At first, this last result may seem a little counterintuitive. The power also decreases when ρ 1 increases. 47):įor larger sample sizes, the power increases with ρ 2, as it should because the larger the ρ 2, the stronger the spatial correlation, and hence, the more easily it can be detected. A suggested probability specification for the Olympian ordered model described above is given in Maddala (1983, p. The interpretation of this is that if x i β is small, then the event will have a low chance of taking place. Recall that in the dichotomous variable model, Pr ( y i = 1 ) = F ( x i β ) = Pr ( y i ⩽ x i β ), and Pr ( y i = 0 ) = 1 − F ( x i β ) = Pr ( y i ⩾ x i β ). Let F ( z ) be the CDF of the ( 0, 1 ) normal distribution, and consider the ordered model for the case of the Olympian described above. In this case we could have y i = 0 if that person does not smoke, y i = 1 if the person smokes, but smokes less than a pack/day, and y i = 2 if more than two packs/day are smoked. As another example, suppose we wish to explain the extent that a person smokes cigarettes.
The vector x i might describe the hours in training, financial support received, etc. In this case the values of y i might be 3 if a gold medal is won, 2 if a silver medal is won, 1 if a bronze medal is won, and 0 if no medal is won. For example, y i could be the medal won by the ith Olympian. However, this is not typically a concern because interest usually only focuses on the coefficients of the regressors, namely γ in this formulation of the model.
It follows that the original intercept b cannot be estimated unless the threshold c is known. Then, in this case the inequality condition in ( B), y i ⁎ ⩾ c would be b 0 + z i γ + η i ⩾ 0, where b 0 = ( b − c ). Clearly, if the purchase is not made y i = 0 as in ( C).Īs a technicality, given that the constant term is one of the regressors in x i, and for purposes of illustration, let x i β = b + z i γ where z i is an observed regressor vector. For instance, in the case of a durable good purchase, ( A) in (11.6.7) would relate to the extent of desired purchases, c in ( B) would be the least expensive of that durable good, and y i would indicate that the extent of the actual purchase, which would be equal to the desired purchase, y i ⁎, if the desired purchase is at least as high as the least expensive durable.
Examples of Tobit models are durable good purchases, length of a worker's “down” time due to injury, length of unemployment, etc. Where y i ⁎ is a latent variable which is not observed, c is a constant threshold value which need not be known, and could be zero, x i is a row vector of regressors which includes the constant term. (11.6.7) ( A ) y i ⁎ = x i β + η i, ( B ) y i = y i ⁎ if y i ⁎ ⩾ c, ( C ) y i = 0 otherwise
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