Chow (1984) Random and Changing Coefficient Models

Summary

Chapter 21 of the Handbook of Econometrics, Vol. II (Griliches and Intriligator, eds.). Provides the foundational treatment of random and changing coefficient models — linear regressions $y_t = x_t'\beta_t + \varepsilon_t$ with coefficient dynamics $\beta_t = M\beta_{t-1} + \eta_t$ . Derives two equivalent estimation methods (Kalman filter and generalized least squares, GLS) for filtered and smoothed estimates $\hat{\beta}_{t|s}$ ; develops maximum likelihood estimation (MLE) of hyperparameters ( $\sigma^2$ , $V$ , $M$ ) via the Kalman innovation likelihood; extends the framework to seemingly unrelated regression (SUR) systems, simultaneous equations, and nonlinear models; and surveys (§10) the classical tests for coefficient constancy.

Key Claims

The filtered estimate $\hat{\beta}_{t|t}$ and smoothed estimate $\hat{\beta}_{t|s}$ are the same under the recursive Kalman approach (§2) and the GLS regression approach (§3); the two paths are mathematically equivalent.
MLE of $\sigma^2$ , $P = \sigma^{-2}V$ , and $M$ requires maximizing the Kalman innovation likelihood; no closed form — gradient-based numerical optimization is required.
Setting $M = I$ and $V \to 0$ recovers fixed-coefficient ordinary least squares (OLS); $M = I$ with $V > 0$ gives the standard random-walk time-varying-parameter (TVP) model; $|\text{eigenvalues}(M)| < 1$ gives stationary mean-reverting coefficients whose initial covariance satisfies the Lyapunov equation $\Gamma_0 = M\Gamma_0 M' + V$ .
§10 surveys tests for $H_0: V = 0$ (coefficient constancy): Chow (1960) F-test, Brown-Durbin-Evans (1975) cumulative sum (CUSUM/CUSUM²), Quandt (1960) maximum likelihood ratio test (LRT), Nyblom (1983) locally most powerful (LMP) test, Pagan (1980) score test.
Direct precursor to the TVP vector autoregression (TVP-VAR): the univariate state-space structure generalizes by stacking $\beta_t = \text{vec}[(A_{1t},\ldots,A_{pt})']$ and applying the Carter-Kohn algorithm.

Concepts Introduced or Extended

Entities Mentioned

Gregory C. Chow

Quotes

"The model reduces to a system with fixed coefficients when $V = 0$ , to a system with random coefficients when $M = 0$ , and to a system with a random walk if $M = I$ ." (§8, paraphrase)

My Take

The most comprehensive pre-1990 treatment of TVP regression. The equivalence of the Kalman filter and GLS methods (§2–§3) is underappreciated — it makes explicit that the Kalman smoother is a Bayesian GLS estimator with the prior implied by the state equation. The §10 testing survey is comprehensive for 1984; the later Zivot-Andrews (1992) endogenous-break and Bai-Perron (1998) multiple-break frameworks extend the program. The extension to simultaneous equations (§6–§7) via linearization anticipates the approach used in dynamic stochastic general equilibrium (DSGE) state-space estimation. The chapter also completes Chow (1973): that paper established the optimal multiperiod predictor assuming known $M$ and $V$ ; this chapter establishes how to estimate them.