73 Patient cost-sharing, mental health care and inequalities (1) Yit =α +f (Age * itm)+Eit[β +g(Age * itm)]+Hd{γ +k(Age * itm)+Eit[δ +l(Age * itm)]}+Year ′ tω + Controlsit ′ κ +εit where Yit is a binary outcome that captures whether individual i uses any basic (outpatient) or specialist (outpatient and inpatient) mental health services in calendar year t, Hd is a binary variable that separates low from high deductible years (taking zero between 2009 and 2011 and one between 2013 and 2014), and Eit is a binary variable that identifies the discontinuity at 18 (taking zero when the individual is a minor and thus exempted from the deductible and one when the individual has to pay it). Age* itm corresponds to age in months defined on December 31st of every year. We use the best fitting polynomial for age imposing that the order is similar for all polynomials (i.e. f ( ), g( ), k( ), and l( ) are all polynomials of order 1, or all polynomials of order 2, and so on) and a bandwidth of twelve months around the discontinuity thresholds. Yeart (with associated coefficient vector ω) are year dummies, Controlsit (with associated coefficient vector κ) include the individual migratory background and household income quartile at age 17 (time-invariant), and dummies of two-digit home address postcode to control for geographical variations in mental health service supply and socioeconomic neighborhood status (time-variant), and εit is an error term. From this model, β represents the RDD estimate of turning 18 in the low deductible period and δ is the coefficient of interest which equals the deviation between the RDD estimate for the low and high deductible periods. This comparison allows disentangling the impact of changes in the deductible from other potential disruptions at age 18. Such discontinuities might be related to changes in the individual life that influence youth mental health care consumption, such as leaving parents´ house or getting a job; or can be the consequence of the transition from CAMHS to AMHS, where a considerable proportion of patients is no longer followed-up [29]. Contrary to a standard RDD, a difference-in-discontinuity approach allows for discontinuous changes in confounding factors at the threshold, under the assumption that other discontinuities at 18 and any manipulation of the running variable are timeinvariant for the study period. Age* itm is defined in months on December 31 st of every year. This means that months 204-215 correspond to individuals that turned 17 in a given calendar year (215 turning 17 in January and 204 turning 17 in December), months 216-227 to those that turned 18, and 228-239 to those turning 19. We exclude the observations corresponding to months 217 to 227 from our model, because they represent youth that pay only partial deductibles in the year they turn 18. These individuals are exempted from the deductible from January till the month of their birthday, and then face deductible amounts that are proportional to the number of months left after turning 18 (i.e. between eleven - birth-month January and one month - birth-month November). Given the annual data on mental health care use, disentangling the age-trends associated with partial deductibles would imply making additional assumptions on the share of annual care utilisation to be assigned to the months leading up to 18, and the share to the months thereafter, when the proportional 3
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