3 Analytical framework

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This paper adopts the methodology employed by Gottschalk and Moffitt (1994: 254), which involves a variance decomposition procedure using longitudinal data. The common starting point is the variance of a worker's (log) earnings over time. Consider the following variables:

y_it = log earnings for person i in year t

T_i = number of years of earnings data observed for person i, i = 1, ..., N

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and

where an over-barindicates a sample average. T is thus the average number of years of earnings

data for the sample of N

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workers. It follows that is average (log) earnings over the earnings-reported years of worker i

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, and is the global, or overall, average level of (log) earnings across all workers in the dataset. The measure of total earnings variation used is then the unbiased estimate for the global variance:

This expression reflects both variation in earnings across time for individual workers and variation in earnings between workers. One can commence the decomposition process by defining a measure of transitory variance or temporary earnings instability as:

The above quantity represents the average across workers of the intertemporal variance of (log) earnings. The measure appearing in square brackets is an (unbiased) estimate of the year-to-year volatility or instability of the (log) earnings of a worker i. The next step is to define a measure of persistent or permanent earnings variance as:

Although this entire Expression (3) is less intuitive than (2), the term on the left essentially captures the variation in earnings—that have already been averaged over time for each worker—across all workers in the sample. It can then be shown that the total variance equals the sum of the transitory variance and the permanent variance, thus providing a convenient decomposition of total variance. Following the same notation as above, we have:

provided that T_i = T for all i, meaning that there are the same number of time series observations for all individuals in the sample. That condition applies throughout this analysis.

In the application of Formulas (1), (2) and (3), y_it is replaced by the life-cycle-adjusted (log) earnings of ln Y_i,t, which is generated as:

where ln Y_it is the actual reported (log) earnings, and estimated ( In Y_it) is predicted log-earnings from an ordinary least squares regression equation of log-earnings on a quartic in age; ya_it is thus generated as log earnings net of life-cycle effects attributable to age. The measure within square brackets in (2), therefore, picks up the life-cycle adjusted variance in (log) earnings, or the variation in (log) earnings about the worker's life-cycle earnings trajectory. The entire Expression (2) captures the average across all workers of this earnings variability. Similarly, Formula (3) essentially captures differences in the levels of life-cycle log-earnings trajectories across workers. Since there is only one life-cycle (log) earnings regression estimated across all workers in each of our samples, high-skilled workers with high-earnings trajectories will have a series of large positive ya_it values, and low-skilled workers with low-earnings trajectories will have a series of large negative ya_it values. The transitory variance captures the volatility of earnings about individuals' life-cycle trajectories, while the permanent variance captures the more persistent and enduring variation in log earnings between workers of different life-cycle profile levels (i.e., between workers of different skill levels).

Formulas (1) to (4) can also be interpreted as a random-effects or error-components model of error structure in the life-cycle equation of log earnings regressed on age (see Johnston 1984: 400). The permanent component of the variation in log-earnings is the 'between (workers) component' of variation, and the transitory component term is the 'within component' of variation (i.e., within the life-cycle for a given worker).