King-Plosser-Stock-Watson (1991) Stochastic Trends and Economic Fluctuations

Summary

King, Plosser, Stock, and Watson (1991) exploit a long-run implication of a broad class of real-business-cycle (RBC) models — that output, consumption, and investment share a single common stochastic trend driven by total factor productivity (TFP) — to build a cointegrated vector autoregression (VAR) in vector error-correction model (VECM) form that is simultaneously unrestricted and nested within the neoclassical growth framework. Applying this framework to postwar U.S. data, they confirm the balanced-growth cointegration prediction but find that permanent productivity shocks explain less than half of business-cycle variability in output once nominal variables are included, casting doubt on single-sector RBC models as complete accounts of the business cycle.

Key Claims

• Balanced-growth theory implies that log output, consumption, and investment are cointegrated with two cointegrating vectors (c − y) and (i − y); data are consistent with this at the 5% level.
• In the three-variable real system, the single balanced-growth shock explains 45–81% of output variance and 88–89% of consumption variance at horizons of 1–24 quarters, but only 12–47% of investment variance.
• When money, prices, and the nominal interest rate are added (six-variable system), the balanced-growth shock accounts for only 35–62% of output variance — less than half — across a variety of model specifications.
• A persistent real-interest-rate permanent shock dominates output and investment fluctuations at short horizons (67% of output forecast-error variance at horizon 1), declining at longer horizons.
• Permanent neutral nominal shocks (constrained to have no long-run effect on real variables) explain a negligible fraction of output and consumption — consistent with long-run monetary neutrality but challenging for monetarist theories of the business cycle.
• The estimated balanced-growth shock is weakly correlated with Hall's (1988) Solow residual (r = 0.19) but more closely with Prescott's (1986) measure (r = 0.48), lending partial support to the TFP interpretation.
• The model provides a multivariate generalisation of the Beveridge-Nelson (1981) trend-cycle decomposition; the implied trend component of output tracks Denison's (1985) estimate of real potential GNP closely.

Model

Theoretical Background

Consider a Cobb-Douglas one-sector neoclassical economy with log-TFP following a random walk:

$$\log(\lambda_t) = \mu_\lambda + \log(\lambda_{t-1}) + \xi_t$$

Balanced growth implies that output, consumption, and investment all share the stochastic trend $\log(\lambda_t)/\theta$, so the ratios $C_t/Y_t$ and $I_t/Y_t$ are stationary. In Engle-Granger (1987) notation, letting $\mathbf{X}_t = (y_t, c_t, i_t)'$:

• Two cointegrating vectors: $\boldsymbol{\alpha}_1 = (-1, 1, 0)'$ and $\boldsymbol{\alpha}_2 = (-1, 0, 1)'$
• One common stochastic trend: $k = n - r = 3 - 2 = 1$

Money-demand stationarity (Fisher equation and quantity theory) adds further cointegrating relations in the six-variable system, implying three common stochastic trends.

Structural Identification

The paper identifies structural permanent shocks by two restrictions (following Blanchard-Quah 1989 extended to multivariate settings):

1. Long-run multiplier restriction: only the $k$ permanent shocks $\boldsymbol{\eta}_t^1$ have non-zero long-run effects. In the three-variable model the $3 \times 3$ long-run multiplier matrix is: $$\boldsymbol{\Gamma}(1) = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}$$ (first column normalised to unit effect on all three variables; columns 2 and 3 are zero.)
2. Orthogonality restriction: the permanent innovation $\eta_t^1$ is uncorrelated with the two transitory shocks $\eta_t^2, \eta_t^3$. This identifies the dynamic response to the balanced-growth shock as the first column of $\boldsymbol{\Gamma}(L)$.

The six-variable extension has $k = 3$ permanent shocks (balanced-growth, neutral-inflation, real-interest-rate). The $6 \times 3$ long-run multiplier matrix $\mathbf{A}$ is lower-triangular (à la Sims 1980) with $\hat{\mathbf{A}}$ and a $3 \times 3$ rotation matrix $\boldsymbol{\Pi}$ determined by the requirement that permanent innovations be mutually uncorrelated.

Estimation

The reduced form is estimated as a VECM with the theoretical cointegrating vectors imposed. Cointegrating vector coefficients are estimated by dynamic OLS (Stock-Watson 1989). The structural moving-average representation and impulse response functions are recovered by inverting the estimated reduced-form polynomial and transforming with $\boldsymbol{\Gamma}(1)$.

Concepts Introduced or Extended

• Common Stochastic Trends
• Cointegration
• Real Business Cycle
• Structural Identification
• Vector Autoregression
• Impulse Response Function
• Variance Decomposition
• Monetary Neutrality

Entities Mentioned

• Robert G. King
• Charles I. Plosser
• James H. Stock
• Mark W. Watson

Quotes

"The U.S. data are consistent with the presence of a common stochastic productivity trend... However, the common trend's explanatory power drops off sharply when measures of money, the price level, and the nominal interest rate are added to the system."

"Permanent productivity shocks typically explain less than half of the business-cycle variability in output, consumption, and investment."

"These results suggest that models that rely solely on permanent productivity or long-run neutral nominal shocks are not capable of capturing important features of the postwar U.S. experience."

My Take

This paper is a landmark precisely because it tests RBC theory at the level of its sharpest implication — cointegration among the great ratios — and finds partial support. The balanced-growth hypothesis survives cointegration testing, but the variance decompositions undercut the strong RBC claim. The paper's real methodological contribution is showing how economic theory can impose exact restrictions on $\Gamma(1)$ rather than $\Gamma(0)$, a shift that proved enormously influential (Blanchard-Quah 1989 independently pioneered this in a bivariate setting; KPSW generalise it fully). The real-interest-rate permanent shock result remains somewhat puzzling — the impulse responses (strong initial positive effect on y, c, i followed by reversal after 2–3 quarters) do not fit standard models — but it points toward credit-market or monetary-transmission channels that the purely real RBC framework ignores. A limitation is the short sample (1949–1988) and the assumption of a fixed number of cointegrating relations throughout; see Sims-Stock-Watson (1990) for inference in the presence of unit roots.