# 4 Estimation method

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Consider now an immigrant *i* who arrived in year
*c* (a member of arrival cohort *c*) at the age of
*j*. The earnings of this person in year *t* can be
described with a fair degree of flexibility by

where *μ** _{cjt}* is the mean log
earnings in each

*cjt*cell. Equation (4) is the first-stage estimation equation that extracts the individual earnings component from the earnings dynamics of the arrival cohort. A two-stage approach is standard in the literature on earnings inequality and earnings instability; however, in some studies

*ŷ*are obtained by regressing log-earnings on an age polynomial (Haider 2001; Beach, Finnie and Gray 2003; Morissette and Ostrovsky 2005). The approach above appears more flexible in the context of this study.

_{cjit}After obtaining *ŷ _{cjit}* from the
first-stage regression, the variance of

*ŷ*can be decomposed into 'between' and 'within' components. In the descriptive part of this study, it is simply assumed (as in Beach, Finnie and Gray 2003; Morissette and Ostrovsky 2005) that

_{cjit}*y*=

_{cjit}*ȳ*+

_{cji}*v*, and both variance components are computed following the formulas in Johnston (1984).

_{cjit}^{3}

As different arrival cohorts are observed for a different number
of years (for instance, the 1980- to-1982 cohort is observed for 22
years, while the 1998-to-2000 cohort is observed for only 4 years)
it would be difficult to make a cross-cohort comparison of
inequality and instability if calculations were made for all
*t*'s in which a cohort is observed. To make results
comparable across cohorts, the decomposition is computed for a
fixed number of post-arrival periods: *t*=4 (all cohorts),
*t*=7 (all cohorts except that of 1998 to 2000) and
*t*=10 (all cohorts except those of 1995 to 1997 and 1998 to
2000). For instance, if *t*=4 then the variance for the
1980-to-1982 arrival cohort is computed based on 1983, 1984, 1985
and 1986; the variance for the 1983-to-1985 arrival cohort is
computed based on 1986, 1987, 1988 and 1989; and so on. The
resulting panels are unbalanced because, for instance, in four-year
panels, those who were present for only two or three periods are
also included; similarly, seven-year panels include those who were
observed for five or six periods and 10-year panels include those
who were observed for eight or nine periods.

As mentioned in the introduction, the goal of this study is not
only to document immigrant earnings inequality and earnings
instability but also to analyse their potential causes, in
particular, the role of pre-arrival education, language ability and
country of birth. The effects of these variables can be estimated
by adding control variables into the first-stage equation,
re-estimating *y _{cjit}* and using the new estimates
of

*y*on the second stage. More specifically, Equation (4) takes the following form

_{cjit}where *X _{cji}* is foreign education measured by
the years of schooling,

*L*is a set of dummy variables reflecting the ability to speak either official language or both, and

_{cji}*B*is the set of dummies related to the place of birth. A model that includes either

_{cji}*X*,

_{cji}*L*,

_{cji}*B*, or the full set can be estimated. Hence, we can not only compare measures of earnings inequality and earnings instability across different arrival cohorts and arrival ages but also see the degree to which the earnings inequality and instability of each cohort are influenced by these variables. In the context of the Canadian immigrant selection process based on a point system

_{cji}^{4}that rewards foreign education and the ability to speak one of the official Canadian languages, such analysis may be particularly useful.

Although this is a very simple and intuitive method of analysing inequality and instability, it has obvious drawbacks. First, and most importantly, it does not allow for over-time changes in either permanent or transitory components. Second, it does not allow for the heterogeneity in earnings growth, as opposed to the heterogeneity in the levels of earnings. Finally, it ignores serial correlation in the transitory component. Hence, we will consider a more flexible model, similar to the models in Haider (2001) and Baker and Solon (2003).

We proceed as follows. Similar to (2), individual earnings of
the members of *c ^{th}* arrival cohort who were

*j-*years old at arrival are assumed to follow

where *u _{cjit}* =

*u*+

_{cji,t−1}*r*and

_{cjit}*ε*=

_{cjit}*ρε*, +

_{cji,t−1}*λ*. Hence, total experience is broken down into two components: (1) 'Canadian experience,'

_{t}v_{cjit}*t*, which is the same for all members of the

_{c}*c*arrival cohort, and (2) potential foreign experience

^{th}*Z*, simply defined as the age at arrival minus 25.

_{cji}From the residuals in (4), a sample auto-covariance matrix is
constructed for each cohort and arrival age. For instance, for
those who arrived during the 1980-to-1982 period at the age of 30,
this will be a 22×22 matrix ( *t=*1983, 1984,…,
2004); for those who arrived during the 1995-to- 1997 period at the
age of 30 this will be a 7×7 matrix ( *t=*1998,
1999,…, 2004). The size of the matrix will depend on both
*c* and *j*; as the total number of arrival cohorts
is seven, then for *j* ∈ [25,49] there will be
7×25=175 auto-covariance matrices
Ω* _{cj}* in total, which will produce 13,615
sample moments.

Let *ω _{cj}* be a vector of unique elements
of Ω

*,*

_{cj}where *M×M* is the size of each
Ω* _{cj}* matrix depending on

*c*and

*j*. All

*ω*can be stacked into a

_{cj}single vector Ω so that each diagonal element
ω* _{cjt}* in Ω

*can be written as*

_{cj}and each off-diagonal element *ω _{cjts}*
as

The transitory variance component
ε_{cjit} =
*ρε _{cji,t−1}* +

*λ*takes the form of

_{t}ν_{cjit}and the covariance takes the form of

As in Baker and Solon (2003),
σ^{2}_{v} can be modelled as a
quadratic or quartic function of *t* and
*Z _{cj}*. In particular, it may be written as

Assuming that Ω^{*} =
*f*(*t,s,Z;θ*) is the population analog of
Ω, we can now estimate the set of model parameters

by the generalized method of moments (GMM) using 13,615 sample moment corresponding to 13,615 elements in Ω

The parameters in (12) can be estimated using a GMM minimum
distance estimator that chooses an optimal set of parameter
estimates *θ* by minimizing

Haider (2001) and Baker and Solon (2003) point out the
advantages of using an identity matrix as a weighting matrix in
place of *W* (see also Altonji and Segal 1996, Clark 1996).
One particular source of efficiency loss in an equal-weighted
minimum distance estimator is that it ignores the fact that
*ω _{cj}* elements of Ω are based on a
different number of observations. A more efficient estimator may be
obtained if sample moments are weighted in proportion to the size
of each

*cj*cell. The estimation results in this study are based on a minimum-distance estimator that uses both an identity matrix as a weighting matrix and a weighting matrix that weights the sample moments according to their sample sizes.

It can be seen from (7) that setting *p*_{1983}=1
(*t*=0) identifies
*σ*^{2}_{α} in a model
with a single
*σ*^{2}_{α}

*t** is the first loading factor for the cohort to which

*i*belongs. For instance, for the 1980-to-1982 cohort

*t**=1983; for the 1983-to-1985 cohort

*t**=1986; and so on. A diagonal element in Ω

_{cjt}can now be expressed as

Hence, assuming *Z _{cj}*

^{
3
} The within variance component is computed
according to

and the between variance component can be computed according to

^{
4
} The 'point system' introduced in
Canada in 1967 rewards applicants with extra points for a higher
education level, knowledge of official languages (English or
French) and younger age.

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