Stark (1998) A Bayesian Vector Error Corrections Model of the US Economy

Summary

Specifies and evaluates a Bayesian Vector Error Correction (BVEC) model for short-term forecasting of U.S. output growth, inflation, and the unemployment rate. The model combines Litterman's Bayesian shrinkage priors with a single error correction (EC) term — the spread between the federal funds rate and the 10-year Treasury bond rate, which is treated as I(0) — estimated via Theil's mixed estimator with a diffuse prior on the EC coefficient. A systematic forecasting experiment (mid-1975–1997Q4) demonstrates that explicit differencing and the EC term both improve forecast performance relative to naive Bayesian Vector AutoRegression (BVAR) alternatives; the Bayesian priors are essential, with Ordinary Least Squares (OLS) substantially worse across all variables; imposing a stationary real rate (Fisher relation) is counterproductive, worsening the inflation forecast notably.

Key Claims

Baseline Model Specification

Seven-variable system in first differences plus one error correction term:

$\text{Model 1: BVEC(5)}\{\Delta\text{RGDP},\; \Delta^2\text{PGDP},\; \Delta\text{RFF},\; \Delta\text{RPIM},\; \Delta U,\; \Delta\text{RM2},\; \Delta\text{RTB}\}$

where RGDP = log real GDP, PGDP = log GDP price index, RFF = federal funds rate/400, RPIM = log real import price, U = unemployment rate/100, RM2 = log real M2, RTB = 10-year Treasury bond rate/400. Each equation also includes one lag of the EC term $\text{RFF}_{t-1} - \text{RTB}_{t-1}$ , which imposes the cointegrating relation $(1,-1)$ between the two rates. Estimated 1960Q4–1997Q4, five lags in first differences.

The model departs from standard Litterman in two ways:

Unit roots imposed outright (differencing) rather than assigned as a prior with finite variance.
Lag decay exponent $\gamma = 0.50$ instead of 1.0, so prior standard deviations decline as $l^{-0.50}$ rather than $l^{-1}$ , reducing aggressive shrinkage at longer lags.

Bayesian Prior (Theil Mixed Estimator)

$\text{SD}(i,j,l) = \lambda \cdot T_{ij} \cdot (\sigma_i/\sigma_j) \cdot l^{-\gamma}$ with hyperparameters $\lambda = 0.20$ , $\gamma = 0.50$ , $T_{ii} = 1.0$ . The EC coefficient $\delta_i$ receives a diffuse prior in all equations, following LeSage (1990) and Joutz-Maddala-Trost (1995). The Theil mixed estimator is used throughout; under normality assumptions this coincides with the Bayesian posterior mean.

Error Correction Significance

The spread $\text{RFF}_{t-1} - \text{RTB}_{t-1}$ enters significantly ( $p < 0.10$ ) in the $\Delta$ RGDP equation ( $p = 0.003$ ), $\Delta$ RFF ( $p = 0.068$ ), $\Delta U$ ( $p = 0.000$ ), and $\Delta$ RTB ( $p = 0.063$ ). The EC coefficient in the GDP equation is stable across subsamples ending 1979–1997, though it shows some instability in magnitude. Consistent with Laurent (1988).

Forecast Evaluation (Rolling Regression, Mid-1975–1997Q4)

Four-quarter-average one- and two-year-ahead Root Mean Square Errors (RMSEs) for the preferred specification (Model 1) against three alternatives:

	Model 1 (BVEC)	Model 2 (levels BVAR)	Model 3 (BVAR, no EC)	Model 4 (no RPIM)
Inflation 1-yr	0.932	1.198	0.908	1.113
Inflation 2-yr	1.523	2.395	1.531	1.684
GDP growth 1-yr	1.820	2.532	1.968	1.826
Unemployment 2-yr	0.805	1.380	1.113	0.779

Key findings:

Differencing: failure to difference (Model 2) causes substantial deterioration on all variables and horizons.
EC term: removing the spread (Model 3) raises the two-year unemployment RMSE from 0.805 to 1.113 and has mixed effects on inflation/GDP.
Import prices: removing RPIM (Model 4) raises the inflation RMSE by ~0.2 pp at both horizons.
Aggregate demand components (consumption, investment, government, stock prices): adding these one at a time yields no improvement in core variable forecast accuracy.
Stationary real rate (adding $\text{RFF}_{t-1} - \Delta\text{PGDP}_{t-1}$ as a second EC term): two-year inflation RMSE rises from 1.52 to 2.36; the Fisher relation appears counterproductive in this specification.
Removing Bayesian priors (OLS): one-year inflation RMSE rises from 0.93 to 1.35, GDP growth from 1.82 to 2.60, unemployment from 0.52 to 0.72. Bayesian shrinkage essential.

Impulse Responses (Cholesky Identification)

Ordering: $\Delta$ RGDP, $\Delta^2$ PGDP, $\Delta$ RPIM, $\Delta U$ , $\Delta$ RTB, $\Delta$ RFF, $\Delta$ RM2. Results for a 0.6 pp shock to the federal funds rate:

Output growth slows over 10 quarters, peak $-0.7$ percentage points (pp) at 2 quarters after shock.
Inflation begins to decline after 5 quarters.
Unemployment rises gradually, peak $+0.3$ pp.
Qualitatively consistent with Christiano, Eichenbaum, and Evans (1996) monetary "facts."

Comparison With Anderson-Hoffman-Rasche (1998)

Non-Bayesian VEC model (St. Louis Fed, 4 cointegrating relations including Fisher relation, real M1). Over 1988Q1–1995Q4 (AHR benchmark period), the BVEC produces smaller step-ahead RMSEs for $\Delta$ PGDP, $\Delta$ RGDP, and RFF at all horizons; RTB RMSEs are comparable.

Concepts Introduced or Extended

Cointegration — BVEC application: single known EC term RFF $-$ RTB; diffuse prior on EC coefficient; Fisher relation (stationary real rate) debate; imposing Fisher worsens inflation forecasts; EC term improves unemployment forecasts; Stark (1998) BVEC as Bayesian VEC forecasting methodology
Minnesota Prior — BVEC modification: $\gamma=0.50$ lag decay (vs standard 1.0); unit root imposition vs random walk prior; Bayesian priors essential (Table 6 comparison with OLS)
Vector Autoregression — BVEC specification; differencing discipline; comparative RMSE 1975–1997

Entities Mentioned

Tom Stark

Quotes

"The results confirm the motivation provided at the beginning of this paper — that Bayesian priors can help to improve the forecast performance of VARs."

My Take

The paper's core contribution is a well-documented example of BVEC forecasting that clarifies two practical questions: (1) explicit differencing beats a unit-root prior for this 7-variable US system, and (2) the EC term helps for unemployment but not uniformly. The most interesting negative result is that imposing real rate stationarity (the Fisher relation) hurts inflation forecasts, confirming that the empirical evidence for a long-run Fisher effect is weak enough that imposing it creates more specification error than it eliminates. The Bayesian priors comparison (Table 6) is a clean demonstration of the BVAR value-add: OLS is substantially worse on all core variables, especially in the 1970s–1980s. The $\gamma=0.50$ modification is noteworthy but rarely discussed in subsequent literature — the implication is that allowing slightly more distant lag information into the model is worth the weaker shrinkage. The model was in active use at the Philadelphia Fed at the time of writing.