Schorfheide (2005) VAR Forecasting Under Misspecification

Summary

Develops an asymptotic theory for multi-step Vector AutoRegressive (VAR) forecasting when the model is potentially misspecified, published in Journal of Econometrics 128 (2005): 99–136. Introduces a local misspecification framework where the Data Generating Process (DGP) drifts toward a VAR($p^*$) at rate $T^{-1/2}$, defines two competing plug-in predictors (Maximum Likelihood Estimator (MLE) and Loss Function Estimator (LFE)), and proposes a Prediction Criterion (PC) — a modification of Shibata's (1980) Final Prediction Error — that jointly selects lag order and estimator type. Monte Carlo results with 100,000 repetitions illustrate when each predictor dominates.

Key Claims

• The MLE predictor (iterates $\hat{F}_t^h Y_t$) and the Loss Function Estimator (LFE) predictor (regresses $y_t$ on $y_{t-h}$ minimizing $h$-step quadratic loss, $\hat{C}_t = S_{t,0h} \cdot S_{t,hh}^{-1} \cdot [\text{restriction}]$) differ only in their cross-lag products: MLE uses lag-1 covariances $S_{t,01} \cdot S_{t,11}^{-1}$, LFE uses lag-$h$ covariances $S_{t,0h} \cdot S_{t,hh}^{-1}$.
• Under local misspecification parameterized by scalar $a$: when $a=0$ (no misspecification), MLE dominates; as $a$ increases, LFE becomes preferable because it tilts the estimation bias favorably toward the $h$-step loss function.
• The Prediction Criterion (PC) provides an asymptotically unbiased estimate of prediction risk; it can jointly select lag order p and estimator type (MLE/LFE). For small misspecification, Schwarz Information Criterion (SIC)+MLE is optimal (MLE(VAR(1)) reduces risk ${\sim}70\%$ over baseline). For large misspecification ($a \geq 5$), PC+LFE dominates.
• MLE and LFE are substitutes in correcting misspecification: using more lags (higher-p MLE) and switching to LFE both reduce bias but increase variance.
• Bayesian VARs (Doan-Litterman-Sims 1984; Litterman 1986) are asymptotically equivalent to MLE predictors but have better finite-sample properties through shrinkage. SIC is a large-sample approximation to the Bayesian marginal data density (Zellner 1971; Chib 1995).

Concepts Introduced or Extended

• Lag-Order Selection Criteria
• Vector Autoregression

Entities Mentioned

• Frank Schorfheide

Quotes

"Under severe misspecification it is preferable to use the multi-step loss function also for parameter estimation."

My Take

A technically rigorous frequentist paper that formalizes an intuition (misspecification $\to$ use direct/multi-step estimators) that practitioners had already acted on. The local misspecification framework is elegant but the $T^{-1/2}$ drift rate is somewhat restrictive. The main practical insight — that PC guards against the wrong predictor choice ex ante — is useful but the selection criterion has substantial sampling noise (Tables 1–4 show the PC standard deviation is often comparable to the risk differential). Connections to Bayesian VAR are acknowledged but not developed; this is primarily a frequentist contribution.