10ISSN 1183-1057
SIMON FRASER UNIVERSITY
Department of Economics
Discussion Papers
07-13
Match Effects
Simon D. Woodcock August 2007Economics
Match E¤ects
1
Simon D. Woodcock
2Simon Fraser University
[email protected]
June 2007
1This document reports the results of research and analysis undertaken by the U.S. Census
Bureau sta¤. It has undergone a Census Bureau review more limited in scope than that given to o¢ cial Census Bureau publications. This document is released to inform interested parties of ongoing research and to encourage discussion of work in progress. This research is a part of the U.S. Census Bureau’s Longitudinal Employer-Household Dynamics Program (LEHD), which is partially supported by the National Science Foundation Grants SES-9978093 and SES-0427889 to Cornell University (Cornell Institute for Social and Economic Research), the National Institute on Aging Grant R01~AG018854, and the Alfred P. Sloan Foundation. The views expressed herein are attributable only to the author(s) and do not represent the views of the U.S. Census Bureau, its program sponsors or data providers. Some or all of the data used in this paper are con…dential data from the LEHD Program. The U.S. Census Bureau supports external researchers’ use of
these data through the Research Data Centers (see www.ces.census.gov). For other questions
regarding the data, please contact Jeremy S. Wu, Manager, U.S. Census Bureau, LEHD Program, Demographic Surveys Division, FOB 3, Room 2138, 4700 Silver Hill Rd., Suitland, MD 20233, USA. ([email protected] http://lehd.dsd.census.gov ).
2Correspondence to: Department of Economics, Simon Fraser University, 8888 University Dr.,
Burnaby, BC V5A 1S6, Canada. I thank John Abowd, Krishna Pendakur, Julia Lane, and
participants at CAFE 2006 for helpful comments and suggestions. I also thank the US Census Bureau and LEHD Program for facilitating data access. This research was partially supported by the SSHRC Institutional Grants program and NSF Grant SES-0339191 to Cornell University.
Abstract
We present an empirical model of earnings that controls for observable and unobservable characteristics of workers (person e¤ects), unmeasured characteristics of their employers (…rm e¤ects), and unmeasured characteristics of worker-…rm matches (match e¤ects). The distinction between these components is important, because they have di¤erent implications for the persistence of individual earnings and the returns to employment mobility. We …nd that match e¤ects, which have been ignored in previous work, are an important determinant of log earnings. They explain about 16 percent of observed variation, and much of the change in earnings when workers change employer. Speci…cations that omit match e¤ects over-estimate the returns to experience by as much as 30 percent, attribute too much variation to person e¤ects and little to …rm e¤ects, and underestimate the correlation between person and …rm e¤ects. Overall, our results suggest that some of the returns previously attributed to general human capital actually re‡ect the returns to sorting into higher-paying …rms and better worker-…rm matches.
JEL Classi…cation: C23, J24
Keywords: linked employer-employee data, earnings dispersion, person and …rm e¤ects, …xed e¤ects, random e¤ects, labor market sorting, human capital
1
Introduction
It is well known that observable characteristics of workers and …rms explain little of the observed variation in employment earnings. However, the empirical nature of residual earn-ings dispersion remains poorly understood. Recently, the advent of longitudinal linked data on employers and employees has opened the door to understand its nature more deeply. A major …nding in this area, now documented for several countries, is that residual earnings dispersion comprises large and persistent worker- and …rm-speci…c components.1 These
het-erogeneity components re‡ect the contribution of unmeasured worker and …rm characteristics to employment earnings, and have come to be known as person and …rm e¤ects. Person and …rm e¤ects have been used to explain a number of well-known sources of earnings di¤erences between otherwise similar workers, including …rm-size and inter-industry wage di¤erentials. Of course person and …rm e¤ects can not capture all salient features of residual variation in earnings. In particular, there are several reasons to expect systematic earnings di¤erences at the level of the worker-…rm match that are distinct from person and …rm e¤ects. For example, matching models such as Jovanovic (1979) suggest there are “good” and “bad” matches between workers and …rms. Production complementarities between workers and …rms, or the presence of match-speci…c human capital, will induce match-speci…c earnings variation that cannot be captured by additively-separable person and …rm e¤ects. These considerations lead us to consider a more general empirical speci…cation that, in addition to person and …rm e¤ects, includes an interaction e¤ect between worker and …rm. We call this the match e¤ect. It measures persistent match-speci…c di¤erences in log earnings between workers with the same measured and unmeasured characteristics, and who are employed in otherwise identical …rms.
The primary contribution of the match e¤ects model is to allow us to measure the rel-ative importance of worker-speci…c, …rm-speci…c, and match-speci…c heterogeneity in labor earnings. The relative magnitude of these three components is of substantive economic inter-est. If residual wage variation primarily re‡ects unmeasured personal characteristics, then individual wages will be highly persistent, largely invariant to where individuals work, and the potential returns to employment mobility will be small. On the other hand, if residual wage variation is primarily …rm- and match-speci…c, then job change will induce substantial variability in earnings. The implied cost of involuntary displacement from high-paying …rms and “good” matches will be large, but so will the potential returns to job search.
We estimate the match e¤ects model on the US Census Bureau’s Longitudinal
Employer-1See Abowd et al. (2003) for US results, Abowd et al. (1999) for France, Andrews et al. (2004a) and
Cornelißen and Hübler (2007) for Germany, Barth and Dale-Olsen (2003) for Norway, Gruetter and Lalive (2004) for Austria, and Maré and Hyslop (2006) for New Zealand.
Household Dynamics (LEHD) database. We …nd that unobserved match-speci…c heterogene-ity is an important component of log earnings: match e¤ects explain about 16 percent of observed variation. This compares to 22 percent explained by unobserved …rm-speci…c het-erogeneity, 36 percent explained by unobserved personal hethet-erogeneity, and 17.5 percent due to observable characteristics. Overall, our results imply considerable persistence in individual earnings, coupled with substantial potential returns to job search.
We use the match e¤ects model to investigate the sources of earnings growth when indi-viduals change employer. It is well known that a large portion of lifetime earnings growth occurs when individuals change job (e.g., Bartel and Borjas (1981), Altonji and Shakotko (1987), Topel and Ward (1992), and others). Workers could experience above-average wage growth when they change employers because they move from lower-paying …rms to higher-paying …rms, or because they move to “better” (i.e., higher-higher-paying) worker-…rm matches. We …nd that the relative importance of these two sources of wage growth depends critically on whether there is an intervening period of non-employment between jobs. In the LEHD data, workers who transit directly from one employer to another experience earnings growth that is nearly 3 times greater than the annual earnings growth of job stayers. Our preferred speci…cation attributes about 60 percent of the excess earnings growth to sorting into higher-paying …rms, and 28 percent to sorting into better matches. In contrast, individuals who experience an intervening period of non-employment have much lower wage growth than individuals who do not change jobs, and the di¤erence is entirely attributable to sorting into lower-paying matches.
A secondary contribution of the match e¤ects model is to correct potential biases in the person and …rm e¤ects model. Omitted match e¤ects will bias the estimated coe¢ cients of observable characteristics that are correlated with the match e¤ect. This will manifest itself, for example, if workers with some characteristics are more successful at …nding good worker-…rm matches than others. We …nd considerable evidence of this bias in the estimated returns to experience. We …nd that the person and …rm e¤ects model over-estimates the returns to 25 years of experience by 30 percent for men and 25 percent for women. Our results imply that some of the returns traditionally attributed to the accumulation of general human capital are actually attributable to how individuals sort into worker-…rm matches, and that workers sort into increasingly good matches over the course of their career.
Estimated person and …rm e¤ects are also biased by the omission of match e¤ects. In fact, we show that estimated person and …rm e¤ects are unbiased only if all excluded match e¤ects are zero. We easily reject this hypothesis. As a consequence, we …nd that the person and …rm e¤ects model overestimates the proportion of variation attributable to person e¤ects, underestimates the proportion of attributable to …rm e¤ects, and underestimates the
correlation between person and …rm e¤ects. The latter result is intriguing. US estimates of the person and …rm e¤ects model consistently yield a near-zero correlation between estimated person and …rm e¤ects (see Abowd et al. (2004), in particular), and the correlation is usually negative in other countries. This puzzling result suggests that “good” workers do not, in general, sort into employment at “good” …rms. In contrast, our estimates based on the match e¤ects model yield a small positive correlation between person and …rm e¤ects. To our knowledge, this is the …rst evidence of positive assortative matching, albeit weak, in the US labor market.
The remainder of the paper is organized as follows. In Section 2, we present the match e¤ects model, develop our estimators, and derive the bias due to omitted match e¤ects. Section 3 describes the data used in the empirical application, and Section 4 presents the estimation results. We conclude with some brief remarks in Section 5.
2
The Match E¤ects Model
We consider the empirical speci…cation
yijt= +x0ijt + i + j + ij +"ijt (1) where yijt is log compensation of worker i at …rm j in period t; is the grand mean; xijt is a vector of time-varying observable characteristics that earn returns ; i is a person e¤ect that measures the returns to time-invariant personal characteristics; j is a …rm e¤ect that measures the returns to time-invariant …rm characteristics; ij is a match e¤ect that measures the returns to characteristics of the worker-…rm match; and"ijtis stochastic error. In general, the person e¤ect measures persistent di¤erences in compensation between individuals, conditional on time-varying characteristics, …rm e¤ects, and match e¤ects. It may include both observed and unobserved components. Here, we consider the case where i = i+u0i ; where ui is a vector of time-invariant observable personal characteristics that earn returns ; and i measures the returns to unmeasured personal characteristics.
The …rm e¤ect measures persistent di¤erences in compensation between …rms, conditional on measured and unmeasured characteristics of their employees and worker-…rm matches. Persistent di¤erences in compensation could arise for a variety of reasons, including pro-ductivity di¤erences between …rms, returns to …rm-speci…c human capital, product market conditions, monopsony power, compensating di¤erentials, or …rm-speci…c compensation poli-cies. In general, the …rm e¤ect may include both observed and unobserved components. In our application, j re‡ects purely unobserved …rm-speci…c heterogeneity.
The match e¤ect measures the returns to time-invariant characteristics of worker-…rm matches, conditional on measured and unmeasured characteristics of workers and their em-ployers. These may include the returns to potentially observable characteristics such as occu-pation, or unobserved characteristics such as match-speci…c human capital, “match quality,” or the value of production complementarities between worker and …rm. In our application
ij re‡ects purely unobserved match-speci…c heterogeneity.
Let N denote the total number of observations; N is the number of individuals; J is the number of …rms; M N J is the number of worker-…rm employment matches; k is the number of time-varying covariates; andqis the number of time-invariant observable personal characteristics. We rewrite the match e¤ects model in matrix notation:
y = +X +D +F +G +" (2)
= +U (3)
whereyis theN 1vector of log compensation; is now the N 1mean vector;X is the N k matrix of time-varying covariates; is a k 1 parameter vector; D is theN N design matrix of the person e¤ects; is the N 1 vector of person e¤ects; F is the N J design matrix of the …rm e¤ects; is theJ 1vector of …rm e¤ects;Gis the N M design matrix of the match e¤ects; is theM 1 vector of match e¤ects; is the N 1vector of unobserved components of the person e¤ect;U is theN q matrix of time-invariant personal characteristics; is aq 1 parameter vector; and " is the N 1error vector.
In the absence of match e¤ects, equation (2) reduces to the person and …rm e¤ects model considered by Abowd et al. (1999), Abowd et al. (2002), and others. The person and …rm e¤ects model is a more parsimonious alternative to the match e¤ects model, so we treat it as a baseline speci…cation in our empirical application. Note this speci…cation impliesM linear restrictions ( ij = 0) on the match e¤ects model. We test these restrictions in the empirical application of Section 4.
2.1
Identi…cation and Estimation
We assume throughout that errors have zero conditional mean and are spherical:
E["ijtji; j; t; xijt] = 0 (4) E["ijt"mnsji; j; t; m; n; s; xijt; xmns] = ( 2 " fori=m; j =n; t=s 0 otherwise. (5)
Identi…cation of the person, …rm, and match e¤ects depends critically on employment mobility. Because the person e¤ect measures the component of earnings that is common to all of an individual’s employment spells (i.e., is portable), identi…cation requires repeated observations on the individual at di¤erent employers. Likewise, because the …rm e¤ect mea-sures the component of earnings that is common to all employees of the …rm, identi…cation requires observations on multiple employees. The match e¤ect measures the component of earnings that is speci…c to an employment spell and distinct from person- and …rm-speci…c components. Hence identifying the match e¤ect requires repeated observations on worker-…rm matches, and separately identifying all three e¤ects requires mobility of workers between …rms.
The zero conditional mean assumption (4) has implications for employment mobility. Speci…cally, it requires that employment mobility is conditionally exogenous. That is, em-ployment mobility may depend only on observable characteristics, person e¤ects, …rm e¤ects, and match e¤ects. We note this is a weaker requirement than in the case of the person and …rm e¤ects model. Whereas the match e¤ects model is robust to employment mobility based on unmeasured match-speci…c heterogeneity (match e¤ects), the person and …rm ef-fects model is not. It is reasonable to expect that mobility due to unmeasured match-speci…c heterogeneity will be empirically relevant in the presence of match-speci…c human capital or production complementarities between worker and …rm.
In principle, it is possible to estimate the match e¤ects model under either …xed or random e¤ects assumptions about the unobserved heterogeneity components. We consider both approaches below. We begin by discussing …xed e¤ect estimation because most prior work based on the person and …rm e¤ects model has taken this approach, and because economists typically prefer …xed e¤ect estimators over random e¤ect alternatives. Unfortunately, a …xed e¤ect estimator is not ideally suited to the match e¤ects model. This leads us to prefer a novel random e¤ects estimator that we present subsequently.
2.1.1 Fixed E¤ect Estimation
Estimating in the presence of …xed person, …rm, and match e¤ects is straightforward. Applying standard results for partitioned regression, the least squares estimator of is the “within-match” estimator:
^ = X0M[D F G]X 1
where MA I A(A0A) A0 projects onto the column null space of A, and A denotes a generalized inverse of A.2 Some algebra veri…es thatM
[D F G] takes deviations from match-speci…c means.3 So we can recover ^ from the regression of y
ijt on xijt, both in deviations from match-speci…c means:
yijt yij = (xijt xij )0 + ijt (7) where yij and xij are sample means of yijt and xijt; respectively, in the match between workeriand …rmj;and ijtis statistical error. Note this simple method to recover the least squares estimate of is only valid when the model includes match e¤ects.4
Separately identifying the person, …rm, and match e¤ects using a …xed e¤ect estimator is less trivial than estimating . At its core, the identi…cation problem is to distinguish “good” workers and …rms (i.e., those with large person/…rm e¤ects) from “good” matches. In the case of the …xed e¤ect estimator, this is complicated by the large number of parameters to estimate (k elements of ; N person e¤ects, J …rm e¤ects, M match e¤ects, and the intercept; in our application, the total number of these parameters exceeds 15 million). A more substantive complication, however, is that a …xed e¤ect speci…cation of the match e¤ects model is over-parameterized. There are N +J+M + 1 person e¤ects, …rm e¤ects, match e¤ects, and a constant term to estimate, but only M worker-…rm matches (“cell means”) from which to estimate them.5 Alternately put, the only estimable functions of the …xed
e¤ects i; j; ij and in equation (1) are theM population cell means ij = + i+ j+ ij (Searle, 1987 p. 331).6
2For simplicity, we assumeX has full column rankk: DandF do not, in general, have full column rank
without additional identifying restrictions, e.g., exclusion of one column per connected group of workers and …rms. See Searle (1987, Ch. 5) for a general statistical discussion of connected data; or Abowd et al. (2002) for a thorough discussion in the context of linked employer-employee data, including a graph-theoretic algorithm to determine connected groups of observations. Brie‡y, observations on di¤erent workers are connected by a common employer, and observations on di¤erent …rms are connected by common employees.
3M
[D F G] projects onto the column null space of[D F G]: It is a block diagonal matrix withN rows
and columns, where the M diagonal blocks correspond to each of the M worker-…rm matches in the data. The diagonal block corresponding to the match between worker i and …rm j is a Tij Tij submatrix
M[D F G]ij =ITij
1 Tij Tij
0
Tij;where Tij is the duration of the match between workeri and …rmj;IA is the
identity matrix of orderA;and A is anA 1vector of ones. EachM[D F G]ij takes deviations from means in
the match between workeriand …rmj:
4That is,M
[D F G] takes deviations from match-speci…c means butM[D F] does not.
5The term “cell mean” is adopted from the statisical literature on estimation of the two-way crossed
classi…cation with interaction, of which the match e¤ects model is an example. It arises from representing the data as a table with rows de…ned by the levels of i (workers), and columns de…ned by the levels of j
(…rms). The entry in rowiand columnj is the mean earnings of worker iat …rmj:
6In practice, there are only M estimable functions of the person, …rm, and match e¤ects, the overall
constant, and a set of group means. The group means are de…ned for connected groups of observations in the sample. When the sample consists of G connected groups of observations, the number of estimable
To see the problem, note that with^ in hand, the least squares estimator of i; j; ij and solves the remaining normal equations from the partitioned regression. This is equivalent to regressing y X^ on D; F; G, and an intercept. Predicted values from this regression are the N 1 vector I M[D F G] y X^ = ^ +D^ +F^ +G^: There are only M distinct elements in the vector of predicted values, the sample cell means:
ij = 1 Tij Ti X t=t1 i yijt x0ijt^ = ^ + ^i+ ^j + ^ij (8)
whereTij is the duration of the match between workeriand …rmj;and we denote the periods that workeriappears in the samplet1
i; t2i; :::; Ti:And yet we are tasked with decomposing the M sample cell means into N +J+M + 1 parameters. There is no unique solution without additional identifying restrictions.
In light of these complications, our approach is to assume that match e¤ects are orthog-onal to person and …rm e¤ects.7 This is a strong assumption, and is the main reason we prefer the random e¤ects alternative discussed below. However it substantially eases estima-tion, since an orthogonal match e¤ect is identi…ed whenever the corresponding person and …rm e¤ects are identi…ed in a model without match e¤ects.8 Our estimation procedure is as
follows.
We …rst estimate using the within-match estimator (6), and calculate the sample cell means (8). Let denote the N 1 vector of sample cell means. The orthogonal match e¤ect estimator is de…ned by the least squares regression of on an intercept,D, andF:The implied estimate of the intercept, ^; is the sample mean of the cell means: ^ = N1 P ij,
functions of the other e¤ects is reduced by a corresponding amount. For clarity of exposition, we abstract from these considerations in the main text, and presume the sample consists of a single connected group.
7An alternative would be to impose (at least)N+J+ 1independent linear restrictions on the estimated
e¤ects. For instance,PNi=1^i= 0; PJj=1^j= 0;
PN
i=1^ij = 08j;and
PJ
j=1^ij = 08i:The main problem
with this approach is that estimated person, …rm, and match e¤ects are not comparable across workers or …rms. This is because only the cell means are estimable, and hence the only estimable linear contrasts are those involving the cell means. For example, in the case of two employees i and m of …rm j; the linear contrast
ij mj= + i+ j+ ij + m+ j+ mj = ( i m) + ij mj (9)
is estimable. However, least squares estimates of linear constrasts like i mand j n are not, because
there is no way to eliminate match e¤ects from (9). This precludes inter-worker or inter-…rm comparisons of person and …rm e¤ects. This result holds for any collection of linear restrictions on the estimated e¤ects, and is the main reason we do not take this approach.
8TheM restrictions
ij = 0 are generally su¢ cient to identify least squares estimates of the person and
and the estimated person and …rm e¤ects solve " D0D D0F F0D F0F # " ^ ^ # = " D0 F0 # ( ^) (10)
subject to the Abowd et al. (2002) grouping conditions.9 The least squares estimator of
the orthogonal match e¤ect is ^ = M[D F]( ^) = ^ D^ F , which is the vector^ of residuals in the regression of on D; F; and an intercept. The only complication here is computational: directly solving the normal equations (10) implies inverting a cross-products matrix with N +J rows and columns – typically a very large number. We use the Abowd et al. (2002) conjugate gradient algorithm to solve (10), which directly minimizes the sum of squared residuals without inverting the cross-products matrix. As a …nal step, we decompose the person e¤ect into its observable and unobservable components via least squares regression of ^i on observable characteristics ui. Residuals from this regression de…ne an estimator of the unobserved component i that is orthogonal to ui:
2.1.2 Mixed E¤ect Estimation
Our preferred estimation strategy is to assume the unobserved heterogeneity components are random. The main advantage of this approach is that it permits meaningful comparisons of estimated person, …rm, and match e¤ects without assuming that match e¤ects are orthogonal to person and …rm e¤ects. Instead, identi…cation relies on assumptions about the conditional moments of the random e¤ects. These are like Bayes prior information on the distribution of the random e¤ects (see Searle et al. (1992) for a Bayesian interpretation of the mixed e¤ect estimator).
In the interest of keeping the identifying assumptions as weak as possible, we estimate a novel “hybrid” mixed e¤ect estimator that combines features of traditional …xed e¤ect and mixed (random) e¤ect estimators. The main advantage of this approach is that it allows correlation between the random e¤ects and time-varying observable characteristics, xijt. It is therefore in the same spirit as the Hausman and Taylor (1981) correlated random e¤ects estimator. Estimation proceeds in three stages, as follows.
In the …rst stage, we estimate under …xed e¤ect identifying assumptions, so that ^ is given by the within-match estimator (6). Second, we estimate the variance of the random e¤ects 2; 2; 2 and the error variance 2
"by Restricted Maximum Likelihood (REML) on
9Abowd et al. (2002) derive necessary and su¢ cient conditions to identify^and ^ in the person and …rm
e¤ects model. They are only identi…ed up to a group mean in each connected group of workers and …rms. A su¢ cient condition for identi…cation of^ and ^ isPi2g^i= 0and Pj2g^j= 0in each group g:
yijt x0ijt^ .10 REML estimates of these variance components are based on the following conditional moment assumptions:
Eh ijui;^ i = Eh jjui;^ i =Eh ijjui;^ i = 0 (11) Cov 2 6 4 i j ij ui;^ 3 7 5 = 2 6 4 2 0 0 0 2 0 0 0 2 3 7 5: (12)
Note that conditional on ^, (11) and (12) do not condition on xijt. Hence the identifying assumptions of our hybrid mixed model are weaker than for a traditional random e¤ect estimator, for which (11) and (12) would condition on xijt instead of ^.
In the …nal stage, we solve for the Best Linear Unbiased Estimator (BLUE) of and Best Linear Unbiased Predictor (BLUP) of the random e¤ects.11 These solve an equation system
based on the Henderson et al. (1959) mixed model equations:
2 6 6 6 6 4 U0U U0D U0F U0G D0U D0D+ ~2"=~2 IN D0F D0G F0U F0D F0F + ~2"=~2 IJ F0G G0U G0D G0F G0G+ ~2 "=~2 IM 3 7 7 7 7 5 2 6 6 6 6 4 ~ ~ ~ ~ 3 7 7 7 7 5 = 2 6 6 6 6 4 U0 D0 F0 G0 3 7 7 7 7 5 y X^ : (13) Note that as ~2"=~2 ! 0; ~2"=~2 ! 0, and ~2"=~2 ! 0; the mixed model equations converge to the least squares normal equations solved by a …xed e¤ect estimator of the match e¤ects model. Thus the least squares estimator is a special case of our hybrid mixed e¤ect estimator. Outside of this limiting case, BLUPs of the random e¤ects can be interpreted as a precision-weighted average of the least squares (…xed e¤ect) estimates and the population mean (zero).
The hybrid mixed e¤ect estimator has the following properties. ^ is consistent and the BLUE of given the minimal assumptions (4) and (5) on":Given the additional stochastic
10REML is a variance component estimator in wide use among statisticians. It is often described as
maximizing the part of the likelihood that is invariant to the values of the regression coe¢ cients, and is akin to partitioned regression. It is equivalent to maximum likelihood on linear combinations of the dependent variable under normally distributed errors. The linear combinations are chosen so that they are invariant to the value of regression coe¢ cients . In our application, where the dependent variable isy =y X^;the linear combinationsK0y satisfyK0U = 0for all values of ;which impliesK0U = 0:ThusK0projects onto the column null space ofU and is of the formK0 =C0M
U for arbitraryC0: We compute REML estimates
using the Average Information algorithm of Gilmour et al. (1995).
11BLUPs arebest in the sense of minimizing the mean square error of prediction among linear unbiased
estimators, andunbiased in the senseE[~] =E[ ],Eh~i=E[ ], andEh~i=E[ ]:See Robinson (1991) for details.
assumptions (11) and (12), the estimated variance components are invariant to the value of , consistent, asymptotically normal, and asymptotically e¢ cient in the Cramer-Rao sense;
~ is consistent and the BLUE of ; and ~; ;~ ~ are BLUPs of the random e¤ects.
2.2
Bias Due To Omitted Match E¤ects
In the empirical application described below, we …nd substantial di¤erences between esti-mates of the match e¤ects model and the person and …rm e¤ects model. Given the growing use of least squares estimates of the person and …rm e¤ects model in applied work, it is important to understand the nature of these di¤erences. Our primary interest is how omit-ted match e¤ects bias least squares estimates of regression coe¢ cients ( ) and person and …rm e¤ects. We derive and interpret this bias here. Of secondary interest is how omitted match e¤ects bias sample estimates of the variance and covariance of person and …rm e¤ects. Expressions for this bias are lengthy and complicated, so we relegate them to the Appendix. When the data generating process is given by equation (1) but the estimated equation excludes the match e¤ect ij;the estimated parameters ; i;and j are biased. Speci…cally, least squares estimates of the mis-speci…ed model satisfy
E[ ] = + X0M[D F]X 1
X0M[D F]G (14)
E[ ] = + D0M[X F]D D0M[X F]G (15)
E[ ] = + F0M[X D]F F0M[X D]G : (16)
In expectation, the estimated returns to observable characteristics, , equal the true vector of returns plus an employment-duration weighted average of the omitted match e¤ects, conditional on the design of the person and …rm e¤ects (Dand F). The sign and magnitude of the bias depends on the conditional covariance between X and G ; given D and F: Intuitively, if workers with particular characteristics (e.g., more experience) sort into better employment matches than others, the estimated returns to those characteristics re‡ect true returns plus the returns to sorting. Our within-match estimator (6) corrects this bias.
There is a simple relationship between D; F; and G that implies the bias terms in (15) and (16) are non-zero, except in the special case where = 0. Speci…cally, the column of G corresponding to the match between workeriand …rmj is the elementwise product of theith column ofD and thejth column ofF:Therefore Gis always correlated with Dand F: This is intuitive: the design of the person e¤ects contains information on worker identities (“who you are”), the design of the …rm e¤ects contains information on …rm identities (“where you work”), and the design of match e¤ects contains information on match identities (“who you
are and where you work”). Because the design matrices are always correlated, estimated person and …rm e¤ects are biased when omitted match e¤ects are nonzero.
The expected value of estimated person e¤ects in the mis-speci…ed model, , equal the vector of true person e¤ects plus the employment-duration weighted average of the omitted match e¤ects, conditional on observable time-varying characteristics and the design of …rm e¤ects. In the simplest case whereX andF are orthogonal to D, so thatD0M[X F]D=D0D andD0M[X F]G=D0G, the omitted variable bias is exactly a vector of employment duration-weighted average match e¤ects, so that
E[ i] = i+ 1 Ti Ti X t=t1 i iJ(i;t) (17)
where the function J (i; t) = j indicates workeri’s employer in periodt.
In similar fashion, the omitted variable bias in is an employment-duration weighted average of the omitted match e¤ects, conditional onX and the design of the person e¤ects. If X and D are orthogonal to F; so that F0M[X D]F = F0F and F0M[X D]G = F0G; the omitted variable bias in is exactly
E j = j + 1
Nj
X
i2Ij
Tij ij (18)
where Ij = fi:J (i; t) =j for some tg is the set of all employees of …rm j, and Nj is the number of observations on …rmj.
3
Data
Identifying the person, …rm, and match e¤ects requires longitudinal linked data on employers and employees. We use data from the US Census Bureau’s Longitudinal Employer-Household Dynamics (LEHD) database. These data span thirty-two states that represent the majority of American employment. We use data from two participating states (whose identity is con…dential) that are broadly representative of the LEHD database.12
The LEHD data are administrative, constructed from Unemployment Insurance (UI) system employment reports. These are collected by each state’s Employment Security agency to manage the unemployment compensation program. Employers are required to report total
12As discussed below, computational complexities dictate that we restrict our analysis to a subset of
obser-vations. In a small sample drawn from all thirty-seven states, work histories are not su¢ ciently connected to precisely estimate the person, …rm, and match e¤ects. Hence we focus on two representative states instead.
payments to all employees on a quarterly basis. These payments (earnings) include gross wages and salary, bonuses, stock options, tips and gratuities, and the value of meals and lodging when these are supplied (Bureau of Labor Statistics (1997, p. 44)).
The coverage of UI data varies slightly from state to state, though the Bureau of Labor Statistics (1997, p. 42) claims that UI coverage is “broad and basically comparable from state to state”and that “over 96 percent of total wage and salary civilian jobs”were covered in 1994. See Stevens (2002) and Abowd et al. (2006) for further details. With the UI employment records as its frame, the LEHD data comprise the universe of employment at …rms required to …le UI reports — that is, all employment potentially covered by the UI system in participating states.
Individuals are uniquely identi…ed in the data by a Protected Identity Key (PIK). Em-ployers are identi…ed by an unemployment insurance account number (SEIN). The UI em-ployment records contain only limited information: PIK, SEIN, and earnings. The LEHD database integrates these with internal Census Bureau data sources to add demographic and …rm characteristics, including sex, race, date of birth, industry, and geography.
The underlying quarterly data are aggregated to the annual level for estimation. The full sample consists of over 49 million annualized employment records on full-time workers between 25 and 65 years of age who were employed at private-sector non-agricultural …rms in our two states between 1990 and 1999.
The conjugate gradient algorithm that we use to estimate of our …xed e¤ect speci…-cations does not solve the least squares normal equations directly. This is a substantial computational advantage and allows us to estimate the …xed e¤ect speci…cations on very large samples. Unfortunately, there is no computational alternative to directly solving the mixed model equations (13). We therefore estimate the hybrid mixed e¤ect speci…cation on a subsample of observations. Sampling from linked employer-employee data is nontrivial because the sample must be su¢ ciently connected to precisely estimate the person, …rm, and match e¤ects. We therefore draw a ten percent subsample of individuals employed in 1997 using the Woodcock (2005) dense sampling algorithm. This algorithm ensures that each worker is connected to at least …ve others by a common employer, but is otherwise represen-tative of the population of individuals employed in 1997. That is, all individuals employed in 1997 have an equal probability of being sampled.13 The dense subsample consists of the
full work history of each sampled individual. To enable direct comparison of results between
13The dense subsample is constructed by sampling …rms with probabilities proportional to 1997
employ-ment, and then sampling workers within …rms with probabilities inversely proportional to …rm employment. A minimum of 5 employees are sampled from each …rm. By careful choice of sampling probabilities, all workers employed in the reference period have an equal probability of being sampled, and each sampled worker is connected to at least 5 others by a common employer.
the …xed and mixed e¤ect speci…cations, we estimate the …xed e¤ect speci…cations on the full work histories of all individuals employed in 1997.
Table 1 presents characteristics of the samples (see Appendix Table 1 for variable def-initions). The sample of individuals employed in 1997 is largely representative of the full sample of observations. Minor di¤erences indicate that individuals employed in 1997 have a slightly stronger labor force attachment than the universe of all individuals employed be-tween 1990 and 1999: males are slightly over-represented, as are individuals with higher educational attainment and individuals who work four full quarters in an average calendar year. The ten percent dense subsample has characteristics virtually identical to the sample of all individuals employed in 1997.
4
Estimation Results
We report results for three speci…cations. Our baseline speci…cation is the person and …rm e¤ects model. In keeping with prior research, we treat person and …rm e¤ects as …xed in the baseline speci…cation, and present least squares estimates computed using the Abowd et al. (2002) conjugate gradient algorithm. We compare these to least squares estimates of the orthogonal match e¤ects model, and hybrid mixed e¤ect estimates of the match e¤ects model. Not reported, but available on request, are estimates from traditional random e¤ect speci…cations of the person and …rm e¤ects model and the match e¤ects model. The former are very similar to the least squares estimates presented here, and the latter are similar to our hybrid mixed model estimates. Also not reported, but available on request, are estimates of the hybrid mixed model from a second (disjoint) ten percent sample. They are virtually identical to the estimates presented here.
Table 2 presents estimated regression coe¢ cients ( ; ) for all three speci…cations. Esti-mates based on the person and …rm e¤ects model (column 1) are consistent with earlier work. There are a number of di¤erences, however, between this speci…cation and our estimates of the match e¤ects model (columns 2 and 3). Most notably, as illustrated in Figures 1 and 2, the person and …rm e¤ects model over-estimates the returns to experience. For example, the person and …rm e¤ects model estimates that a male worker with 25 years of labor market experience earns 0.78 log points (118 percent) more than a labor market entrant, all else equal. The comparable di¤erential for women is 0.59 log points (80 percent). The within-match estimator (6), on which both within-match e¤ects speci…cations are based, yields a much ‡atter experience pro…le. Here, the earnings gap associated with 25 years of experience is 0.52 log points (68 percent) for men and 0.36 log points (43 percent) for women. Hence the baseline speci…cation over-estimates the returns to 25 years of experience by 0.26 log points
(30 percent, or 50 percent of the earnings of a labor market entrant) for men, and 0.23 log points (25 percent, or 36 percent of an entrant’s earnings) for women.
The discrepancy arises because the match e¤ects estimator of is based entirely on within-match variation. In contrast, the person and …rm e¤ects model attributes some of the earnings growth that occurs when individuals change employer to labor market experience. Thus traditional estimates of the returns to labor market experience (i.e., the accumulation of general human capital) partly re‡ect wage growth accruing to employment mobility.14 Intuitively, workers could experience higher wage growth when they change jobs because they sort into higher-paying …rms or higher-paying matches. Of course sorting into higher-paying …rms can not account for bias in the estimated returns to experience, because our baseline speci…cation includes …rm e¤ects. Rather, expression (14) makes clear that the bias is due to conditional covariation between experience and omitted match e¤ects. The implication is that the person and …rm e¤ects model over-estimates the returns to experience because individuals sort into better matches over the course of their career (on average). When match e¤ects are omitted, the returns to sorting are attributed to labor market experience.
Table 3 presents the estimated variance of log earnings components. In all three spec-i…cations, person e¤ects exhibit the greatest dispersion and the returns to time-varying characteristics exhibit the least. This is consistent with prior research based on the person and …rm e¤ects model, e.g., Abowd et al. (1999) and Abowd et al. (2002). The person and …rm e¤ects model and the orthogonal match e¤ects model deliver nearly identical estimates of the variance of person and …rm e¤ects: about 0.27 squared log points and 0.065 squared log points, respectively. The hybrid mixed model speci…cation of the match e¤ects model, in contrast, attributes less variation to person e¤ects (0.198 squared log points) and more to …rm e¤ects (0.102 squared log points). Both match e¤ects speci…cations attribute greater variation to match e¤ects than to time-varying covariates: 0.022 squared log points in the orthogonal match e¤ects speci…cation, and 0.079 squared log points for the hybrid mixed model.
Taken together, estimates from the hybrid mixed model imply that a one standard devi-ation increase in the value of the person e¤ect increases earnings by 0.44 log points, a one standard deviation increase in the …rm e¤ect increases earnings by 0.32 log points, and a one standard deviation increase in the match e¤ect increases earnings by 0.28 log points. Comparable estimates for the orthogonal match e¤ects speci…cation are 0.52 log points for the person e¤ect, 0.26 log points for the …rm e¤ect, and 0.15 log points for the match e¤ect. Both speci…cations agree that all three sources of heterogeneity are important components
14A speci…cation that excludes person, …rm, and match e¤ects (full results available on request) estimates
of log earnings dispersion.
For completeness, we test for the presence of match e¤ects. The results are also in Table 3. For the orthogonal match e¤ects estimator, the null hypothesis isH0 : ij = 0for eachi; j pair in the data, i.e., that all match e¤ects are zero. This implies M N J = 4;176;870
linear restrictions,15 which we test using a conventional Wald test. Given the huge number of
restrictions, it is no surprise that we easily reject the null of no match e¤ects at conventional signi…cance levels.16
In the hybrid mixed model speci…cation, the null of no match e¤ects is H0 : 2 = 0: We test this hypothesis with a likelihood ratio test based on the REML log-likelihoods of speci…cations with and without match e¤ects. Because the null hypothesis places 2 on the boundary of the parameter space, the test statistic has a non-standard asymptotic distribution. Stram and Lee (1994) show its asymptotic distribution is a 50:50 mixture of a 2
0 and a 21: Once again, we easily reject the null of no match e¤ects at conventional signi…cance levels.17
Table 4 presents a proportional decomposition of the variance of log earnings into com-ponents attributable to time-varying observable characteristics, person e¤ects, …rm e¤ects, match e¤ects, and a residual component. Gruetter and Lalive (2004) present a similar de-composition for the person and …rm e¤ects model. The dede-composition is based on:
V ar(yijt) = Cov(yijt; yijt)
= Cov yijt;^ +x0ijt^ + ^i + ^j + ^ij +eijt
= Cov yijt; x0ijt^ +Cov yijt;^i +Cov yijt;^j +Cov yijt;^ij
+Cov(yijt; eijt) (19)
where ^;^i;^j;^ij are sample estimates from one of our speci…cations, and eijt is the corre-sponding residual. Dividing both sides of the equality byV ar(yijt) provides a proportional decomposition of the variance of log earnings into the aforementioned components.
The person and …rm e¤ects model attributes nearly 64 percent of the variance of log earn-ings to person e¤ects, 16 percent to …rm e¤ects, and 6.7 percent to time-varying observable
15When the data consist of
Gconnected groups of observations, there areN +G N J k 1 degrees of freedom in the person and …rm e¤ects model, andN +G M k 1 degrees of freedom in the match e¤ects model. The person and …rm e¤ects model therefore imposes
(N +G N J k 1) (N +G M k 1) =M N J
linearly independent restictions on the estimated e¤ects.
16The value of the Wald statistic is around 18 million.
characteristics. More than 13 percent of the variance of log earnings remains unexplained. Results for the orthogonal match e¤ects model are very similar, which is not surprising given the similarities between these speci…cations we noted in Table 3. The orthogonal match ef-fects speci…cation attributes about 5 percent of the variance of log earnings to match e¤ects, which is comparable to the proportion it attributes to time-varying observable characteris-tics, and reduces the unexplained component by a corresponding amount.
Results for the hybrid mixed model are quite di¤erent. It attributes considerably less variation to person e¤ects (48 percent), and considerably more to …rm e¤ects (22 percent) and match e¤ects (16 percent). The component attributed to match e¤ects is now about three times the variation explained by time-varying observables. The unexplained component falls below 9 percent of the variance of log earnings, which implies that about one quarter of the variance explained by match e¤ects was unexplained by the person and …rm e¤ects model. The remaining three quarters were incorrectly attributed to other components of earnings.
Table 5 presents sample correlations between the estimated e¤ects (BLUPs, in the case of the hybrid mixed model). Our main interest is the correlation between estimated person and …rm e¤ects, which others have used to measure the extent to which “good”workers sort into employment at “good”…rms (see Abowd et al. (2004) in particular). Like prior studies on US data, we …nd a near-zero correlation in both …xed e¤ect speci…cations. In contrast, we …nd a small positive correlation between estimated person and …rm e¤ects (0.185) in the hybrid mixed model.18 In the Appendix, we show that when match e¤ects are omitted, the
estimated correlation between person and …rm e¤ects is biased.19 However we cannot, in
general, sign the bias. Our hybrid mixed model estimates suggest that the bias is toward zero in the LEHD data. In the Appendix, we give su¢ cient conditions for the bias to take this form.
We note that a non-zero sample correlation between estimated person and …rm e¤ects seems contrary to the assumed conditional moments of the random e¤ects, (11) and (12). By way of explanation, we note that these moment assumptions are conditional onui and ^;but the sample correlations presented in Table 5 are unconditional. A Bayesian interpretation of the hybrid mixed model is also helpful for understanding this result. Solutions to the mixed model equations (13) can be interpreted as the posterior expectation of ( ; ; ; ) under a normal likelihood for y; an uninformative prior for ; and a normal prior for the
18The non-zero correlation between estimated person and …rm e¤ects does not appear to be a feature of
random e¤ect estimation versus …xed e¤ect estimationper se. That is, random e¤ect estimates of the person and …rm e¤ects model also yield a near-zero correlation between estimated person and …rm e¤ects.
19Andrews et al. (2004b) show that the estimated correlation is biased even in the absence of match
random e¤ects with mean and variance given by (11) and (12). Thus, a non-zero sample correlation between estimated person and …rm e¤ects can be interpreted as evidence that the data swamp the prior distribution of the random e¤ects.
4.1
Earnings Growth and Employment Mobility
We use the match e¤ects model to decompose the sources of earnings growth when individuals change employer. For an individual i who changes employer (from employer j to employer n in periods t and s, respectively), the total change in log earnings is
y = yins yijt
= x0ins x0ijt ^ + ^n ^j + ^in ^ij + (eins eijt): (20) This de…nes a simple decomposition of earnings changes into components attributable to the change in time-varying observables, …rm e¤ects, match e¤ects, and a residual component. The decomposition aggregates linearly over job transitions, so we use (20) to decompose the mean change in log earnings when individuals change employers into its respective compo-nents.
Table 6 presents the decomposition. The average annual change in real log earnings is 0.03 log points. Year-over-year wage growth is slightly lower when individuals do not change employer: 0.027 log points in the full sample. In total, we observe over 4.8 million employ-ment transitions in the full sample (Panel A of the Table). When they change employer, individuals experience wage growth about 50 percent greater than the annual average (0.045 log points in the full sample). The person and …rm e¤ects model attributes 73 percent (0.033 log points) of this growth to the change in time-varying characteristics, 24 percent (0.011 log points) to the change in …rm e¤ects, and the remainder is unexplained. The orthogonal match e¤ects speci…cation attributes slightly less to the change in …rm e¤ects (0.009 log points, or 20 percent of the total), and the hybrid mixed model attributes slightly more (0.016 log points, or 33 percent of the total). All three speci…cations agree, however, that most of the excess wage growth experienced by job changers is due to sorting into higher-paying …rms. The two match e¤ects speci…cations disagree on the proportion attributable to match e¤ects: the orthogonal match e¤ect speci…cation attributes about 6.7 percent (0.003 log points) of wage growth to match e¤ects, versus 6:1 percent ( 0:003 log points) in the hybrid mixed model.
However, the decomposition in Panel A masks considerable heterogeneity in wage growth among job changers. Those who experience a spell of non-employment have far worse out-comes than those who transition directly from one employer to another. In Panels B and C,
we divide employment transitions into two groups: job-to-job transitions, and those that in-clude a period of non-employment. Because we do not observe the exact start and end dates of employment spells, we de…ne job-to-job transitions as employment spells that overlap in at least one quarter.20 About 2.2 million employment transitions satisfy this de…nition. With
the exception of employment transitions where the start/end dates of employment spells coincide exactly with the beginning/end of quarters, the remaining 2.6 million employment transitions will include a period of non-employment. In all likelihood, most of these are unemployment spells. We are reluctant to uniquely characterize them as such, however, because observed non-employment could be voluntary, could re‡ect employment not covered by the UI reporting system, or re‡ect employment in a state outside our sample.
Average log earnings growth in the subset of job-to-job transitions is large: 0.08 log points, which is nearly triple the annual wage growth of job stayers. Of this, our baseline speci…cation attributes about 40 percent (0.031 log points) to the change in time-varying covariates, 32 percent (0.025 log points) to the change in …rm e¤ects, and leaves the remaining 29 percent unexplained. The orthogonal match e¤ects speci…cation attributes similar components to covariates and …rm e¤ects, and virtually all of the remainder (0.026 log points, or 33 percent) to match e¤ects. The hybrid mixed model attributes a greater proportion to the change in …rm e¤ects (0.032 log points, or 39 percent of the total), and a smaller proportion to the change in match e¤ects (0.015 log points, or 18 percent of the total). Of the 0.053 log point di¤erence between earnings growth of job-to-job movers and job stayers, the hybrid mixed model attributes 60 percent to the change in …rm e¤ects and 28 percent to the change in match e¤ects. When individuals move from one job to another, they evidently sort into higher-paying …rms and matches on average, with the former accounting for about twice as much earnings growth.
In contrast, individuals who experience a spell of non-employment have much lower log earnings growth when they change jobs: 0.015 log points, or only 55 percent of the annual wage growth of job stayers. All three speci…cations agree that the lower earnings growth is not attributable changes in …rm e¤ects. The person and …rm e¤ects model attributes it entirely to the residual component ( 0:017log points), and both match e¤ects speci…cations attribute almost all of the lower earnings growth to changes in match e¤ects. The consensus across speci…cations is that when individuals experience a spell of non-employment, their wage growth su¤ers because they sort into lower-paying matches, and not because they sort into lower-paying …rms.
20That is, the new spell begins in the same quarter that the previous spell ended, or earlier. We note that
5
Conclusion
Our empirical results make clear that match e¤ects are an important component of log earn-ings. They explain about 16 percent of observed variation, and much of the wage change that occurs when individuals change job. Person e¤ects and observable characteristics to-gether explain more than half of the variation in log earnings, which implies considerable persistence in individual earnings. However, …rm and match e¤ects are also important. The potential returns to job search are large, as is the loss associated with displacement from a high-paying …rm or a good match: 0.32 log points for a …rm one standard deviation above the mean, and 0.28 log points for a match one standard deviation above the mean.
We have also shown that omitted match e¤ects cause substantial bias in parameter esti-mates. Our orthogonal match e¤ects estimator provides a partial solution. It corrects bias in ; which we found was substantial in the case of the estimated returns to experience. However, its ability to correct bias in the estimated person and …rm e¤ects is limited: if match e¤ects are truly orthogonal to person and …rm e¤ects, then there is no bias to correct; otherwise, the orthogonal match e¤ects estimator is mis-speci…ed. The hybrid mixed model is therefore a compelling alternative. However, the positive correlation between estimated person and …rm e¤ects that we found empirically di¤ers from the assumed conditional mo-ments of the random e¤ects. This points to an important avenue for future research, namely, to explicitly model the correlation between random person and …rm e¤ects. There has been surprisingly little research on the subject of cross-correlated random e¤ects. To our knowl-edge, the only econometric research in this area is Conway and Houtenville (2001), who model migration ‡ows.
We foresee numerous applications of the match e¤ects model. Of particular interest is how the relative importance of person, …rm, and match heterogeneity in wages varies over the business cycle, between industries, and geographically. Are match e¤ects more important in some industries than others? During layo¤ events, do …rms terminate the “bad”matches …rst? We leave these and other important questions for future research.
6
Appendix: Bias in the Estimated Variance and
Co-variance of Person and Firm E¤ects
In this appendix, we give expressions for bias in the sample variance and covariance of estimated person and …rm e¤ects when match e¤ects are omitted. A full derivation of these expressions is available on request. Let A = IN N1 N 0N denote the matrix that takes deviations from sample means. When match e¤ects are omitted, the expected value of the sample variance of least squares estimates of person e¤ects is:
EhV ard ^ i = 1 N 1E h ^D0AD^i = 1 N 1 D 0AD + 2" N 1tr h D0M[X F]D D0AD i + 1 N 1tr h 0G0M [X F]D D0M[X F]D D0AD D0M[X F]D D0M[X F]G i + 2 N 1tr h 0D0AD D0M [X F]D D0M[X F]G i : (21)
The …rst term is the true sample variance of person e¤ects. The second term is least squares estimation error. That is, as shown by Andrews et al. (2004b), this bias arises even in the absence of match e¤ects. It is positive, proportional to the average number of observations on each person, and approaches zero as N1 PiTi ! 1: The two remaining terms are bias due to omitted match e¤ects. The third term is the average within-person variance of match e¤ects, weighted by employment duration and conditional onX and F. It is positive. The …nal term is twice the duration-weighted covariance between person e¤ects and match e¤ects, conditional on X and F. Its sign is indeterminate. Thus we cannot, in general, sign the bias in the estimated variance of person e¤ects. However, we can conclude that when the covariance between person e¤ects and match e¤ects is zero, which is a reasonable parameterization, the estimated variance of person e¤ects is biased upward.
The expected value of the sample variance of least squares estimates of …rm e¤ects is likewise: EhV ard ^ i = 1 N 1E h ^ F0AF^i = 1 N 1 0F0AF + 2" N 1tr h F0M[X D]F F0AF i + 1 N 1tr h 0G0M [X D]F F0M[X D]F F0AF F0M[X D]F F0M[X D]G i + 2 N 1tr h 0F0AF F0M [X D]F F0M[X D]G i : (22)
Again, the …rst term is the true sample variance of …rm e¤ects and the second term is estimation error that arises even in the absence of match e¤ects. The estimation error is positive, proportional to the average number of observations on each …rm, and approaches zero as 1JPjNj ! 1: The two remaining terms are bias due to omitted match e¤ects. The third term is the average within-…rm variance of match e¤ects, weighted by employment duration and conditional onX andD; and is positive. The …nal term is twice the duration-weighted covariance between …rm e¤ects and match e¤ects, conditional on X and D: Its sign is indeterminate. However, as in the case of person e¤ects, we can conclude that the estimated variance of …rm e¤ects is biased upward when …rm e¤ects are uncorrelated with match e¤ects.
When match e¤ects are omitted, the expected value of the sample covariance between least squares estimates of person and …rm e¤ects is:
EhCovd ^;^ i = 1 N 1E h ^D0AF^i = 1 N 1 D 0AF 2" N 1tr h IN M[X D] AF F0M[X D]F F0 i + 1 N 1tr h 0G0M [X F]D D0M[X F]D D0AF F0M[X D]F F0M[X D]G i + 1 N 1tr h 0D0AF F0M [X D]F F0M[X D]G i + 1 N 1tr h 0F0AD D0M [X F]D D0M[X F]G i : (23)
Once again, the …rst term is the true sample covariance between person and …rm e¤ects, and the second term is least squares estimation error that arises even in the absence of match e¤ects. The estimation error can be interpreted as the conditional variance of employment duration, weighted by the inverse of the product of the number of observations on each match participant. It is zero whenDandF are orthogonal, and in the balanced data case where each worker works at each …rm and all employment spells have the same duration. Otherwise, the trace of the term in square brackets is positive and approaches zero as N1 J1 PiTi
P
jNj ! 1: This term, as noted by Andrews et al. (2004b), biases the estimated covariance between person and …rm e¤ects downward.
The remaining three terms in (23) are bias due to omitted match e¤ects. The third term is the duration-weighted covariance between person-average match e¤ects (T1
i
P
jTij ij) and …rm-average match e¤ects ( 1
Nj
P
iTij ij), conditional on X: When match e¤ects are uncorrelated with one another, this reduces to the conditional variance of match e¤ects, and thus imparts upward bias to the estimated covariance between person and …rm e¤ects. The fourth term is the duration-weighted covariance between …rm-average person e¤ects
(N1
j
P
iTij i) and match e¤ects, conditional on X. The …nal term is likewise the duration-weighted covariance between person-average …rm e¤ects (T1
i
P
jTij j) and match e¤ects, conditional onX. In general, both are of indeterminate sign. However, they are zero when match e¤ects are uncorrelated with person and …rm e¤ects.
Thus we cannot, in general, sign the bias in the estimated covariance between person and …rm e¤ects when match e¤ects are omitted. Consequently we cannot sign the bias in the estimated correlation between person and …rm e¤ects either. We can, however, conclude that a su¢ cient condition for the estimated correlation to be biased toward zero is that match e¤ects are uncorrelated with one another, person e¤ects, and …rm e¤ects; and the weighted conditional variance of employment duration exceeds the conditional variance of match e¤ects.
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TABLE 1 SUMMARY STATISTICS
(Sample Proportions Unless Otherwise Stated)
FULL SAMPLE
Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.
Demographic Characteristics Male .56 .50 .58 .49 .57 .50 Age (Years) 40.6 10.2 40.3 9.6 40.3 9.6 Men Nonwhite .21 .57 .20 .55 .20 .56 Race Missing .04 .25 .03 .24 .03 .24 Less Than High School .12 .45 .11 .43 .11 .43 High School .30 .67 .30 .65 .29 .66 Some College .23 .60 .23 .59 .23 .59 Associate or Bachelor's Degree .25 .62 .25 .61 .25 .62 Graduate or Professional Degree .10 .42 .11 .42 .11 .42 Women Nonwhite .24 .69 .24 .71 .25 .72 Race Missing .02 .22 .02 .22 .02 .22 Less Than High School .09 .45 .09 .45 .09 .44 High School .31 .78 .30 .79 .30 .78 Some College .25 .71 .25 .73 .25 .72 Associate or Bachelor's Degree .26 .72 .27 .75 .27 .75 Graduate or Professional Degree .08 .42 .09 .44 .09 .44 Work History Characteristics Real Annualized Earnings (1990 Dollars) 41,107 38,849 43,183 39,324 43,528 38,782 Men Labor Market Experience (Years) 11.8 13.1 11.9 12.7 11.8 12.7 Worked 0 Full Quarters in Calendar Year .08 .36 .06 .32 .06 .32 Worked 1 Full Quarter in Calendar Year .15 .49 .12 .44 .12 .44 Worked 2 Full Quarters in Calendar Year .13 .47 .12 .44 .12 .44 Worked 3 Full Quarters in Calendar Year .14 .48 .13 .46 .14 .47 Worked 4 Full Quarters in Calendar Year .50 .80 .56 .81 .57 .00 Women Labor Market Experience (Years) 9.5 13.0 9.0 12.5 9.2 12.6 Worked 0 Full Quarters in Calendar Year .07 .39 .06 .36 .05 .35 Worked 1 Full Quarter in Calendar Year .14 .54 .11 .50 .11 .50 Worked 2 Full Quarters in Calendar Year .13 .53 .12 .51 .11 .50 Worked 3 Full Quarters in Calendar Year .14 .55 .13 .54 .13 .54 Worked 4 Full Quarters in Calendar Year .52 .96 .58 1.02 .59 1.01 Year 1990 .09 .29 .07 .26 .07 .26 1991 .09 .29 .08 .27 .08 .27 1992 .09 .29 .08 .27 .08 .28 1993 .10 .29 .09 .28 .09 .28 1994 .10 .30 .10 .29 .10 .29 1995 .10 .30 .10 .31 .10 .31 1996 .10 .31 .11 .32 .11 .32 1997 .11 .31 .14 .35 .14 .34 1998 .11 .31 .12 .32 .12 .32 1999 .11 .31 .11 .31 .11 .31 49,291,205 37,688,492 3,652,544 Number of Workers (N) 9,272,529 5,235,887 503,179 Number of Firms (J) 573,307 476,745 121,227 Number of WorkerFirm Matches (M) 15,309,134 9,889,502 947,883 Number of Connected Groups 84,748 46,829 1,460 ALL INDIVIDUALS EMPLOYED IN 1997 TEN PERCENT DENSE SUBSAMPLE Number of Observations (N*)
TABLE 2
ESTIMATED RETURNS TO OBSERVABLE CHARACTERISTICS
(1) (2) (3)
Estimate SE Estimate SE Estimate SE
Male x Experience .074 .000 .058 .000 .056 .001 .243 .001 .215 .002 .194 .005 .036 .000 .033 .000 .027 .001 .002 .000 .002 .000 .002 .000 Male x Worked 0 Full Quarters .035 .000 .053 .000 .053 .001 Male x Worked 1 Full Quarters .004 .000 .007 .000 .008 .001 Male x Worked 2 Full Quarters .013 .000 .007 .000 .007 .001 Male x Worked 3 Full Quarters .015 .000 .013 .000 .014 .001 Female x Experience .031 .000 .018 .000 .019 .001 .020 .001 .004 .002 .000 .005 .006 .000 .010 .000 .009 .001 .001 .000 .001 .000 .001 .000 Female x Worked 0 Full Quarters .010 .000 .029 .000 .027 .001 Female x Worked 1 Full Quarters .006 .000 .008 .000 .006 .001 Female x Worked 2 Full Quarters .014 .000 .009 .000 .012 .001 Female x Worked 3 Full Quarters .020 .000 .017 .000 .020 .001 Male x High School .075 .000 .054 .000 .051 .004 Male x Some College .168 .000 .143 .000 .131 .004 Male x Associate or Bachelor's Degree .329 .000 .294 .000 .265 .004 Male x Graduate or Professional Degree .526 .000 .491 .001 .437 .005 Male x Nonwhite .326 .000 .339 .000 .360 .003 Male x Race Missing .006 .001 .003 .001 .062 .006 Male x First Period Potential Experience <0 .074 .001 .196 .001 .186 .006 Female .254 .001 .222 .001 .236 .005 Female x High School .180 .000 .102 .000 .073 .004 Female x Some College .286 .000 .202 .000 .155 .004 Female x Bachelor or Associate's Degree .466 .000 .368 .000 .299 .004 Female x Graduate or Professional Degree .650 .001 .545 .001 .463 .005 Female x Nonwhite .121 .000 .131 .000 .130 .003 Female x Race Missing .004 .001 .009 .001 .038 .008 Female x First Period Potential Experience <0 .092 .001 .021 .001 .025 .007 Intercept 9.84 .001 10.00 .001 9.86 .003
Year Effects YES YES YES
PERSON AND FIRM EFFECTS ORTHOGONAL MATCH EFFECTS HYBRID MIXED MODEL TimeVarying Characteristics ( Male x Experience2 / 100 Male x Experience3 / 1000 Male x Experience4 / 10000 Female x Experience2 / 100 Female x Experience3 / 1000 Female x Experience4 / 10000 TimeInvariant Characteristics( Notes: Columns (1) and (2) are estimates from fixed effect specifications estimated on the sample of all individuals employed in 1997. Column (3) estimates are based on the ten percent dense subsample.
TABLE 3 VARIANCE OF ESTIMATED COMPONENTS OF LOG EARNINGS (1) (2) (3) Variance of Log Real Annualized Earnings (y) .422 .422 .410 .031 .017 .017 .274 .273 .198 .043 .041 .039 .232 .233 .159 .065 .066 .102 .022 .079 .066 .040 .036 <0.00001 <0.00001 .867 .919 .933 Model Degrees of Freedom 32,022,663 27,845,793 3,652,500 PERSON AND FIRM EFFECTS* ORTHOGONAL MATCH EFFECTS* HYBRID MIXED MODEL† Variance of Returns to TimeVarying Covariates (X) Variance of Pure Person Effect () Returns to TimeInvariant Covariates (U) Unobserved Heterogeneity () Variance of Firm Effect () Variance of Match Effect () Error Variance () H0: No Match Effects (pvalue) R2 * Estimates in columns 1 and 2 are based on the full sample of individuals employed in 1997. Values in these columns are sample variances of the estimated effects. The estimated error variance is corrected for degrees of freedom.
† Estimates in column 3 are estimated on the ten percent dense subsample of individuals employed in 1997. For the rows labeled y, X U values in
