llmnp evaluates the log-likelihood for the multinomial probit model.
llmnp(beta, Sigma, X, y, r)k x 1 vector of coefficients
(p-1) x (p-1) Covariance matrix of errors
X is n*(p-1) x k array. X is from differenced system.
y is vector of n indicators of multinomial response (1, …, p).
number of draws used in GHK
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.
X is (p-1)*n x k matrix. Use createX with DIFF=TRUE to create X.
Model for each obs: \(w = Xbeta + e\). \(e\) \(\sim\) \(N(0,Sigma)\).
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, \(A_jw = A_jmu + A_je > 0\) 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 \(j=p\), \(e < - mu\).
For further discussion, see Bayesian Statistics and Marketing by Rossi, Allenby and McCulloch, Chapters 2 and 4. http://www.perossi.org/home/bsm-1