Hamilton (1989) A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle

Summary

James Hamilton's landmark paper introduces the Markov-switching autoregression — a tractable way to let a time series' parameters change across discrete, unobserved regimes governed by a first-order Markov chain. The trend growth rate switches between two states (e.g., expansion vs. recession), and an AR process is layered on the deviations from this stochastically shifting trend. Hamilton derives a nonlinear filter (the "Hamilton filter") that computes, by a single recursive forward pass, the conditional probability that the economy is in each regime given observed data, and from it the exact sample likelihood; a backward pass yields smoothed regime probabilities. Applied to postwar U.S. real GNP, the low-growth state lines up cleanly with NBER recession dates, delivering an objective, model-based business-cycle dating method and reinforcing Neftci (1984) / Sichel (1987) evidence on business-cycle asymmetry. Econometrica 57(2): 357–384.

Key Claims

• Markov model of trend (eqs. 2.1–2.4): the trend $n_t$ grows at a regime-dependent rate $\Delta n_t = \alpha_0 + \alpha_1 S_t$, where the state $S_t \in \{0,1\}$ follows a first-order Markov chain with $\Pr[S_t=1\mid S_{t-1}=1]=p$ and $\Pr[S_t=0\mid S_{t-1}=0]=q$. The differenced trend is itself an AR(1)/MA process whose persistence root is $\lambda = -1 + p + q$.
• Layered AR cycle: observed $y_t = n_t + z_t$, where $z_t$ is a stationary AR($r$) cycle independent of the regime shifts; in first differences the model couples a Markov-switching mean with autocorrelated Gaussian innovations.
• Hamilton's nonlinear filter (Section 4): treats the regime path as unobserved and computes $\Pr[S_t=s_t,\ldots,S_{t-r}=s_{t-r}\mid y_t,\ldots,y_1]$ recursively. Each step (i) forms the joint density of the next observation and the regimes, (ii) sums over the oldest regime to obtain the predictive likelihood (a by-product that gives the exact sample log-likelihood), and (iii) updates by Bayes' rule. A smoother (Section 4.3) gives full-sample probabilities $\Pr[S_t=j\mid y_1,\ldots,y_T]$.
• Likelihood-based, not "turning-point": by maximizing the exact conditional likelihood with respect to $(\alpha_0,\alpha_1,p,q,\sigma,\phi_1,\ldots,\phi_r)$ Hamilton makes the regime inference and parameter estimation a single maximum likelihood estimation (MLE) problem — contrasting with Neftci's (1982) and Wecker's (1979) turning-point predictors, which take the data-generating process as given.
• U.S. GNP application (Table I; real GNP, 1952:II–1984:IV, $y_t = 100\times\Delta\log\mathrm{GNP}$, AR(4)): $\hat\alpha_1 = 1.522$, $\hat\alpha_0 = -0.358$ (state-1 mean $\alpha_0+\alpha_1 \approx +1.2\%$/qtr expansion; state-0 mean $\alpha_0 \approx -0.4\%$/qtr recession); $\hat p = 0.905$, $\hat q = 0.755$; $\hat\sigma = 0.769$; AR lags $\hat\phi_3 = -0.247$, $\hat\phi_4 = -0.213$ significant.
• NBER concordance: the smoothed probability $\Pr[S_t=0]$ of the low-growth state reproduces the NBER business-cycle chronology of postwar recessions, providing an objective alternative to the NBER's judgmental dating.
• Permanent-income implication (Section 8): with a quarterly real discount factor $\beta = 0.99$, the certain knowledge that the economy has entered a recession is associated with roughly a 3% drop in the long-run forecast level of GNP — recessions carry a permanent, not merely transitory, component.
• Testing caveat: testing one regime vs. two is non-standard because $p,q$ are unidentified nuisance parameters under the single-regime null (Davies 1977; the score is identically zero, Lee-Chesher 1986).

Concepts Introduced or Extended

• Markov-Switching Model
• Markov-Switching VAR
• Plucking Model
• Great Moderation
• Kalman Filter
• State-Space Representation

Entities Mentioned

• James D. Hamilton

Quotes

"A positive growth rate is associated with normal times, and a negative growth rate associated with recessions. Indeed, the best statistical estimates of which quarters were historically characterized by negative growth correspond remarkably well with NBER dating of business cycles."

"The certain knowledge that the economy has gone into a recession is associated with a 3% drop in permanent income."

My Take

This is the foundational paper of the regime-switching literature — the single most-cited source behind the Markov-Switching Model hub and a direct ancestor of the Markov-Switching VAR, Plucking Model, and Great Moderation applications already in the wiki. Its enduring contributions are conceptual (recessions as a distinct stochastic state with permanent effects) and computational (the Hamilton filter, the discrete-state analogue of the Kalman filter). The chief limitation, acknowledged in the paper, is the non-standard inference on the number of regimes — later addressed by Bayesian approaches (Albert-Chib 1993; Chib 1996) that integrate out the transition probabilities.