Models with Homogeneous Slopes

We start with the “standard” linear models considered in such courses often assume homogeneity in individual responses to covariates. A common cross-sectional specification is:

\(\begin{equation}\label{eq-standard_linear} y_i = \bbeta'\bx_i + u_{i}, \end{equation}\) 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 $\bbeta$:

\[\begin{equation}\label{eq-panel_linear} y_{it} = \alpha_i + \delta_t + \bbeta'\bx_{it} + u_{it}. \end{equation}\]

Models with Heterogeneous Slopes

Modern economic theory rarely supports the assumption of homogeneous slopes $\boldsymbol{\beta}$. 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:

  1. Cross-sectional model ($\ref{eq-standard_linear}$) generalizes to

    \[\begin{equation}\label{eq-cross_sectional_hetero} y_i = \boldsymbol{\beta}_{i}'\boldsymbol{x} + u_i. \end{equation}\]

  2. Panel data model (\ref{eq-panel_linear}) generalizes to

    \[\begin{equation}\label{eq-panel_hetero} y_{it} = \boldsymbol{\beta}_{it}'\boldsymbol{x}_{it} + u_{it}, \end{equation}\]

    or assuming time-invariant coefficients

    \[\begin{equation}\label{eq-time_invariant} \color{red} y_{it} = \boldsymbol{\beta}_{i}'\boldsymbol{x}_{it} + u_{it}. \end{equation}\]

For identification purposes, we will generally focus on the case where the number $N$ of units is large, while the number $T$ of observations per unit is fixed and not necessarily large.

Note that model (\ref{eq-time_invariant}) includes a particular special case — the random intercept model (confusingly also called the “fixed effects model”).

In the panel data literature, approaches that do not restrict the dependence between the unobserved and the observed components are called “fixed effects”.

The random intercept model imposes homogeneity on all parameters except the intercept term. In the one-way case, the model takes the form:

\[\begin{equation}\label{eq-random_intercept} y_{it} = \alpha_i + \bbeta'\bx_{it} + u_{it}. \end{equation}\]

Model (\ref{eq-random_intercept}) is one of the oldest ways of including unobserved heterogeneity in linear models and goes back at least to (Mundlak, 1961).

References:

  1. Mundlak, Y. (1961). Empirical Production Function Free of Management Bias. Journal of Farm Economics, 43(1), 44. https://doi.org/10.2307/1235460