Error Bands

varimpulse-responsebayesianuncertaintynormalizationbootstraplikelihood-principle

Definition

Error bands are probability intervals around estimated Impulse Response Functions (IRFs) that characterize posterior uncertainty. Two issues are central: (1) normalization of the contemporaneous matrix A0A_0 — which determines whether the posterior is unimodal — and (2) pointwise vs. joint bands — which determine whether shape uncertainty is correctly represented.

Key Ideas

How It Works

Pointwise Bayesian Error Bands

Given the Gibbs sampler (see Marginal Data Density), each draw {bj(m),gj(m)}j=1n\{b_j^{(m)}, g_j^{(m)}\}_{j=1}^n is transformed to structural parameters A0(m),A1(m),…,Ap(m)A_0^{(m)}, A_1^{(m)}, \ldots, A_p^{(m)}, from which impulse responses Φs(m)\Phi_s^{(m)} are computed via the recursion:

Φs(m)=∑ℓ=1pΦs−ℓ(m)Bℓ(m),Φ0(m)=(A0(m))−1\Phi_s^{(m)} = \sum_{\ell=1}^p \Phi_{s-\ell}^{(m)} B_\ell^{(m)}, \quad \Phi_0^{(m)} = (A_0^{(m)})^{-1}

The empirical distribution of each element Φs(m),(i,j)\Phi_s^{(m),(i,j)} yields the equal-tail 68%68\% and 90%90\% probability bands. These are exact small-sample Bayesian credible intervals.

The Normalization Problem

Because the structural form is invariant to sign-flipping columns of A0A_0 (replacing aj→−aja_j \to -a_j and εjt→−εjt\varepsilon_{jt} \to -\varepsilon_{jt} leaves the likelihood unchanged), the posterior is bimodal in the direction of each aja_j unless a normalization rule is imposed.

Conventional normalization: keep the diagonal of A0A_0 positive. This works for recursive VARs but fails for non-recursive ones: if a diagonal element [A0]jj[A_0]_{jj} is not significantly different from zero, the two modes overlap in the interior, creating a spuriously wide and flat posterior.

Waggoner-Zha (2003b) normalization: for each posterior draw aja_j, compute:

sign(A^jj−1aj)\text{sign}(\hat{A}^{-1}_{jj} a_j)

where A^\hat{A} is the posterior mode (or any fixed reference point). Keep aja_j if ej′A^−1aj>0e_j' \hat{A}^{-1} a_j > 0; otherwise replace aj←−aja_j \leftarrow -a_j. Formally:

ajnorm=sign(ej′A^−1aj)⋅aja_j^{\text{norm}} = \text{sign}(e_j' \hat{A}^{-1} a_j) \cdot a_j

This rule selects the mode nearest to A^\hat{A}, preventing the normalized posterior from being multi-modal. It applies to both short-run and long-run restrictions.

Joint Error Bands (Sims-Zha 1999)

Pointwise bands treat Φs(i,j)\Phi_s^{(i,j)} as independent across ss and ii, ignoring that:

Procedure:

  1. Stack the relevant impulse responses into a column vector c~(m)\tilde{c}^{(m)} for each draw mm.
  2. Compute the posterior mean cˉ\bar{c} and covariance Ω=Cov(c~)\Omega = \text{Cov}(\tilde{c}).
  3. Eigendecompose: Ω=WΛW′\Omega = W \Lambda W', where wkw_k is the kkth eigenvector (ordered by λk\lambda_k descending).
  4. For each draw, project onto the kkth principal direction:

γk(m)=(c~(m)−cˉ)′wk\gamma_k^{(m)} = (\tilde{c}^{(m)} - \bar{c})' w_k

  1. Tabulate quantiles: γk,0.16\gamma_{k,0.16} and γk,0.84\gamma_{k,0.84} (for 68% bands).
  2. The kkth joint error band:

ck,α=cˉ+γk,α wkc_{k,\alpha} = \bar{c} + \gamma_{k,\alpha}\, w_k

If only one or two eigenvalues dominate Ω\Omega, a small set of "connecting-dots" paths (parameterized by α\alpha) captures the majority of posterior shape uncertainty. This method is particularly informative when the entire path of responses matters (e.g., characterizing whether a policy shock is followed by a hump-shaped output response or a monotone one).

Empirical Application: Inflation and GDP Forecasting (Zha 1997/1998)

The 6-variable monthly Bayesian VAR (BVAR) (Consumer Price Index (CPI), commodity prices, federal funds rate, real gross domestic product (GDP), M2, unemployment; 1959:1 start; 13-month lags) from Zha (1997/1998) illustrates error bands in a policy setting:

The joint error region generalizes the univariate error band by capturing the covariance structure across forecast variables — e.g., whether high GDP growth and low inflation are jointly plausible given the model's estimated dynamics.

Likelihood Principle and Bootstrap Critique (Sims-Zha 1999)

Sims and Zha (1999) make the conceptual case that classical confidence intervals (CIs) for impulse responses should be replaced by Bayesian posterior probability bands. The argument rests on the likelihood principle: inference should depend only on the likelihood function evaluated at the observed data. Classical CIs are constructed by inverting test statistics — they report the set of parameter values that a test would fail to reject — which mixes information about where the parameters are likely with information about how well the model fits. The Bayesian posterior, by contrast, concentrates directly on the plausible parameter region given the data.

Practical consequences:

Why It Matters

Misreported or misinterpreted error bands are a common source of overconfidence in VAR results. The two fixes here — normalization and joint bands — are often overlooked in applied work, but they can substantially change the qualitative conclusions about whether responses are "precisely estimated."

Open Questions

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