Regression models

Land use and land cover change research is oftentimes based on modelling ap-proaches, because models allow for understanding key processes and for their quanti-fication. We employ empirical-statistical models using multiple linear regression techniques to analyse the driving factors of cocoa area extension and intensification.

Modelling the effects on cocoa area extension

In order to identify determinants of cocoa expansion, we attempt to identify determi-nants not only for the probability of households to extend their cocoa area, but also

of the extent of area expansion conditional to expansion. For this purpose, we apply a Tobit model which is appropriate when the same explanatory variables influence both the probability of adoption and its extent, because conventional regression methods fail to take into account the qualitative difference between zero and con-tinuous observations (Wooldridge 2003). Input data are derived from 166 cocoa pro-ducing households of the socioeconomic panel survey. The dependant variable is the extension of cocoa cropping area between 2001 and 2006, which includes a substan-tial share of zero values as many households did not expand cocoa area at all. For explanatory factors lag-variables from 2001 were used to adhere to a proper cause-effect relationship.

The Tobit model is an extension of the Probit model, and it was originally developed by James Tobin (Tobin 1958). Tobit models are explicitly developed for censored dependant variables (with upper and/or lower limits), that comprise a substantial amount of zero values (Godoy et al. 1997, Godoy et al. 1998, Dolisca et al. 2007).

The error term is assumed to follow a truncated normal distribution. When determin-ing both, the probability and the intensity of adoption, elasticities measured at the means can be decomposed into an elasticity of adoption and an elasticity of effort when adoption occurs. Tobit models have been rarely applied in agroforestry studies so far (Mercer 2004).

We apply a model of the type Tobit 1 for dependant variables censored at zero:

) distributed with mean at zero and variance σ² (Wooldridge 2003).

The observed value yi is censored at zero:



yi is the observed censored variable, which is equal to the unobserved latent variable yi*, when yi* is bigger than zero. In all other cases yiis equal to zero.

The coefficients are calculated by maximum likelihood estimators (MLE), whereas the likelihood consists of the product of expressions for the probability of obtaining each observation. For each observation greater than zero this expression is just the height of the appropriate density function representing the probability of getting that particular observation.

Hence, the log likelihood function can be defined by

{ } { } 

β and σ are estimated in an iterative numerical procedure. The ML estimator requires homoscedasticity and the normal distribution of the error term (Schmidheiny 2007).

In order to avoid heteroscedasticity, robust standard errors as proposed by White (1980) were applied in the analysis (Gujarati 2004). Standard errors of fitted values are tested for normality, using kernel density plots and Shapiro-Wilk tests.

The direct interpretation of β coefficients would reveal information only on the latent variable y_i*, which mostly is of minor interest. In order to interpret the effects on the expected value of the observed (censored) value, marginal effects should be analysed (cf. Wooldridge 2003, Cong 2000, McDonald and Moffitt 1980).

Two distinct effects are of major interest in our study:

1.) The changes in the probability of being uncensored, hence the probability of co-coa area extension:

and 2.) The changes in the conditional expected value of the dependant variable, hence the effects on the extent of area expansion, conditional on expansion:

( )

The probability of cocoa area extension (equation (4)) is, in addition, calculated with a Probit model in order to test the consistency of Tobit regression results. Tobit and Probit regressions are calculated in Stata 9.2.

Modelling the effects on intensification

For the analysis of intensification determinants, we have to consider that the MI is bound between 0 and 3. For variables with a lower and upper bound, the beta distri-bution can be a suitable model. Particularly, when the boundaries are fixed (i.e., there are no out-of-domain scores) and when boundary scores are qualitatively equal to interior scores, a beta regression should be preferred to a Tobit regression (Smithson and Verkuilen 2006). In contrast, Tobit models treat boundary cases as qualitatively distinct from cases in the interior (see previous paragraph).

Ferrari and Cribari-Neto (2004) propose a regression model that is tailored for situa-tions where the dependent variable (y) is measured continuously on the standard unit interval, i.e. 0 < y < 1. Fitting a bounded dependant variable which exceeds this range can be realized by just rescaling this variable (cf. Smithson and Verkuilen 2006). This type of regression model is based on the assumption that the response is beta distributed. The beta distribution is very flexible for modelling proportions since its density can have quite different shapes depending on the values of the two pa-rameters that index the distribution (Paolino 2001). The model of Ferrari and Cribari-Neto (2004) is defined by only two shape parameters that do not correspond directly to either the mean or variance of the distribution. Rather, the mean and variance of a (standard) two parameter beta distribution are functions of the two shape parameters, α and β. It is considered to be flexible enough to handle a wide range of applications (Paolino 2001).

Instead of using the conventional parameterization approach with two shape parame-ters (α and β), Buis (2006) proposes an alternative parameterization with one location and one scale parameter (φ and μ). We here apply the alternative parameterization of the beta regression which is useful if covariates are present (Ferrari and Cribari-Neto 2004, Paolino 2001, Smithson and Verkuilen 2006) and which corresponds to the conventions of Generalized Linear Models (GLM) (Buis 2006).

The probability density for the beta distribution in the alternative parameterization is given by

The likelihood function is then defined by

( )

[ ( ) ]

[ ( (

) ) ] [

]

( ) ( [

)

]

(

)

We use a statistical package for Stata 9.2 (‘betafit’), developed by Cox, Jenkins and Buis (2006), which fits by maximum likelihood a two-parameter beta distribution to a distribution of the dependant variable, using the alternative parameterization ap-proach. μ is reported on the logit scale, hence it ranges from 0 to 1. φ is reported on the logarithmic scale to ensure that it remains positive. The postestimation command

‘dbetafit’ calculates various types of marginal effects. We consider marginal effects

of each continuous explanatory variable, while keeping all variables at their specified values. The marginal effect is the change in predicted dependant variable for a unit change in the explanatory variable, assuming that the effect does not change over that interval. For dummy variables, a discrete change effect will be estimated, which shows the changes in predicted dependant variable when the explanatory variable changes from its minimum value to its maximum value, while keeping all other vari-ables at their specified values (mean values). Robust standard errors are obtained by using a Huber/White/sandwich estimator of variance. Model consistency was tested by applying post-estimation tests for OLS estimators in linear regression analysis.

Influential values were detected and cleaned using Cook’s distance measures (d-values >0.5 were excluded). Multicollinearity was checked with VIF and 1/VIF (=tolerance). Input data are derived from the cocoa agroforestry study as well as from the socioeconomic panel survey of 2007.

2.3 Results