Robertson-Tallman (1999) Vector Autoregressions: Forecasting and Reality

Summary

A practitioner's guide to Vector AutoRegressive (VAR)-based macroeconomic forecasting at the Federal Reserve Bank of Atlanta, structured around a live 6-variable monthly model (real Gross Domestic Product (GDP), Consumer Price Index (CPI), unemployment, fed funds rate, M2, commodity prices). The paper covers three practical problems: constructing monthly GDP via Chow-Lin temporal disaggregation, generating conditional forecasts under the Waggoner-Zha (1998) minimum-Mean Square Error (MSE) procedure when data releases are staggered, and comparing six prior specifications on Root Mean Square Error (RMSE) from 1986–1997. The decisive finding: a Modified Litterman model (base Minnesota + sum-of-coefficients $\lambda_5$ + cointegration dummy $\lambda_6$) essentially matches the full Sims-Zha (ZVAR) prior in forecast accuracy; the long-run restrictions $\lambda_5$ and $\lambda_6$ explain the gain over plain Litterman, not the systemwide Normal-Wishart covariance structure.

Key Claims

Variable Set

Six-variable monthly VAR: real GDP (seasonally adjusted, chain-weighted 1992 dollars; distributed from quarterly via Chow-Lin with industrial production, nonagricultural employment, and real personal consumption as monthly indicators), CPI for all urban consumers (not seasonally adjusted), civilian unemployment rate (seasonally adjusted), effective federal funds rate, M2 money stock (seasonally adjusted), and Commodity Research Bureau (CRB) spot raw industrial commodity price index. Estimation period 1959:2–1997:12 with $p = 13$ monthly lags.

Chow-Lin Temporal Disaggregation (Chow-Lin 1971)

Quarterly real GDP is distributed to monthly frequency using the Chow-Lin Generalized Least Squares (GLS) procedure. Let $y_m$ be the $T$-vector of unknown monthly values, $X_m$ the corresponding matrix of monthly indicator variables, and $C$ the $T/3 \times T$ temporal aggregation matrix. The minimum-variance linear unbiased estimator of $y_m$ is:

$$\hat{y}_m = X_m \hat\beta + \hat{P}_m C'(C\hat{P}_m C')^{-1} \hat{u}_q$$

where $\hat\beta$ is the GLS estimate of the regression of quarterly aggregates on quarterly-averaged indicators, $\hat{u}_q = y_q - C X_m \hat\beta$ are quarterly residuals, and $\hat{P}_m$ is the monthly covariance matrix. The autocorrelation coefficient of the monthly residuals is estimated from the implied quarterly AR(1) coefficient via a polynomial identity.

Staggered Data Release and Conditional Forecasting

At most forecast origin dates, the latest quarterly GDP observation lags the available monthly indicators by one to three months. The solution is the Waggoner-Zha (1998) minimum-MSE conditional forecasting procedure. If $\hat{e}$ is the $nH$-vector of stacked forecast residuals and $R \hat{e} = r$ encodes $n$ affine constraints (e.g., "set the fed funds rate equal to its January value"), the minimum-norm residuals satisfying the constraint are:

$$\hat{e}^* = R'(RR')^{-1}r$$

The constrained forecast error bands have singular covariance $I - R'(RR')^{-1}R$, requiring simulation from a reduced-rank distribution.

Prior Taxonomy (6 Specifications)

Label	Description
OLS	Ordinary Least Squares (OLS), unrestricted; baseline
DOLS-AIC	OLS with Akaike Information Criterion (AIC) lag selection (1–3 lags)
Litterman	Base Minnesota: $\lambda_1=0.2$, $\lambda_2=0.2$, $\lambda_3=1$, $\lambda_4=0.3$
Modified Litterman	Litterman + sum-of-coefficients ($\lambda_5=5$) + cointegration dummy ($\lambda_6=5$)
ZVAR	Full Sims-Zha: $\lambda_0=0.6$, $\lambda_1=0.1$, $\lambda_2=1$, $\lambda_3=1$, $\lambda_4=0.1$, $\lambda_5=5$, $\lambda_6=5$
Partial ZVAR	ZVAR with $\lambda_5=0$ and $\lambda_6=0$

Monthly harmonic lag decay is approximated by $\exp(c(\ell-1))$ with $c = -0.13412$, matching the quarterly harmonic $(1, 1/2, \ldots, 1/13)$ at $p = 13$ monthly lags.

RMSE Results (1986–1997, Unemployment / CPI Inflation / GDP Growth)

Modified Litterman produces the lowest or near-lowest RMSE for most variables and horizons. ZVAR is within 1–3% of Modified Litterman in nearly all cases. OLS is worst by 25–132% depending on variable and horizon. Partial ZVAR closely tracks plain Litterman. The key decomposition: ZVAR minus Partial ZVAR $\approx$ Modified Litterman minus Litterman, establishing that $\lambda_5$ and $\lambda_6$ account for essentially all of ZVAR's improvement over plain Litterman.

Concepts Introduced or Extended

• Conditional Forecasting — Waggoner-Zha minimum-MSE procedure; $\hat{e}^* = R'(RR')^{-1}r$; singular covariance $I - R'(RR')^{-1}R$; application to staggered GDP release
• Minnesota Prior — Modified Litterman prior: $\lambda_5$ sum-of-coefficients and $\lambda_6$ cointegration dummy extensions; RMSE evidence attributing ZVAR's advantage to these long-run restrictions; monthly harmonic lag decay approximation; prior taxonomy compared at 6-variable monthly scale
• Sims-Zha Prior — Empirical validation: ZVAR $\approx$ Modified Litterman in RMSE; $\lambda_5$ and $\lambda_6$ are the operative components of ZVAR's forecast advantage
• Vector Autoregression — Practical 6-variable monthly macro VAR construction; Chow-Lin disaggregation of quarterly GDP; staggered data release handling; comparative RMSE evaluation 1986–1997

Entities Mentioned

• John C. Robertson
• Ellis W. Tallman

Quotes

"This improvement is largely explained by the incorporation of reasonably tight priors on the long-run properties of the VAR. This long-run aspect of the specification appears to matter more for the improvements in accuracy than the systemwide nature of the formulation does."

My Take

The paper's most important contribution is the decomposition of the ZVAR prior's performance advantage: $\lambda_5$ (sum-of-coefficients, near-unit-root shrinkage) and $\lambda_6$ (cointegration dummy) explain the gain over base Litterman; the systemwide Normal-Wishart covariance structure is largely irrelevant. This is a practically decisive result — practitioners don't need the full Sims-Zha machinery to capture most of the long-run accuracy gains. The Chow-Lin section is a useful reference for any practitioner distributing quarterly aggregates to monthly frequency; the GLS formula is correct and the autocorrelation estimation via the polynomial identity is clean. The conditional forecasting treatment accurately summarizes Waggoner-Zha (1998). One caveat: the ZVAR hyperparameters were chosen post-sample, so the comparison is not a clean real-time experiment.