Bayesian Multiperiod Prediction

Definition

Bayesian multiperiod prediction is the formation of $k$ -step-ahead forecasts from autoregressive (AR) models by integrating over the posterior distribution of the parameters, rather than substituting a point estimate and iterating. The key result (Chow 1973) is that the optimal Bayesian predictor uses $E[A^k \mid \text{data}]$ — the expected $k$ th power of the coefficient matrix under the posterior — which is not equal to $(E[A \mid \text{data}])^k$ , the $k$ th power of the posterior mean.

Key Ideas

The classical $k$ -step predictor substitutes $\hat{A}$ and iterates: $\hat{y}_{t+k} = \hat{A}^k y_t$ . This is suboptimal because it treats $A$ as known.
The Bayesian optimal predictor (minimum mean-squared-error) is $E[y_{t+k} \mid y_1,\ldots,y_t]$ , which requires integrating over $p(A \mid \text{data})$ .
For $k \geq 2$ : $E[A^k \mid \text{data}] \neq (E[A \mid \text{data}])^k$ . The gap is largest in small samples and at long horizons; it vanishes as $T \to \infty$ at rate $O(T^{-1})$ .
Under a normal-gamma (univariate) or normal-Wishart (multivariate) prior, $E[A^k \mid \text{data}]$ is analytically tractable via moments of the (matrix-) $t$ posterior.

How It Works

Univariate AR(1) (Chow 1973, §2)

Let $y_t = A y_{t-1} + \varepsilon_t$ with $\varepsilon_t \sim \mathcal{N}(0, \sigma^2)$ and a normal-gamma prior on $(A, \sigma^2)$ . The posterior $p(A \mid y_1,\ldots,y_T)$ is a univariate $t$ -distribution with posterior mean $\bar{A}$ and posterior variance $v$ .

The optimal $k$ -step predictor is:

$\hat{y}_{T+k|T} = E[A^k \mid \text{data}] \cdot y_T$

$k = 1$ : $E[A \mid \text{data}] = \bar{A}$ — Bayesian and classical predictors coincide.
$k = 2$ : $E[A^2 \mid \text{data}] = \bar{A}^2 + v$ — the Bayesian predictor exceeds the classical $\bar{A}^2 y_T$ by $v \cdot y_T$ .
General $k$ : $E[A^k]$ depends on all even moments of the posterior; in the $t$ case these have closed forms.

Companion-Form VAR (Chow 1973, §3)

Write the vector autoregression (VAR) in companion form: $z_t = M z_{t-1} + u_t$ where $M$ is the $nk \times nk$ companion matrix. The $k$ -step predictor is:

$\hat{z}_{T+k|T} = E[M^k \mid \text{data}] \cdot z_T$

Under a normal-Wishart prior, the posterior of $M$ is matrix- $t$ , and $E[M^k \mid \text{data}]$ is computable from its moments. The result generalizes: the matrix analogue of the univariate $E[A^2] = E[A]^2 + \text{Var}(A)$ includes cross-moment terms from all entries of $M$ .

Diffuse Prior Limit

Under a diffuse prior and large $T$ , the posterior of $M$ concentrates around the ordinary least squares (OLS) estimate $\hat{M}$ , and $E[M^k \mid \text{data}] \to \hat{M}^k$ . Classical and Bayesian predictors converge. The correction for parameter uncertainty is $O(T^{-1})$ .

Why It Matters

Provides the theoretical justification for why informative priors improve multiperiod forecast accuracy: tighter posteriors reduce the wedge between $E[A^k]$ and $\hat{A}^k$ .
Explains why Bayesian VAR (BVAR) improvements are larger at longer horizons: the $E[A^k] - \hat{A}^k$ gap grows with $k$ but shrinks when the posterior is tight.
Foundational for the Bayesian VAR forecasting tradition: Litterman (1986b) operationalized Bayesian prediction via the Minnesota Prior; Chow (1973) established why it is the theoretically correct approach.
In practice, $E[M^k]$ is computed via Markov chain Monte Carlo (MCMC) integration over posterior draws rather than analytically — the theoretical result motivates the practice.

Frequentist Baseline: Yamamoto (1981)

Yamamoto (1981) establishes the frequentist counterpart to Chow's result for multivariate AutoRegressive Moving Average (ARMA(p,q)) models. He derives the closed-form Minimum Mean-Squared Error (MMSE) predictor via a companion-form recursion: denoting the AR companion matrix $A$ (of dimension $mp \times mp$ ) and the MA companion matrix $B$ ( $mq \times mq$ ), the $h$ -step prediction weight for the $k$ -th past innovation is

$\pi_{h,k} = H' A^{h-1} B^k (A - B) H$

with compact predictor $\hat{x}_{n,h} = H' A^{h-1} P X_n$ . The key Theorem 2 result shows that when parameters are estimated from $n$ observations, the asymptotic Prediction Mean Squared Error (PMSE) is

$\text{AMSE}(\hat{x}_{n,h}) = \Omega_h + n^{-1} E(Y_n' U_h' V U_h Y_n)$

where AMSE denotes the asymptotic PMSE, $V$ is the inverse Fisher information, and $U_h$ is the gradient of the prediction weights. This $O(n^{-1})$ inflation term is the frequentist analogue of the Bayesian gap $E[M^k \mid \text{data}] - \hat{M}^k$ : both grow with horizon $h$ and shrink with sample size $n$ . In large samples the two corrections are asymptotically equivalent. The Yamamoto result shows that the multiperiod PMSE inflation is not a prior-choice artefact but a fundamental statistical phenomenon under any asymptotically efficient estimator.

Open Questions

Computing $E[M^k]$ analytically is expensive for large $n$ and $k$ ; MCMC posterior integration is the standard workaround.
Appropriate loss function: squared-error loss motivates the posterior mean predictor, but other losses may favor different predictors.