Education, Employment and Earnings of Secondary School-Leavers ...

In order to understand how the model is implemented, responses are coded 1,2,…..,8to capture the eight distinct earnings categories in our application for employees. 15Let y i denote the observable ordinal variable coded in this way and let y * idenote anunderlying variable that captures the earnings of the i th individual. This can beexpressed as a linear function of a vector of explanatory variables (x i ) using thefollowing relationship:y = + u i where u i ~ N(0, σ 2 ) [1]*ix ' iIt is assumed that y * i is related to the observable ordinal variable y i as follows:y i = 1 if a 0 < y * < a i 1y i = 2 if a 1 ≤ y * < a i 2y i = 3 if a 2 ≤ y * < a i 3y i = 4 if a 3 ≤ y * < a i 4y i = 5 if a 4 ≤ y * < a i 5y i = 6 if a 5 ≤ y * < a i 6y i = 7 if a 6 ≤ y * < a i 7y i = 8 if a 7 ≤ y * < +∞iwhere the a j for j=1,…8 denote the interval boundaries. Following Stewart (1983), wetreat the first and the last intervals as open-ended in this case so for j=0, Φ(a j ) = Φ(–∞) = 0 and for j=8, Φ(a j ) = Φ(+∞) = 1, where Φ(·) denotes the cumulative distributionfunction for the standard normal. 16The exact knowledge of the thresholds allows the likelihood function to be specifiedin a fairly straightforward manner. The variable y * iis best interpreted not as a latentmeasure but one with a quantitative interpretation. In implementing the procedure thestandard normal assumption conventionally invoked for the ordered probit model is15 The focus on the employees is expositional and the j (or sectoral subscript) is ignored forconvenience.16 In the case of the self-employed sample, there are ten categories and thus nine known thresholdvalues.12