Estimation of the probit model
Estimation of the Tobit model
Estimation of the probit–Tobit model
More formally, with the usual, non-informative prior over the unknown quantities, p(z, a) µ constant, the full conditional distributions of the two vector blocks z and a are easily established. Respectively, the conditional distribution of z given a is normal with mean the prediction vector Ez º xa and covariance the N × N identity matrix Vz º IN. The conditional distribution of a given z is also normal, but with mean the k-vector Ea º (x¢ x)–1x¢ z and covariance Va º (x¢ x)–1. Accordingly, a Gibbs-sampling, data-augmentation algorism can be constructed to simulate draws from the joint posterior distribution p( z, a | y ). There are five sequential steps:
Step 1: Select starting values a(s).
Step 2: Draw z(s) from the multivariate normal distribution N(xa(s), IN), where a(s) implies conditioning on a(s) from Step 1 and the draws are made such that zi £ 0 when y = 1 is observed, and zi < 0 otherwise.
Step 3: Draw a(s
+ 1)
from the multivariate normal distribution
N((x¢
x)–1x¢z(s),
(x¢x)–1), where z(s) denotes conditioning on z(s) from step 2
(4)
Step 4: Repeat steps 1–3 many times, S1, until convergence is attained.
Step 5: Repeat steps 1–3 many times, S2, and collect samples {z(s) s = 1, 2, ..., S} and {a(s) s = 1, 2, ..., S}.
The draws in the last step can be used to compute means, standard errors or, indeed, to plot histograms of any characteristic of interest in the posterior (Gelman et al. 1995). In the reports of the empirical results that follow, the algorism is run for a burn-in phase of S1 = 5000 observations followed by a collection phase of S2 = 5000 and means and implied standard errors are used to infer confidence intervals. The procedures are implemented on a DELLTM LATITUDETM laptop machine running a PentiumTM III processor at 600 megahertz with commands executed in MATLABTM version 5.1.0.421. All computer codes are available upon request.
In comparison with the algorism in (4) above, Tobit estimation is executed through the following steps:
Step 1: Select starting values a(s) and z(s).
Step 2: Draw s(s), a scalar, from the inverse-gamma distribution ig(s2, v) (Zellner 1971, pp. 371–373), with s2 º (z(s) – x a(s) )¢(z(s) – x a(s))/v, v º N – k and where z(s) and a(s) denote conditioning on z(s) and a(s) from Step 1.
Step 3: Construct z(s) by combining the observed, positive sales quantities
with a
draw
for the latent quantities
zc from the normal distribution
N(xca(s), s(s)IN), where a(s)
and s(s) denote conditioning on a(s)
from Step 1 and
s(s) from step 2 and the draws
z(s) are made
such that zi < 0 when y = 0 is observed
(5)
Step 4: Draw a(s + 1) from the multivariate normal distribution N((x¢ x)–1x¢ z(s), s(s) (x¢ x)–1), where z(s) denotes conditioning on z(s) from step 3 and s(s) implies conditioning on s(s) from step 2.
Step 5: Repeat steps 1–4 many times, S1, until convergence is attained.
Step 6: Repeat steps 1–4 many times, S2, and collect samples {z(s) s = 1, 2, ..., S} and {a(s) s = 1, 2, ..., S}.
Step 1: Select starting values a(s) and z(s).
Step 2: Draw S(s), (a 2 x 2 square, symmetric, positive definite matrix), from the inverse-gamma distribution iW(W, v) (Zellner 1971, pp. 371–373), with W º (z(s) – x a(s))¢(z(s) – x a(s))/v, v º N + m + k + 1 and where z(s) and a(s) denote conditioning on z(s) and a(s) from Step 1.
Step 3: Construct the latent components z(s) by combining the observed,
positive sales quantities with draws for the censored observations
and draws for the latent data in the probit equation, as in steps 2 and 3
in the algorisms (4) and (5) above, respectively
(6)
Step 4: Draw a(s + 1) from the multivariate normal distribution N((x¢ x)–1x¢ z(s), s(s) (x¢x)–1), where z(s) denotes conditioning on z(s) from step 3 and s(s) implies conditioning on s(s) from step 2.
Step 5: Repeat steps 1–4 many times, S1, until convergence is attained.
Step 6: Repeat steps 1–4 many times, S2, and collect samples {z(s) s = 1, 2, ..., S} and {a(s) s = 1, 2, ..., S}.
