High Wage Workers Match with High Wage Firms: Clear Evidence of the Effects of Limited Mobility Bias

Positive assortative matching implies that high productivity workers and firms match together. However, there is almost no evidence of a positive correlation between the worker and firm contributions in two-way fixed-effects wage equations. This could be the result of a bias caused by standard estimation error. Using German social security records we show that the effect of this bias is substantial in samples with limited inter-firm movement. The correlation between worker and firm contributions to wage equations is unambiguously positive.


Introduction
Positive assortative matching (PAM) implies that high productivity workers and firms match together. This intuitively plausible idea goes back to Roy (1951), Becker (1973, Sattinger (1975), but more recent contributions include Kremer (1993) and Shimer & Smith (2000). The extent to which PAM is actually observed in the labour market sheds light on the mechanisms which determine matching, and has important policy implications, not least because PAM is related to the degree of wage inequality.
Following the publication of Abowd, Kramarz & Margolis (1999) a number of papers have attempted to find evidence for PAM by estimating wage equations with worker and firm fixed effects. However, the majority of the literature has found small or even negative correlations between the worker and firm effects. 1 There are three possible explanations for this stylised fact. First, there are a number of highly structural models that attempt to model this prediction. In particular, some papers suggest that it is difficult to identify assortative matching from wage data only (de Melo 2008, Eeckhout & Kircher 2011. See also Bagger & Lentz (2008).
This prompted Mendes, van den Berg & Lindeboom (2010) to take a more direct approach and estimate plant-level production functions using observable proxies for worker and firm productivities. They find clear evidence of PAM.
Second, it is possible that two-way fixed-effects wage equations are misspecified because they ignore the contribution of additional worker-firm match effects. Woodcock (2008) estimates a wage equation which allows for such match effects, and finds that the estimated correlation between worker and firm effects increases from zero to 0.185.
The third explanation is that there is a limited mobility bias in the estimated cor-relation caused by estimation error. This was noted originally by Abowd, Kramarz, Lengermann & Perez-Duarte (2004) but was developed formally by Andrews, Gill, Schank & Upward (2008). Andrews et al. develop formulae that show that the estimated correlation is biased downwards if there is true PAM. Moreover, this bias is bigger the fewer workers who move between firms in the data, which is why it is labelled limited mobility bias.
In this paper we show empirically that limited mobility bias matters a lot for the estimated correlation between worker and firm effects. Using the employment statistics register of the German Federal Office of Labour we alter the amount of interfirm mobility by sampling a varying fraction of workers from the population. We show that the estimated correlation between worker and firm contributions to wage equations is negative when inter-firm mobility is small, but the correlation becomes unambiguously positive for larger samples with more inter-firm mobility.

Methodology and Limited Mobility Bias
Using linked employer-employee panel data, the literature typically estimates There are i = 1, . . . , N workers, j = 1, . . . , J firms and t = 1, . . . , T years. y it is wages; z 1it is a vector of observable time-varying worker covariates and z 2jt is a vector of observable time-varying firm covariates. θ i and ψ j are time-invariant (scalar) unobserved heterogeneities, potentially correlated with each other, but also with z 1it and z 2jt . Workers may move between firms; there are M movers in total.
It is standard to assume strict exogeneity: This implies workers' mobility decisions are independent of ε it , but can be a function of the unobservables θ i and ψ j . Because the θ i and ψ j are correlated with the observed covariates, random effects methods are biased and inconsistent, and so two-way fixed-effects methods are needed to estimate θ i and ψ j . 2 Evidence for PAM comes from seeing whether or not the correlation between the worker and firm components of Equation (1) is positive: These two correlations each comprise a covariance and two variances, which in turn depend on θ i and ψ j . As noted by Krueger & Summers (1988) both Var(θ i ) and Var(ψ j ) are biased upwards; this is because every θ i and every ψ j are subject to estimation error. Andrews et al. (2008) show that Cov(θ i ,ψ j ) is also biased, because of estimation error and because the estimates of θ i and ψ j are related by: where " " averages a variable over t, and "ˆ" denotes an estimate from Equation (1).
Conditional on the observed covariates, if a ψ j is over-estimated, then, on average, the corresponding θ i is under-estimated, and vice versa. Thus, if the true correlation is positive, then the estimated correlation is biased downwards. Further, Andrews et al. (2008) show formally, and by simulations, that the bias can be sizeable, and reduces as the the number of workers who move between firms, M , increases.

Data and Empirical Strategy
The data come from the employment statistics register of the German Federal Office of Labour (Beschäftigtenstatistik ), which covers all workers or trainees registered by the social insurance system (Bender, Haas & Klose 2000). Each observation has a unique establishment identification number. 3 We select all workers in the employment register who were employed on June 30th each year to create a simple annual The dependent variable is daily gross wages, which are censored at the social security contribution ceiling. 7 These censored observations will also attenuate the estimated 3 Typically the literature refer to firms when discussing PAM; our data, in fact, comprise establishments.
4 Splitting the sample means that we lose inter-establishment mobility which occurs between states. However, the degree of inter-state mobility is extremely low. Between 2006 and 2007 the proportion of workers remaining in the same state is 98.5% (Bavaria), 98.5% (North-Rhine Westphalia) and 97.6% (Saxony).
5 In our estimates of Equation (1) z 1it comprises tenure and a set of occupation dummies, z 2it comprises log establishment size. We also include year dummies which capture the effects of time, age and experience. To get a clean experiment that allows us to increase the number of movers, but keep the sample of establishments constant, we: 1. Take a 10% random sample of workers, and define p as the proportion of workers sampled (p = 0.1); 2. Record the identities of all establishments which employ those workers; 3. Holding this sample of establishments constant, increase p to 0.2, 0.3, 0.5 and 1.
We do not take a random sample of establishments and vary p, because we would lose all inter-establishment mobility to and from establishments outside the sample.

Results
Our basic results are reported in Figure  This demonstrates that there is a positive correlation between worker and estab-8 lishment effects in German data. 8 These correlations are only slightly smaller than those estimated by Mendes et al. (2010) for Portugal using data on productivity.
Our results also explain why many studies do not find such a correlation, because any given dataset could have been sampled anywhere along the M/J-axis. Indeed, our results also explain why some studies, with very few movers, estimate negative correlations.
The inclusion of observable characteristics in the correlation (see Equation 2) does not change our conclusion because the observable components are not subject to limited mobility bias. When we estimate Corr(z 1it β 1 +θ i , z 2jt β 2 +ψ j ), the correlation increases by only 0.04 (Bavaria), 0.03 (North-Rhine Westphalia) and 0.02 (Saxony) when p = 1.0.
The increase in Corr(θ i ,ψ j ) is not simply a result of increasing the sample size (worker-years). To show this, we repeat the experiment of increasing p, but now we only keep additional workers if they do not join or leave their establishment during the sample period. This ensures that the number of movers per establishment is held fixed. The result is shown in Figure 2, which plots the resulting proportion of workers sampled (which is necessarily less than p) against the estimated correlation.
The (fixed) number of movers per establishment is also indicated at each point.
Because M/J is held constant there is no increase at all in Corr(θ i ,ψ j ).

Conclusion
The existing empirical literature has generally failed to find a positive correlation between worker and firm components of wage equations, a result often seen as evidence against PAM. We show that limited mobility bias can have a large effect on the estimated correlation if the data include only a small number of worker movements 8 The estimate for p = 1.0 is still likely to be a lower bound because wages are top-censored. per firm. In our data, when the number of movers per establishment is small (< 5) the estimated correlation is consistently negative. When the number of movers per establishment is large (> 25), the estimated correlation is consistently positive and in the range 0.2 − 0.3. This strongly suggests that, for Germany, the true correlation is positive.