User’s Guide : Advanced Single Equation Analysis : The Log Likelihood (LogL) Object : Limitations
Simple One-Sided Restrictions
Simple Two-Sided Restrictions
The likelihood object can be used to estimate parameters that maximize (or minimize) a variety of objective functions. Although the main use of the likelihood object will be to specify a log likelihood, you can specify least squares and minimum distance estimation problems with the likelihood object as long as the objective function is additive over the sample.
You should be aware that the algorithm used in estimating the parameters of the log likelihood is not well suited to solving arbitrary maximization or minimization problems. The algorithm forms an approximation to the Hessian of the log likelihood, based on the sum of the outer product of the derivatives of the likelihood contributions. This approximation relies on both the functional form and statistical properties of maximum likelihood objective functions, and may not be a good approximation in general settings. Consequently, you may or may not be able to obtain results with other functional forms. Furthermore, the standard error estimates of the parameter values will only have meaning if the series describing the log likelihood contributions are (up to an additive constant) the individual contributions to a correctly specified, well-defined theoretical log likelihood.
Currently, the expressions used to describe the likelihood contribution must follow the rules of EViews series expressions. This restriction implies that we do not allow matrix operations in the likelihood specification. In order to specify likelihood functions for multiple equation models, you may have to write out the expression for the determinants and quadratic forms. Although possible, this may become tedious for models with more than two or three equations. See the multivariate GARCH sample programs for examples of this approach.
Additionally, the logl object does not directly handle optimization subject to general inequality constraints. There are, however, a variety of well-established techniques for imposing simple inequality constraints. We provide examples below. The underlying idea is to apply a monotonic transformation to the coefficient so that the new coefficient term takes on values only in the desired range. The commonly used transformations are the @exp for one-sided restrictions and the @logit and @atan for two-sided restrictions.
You should be aware of the limitations of the transformation approach. First, the approach only works for relatively simple inequality constraints. If you have several cross-coefficient inequality restrictions, the solution will quickly become intractable. Second, in order to perform hypothesis tests on the untransformed coefficient, you will have to obtain an estimate of the standard errors of the associated expressions. Since the transformations are generally nonlinear, you will have to compute linear approximations to the variances yourself (using the delta method). Lastly, inference will be poor near the boundary values of the inequality restrictions.
Simple One-Sided Restrictions
Suppose you would like to restrict the estimate of the coefficient of X to be no larger than 1. One way you could do this is to specify the corresponding subexpression as follows:
' restrict coef on x to not exceed 1
res1 = y - c(1) - (1-exp(c(2)))*x
Note that EViews will report the point estimate and the standard error for the parameter C(2), not the coefficient of X. To find the standard error of the expression 1-exp(c(2)), you will have to use the delta method; see for example Greene (2008).
Simple Two-Sided Restrictions
Suppose instead that you want to restrict the coefficient for X to be between -1 and 1. Then you can specify the expression as:
' restrict coef on x to be between -1 and 1
res1 = y - c(1) - (2*@logit(c(2))-1)*x
Again, EViews will report the point estimate and standard error for the parameter C(2). You will have to use the delta method to compute the standard error of the transformation expression 2*@logit(c(2))-1.
More generally, if you want to restrict the parameter to lie between L and H, you can use the transformation:
(H-L)*@logit(c(1)) + L
where C(1) is the parameter to be estimated. In the above example, L=-1 and H=1.