where the rate of change of service access by district (hence with Si in logarithmic
form, with i = b,w)5 is modelled linearly in terms of a constant, a component of
observable characteristics expressed in a vector of variables Xi, an eligibility binary
variable ri based on an access threshold (ri = 0 for Si > S∗, ri = 1 for Si ≤ Sˆ),
a proxy Ni for the numbers effect, and a component of unobservable factors (given
by the zero-mean error term ϵi). If the distribution objective matters, the gain in
mean access for worse off districts is measured by the parameter δ, while γ is an
ex post parameter associated with N in (2.3) (γ = 0 − αN = −αN = −α/Pi, with
0 corresponding to an ideal outcome without uncompensated losers). Eq. (2.5)
can be seen as a restricted version of a switching regression model ([36]: 262),
given two regimes, that is in this case public access improvement vs. lack thereof.
Alternatively, if spatial heterogeneity is present with no well-defined threshold, this
would justify GWR with no use of dummies, and ri in (2.5) would be replaced with a
proxy for initial access conditions, which would also capture geographically varying
spatial diffusion (Table 1: lnhca, lnsch).