0.995
Asterisks indicate the significance level of the estimate: “nnn” 1% level, “nn” 5% level; “n” 10% level.
with time invariant efficiency takes the following form: lnyit ¼ β0 þ β1 lnxit þ vit −ui This time invariant panel data version can be estimated using the Schmidt and Sickles [34] fixed effects model – as is done in the body of this paper – or using random effects – as we do in this appendix. This random effects model was used by Pitt and Lee [31] assuming a half normal distribution for the inefficiency term and by Battese and Coelli [4] assuming a truncated normal distribution for the inefficiency term. After assuming that that the inefficiency component is constant across time for each firm and assuming the inefficiency term is a random draw from some specific positive distribution, we can estimate this model using maximum likelihood techniques. The results displayed in Table A2 were estimated with maximum likelihood techniques assuming a truncated normal distribution for the inefficiency term (as in [4]).17 We list the coefficient estimates from the fixed effects model below these maximum likelihood coefficient estimates and display the correlations between these two estimates in the last column of the table. Every correlation is above or equal to 0.984. Thus we only reported the fixed effects estimates in the body of the paper.
17 The maximum likelihood standard errors are bootstrapped with 100 replications, except for 2007 ( based on 10), 2008 (based on 5), and 2009 (based on 10)
References [1] Afriat S. Efficiency estimation of production functions. International Economic Review 1972;13:568–598. [2] Aigner DJ, Lovell CAK, Schmidt P. Formulation and estimation of stochastic frontier production function models. Journal of Econometrics 1977;6: 21–37. [3] Banker R, Charnes A, Cooper WW. Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 1984;30:1078–1092. [4] Battese GE, Coelli TJ. Production of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics 1988;38:387–399. [5] Cazals C, Florens J, Simar L. Nonparametric frontier estimation: a robust approach. Journal of Econometrics 2002;106:1–25. [6] Chambers R, Chung Y, Färe R. Profit, Directional distance function and Nerlovian efficiency. Journal of Optimization Theory and Application 1998; v.98(2):351–364. [7] Charnes A, Cooper WW, Rhodes E. Measuring the inefficiency of decision making units. European Journal of Operational Research 1978;2: 429–444. [8] Chen Y, Du J, Huo J. Super-efficiency based on modified directional distance function. Omega 2013;41(3):632 625. [9] Collier T, Johnson AL, Ruggiero J. Technical efficiency estimation with multiple inputs and multiple outputs using regression analysis. European Journal of Operational Research 2011;2:153–160. [10] Dario C, Simar L. Advanced robust and nonparametric methods in efficiency analysis. NY: Springer; 2007. [11] Dupont D. Testing for input substitution in a regulated fishery. American Journal of Agricultural Economics 1991;73(1):155–164. [12] Dupont D, Grafton RQ, Kirkley J, Squires. D. Capacity utilization measures and excess capacity in multi-product privatized fisheries. Resource and Energy Economics 2002;24(3):193–210. [13] Esmaeili A. Technical efficiency analysis for the Iranian fishery in the Persian Gulf. ICES Journal of Marine Science 2006;63:1759–1764. [14] Färe R, Grosskopf S. Theory and application of directional distance functions. Journal of Productivity Analysis 2000;13(2):93–103. [15] Färe R, Grosskopf S. DEA, directional distance functions and positive affine data transformations. Omega 2013;41(1):28–30. [16] Färe R, Kirkley J, Walden J. Adjusting technical efficiency to reflect discarding: the case of the U.S. Georges Bank multi-species otter trawl fishery. Fisheries Research 2006;78:257–265. [17] Farrell MJ. The measurement of productive efficiency. Journal of the Royal Statistical Society. Series A (General) 1957;120(3):253–290. [18] Felthoven RG. Effects of the American Fisheries Act on capacity, utilization and technical efficiency. Marine Resource Economics 2002;17(3):181–205. [19] Felthoven RG, Morrison Paul CJ. Multi-output, nonfrontier primal measures of capacity and capacity utilization. American Journal of Agricultural Economics 2004;86(3):619–633. [20] Felthoven RL, Horrace WC, Schnier KE. Estimating heterogeneous capacity and capacity utilization measures in a multi-species fishery. Journal of Productivity Analysis 2009;32:173–189. [21] Kirkley J, Morrison Paul CJ, Cunningham S, Catazano. J. Embodied and disembodied technical change in fisheries: an analysis of the Sete Trawl Fishery 1985–1999. Environmental and Resource Economics 2004;29: 191–217. [22] Kompas T, Che TN, Grafton. RQ. Technical efficiency effects of input controls: evidence from Australia's banana prawn fishery. Applied Economics 2004;36 (15):1631–1641. [23] Liu J, Lu L, Lu W-M, Lin B. Data Envelopment Analysis 1978–2010: a citation based literature survey. Omega 2013;41(1):3–15. [24] Liu J, Lu L, Lu W-M, Lin B. A survey of DEA applications. Omega 2013;41 (5):893–902. [25] Mason J., Kosaka R., Mamula A., Speir C. Effort changes around a marine reserve: the case of the California Rockfish Conservation Area, Working paper; 2011. [26] Meeusen W, van den Broeck J. Efficiency estimation from Cobb-Douglas production functions with composed error. International Economic Review 1977;18:435–444. [27] Morrison-Paul CJ, Johnston W, Frengley G. Efficiency in New Zealand sheep and cattle farming: the impacts of regulatory reform. Review of Economic and Statistics 2000;82:325–337. [28] Murawski S. Mixed-species yield-per-recruitment analysis for technological interactions. Canadian Journal of Fisheries and Aquatic Sciences 1984;41 (6):897–916. [29] Olesen O, Petersen NC. Imposing the regular Ultra Passum law in DEA models. Omega 2013;41(1):16–27. [30] Pascoe S, Robinson C. Input controls, input substitution and profit maximization in the English Channel beam trawl fishery. Journal of Agricultural Economics 2008;49(1):16–33. [31] Pitt MM, Lee L-F. Measurement and sources of technical inefficiency in the Indonesian weaving industry. Journal of Development Economics 1981;9: 43–64. [32] Primont D, Primont D. Homothetic non-parametric production models. Economics Letters 1994;45:191–195.
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[33] Rogers J, Pikitch E. Numerical definition of groundfish assemblages caught off the coast of Oregon and Washington using commercial fishing strategies. Candadian Journal of Fisheries and Aquatic Sciences 1992;94: 2648–2656. [34] Schmidt P, Sickles R. Production frontiers and panel data. Journal of Business and Economic Statistics 1984;2(4):367–374. [35] Segerson K, Squires D. On the measurement of economic capacity utilization for multiproduct industries. Journal of Econometrics 1990;44(3):81–95. [36] Shephard RW. Theory of cost and production functions. Princeton, N.J.: Princeton University Press; 1970.
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Contents lists available at SciVerse ScienceDirect
Omega journal homepage: www.elsevier.com/locate/omega
Estimation of multi-output production functions in commercial fisheries$ Trevor C. Collier a, Aaron Mamula b, John Ruggiero c,n a
University of Dayton, USA NOAA Fisheries, Southwest Fisheries Science Center, Santa Cruz, CA 95060, USA c University of Dayton, 509 Miriam Hall, Dayton, OH 45469-2251, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 13 January 2012 Accepted 6 May 2013 Processed by B. Lev Available online 21 May 2013
Measuring the productivity of vessels in a multi-species fishery can be problematic. Typical regression techniques are not capable of handling multiple outputs while Data Envelopment Analysis (DEA) tends to ignore the stochastic nature of production. Applied economists have devoted considerable time to this problem and have developed several methods of dealing with the issue of multiple output technologies in commercial fisheries. Our paper contributes to this literature by providing another method for estimating production functions of vessels operating in multi-species fisheries. We utilize a two-stage model – with data from the West Coast Limited Entry Groundfish Trawl Fishery – using DEA to aggregate output in the first stage. This aggregate index is then used as the dependent variable in a regression framework, allowing for the estimation of the return to different inputs in fisheries production. This provides information that may be particularly important for fisheries managers. & 2013 Elsevier Ltd. All rights reserved.
Keywords: DEA Stochastic frontier Panel data Fisheries
1. Introduction The Magnuson-Stevens Fisheries Conservation and Management Act (MSA) established a legislative mandate for fisheries managers in the U.S. to consider the economic implications of management actions. In response, resource economists have built a large body of research around the issue of developing metrics to assess the economic performance of commercial fishing operations. The frequent omission of reliable input prices from fisheries data has led to the emergence of technical efficiency and related primal productivity measures as important performance metrics for commercial fisheries. The recent move by NOAA to adopt a policy of promoting catch-share management wherever feasible has renewed interest in the topic of performance metrics. In order to fulfill their responsibilities under MSA, and evaluate impacts of newly instituted catch-share programs on producers, it is important for policy makers to have reliable estimates of key performance indicators, such as technical efficiency. Evaluation of technical efficiency for fisheries in which production technologies are characterized by multiple outputs can involve an uncomfortable choice between deterministic methods (such as Data Envelopment Analysis (DEA)), which remain faithful to the multi-output nature of the technology but lack a statistical
☆ The authors want to acknowledge support from NOAA Fisheries and thank Janet Mason for data support and key insights about the fishery. n Corresponding author. Tel.: +1 937 229 2550; fax: +1 937 2293477. E-mail address: [email protected] (J. Ruggiero).
0305-0483/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.omega.2013.05.001
foundation addressing the inherent stochasticity of commercial fishing, and regression-based models (such as Stochastic Frontier Analysis (SFA)), which respects the uncertainty underlying fishing but is ill-equipped to model production of multiple outputs. Our study reconciles this dichotomy by leveraging a newly developed empirical technique [9] to estimate efficiency among harvesters in California's multi-species Groundfish Trawl Fishery. The CJR model has been shown to recover the true inefficiency from simulated data. Moreover, the method permits identification and estimation of key parameters of the production function. This is a potentially valuable development for fisheries managers as estimated marginal input productivities and output elasticities can have important implications for fisheries managers.1 This study provides the first policy-relevant, empirical application of the CJR model.
2. Literature review Measuring vessel performance in a multi-species fishery is complicated by the frequent lack of input price data and the multi-output nature of the production function. Typical regression techniques are not capable of handling multiple outputs. Thus, regression analysis struggles to accurately measure the productivity 1 One way that marginal productivities can be informative for fisheries managers is in the context of input controls (see [11,22,30] for discussion of input controls in fisheries management). If regulated and unregulated inputs can be easily substituted for one another, this may erode the efficacy of input controls.
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of vessels in a multi-species fishery, unless input or output prices are known. The issue of estimating productivity, technical efficiency and/ or harvesting capacity in a multi-species fishery in the absence of input or output prices is a familiar one and the one in which applied economists have invested considerable resources [35,39,19,20]. One approach to handle multiple outputs is to specify an output aggregate that can then be incorporated as the dependent variable in a regression model. One simple measure of aggregate output is the average (or sum) of all relevant species in a fishery (see e.g. [13]). Such an aggregate assumes the marginal rate of technical substitution is constant (and equal to one), implying a constant opportunity cost regardless of the output mix. This measure violates standard microeconomic principles. A popular alternative to the regression based approaches is Data Envelopment Analysis (DEA), a linear programming model that evaluates each production possibility relative to a piecewise linear frontier. DEA, coined by Charnes et al. [7], built on the pioneering work of Farrell [17] by allowing multiple inputs and outputs assuming constant returns to scale. Banker et al. [3] extended DEA to the variable returns to scale technology of Afriat [1]. See [23,24] for recent survey articles. In addition to allowing multiple outputs, the approach is axiomatic and hence does not require a priori specification of the production function. One limitation; however, is that the approach is deterministic and does not allow statistical noise. In this paper, we apply the twostage model of Collier et al. [9]. In the first stage, DEA is applied to obtain an index of aggregate output. In the second stage, regression is used. This approach allows consideration of statistical noise and multiple outputs.2 Finally, approaches based on the directional technology distance function [6,14,8,15] have emerged in recent years as popular methods for representing multi-output production technologies. Using a directional distance function approach, multi-output production functions can be estimated with standard econometric techniques. Applications of the directional distance function ([27,18]); however, have highlighted some important practical challenges, such as the difficulty of dealing with zeros in the output vector with a directional distance function model. This issue can be particularly limiting in the context of commercial fishing, where it can be common for fishermen to target particular species or species aggregates within a multi-species fishery. Our paper contributes to the applied production economics literature by providing another method for estimating production functions of vessels operating in multi-species fisheries. We utilize a recently developed DEA technique that transforms multiple outputs into an aggregate output index without forcing the production frontier to have a constant marginal rate of technical substitution [9]. This aggregate index is used as the dependent variable in a regression framework to estimate the return to effort and technical efficiency in fisheries production. We apply this model to data from the West Coast Limited Entry Groundfish Trawl Fishery. The paper proceeds as follows: in Section 2 we discuss the data and some background on the West Coast Limited Entry Groundfish Trawl Fishery, Section 3 explains the methodology, Section 4 discusses the results and Section 5 concludes.
3. Methodology Assume that each of n vessels employs a vector x ¼ ðx1 ; …; xm Þ of m inputs to produce a vector y ¼ ðy1 ; …; ys Þ of s outputs according to the technology set T ¼ ðx; yÞ : x can produce y: Input 2 An interesting extension beyond the scope of this paper would be to consider imposing the Ultra Passum law [29].
and output vectors for vessel j ðj ¼ 1; …; nÞ are given by ðx1j ; …; xmj Þ and ðy1j ; …; ysj Þ. The output set PðxÞ is defined as PðxÞ ¼ y : ðx; yÞ∈T: An output set consistent with microeconomic theory is presented in Fig. 1, where for convenience we assume only two outputs are produced. We observe a convex output set consistent with increasing opportunity costs. Three possibilities A–C are observed producing maximum output (with different mixes) given input usage. In the case of different types of catch, the increasing costs could reflect costs of relocation and/or that some inputs are better suited for certain species of fish. Past research has aggregated output by adding (or averaging) the various outputs. This implicitly assumes constant costs with an arbitrary weighting scheme. The problem of aggregating with this approach is revealed in Fig. 2, where the production set from Fig. 1 is replicated. As shown, it appears that production possibility A is producing less aggregate output than possibility C, which is itself observed producing less output than production possibility B. This contradicts the assumption that each observation is technically efficient, producing the maximum feasible output given the same input usage. This arises because of the improper linear aggregation. An alternative aggregation consistent with microeconomic theory was recently developed by Collier et al. [9]. Instead of assigning arbitrary weights, aggregate output is estimated nonparametrically using linear programming.3 Aggregate production is measured relative to a fixed isoquant. We illustrate the approach in Fig. 3, where we now assume that 5 vessels A–E are observed producing (catching) two species of fish, y1 and y2. Since the approach is nonparametric, it is not necessary to choose a particular functional form nor assign weights a priori. Similar to our previous figures, vessels A–C are operating efficiently on the isoquant of the output set. Two vessels, D and E are observed producing less output than vessels A–C. Vessel D (E) is producing 100y1D =y1G ð100y1E =y1G Þ percent as much aggregate output. There are two possibly explanations for why D and E produce less output: inefficiency or smaller input usage. Collier et al. [9] measured aggregate output ðAY i Þ for vessel i (i¼1 to n) using the following linear programming model: AY −1 i ¼ Max θ s:t: ∑nj¼ 1 λj ykj ≥θyki
∀ k ¼ 1; …; s
ð1Þ
∑nj¼ 1 λj ¼ 1 λj ≥0
∀ j:
The model seeks the maximum radial expansion (theta) of vessel i's output while maintaining the assumption of convexity. This is achieved via convex combinations, where λJ is the weight placed on observation j's output vector and convexity is defined by maintaining that each λJ is non-negative and they sum to one. The model is equivalent to a data envelopment model without constraints on inputs. One can view this as an additional assumption that all vessels are using the same amount of each input. Aggregate output is the inverse of the maximum radial expansion; for a given output mix, lower levels of output will lead to higher expansions. The resulting index AY i ≤1: Solution of (1) using the data from Fig. 3 results in AY A ¼ AY B ¼ AY C ¼ 1; implying that vessels A–C are achieving the highest aggregate output. Unlike the simple linear aggregation methods, this approach properly evaluates aggregate output. 3 This necessarily assumes that production is output homothetic, which allows us to generate any output isoquant from a base isoquant. The distance between isoquants is only a function of the inputs. See Shephard [36] for a formal proposition and proof.
T.C. Collier et al. / Omega 42 (2014) 157–165
y2
159
Grouping the intercept and the technical inefficiency term, Eq. (2) may be re-written as
A
yit ¼ ðα–ui Þ þ xit β þ εit ¼ αi þ xit β þ εit :
B
ð3Þ
Given the above assumption concerning the error term, ε, Eq. (3) may be estimated using the standard fixed effects (‘within’) estimator6. Estimates of ui that are strictly non-negative are then given by the deviation between each vessel-specific intercept and the maximum intercept:
P(x) C
u^ i ¼ max α^ i −α^ i ≥0:
y1
i
ð4Þ
By construction, the most efficient vessel is deemed completely efficient. The technical efficiency measure is defined as ^ i ¼ expð−u^ i Þ, which is bound by zero and unity. As shown in te ^ i is a consistent estimator as N,T-∞. In the present context [34], te N refers to the number of vessels, and T refers to the number of trips. Two potential drawbacks to the Schmidt and Sickles' [34] approach have been documented in the literature, and are worth mentioning. First, technical inefficiency is assumed to be timeinvariant. In the present context this means that vessels are assumed to have the same level of efficiency across multiple trips in the same year. Second, all heterogeneity across observations is counted as inefficiency. In other words, in excluding the inefficiency term in Eq. (2) there is no other source of individual heterogeneity. We attempt to control for time varying individual variables to minimize this potential issue.
Fig. 1. Standard production set.
y2 A B
C
y1 Fig. 2. Linear aggregation of output.
y2
4. Empirical analysis
A B
D E
y1E y1D y1G
C
y1
Fig. 3. Nonparametric estimation of output aggregate.
Likewise, we observe AY D ¼ y1D =y1G 4 AY E ¼ y1E =y1G : Hence, the method properly evaluates that vessel D produces more than vessel E, but less than A–C. With the aggregate output estimated using the method described above, we then specify a panel production function of the form:4 AY it ¼ α þ xit β–ui þ εit
ð2Þ
where AY it is our estimate of aggregate output for vessel i on trip t, xit is a vector of inputs, ui is the level of technical inefficiency in vessel i, and εit represents purely idiosyncratic shocks which are uncorrelated with the choice of inputs.5 Consistent with the interpretation of u as an inefficiency term, it is assumed that ui ≥0 for all i.
4
The assumption of output homotheticity allows us to specify this two-stage approach. See Primont and Primont [32] for a discussion in terms of distance functions. 5 Although we perform the analysis for one year at a time, we still have a panel of multiple vessels that each take multiple trips within a given year.
This study analyzes harvesting efficiency in the California Limited Entry Groundfish Trawl fishery. The fishery frequently ranks as one of the most significant commercial fisheries in the state in terms of both volume (pounds landed) and value and has been the subject of previous economic inquiry. We provide only a short background; curious readers are directed to Squires and Kirkley [37] and Mason et al. [25] for more depth. The fishery is regulated by species-specific catch limits, gear restrictions, and area closures. Each vessel may land a limited weight of each species over each two-month period throughout the year. Limits may differ by season and be adjusted during the year so that the total allowable catch is not exceeded by the end of the year. The data for the analysis consist of trawl logbook reports collected by California Department of Fish and Game for boats with limited entry groundfish permits from 2004 through 2009. Our study utilizes only those observations on trips returning to 0 California ports and only those vessels fishing south of 40∘ 10 N latitude7. Logbooks contain trawl locations defined by start and end points of each tow, hours towed, the weight and market category of fish landed from the tow, and landing port. Although our data provide an extremely fine breakdown of species harvested, we aggregate into a subset of popular species targets according to convention [38,16,12]. There are over 90 distinct species harvested by West Coast groundfish trawlers. From the landings data we identified several distinct species or species targets comprising at least 1% of the 6 At the suggestion of an anonymous referee, we chose to also estimate a random effects (maximum likelihood) model, assuming a truncated normal distribution for the efficiency term. Since the results are similar, we present the random effects model and the corresponding results in an appendix. 7 Since harvest limits differ significantly north and south of this parallel we confine our analysis to the southern portion of the fishery.
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total landings during the study period. Major species included were Dover sole, sablefish, thornyheads, California halibut, English sole, petrale sole, rex sole, sanddab, and several species of rockfish. In order to reduce the problem to a tractable dimensionality we followed convention and collapsed outputs to a group of four commonly targeted species groupings.8 The four output assemblages used in our model are: (1) a deepwater grouping consisting of Dover sole, sablefish, and thornyheads (DTS), (2) a nearshore mix (NSM), consisting of English, petrale, and rex sole as well as sanddabs, (3) California halibut and (4) a rockfish mix. These four species assemblages accounted for just over 96% of total groundfish harvest among vessels in our sample. This basic level of output aggregation is both practically attractive (a multi-species output model with 90 outputs would almost certainly contain too many zeros to carry out the appropriate matrix inversions) and intuitively reasonable (since trawl gear is necessarily unselective it sometimes makes more sense to think of outputs as co-occuring species complexes – such as DTS – rather than as individual species). Summary statistics for inputs and outputs used in our analysis are provided in Table 1. Table 2 displays the results of our CobbDouglas production function estimates for the years 2004– 2009.9,10 The table shows coefficient estimates and standard errors for production functions with two different dependent variables: (i) natural log of average catch over the four target species and (ii) natural log of the aggregate index calculated as described in linear programming model (1). The coefficient estimates on our inputs are fairly similar across the specifications with different dependent variables within a given year. The coefficient estimates on two hours and days at sea, both of which are measured in natural logs are positive in every instance. The coefficient estimate on “log of two hours” is statistically significant at the one-percent level in every year – except 2009—when using the aggregate index as the dependent variable. However, it is not statistically significant at the one-percent level in any of the years when using average catch as the dependent variable. The coefficient estimate on “log of days at sea” is statistically significant at better than the five-percent level in every year when using the average catch as the dependent variable and when using the aggregate index as the dependent variable. Although the present study focuses primarily on analyzing efficiency scores for vessels in our data sample, it deserves some emphasis that the CJR model provides these estimates while also allowing identification of important, policy-relevant quantities such as the return on fishing effort. The output elasticities, represented by estimated parameters of the production function can aid fisheries managers in a few ways. Notably, the nature of
8 Walden et al. [39] and Färe et al. [16] estimate harvesting capacity and technical efficiency for multi-species fisheries, restricting attention to a subset of total species landed. In addition, it should be noted that our species groupings are consistent with prior biological work on mixed-species modeling [28,33]. In particular Rogers and Pikitch find six assemblages to span the space of harvested species for the Oregon/Washington multi-species groundfish trawl fishery. Our groupings contain one assemblage dominated by a species not harvested outside California (California halibut). The remaining three species groupings are consistent with the Rogers and Pikitch assemblages. 9 We also estimated translog production functions for these same years. The coefficient estimates on the translog model were generally statistically insignificant, so we chose to discuss the Cobb-Douglas model here. The estimates of technical efficiency obtained using the translog model and the Cobb-Douglas model had correlations of greater than.99 for each of the years. These results are available from the authors upon request. 10 Appendix 2 presents the results of a maximum likelihood model where the efficiency term is assumed to have a truncated normal distribution. All technical efficiency estimates from the maximum likelihood model have correlations of greater than or equal to 0.935 with the technical efficiency estimates from the fixed effects model.
technical change in fisheries is of great interest to managers. Technical progress, which increases the effectiveness of effort, can erode the efficacy of input controls. Moreover, changes in the production function resulting from regulatory changes can inform managers regarding the economic consequences of management policy. Kirkley et al. [21] illustrate how estimated parameters of a Cobb-Douglas production function can be used to investigate embodied and disembodied technical change among commercial fishing operations. Although the present analysis does not address technical change (as our data do not include many of the potentially key drivers of such change11), the methodology employed here provides a framework into which drivers of technical change could be easily introduced. Although the coefficient estimates are generally similar across specifications with the two different dependent variables, the corresponding estimates of technical efficiency are not. Table 3 displays summary measures for the technical efficiency estimates using the two different dependent variables across the years 2004–2009, respectively. T-tests of the similarity of the means of technical efficiency were generated using the two different dependent variables. The hypotheses that these means are equal are rejected at the one percent level in every year, 2004–2009. Although there appears to be significant differences in the means of these two distributions of efficiency, KomolgorovSmirnov tests for the equality of distributions fail to reject the null hypothesis that the two distributions of efficiency are equal in each year, 2004–2009. Thus, we have generated two different estimates of efficiency using: (i) a new technique that estimates aggregate output and (ii) using average catch over multiple species. These estimates of technical efficiency have different means, but similar distributions. The question now becomes: does the ranking of vessels in terms of their efficiency score change depending on which measure of output is used? Theoretically speaking, the answer to this question is a resounding yes. Empirically, our models result in correlation coefficients between the efficiency scores estimated using the two different outputs ranging from 0.753 in 2006 to 0.937 in 2005. To illustrate some important differences between our model and the naïve average output, we isolate the subsample of vessels and trips which were estimated to be on the frontier of our measure of aggregate output. Since 2006 shows the lowest correlation, we will focus our discussion on how the estimates of efficiency differ using the two measures of output for specific vessels in 2006. Five trips taken by four different vessels were determined to be on the frontier of our measure of aggregate output in 2006, giving them aggregate output estimates of 100. Table 4 displays the amount of catch for each species group, the average catch, and technical efficiency scores estimated using each of the two measures of output, for each of these trips. You can see that one of the vessels (the one we are calling Vessel #36) was determined to have two trips that were on the frontier of our aggregate output measure. These two trips resulted in average catch of much greater magnitude than the other trips on our frontier and, not surprisingly, this vessel is estimated to be the most efficient vessel when average catch is used as the dependent variable in our production function. Conversely, Vessel #29 has the lowest average catch among the vessels on the frontier, because this vessel chose to target only one species group on this particular trip. In fact, this vessel caught the largest quantity of this species group on any one trip in 2006. 11 Notably, our data do not contain information on harvest limits, which one would suspect influence the production function. Extensions to the current analysis will work to address this data gap and investigate the impact of output constraints on production.
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161
Table 1 Summary statistics for inputs and outputs. Variable
Min
Mean
Max
St. Dev.
Outputs DTS (lbs) NSM (lbs) CA Halibut (lbs) Rockfish (lbs)
0 0 0 0
6.795 2.135 334 1.102
87.400 50.214 16.852 76.090
11.345 3.927 1.126 3.924
Inputs Tow hours Days at sea
0 0
17.95 1.70
75.23 6.29
14.16 1.08
Table 2 Regression results by year.a Two hours (ln)
2004 Avg. output Aggregate output index 2005 Avg. output Aggregate output index 2006 Avg. output Aggregate output index 2007 Avg. output Aggregate output index 2008 Avg. output Aggregate output index 2009 Avg. output Aggregate output index
a
Days at sea (ln)
Constant
R-square within
between
overall
0.165 (0.165) 0.409nnn (0.117)
0.861nnn (0.137) 0.559nnn (0.120)
6.807nnn (0.394) 1.431nnn (0.280)
0.211
0.791
0.582
0.281
0.767
0.562
0.347n (0.199) 0.593nnn (0.146)
0.822nnn (0.248) 0.415nn (0.168)
5.376nnn (0.451) 0.783nn (0.336)
0.274
0.592
0.626
0.345
0.692
0.678
0.232n (0.135) 0.400nnn (0.068)
1.102nnn (0.300) 0.575nnn (0.117)
5.563nnn (0.295) 1.351nnn (0.162)
0.294
0.784
0.692
0.279
0.757
0.638
0.242 (0.147) 0.590nnn (0.093)
0.915nn (0.355) 0.135 (0.120)
5.801nnn (0.289) 1.194nnn (0.214)
0.190
0.713
0.630
0.248
0.703
0.624
0.479nn (0.179) 0.576nnn (0.155)
0.550n (0.317) 0.312nn (0.144)
5.417nnn (0.367) 1.360nnn (0.366)
0.272
0.740
0.675
0.323
0.873
0.680
0.094 (0.180) 0.245 (0.160)
1.00nnn (0.269) 0.665nnn (0.145)
6.293nnn (0.393) 2.042nnn (0.382)
0.253
0.776
0.670
0.246
0.836
0.634
Asterisks indicate the significance level of the estimate: “nnn” −1% level, “nn” −5% level; “n” −10% level. Standard errors are in parentheses.
There are a potentially large number of reasons why a vessel may target a particular species or species group12 (existing contracts with buyers, unobserved characteristics of a vessel or skipper knowledge of spatial distribution of certain species/species groups for example). Since many of these will generally be unobservable (to the researcher) it is important to have a method capable of accurately assessing the performance of many different types of operators. When estimating a production function with average catch as the dependent variable, one is assuming that all species of fish are perfect substitutes for one another. This type of faulty assumption would lead to an estimate of technical efficiency of
12 It should also be noted that an individual vessel may pursue both generalist and specialist strategies on different trips throughout the year.
0.084 for Vessel #29, whereas using our aggregate output measure leads to a technical efficiency estimate of 0.362. This is one example of how this method of estimating aggregate output is useful when data on market prices is unavailable. An important result here that warrants some emphasis is that specialists – vessels harvesting a small number of the available species – are disproportionately punished by the average catch method (in estimated efficiency terms) relative to generalists. Note the large discrepancies between efficiency scores calculated by the CJR method and the average catch method for vessel #29 in all years. This can have implications in practical settings such as fisheries management where regulators frequently concern themselves with the harvesting efficiency of commercial fishing operations. It is important for managers to be able to evaluate efficiency for a heterogeneous group of producers in a multi-output production setting.
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Table 3 Mean estimated technical efficiency.
Agg. output index Avg. output Difference t-score N
2004
2005
2006
2007
2008
2009
0.358 0.283 0.075 4.752 34
0.384 0.284 0.100 6.699 31
0.476 0.312 0.164 4.699 23
0.414 0.291 0.123 6.473 23
0.441 0.367 0.074 2.908 21
0.436 0.355 0.081 3.080 17
Table 4 Vessels with aggregate output index ¼100 in each year. Year
2004 2004 2004 2004 2004 2005 2005 2005 2005 2005 2005 2006 2006 2006 2006 2006 2007 2007 2007 2007 2007 2007 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2009 2009 2009 2009 2009 2009 2009 2009
Vessel
9 9 29 36 37 9 27 27 36 36 36 9 29 34 36 36 21 26 33 33 33 36 9 21 21 21 21 26 26 30 30 30 36 36 21 30 36 36 36 36 36 36
Trip
1 2 1 1 1 1 1 2 1 2 3 1 1 1 1 2 1 1 1 2 3 1 1 1 2 3 4 1 2 1 2 3 1 2 1 1 1 2 3 4 5 6
y1
10.292 30.188 – 52.748 87.400 689 – 45.152 40.174 12.710 38.973 786 – 9.220 55.574 46.002 – 17.750 41.324 1.166 10.795 25.046 33.932 4.083 124 – 656 20.560 9.620 31.650 30.975 28.477 24.896 480 – 26.333 16.615 31.813 18.785 11.506 15.661 23.816
y2
50.214 46.986 – 3.660 3.708 29.704 10 – 818 1.852 4.555 20.048 – 2.362 1.705 1.878 – 3.568 13.334 23.269 307 1.618 – 23.460 2.139 10 9.916 379 14.714 2.566 4.920 9.352 304 24.302 – 7.614 108 1.824 84 19.588 5.646 2.025
y3
5.132 4.862 – 76.090 6.864 3.408 – 720 19.914 26.595 – 951 – 16.957 6.645 17.148 – 16.284 498 22.888 7.246 13.123 70 828 12.811 – 1.055 17.601 13.021 3.069 2.750 1.791 14.545 10.879 – 933 16.916 7.439 16.508 6.967 14.560 14.251
5. Conclusions Fisheries managers and policy makers are responsible for monitoring performance of commercial fishing fleets and have a legislative mandate to consider the efficiency implications of fisheries management policy.13 A key complication in the measurement of economic performance among fishing vessels is the lack of available data on
13 See NOAA Fisheries' Office of Science and Technology (http://www.st.nmfs. noaa.gov/st5/ExecutiveLegislativeMandates.html) for detail on these responsibilities under the Magnuson-Stevens Fisheries Conservation and Management Act.
y4
– – 16.852 – – – 9.499 – – – – – 8.005 – – – 3.720 – – – 3.044 – – 78 1.820 3.010 2.862 – – – – – – – 4.964 – – – – – – –
Avg. catch
16.410 20.509 4.213 33.125 24.493 8.450 2.377 11.468 15.227 10.289 10.882 5.446 2.001 7.135 15.981 16.257 930 9.401 13.789 11.831 5.348 9.947 8.501 7.112 4.224 755 3.622 9.635 9.339 9.321 9.661 9.905 9.936 8.915 1.241 8.720 8.410 10.269 8.844 9.515 8.967 10.023
Technical efficiency Aggregate output
Avg. catch
0.415 0.415 0.320 1.000 0.582 0.577 0.349 0.349 1.000 1.000 1.000 0.605 0.362 1.000 0.973 0.973 0.470 0.664 0.679 0.679 0.679 1.000 0.460 0.607 0.607 0.607 0.607 0.612 0.612 0.625 0.625 0.625 1.000 1.000 0.624 0.663 1.000 1.000 1.000 1.000 1.000 1.000
0.399 0.399 0.102 1.000 0.511 0.483 0.087 0.087 1.000 1.000 1.000 0.427 0.084 0.705 1.000 1.000 0.131 0.477 0.399 0.399 0.399 1.000 0.437 0.276 0.276 0.276 0.276 0.624 0.624 0.627 0.627 0.627 1.000 1.000 0.229 0.671 1.000 1.000 1.000 1.000 1.000 1.000
input prices. In the absence of such data, technical efficiency has emerged as an important performance metric. We utilize a recently developed DEA technique to transform multiple outputs into an aggregate output index, without forcing the production frontier to have a constant marginal rate of technical substitution [9]. This aggregate index is used as the dependent variable in a regression framework to estimate the return to effort and technical efficiency in fisheries production. We apply this model to data from the West Coast Limited Entry Groundfish Trawl Fishery. Our research addresses an important issue in efficiency measurement among fishing vessels: modeling the multi-output production technology present in multi-species fisheries while
T.C. Collier et al. / Omega 42 (2014) 157–165
163
Table A1 Regression results by year using CJR method and order M method.a Two hours (ln)
2004 Order M index Aggregate output index 2005 Order M index Aggregate output index 2006 Order M index Aggregate output index 2007 Order M index Aggregate output index 2008 Order M index Aggregate output index 2009 Order M index Aggregate output index
a
Days at sea (ln)
Constant
R-square
Correlation between efficiency estimates
within
between
overall
0.422nnn (0.114) 0.409nnn (0.117)
0.522nnn (0.122) 0.559nnn (0.120)
−2.220nnn (0.272) 1.431nnn (0.280)
0.275
0.741
0.54
0.281
0.767
0.562
0.595nnn (0.137) 0.593nnn (0.146)
0.398nn (0.179) 0.415nn (0.168)
−2.855nnn (0.305) 0.783nn (0.336)
0.337
0.710
0.686
0.345
0.692
0.678
0.399nnn (0.068) 0.400nnn (0.068)
0.595nnn (0.125) 0.575nnn (0.117)
−2.425nnn (0.161) 1.351nnn (0.162)
0.279
0.762
0.657
0.279
0.757
0.638
0.513nnn (0.082) 0.590nnn (0.093)
0.273nn (0.128) 0.135 (0.120)
−2.280nnn (0.188) 1.194nnn (0.214)
0.250
0.707
0.643
0.248
0.703
0.624
0.563nnn (0.155) 0.576nnn (0.155)
0.323nn (0.140) 0.312nn (0.144)
−2.402nnn (0.368) 1.360nnn (0.366)
0.314
0.854
0.656
0.323
0.873
0.624
0.212 (0.167) 0.245 (0.160)
0.686nnn (0.151) 0.665nnn (0.145)
−1.614nnn (0.398) 2.042nnn (0.382)
0.228
0.823
0.610
0.246
0.836
0.634
0.984
0.988
0.987
0.99
0.991
0.996
Asterisks indicate the significance level of the estimate: “nnn” −1% level, “nn” −5% level; “n” −10% level. Standard errors are in parentheses.
respecting the inherent stochasticity of commercial capture fisheries. Moreover, our method permits recovery of important, economically meaningful parameters. The estimated coefficients from the translog and Cobb-Douglas production functions represent marginal productivities and output elasticities of the production function. Given the emphasis that has been placed on input controls as a fisheries management tool, these quantities provide potentially valuable information. For instance welfare effects of input controls can be approximated using estimated marginal value product of affected inputs. Additionally, in cases where managers are interested in imposing input controls to limit output, it should be recognized that not all inputs influence fish production identically at the margin. Our method, in contrast to other multi-output methodologies, permits estimation of these important quantities and their associated confidence bounds. Our analysis highlights the potential problems associated with aggregating multiple outputs by taking a simple average. In the absence of price data, this type of aggregation has been used in previous research and can lead to poor estimates of technical efficiency. In particular this is problematic when some outputs are more difficult to produce (species are more difficult to catch). We show that using the two-stage approach developed by Collier et al. [9], we can provide more reasonable estimates of technical efficiency without the need for input or output prices.
the outer frontier to estimate aggregate output. If this outer frontier is influenced by outliers then our estimate of aggregate output will be biased. As a check for robustness, we consider an order-m frontier introduced in [5]. As discussed in [10], the orderm frontier does not envelop all data and hence, is less sensitive to potential outliers.14 Because we are interested in the outermost output isoquant, the procedure involves sampling from all data points with replacement. For each sample, we estimate the aggregate output. This process is repeated a large number of times (2000 in our application) and the average level of aggregate output is used as our measure of aggregate output. In Table A1, we report the stochastic frontier results using the order-m approach (and the approach used in our paper for comparison purposes). The results are similar across approaches. The standard stochastic production frontier model of Aigner et al. [2] and Meeusen and van den Broeck [26] took the following form: lnyi ¼ β0 þ β1 lnxi þ vi −ui
where yi is the output of firm i, xi is an input for firm i, vi is a random error term for firm i from a standard normal distribution and ui is an inefficiency component for firm i from that must be positive. A panel data version of the stochastic frontier model,
Appendix In this appendix, we address some of the concerns raised by two anonymous referees. A potential concern is with our use of
14
Technical details can be found in [10].
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Table A2 Results by year using fixed effects and maximum likelihood.
2004 ML truncated normal Fixed effects 2005 ML Truncated Normal Fixed Effects 2006 ML truncated normal Fixed effects 2007 ML truncated normal Fixed effects 2008 ML truncated normal Fixed effects 2009 ML truncated normal Fixed effects
Two Days at hours (ln) sea (ln)
Constant
Correlation between efficiency estimates
0.412nnn (0.112) 0.409nnn (0.117)
0.617nnn (0.146) 0.559nnn (0.120)
2.145nnn (0.406) 1.431nnn (0.280)
0.601nnn (0.176) 0.593nnn (0.146)
0.446nn (0.151) 0.415nn (0.168)
−1.938nnn (0.463) 0.996 0.783nn (0.336)
0.398nnn (0.071) 0.400nnn (0.068)
0.629nnn (0.112) 0.575nnn (0.117)
2.313nnn (0.208) 1.351nnn (0.162)
0.591nnn (0.069) 0.590nnn (0.093)
0.186 (0.115) 0.135 (0.120)
2.059nnn (0.336) 1.194nnn (0.214)
0.600nnn (0.067) 0.576nnn (0.155)
0.350nnn (0.049) 0.312nn (0.144)
2.351nnn (0.335) 1.360nnn (0.366)
0.244nnn (0.088) 0.245 (0.160)
0.716nnn (0.093) 0.665nnn (0.145)
3.108nnn (0.377) 2.042nnn (0.382)
0.935
0.985
0.987
0.99
0.995
Asterisks indicate the significance level of the estimate: “nnn” 1% level, “nn” 5% level; “n” 10% level.
with time invariant efficiency takes the following form: lnyit ¼ β0 þ β1 lnxit þ vit −ui This time invariant panel data version can be estimated using the Schmidt and Sickles [34] fixed effects model – as is done in the body of this paper – or using random effects – as we do in this appendix. This random effects model was used by Pitt and Lee [31] assuming a half normal distribution for the inefficiency term and by Battese and Coelli [4] assuming a truncated normal distribution for the inefficiency term. After assuming that that the inefficiency component is constant across time for each firm and assuming the inefficiency term is a random draw from some specific positive distribution, we can estimate this model using maximum likelihood techniques. The results displayed in Table A2 were estimated with maximum likelihood techniques assuming a truncated normal distribution for the inefficiency term (as in [4]).17 We list the coefficient estimates from the fixed effects model below these maximum likelihood coefficient estimates and display the correlations between these two estimates in the last column of the table. Every correlation is above or equal to 0.984. Thus we only reported the fixed effects estimates in the body of the paper.
17 The maximum likelihood standard errors are bootstrapped with 100 replications, except for 2007 ( based on 10), 2008 (based on 5), and 2009 (based on 10)
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