We consider independent estimation of the two equations and, subsequently, discuss joint estimation. Following Albert and Chib (1993), Bayes estimates of the probit model, equation (1), are simplified considerably by exploiting the restrictions on the latent dependent variable, z ≡ (z1, z2, ..., zn)¢. The equations depicting participation across the N households can be arranged in matrix form into the system as:
z = xa + u (3)
where z ≡ (z1, z2, ..., zN)¢ is a vector of latent effects constrained to the relevant quadrant (positive if the corresponding element of y ≡ (y1, y2, ..., yn)¢ equals one and negative if the corresponding element is zero; x º (x1¢, x2¢ .., xN¢)¢ is an N x k matrix of household-specific covariates; a º (a1, a2, ..., ak)¢ is the vector depicting impacts of the covariates on the latent effects; and vector u º (u1, u2, .., uN)¢ contains the random errors. Assuming independence across households, each element of u has a normal distribution with mean zero and variance (due to identification) constrained to equal one. With these details at hand, estimation difficulties (arising from the need to evaluate integrals appearing in the likelihood function) are easily overcome by following developments in Albert and Chib (1993). Specifically, by noting that, although the joint distribution of the unknown quantities z and a is intractable, the full conditional distributions comprising the posterior have well known forms. Moreover, these well-known forms are easy to sample from and, hence, a Gibbs-sampling, data-augmentation algorism can be constructed for the purpose of simulating draws from the joint posterior. A detailed description of the estimation algorism is presented in Annex I.
With the probit specification at hand, Tobit estimation can be outlined easily with respect to the matrix model in equation (3) and the algorism in (4) (Annex I). In terms of equation (3), z now contains observed and censored sales data; x contains covariates and a contains the effects of the covariates on the sales quantities. Obviously, and important for policy making, the components of x affecting the sales decision may be different from the covariates affecting the participation decision, and the magnitudes of the effects of a in each case will, in general, also differ.
In the censored regression framework the econometrician observes x and, for i = 1, 2, ..., N, the maximum of zero or yi. This model is investigated in detail in Chib (1992). As noted by Chib (1992), censoring implies complications for both maximum-likelihood and traditional Bayesian approaches to estimation. However, when the observed data y are augmented by the latent data z, the estimation task is simplified because no censoring is involved in the latter formulation. The latent quantities appearing in z are the distance measures appearing in Figure 1 as d. As such, they are important for deriving estimates of the extent to which the households are deficient in sales quantities and, as such, provide important measures for policy formation. However, our main interest presently is the impact that these latent values have on the estimation problem. Developments in Chib (1992) show that these quantities considerably simplify estimation. In fact, the algorism for Tobit estimation of the supply decision has a very similar structure to the algorism for probit estimation of the participation decision, as described in Annex I. In fact, only two changes are required in the notation applied in (4). In step 2, the draws for the latent components of the vector z, namely zi for each i Î c Î{i | yi = 0}, are constrained to be negative. The second modification permits estimation of the nuisance parameter s, which arises due to the fact that, unlike the error variance in the probit model, no such restriction is required in the context of Tobit estimation. A detailed description of the estimation algorism for the Tobit model is presented in Annex I.
Finally, with the probit and Tobit algorisms at hand, it seems natural to discuss a third specification which relies on the idea that the participation and sales decisions, although they may depend on different covariates, are intimately related. In particular, it is natural to presuppose that errors in the two latent specifications (equations (1) and (2)) will be correlated. Where these correlations may be important it seems natural, albeit prudent, to take account of this potentially important feature of the decision-making process. Hence, we now consider the construction of a model in which errors about participation are linked to errors in supply of the livestock products. This issue can now be handled almost trivially in terms of the matrix equations (4) and the algorism in (5) described in Annex I, and with the aid of a little additional notation. Now, let z º (zp, zt)¢, zp º (zp1, zp2, ..., zpN)¢, zt º (zt1, zt2, ..., ztN)¢ denote an N x 2 matrix of observations on the dependent variables in the probit and Tobit models (equations (1) and (2)). In addition, in (3), assume that the same set of covariates are used across both equations but permit the impacts to be different so that a º (ap¢, at)¢, ap º (ap1, ap2, .., apk)¢, at º (at1, at2, .., atk)¢ denotes a k x 2 matrix of effects and the corresponding N x 2 error matrix u º (up, ut), up º (up1, up2, ..., upN)¢, ut º (ut1, ut2, ..., utN)¢ is assumed to have a multivariate normal distribution with mean the 2N null vector 02N and covariance SÄIN. The parameters of the 2 x 2 covariance matrix S are important because they are these parameters that indicate the extent to which the participation and sales decisions will be correlated.
Finally, to complete the outline of the estimation, it remains to determine the form of the joint posterior for the parameters a (a k x 2 matrix of unknown quantities) and the unknown or latent components of the dependent data matrix z (which has dimensions N x 2). As above, the joint distribution is complicated, but its component conditional distributions have the same well-known forms that appear in the single-equation specifications. Relegating the full definitions until later, the fully conditional distributions for the latent components of the first column in matrix z are normal with mean Ezp and variance Vzp dependent on a and with truncation corresponding to the observed participation data (binary, zero or one). The latent components of the second column in matrix z are normally distributed N(Ezt, Vzt) truncated to the negative quadrant. The parameter matrix a has a multivariate-normal distribution N(Ea, Va) and the variance–covariance matrix S has an inverse-Wishart distribution iW(W, v) (Zellner 1971, pp. 395–396). These distributions depend, in turn, on the parameters, Ezp º x ap + Spt Stt–1 (zt – x at), Vzp º Spp – Spt Stt–1Stp; Ezt º x at + Stp Spp–1 (zp – x ap), Vzt º Stt – Stp Spp–1Spt; Ea º (x¢x)–1z, Va º S Ä (x¢x)–1; W º (z – x a)¢(z – x a), v º N – k + m + 1; and the 2 ´ 2 matrix S has (scalar) components Spp, Spt, Stp and Stt. Consequently, simulations from the joint posterior can be undertaken by applying the algorism described in Annex I.
