2. Methods
2.1 Study area descriptions
This study was conducted in the southern foothills of the Churiahills of Dhanusha District (350-27.50 N and 85.50-86.20 E) from May to August 2014. The Dhanusha District is in the central development region of Nepal, 350 km south-east of the capital city, Kathmandu. The district abuts India in the south. The elevation is approximately 95 m and the climate is sub-tropical with three distinct seasons: spring, monsoon and winter. The mean monthly minimum/maximum temperature is 9.3/21.40 C in January and 26.7/39.60 C in April. The average annual rainfall is 2199 mm. The District covers an area of 119,000 ha, out of which 76,792 ha is used for agriculture. Administratively the district consists of one sub-metropolitan city, eleven urban municipalities and six rural municipalities. The Terai Private Forest Development Association (TPFDA), a local NGO, has worked to promote a tree-based farming practice in then nine Village Development Committees (VDCs11Now, VDCs are a part of either urban or rural municipalities after restructuring the state.) covering 10,500 hectares (Figure 1). Therefore, these nine VDCs were selected for this study. After the state is restructured, some parts of the study site fall in the urban municipality while most parts are still VDCs, now known as rural municipalities.
Figure 1: Study Area
2.2 Household survey
A two-stage sampling approach was adopted for this study. First, one ward22Ward is the lowest administrative unit. from each VDC was selected through purposive sampling. This means a total of nine wards were selected. Second, thirty-two households from each ward were selected randomly. This means 288 sample households were selected. In-person interviews were conducted with the head of the sample households using a questionnaire.
The questionnaire contains detail information on agroforestry practices adopted by farmers and the data on adoption variables including socio-economic, demographic, institutional, and biophysical. The questionnaire was pre-tested through a pilot survey in a village of the study area. A few modifications were made following the pre-testing. A total of 18 households were dropped out of the analysis since these households were practicing a combination of two or more agroforestry practices, agroforest/woodlot, boundary plantation and alley cropping.
2.3 Analytical model
There are four types of agroforestry practices in the study area. These are (i) agroforest/woodlot system (AFS), (ii) alley cropping system (ACS), (iii) combination of two or more AF practices, and (iv) conventional agricultural system (CAS). Since the third system is very rarely practiced in the study area, this has been dropped off from the analysis and we considered the rest three practices only as major farm practices for this study. Farmers can choose one they prefer most from the three alternatives. That means their choice is discrete and mutually exclusive. Therefore, the choice model for this study is either a multinomial probit (MNP) or a multinomial logit (MNL) model. We considered the MNL model the best fit because it gives more precise parameter estimation (Kropko, 2007). The other reason for choosing this model is that this has been more commonly used in recent studies (Deressa et al., 2009; Hassan and Nhemachena, 2008; Kurukulasuriya and Mendelsohn, 2007). Besides, the MNP model is not usually used largely because of the practical difficulty involved in its estimation process (Cheng and Long, 2007).
According to the random utility theory, consumers generally choose what they prefer from among the alternatives available. More precisely, they are assumed to select the alternative that has the highest utility. In this study too, a farmer has three choices and selects a practice from these choices. We assume that the selection of one of the practices is independent of other practices. The choice of one practice is characterized by various factors such as age, education, tenure status, and extension services. Under the random utility theory, the utility of each alternative is modelled as a linear function of observed characteristics (farmer and/or alternative specific) plus an additive error term. More particularly, the utility a farmer i associating to alternatives j and k is given by
\begin{equation} U_{\text{ij}}=V_{\text{ij}}+\varepsilon_{\text{ij}}\ldots\ldots\ldots\ldots\ldots\ldots.\ (1)\nonumber \\ \end{equation}\begin{equation} U_{\text{ik}}=\ V_{\text{ik}}+\varepsilon_{\text{ik}}\ldots\ldots\ldots\ldots\ldots..\ (2)\nonumber \\ \end{equation}
where V implies the deterministic or systematic component of the utility and represents the stochastic component which represents the uncertainty. According to utility maximisation, farmer i will, thus, only chooses a particular alternative j ifUij > Uik for all k ≠ j.
A common formulation of equations (1) and (2) is as follows, assuming V ( ) is a linear function of xi , observed factors to the farmer’s utility:
\begin{equation} U_{\text{ij}}=\ x_{i}\beta_{j}+\ \varepsilon_{\text{ij}}\ldots\ldots\ldots\ldots.\ (3)\nonumber \\ \end{equation}\begin{equation} U_{\text{ik}}=\ x_{i}\beta_{k}+\ \varepsilon_{\text{ik}}\ldots\ldots\ldots\ldots(4)\nonumber \\ \end{equation}
Then, if we denote Yi = j and the farmer’s choice of alternative j , it can be written that
\begin{equation} \text{Prob\ }\left[Y_{i}=j\left|x\right.\ \right]=Prob\ [U_{\text{ij}}>\ U_{\text{ik}}]\nonumber \\ \end{equation}
\(=Prob[x_{i}\beta_{j}+\ \varepsilon_{\text{ij}}\ -\ x_{i}\beta_{k}-\ \varepsilon_{\text{ik}}\ >0\left|x\right.\ \)] =\(Prob\ [x_{i}\left(\beta_{j}-\ \beta_{k}\right)+\ \varepsilon_{\text{ij}}-\ \varepsilon_{\text{ik}}>0\left|x\right.\ ]\)= \(Prob\ [x_{i}\beta+\ \epsilon>0\left|x\right.\ ]\)
Where,
 β is a vector of unknown coefficients that can be explained as the net impacts of a vector of explanatory variables influencing the choice of farming practice and \(\epsilon\) is a random error term.
Assuming that \(\epsilon\) for all alternatives is independent and identically distributed (i.i.d) conditional onxi , with the Type I extreme value distribution, the probability that a farmer will choose alternative j is given by Equation (5):
\begin{equation} \text{Prob\ }\left(Y=j\right)=\frac{e^{\beta_{j}x_{i}}}{\sum_{k=1}^{3}e^{\beta_{k}x_{\text{i\ }}}}\ldots\ldots\ldots\ldots\ldots(5)\nonumber \\ \end{equation}
Equation 5 is the MNL model (Greene 2003). The MNL model significantly requires to hold the assumption of independence of irrelevant alternatives (IIA) in order to obtain unbiased and consistent parameter estimates. The IIA assumption necessitates that the probability of adopting a farming practice by a given farmer requires independence from the probability of selecting another farming practice.
The numerator is the utility (i.e., net benefit) from choice ‘j’ and the denominator is the sum of utilities of all alternative choices. The probability of selecting a specific farming practice is equal to the probability of that specific alternative being higher than or equal to, the utilities of all other alternatives in the set of farming practices. The parameters of this model can be estimated using maximum likelihood methods. However, the parameter estimates of the MNL model merely show the direction of the impact of the explanatory variables on the dependent variable. The real extent of changes or probabilities is not represented by the estimates. Moreover, parameter estimates are hard to interpret since they are derived from non-linear estimates (Greene, 2003). Therefore, the MNL model parameters are transferred to relative risk ratios (RRR). This RRR measures the effects on the relative odds of one outcome being selected relative to the baseline outcome for a unit change in any of the explanatory variables.
2.4 Test of multicollinearity
The model was tested for multicollinearity using the variance inflation factor (VIF). The VIFs for all variables are less than 10 (1.09– 2.03), which indicates that multicollinearity is not a serious problem in this model. Finally, the model was tested for the validity of the IIA assumptions by using the Hausman test for IIA. The test failed to reject the null hypothesis of independence of the farming practices, suggesting that the multinomial logit MNL specification is appropriate to model these practices of smallholder farmers (ranged from -4.63 to 40.73, with probability values ranging from 0.85 to 1.00).
The estimation of the MNL model for this study was undertaken by selecting CAS as the reference state or base category. The odds of two other farming systems namely AFS and ACS to be adopted by farmers with respect to the CAS are estimated in this study. Since the CAS is the base category, most predictor variables will have a positive impact on the adoption of the tree-based farming practices i.e. one unit increase in the predictor variable will increase the likelihood of AFS and ACS adoption.
2.5 Variables used in the model
The dependent variable in the empirical estimation is the choice of a farming option from the three farming practices. The choice of explanatory variables is based on data availability and literature. The explanatory variables for this study include socio-economic, biophysical and institutional characteristics including gender (sex of household heads), age and education of the household head; household size (15 to 60 years); off-farm income; landholding size; risk-taking attitude; level of awareness; livestock herd size; extension services; home to nearest government forest distance, irrigation facility; availability of transport means; membership with farmers’ groups and agricultural organizations, and types of household (native or migrated) (Table 1). Some variables were not included in the model such as farmers’ perception of agroforestry, slope gradient, and access to the credit facility. The variable ‘farmers’ perception on agroforestry’ was dropped off the model because several studies showed that this variable had no relationship with adoption (Alavalapati et al., 1995; Anley et al., 2007; Carlson et al., 1994; Thangata and Alavalapati, 2003) and there is a methodological challenge measuring it (Roberts et al., 1999). The second variable ‘slope gradient’ was not applicable because the study area has little altitudinal variation across the sampled households. The third variable ‘access to credit facility’ was not included because the financial institutions in the study area are reluctant to release loans for agroforestry.