In this section, we provide extended examples of working with the logl object to estimate a multinomial logit and a maximum likelihood AR(1) specification. Example programs for these and several other specifications are provided in your default EViews data directory. If you set your default directory to point to the EViews data directory, you should be able to issue a RUN command for each of these programs to create the logl object and to estimate the unknown parameters.

In this example, we demonstrate how to specify and estimate a simple multinomial logit model using the logl object. Suppose the dependent variable Y can take one of three categories 1, 2, and 3. Further suppose that there are data on two regressors, X1 and X2 that vary across observations (individuals). Standard examples include variables such as age and level of education. Then the multinomial logit model assumes that the probability of observing each category in Y is given by:

The log likelihood function for the multinomial logit can be written as:

We have provided, in the Example Files subdirectory of your default EViews directory, a workfile “Mlogit.WK1” containing artificial multinomial data. The program begins by loading this workfile:

from the EViews example directory.

Next, we declare the coefficient vectors that will contain the estimated parameters for each choice alternative:

As an alternative, we could have used the default coefficient vector C.

We then set up the likelihood function by issuing a series of append statements:

Since the analytic derivatives for the multinomial logit are particularly simple, we also specify the expressions for the analytic derivatives to be used during estimation and the appropriate @deriv statements:

Note that if you were to specify this likelihood interactively, you would simply type the expression that follows each append statement directly into the MLOGIT object.

This concludes the actual specification of the likelihood object. Before estimating the model, we get the starting values by estimating a series of binary logit models:

To check whether you have specified the analytic derivatives correctly, choose View/Check Derivatives or use the command:

If you have correctly specified the analytic derivatives, they should be fairly close to the numeric derivatives.

We are now ready to estimate the model. Either click the Estimate button or use the command:

Note that you can examine the derivatives for this model using the Gradient Table view, or you can examine the series in the workfile containing the gradients. You can also look at the intermediate results and log likelihood values. For example, to look at the likelihood contributions for each individual, simply double click on the LOGL1 series.

In this example, we demonstrate using the logl to compute full maximum likelihood estimates of an AR(1). This logl example replicates the ML estimator that is built-into the least squares estimator for an equation ( “Time Series Regression”).

To illustrate, we first generate data that follows an AR(1) process:

The exact Gaussian likelihood function for an AR(1) model is given by:

Since the likelihood function evaluation differs for the first observation in our sample, we create a dummy variable indicator for the first observation:

Next, we declare the coefficient vectors to store the parameter estimates and initialize them with the least squares estimates:

We then specify the likelihood function. We make use of the @recode function to differentiate the evaluation of the likelihood for the first observation from the remaining observations. Note: the @recode function used here uses the updated syntax for this function—please double-check the current documentation for details.

The likelihood specification uses the built-in function @dnorm for the standard normal density. The second term is the Jacobian term that arises from transforming the standard normal variable to one with non-unit variance. (You could, of course, write out the likelihood for the normal distribution without using the @dnorm function.)

The program displays the MLE together with the least squares estimates:

The following additional example programs can be found in the “Example Files” subdirectory of your default EViews directory.