Appendix C. Estimation and Solution Options
EViews estimates the parameters of a wide variety of nonlinear models, from nonlinear least squares equations, to maximum likelihood models, to GMM specifications. These types of nonlinear estimation problems do not have closed form solutions and must be estimated using iterative methods. EViews also solves systems of non-linear equations.
Again, there are no closed form solutions to these problems, and EViews must use an iterative method to obtain a solution.
Below, we provide details on the algorithms used by EViews in dealing with nonlinear estimation and solution, and the optional settings that we provide to allow you to control estimation.
Our discussion here is necessarily brief. For additional details, we direct you to the quite readable discussions in Press, et al. (1992), Quandt (1983), Thisted (1988), and Amemiya (1983).
Setting Estimation Options
When you estimate an equation in EViews, you enter specification information into the Specification tab of the Equation Estimation dialog. Clicking on the Options tab displays a dialog that allows you to set various options to control the estimation procedure. The contents of the dialog will differ depending upon the options available for a particular estimation procedure.
The default settings for the options will be taken from the global options (“Estimation Defaults” on page 941), or from the options used previously to estimate the object.
The Options tab for binary models is depicted here. For other estimator and estimation techniques (e.g. systems) the dialog will differ to reflect the different estimation options that are available.
Starting Coefficient Values
Iterative estimation procedures require starting values for the coefficients of the model. There are no general
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•The objective function must be defined at the starting values. For example, if your objective function contains the expression 1/C(1), then you cannot set C(1) to zero. Similarly, if the objective function contains LOG(C(2)), then C(2) must be greater than zero.
•A poor choice of starting values may cause the nonlinear least squares algorithm to fail. EViews begins nonlinear estimation by taking derivatives of the objective function with respect to the parameters, evaluated at these values. If these derivatives are not well behaved, the algorithm may be unable to proceed.
If, for example, the starting values are such that the derivatives are all zero, you will immediately see an error message indicating that EViews has encountered a “Near Singular Matrix”, and the estimation procedure will stop.
•Unless the objective function is globally concave, iterative algorithms may stop at a local optimum. There will generally be no evidence of this fact in any of the output from estimation.
If you are concerned with the possibility of local optima, you may wish to select various starting values and see whether the estimates converge to the same values. One common suggestion is to estimate the model and then randomly alter each of the estimated coefficients by some percentage, then use these new coefficients as starting values in estimation.
Iteration and Convergence Options
There are two common iteration stopping rules: based on the change in the objective function, or based on the change in parameters. The convergence rule used in EViews is based upon changes in the parameter values. This rule is generally conservative, since the change in the objective function may be quite small as we approach the optimum (this is how we choose the direction), while the parameters may still be changing.
The exact rule in EViews is based on comparing the norm of the change in the parameters with the norm of the current parameter values. More specifically, the convergence test is:
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where θ is the vector of parameters, 
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2 is the 2-norm of x , and tol is the specified tolerance. However, before taking the norms, each parameter is scaled based on the largest observed norm across iterations of the derivative of the least squares residuals with respect to that parameter. This automatic scaling system makes the convergence criteria more robust to changes in the scale of the data, but does mean that restarting the optimization from the final converged values may cause additional iterations to take place, due to slight changes in the automatic scaling value when started from the new parameter values.
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(probit, logit, etc.), count models, censored (tobit) models, and ordered models all fall into this category.
•The derivatives with respect to the AR coefficients in an ARMA specification are always computed analytically while those with respect to the MA coefficients are computed numerically.
•In a limited number of cases, EViews will always use numeric derivatives. For the moment, GARCH and state space models always use numeric derivatives. As noted above, MA coefficient derivatives are always computed numerically.
•Logl objects always use numeric derivatives unless you provide the analytic derivatives in the specification.
Where relevant, the estimation options dialog allows you to control the method of taking derivatives. For example, the options dialog for standard regression allows you to override the use of EViews analytic derivatives, and to choose between favoring speed or accuracy in the computation of any numeric derivatives (note that the additional LS and TSLS options are discussed in detail in Chapter 16, “Additional Regression Methods”, beginning on page 461).
Computing the more accurate numeric derivatives requires additional objective function evaluations. While the algorithms may change in future versions, at present, EViews computes numeric derivatives using either a one-sided finite difference (favor speed), or using a four-point routine using Richardson extrapolation (favor precision). Additional details are provided in Kincaid and Cheney (1996).
Analytic derivatives will often be faster and more accurate than numeric derivatives, especially if the analytic derivatives have been simplified and carefully optimized to remove common subexpressions. Numeric derivatives will sometimes involve fewer floating point operations than analytic, and in these circumstances, may be faster.