Parametric and Semiparametric Efficiency Frontiers in Fishery Analysis: Overview and Case Study on the Falkland Islands

Abstract

Provision of adequately valued individual transferable quotas and effort quotas is essential for sustainability and profitability of a fishery. Despite possible misleading consequences for policy-making, the extent to which fishery inefficiency estimates and rankings may depend on the model used, has received less attention. This paper first reviews determinants of fishers’ behaviour under regulated harvesting, with the Falkland Islands as focus case. Next, a ‘best scenario’ long-term equilibrium framework is outlined, under a regime of transferable effort quotas and fishing seasons as implemented in the Islands, followed by an overview of panel data stochastic frontier models, with specific regard to fisheries. To test hypotheses and impact of a mainly ITEQ-based regime for Falkland fisheries, two parametric and one semiparametric model rely on different assumptions on frontiers and inefficiency scores. Relative to companies operating in Falkland seas, regression estimates highlight the relevance of economies of scale, vessel ownership, and climatic factors among others, with improved cost effectiveness, and revenue efficiency frontier-enhancing/inefficiency-reducing effects, following the implementation of the new regime. Within either modelling approach, inefficiency differs marginally across regression specifications, but mismatches in levels and rankings emerge between parametric and semiparametric models. Relative to southern hake catches by Falkland trawlers, the semiparametric approach suggests upward shifts in output frontiers under the new fishery regime, with inefficiency scores substantially unaltered between two functional specifications.

Appendix

1.1 Stochastic Frontiers, Unobserved Heterogeneity, and Inefficiency

In SF regression analysis, an outcome variable (y_i) is regressed on deterministic and stochastic parts of the efficiency frontier (α + β′x_i and v_i, respectively), with v_i included in an asymmetric compound error ε_i. Besides the independently and identically normally distributed symmetric random disturbance v_i (~ N[0,σ_v²]), a second, independent component of the compound error is given by a skewed stochastic (inefficiency) term u_i (~ N⁺[0,σ_u²]), with ε_i = (v_i − u_i) in production, revenue and profit frontiers, and ε_i = (v_i + u_i) in cost frontiers. In the seminal study by Aigner et al. (1977), x_i, v_i and u_i were assumed to be mutually independent, and firm-specific effects were absorbed into the inefficiency term. Since the mid-1980s, SF panel data (PD) models have striven, each with its own advantages and shortcomings, to distinguish inefficiency from unobserved heterogeneity (Table A [online]). Apart from an idiosyncratic random error v_it, individual-specific heterogeneity (α_i(t)) and inefficiency (u_i(t)) can have time-invariant or time-varying unobserved elements. Without prior information, disentangling the two components through SF PD models, based on observed data, is subject to empirical uncertainty.

Conventional fixed effects (FE) and random effects (RE) estimators redress either one or the other of the restrictive assumptions underlying the cross-section SF model. FE SF measures inefficiency as a distribution-free gap from maximum (/minimum in SF cost functions) firm-specific parameter (Table A: FE-SF), thus not complying with an absolute yardstick of efficiency. In the FE SF model by Schmidt and Sickles (1984), all time-invariant effects are part of inefficiency and the frontier is likely to have an upward bias in small-T panels, thus overestimating inefficiency scores (Kim and Schmidt 2000: 96). Conversely, in RE SF the inefficiency component complies with a unit-value yardstick, but it is tightly parameterised, and assumed to be stochastically independent from the explanatory variables. This is often unrealistic, since inefficiency can vary with quality and use of inputs, and persistent inefficiency will be ‘learnt’ by firms, thus leading to adjustments in input choices. Alternative RE SF specifications relax this assumption, by testing for partial dependence on firm-specific effects, including the intermediate case that only some inefficiency stays with a firm (Pitt and Lee 1981).

Beyond these differences, both SF PD estimators treat inefficiency as structural, with no substantial learning through rethinking on past decisions. To better distinguish unobserved heterogeneity from inefficiency, SF PD estimation methods have undergone substantial revisions. Battese and Coelli (1992/1995) formulate two parametric RE models: one based on monotone time dependency of half-normally or truncated normally distributed inefficiency (u_it), which declines, remains constant, or increases given a parameter η > 0, ≈ 0 or < 0, respectively (Eq. [1b₁]), and another where inefficiency depends on a vector of firm-specific determinants (Eq. [1b₂]).

$$ y_{it} = \upalpha +\upbeta ^{{\prime }} x_{it} + \varepsilon_{it} $$

(1a)

$$ \varepsilon_{it} = v_{it} - u_{it} \quad \left( {v_{it} \sim {\text{N}}\left[ {0,\upsigma _{v}^{2} } \right],u_{it} = exp\left[ { -\upeta \left( {t - T} \right)} \right] \sim {\text{N}}^{ + } \left[ {\upmu _{u|t} ,\upsigma _{u}^{2} } \right]} \right) $$

(1b1)

$$ \varepsilon_{it} = v_{it} - u_{it} \quad \left( {v_{it} \sim {\text{N}}\left[ {0,\upsigma _{v}^{2} } \right],u_{it} = exp\left[ {\upeta ^{{\prime }} {\text{z}}_{{{\text{it}}}} } \right] \sim {\text{N}}^{ + } \left[ {\upmu _{u|z} ,\upsigma _{u}^{2} } \right]} \right) $$

(1b2)

Other SF PD models have focused on groupwise heterogeneity, modelled through multivariate truncated normal distributions and/or prior information on sample separation in switching regression (Table A [online]: HET-SF; SW-SF), and hence fall in between parametric and fully-fledged semiparametric approaches. Relative to the latter, true FE and RE models assume unobserved heterogeneity to be time-invariant (Greene 2007: 154; the term ‘true’ just serves to define concisely the newly formulated models). In a true FE (TFE) model, individual-specific dummies capture heterogeneity as shifts in production (/cost) or inefficiency. In a true RE (TRE) model, heterogeneity affects frontier locations and/or inefficiency distributions, thus being a special case of hierarchical random-parameter model. A TRE SF model (also defined as random constants model and estimated by MSL; Greene 2005a, 2012: R24) is written as follows:

$$ y_{it} = \upalpha _{i} +\upbeta ^{{\prime }} x_{it} + \varepsilon_{it} $$

(2a)

$$ \varepsilon_{it} = v_{it} - u_{it} \quad \left( {v_{it} \sim {\text{N}}\left[ {0,\upsigma _{v}^{2} } \right],u_{it} \sim {\text{N}}^{ + } \left[ {0,\upsigma _{u}^{2} } \right]} \right) $$

(2b)

$$ \upalpha _{{\text{i}}} = {\upalpha } + w_{i} \quad \left( {w_{i} \sim {\text{N}}\left[ {0,\upsigma _{w}^{2} } \right]} \right) $$

(2c)

Greene (2007: 157) argues that, with an unsolved identification problem, the ‘truth’ will lie somewhere between the conventional and the thus modified approaches. Possible pitfalls of TFE and TRE mirror those observed for conventional FE/RE estimators: if stable and persistent over time, inefficiency, or its latent effects, can be ‘masqueraded’ as time-invariant heterogeneity, thus being systematically underestimated (Agrell et al. 2014).

1.2 Distributional Assumptions and Panel-Data Estimator Consistency

In SF models, shortfalls from a stochastic efficiency frontier (g(x_i,β) + v_i) are measured by the conditional mean of u_i, i.e. E[u_i|ε_i], with u_i usually assumed to follow a zero-truncated half-normal distribution (0 ≤ u_i < ∞). Define μ* =  − ε[σ_u²/(σ_v² + σ_u²)], σ*² = [σ_v²σ_u²]/[σ_v² + σ_u²], and an asymmetry parameter λ = σ_u/σ_v (i.e. ‘signal-to-noise’ ratio, which reflects the relative importance of inefficiency over random disturbances). Similar to a truncated-from-below distribution for Y ≥ c (for a stardard normal variable Y), the conditional mean of inefficiency is written as follows (with φ(.) and Φ(.) denoting standard normal density and cumulative distribution functions; Verbeek 2012: 456; Kumbhakar and Lovell 2000: 78; Jondrow et al. 1982):

$$ {\text{E}}\left[ {u_{i} |\varepsilon_{i} } \right] =\upmu ^{*} +\upsigma ^{*} \left[ {{\varphi }\left( { -\upmu ^{*} /\upsigma ^{*} } \right)} \right]/\left[ {1 -\Phi \left( { -\upmu ^{*} /\upsigma ^{*} } \right)} \right] =\upsigma ^{*} \left\{ {\left[ {{\varphi }\left( {{\upvarepsilon \lambda /\sigma }} \right)/\left( {1 -\Phi \left( {{\upvarepsilon \lambda /\sigma }} \right)} \right.} \right]{-}\left( {{\upvarepsilon \lambda /\sigma }} \right)} \right\} $$

(3)

The unconditional mean and variance of u are σ_u√(2/π) (= − E(ε)) and [(π − 2)/π]σ_u², respectively (Greene 2007: 117; Jondrow et al. 1982: 234–235). In subsequent extensions of the model to a PD framework, Eq. (3) holds except for ε being replaced with its sample average, and σ_v² with σ_v²/T (Kim and Schmidt 2000: 94). Among distributions other than the half-normal, i.e. non-zero restricted truncated normal, gamma, and (negative) exponential, under exponentially distributed u_i(t) the range of inefficiency scores can be wider, but the frequency of high scores tends to be lower than under half- or truncated-normal assumptions, with tighter clustering of u_i(t) near zero. Relative to a gamma distribution with shape parameter r > 0 and inverse scale parameter (or rate parameter) θ > 0 (u = f(z_i) ~ Γ[r/θ_u|z, r/(θ_u|z)²]), the exponential distribution is a nested case with r = 1, while for large r and θ < 1 the gamma density approximates a normal shape with near-zero skewness (Mood et al. 1974: 112–113).

In PD models, parameter consistency relies on cross-sectional asymptotic (N → ∞, with T fixed) in RE estimates, and on time asymptotic (T → ∞, with N fixed) in FE estimates. The latter corrects the bias associated with the incidental parameters problem—even if structural parameters prove to be less sensitive to this bias in SF models than in binary choice models, among others—(Greene 2005a). Hence, RE models—eventually including fixed time-effects—are more suited for panels with large N and small T, with scope for population inference, and FE models for large T and small N, where the analysis focuses more on explaining a sample (Hsiao 2007; Greene 2003; Cameron and Trivedi 1998: 291). Wooldridge (2002: 287) shows that the RE estimator is a quasi-time demeaning, that is removing from dependent and explanatory variables, at each time t, a fraction of their time average: in the case of Eq. [2a/c], this fraction is given by τ = 1 − {1/[1 + T[(σ_w)²/(σ_ε)²]]}^0.5. As T → ∞ or [σ_w²/σ_ε²] → ∞ (i.e. with heterogeneity overshadowing the conditioning composed error variance), τ → 1 and RE approaches FE, and the effects of nearly time-constant regressors become more difficult to estimate in either case.