Chow (1973) Multiperiod Predictions from Stochastic Difference Equations by Bayesian Methods

Summary

Chow (1973) establishes that the optimal Bayesian $k$-step-ahead predictor from an autoregressive model is $E[A^k \mid \text{data}] \cdot y_t$, not the classical $\hat{A}^k y_t$. Since $E[A^k] \neq (E[A])^k$ for $k \geq 2$ when the posterior of $A$ has nonzero variance, the classical multiperiod predictor is generally suboptimal. The paper derives closed-form predictors for the univariate first-order autoregression (AR(1)) under a normal-gamma prior (§2) and extends the result to systems of stochastic difference equations under a normal-Wishart prior (§3).

Key Claims

• Classical $k$-step prediction (substitute $\hat{A}$, iterate) is suboptimal because it treats parameters as known.
• The Bayesian optimal predictor under squared-error loss is $\hat{y}_{T+k|T} = E[A^k \mid \text{data}] \cdot y_T$ (univariate) or $E[M^k \mid \text{data}] \cdot x_T$ (companion-form vector autoregression, VAR).
• For $k = 2$: $E[A^2 \mid \text{data}] = \bar{A}^2 + v$ where $v$ is the posterior variance of $A$ — the Bayesian 2-step predictor exceeds the classical one whenever $v > 0$.
• Under normal-Wishart priors, $E[M^k \mid \text{data}]$ is analytically tractable via moments of the matrix-$t$ posterior.
• The correction vanishes as $T \to \infty$; it is largest in small samples and at long horizons.

Concepts Introduced or Extended

• Bayesian Multiperiod Prediction
• Vector Autoregression

Entities Mentioned

• Gregory C. Chow

Quotes

"The predictor could not be computed by taking the mathematical expectation of $A$ and raising it to the $k$th power. The entire distribution of $A$ is required." (p. 111, paraphrase)

My Take

A clean theoretical paper that identifies a fundamental gap between classical and Bayesian multiperiod forecasting. The main practical implication — that posterior variance of parameters matters for multi-step predictions — is exactly why Minnesota-prior shrinkage helps: a tighter posterior means $E[A^k]$ is closer to $\bar{A}^k$, making the Bayesian VAR (BVAR) predictor more robust. The limitation is scope: the result assumes linear AR models with conjugate priors; nonlinear or time-varying models require different treatment. Chow (1984) extended the framework to random and changing-coefficient models.