FAIR CT96-1778
The Management of High seas Fisheries
Partner: Fisheries Research Institute, University of Iceland, Reykjavik, Iceland
Estimation of Growth Functions for
the Norwegian Spring-Spawning Herring*
by
Sveinn Agnarsson
M-3.99
This document does not necessarily reflect the views of the
Commission of the European Communities and in no case
anticipates the Commission’s position in this domain.
*
Acknowledgements: This document has been produced as a part of the European Commission
research project: FAIR-CT-1778. Participants in this project are: The Foundation for Research in
Economics and Business Administration and Centre for Fisheries Economics, Norwegian School of
Economics, Bergen, Norway; Helsinki University of Technology, Helsinki, Finland; Fisheries Research
Institute, University of Iceland, Reykjavik, Iceland; Universidade Nova de Lisboa, Lisbon, Portugal.
The financial support of the European Commission FAIR programme is hereby gratefully
acknowledged.
1.
Growth functions
The exploitation of biological resources can be described by a differential equation of the
form
(1)
dx
 x  F ( x)  h(t )
dt
where x = x(t) denotes the size of the resource population at time t, F(x) represents the
natural growth rate of the population and h(t) is the harvesting rate. From (1) it is obvious
that the population level will decline whenever the harvesting rate exceeds the growth rate
and visa verse.
It is often assumed that both the birth rate, b, and mortality rate, m, are proportional to the
population size. The differential equation in (1) can then be rewritten as
(2)
dx
 bx  mx  rx
dt
where r = (b - m) stands for the net proportional growth rate of the population.
Environmental limitations may, however, force the growth rate to decline as the population
becomes larger. Thus, r may be a function of the population size, r = r(x). The proportional
growth rate of x can then be written as
(3)
r ( x) 
F ( x)
.
x
This model describes a process of feedback, or compensation, when r(x) is a decreasing
function of x.
A useful and much used definition of r(x) is the logistic equation
(4)
dx
1
 rx (1  x)  F ( x)
dt
K
2
where r, assumed positive, is called the intrinsic growth rate, since the proportional growth
rate for small x approximately equals r, and the positive constant K is referred to as the
environmental carrying capacity or saturation level.
Rewriting (4) and assuming a linear harvesting function, i.e. h(t) = y, we have
(5)
x  y  rx (1 
1
x)
K
A more general version of (5) can be written in discrete form as
(6)
( x t  x t 1 )  y t  rx t 1 (1 
1 
x t 1 )
K
which collapses to the logistic curve in (4) when =1.
2.
Data
The data used here is for the Norwegian spring-spawning herring from 1950 to 1995.
During this period the size of the stock varied greatly. It peaked at 11.5 million tons
in 1951 but in the 16 years following 1956 the stock showed an almost uninterrupted
decline, finally shrinking to 4.2 thousand tons in 1972, at which time the Norwegian
spring-spawning herring was at the brink of extinction. The spawning stock remained
very low until the mid 1980’s when the herring stock began to increase again and
measured 5 million tons in 1995, almost half of it’s size in 1951.
The bulk of the spring-spawning herring was traditionally caught by Norwegian boats,
but the 1950’s witnessed a drastic increase in the landings by Russian boats. Iceland
and the Faroe Islands have also been engaged in the fishing, as well as EU-countries.
3
Figure 1.
Development of the spawning stock and total landings of the
Norwegian spring-spawning herring 1950-1995 (thousand tons).
spawning stock
landings
14000
12000
10000
8000
6000
4000
2000
0
1950 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994
3.
Results
The growth function described in (6) was estimated using both linear and non-linear
techniques. The estimated equation was
(7)
( xt  xt 1 )  y t   1 xt 1   2 xt1
where  1 equals r in (6) and  2 equals

r
. Consequently, K can be expressed as 1 .
K
2
Equation (7) was first estimated using ordinary least squares (OLS) but as tests indicated the
presence of hetereoskedasticity - the White test statistic was 18.18 - equation (7) was reestimated using the White heteroskedasticity consistent covariance matrix estimator (OLS in
Table 1). This estimator provides correct estimates of the coefficient covariances in the
presence of heteroskedasticity of unknown form. The estimated intrinsic value was -0.00005
and estimated carrying capacity was 9.2 million tons.
Using OLS may lead to simultaneity bias, since we have lagged values of the herring stock on
both sides of equation (7). Further, the estimates of herring stock may be subject to
measurement error. Thus, (7) was also estimated using the method of instrumental variables
(IV) with two different sets of instruments, lagged values of the annual catch (IV-1 in Table
4
1) and second lags of the herring stock (IV-2). The carrying capacity in the former case is
estimated at 12.9 million tons and 8.1 million tons in the latter. Note, however, that the IV-2
estimate appears to be plagued by autocorrelation since the Q statistic for serial correlation is
highly significant.
Finally, equation (7) was also estimated as a first-order autoregressive conditional
heteroskedasticity process (ARCH). The length of the process was determined by ARCH
tests on the OLS version of (7). The intrinsic value is now -0.00005 and the carrying capacity
7.3 million tons. However, the ARCH model appears to still suffer from serial correlation.
All the methods used to estimate (7) reveal that the residuals from each estimate are nonnormal; the Jarque-Bera test consistently yields a high statistic. This is quite serious, since all
the tests undertaken on the residuals are based on the normality assumption. However, the
assumption of normal residuals can not be rejected at the 1% level when the ARCH process
is used, leading us to prefer that model to the others.
Table 1.
Estimates of logistic curves (equation (7)) for the Norwegian spring
spawning herring. Standard errors in parenthesis.
__________________________________________________________
1
2

K
OLS
0.4709 **
(0.1674)
-0.00005 *
(0.00003)
2
9216
IV-1
0.2991 **
(0.237)
-0.0000 *
(0.0000)
2
12892
IV-2
ARCH(0,1)
0.4855 **
0.4630 **
(0.1556)
(0.0714)
-0.0001 **
-0.0001 **
(0.0000)
(0.0000)
2
2
8118
5571
logL
-376.879
-358.881
2
0.163
0.073
0.296
-0.095
R
Jarque-Beraa 19.109 ** 31.341 ** 20.736 **
4.736
Q
1.147
0.150
12.783 **
9.230
**
White
34.211 **
7.885
ARCH1
5.376 **
2.256
0.729
0.282
__________________________________________________________
logL denotes the value of the log likelihood function, R2 the adjusted R2,
Jarque-Bera the Jarqua-Bera test for normally distributed residuals
(2-distributed with 2 degrees of freedom), Q is the Ljung-Box test for
first order serial correlation (2-distributed with 1 degree of freedom),
White the White test for heteroskedasticity (2-distributed with 4 degrees
of freedom) and ARCH1 is the Lagrange multiplier test for autoregressive
conditional heteroskedasticity ( 2-distributed with 1 degree of freedom).
5
** and * denote significance at the 1% and 5% levels respectively.
The growth equation in (7) can also be expressed in a slightly different manner by
dividing through both sides by xt-1. This variant therefore takes the form
( xt  xt 1 )  yt
 1   2 xt1
xt 1
(8)
where  is expected to take on a value of unity or greater.
Results from estimating this equation are rather different from that obtained earlier
(see Table 2). Although all three models appear to be free from autocorrelation and
heteroskedasticity, the residuals are still not normally distributed, in fact the JarqueBera test statistics are much higher than in the previous cases. The estimated carrying
capacity is between 6,275 and 7,425 million tons, which is somewhat higher than was
obtained from the ARCH model above.
Table 2.
Estimates of logistic curves (equation (8)) for the Norwegian
spring spawning herring. Standard errors in parenthesis.
_______________________________________________
1
2

K
OLS
IV-1
0.9801 ** 1.1357
(0.3562)
(0.4887)
-0.0001
-0.0002
(0.0001)
(0.0001)
1
1
7425
6275
*
IV-2
1.0400
(0.3746)
-0.0002
(0.0001)
1
**
6667
logL
-86.250
0.038
0.029
0.040
R2
Jarque-Beraa1911.457 ** 1616.251 ** 1717.047 **
Q
0.009
0.002
0.001
White
1.981
2.062
1.945
ARCH1
0.024
0.019
0.023
______________________________________________
logL denotes the value of the log likelihood function, 
R2 the adjusted R2, Jarque-Bera the Jarqua-Bera test for
normally distributed residuals (2-distributed with 2
degrees of freedom), Q is the Ljung-Box test for first
order serial correlation (2-distributed with 1 degree of
freedom), White the White test for heteroskedasticity
(2-distributed with 4 degrees of freedom) and ARCH
is the Lagrange multiplier test for autoregressive
conditional heteroskedasticity ( 2-distributed with 1
degree of freedom).
6
** and * denote significance at the 1% and 5% levels
respectively.
Neither the estimates of equation (7) nor (8), presented in Tables 1 and 2, are valid
statistical descriptions of the data generating process.
However, they may be
acceptable for forecasting purposes. Since we would prefer to base our forecasts on a
model with a simple error structure, we have chosen the OLS model without
correction for heteroskedasticity. Heteroskedasticity only affects the standard errors
but not the parameter estimates themselves so that the parameter estimates of the
model are the same as those reported for OLS in Table 1.
Figure 2.
Recursive residuals from an OLS estimate of equation 7.
8000
4000
0
-4000
-8000
55
60
65
70
75
Recursive Residuals
80
85
90
95
± 2 S.E.
As is evident from Figure 2, the parameter estimates have been reasonably stable
throughout our period of study, although the results show that the estimates were wide
of the mark in 1988 when the herring spawning stock increased from 1.2 to 4.0
7
million tons. The parameter estimates appear also to have been quite shaky in the
following years.
4.
Conclusion
In this paper we have estimated logistic growth functions for the Norwegian springspawning herring, assuming both that the growth function was liner and non-linear.
Neither model yielded satisfactory results, with statistical tests always showing that
the error term was not normally distributed. In only one case, the ARCH model of the
non-linear equation, could the theory of normal residuals not be rejected, and then
only at the 1% level.
Bearing in mind the poor statistical properties of these models estimated here, it may
appear heroic to use these models for any meaningful purposes. However, we believe
that a simple model, such as the OLS estimates of equation (7), can be used for
forecasting.
8
References
Clark, C.W. (1990): Mathematical Bioeconomics: The Optimal Management of
Renewable Sources. John Wiley & Sons, New York.
White, K. J. (1993): Shazam User’s Reference Manual. McGraw-Hill, New York.
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