Labour market integration of immigrants: estimating local authority effects

Abstract

We propose a method for computing indicators of the relative success of local authorities at integrating new immigrants in the labour market, taking account of differences in characteristics of immigrants and local labour markets. The indicator for integration success is based on mean duration from date of residence permit to start of an employment spell. We apply this method to a Danish administrative dataset. Correcting for differences in immigrant characteristics and local labour market conditions has an important impact on the estimated ranking of local authorities. We investigate how robust the ranking is with respect to changes in the specification of criteria for integration success and controls.

Appendix

1.1 Variance of local authority effects

The variance of the Kaplan-Meier restricted mean estimate is

$$Var\left( {\hat E_{KM,t_{\max } } \left( T \right)} \right) = \sum\limits_{t_i \leqslant t_{\max } } {\left( {\int\limits_{t_i }^{t_{\max } } {\hat S_{KM} \left( t \right)dt} } \right)} ^2 \frac{{\Delta N\left( {t_i } \right)}}{{Y\left( {t_i } \right)\left( {Y\left( {t_i } \right) - \Delta N\left( {t_i } \right)} \right)}}$$

where t _i , i = 1,...,n _i , are discrete durations between 0 and t _max (in our application months, i.e. t _i  = 0, 1, 2, ... months), $\hat S_{KM} \left( t \right)$ is the estimated Kaplan-Meier survival function, Y(t _i ) is the number of spells ‘at risk’ (i.e. the number of spells not completed or right-censored at duration t _i ), and ΔN(t _i ) is the number of spells which are completed in the time interval after t _i (i.e. in [t _i , t _{i  + 1}]); see Klein and Moeschberger (2003, chapter 4).

The variance of the restricted mean of the survival function of the estimated model, $\hat E_{t_{\max } } \left( {T\left| x \right.} \right)$ , is estimated as follows. Given the estimates of the parameters and their covariance matrix, we draw randomly 500 parameter vectors. For each of these, we calculate for each spell its expected survival curve (given the model), then the average survival curve of all spells in each municipality and from this, the restricted mean duration of each municipality. Thus, for each municipality, we have 500 different values of the restricted mean duration. From these, we calculate for each municipality the variance $Var\left[ {\hat E_{t_{\max } } \left( {T\left| x \right.} \right)} \right]$.

The variance of the estimated ‘residuals’ $R_{t_{\max } }$ is estimated as the sum of the variances of the restricted mean durations of the Kaplan-Meier and model-based survival functions, respectively:

$$Var\left( {\hat R_{t_{\max } } } \right) = Var\left[ {\hat E_{KM,t_{\max } } \left( T \right)} \right] + Var\left[ {\hat E_{t_{\max } } \left( {T\left| x \right.} \right)} \right]$$

Thus, it is assumed that the covariance is equal to zero. Because it is probably positive, our variance estimate is likely to be ‘conservative’.

1.2 Descriptive statistics and estimation results for women

Table A1 shows descriptive statistics and estimation results for females.

Table A1 Descriptive statistics for controls, and estimation results; females