Estimating Heterogeneous Production in Fisheries

Abstract

Stochastic production frontier models are used extensively in the agricultural and resource economics literature to estimate production functions and technical efficiency, as well as to guide policy. Traditionally these models assume that each agent's production can be specified as a representative, homogeneous function. This paper proposes the synthesis of a latent class regression and an aagricultural production frontier model to estimate technical efficiency while allowing for the possibility of production heterogeneity. We use this model to estimate a latent class production function and efficiency measures for vessels in the Northeast Atlantic herring fishery. Our results suggest that traditional measures of technical efficiency may be incorrect, if heterogeneity of agricultural production exists.

Full text

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Syracuse University
SUrface
Center for Policy ResearchMaxwell School of Citizenship and Public Affairs
1-1-2006
Estimating Heterogeneous Production in Fisheries
Kurt E. Schnier
University of Rhode Island, Department of Environmental and Natural Resource Economics, SCHNIER@URI.EDU
Christopher M. Anderson
University of Rhode Island, Department of Environmental and Natural Resource Economics
William Clinton Horrace
Syracuse University, Maxwell School, Center for Policy Research, whorrace@maxwell.syr.edu
This Working Paper is brought to you for free and open access by the Maxwell School of Citizenship and Public Affairs at SUrface. It has been
accepted for inclusion in Center for Policy Research by an authorized administrator of SUrface. For more information, please contactsurface@syr.edu.
Recommended Citation
Schnier, Kurt E.; Anderson, Christopher M.; and Horrace, William Clinton, "Estimating Heterogeneous Production in Fisheries"
(2006).Center for Policy Research.Paper 84.
http://surface.syr.edu/cpr/84

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ISSN: 1525-3066




Center for Policy Research
Working Paper No. 80

ESTIMATING HETEROGENEOUS
PRODUCTION IN FISHERIES

Kurt E. Schnier, Christopher M. Anderson,
and William C. Horrace






Center for Policy Research
Maxwell School of Citizenship and Public Affairs
Syracuse University
426 Eggers Hall
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March 2006


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__________

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Abstract
Stochastic production frontier models are used extensively in the agricultural and resource
economics literature to estimate production functions and technical efficiency, as well as to guide
policy. Traditionally these models assume that each agent’s production can be specified as a
representative, homogeneous function. This paper proposes the synthesis of a latent class
regression and an agricultural production frontier model to estimate technical efficiency while
allowing for the possibility of production heterogeneity. We use this model to estimate a latent
class production function and efficiency measures for vessels in the Northeast Atlantic herring
fishery. Our results suggest that traditional measures of technical efficiency may be incorrect, if
heterogeneity of agricultural production exists.

Page 5

Introduction
Production function estimation is important to the development and analysis of a wide range of
agricultural and environmental policies. It can be used to identify areas of improvement in
agricultural processes, to measure the value of production or input technology changes, or to
assess producer response to new regulation or opportunities. Recent studies have focused on the
role of agricultural policy (Paul et al. 2000), the accessibility to credit markets, and the use of new
agricultural practices in developing nations (Bayarsaihan and Coeilli 2003; Hazarika and Alwang
2003; Kudaligama and Yanagida 2000; Liu and Zhuang 2000 to cite a few). In many applications,
production function estimation is supplemented by producer-level technical efficiency estimates,
which are used to identify the extent to which producers select inputs to make effective use of
fixed resources. In many agricultural applications, efficiency analyses help extension agents
identify resources that might aid farmers and help policymakers target resources for subsidy
(Khairo and Battese 2004).1

When a production technology is used to exploit a common pool resource (such as a fishery)
accurate characterization is particularly important. In this case, production estimates are often
used to guide management policies aimed as reducing pressure on the resource and ensuring its
future viability. For example, buyback programs are used in many over-exploited fisheries to
reduce the amount of capital being applied to a dwindling stock. Buybacks have also been
utilized in rationalizing so-called "derby fisheries," where the fishing season is open only until a
set quantity of fish is harvested, providing an incentive to overcapitalize and catch as much as
possible before others catch the limit. Therefore, an accurate picture of a fleet’s production
profile aids in identifying likely participants in buyback programs and in developing estimates of
reservation prices that may be used to establish budgets for a successful program (Guyader et al.
2004).

Other fisheries are managed by input restrictions, such as maximum days-at-sea, gear restrictions
or limits on the quantity of fixed gear (e.g., traps). However, experience shows that fishermen
often respond to these restrictions by substituting unrestricted inputs, in some cases using more
variable inputs (e.g., increasing crew size), or by investing in more fixed capital (e.g., purchasing
a larger engine to reduce steam time or using a larger trawl device). Estimating production input
elasticities helps managers predict the extent to which new input restrictions are likely to result in
decreased stock pressure, or simply a substitution of other unrestricted inputs (Kompas et al.
2004). With both types of management measures, policymakers can use production functions to

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determine the technical efficiency of each operation. Furthermore, measures of technical
efficiency can be used to determine the efficiency gains of switching from input regulations to
property right management regimes, such as individual transferable quotas (Kompas and Che
2005; Weninger and Waters 2003).

Technical efficiency indicates how well vessels perform relative to the optimal use of their inputs,
and provides a measure of excess harvesting capacity resulting from inefficiently managed or
underutilized capital (Kirkley et al. 2002). This is of particular interest, because the excess
capacity could be put into use in the event that part of the fleet were to be bought out, or it could
reflect the potential for capital substitution in the event of new input restrictions.2 Although
production function and technical efficiency estimates are critical to assessing the likely effect of
new policy, the traditional approach to production analysis develops a representative
(homogenous) producer model. This approach is commonly employed even though there may be
production heterogeneity among producers. In fisheries, this heterogeneity is often explained by
the "good captain" hypothesis, which states that some captains possess skills, usually not directly
measurable, which allow them to consistently outperform other captains with similar capital in
the same fishery. This has lead economists to estimate the degree of technical efficiency
possessed by captains within a number of different fisheries in an effort to determine the captain
specific factors which determine their relative rates of inefficiency (Kirkley et al. 1998; Pascoe
and Coglan 2002; Sharma and Leung 1998; Squires and Kirkley 1999; Viswanathan et al. 2002).3

The presence of "good" captains introduces latent heterogeneity in the production capabilities
possessed by fishermen within a fishery because the determinants of a "good" captain are often
unobservable. In addition, these differences may not be completely explained by differences in
technical efficiency and the captain’s managerial skill. For instance, Kirkley, Squires and Strand
(1998) observed that two captains, using nearly identical vessels and possessing similar
experience levels and backgrounds, possessed different measures of technical efficiency. Perhaps
these differences were due to latent heterogeneity in the vessels' production functions, which
when controlled for generates similar measures of technical efficiency. Recognizing this
heterogeneity may not only improve the accuracy of production estimates, but it may also support
more refined analysis and better-targeted policies.

In this paper, we use a latent class stochastic production frontier estimator to investigate the
presence of latent heterogeneity in fisheries. The estimator is a statistical model that

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simultaneously estimates a set of distinct production functions and selects which producers use
which function. The model generates a set of estimated production functions, along with a
likelihood-based assignment of producers to each function. These different functions have the
natural interpretation of reflecting different marginal productivities of both their fixed capital
(e.g., vessel characteristics) and variable inputs (e.g., number of crew members and hours fished).
These differences are not identified if we restrict the specification to a representative
(homogenous) producer. Therefore, we employ latent class modeling to separate vessel
production via explicit differences in their elasticities of input utilization.4

This latent class methodology has been used to investigate inefficiency heterogeneity in Turkish
banking (El-Gamal and Inanoglu 2005), but to the best of our knowledge this is the first known
application of this model to agricultural and resource production modeling. We demonstrate the
value of latent class modeling in agricultural and resource production with an application to the
Northeastern US Atlantic herring fleet. We identify three economically and statistically different
production functions within the fishery. Technical efficiency analysis indicates that, within each
production function, captains have a range of aptitude for selecting variable inputs to efficiently
catch fish. However, relative to the traditional homogeneous model, they suggest dramatically
different technical efficiency measures and marginal products of input utilization. These results
highlight the importance of utilizing latent class modeling when heterogeneity is suspected.

The next section of the paper presents the latent class stochastic production frontier model. We
then describe the data set, and present a three segment (heterogeneous) estimate of the fleet
production function. We use these production functions to generate technical efficiency
estimates, and compare the results across homogenous and heterogeneous specifications. These
comparisons highlight the utility of estimating heterogeneous productions and provide some
insight into the paired trawling practices within the Northeast Atlantic herring fishery. Finally,
we discuss the pitfalls and policy implications of using a homogeneous analysis to interpret
production measures in a heterogeneous environment.

Methodology
Latent class models posit that the population consists of several distinct types of producers with
similar production functions, based on unobserved characteristics of the producers. The statistical
task is to identify both which producers are of the same (unobserved) type and the parameters that
represent each type’s production function. Developing a latent class model requires two steps:

3

Page 8

(1) specification of a parametric form for the production function of each type and (2)
implementation of a method for determining the combination of parameters for each segment and
assigning producers to each segment. We specify each type as having a partial trans-log
functional form estimated within a stochastic production frontier model.5 We then use El-Gamal
and Grether’s (1995; 2000) estimation-classification (EC) algorithm to simultaneously group
producers into types and to estimate parameter values for each segment.

A stochastic frontier model has a composed error (Aigner et al. 1977; Meeusen and van den
Broeck 1977), which is decomposed into an conventional random noise term and a random, firm-
specific technical inefficiency term. The stochastic frontier model is specified as follows,

}exp{);(
ititit
XfY
εβ=
(1)

where
is the production of producer
it Y
Ni
,...,1
=
in period
iTt
,...,1
=
. The
is the level of
inputs used in the production process,
it
X
β is a parameter vector, and
it
ε is a composed error term.
The error term is linearly specified as

iitit
v
ηε−=
(2)

where is an independently and identically distributed , and
it v
), 0 (
N
2
v
σ
i η is a non-negative,
vessel-specific error term, distributed as the truncation below zero of a random
variable. Further, the random variables and the
),(
2
μ
σμ
N
it v
i η are assumed to be independent of the
inputs and of each other. For an unbalanced panel, the log-likelihood function is (Battese et al.
1989; Battese and Coelli 1995),

∑
=
i
∑
=
i
∑
=
i
∑
=
i
∑
=
i
+−−−−−−Φ−+−Φ−−
−+−−−−
⎟
⎠
⎞
⎜
⎝
⎛
−=
N
i
N
Si
NN
iSSi
N
i
zxyxy NzzzN
TTTyL
1
2*
1
22*
11
22
1
2
1
])1)[(()'(
2
1
2
1
)](1log[)](1log[
]}) 1( 1 [log{
2
1
])1log[() 1(
2
1
)2log(
2
1
);(
σγββ
γσσγπθ


where,

4

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5 . 02)
γ
S
(
σ
μ
z =
,
5 . 02
S
*
i
])) 1
−
( 1 [)1 (
γ
(
)() 1 (
−γσγ
βγ
+
γμ−−−
=
i
iii
T
xyT
z
,
i
i
i
T
y
y =
, and
.
i
i
i
T
x
x =
(3)

iT is the number of observations for each agent i and θ is the parameter vector to be estimated
which consists of the coefficients for each segment, βi , γ = σμ
2/ σS
2, μ and σS
2 = (σμ
2 + σV
2).

Using this stochastic production frontier model as the functional form for each type, El-Gamal
and Grether’s estimation-classification (EC) algorithm performs the task of grouping producers
into a pre-specified number of types, H, and estimating the unknown parameters, θh, for each
segment. Each producer’s contribution to the likelihood function is the maximum across the H
segments of the joint log-likelihood of all their observations given Θ=(θ1,… θh) and may be
expressed as6,

ln[L(Yit;Zit|Θ,H)]=
argmaxh
ln(L(Yit;Zit|θh))
t=1
Ti
∑
i=1
N
∑
(4)

where is the single-segment likelihood function. This method can be applied with a
different number of segments, H, and statistical tests can be performed to determine the number
of segments in the population (e.g., El-Gamal and Grether 1995). As mentioned earlier, El-
Gamal and Inanoglu (2005) have used this methodology to investigate inefficiency in Turkish
banking. This technique has also been used to analyze experimental data on individual and public
good decision making problems (Anderson and Putterman 2006; El-Gamal and Grether 1995,
2000; Schnier and Anderson in press). This research illustrates the benefits of using this
methodology within the agricultural and natural resource economics literature.
(.)
L

Technical efficiency of each vessel within a fishery is defined by the vessel ability to generate the
maximum level of output (harvest) possible given a fixed level of inputs, the present stock level,
and all other exogenously determined production factors. Measurements of technical efficiency
are obtained from the vessel specific errors resulting from the stochastic frontier estimation.
Since output is in logarithms, technical efficiency is expressed as
}exp{
i η−
for each vessel i.
However, since
i η is unobserved, so we can only estimate its distribution (conditional on
it
ε )

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Page 10

and the mean of this distribution. The latter serves as an estimate of vessel i's technical
efficiency. In particular (per Battese et. al, 1989; Battese and Coelli 1992,1993),

⎭⎬⎫ ⎩⎨⎧
⎟
⎠
⎟
⎟
⎞
⎜
⎝
⎜
⎜
⎛
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
⎟
⎠
⎟
⎟
⎞
⎜
⎝
⎜
⎜
⎛
+−
−
Φ−
−Φ−
=−=
2*
i
2
1
*
i
exp
*
i
*
i
1
*
i
*
i
*
i
1
] | }
i
[exp{
it
E
i
TE
σμ
σ
μ
σ
μ
σ
εη
(5)

where,

2
U
2
V
2
U
σ
2
V
*
i
i
ii
T
ET
σ
σ
+
μσ
μ
−
=
(6)

∑
=
t
=
iT
it
i
i
T
E
1
1
ε (7)

.
2
U
2
V
22
U
2*
i
i
V
Tσσ
σ
+
σ
σ
=
(8)

These are the estimates used in the empirical section of the paper, but with the residual of the
maximum likelihood estimation substituted for the composed error,
it
ε , in the above formulae.
The next section describes the data and provides a brief description of the Northeast Atlantic
herring fleet.

Data Description and Fishery Background
We analyze production data from the Northeast Atlantic herring fishery. The data set utilized for
this study was obtained from the National Marine Fisheries Service and consists of 2894 logbook
entries for 39 vessels participating in the herring fishery during the years 2000 through 2003.7
Each entry represents a single trip made by a vessel, and is considered the best available data for
analyzing this fishery.8 Each entry indicates the reporting vessel ID, tons of herring landed, the
gear used, the crew size, the vessel characteristics (length, gross-tons, horsepower and hold

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capacity), home port, the time and date of departure and return to port, and the statistical
reporting area fished (see Table 3 for descriptive statistics).

The Atlantic herring, Clupea herengus, is a pelagic species targeted by fishermen from Maine
down to New Jersey along the New England seaboard. The primary products produced within the
herring fishery are sardines (juvenile herring ranging from 1 to 3 years old), bait for the Maine
lobster fishery, fishmeal used for livestock and aquaculture, smoked herring and a small market
for large flavored and filleted "kippers." While landings were as much as 470,000 metric tons in
the late 1960s, due to both stock and demand effects, current annual landings range from 81,000
to 124,000 metric tons. While the fishery is not currently heavily managed, there is discussion of
implementing new management measures which divide the fishery into inshore and offshore
management areas.9

The fishing fleet targeting Atlantic herring is relatively small, yet herring are captured in small
quantities as bycatch by vessels targeting groundfish. However, we focus solely on those trips
which herring were directly targeted. In addition, the fishery is primarily a single-species fishery
with herring dominating the catch composition on those trips where fishing vessels are directly
targeting herring.10 There are three primary methods used to capture herring: mid-water trawl,
purse seine, and paired mid-water trawl. Figure 1 contains a histogram of the landings for each of
these gear types over the four years analyzed.11 Our production estimates include only the purse
seiners and mid-water trawlers. We do not investigate the production frontiers possessed by the
paired-trawlers because we are interested in obtaining vessels specific measures of technical
efficiency and paired-trawling involves two vessels towing a single trawl. However, because we
often observe mid-water trawlers fishing alone and with another mid-water trawler while paired-
trawling, we are able to determine with whom each vessel decides to pair up with and how this
relates to the H production classes estimated. This is discussed further in the sequel.

There are five primary reasons why we suspect that heterogeneity may exist in the production
technology of the Northeast Atlantic herring fishery: (i) purse seine and mid-water trawlers may
possess different elasticities of input utilization, (ii) there exists a substantially large variance in
the vessel characteristics, landings and trips conducted within the fleet targeting herring (see
Table 3), (iii) landings are often pre-contracted before fishing and some vessels are predominately
order filling vessels, (iv) herring is supplied to a number of different markets (e.g., bait and
consumption markets) with vessels often pre-determining their intended market (v) the "good"

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captain hypothesis may be present. The later three forms of heterogeneity are truly unobserved
within our data set. Determining the amount of pre-contracted fishing and the target market for
each vessel would require contract data, which we do not possess. The "good captain" hypothesis
could be investigated through inefficiency regressions but a rigorous investigation would require
captain socioeconomic data (e.g., educational background, experience, see Kirkley et al., 1998)
which we also do not possess. However, given that we suspect heterogeneity in the data (some of
which we can directly observe and some which we cannot), the herring fishery data are well-
suited to the investigation of heterogeneous production using the EC algorithm.

Estimation Procedure
The stochastic frontier model estimated for each
Hh
,...,1
=
segment is specified as follows,

i itith
ith ith
ithit ithitih
ith
β
ithih
+
ihh
*
it
v OffshoreDumSumFall
fshoreDumSpWntOf
η−
shoreDumSpWntIn
DumNoCrewHoursCrewCrewGRT
HoursCrew HPGRTC
β
ββ
β
+
β
+
β
+
)
ββββ
++
+++++=
*
**
*ln(*)ln(*) ln(*)ln(
)ln(*) ln(*)ln(*)ln(*)ln(
|10
| 9| 8
| 7| 6| 5
| 4| 3 | 2| 1| 0
.
(10)
Cit represents the catch for vessel i on trip t expressed in metric tons of herring harvested. GRTi
and HPi capture the fixed inputs of production for vessel i and represent the vessel’s gross-
registered tonnage and engine horsepower respectively.12 Crewit and Hoursit represent the number
of crew members on board the vessel and the hours spent fishing on trip t respectively. We
constructed the hours fished by calculating the difference in the departure and arrival time for
each trip, and subtracting steam time.13 Steam time was calculated by determining the distance
between port and the centroid of the reported area fished and assuming a typical speed of 12
knots steaming to and from the fishing grounds.14 DumNoCrewit is a dummy variable indicating
whether or not the number of crew members on board the vessel was observed on trip t. In the
case that no crew members were observed we substituted the mean number of crew members
utilized by vessel i within the data set for the missing value.15 The other three remaining
variables, DumSpWntInshoreit, DumSpWntOffshoreit, and DumSumFlOffshoreit are dummy
variables indicating whether vessel i fished inshore during the Spring or Winter, offshore during
the Spring or Winter or offshore during the Summer or Fall in time period t respectively.16 The
peak seasons for the inshore fishery are Summer and Fall, while they are the Spring and Winter
for the offshore fishery. These peaks in the inshore and offshore activity correspond with the

8

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seasonal migration of herring from the inshore northern latitudes to the offshore southern
latitudes within the year. Therefore, these dummy variables control not only for the respective
inshore and offshore seasons but for the stock abundances present during these time periods.

The production function was determined by specifying the full trans-log production specification
and then removing variables which were highly collinear. Any variable which possessed a linear
correlation with GRT, HP, Crew or Hours greater then 0.90 was removed from the specification
of the production function. Although this is an arbitrary rule for determining the specification of
the model, it facilitates the estimation of the H production classes by reducing the probability of
within segment multicollinearity.17 For computational parsimony, we select the same number of
production parameters for each segment.18 Therefore, there are eleven parameters for each of the
H segments within the latent class model. Denote each segment's
) 1( ×
J
parameter vector as
],...,,[
|11 | 1| 0
hhhh
′
βββλ =
, then there are H*J+3 parameters, , to
estimate.
],,, ,...,[
2
s
1
′
H
′
σμγλλ
=Θ′
19

The production function estimated for each segment was carefully selected to control for within
segment multicollinearity. As mentioned earlier there exist two different fishing technologies in
the North Atlantic herring fleet, purse seiners and mid-water trawlers. One obvious way to
control for these technological differences would be to construct a dummy variable for one of the
technologies. This dummy variable could also be interacted with the other variables within
equation (10) to obtain two representative production functions within each of the H segments.
This would imply that we would be (in essence) estimating 2*H segments within the fishery.
However, the EC algorithm aggregates agents into the H segments based on latent similarities in
their production technology and one of these latent similarities is the marginal products possessed
by these different fishing technologies. Therefore, this may increase the probability of within-
segment multicollinearity, which could produce highly variable parameter estimates. Hence, we
do not use a dummy variable to control for fishing technologies so as to obtain reliable parameter
estimates and to investigate whether or not the EC algorithm is capable of partitioning the vessels
according to their production similarities.20 This serves as an ex post justification for using the
EC algorithm in future production modeling where latent heterogeneity is believed to exist but is
truly unobserved by the researcher.



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Model Selection and Empirical Results
To select the appropriate number of segments, H, we appealed to likelihood ratio tests, the
Bayesian Information Criteria (BIC), and the Akiake Information Criteria (AIC) tests. The
likelihood ratio test in the context of the latent class regressions is LR = -2[ln(L|H-1)-ln(L|H)],
where the degrees of freedom for the test statistic is equal to J. The BIC and AIC tests are
specified as follows, BIC = -2ln(L)+J(ln(N)) and AIC = -2ln(L)+J2 respectively, with lower
values for the BIC and AIC supporting the further segmentation of the latent class model. All
three test statistics support the segmentation of the model and indicate that the preferred number
of segments is H=3.21 Further segmentation of the data set was explored, H=4, however the log-
likelihood estimates produced ill-conditioned Hessians, suggesting a high degree of within-
segment multicollinearity. We, therefore, focus on the H=3 segmentation results in what follows.
Additionally, given the small number of vessels on the data (39) and the estimated number of
vessels within each of three segments, we believe that the four segment model is potentially
asking too much of the data. Therefore, we present results for the one and three segment models
to compare the model results under homogeneous and heterogeneous production assumptions.

In both the homogeneous and heterogeneous production models the variance parameter, γ, is
close to one which indicates that the inefficiency effects are significant in our model (Battese and
Coelli 1995). This was confirmed using a likelihood-ratio test on the null hypothesis that the
inefficiency effects were absent in the homogeneous model, γ =0.22 The magnitude of the
variance parameter, γ, is reduced as we increase the number of segments in the model. This
suggests that the heterogeneous production profiles generated by the EC algorithm reduce the
explanatory power of the inefficiency effects relative to the homogeneous production model.
However, the significance of this parameter across all the models estimated indicates that the
inefficiency effects prevail.

The EC algorithm induces a small-sample classification bias in determining each vessel’s
segment membership (El-Gamal and Grether 1995; El-Gamal and Inanoglu 2005). To
characterize the extent of this bias, El-Gamal and Grether (1995) introduce an "average
normalized entropy" (ANE) statistic as a reliability measure of the segmentation algorithm. The
ANE is based on the posterior probabilities for each vessel and is expressed as,


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Page 15

∑
=
j
=
H
ji
ki
ik
l
l
1
)(
)(
θ
θ
κ
(12)

where li(θk) is vessel i’s likelihood function value given parameter vector θk and j indicates the
segment within the H segmented model. Using these posterior probabilities the ANE test statistic
is constructed as,

∑∑
=
i
1
=
−=
NH
j
ij ij
N
H ANE
1
2
)(log
1
)(
κκ
. (13)

The test statistic is bounded between zero and one, with lower values indicating a reliable
segmentation of the data. El-Gamal and Inanoglu cite ANE values of 0.065 and 0.09 within their
study of the Turkish banking as reliable measures of segmentation (El-Gamal and Inanoglu
2005). The ANE(3) value obtained in our research was 0.226. A four segment model could
conceivably improve the statistic, but given the empirical intractability of the four segment
model, this ANE value was the best we could obtain.

The estimation results are contained in Table 2. The first column of Table 2 shows the estimated
homogeneous stochastic production frontier for the Northeast Atlantic herring fleet. This model
indicates that production is primarily determined by the vessel’s size and horsepower as well as
the season and location. In addition, the elasticities of input utilization are all positive and satisfy
the traditional monotonicity assumptions. Columns two through four of Table 2 show the
estimated parameters for the three segment stochastic production frontier model. The results
from the heterogeneous production frontier indicate that there exists substantial variation in input
utilization across the segments (i.e., there are substantial differences in the elasticities across
segments). Not all variables significant in the homogeneous model are significant in each of the
heterogeneous segments, and visa versa. The most pronounced being the change in significance
of Hours across the models. Additionally, there do exist a few statistically significant curvature
violations (sign violations) within the heterogeneous model, which will be discussed in the next
section. Fortunately, as we shall see, each segment possesses a set of unique features which may
explain these violations once they are taken into consideration.


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