2.5 Replication of econometric studies  (Page 6/10)

In an ordered probit, an underlying, normally distributed, latent variable has a mean which is a function of observable variables. The latent variable gives rise to a set of observed dummy variables for ordered categories based on ranges between unobserved but estimable truncation points which correspond to levels of effort, ability, or other factors reflected in the explanatory variables. If L categories are observed, there are $L-1$ truncation points, of which the first is normalized to be zero, so that $L-2$ truncation points are estimated and reported in the table. The values correspond to standard deviations of the latent normally distributed variable.

The key idea is that the values of cutoffs are relative and can be normalized around any value. Notice that the Stata results do not report an intercept term but do report six cutoff values. Moreover, the difference between the estimate by Stata for the first cutoff (3.08402) and the estimate for the second cutoff (3.356916) is equal to 0.272896, which is itself equal to the first truncation point reported by BFS (1998: 193). Use Table 5 to report the difference between the first cutoff value and each of the cutoff points reported by Stata .

The second part of the reconciliation of the two sets of results is to compute the t-ratios. To do this we need to compute the standard deviation of the estimates of the cutoff points reported by Stata . To do this we need to retrieve the variance-covariance matrix from the regression. First, let's see what we are interested in computing. Let ${\widehat{\beta }}_i$ be the estimate of the i ^th cutoff point. In column 3 of Table 5 you computed ${\widehat{\alpha }}_i={\widehat{\beta }}_i-{\widehat{\beta }}_1$ for $i=2,\dots ,6$ . The variance of the new variable is:

The variance-covariance matrix will give us estimates of these variances and covariances. When there are j parameters in a regression equation, this matrix is defined to be:

$$\widehat{\Sigma }=\left[\begin{array}{cccc}{\widehat{\sigma }}_{{\beta }_1}^2& {\widehat{\sigma }}_{{\beta }_1{\beta }_2}& \cdots & {\sigma }_{{\beta }_1{\beta }_k}\\ {\widehat{\sigma }}_{{\beta }_2{\beta }_1}& {\widehat{\sigma }}_{{\beta }_2}^2& \cdots & {\widehat{\sigma }}_{{\beta }_2{\beta }_k}\\ ⋮& ⋮& \ddots & \\ {\sigma }_{{\beta }_k{\beta }_1}& {\sigma }_{{\beta }_k{\beta }_2}& \cdots & {\widehat{\sigma }}_{{\beta }_k}^2\end{array}\right].$$

If you type the command .vce , Stata will report $\widehat{\Sigma }$ as shown in Figure 4. We need the section of this matrix shown in Part A of Table 6. Use equation (5) to estimate the standard errors of the estimates of the cutoff points and complete Part B of Table 6 and compares the t-ratios with the values reported by Butler, et al. (and shown in the last column 4 of Table 6). Are you satisfied that we have been able to come reasonably close to the results reported in the article?