Adoption and intensity of adoption of conservation farming practices in Zambia

the [0,1] interval. We use three different models to analyse the intensity of adoption to ensure
the robustness of our results: random effects tobit, pooled fractional probit, and unobserved
linear fixed effects models (Papke and Wooldridge 2008).
Correlated Random Effects Tobit Model: Let the share of land that is allocated to CF by
farmer i at time t be it S . The two-limit random effects tobit model for Sit can be specified as
follows:
*
S = X β + u + v
it it it i
⎧ ≤
*
0 if Sit
0
⎪ * *
it ⎨ it it
⎪
*
1 if Sit
≥ 1
S = S if 0< S < 1
⎩
where the dependent variable takes the values of 0 and 1 with positive probabilities (i.e., the
choices to allocate all or no land to CF are legitimate corner solutions). We allow i v and i X
to be correlated using a Chamberlain-like model by assuming
v X N X
2
i | i ~ ( ψ+ iξσ , a)
where
2
σ a is the variance of i
T
−1
i it
t=
1
a in the equation vi = ψ + Xiξ + ai,
and X ≡ T ∑ X is the
1xK vector of time averages (Chamberlain 1980). We can write Sit in the [0,1] interval as:
S = ψ + X β + X ξ + a + u
it it i i it
it | i, i ~
2
(0, σ u )
i | i ~
2
(0, σ a)
u X a N
a X N
The addition of X i on the right hand side of the random effects tobit model takes care of the
unobserved heterogeneity problem and allows us to estimate N -consistent estimates of
2 2
ψ, βξσ , , , and σ (Wooldridge, 2002, Ch. 16).
u a
Pooled Fractional Probit Model: The share of land cultivated with CF is by definition a
fractional response variable, whose properties are not fully taken into account by the random
effects tobit model described above. Papke and Wooldridge (1996) used a quasi-maximum
likelihood estimation (QMLE) to obtain robust and efficient estimators in their seminal
contribution to the empirical literature on fractional response variables. Similar to the tobit
model described above, this model allows for corner solutions. We use pooled QMLE
suggested by Papke and Wooldridge (2008) for panel data applications. More specifically, we
assume,
ES ( | X, v) =Φ ( Xβ + v), t= 1,..., T
it it i it i
where Φ (.) is the standard normal cumulative distribution function (that leads to
computationally simple estimators with unobserved heterogeneity). In order to consistently
estimate the β and the average partial effects (APE) we need to assume that conditional on v i ,
X it is strictly exogenous and that i
15
v , conditional on X it is distributed normally: