Fund liquidation, self-selection and look-ahead bias in the hedge ...

conditional distribution of returns r i,t+1,...,r i,t+s+1
given Ω t
, which we
the
by denote
f (r i,t+1, ..., r i,t+s+1|Ω t
), (3)
f is generic notation for a (conditional) density function. Empirically,
where
can only obtain full information about this joint distribution if the fund
we
us denote this by Y it
f(r i,t+1,...,r i,t+s+1|Ω t
,Y it
=1). (4)
(3) and (4) are identical, liquidation and self-selection is exogenous and
If
biases arise if the sample selection process is ignored. However, as we
no
seen in the previous sections, it is likely that both liquidation and selfselection
have
are determined by historical performance and other characteristics
may have a relation with returns during the period t +1 to t+s+1. For
that
funds that have high levels of (idiosyncratic) risk are more likely
example,
have extreme returns and are typically less likely to survive (see Brown et
to
1992, or Hendricks, Patel and Zeckhauser, 1997). The difference between
al.,
Theroleofz it
has not liquidated or self-selected during the period t +1to t + s +1. Let
=1. This means we can empirically identify
and (4) drives the look-ahead bias in performance measures.
(3)
distribution of interest in (3) can be derived from the joint distribu-
The
of r i,t+1, ..., r i,t+s+1
and z it
, conditional upon Ω t
and Y it
=1,wherez it
tion
a vector of observable fund characteristics and other variables that
denotes
are relevant for fund liquidation and self-selection from t +1to t + s +1.
will become clear below. First, note that
i,t+1,...,.r
i,t+s+1,z
it
|Ω t
,Y it
=1)= f(r i,t+1,...,r i,t+s+1,z it
,Y it
=1|Ω t
)
f(r
{Y it
=1|Ω t
}
P
P {Y it
=1|r i,t+1, ..., r i,t+s+1,z it
, Ω t
}f (r i,t+1, ..., r i,t+s+1,z it
|Ω t
)
=
{Y it =1|Ω t }
P
(5)
f i,t+1, ..., r i,t+s+1,z it
|Ω t
)
= ,
(r
it w
where
P {Y it =1|Ω t }
(6)
w it
=
P {Y it =1|r i,t+1, ..., r i,t+s+1, Ω t ,z it }
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