There is a rapidly-growing empirical literature which uses linked employer-employee data to estimate the contribution of worker and firm heterogeneity to outcomes in the labour market. Much of this literature stems from Abowd et al. (1999) (henceforth AKM) and related papers. (See also Abowd and Kramarz (1999) and Haltiwanger et al. (1999) for early surveys of the wide range of issues covered in this literature.) An important issue in the literature is the relationship between the unobserved worker and firm-components of wages. Both economic theory and common sense suggest that there should be a positive correlation between worker and firm productivities.
In the words of AKM: ‘high-wage workers and high-wage firms’ match together. This is known as positive ‘assortative matching’ in the economics literature (see Atakan (2006), for example, and references within). (‘Assortative matching’ means that the matching is non-random.)
However, a puzzle has emerged, in that the unobserved component of workers’ wages appears to be negatively correlated with the unobserved component of firms’ average wages. Apart from AKM’s original study, which reports a positive sample correlation, all subsequent work has reports negative correlations. Abowd et al. (2004) re-estimate these models using the exact solution they developed subsequently, and report correlations of ?0.24 for French data and 0.02 for data from Washington State, whereas Goux and Maurin (1999), using different French data, find a correlation ranging from ?0.32 to 0.01 depending on the time period chosen. Gruetter and Lalive (2004) report a correlation of ?0.27 for Austrian data. All of these are weaker than Barth and Dale-Olsen’s (2003) correlations of between ?0.47 and ?0.55 for Norwegian data. In other words, when focussing on unobserved components, low wage workers tend to work in high wage firms, and vice versa. This seems counter intuitive in the light of theories of assortative matching.
There are two possible explanations for this emerging stylised fact. The first, suggested by Barth and Dale-Olsen (2003) and Abowd et al. (2004), is that the observed negative sample correlation is simply the result of using standard econometric techniques. Because the estimates of the worker and firm dummies are estimated with error, it is possible that the estimated correlation between them is biased downwards. It is not immediately obvious why this is so, but an over-estimate of a worker effect leads to, on average, an under-estimate of a firm effect. The second explanation focuses on whether there any genuine economic explanations for why there might be negative assortative matching. Again, see Abowd et al. (2004).
In this paper, we analyse the first explanation. In what we label a fixed effects data generation process, we consider the standard fixed-effects estimator of the model; we then derive formulae for the bias in the sampling distribution of the sample covariance between the unobserved worker- and firm-components of wages, and the biases in the sample variances of both components. When there are no conditioning covariates in the model, or when these covariates are not correlated with the worker and firm dummies, we show that the bias in the sample correlation is unambiguously negative when there is positive assortative matching.
However, it is possible, but unlikely, that the bias can become positive when there is a strong correlation between the observed covariates and the worker and firm dummies. We also show that correcting the biases in the sample covariance and two sample variance terms, leads to consistent estimators and, by Slutsky’s theorem, the correlation can also be consistently estimated. By simulating the data generation process, we show that the corresponding bias-corrected estimator of the correlation is also approximately unbiased. Subject to possible size constraints, the bias corrections can be computed for any given dataset.
