Appendix I: Appendix I
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In model 2,
measures the gap in outcome Y between the 1960s
cohort of childhood immigrants and the Canadian-born comparison group in 1985 after
adjusting for differences in Xi. In other words, the difference
between
(coefficient on the childhood immigrant indicator in Model 1)
and
(coefficient on the childhood immigrant indicator in Model 2),
represents
the portion of the observed gap in outcome Y between the 1960s cohort
of childhood immigrants and their Canadian-born counterparts in 1985 that
can be accounted for by differences in the observed individual characteristics.
It can be demonstrated that
when immigrant-specific characteristics are
transformed as the deviation from the means of the 1960s cohort, where
are
means of Xi among the 1960s cohort of childhood immigrants
in 1986 and
are means of Xi among the Canadian-born comparison
group in 1986. From equation 1, one can state that
where
is
the mean outcome of the 1960s cohort of childhood immigrants in 1986 and
is
the mean outcome of the Canadian-born comparison group in 1986. From
equation 2, one can state that
where
are means of immigrant-specific variables.
Since all immigrant-specific variables are transformed as the deviation from
their respective means (in 1986), the means of these transformed variables
equal 0. Accordingly,
and
Therefore
or
Similarly,
reflects changes in the relative (to the comparison group)
outcomes between the 1960s and 1970s cohorts after adjusting for
compositional changes. The difference
captures the portion of the change in the
relative outcome between the 1960s and 1970s cohorts that can be
accounted for by compositional changes. More specifically, it can be shown
that
where
is a vector of means of Xi among the 1960s
cohort of childhood immigrants in 1986 and
is a vector of means of Xi among the Canadian-born
comparison group in 1986;
is a vector of means of Xi among
the 1970s cohort of childhood immigrants in 1996 and
is
a vector of means of Xi among the Canadian-born comparison group in 1996;
are
differences in means of immigrant-specific variables between the 1960s
and 1970s cohorts. On the basis of this equation, one can decompose
into
contributions due to factors common to immigrants and the Canadian-born comparison
group (the sum of the first two terms) and due to immigrant-specific factors
(the third term). Further, one can evaluate the relative contribution of each
explanatory variable to
. The interpretation of
and
also applies to
and
.
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