Sims-Zha Prior

Definition

The Sims-Zha prior is a conjugate Gaussian prior over the structural VAR parameters, proposed in Sims and Zha (1997). It extends Litterman's (1986) Minnesota Prior to identified (structural) VARs, adds a cointegration/unit-root component via dummy observations, and is designed to prevent overfitting in large VARs without truncating lag length.

Key Ideas

Two components: (1) a Gaussian coefficient prior encoding the random walk belief, (2) a dummy-observation component encoding unit-root and cointegration beliefs.
Five shrinkage hyperparameters ( $\lambda_0$ – $\lambda_4$ ) and two cointegration hyperparameters ( $\mu_5$ , $\mu_6$ ), all set before estimation.
Applies to both reduced-form and identified structural VARs.
The resulting prior has a closed-form Gaussian pdf, enabling efficient Gibbs sampling and marginal data density (MDD) computation.

How It Works

Structural VAR Notation (Sims-Zha 1998)

Write the model as $A(L)y(t) + C = \varepsilon(t)$ , or in matrix form stacking $T$ observations:

$Y A_0 - X A_+ = E \tag{eq. 4}$

where $Y$ is $T \times m$ , $A_0$ is $m \times m$ (contemporaneous), $X$ is $T \times k$ , $A_+$ is $k \times m$ (lag coefficients; $k = mp + 1$ ), and each column of $E$ is i.i.d. $\mathcal{N}(0,I)$ . Defining $Z = [Y\; -X]$ and $A = [A_0'; A_+']'$ :

$L(Y \mid A) \propto |A_0|^T \exp\!\left[-\tfrac{1}{2} \operatorname{tr}(ZA)'(ZA)\right] = |A_0|^T \exp\!\left[-\tfrac{1}{2} a'(I \otimes Z'Z)a\right] \tag{eq. 6}$

The prior takes the form:

$\pi(a) = \pi_0(a_0)\; \phi\!\left(a_+ - \mu(a_0);\, H(a_0)\right) \tag{eq. 7}$

where $\pi_0$ is the marginal prior on $a_0$ , $\mu(a_0)$ centers $a_+$ conditional on $A_0$ , and $H(a_0)$ is the lag-coefficient prior covariance. The resulting posterior is:

$q(a) \propto \pi_0(a_0)\,|A_0|^T\,|H(a_0)|^{-1/2} \exp\!\left[-\tfrac{1}{2}(\cdots \text{quadratic in } a_+\cdots)\right] \tag{eq. 8}$

Conditional posterior $q(a_+ \mid a_0)$ is Gaussian (eq. 9):

$a_+^* = \left(I \otimes X'X + H(a_0)^{-1}\right)^{-1}\!\left[(I \otimes X'Y)a_0 + H(a_0)^{-1}\mu(a_0)\right]$

Marginal posterior $q(a_0)$ integrates out $a_+$ analytically (eq. 10), leaving a determinantal and quadratic form in $a_0$ that can be evaluated equation-by-equation.

Kronecker Symmetry and Computational Speedup (eq. 12)

The bottleneck in evaluating $q(a_0)$ is inverting or decomposing $(I \otimes X'X) + H(a_0)^{-1}$ , which is $(mk) \times (mk)$ . For $m = 20$ variables and $p = 6$ lags ( $k = 121$ ), this is a $2420 \times 2420$ problem — roughly $2420^3 \approx 10^{10}$ operations.

Preserving Kronecker product structure requires:

$H(a_0) = B \otimes G, \quad B = c\,I_m \tag{eq. 12}$

With equation-specific prior covariances $G_i$ (eq. 13) this becomes $\operatorname{diag}(G_1, \ldots, G_m)$ , and the combined term reduces to:

$(I \otimes X'X) + \operatorname{diag}(G_1^{-1}, \ldots, G_m^{-1}) = \operatorname{diag}(G_1^{-1} + X'X,\; \ldots,\; G_m^{-1} + X'X)$

Each block is $k \times k$ and can be decomposed independently. For $m = 20$ , $k = 121$ : twenty $121^3 \approx 1.77 \times 10^6$ operations versus one $2420^3 \approx 1.42 \times 10^{10}$ — a ~400× speedup in cubic-order computation. This is the central practical contribution of SZ98.

Random Walk Prior (Section 4.1)

The prior mean for lag coefficients encodes the random walk belief: $E[A_+ \mid A_0] = [A_0;\; 0;\; \ldots;\; 0]$ (eq. 16), i.e., the reduced-form companion matrix $B_1 = A_0^{-1} A_1 \approx I_m$ and higher lags shrink toward zero. The conditional standard deviation on lag $\ell$ of variable $j$ in equation $i$ is:

$\frac{\lambda_0 \lambda_1}{\sigma_j \ell^{\lambda_3}} \tag{eq. 17}$

where $\sigma_j$ is the residual std. dev. from a univariate AR fit to series $j$ . Key differences from Litterman (1986b): the scale factor is $\sigma_j$ alone (not the $\sigma_i/\sigma_j$ ratio of Litterman), and there is no separate own/other distinction in eq. 17 — cross-equation tightness enters through $\lambda_2$ in the full hyperparameter set $(\lambda_0, \lambda_1, \lambda_2, \lambda_3, \lambda_4)$ .

Component 1 — Coefficient Prior

Write the structural VAR in compact form:

$y_t'\, A_0 = x_t'\, F + \varepsilon_t', \quad x_t = [y_{t-1}',\ldots,y_{t-p}',z_t']'$

For $j = 1,\ldots,n$ , let $a_j$ be the $j$ th column of $A_0$ (contemporaneous) and $f_j$ the $j$ th column of $F$ (lag coefficients + deterministic). The prior is:

$a_j \sim \mathcal{N}(0,\, S)$

$f_j \mid a_j \sim \mathcal{N}(P\, a_j,\, H_j)$

where $P = [I_n,\, 0,\, \ldots,\, 0]^{n \times k}$ encodes the random walk belief: $f_j$ 's first $n$ components (the lag-1 coefficients) are centered at $a_j$ , i.e., $A_1 \approx A_0$ , or equivalently $B_1 = A_0^{-1} A_1 \approx I_n$ .

Hyperparameter roles:

Symbol	Controls
$\lambda_0$	Overall scale of $S$ and $H$ ; global tightness
$\lambda_1$	Tightness of random-walk belief on lagged coefficients
$\lambda_2$	Cross-variable influence ( $i \neq j$ shrunk relative to own lags)
$\lambda_3$	Lag-decay rate: coefficient on lag $\ell$ shrunk by $\ell^{-\lambda_3}$
$\lambda_4$	Tightness of prior on constant toward zero

The $i$ th diagonal element of $S$ is:

$S_{ii} = \frac{\lambda_0}{\sigma_i}$

where $\sigma_i$ is the residual standard deviation from a univariate AR fit to series $i$ (Litterman's scale normalization). The diagonal element of $H_j$ for the coefficient on lag $\ell$ of variable $i$ in equation $j$ reflects $\lambda_0$ , $\lambda_1$ , $\lambda_2$ , and the lag decay $\lambda_3$ .

Typical hyperparameter values:

Frequency	$\lambda_0$	$\lambda_1$	$\lambda_2$ – $\lambda_4$	$\mu_5$ , $\mu_6$
Quarterly	1.0	0.2	1.0	1.0
Monthly	0.6	0.1	1.0, -, 0.1	5.0

Explicit Standard Deviation Formulas (Minnesota Parametrization)

Meseguer (2010) states the prior standard deviations explicitly in the Minnesota parametrization. For equation $j$ , the prior standard deviation on the coefficient of lag $p$ of variable $i$ is:

Own lags ( $i = j$ ): $\sigma_{ijp} = \frac{\lambda_1}{p^{\lambda_3}}$

Cross lags ( $i \neq j$ ): $\sigma_{ijp} = \frac{\lambda_1 \,\lambda_2(i,j)}{p^{\lambda_3}} \cdot \frac{s_i}{s_j}$

Intercepts / deterministic components: $\sigma_i = \lambda_4 \, s_i$

where $s_i$ is the residual standard deviation from a univariate AR fit to series $i$ (Litterman's scale normalization). The ratio $s_i/s_j$ rescales cross-lag coefficients for the difference in units between series $i$ and $j$ .

Correlation-adaptive $\lambda_2$ (Meseguer 2010). Rather than setting $\lambda_2$ to a single scalar, Meseguer makes it a function of the empirical correlation between the two series:

$\lambda_2(i,j) = 0.8 \times \mathrm{Corr}[\Delta\log y_i,\, \Delta\log y_j]$

This is particularly effective in demographic applications where adjacent age groups have very high pairwise correlations — it automatically allows strong cross-age shrinkage where the data support it and enforces tighter shrinkage where they do not. The factor 0.8 caps the effective cross-variable weight below 1 even for perfectly correlated series, preserving some shrinkage.

Component 2 — Unit-Root / Cointegration Prior (Dummy Observations)

Adds $n+1$ dummy observations to the system, expressed as:

$Y_d\, A_0 = X_d\, F + E$

where $Y_d$ and $X_d$ are constructed from pre-sample means $\bar{y}_{0,i}$ (sample average of the $p$ initial conditions for variable $i$ ) and hyperparameters $\mu_5$ , $\mu_6$ . The first $n$ dummies express the belief that variables are stationary around their pre-sample means or that cointegration holds. The $(n+1)$ th dummy expresses the belief that the system's intercept is zero relative to the pre-sample level.

Larger values of $\mu_5$ and $\mu_6$ strengthen these beliefs. The dummy observations can be incorporated by augmenting the data matrix, so no change in the likelihood calculation is needed.

Dummy Observation Taxonomy (Sims-Zha 1998, Tables 1–3)

All three dummy types are implemented by augmenting $[Y, X]$ with artificial rows, so no modification of the estimation code is needed. Let $\bar{y}_{0i}$ denote the pre-sample mean of variable $i$ and $\sigma_r$ the univariate-AR residual std. dev. for variable $r$ .

Table 1 — Equation-specific dummies (one block of $k-1$ dummies per equation, hyperparameters $\mu_0\text{–}\mu_4$ ):

Artificial observation $(r, j)$ sets $y_{i}(r,j) = \mu_0 \mu_1 \mu_2^{\delta(i,j)} \sigma_r / \ell^{\mu_3}$ and the corresponding lag blocks in $x$ to the same value.
These are not symmetric across equations — each equation gets its own block — which is the origin of the $\mu_2$ cross-equation scaling.

Table 2 — Sums-of-coefficients (m system-wide dummies, hyperparameter $\mu_5$ ):

Artificial $y(i,j) = \mu_5 \bar{y}_{0i}$ for $i=j$ , 0 otherwise; artificial $x$ set similarly.
As $\mu_5 \to \infty$ : forces the sum of each variable's own lag coefficients to 1 and all cross-variable lag sums to 0, effectively differencing all data and eliminating cointegration.

Table 3 — Dummy initial observation (one system-wide dummy, hyperparameter $\mu_6$ ):

A single observation with $y(i,j) = \mu_6 \bar{y}_{0i}/m$ for all $i,j$ and $x$ set similarly.
As $\mu_6 \to \infty$ : imposes $(I - B(1))\bar{y}_0 + A_0^{-1}C = 0$ , the condition for the pre-sample mean to be a steady state — compatible with both unit roots and cointegration simultaneously.

The two cointegration hyperparameters $\mu_5$ and $\mu_6$ are independent and can be tuned separately: large $\mu_5$ alone aggressively differences the data; large $\mu_6$ alone allows cointegrated behavior around the pre-sample mean.

Prior on $A_0$ (Section 4.3)

For structural identification, $A_0$ is not triangular in general. The prior on $A_0$ encodes identifying restrictions (zero entries forced to zero, non-zero entries given informative priors). Under a triangular normalization within blocks of equations:

The reduced-form corresponds to $A_0$ lower-triangular, in which case the prior on $A_0$ is equivalent to a prior on $\Sigma^{-1}$ (the inverse error covariance).
The Wishart form is $|{\Sigma}^{-1}|^{(\nu-m-1)/2} \exp[-\tfrac{1}{2}\operatorname{tr}(S\Sigma^{-1})]$ (eq. 24).
The change-of-variable Jacobian from $\Sigma^{-1}$ to $A_0$ is $|{\partial\Sigma^{-1}}/{\partial A_0}| = 2^m \prod_j a_{jj}^j$ (eq. 25), which must be included when working in $A_0$ coordinates.

$A_+|A_0$ vs. $B|A_0$ Prior: Empirical Comparison (Section 5)

SZ98 demonstrate the two approaches on a 6-variable US quarterly model (interest rate $R$ , M1, output $y$ , price level $P$ , unemployment $U$ , investment $I$ ; 1948:1–1982:4, $p = 6$ ):

Approach	Prior on	Mode computation	MCMC draws	Posterior quality
$A_+\\|A_0$ (structural)	Lag coefficients given $A_0$	No mode needed	31 sec/1,000	68% bands cover data (Fig. 2)
$B\\|A_0$ (reduced-form)	Reduced form $B$ given $A_0$	2.8 hours	16 min/1,000	Unemployment bands miss data; inflation badly overpredicted (Fig. 3)

The $B|A_0$ prior (eq. 26: $H(a_0) = (A_0 \otimes I)\operatorname{diag}(\{G_i\})(A_0 \otimes I)'$ ) loses the Kronecker symmetry and implicitly imposes cross-equation restrictions that are not economically motivated. The $A_+|A_0$ approach exploits the block-diagonal decomposition and produces well-calibrated posteriors with dramatically lower computational cost.

Combined Prior

With both components, the overall prior retains the Gaussian form (Zha 2005, result (7)):

$a_j \sim \mathcal{N}(0,\, \bar{S}), \qquad f_j \mid a_j \sim \mathcal{N}(P a_j,\, \bar{H}_j)$

where $\bar{H}_j^{-1} = H_j^{-1} + X_d' X_d$ (the dummy observations tighten the prior on $F$ ). This closed-form structure is critical for enabling the Chib (1995) MDD algorithm via Gibbs sampling.

Canonical Application: 6-Variable Monthly Model

The standard policy application (Zha 1997/1998) uses: consumer price index (CPI), commodity prices (International Monetary Fund (IMF) index), federal funds rate, real gross domestic product (GDP) (Leeper-Sims-Zha monthly interpolation), M2, unemployment. Data begin 1959:1; lag length = 13 months. The Bayesian priors resolve three ordinary least squares (OLS) failure modes: (1) in-sample overfitting with near-perfect fit but poor out-of-sample performance; (2) erratic large coefficients from low degrees of freedom; (3) resulting explosiveness at long horizons. Regime-shift robustness: dropping the pre-1983 Volcker data worsens out-of-sample inflation forecasting, implying the prior successfully extracts information from that structural episode without being overwhelmed by it.

Why It Matters

Prevents overfitting in large VARs (up to 18 variables demonstrated in Leeper-Sims-Zha 1996) without sacrificing lag length.
Improves out-of-sample forecasting and produces robust impulse responses across identification schemes.
The closed-form Gaussian structure is the prerequisite for efficient Marginal Data Density computation.

Contrast with Noninformative Priors

The Sims-Zha prior is an informative prior — it encodes substantive economic beliefs (random walk, cointegration) via carefully chosen hyperparameters. An alternative approach uses noninformative priors that encode as little prior information as possible. The most common noninformative choice is the constant-Jeffreys prior: flat on $\Phi$ , Jeffreys ( $\propto |\Sigma|^{-(p+1)/2}$ ) on $\Sigma$ . Sun and Ni (2005) show this prior systematically over-estimates $\Sigma$ variances by a factor of $p/(T-(L+1)p-1)$ , which is substantial in typical macro VAR settings. The Yang-Berger (1994) reference prior $\pi_R(\Sigma) \propto |\Sigma|^{-1}\prod_{i<j}(d_i-d_j)^{-1}$ corrects this bias while remaining noninformative. See Noninformative Prior for VAR.

Open Questions

Hyperparameters are typically fixed by convention rather than estimated; hierarchical estimation is possible but rarely done in practice.
The prior may not be appropriate when structural breaks or nonlinearities are present — see Markov-Switching VAR.
Neither component addresses the unconditional mean (steady state) of the process. In standard form $\mu = \Pi^{-1}(1)Fc$ , the implied prior on $\mu$ is a complex non-linear function of $\Pi_1,\ldots,\Pi_k$ and is highly diffuse even when the coefficient prior is tight. Villani (2008) fills this gap via the Steady State VAR reparametrization.