Evaluate Log Likelihood for Multinomial Probit Model
llmnp evaluates the log-likelihood for the multinomial probit model.
llmnp(beta, Sigma, X, y, r)
beta |
k x 1 vector of coefficients |
Sigma |
(p-1) x (p-1) covariance matrix of errors |
X |
n*(p-1) x k array where X is from differenced system |
y |
vector of n indicators of multinomial response (1, ..., p) |
r |
number of draws used in GHK |
X is (p-1)*n x k matrix. Use createX with DIFF=TRUE to create X.
Model for each obs: w = Xbeta + e with e ~ N(0,Sigma).
Censoring mechanism:
if y=j (j<p), w_j > max(w_{-j}) and w_j >0
if y=p, w < 0
To use GHK, we must transform so that these are rectangular regions e.g. if y=1, w_1 > 0 and w_1 - w_{-1} > 0.
Define A_j such that if j=1,…,p-1 then A_jw = A_jmu + A_je > 0 is equivalent to y=j. Thus, if y=j, we have A_je > -A_jmu. Lower truncation is -A_jmu and cov = A_jSigmat(A_j). For j=p, e < - mu.
Value of log-likelihood (sum of log prob of observed multinomial outcomes)
This routine is a utility routine that does not check the input arguments for proper dimensions and type.
Peter Rossi, Anderson School, UCLA, perossichi@gmail.com.
For further discussion, see Chapters 2 and 4, Bayesian Statistics and Marketing by Rossi, Allenby, and McCulloch.
http://www.perossi.org/home/bsm-1
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