The COPULA Procedure

Displayed Output

Subsections:

PROC COPULA produces displayed output described in the following sections.

Optimization Start and Resulting Parameter Estimates

If you specify the ITPRINT option in the PROC COPULA statement, PROC COPULA displays two tables, “Optimization Start Parameter Estimates” and “Optimization Results Parameter Estimates.” Each table contains the following information for each model parameter:

• parameter number

• parameter name

• parameter estimate

• gradient of the objective function at the initial parameter values

In addition to this information, the table “Optimization Start Parameter Estimates” contains the following columns:

• lower-bound constraint

• upper-bound constraint

The value of the objective function at the parameter values is displayed below each table.

Iteration History for Parameter Estimates

If you specify the ITPRINT option in the PROC COPULA statement, PROC COPULA displays a table that contains the following information for each iteration. Note that some information is specific to the model-fitting method chosen (for example, Newton-Raphson, trust region, or quasi-Newton method).

• iteration number

• number of restarts since the fitting began

• number of function calls

• number of active constraints at the current solution

• value of the objective function (–1 times the log-likelihood value) at the current solution

• change in the objective function from previous iteration

• value of the maximum absolute gradient element

• step size (for Newton-Raphson and quasi-Newton methods)

• slope of the current search direction (for Newton-Raphson and quasi-Newton methods)

• lambda (for trust region method)

• radius value at current iteration (for trust region method)

Model Fit Summary

The "Model Fit Summary" table contains the following information:

• number of observations used

• number of missing values in data set, if any

• data set name

• type of model that was fit

• log-likelihood value at solution

• maximum absolute gradient at solution

• number of iterations

• optimization method

• value of Akaike’s information criterion (AIC) at the solution (a smaller value indicates better fit)

• value of Schwarz-Bayesian criterion (SBC) at the solution (a smaller value indicates better fit)

Under the "Model Fit Summary" is a statement about whether the algorithm successfully converged.

Parameter Estimates

The “Parameter Estimates” table contains the estimates of the model parameters. For the normal copula, this table is not displayed because the only parameters are in the correlation matrix, which is displayed in the “Correlation Matrix” table. For the t copula, the parameter is the number of degrees of freedom; in the table it is called "DF." For Archimedean copulas such as Clayton, Frank, and Gumbel, the parameter is called "theta."

Correlation Matrix

The “Correlation Matrix” table contains the estimates of the model correlation matrix. This table is displayed only for elliptical copulas such as the normal and t copulas. Row and column names come from the list of variables defined in VAR statement.