638—Chapter 21. Discrete and Limited Dependent Variable Models

Then run the artificial regression by clicking on Quick/Estimate Equation…, selecting

Least Squares, and entering:

brmr_y brmr_x (psi*(-xb)*fac)

You can obtain the fitted values by clicking on the Forecast button in the equation toolbar of this artificial regression. The LM test statistic is the sum of squares of these fitted values. If the fitted values from the artificial regression are saved in BRMR_YF, the test statistic can be saved as a scalar named LM_TEST:

scalar lm_test=@sumsq(brmr_yf)

which contains the value 1.5408. You can compare the value of this test statistic with the critical values from the chi-square table with one degree of freedom. To save the p-value as a scalar, enter the command:

scalar p_val=1-@cchisq(lm_test,1)

To examine the value of LM_TEST or P_VAL, double click on the name in the workfile window; the value will be displayed in the status line at the bottom of the EViews window. The p-value in this example is roughly 0.21, so we have little evidence against the null hypothesis of homoskedasticity.

Ordered Dependent Variable Models

EViews estimates the ordered-response model of Aitchison and Silvey (1957) under a variety of assumptions about the latent error distribution. In ordered dependent variable models, the observed y denotes outcomes representing ordered or ranked categories. For example, we may observe individuals who choose between one of four educational outcomes: less than high school, high school, college, advanced degree. Or we may observe individuals who are employed, partially retired, or fully retired.

As in the binary dependent variable model, we can model the observed response by considering a latent variable yi that depends linearly on the explanatory variables xi :

where is i are independent and identically distributed random variables. The observed yi is determined from yi using the rule:

640—Chapter 21. Discrete and Limited Dependent Variable Models

To estimate this model, select Quick/Estimate Equation… from the main menu. From the Equation Estimation dialog, select estimation method ORDERED. The standard estimation dialog will change to match this specification.

There are three parts to specifying an ordered variable model: the equation specification, the error specification, and the sample specification. First, in the Equation specification field, you should type the name of the ordered dependent variable followed by the list of your regressors. In our example, you will enter:

danger body brain sleep

Ordered estimation only supports specification by list so you may not enter an explicit equation.

Also keep in mind that:

•A separate constant term is not separately identified from the limit points γ , so EViews will ignore any constant term in your specification. Thus, the model:

danger c body brain sleep

is equivalent to the specification above.

•EViews requires the dependent variable to be integer valued, otherwise you will see an error message, and estimation will stop. This is not, however, a serious restriction, since you can easily convert the series into an integer using @round, @floor or @ceil in an auto-series expression.

Next, select between the ordered logit, ordered probit, and the ordered extreme value models by choosing one of the three distributions for the latent error term.

Lastly, specify the estimation sample.

Now click on OK, EViews will estimate the parameters of the model using iterative procedures.

Once the estimation procedure converges, EViews will display the estimation results in the equation window. The first part of the table contains the usual header information, including the assumed error distribution, estimation sample, iteration and convergence information, number of distinct values for y , and the method of computing the coefficient covariance matrix.

Estimating Ordered Models in EViews—641

Dependent Variable: DANGER

Method: ML - Ordered Probit

Date: 09/13/97 Time: 10:00

Sample(adjusted): 1 61

Included observations: 58

Excluded observations: 3 after adjusting endpoints

Number of ordered indicator values: 5

Convergence achieved after 5 iterations

Covariance matrix computed using second derivatives

Below the header information are the coefficient estimates and asymptotic standard errors, and the corresponding z-statistics and significance levels. The estimated coefficients of the ordered model must be interpreted with care (see Greene (1997, section 19.8) or Johnston and DiNardo (1997, section 13.9)).

ˆ

The sign of βj shows the direction of the change in the probability of falling in the endpoint rankings (y = 0 or y = 1 ) when xij changes. Pr(y = 0 ) changes in the oppo-

ˆ

site direction of the sign of βj and Pr(y = M ) changes in the same direction as the sign

ˆ

of βj . The effects on the probability of falling in any of the middle rankings are given by:

for k = 1, 2, …, M − 1 . It is impossible to determine the signs of these terms, a priori.

The lower part of the estimation output, labeled “Limit Points”, presents the estimates of the γ coefficients and the associated standard errors and probability values:

LimitPoints

Note that the coefficients are labeled both with the identity of the limit point, and the coefficient number. Just below the limit points are the summary statistics for the equation.