22 Linear Models with Heterogeneous Coefficients
22.1 Linearity and Heterogeneity
22.1.1 Models with Homogeneous Slopes
We begin our journey where standard textbooks and first-year foundational courses in econometrics leave off. The “standard” linear models considered in such courses often assume homogeneity in individual responses to covariates (e.g., Hansen (2022)). A common cross-sectional specification is:
\[ y_i = \bbeta'\bx_i + u_{i}, \tag{22.1}\] where \(i=1, \dots, N\) indexes cross-sectional units.
In panel data, models often include unit-specific \((i)\) and time-specific \((t)\) intercepts while maintaining a common slope vector \(\bbeta\):
\[ y_{it} = \alpha_i + \delta_t + \bbeta'\bx_{it} + u_{it}. \tag{22.2}\]
22.1.2 Heterogeneity in Slopes.
However, modern economic theory rarely supports the assumption of homogeneous slopes \(\bbeta.\) Theoretical models recognize that observationally identical individuals, firms, and countries can respond differently to the same stimulus. In a linear model, this requires us to consider more flexible models with heterogeneous coefficients:
Cross-sectional model (22.1) generalizes to
\[ y_i = \bbeta_{i}'\bx + u_i. \tag{22.3}\]
Panel data model (22.2) generalizes to
\[ y_{it} = \bbeta_{it}'\bx_{it} + u_{it}. \tag{22.4}\]
Such models are worth studying, as they naturally arise in a variety of contexts:
Structural models with parametric restrictions: Certain parametric restrictions yield linear relationships in coefficients. An example is given by firm-level Cobb-Douglas production functions where firm-specific productivity differences induce heterogeneous coefficients (Combes et al. (2012); Sury (2011)).
Binary covariates and interaction terms: if all covariates are binary and all interactions are included, a linear model encodes all treatment effects without loss of generality (see, e.g., Wooldridge (2005)).
Log-linearized models: Nonlinear models may be approximated by linear models around a steady-state. For example, Heckman and Vytlacil (1998) demonstrate how the nonlinear Card (2001) education model simplifies to a heterogeneous linear specification after linearization.
References
- Vladislav Morozov, Econometrics with Unobserved Heterogeneity, Course material, https://vladislav-morozov.github.io/econometrics-heterogeneity/linear/linear-introduction.html
- Vladislav Morozov, GitHub course repository, https://github.com/vladislav-morozov/econometrics-heterogeneity/tree/main/src/linear