Appendix C. Gradients and Derivatives

Many EViews estimation objects provide built-in routines for examining the gradients and derivatives of your specifications. You can, for example, use these tools to examine the analytic derivatives of your nonlinear regression specification in numeric or graphical form, or you can save the gradients from your estimation routine for specification tests.

The gradient and derivative views may be accessed from most estimation objects by selecting View/Gradients and Derivatives or, in some cases, View/Gradients, and then selecting the appropriate view.

If you wish to save the numeric values of your gradients and derivatives, you will need to use the gradient and derivative procedures. These procs may be accessed from the main Proc menu.

Note that all views and procs are not available for every estimation object or every estimation technique.

Gradients

EViews provides you with the ability to examine and work with the gradients of the objective function for a variety of estimation objects. Examining these gradients can provide useful information for evaluating the behavior of your nonlinear estimation routine, or can be used as the basis of various tests of specification.

Since EViews provides a variety of estimation methods and techniques, the notion of a gradient is a bit difficult to describe in casual terms. EViews will generally report the values of the first-order conditions used in estimation. To take the simplest example, ordinary least squares minimizes the sum-of-squared residuals:

The first-order conditions for this objective function are obtained by differentiating with respect to b , yielding

EViews allows you to examine both the sum and the corresponding average, as well as the value for each of the individual observations. Furthermore, you can save the individual values in series for subsequent analysis.

The individual gradient computations are summarized in the following table:

Gradients—765

estimation object that has not been successfully estimated, EViews will compute the gradients at the current parameter values and will note this in the table. This behavior allows you to diagnose unsuccessful estimation problems using the gradient values.

Second, you will note that EViews informs you that the gradients were computed using analytic derivatives. EViews will also inform you if the specification is linear, if the derivatives were computed numerically, or if EViews used a mixture of analytic and numeric techniques. We remind you that all MA coefficient derivatives are computed numerically.

Lastly, there is a table showing the sum and mean of the gradients as well as a column labeled “Newton Dir.”. The column reports the non-Marquardt adjusted Newton direction used in first-derivative iterative estimation procedures (see “First Derivative Methods” on page 757).

In the example above, all of the values are “close” to zero. While one might expect these values always to be close to zero when

evaluated at the estimated parameters, there are a number of reasons why this will not always be the case. First, note that the sum and mean values are highly scale variant so that changes in the scale of the dependent and independent variables may lead to marked changes in these values. Second, you should bear in mind that while the Newton direction is related to the terms used in the optimization procedures, EViews’ test for convergence does not directly use the Newton direction. Third, some of the iteration options for system estimation do not iterate coefficients or weights fully to convergence. Lastly, you should note that the values of these gradients are sensitive to the accuracy of any numeric differentiation.

Gradient Table and Graph

There are a number of situations in which

you may wish to examine the individual contributions to the gra-

dient vector. For example, one common source of singularity in nonlinear estimation is the presence of very small combined with very large gradients at a given set of coefficient values.

EViews allows you to examine your gradients in two ways: as a spreadsheet of values, or as line graphs, with each set of coefficient gradients plotted in a separate graph. Using these tools, you can examine your data for observations with outlier values for the gradients.

Derivatives—767

Derivative Description

The Derivative Description view provides a quick summary of the derivatives used in estimation.

Following estimation of this single equation, we can display the description view by selecting View/Gradients and Derivatives.../Derivative Description.

Derivatives of the Equation Specification

Equation: EQ02

Method: Least Squares

Specification: RESID = Y - ((C(1)*(1-EXP(-C(2)*X))))

Computed using analytic derivatives

There are three parts to the output from this view. First, the line labeled “Specification:” describes the equation specification that we are estimating. You will note that we have written the specification in terms of the implied residual from our specification.

The next line describes the method used to compute the derivatives used in estimation. Here, EViews reports that the derivatives were computed analytically.

Lastly, the bottom portion of the table displays the expressions for the derivatives of the regression function with respect to each coefficient. Note that the derivatives are in terms of the implied residual so that the signs of the expressions have been adjusted accordingly.

In this example, all of the derivatives were computed analytically. In some cases, however, EViews will not know how to take analytic derivatives of your expression with respect to one or more of the coefficient. In this situation, EViews will use analytic expressions where possible, and numeric where necessary, and will report which type of derivative was used for each coefficient.

where f is the standard normal density function. The derivative view of this equation is

Derivatives—769

Derivatives of the Equation Specification

Equation: EQ02

Method: Least Squares

Specification: [AR(1)=C(3)] = Y - (C(1)-EXP(-C(2)*X))

Computed using analytic derivatives

*Note: derivative expressions do not account for ARMA components

Recall that the derivatives of the objective function with respect to the AR components are always computed analytically using the derivatives of the regression specification, and the lags of these values.

One word of caution about derivative expressions. For many equation specifications, analytic derivative expressions will be quite long. In some cases, the analytic derivatives will be longer than the space allotted to them in the table output. You will be able to identify these cases by the trailing “...” in the expression.

To see the entire expression, you will have to create a table object and then resize the appropriate column. Simply click on the Freeze button on the toolbar to create a table object, and then highlight the column of interest. Click on Width on the table toolbar and enter in a larger number.

Derivative Table and Graph

Once we obtain estimates of the parameters of our nonlinear regression model, we can examine the values of the derivatives at the estimated parameter values. Simply select View/Gradients and Derivatives... to see a spreadsheet view or line graph of the values of the derivatives for each coefficient: