Swamy-Mehta (1975) Bayesian and Non-Bayesian Analysis of Switching Regressions and of Random Coefficient Regression Models

Summary

Shows that maximum likelihood (ML) fails for Quandt's (1972) switching regression — the likelihood is almost always unbounded — and that the exact Bayesian posterior is computationally intractable even asymptotically. Proposes substituting a random coefficient model in which the regression coefficient vector $\beta_t$ is treated as an i.i.d. draw from a continuous multivariate distribution with mean $\bar{\beta}_1$ and unrestricted $K\times K$ covariance matrix $\Delta_1$. Derives minimum average-risk linear estimators and Minimum Norm Quadratic Unbiased Estimator (MINQUE) estimators for $\bar{\beta}_1$ and $\Delta_1$, provides a Bayesian interpretation under normality, extends the model to panel data, and applies it to a demand function for U.S. bank deposits.

Key Claims

• ML pathology for switching regression: The likelihood supremum over the parameter space $(\beta_1, \beta_2, \sigma_1^2, \sigma_2^2, \lambda)$ is almost always infinite — as $\sigma_i^2 \to 0$ with the regime mean $\bar{y}_{ti}$ held at an observed value, the likelihood diverges. Bayesian methods with proper priors avoid this but produce posteriors with no closed form and no normal large-sample approximation (Johnson regularity conditions fail).
• Random coefficient model (eq. 2.2): $y_t = \sum_{j=1}^K x_{jt}\beta_{jt}$, where $\beta_t' = (\beta_{1t},\ldots,\beta_{Kt})$ are i.i.d. draws with $E\beta_t = \bar{\beta}_1$ and $\text{Cov}(\beta_t) = \Delta_1$ (full $K\times K$, not restricted to diagonal). The observation variance-covariance is $\Sigma_1 = \text{diag}[x_t'\Delta_1 x_t]$ — heteroskedastic across $t$ even though $\Delta_1$ is homogeneous.
• Bayesian/minimum average-risk estimator of $\bar{\beta}_1$ (eq. 2.10): With prior mean $r_1$ and variance-covariance $\psi_1$: $$\bar{\beta}_1^* = (X'\Sigma_1^{-1}X + \psi_1^{-1})^{-1}(X'\Sigma_1^{-1}y + \psi_1^{-1}r_1)$$ Under normality this equals the mean of the conditional posterior for $\bar{\beta}_1$ given $y$, $X$, $r_1$, $\psi_1$, and $\Delta_1$. Under diffuse prior ($\psi_1^{-1} \to 0$) it collapses to the Minimum Variance Linear Unbiased Estimator (MVLUE) $\bar{b}_1 = (X'\Sigma_1^{-1}X)^{-1}X'\Sigma_1^{-1}y$ (eq. 2.12).
• MINQUE of $\Delta_1$ (eq. 2.24/2.26): The minimum norm quadratic unbiased estimator solves $\bar{s}_1 = \bar{Z}_1\delta_2$ where $\delta_2$ stacks the $K(K+1)/2$ unique elements of $\Delta_1$ and $\bar{s}_1$ contains sums of squares and cross-products of Generalized Least Squares (GLS) residuals. A Bayesian analogue with prior $(\bar{\delta}_2, \Psi_2)$ gives regularised estimates (eq. 2.26); negative variance estimates are truncated.
• Panel decomposition (eq. 3.2): Coefficients decomposed as $\beta_{it} = \bar{\beta} + \alpha_i + \xi_{it}$, where $\alpha_i$ is individual-specific (constant over time) and $\xi_{it}$ is time-varying and random. Mean coefficient $\bar{\beta}$ estimated by a panel GLS/Bayes estimator (eq. 3.8); $\Delta$ (cross-section dispersion) estimated by $(n-1)^{-1}S$ minus a precision-matrix correction (eq. 3.12).
• Key conceptual distinction: The paper draws a sharp line between (a) $\beta_t$ random as part of the data-generating process and (b) $\beta$ random as a subjective Bayesian prior. Both yield similar estimator forms but have different interpretations — in (a), the distribution is a feature of the world; in (b), it encodes the analyst's uncertainty.

Concepts Introduced or Extended

• Random Coefficient Model

Entities Mentioned

• P.A.V.B. Swamy

Quotes

"The use of a random coefficient method can result in fruitful and meaningful econometric analysis of data."

"In the former [random coefficient DGP] a distribution of coefficients is introduced as part of the data-generating process, while in the latter [Bayesian] a distribution of coefficients is introduced to represent subjective prior information."

My Take

The ML-unboundedness result for switching regression is genuinely important and under-appreciated: it explains why naive maximum likelihood on mixture models is unreliable without regularization. The random coefficient model proposed is essentially what became the "Swamy estimator" in panel econometrics — the random-effects GLS approach with estimated $\Delta$. The key advance over Swamy (1971) is relaxing the diagonal restriction on $\Delta_1$, which had no theoretical justification. The paper is less a Bayesian paper than a frequentist GLS paper with a Bayesian interpretation bolted on — the estimator (2.10) is derived from average risk criteria first, and normality is only invoked later to justify the posterior interpretation.