Basic Maximum Likelihood Estimation¶
This GAUSS maximum likelihood example demonstrates the use of CMLMT to estimate parameters of a tobit model.
Key features¶
Usages of data from the file cmlmttobit.dat (included with cmlmt).
- User defined likelihood function,
lprwith four inputs: The required parameters passed as i) a parameter vector, ii) a PV structure.
Additional X and y data matrices, which are passed to
cmlmt()as optional arguments.The required ind input.
- User defined likelihood function,
Comparison of parameter vector versus PV structure to pass parameters.
Case One: Use of parameter vector¶
/*
** 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, res, u;
// 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;
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 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
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 0.067007218 -0.24418499 0.05530801 -0.1086616
0.067007218 1 -0.30495203 -0.061964254 0.058089917
-0.24418499 -0.30495203 1 -0.31656527 0.067029865
0.05530801 -0.061964254 -0.31656527 1 0.044663539
-0.1086616 0.058089917 0.067029865 0.044663539 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
Case Two: Use of PV Structure¶
The cmlmt() also allows the use of the PV parameter structure to pass parameter values to the likelihood function.
While the parameter vector is generally a simpler method, the PV structure can be useful in certain cases:
It allows you to name parameters for easier interpretation of results.
It can be used to fix certain parameters at their start values with
pvPackM().It can be used to specify that parameters are a symmetric matrix with
pvPackSM().
The code below performs the same estimation as the first example but uses the PV structure, in combination with the pack procedures to pass parameters.
new;
library cmlmt;
// Tobit likelihood function with 4 inputs
// i. p - The PV parameter structure
// ii-iii. x and y - Extra data needed by the objective procedure
// ii. ind - The indicator vector
proc lpr(struct PV p, x, y, ind);
local s2, b0, b, yh, u, res;
// Declare 'mm' to be a modelResults
// struct local to this procedure
struct modelResults mm;
// Unpack parameters from PV structure
b0 = pvUnpack(p, "b0");
b = pvUnpack(p, "b");
s2 = pvUnpack(p, "variance");
// 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;
// Return modelResults struct
retp(mm);
endp;
// Declare instance of PV structure
struct PV p0;
p0 = pvCreate;
// Pack parameters into PV structure
// note that first call to pvPack
p0 = pvPack(p0, 1, "b0");
p0 = pvPack(p0, 1|1|1, "b");
p0 = pvPack(p0, 1, "variance");
// 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¶
For the sake of brevity, we won’t separate the sections of the 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
---------------------------------------------------------------------
b0[1,1] 1.4253 0.0376 37.925 0.0000 0.0000
b[1,1] 0.4976 0.0394 12.642 0.0000 0.0000
b[2,1] 0.4992 0.0458 10.889 0.0000 0.0000
b[3,1] 0.4141 0.0394 10.506 0.0000 0.0000
variance[1,1] 0.1231 0.0196 6.284 0.0000 0.0000
Correlation matrix of the parameters
1 0.067007218 -0.24418499 0.05530801 -0.1086616
0.067007218 1 -0.30495203 -0.061964254 0.058089917
-0.24418499 -0.30495203 1 -0.31656527 0.067029865
0.05530801 -0.061964254 -0.31656527 1 0.044663539
-0.1086616 0.058089917 0.067029865 0.044663539 1
Wald Confidence Limits
0.95 confidence limits
Parameters Estimates Lower Limit Upper Limit Gradient
----------------------------------------------------------------------
b0[1,1] 1.4253 1.3507 1.4999 0.0000
b[1,1] 0.4976 0.4195 0.5757 0.0000
b[2,1] 0.4992 0.4082 0.5903 0.0000
b[3,1] 0.4141 0.3358 0.4923 0.0000
variance[1,1] 0.1231 0.0842 0.1620 0.0000
Number of iterations 20
Minutes to convergence 0.00065
The notable feature of these results, is that parameter names are now included in the output tables. This is because they were provided to the PV structure when the starting values were packed.