Estimations of generic drug entry in Brazil using count ... - EPGE/FGV

Zero-in‡ated modelsLambert [17] introduced the Zero-In‡ated Poisson in order toaccount for the excess number of zeros usually found in count regressions, which implyoverdispersion. It can be interpreted as a mixture of the Poisson distribution f 2 () witha degenerate distribution whose mass is concentrated at zero. A binary process withdensity f 1 () sorts the outcomes between the two distributions. Thus, a zero can bean outcome either of the degenerate distribution or of the Poisson distribution, and theresulting probabilities are:Pr [y i = 0] = f 1 (0) + (1 f 1 (0)) f 2 (0) (15)Pr [y i = r] = (1 f 1 (0)) f 2 (r) ; r = 1; 2::: (16)The binary distribution is ordinarily speci…ed as a logit with regressors z i and parameters.The log-likelihood then becomes:nXnXln L () = 1 (y i = 0) ln (exp (z 0 i) + exp ( exp (x 0 i)))+ (1 1 (y i = 0)) (y i x 0 i exp ( exp (x 0 i)i=1i=1The zero-in‡ated negative binomial follows the same reasoning, just substituting thenegative binomial for f 2 () .4.1.2 Ordered multinomial modelsIn such models a …xed number of outcomes, say m, can be obtained, which can be thenordered into integer categories. Let the latent variable be:y i = x 0 i + u i (17)Then de…ne the observed outcomey i = j if j1 < y i jwhere the a j are the thresholds, or cuto¤ points, to be estimated along with the parameters by MLE. It is conventioned that 0 = 1 and m = +1. Then16