Maximum Likelihood Estimation with Analytic Gradients ====================================================== This **GAUSS** maximum likelihood example demonstrates the use of **CMLMT** to estimate parameters of a tobit modelwith analytic first derivatives. Key example features ++++++++++++++++++++++ - Usages of data from the file *cmlmttobit.dat* (included with **cmlmt**). - User defined likelihood function, :class:`lpr` with four inputs: - A parameter vector. - Additional *X* and *y* data matrices, which are passed to :func:`cmlmt`` as optional arguments. - The required *ind* input. - The inclusion of analytic gradient computations, as specified in the :class:`lpr` function. Code for estimation ---------------------- :: /* ** Maximum likelihood tobit model */ new; library cmlmt; // Tobit likelihood function with 4 inputs // i. p - The parameter vector // ii-iii. x and y - Extra data needed by the objective procedure // ii. ind - The indicator vector proc lpr(p, x, y, ind); local s2, b0, b, yh, u, res, g1, g2; // Declare 'mm' to be a modelResults // struct local to this procedure struct modelResults mm; // Parameters b0 = p[1]; b = p[2:4]; s2 = p[5]; // Function computations yh = b0 + x * b; res = y - yh; u = y[., 1] ./= 0; // If first element of 'ind' is non-zero, // compute function evaluation if ind[1]; mm.function = u.*lnpdfmvn(res, s2) + (1 - u).*(ln(cdfnc(yh/sqrt(s2)))); endif; // If second element of 'ind' is non-zero, // compute function evaluation if ind[2]; yh = yh/sqrt(s2); g1 = ((res~x.*res)/s2) ~ ((res.*res/s2) - 1)/(2*s2); g2 = ( -( ones(rows(x), 1) ~ x )/sqrt(s2) ) ~ (yh/(2*s2)); g2 = (pdfn(yh)./cdfnc(yh)).*g2; mm.gradient = u.*g1 + (1 - u).*g2; endif; // Return modelResults struct retp(mm); endp; // Set parameter starting values p0 = {1, 1, 1, 1, 1}; // Load data z = loadd(getGAUSSHome("pkgs/cmlmt/examples/cmlmttobit.dat")); // Separate X and y y = z[., 1]; x = z[., 2:4]; // Declare 'out' to be a cmlmtResults // struct to hold optimization results struct cmlmtResults out; out = cmlmtprt(cmlmt(&lpr, p0, x, y)); Results ----------- The :func:`cmlmtprt` procedure prints three output tables: - Estimation results. - Correlation matrix of parameters. - Wald confidence limits. Estimation results ++++++++++++++++++++ :: =============================================================================== CMLMT Version 3.0.0 =============================================================================== return code = 0 normal convergence Log-likelihood -43.9860 Number of cases 100 Covariance of the parameters computed by the following method: ML covariance matrix Parameters Estimates Std. err. Est./s.e. Prob. Gradient --------------------------------------------------------------------- x[1,1] 1.4253 0.0376 37.925 0.0000 0.0000 x[2,1] 0.4976 0.0394 12.642 0.0000 0.0000 x[3,1] 0.4992 0.0458 10.889 0.0000 0.0000 x[4,1] 0.4141 0.0394 10.506 0.0000 0.0000 x[5,1] 0.1231 0.0196 6.284 0.0000 0.0000 The estimation results reports: - That the model has converged normally with a return code of 0. Any return code other than 0, indicates an issue with convergence. The :func:`cmlmt` documentation provides details on how to interpret non-zero return codes. - The log-likelihood value and number of cases. - Parameter estimates, standard errors, t-statistics and associated p-values, and gradients. Parameter correlations +++++++++++++++++++++++ :: Correlation matrix of the parameters 1 0.067006788 -0.24418626 0.05530654 -0.10868104 0.067006788 1 -0.30495236 -0.061965451 0.05808199 -0.24418626 -0.30495236 1 -0.3165649 0.067030893 0.05530654 -0.061965451 -0.3165649 1 0.04466025 -0.10868104 0.05808199 0.067030893 0.04466025 1 Confidence intervals +++++++++++++++++++++++ :: Wald Confidence Limits 0.95 confidence limits Parameters Estimates Lower Limit Upper Limit Gradient ---------------------------------------------------------------------- x[1,1] 1.4253 1.3507 1.4999 0.0000 x[2,1] 0.4976 0.4195 0.5757 0.0000 x[3,1] 0.4992 0.4082 0.5903 0.0000 x[4,1] 0.4141 0.3358 0.4923 0.0000 x[5,1] 0.1231 0.0842 0.1620 0.0000