2.2. Empirical Model and Data
Since we introduced the damage function proposed by [
5], our empirical damage function of the pine wilt disease can be expressed by the following Equation (5):
where
D is the pine wilt disease damage rate,
z is the host tree health,
P is the vector beetle density.
The dependent variable, damage rate
(D), is defined as the portion of the damaged area (ha) by pine wilt disease in terms of total conifer forest areas (ha) in Equation (6). The damaged area was calculated from the number of trees killed by pine wilt disease × basal area. The factors affecting the vector beetle density
(P) are expressed by Equation (7):
where MinWT
(it-1) is the average minimum winter temperature the previous year, SnowWT
(it-1) is the average winter snow fall the previous year, SPT
(it) is the average spring temperature, SMT
(it) is the average summer temperature, SMT
2(it) is the square term of average summer temperature, SMP
(it) is the average summer precipitation, MinFA
(it) is the average minimum fall temperature, RHUM_SP
(it) is the average spring relative humidity, RHUM_FA
(it) is the average fall relative humidity, POP
(it) is the population, t is the year(2010–2017), i represents the 230 cities and counties.
We included the climate variables which affect vector’s development and survival in each life stage in the model. Several climate variables such as MinWT
(it-1), and SnowWT
(it-1) are likely to affect larvae development and survival. The average winter snowfall was included because the warming effect of the snow can prevent the larvae from being exposed to the cold temperatures [
24]. Several climate variables (SPT
it, SMT
it, SMT
2it, SMP
it, MinFA
it, RHUM_SP
it, RHUM_FA
it, SMP
it) are related to the adult flying and oviposition periods. The human population variable represents the infestation by human activities.
The factors influencing host health
(z) during the summer, when newly emerging adults carry the nematodes to healthy trees, were determined by the following equation:
where SMT
(it) is the average summer temperature, SMT
2(it) is the square term of average summer temperature, SMP
(it) is the average summer precipitation.
Combining
fp with
fz, the damage function of pine wilt disease is derived in Equation (9). Here, ct
(2013) implies a catastrophic dummy variable. In 2013, the epidemic of pine wilt disease sharply increased in Korea. The infestation area has decreased after 2013, but the causes of that temporal large outbreak have not yet been clarified. Some experts indicate drought and high summer temperatures in 2013 as the main cause of the large outbreak. The data periods (t) examined were from 2013 to 2017, and the number of panels was 230 counties and cities across the country. In total, 1839 observation were collected from 230 panels in eight-year time periods.
The forest-related data (number of killed trees and forest areas occupied by different species) by county and city were obtained from the Korean Forest Service. The population data by city and county were obtained from Statistics Korea. The climate data was provided by the Korea Meteorological Administration. We obtained the average daily climate data by city and county to derive the average monthly values and then calculated the seasonal mean values. Spring was from March to May, summer from June to August, fall from September to November, and winter from December to the following February.
Table 1 shows the climatic factors that affect insect vector and host trees at each growth stage based on the previous studies. As we see in the
Table 1, the average temperature, humidity, and maximum and minimum temperatures in each season are closely related to the expansion of the pine wilt disease.
2.3. Estimation Method
In this study, the damage rate
(D) is the proportional response variable with a value between 0 and 1. Therefore, by applying the random or fixed-effect method with log transformation, we could omit the observations with a value of 0 during the log conversion process. Also, the effect of any independent variable
could not be constant throughout the range of X [
31]. The application of the logit method has also a drawback, since the log-odd ratio cannot be true if the dependent variable takes on the value of 0 or 1. To mitigate this problem, it is possible to add a small value to zeros, but this may generate a bias estimation due to changes in the distribution of the variables.
To take into account the characteristics of the proportional response variable, we applied Papke and Wooldridge’s method [
32]. For cross-sectional observation
i and time period
t for the response variable,
, and outcomes at zero and one are available. We assumed a set of
vector explanatory variables X
it. The conditional average of the fractional response variable can be expressed as form of the following nonlinear panel probit function, where
is the standard normal cumulative distribution function (CDF), and
Ci is an unobserved effect:
With strictly exogeneity of {x
it:
t = 1,…,T} and conditional normal distribution of
(Equation (11) [
7], Equation (10) can be expressed as Equation (12):
To estimate Equation (12), we can apply the generalized linear model (GLM) with probit link function using quasi-maximum likelihood estimation (QLME). However, this method tends to ignore the serial dependence that may exist in the joint distribution, which may result in inefficiency. MWNLS (multivariate weighted nonlinear least square) is appropriate for estimating conditional means for panel data with strictly exogenous regressor when serial correlation and heteroskedasticity are present [
32]. However, this method requires the parametric model of
. Y
i, here, is the
Tⅹ1 vector of response variables, and obtaining the covariance
is difficult even if
has a fairly simple form. A study [
32] proposed the GEE approach to overcome this problem. GEE uses exchangeable correlation rather than finding a parametric model of
, and GEE and MWNLS are asymptotically equivalent whenever they use the same estimates of the matrix
[
33].
In this study, we used the GEE model with probit link function to estimate the damage function. The conditional average of the damage rate with N number of panels
(i = 1,…N) and T
(i = 1,...,T) years could be expressed by the following Equation (13)
where
Dit is the pine wilt disease damage rate,
Xit is the vector of the explanatory variables,
is the average of across panels.
here is the Mundlak device [
6] which reflects the unobserved effect within the panel. Since the estimated coefficient
cannot explain the estimation result directly in the fractional response model, we should calculate the average partial effects (APEs). Let
be the conditional mean function for the vector of response variable
. Given any consistent estimator
, the APEs can be estimated by taking derivatives or changes with respect to the element X
t of the following Equation (14):
2.5. Economic Evaluation of Pine Wilt Disease
This study evaluated the economic impacts of pine wilt disease by applying the Macpherson’s method [
33] that introduced the concept of green payment to include non-timber benefits, such as carbon sequestration, biodiversity, and wildlife habitats in the model. To consider the impacts of pest inspection and control, we included the costs of inspection and control in the objective function. The objective function also considered the amount of changes in timber production due to damage by pine wilt disease.
The objective function with inspection and control under disease outbreak can be expressed as in Equation (15), which includes timber and non-timber benefit. Through the objective function, we could find the optimal rotation age (
t), which generates the greatest net present value from harvest considering timber and non-timber benefits:
P represents the market price of timber, V(t) is the timber production per unit of land, L is the land area, C is the harvest and establishment cost, r is the discount factor, a refers to the maximum income from the forest each year after logging (after the year t), G(L) denotes the green payment to include non-timber benefits and is assumed to be a linear function of the land area (L), D(∙) denotes the inspection and control cost and is a function of the damaged area, I(t).
Macpherson et al. also assumed that the total area L can be divided into N small areas, while
LiTB(
t) represents the sum of small areas considered for the condition of pest infestation. In this case,
LiTB(
t) is the area that produces timber:
i indicates the rate of infection of the pest and has a value between 0 and 1. The value of one indicates the uninfected condition in which timber production is not affected by pests. Small areas with zero values of
i are excluded from the timber production area. Therefore, the value of
i varies depending on the type of pests and can be expressed as zero for pine wilt disease.
Similarly, the area that affects the non-timber benefits is represented by
LiNTB as follows:
When it is assumed that G(L
iNTB(t)) representing green payments is proportional to the area that generates non-timber benefits, the value can be calculated by multiplying the amount paid per unit area(g) and L
iNTB:
We can derive Equation (19), the first-order conditions for the optimal rotation age, by differentiating Equation (15) with respect to time (
t). The optimal rotation age occurs when the growth rate subtracting the discount rate is the same as the current value of the changes in land area and rent plus the changes in timber benefits and green payments:
We set some scenarios to compare the economic impacts of disease outbreaks. The baseline scenario assumes no disease infestation. If there is no infestation, the objective function including the timber and non-timber benefits can be expressed by Equation (20) [
33]:
Also, a scenario for disease outbreak with no control is established. In that case, the damage rate is the same as in Equation (15) except for the action term. Therefore, the objective function and the condition for optimal rotation age are expressed by the following equations (21) and (22):
If no action has been taken in the infected area, the forest area can be divided into two categories: the susceptible area (S(t)) and the infected area (I(t)). That is, the total area L is the sum of S(t) and I(t) (L = S(t) + I(t)). If the forest area L
iTB, which is affected by pests and affects wood production, is represented by the area where the same timber is produced without pests, it can be described as follows [
33]:
where (L-S(t)) in the right-hand side indicates the infected area (I(t)). In contrast, when the control and inspection are applied, timber areas can be divided into three categories: the susceptible area (S(t)), the controlled area (T(t)), and the infected area (I(t)). Therefore, the timber production area including the control and prevention of pests can be expressed as in Equation (24):
Here, it is assumed that the controlled area (T(t)) is free from pest infestation and does not affect timber production. In addition, assuming that the controlled area is linearly proportional to the area of infection (T(t) = αI(t)), it can be expressed as follows (α(alpha) here implies the control rate):
where the timber production area is the sum of the susceptible area and the controlled area if
ρ = 0.
The areas affecting the non-timber benefits can be represented as in Equations (26) and (27).
The data required for a numerical analysis using the above model include timber volume production function, changes in the area of pest infestation over time, damage rates, control and prevention costs, annual land area, timber prices, and logging and afforestation costs. Most of the data are publicly available, but the area of pest infection over time can be obtained by using the SI model (Susceptible–Infected Model), which is mainly used in studies of pest spreading. The SI model for the no-action model can be described as follows [
34]:
Here, p represents the infected area at the initial stage, and δ indicates the secondary infection rate, i.e., the rate of pests spreading in a forest.
As discussed in the theory section, if no action is taken against pests, the total forest area (L) is the sum of S(t) and I(t), and changes in S(t) over time are indicated as follows:
When the variable separation method is applied to obtain the solution of the above differential equation, S(t) can be expressed as shown in the following expression:
On the other hand, for models performing control and prevention (L is divided into S(t), T(t), and I(t)), changes in S(t) over time are indicated as follows:
Similarly, to obtain S(t), the above expression is solved by applying the variable separation method, which appears as follows:
Substituting Equation (30) for Equations (23) and (26), we can solve
and
. Similarly, substituting Equation (32) for Equations (25) and (27),
and
can be solved. If
ρ = 0 (the timber production is zero in the infected area),
is as follows:
Instead of using the timber volume production function, we used the actual data for age-specific volumes in the National Institute of Forest Science [
35]. The annual average growth (m
3/ha) was derived from the five-year average growth rate (m
3/ha). Data for timber prices (1000 KRW/m
3), afforestation costs (1000 KRW/m
3) and forest management costs (1000 KRW/m
3) were obtained from reference [
36]. It was assumed that pests infest trees after the age of 10 because a trunk injection for disease prevention is targeted to trees over 10 cm in diameter at breast height (DBH), and the age of the trees is approximately 15 to 20 years [
4,
35].
We chose three target species: the Gangwon regional pine
(Pinus densiflora), the central regional pine
(Pinus densiflora), and the Korean pine
(Pinus koraiensis) for our economic analysis. The parameters for the baseline scenario were set as shown in
Table A2 in the
Appendix A. It was assumed that the discount rate was 3% and the green payment was 100,000 KRW/ha. As introduced in the SI model, p represented the infected area, and δ represented the secondary infection rate, i.e. the rate of pest spreading in forests. We estimated the value of p per hectare in 2010 as 0.00054 where the data in the infected area exist. Using the SI model in Equation (28) and the data of the infected area (I(t)) and the susceptible area (S(t)) from 2011 to 2017, we estimated the averaged value of δ as 0.002. The variables
and σ shown in Equations (23) and (26) represented the rates of timber and non-timber available in the infected area, respectively. It was assumed that damaged trees do not generate any timber and non-timber value because all infected trees were cut and incinerated to prevent secondary infection. The subject area was set to 100ha which corresponds to the minimum area owned by corporate forest owners among the members of the Korea Forest Managers Association.