112 Chapter 4 To mitigate bias in we used an instrumental variable approach that exploits exogenous variation in eligibility for supported housing arising from differences between assessors’ leniency (see Maestas et al., 2013, Dahl et al., 2014, Bakx et al., 2020 for prior applications). That is, we exploit the fact that (a) the application’s eligibility status for supported housing was determined by an assessor of the independent needs-assessment agency; (b) the allocation of applications to assessors was non-systematic within the regional office and each type of assessment procedure; and (c) assessors had differential discretionary power, with some being systematically more likely to judge the similar applications as eligible than others. Leniency instrument We estimate the assessors’ leniency by calculating the share of other applications that they approved for the period of 2009 to 2013. This leave-one-out measure of leniency for application i handled by each assessor within a regional office j and using the type of assessment procedure a is calculated as the proportion of applications for which eligibility was granted (2) Leniencyija = Number of approval sja −Ii(approved) number of applicationsja −1 Where each approved application Ii(approved) is excluded to guarantee that the instrument is exogenous. We further calculate a residualised measure of leniency to ensure that our instrument does not reflect average differences between regional offices, type of assessment procedure, or time trends in application characteristics, disorder prevalence or clinical practice. We obtain the residuals from an ordinary-least squares regression at the assessor level, with the leave-one-out leniency defined in (2) as dependent variable and regional office, type of assessment procedure and half-year period as independent variables [14, 16, 32]. Regional office and type of assessment procedure are included as dummies, while time is included as the proportion of applications assessed in each halfyear period. The distribution of residualised leniency is presented in Figure 1, together with the first-stage results (discussed in section Relevance of the Instrument).
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