The Chain Rule of Forecasting

The law of iterated projections implies a recursion relationship that is sometimes very useful in a forecasting context. The relationship is known as Wold’s “chain rule of forecasting”. It shows how projections \(P_t x_{t+k}\) for all \(k \geq 2\) can be calculated from knowledge of the form of \(P_t x_{t+1}\) alone.[1]

Suppose that \(\{x_t\}\) is a linearly indeterministic covariance stationary stochastic process for which

\[ P_t x_{t+1} = \sum_{j=0}^\infty h_j x_{t-j}, \quad \sum_{j=0}^\infty h_j^2 < \infty \]

It follows that

\[ P_{t+k} x_{t+k+1} = h_0 x_{t+k} + h_1 x_{t+k-1} + \ldots + h_k x_t + h_{k+1} x_{t-1} + \ldots. \]

Projecting both sides of this equation on \((x_t, x_{t-1},\ldots)\) gives, via the law of iterated projections,

(284)\[P_t x_{t+k+1} = h_0 P_t x_{t+k} + h_1 P_t x_{t+k-1} + \ldots + h_{k-1} P_t x_{t+1} + \sum_{i=0}^{\infty} h_{k+i} x_{t - i}.\]

This recursion relationship is the “chain rule of forecasting” which shows how to build up projections of \(x_t\) arbitrarily far into the future from knowledge of the formula for the one-step-ahead projection alone.

To take an example, suppose that \(\{x_t\}\) is a first-order Markov process so that

\[ P_t x_{t+1} = \lambda x_t, \quad |\lambda| < 1. \]

From application of (284) it follows that \(P_t x_{t+j} = \lambda^j x_t\), \(j \geq 1\).

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[1]

Interesting applications of the chain rule of forecasting occur in Shiller (1972).