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

eplaced by the assumption that y * i | xi ~ N(the log likelihood function as follows: 17x ' i ,σ 2 ). This then allows specification ofJL = j=0yi= jlog e [Φ(aj− x i' σ) – Φ(aj - 1− x i' σ)] [2]where the first summation operator sums across individuals within the given jcategory and log e (·) denotes the natural logarithmic operator. The maximumlikelihood procedure involves the estimation of the β parameter vector and theancillary standard error parameter σ. Given that the introduction of the knownthresholds fixes the scale of the dependent variable, the estimated coefficients areamenable to a direct and more intuitive OLS-type interpretation. The estimatescontained in the β parameter vector are interpretable on the assumption that we haveactually observed the y * ioutcome for each of the i individuals in the sample.It is important to evaluate our empirical models in regard to certain key econometricassumptions. The adequacy of the estimated models is assessed using the efficientscore tests suggested by Chesher and Irish (1987). These tests require computation ofthe models’ pseudo-residuals. In general terms, the pseudo-residual for the i thindividual is defined for the interval regression model as:u i =aj- 1−'φ(xi) −φσaj−'σ[Φ(xi) −Φσ((aj− x'i)σaj- 1−x'iσ)][3]where φ(·) denotes probability density function operator for the standard normal. 1817 The only difference between this log-likelihood function and that of the ordered probit is that in ourcase the threshold values are known and σ is estimated and not constrained to unity. The exactknowledge of the thresholds allows identification of the scale of the dependent variable and estimationof σ.18 The pseudo-residuals can be obtained by differentiating the log-likelihood expression in [2] withrespect to the constant term implicit in the x vector.13