Plucking Model

Definition

Friedman's (1964, 1993) plucking model asserts that output is bounded above by a trend ceiling representing maximum feasible output and is occasionally "plucked" downward by recessions — transitory departures that always rebound to the ceiling. The key asymmetry: the amplitude of a contraction predicts the amplitude of the subsequent expansion, but expansion amplitude does not predict the next contraction. Kim and Nelson (1999a) provide the first formal state-space econometric model that encompasses both the plucking model and the symmetric Clark (1987) trend-plus-cycle model as special cases.

Key Ideas

Ceiling output: The stochastic trend $\tau_t$ is an upper limit set by available resources and technology; output operates on or near this ceiling during normal times.
Asymmetric shock structure: The transitory component $c_t = \phi_1 c_{t-1} + \phi_2 c_{t-2} + u_t^*$ has shock $u_t^* = \pi S_t + u_t$ — a mixture of a regime-dependent asymmetric term ( $\pi S_t$ , with $\pi < 0$ during recessions $S_t=1$ ) and a symmetric Gaussian term $u_t \sim \mathcal{N}(0, \sigma_{u,S_t}^2)$ .
Markov-switching regime: $S_t \in \{0,1\}$ follows a two-state first-order Markov chain with $\Pr[S_t=1|S_{t-1}=1]=p$ and $\Pr[S_t=0|S_{t-1}=0]=q$ .
Stochastic trend ceiling: $\tau_t = g_{t-1} + \tau_{t-1} + v_t$ , $g_t = g_{t-1} + w_t$ — a random walk with stochastic drift allowing for productivity slowdowns.
Three phases: normal (near ceiling, permanent shocks dominate), recessionary (plucked below by transitory shock), high-growth recovery (fast reversion as transitory effects dissipate).

How It Works

The model is written in state-space form with state vector $\xi_t = (\tau_t, c_t, c_{t-1}, g_t)'$ and estimated via Kim's (1994) approximate maximum likelihood estimator (MLE). At each step the Kalman filter is run conditional on $(S_t = j, S_{t-1} = i)$ , producing $2 \times 2$ branches; these are collapsed back to 2 via the Harrison-Stevens (1976) weighted-average approximation. The log-likelihood is maximized numerically. This contrasts with the Gibbs-sampling approach of Kim-Nelson (1998), which is approximation-free.

Nested special cases:

$\pi = 0$ , $\sigma_{v0}^2 = \sigma_{v1}^2$ , $\sigma_{u0}^2 = \sigma_{u1}^2$ : Clark (1987) symmetric linear unobserved components (UC) model
Asymmetry in trend growth only: Hamilton (1989) / Lam (1990) style models
This model uniquely places asymmetry in the transitory component

Empirical results (U.S. real gross domestic product (GDP), 1951:1–1995:3):

$\hat{\pi} = -0.011$ (std 0.003); $\hat{p} = 0.711$ , $\hat{q} = 0.933$ (recession duration $\approx 3.5$ qtrs; normal duration $\approx 14.7$ qtrs)
$\phi_1 + \phi_2 = 0.797$ (vs. 0.946 in Clark UC) — asymmetry markedly reduces apparent persistence
Symmetric shock $u_t$ essentially zero (likelihood ratio (LR) accepts $\sigma_{u0} = \sigma_{u1} = 0$ )
Log-likelihood: 585.71 vs. 577.83 for Clark UC

Why It Matters

The plucking model has concrete structural implications: recessions are transitory negative shocks while normal times are driven by permanent shocks. This means real business cycle (RBC) models are most relevant for normal times and monetary/demand-oriented models are most relevant for recessions and recovery — a regime-dependent theory of structural relevance. The asymmetric shocks also mean standard unit root tests are biased toward non-rejection: Perron (1990) showed that such tests have low power against a stationary process with a switching mean, which explains the apparent high persistence in linear UC models.

Open Questions