Minnesota Prior

Definition

The Minnesota prior (also called the Litterman prior) is the original Bayesian prior specification for vector autoregressions (VARs), introduced in Litterman (1986b). It encodes two beliefs: (1) each variable is likely to follow a random walk — the first own lag has prior mean 1 and all others have mean 0; (2) longer lags and lags of other variables are a priori less important — prior standard deviations decay as 1/lag and cross-variable lags are shrunk by an additional factor $\theta < 1$ . The result is a ridge-type estimator that regularizes the otherwise overparameterized VAR without hard exclusion restrictions.

Key Ideas

Overparameterization problem: A VAR with $n$ variables and $p$ lags has $n^2 p + n$ parameters; standard approaches must either exclude variables (exact-zero prior) or include them without a prior (flat prior). Both misrepresent actual beliefs.
Random walk belief: Prior mean of first own lag = 1; prior mean of all other lags = 0. Encodes the view that the best single-step forecast of a macro variable is its current value.
Lag decay: Prior standard deviation on lag $\ell$ shrinks as $1/\ell$ , reflecting the belief that longer lags carry less information.
Cross-variable shrinkage: Lags of variable $j$ in equation $i$ ( $i \neq j$ ) receive an additional dampening factor $\theta = 0.2$ , reflecting the assumption that own lags account for most of each variable's variation.
Scale normalization: Standard deviations are scaled by $s_i/s_j$ (ratio of univariate autoregressive (AR) residual standard errors), making the prior invariant to the units of measurement.
Hyperparameter: $\lambda$ (overall tightness) is the single most important tuning parameter; smaller $\lambda$ = more shrinkage. Calibrated to $\lambda = 0.2$ for a 7-variable quarterly U.S. system.

How It Works

Prior Specification (Litterman 1986b, eq. 10)

Let $\sigma_{ij\ell}$ be the prior standard deviation on the coefficient of lag $\ell$ of variable $j$ in equation $i$ , and let $s_i$ denote the residual standard error from a univariate AR fit to series $i$ :

$\sigma_{ij\ell} = \begin{cases} \dfrac{\lambda}{\ell} & \text{if } i = j \quad \text{(own lag)} \\[6pt] \dfrac{\lambda\,\theta\,(s_i/s_j)}{\ell} & \text{if } i \neq j \quad \text{(cross lag)} \end{cases}$

Default: $\theta = 0.2$ ; deterministic terms receive a flat (non-shrinking) prior.

Estimator (eq. 8)

The posterior mean takes the ridge form:

$\hat{\beta} = (X'X + \kappa R'R)^{-1}(X'Y + \kappa R'r)$

where $R$ encodes prior precision, $r$ encodes prior means (zero for all lags except first own lag = 1), and $\kappa = 1$ in standard parameterization.

Simulation Validation

Litterman (1986b) validates the prior via Monte Carlo: draw $b_j \sim \mathcal{N}(0, j^{-1})$ , 3,000 repetitions. The Bayesian posterior mean dominates:

ordinary least squares (OLS) with any fixed variable set
Stepwise variable selection

Dominance is strongest when observations are few relative to regressors or when $R^2$ is low. Even a prior misspecified by a factor of 4 outperforms standard methods.

Calibration

Seven-variable quarterly system (real GNP (RGNP), inflation, unemployment, M1, investment, CP rate, change in business inventories), 1971:1–1975:4. Theil coefficients relative to a no-prior OLS benchmark:

$\lambda = 0.2$ minimizes average Theil coefficients across variables and horizons.
OLS (no prior) has Theil > 1.0 on real GNP and unemployment — worse than a naive random walk forecast.

Extensions: Sum-of-Coefficients and Cointegration Dummy

Two dummy-observation extensions of the base Minnesota prior, incorporated into the Sims-Zha Prior and empirically evaluated in Robertson-Tallman (1999):

Sum-of-coefficients ( $\lambda_5$ ): adds one dummy observation per equation with all dependent and lagged variables set to their pre-sample means $\bar{y}_i$ . This shrinks the sum of each variable's own-lag coefficients toward 1, reinforcing the unit-root belief beyond the harmonic decay of the base prior. Typical calibration: $\lambda_5 = 5$ .

Cointegration dummy ( $\lambda_6$ ): adds a single system-wide dummy observation with all variables simultaneously set to their pre-sample means. This imposes a tendency toward common long-run comovement across equations. Typical calibration: $\lambda_6 = 5$ .

Empirical relevance (Robertson-Tallman 1999, 6-variable monthly U.S. macro VAR, root mean squared error (RMSE) 1986–1997): A Modified Litterman model (base Litterman + $\lambda_5$ + $\lambda_6$ ) achieves essentially the same forecast accuracy as the full Sims-Zha (ZVAR) prior. A partial ZVAR (full Sims-Zha without $\lambda_5$ and $\lambda_6$ ) closely tracks base Litterman. The implication: the long-run restrictions $\lambda_5$ and $\lambda_6$ account for essentially all of ZVAR's forecast advantage over base Litterman; the systemwide Normal-Wishart covariance structure adds little.

Bayesian Vector Error Correction (BVEC) Modifications (Stark 1998)

Stark (1998) modifies the standard Litterman specification in two ways when embedding the Minnesota prior within a vector error correction model (VECM) framework:

1. Unit roots imposed by differencing. The standard Minnesota prior approximates a unit root by setting the prior mean of the first own lag to 1 with finite variance. Stark imposes unit roots outright — all variables enter in first differences (or second differences for the price level). This hard-differencing discipline ensures the model does not inadvertently treat near-unit-root dynamics as stationary, and is shown to substantially outperform levels Bayesian VAR (BVAR) estimation (Model 2) in the rolling RMSE comparison.

2. Reduced lag decay: $\gamma = 0.50$ . Standard Litterman sets $\sigma_{ij\ell} \propto \ell^{-1}$ ( $\gamma = 1.0$ ), halving prior standard deviations from lag 1 to lag 4. Stark uses $\gamma = 0.50$ , a slower decay that preserves more prior variance at longer lags. The empirical motivation is that quarterly U.S. business cycle dynamics extend over several years, and aggressive shrinkage at longer lags discards information that improves medium-horizon forecast accuracy.

Bayesian priors essential (Table 6, Stark 1998). Removing all Minnesota priors and estimating by OLS substantially degrades forecast accuracy across all core variables:

Variable	BVEC 1-year RMSE	OLS 1-year RMSE	Degradation
Inflation	0.93%	1.35%	+45%
gross domestic product (GDP) growth	1.82%	2.60%	+43%
Unemployment	0.52%	0.72%	+38%

The deterioration is concentrated in the volatile 1970s–1980s period, confirming that Bayesian shrinkage provides the most stabilizing regularization precisely when macroeconomic data are noisiest.

Real/Price Block Hyperparameterization (Felix-Nunes 2003)

Felix and Nunes (2003) extend the standard Minnesota prior to a system with variables naturally partitioned into real and price blocks. The scalar tightness parameter $\lambda$ is replaced by:

$\lambda_1$ : own-lag tightness for real variables (GDP, unemployment, interest rates, exchange rate)
$\lambda_2$ : own-lag tightness for price variables (Consumer Price Index (CPI), wages, external price index)
$\theta_1$ : cross-variable shrinkage, same block ( $i \neq j$ , both real or both price)
$\theta_2$ : cross-variable shrinkage, different block (real $\to$ price or price $\to$ real)
$\theta_3$ – $\theta_4$ : tightness on exogenous variable lags (same-block, cross-block)
$\theta_5$ – $\theta_6$ : tightness on monetary instrument lags in real/price equations
$\Omega$ : prior variance on the error-correction factor loadings $\alpha$ when embedded in a Bayesian error correction model (BECM)

The parameter $\Omega$ controls the degree to which the error correction model (ECM) correction is trusted: $\Omega = 0$ suppresses the error-correction term entirely (pure BVAR), while $\Omega \to \infty$ recovers a flat (diffuse) prior on $\alpha$ . In a six-variable quarterly euro area system (1977:1–1997:4), the optimal $\lambda_1 \approx 0.2$ , $\lambda_2 \approx 0.5$ (price variables are shrunk less than real variables), and the best overall model is the BVAR in levels (avg RMSE ratio 0.731 vs. random walk (RW) = 1.000). The key finding is that the choice of $\Omega$ is critical when using Johansen-based cointegrating vectors: a flat prior ( $\Omega \to \infty$ ) with four cointegrating vectors produces an avg RMSE of 1.278 — 28% worse than a random walk. A finite, informative $\Omega$ partially rescues the BECM. See Cointegration for the theoretical reason ( $\alpha$ estimates converge only at the $O_p(T^{-1/2})$ rate).

Bayesian VAR for Irish HICP Inflation (Kenny-Meyler-Quinn 1998)

Kenny, Meyler, and Quinn (1998) apply the Minnesota prior to forecasting Irish Harmonized Index of Consumer Prices (HICP) inflation ahead of European Monetary Union (EMU) accession (January 1999). Their implementation uses Theil mixed estimation, following the Doan-Litterman-Sims framework, with prior standard deviations $S(i,j,l) = [\gamma \cdot g(l) \cdot f(i,j)] \cdot s_i/s_j$ where $g(l) = l^{-1}$ (harmonic decay, $d=1$ fixed).

Indicator pre-screening (Cecchetti 1995). Before model construction, each candidate indicator is screened via Cecchetti-type regressions: $\pi_{t+l,t+k} = a(L)\pi_t + b(L)X_t + \varepsilon_t$ using Newey-West Wald tests of $H_0: b(L) = 0$ at five horizons. Key findings: import-weighted and trade-weighted foreign price indices are significant at all horizons; the nominal effective exchange rate is significant from 1Q through 3 years (implying near-complete pass-through within three years); M3 is not significant at any horizon.

Three BVAR models. Hyperparameters tuned by grid search:

Model	Variables	$\gamma$	$w$
BVAR1	{P, P*, E}	0.4	0.8
BVAR2	{P, P*, E, W, RS}	0.3	0.3
BVAR3	{P, P*, E, r^s, DC}	0.2	0.4

The monetary BVAR requires the most shrinkage ( $\gamma=0.2$ ); the parsimonious small open economy (SOE) model tolerates looser priors ( $\gamma=0.4$ ).

Forecast results (Theil U, T=24 rolling forecasts, 1992Q1–1998Q1). BVAR1 wins overall (0.71 short, 0.55 medium). Unrestricted 5-variable OLS: Theil U = 1.21 (1–4Q) — worse than naïve. Bayesian priors reduce BVAR3 error by $\approx 44\%$ vs. its OLS counterpart. The general prior (forcing P* and E to univariate ARs, Artis-Zhang 1990) does not improve any BVAR. Best achievable 95% confidence interval (CI): $\approx \pm 1.6\%$ per quarter.

Key lesson. The optimal tightness increases with model dimension ( $\gamma$ decreases from 0.4 to 0.2 as variables go from 3 to 5), and the simplest economically motivated specification — external prices plus exchange rate — robustly dominates augmented models.

Spatial Generalization (LeSage-Krivelyova 1999)

The standard prior uses a single cross-variable weight $\theta = 0.2$ for all $i \neq j$ pairs, regardless of geography. LeSage and Krivelyova (1999) replace $\theta$ with equation-specific weights from the row-standardized contiguity matrix $W = \mathrm{diag}(C\mathbf{1})^{-1}C$ (where $C_{ij} = 1$ iff regions $i$ and $j$ share a border). The prior mean on the first lag of variable $j$ in equation $i$ becomes:

$\mu_{ij,1} = \begin{cases} 0.5 & j = i \\ 0.5 \cdot w_{ij} & j \in \mathcal{N}(i) \\ 0 & \text{otherwise} \end{cases}$

This is a spatial random walk belief: the best forecast of region $i$ is half its own current value plus half the spatial average of contiguous neighbors. Non-contiguous lags receive tighter prior standard deviations than under standard Minnesota. Estimation remains Theil-Goldberger mixed estimation — no MCMC needed. In a 20-industry × 8-state Midwest employment forecasting experiment, the spatial BVAR (SVAR) outperforms standard Minnesota BVAR at horizons 4–12 and has smaller forecast error dispersion. See Spatial VAR.

Why It Matters

Founded the BVAR forecasting tradition at the Minneapolis Fed and subsequently in central banks worldwide.
Five-year real-time comparison (1980:2–1985:1) against DRI, Wharton, Chase commercial forecasters: BVAR closest to actual 34.8% of 1,604 forecasts vs. 16–27% for each commercial service.
Demonstrated that regularization via an informative prior outperforms data-driven model selection methods.
Directly preceded and motivated the Sims-Zha Prior, which extends the Minnesota prior to identified (structural) VARs and adds unit-root/cointegration components.

Known Limitations

The prior does not account for structural breaks; BVAR systematically over-forecast U.S. inflation during the disinflationary 1980s because the random-walk prior assigns no weight to level-shifting dynamics. Acknowledged explicitly by Litterman (1986b).

Equation-independence and fixed $\Psi$ (Kadiyala-Karlsson 1993): The Minnesota prior treats each equation independently and fixes the residual covariance matrix $\Psi$ as known and diagonal, typically at $\psi_{ii} = 0.81 s_i^2$ from univariate AR residuals. Kadiyala and Karlsson argue this is the wrong tradeoff: practitioners can form prior beliefs about regression coefficients more easily than about $\Psi$ , yet the Minnesota prior imposes certainty on the harder object and ignores cross-equation information. Empirically, priors that allow inter-equation dependence (Normal-Wishart, Normal-Diffuse, extended natural conjugate (ENC)) consistently match or beat the Minnesota prior, and in small samples the Minnesota prior is significantly worse. OLS performs worst in every experiment.

Computational advantage (Kadiyala-Karlsson 1997): Despite its forecasting inferiority, the Minnesota prior's forced equation-independence makes posterior computation $O(mk)$ — direct, with no Markov Chain Monte Carlo (MCMC) required. The richer Normal-Diffuse and ENC priors cost $O((mk)^3)$ and $O(m(mk)^2)$ per Gibbs step respectively, with a $\approx 140\times$ CPU overhead at 7 variables. This computational gap explains why the Minnesota prior (and its Normal-Wishart generalization, which also has a closed-form posterior at $O(mk)$ ) dominated central-bank BVAR practice even after the forecasting comparisons showed it was inferior. See Gibbs Sampler for details of the Normal-Diffuse and ENC Gibbs samplers.

Open Questions

Optimal selection of $\lambda$ and $\theta$ : most applications use historical values ( $\lambda = 0.2$ , $\theta = 0.2$ ) or grid search on a hold-out period rather than formal Bayesian hyperparameter estimation.
How to incorporate cointegration or known long-run equilibrium relationships — addressed by Steady State VAR (Villani 2008) and Sims-Zha Prior.
Whether the scale normalization by univariate AR residuals (rather than full-system residuals) is optimal in all settings.