10  Conclusion: Linear Models

10.1 Summary

In this block, we have examined the linear panel data model (2.7) with time-invariant heterogeneous coefficients. Our primary focus was on identifying the moments (mean, variance, etc.) and the full distribution of these coefficients:

  • For the mean:
    • We discovered that traditional within and IV estimators are generally inconsistent under coefficient heterogeneity.
    • To address this, we introduced the mean group estimator, which is robust to any dependence structure between the coefficients and the covariates Pesaran and Smith (1995).
  • For the variance and distribution, we employed a constructive approach from Arellano and Bonhomme (2012), which relies on assumptions about the idiosyncratic error components. As with the mean, the variance and distribution are identified without restricting the dependence between the coefficients and the covariates.

10.2 Some Further Results on Heterogeneous Linear Models

We have barely scratched the surface of the literature on linear models with unobserved heterogeneity is extensive. If you are interested, here is a selection of some further results:

  • Further results in model (2.7): For example, it is possible to identify the average effect even when the number of periods equals the number of covariates (Graham and Powell (2012)). Additionally, endogeneity can be permitted in the equation (Laage (2024)).
  • Time-varying coefficients: Models (2.4) with time-varying heterogeneous coefficients have also received some recent attention, particularly in the “grouped fixed effect” literature. Notable papers include Bonhomme and Manresa (2015) and Lumsdaine, Okui, and Wang (2023) (also see Bonhomme, Lamadon, and Manresa (2022)).
  • Unobserved Factors: Models with unobserved factors allow for common shocks to impact individuals differently and provide a parsimonious way to include cross-sectional dependence. Key references include Pesaran (2006) and Bai (2009).
  • Cross-Sectional Data: Some identification is possible with cross-sectional data, typically assuming that coefficients are independent from the covariates (a randomized experiment framework). See Hoderlein, Klemelä, and Mammen (2010) and Masten (2018) for identifying the full distribution of coefficients in single-equation and system of equations frameworks, respectively.
  • Deconvolution Applications: Deconvolution is a versatile technique with broad applications. For examples, see Bonhomme and Robin (2009) and Bonhomme and Robin (2010).

Next Section

In the next section we will move beyond linearity and start our discussion of nonparametric models with unobserved heterogeneity.