Yamamoto (1981) Predictions of Multivariate Autoregressive-Moving Average Models

Summary

Yamamoto (1981) derives closed-form minimum mean squared error (MMSE) predictors for stationary invertible multivariate AutoRegressive Moving-Average (ARMA)$(p,q)$ processes and establishes a first-order asymptotic expression for prediction mean squared error (PMSE) under estimated parameters. The main result (Theorem 1) expresses h-step-ahead prediction weights as a recursive companion-form product involving eigenvalue matrices $A$ (AR companion, $mp \times mp$) and $B$ (MA companion, $mq \times mq$), yielding a compact predictor expressible entirely in terms of the current and past observable data. Theorem 2 shows that the asymptotic PMSE under estimated parameters exceeds the population optimum by an $O(n^{-1})$ correction that grows with horizon $h$, providing the frequentist counterpart to Chow (1973)'s Bayesian result.

Key Claims

• Theorem 1 (prediction weights): For a stationary invertible ARMA(p,q) process with AR companion matrix $A$ and MA companion matrix $B$, the h-step prediction weight for the k-th past innovation is
  $$\pi_{h,k} = H' A^{h-1} B^k (A - B) H$$
  where $H$ is a selector vector. The compact predictor is $\hat{x}_{n,h} = H' A^{h-1} P X_n$ with $P = (p_0, p_1, \ldots)$, $p_k = B^k(A-B)H$, $X_n = (x_n', x_{n-1}', \ldots)'$.
• Lemma 1 (AR/MA weight identification): AR weights satisfy $\mu_k = H'A^{k-1}(A-B)H$ and MA weights satisfy $\theta_k = -H'B^{k-1}(A-B)H$, linking the companion-form recursion directly to the ARMA parameters.
• Special cases: For a pure MA(s) process, $\pi_{h,k} = 0$ for $h > s+1$; prediction beyond that horizon is zero. For a pure AR(s), the predictor reduces to the standard companion-form recursion. The univariate ARMA(1,1) case yields an explicit closed-form asymptotic MSE (AMSE).
• Theorem 2 (asymptotic PMSE with estimated parameters): When parameters are estimated from $n$ observations,
  $$\text{AMSE}(\hat{x}_{n,h}) = \Omega_h + n^{-1} E(Y_n' U_h' V U_h Y_n)$$
  where $\Omega_h$ is the population MMSE, $V$ is the inverse Fisher information matrix for the ARMA parameters, and $U_h$ is the matrix of partial derivatives of the prediction weights with respect to parameters. The $O(n^{-1})$ correction grows with horizon $h$.
• Connection to Chow (1973): Both results show parameter uncertainty inflates multiperiod PMSE. Yamamoto (1981) gives the frequentist first-order asymptotic correction; Chow (1973) gives the exact Bayesian decomposition via $E[A^k \mid \text{data}] \neq (E[A \mid \text{data}])^k$. In large samples the two corrections are asymptotically equivalent.

Concepts Introduced or Extended

• Bayesian Multiperiod Prediction
• Vector Autoregression

Entities Mentioned

• Taku Yamamoto

Quotes

"The prediction weights can be expressed in terms of two companion matrices corresponding to the autoregressive and moving-average parts of the model." (p. 485)

My Take

A clean reference-class result: the companion-form recursion unifies AR, MA, and ARMA prediction in one formula. The $O(n^{-1})$ PMSE inflation result is the frequentist analogue of the Bayesian shrinkage motivation — it shows that parameter uncertainty degrades multiperiod forecast accuracy even under asymptotically efficient estimators, not only under flat priors. Worth citing alongside Chow (1973) whenever the claim is made that Bayesian priors improve long-horizon forecasts: Yamamoto shows the underlying phenomenon is not prior-choice-specific. The paper is purely technical; no empirical application is given.