# Basic Maximum Likelihood Estimation#

This GAUSS maximum likelihood example demonstrates the use of MAXLIKMT to estimate parameters of a tobit model.

## Key features#

• Usages of data from the file maxlikmttobit.dat (included with maxlikmt).

• User defined likelihood function, `lpr` with 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 `mlmt()` as optional arguments.

• The required ind input.

• Comparison of parameter vector versus PV structure to pass parameters.

### Case One: Use of parameter vector#

```/*
**   Maximum likelihood tobit model
*/
new;
library maxlikmt;

// 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 = ones(5, 1);

// Separate X and y
y = z[., 1];
x = z[., 2:4];

// Declare control structure
struct maxlikmtControl c0;
c0 = maxlikmtcontrolcreate;

// Print Iterations to screen
c0.printiters = 1;

// Change descent algorithm to use BHHH
c0.algorithm = 4;

// Set tolerance level
c0.tol = 1e-6;

// Place bounds on coefficients
// -10 < b0 < 10
//- 10 < b1, b2, b3 < 10
// 0.1 < s2 < 10
c0.Bounds = { -10 10,
-10 10,
-10 10,
-10 10,
.1 10 };

// Declare 'out' to be a maxlikmtResults
// struct to hold optimization results
struct maxlikmtResults out;
out = maxlikmtprt(maxlikmt(&lpr, p0, x, y, c0));
```

### Results#

The `maxlikmtprt()` procedure prints three output tables:

• Estimation results.

• Correlation matrix of parameters.

• Wald confidence limits.

## Estimation results#

```===============================================================================
MaxlikMT Version 3.0.1
===============================================================================

return code =    0
normal convergence

Log-likelihood        -44.8988
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]    1.4303        0.0338      42.348   0.0000      0.0000
x[2,1]    0.4948        0.0355      13.953   0.0000      0.0000
x[3,1]    0.4955        0.0413      12.011   0.0000      0.0000
x[4,1]    0.4119        0.0355      11.596   0.0000      0.0000
x[5,1]    0.1000        0.0132       7.587   0.0000     90.9995
```

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 `maxlikmt()` 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      0.069139903      -0.24058032      0.056496838     -0.088475195
0.069139903                1      -0.30744539     -0.060910219      0.047141863
-0.24058032      -0.30744539                1      -0.31863971      0.054593865
0.056496838     -0.060910219      -0.31863971                1      0.036708978
-0.088475195      0.047141863      0.054593865      0.036708978                1
```

## Confidence intervals#

```Wald Confidence Limits

0.95 confidence limits
Parameters    Estimates     Lower Limit   Upper Limit   Gradient
----------------------------------------------------------------------
x[1,1]          1.4303        1.3632        1.4973        0.0000
x[2,1]          0.4948        0.4244        0.5652        0.0000
x[3,1]          0.4955        0.4136        0.5774        0.0000
x[4,1]          0.4119        0.3414        0.4824        0.0000
x[5,1]          0.1000        0.0738        0.1262       90.9995
```

### Case Two: Use of PV Structure#

The `maxlikmt()` 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 maxlikmt;

// 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");

// Separate X and y
y = z[., 1];
x = z[., 2:4];

//Declare control structure
struct maxlikmtControl c0;
c0 = maxlikmtcontrolcreate;

// Print Iterations to screen
c0.printiters = 1;

// Change descent algorithm to use BHHH
c0.algorithm = 4;

// Set tolerance level
c0.tol = 1e-6;

// Place bounds on coefficients
// -10 < b0 < 10
//- 10 < b1, b2, b3 < 10
// 0.1 < s2 < 10
c0.Bounds = { -10 10,
-10 10,
-10 10,
-10 10,
.1 10 };

// Declare 'out' to be a cmlmtResults
// struct to hold optimization results
struct maxlikmtResults out;
out = maxlikmtprt(maxlikmt(&lpr, p0, x, y, c0));
```

### Results#

For the sake of brevity, we won’t separate the sections of the results.

```===============================================================================
MaxlikMT Version 3.0.1
===============================================================================

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.