5    Tobit estimation of milk sales decisions

5.1 Conceptual model

5.2 The Tobit algorithm

5.3 Results

5.4 Conclusions

Although probit estimation of the desired ‘distance’ measures seems to be a fruitful avenue for initial investigation, it suffers a number of limitations. One limitation that seems important here in the context of examining marketable surplus data from the households is that the probit model ignores potentially important information contained in the sales data. In this section we explore the uses of the milk sales data for deriving inferences about entry levels and critical levels of the three key covariates—crossbreed animals, local breed animals and visits by extension agents—as the key precipitators for promoting entry among the subsistence households.

The Tobit procedure is motivated in three steps. First, household maximisation is formalised. Second, relaxing the non-negativity restriction on marketable surplus, a set of latent values are implied for the non-participating households. Third, because we observe the value zero for these households rather than the latent quantities, the data are censored and Tobit estimation is relevant. Here we present the main features of the estimation procedure; details of the procedure are presented in Chib (1992).

5.1 Conceptual model

Let Ф_i ( · )  denote the level of a maximand (a quantity which is to be maximised) of interest in household i (say, the level of expected utility); let φ_i ( · ) denote its first-order partial derivative with respect to variable, v_i (the level of marketable surplus from the household); and let x_{i ≡}(x_1i, x_2i,......x_qi) denote the vector of covariates in question. Across each of the households i = 1, 2, ..., N, we are concerned with the problem:

Ignoring the restriction in (11) for the moment and assuming strict equality in (10), a first-order MacLaurin-series expansion in the left-hand side yields:

where the function φ_i and the partial derivatives φ_i and φ_xki, k = 1, 2 …, q, are evaluated at the point v_i = 0, x_i = 0. Accordingly, we have a (locally) valid expression relating the household’s choice of v_i to the levels of the covariates, x_ki, k = 1, 2, ..., q, in the linear equation:

where β_{0 ≡ –}φ_iφ _vi –1 and  β_{k ≡}–φ_xki φ_vi –1, k=1, 2 …, m. But, when v_i is negative we actually observe zero and, therefore, the relevant statistical framework is the censored regression model:

where ε_i ~ N(0,σ²) and we observe y_i = max{z_i,0}.

Once again, although interest resides with the parameters in (15), fundamental concern lies with the levels of the covariates that are required for participation in the market, that is, the measures beyond which positive marketable surplus is implied for the non-participants in the (censor) set c_≡ {i|z_i ≤ 0}. Just as we did for the probit specification in (7), we can derive estimates of the quantities of interest through a transformation of the estimation equation. In terms of the model in (15), we have ‘distance’ estimates as follows:

The covariates upon which we focus attentions include those considered in the participation exercise in the previous chapter, namely a modern production practice (crossbred cow use), a traditional production practice (indigenous cow use), three intellectual-capital-forming variables (experience, education, extension), and the provision of infrastructure (as measured by time to transport milk to market).

5.2 The Tobit algorithm

In presenting the estimation algorithm for the Tobit model, we note that the Tobit set-up is very similar to the probit model. But this assertion is also true of each of the models to follow. Hence, to conserve notation and remove attention from mathematical detail, we will henceforth note only the dependence of the full conditional densities on latent data and parameters and use the symbols z and θ, respectively, to signify these groups of unknown quantities in the joint posterior of interest.

In the case of the Tobit model, the latent data are the vector arrangement of non-positive quantities z_i, for each household  i ε c {i|y_i = 0}. The parameters, on the other hand, are θ _≡ (β, σ)', with the first vector, β, specifying a column vector of coefficients and the second parameter, σ, specifying the standard deviation of the error variance.

Here the latent data, arranged in the column vector z are important for two reasons. First, as demonstrated incisively by Chib (1992), by augmenting the joint posterior density with these unknown quantities, integrations implied by the Tobit truncation are no longer required, lending the posterior to evaluation simply and, straightforwardly, through data augmentation and Gibbs sampling. Second, because these latent quantities are restricted to lie in the negative segment of the real line, a negative level of marketable surplus is implied. Occasionally, and presently, with a slight abuse of notation, we will use the symbol z to denote both observable and latent data and occasionally, as previously in the case of probit estimation, we will use z to denote only latent quantities. The specification in question should be clear from its context.

With these conventions at hand, the problem of deriving inferences for both z and q can be considered in terms of the regression model:

where z _≡ (z₁, z₂, ..., z_N)' are the observed and latent data on marketed surplus; x _≡ (x₁', x₂', …, x_N'), x_{1 ≡} (x₁₁, x₁₂, …, x_1q), x_{2 ≡} (x₂₁, x₂₂, …, x_2q), …, x_{N ≡} (x_N1, x_N2, ..., x_Nq) are observations on the covariates; β_≡ (β₁, β₂, …, β_q)' are the parameters depicting the effects of changes in the covariates on marketed surplus; and the error vector u _≡ (u₁, u₂, ..., u_N)' is assumed to have the normal distribution N(0_N, σ²I_N), where 0_N denotes the N-dimensional null vector and I_N is the N × N identity matrix. Compared to the set-up in the previous section, we have introduced one additional parameter, namely the error variance σ, and so the estimation algorithm contains one additional step.

We use a conventional non-informative prior for the unknown parameters, namely π (β, σ, z) µ σ^–1. It follows, as demonstrated in Chib (1992), that the introduction of the latent data generates a posterior which, while intractable, has component conditional distributions with well-known forms.

In particular, in terms of the current notation, these conditional distributions can be written as:

where E_{z  ≡} x β, V_{z ≡}  σ²I_N; Eβ_≡ (x' x)^–1x' z, V β _≡ σ² (x' x)^–1; v _≡ N + 1 – K and s² _≡ (z – x β)' (z – x β)/v. Note, once again, that it is simple to sample from multivariate-normal, truncated-normal and inverted-gamma distributions. Consequently, simulations from the joint posterior can be undertaken by sampling sequentially from the respective distributions. The algorithm is very similar to the probit algorithm presented in chapter 4, with the alteration that one additional step must be included in order to simulate the draw from the inverse-gamma distribution, and one additional starting value must be inserted into the first step in the algorithm.

Once again, the resulting outputs, {σ^(s) s = 1, 2, …, S²} {β^(s) s = 1, 2, …, S²} and {z^(s) s = 1, 2, …, S²}, where S² is a number of reasonable magnitude, can be used to plot densities or draw inferences about the likelihood that any of the unknown quantities lie within a specified interval. In the results reported below, the algorithm is run for a ‘burn-in phase’ of S¹ = 2000 observations followed by a ‘collection phase’ of S² = 2000 observations.

The key aspect of the Tobit algorithm that is worth re-emphasising is its production of the latent quantities for the marketable surpluses for the non-participating households. By definition of the Tobit specification, these quantities are negative random variables that specify the amount of marketable surplus (in this case daily sales of fluid milk) by which the households in question are deficient. Unlike the latent specification in the probit model, the dependent variable in (17) takes on positive and zero values. When a zero value is observed, we assume this to imply that the household in question, rather than possessing an excess of the marketable product, actually has a demand for the commodity (that is, a negative supply). Hence, sales quantities are left-censored at zero. This simple observation is developed further in Figure 6.

Figure 6 depicts the utility-maximising household-supply decision. Utility (which is, of course, latent or unobservable) is depicted on the vertical axis and the potential sales quantity is depicted on the horizontal axis. For two households (households i and j) one household maximises utility by producing a positive sales quantity (q_j) whereas the second finds utility maximised in the negative segment of the real line over the supply quantity (q_i). Unlike the first household, the second household’s implicit supply quantity is unobserved and latent. In Figure 6 the quantity z_i is used to represent this latent value. This value is very important for policy purposes because it provides a simple and highly intuitive quantity with which to measure a household’s distance from market (d_i). As such, the values z_i = δ_i, for i є (the censor set) c _≡ { i|q_i = 0 } are an important part of the estimation exercise. In the section that follows we show how they can be used to simplify the estimation problems arising due to censoring in the sales data and latency arising in the probit regression. Hence, these latent data represent another metric by which we may measure a deficiency in the non-participating households.

Figure 6. Rationalising distance to market in the Tobit regression.

5.3 Results

Table 4 reports results of the estimation. All but one of the covariates (experience) is significant at the 5% level. Thus, each of the other covariates has a significant impact on marketable surplus and, therefore, entry into the milk market. Focusing on the parameter estimates themselves, the addition of one crossbred cow raises surplus by about 4.4 litres of milk per day and the addition of one local cow increases surplus by about 1.8 litres—a clear and obvious difference between the modern and the traditional production techniques. Distance to market on the other hand causes surplus to decline, and we estimate that for each one-hour reduction in return time to walk to the milk group, marketable surplus increases by about 3.5 litres. Of the capital-forming variables, (experience, education and extension) education and visits by an extension agent are significant, but surplus is unresponsive to farm experience. The estimates of the responses to education and extension are, perhaps, more important for our study because these variables are potentially more likely to be directly affected by policy. For each additional year of formal schooling of the farm decision-maker, daily marketable surplus increases by about 0.30 litres and, for each additional visit by an extension agent, increases by almost 1.0 litre. The summary statistics suggest a reasonable amount of fit given the high proportion of censoring in the sample—approximately 85% are non-participants.

Table 4. Marketable surplus Tobit-equation estimates.

Turning to the distance measures, Table 5 reports point estimates of the ‘distance’ statistics (equation 16). In order to effect entry, the representative non-participant must increase surplus by 9.8 litres of milk per day. Such an increase, it appears, could be effected by a variety of techniques, including additions to the milking herd of 2.2 crossbred animals or, instead, by 6.4 local cows, a feasible but nonetheless substantial increase in productive assets. Of the remaining covariates for which the distance estimates are significant, entry could also be effected by reducing transport time by almost two hours or by increasing the frequency of extension visits to around 10/household per year.

Table 5. Distance estimates.

5.4 Conclusions

The results of the current investigation, as well as those of the probit specification in the previous section, suggest a clear message: Institutional innovations by themselves are insufficient to catalyse entry; they must be accompanied by a mix of other inputs including infrastructure, knowledge and asset accumulation in the household. Although it is not surprising that milk groups increase the participation of smallholders in fluid milk markets in Ethiopia’s highlands, our empirical results provide insights about how to promote further market participation by smallholder producers. Locating groups so as to minimise the time required to market milk increases the number of participating producers and the level of marketable surplus. Given the difficulty and cost of providing crossbred animals (as experienced by such heifer-loan schemes as Heifer Project International in other parts of Africa), investment in infrastructure such as the milk groups provides a low cost mechanism for increasing smallholder participation and furthering the integration of traditional producers into agro-industrial systems. These results are likely to hold relevance for other perishable and time-constrained agricultural products, such as winter vegetables, cut flowers etc. and, perhaps, for a wider and broader set of circumstances.