Optimal Prediction: Compact Notation

Optimal Prediction: Compact Notation#

Using the fact that \(\epsilon_t\) in (296) is a serially uncorrelated vector process, it is straightforward to deduce from (297) that the projection of \(x_{t+\tau}\) against \(x_t\) is given by

(300)#\[P[x_{t+\tau}|x_t] = A^{\tau}x_t.\]

This is a compact formula for linear least squares predictors of a vector governed by a finite order stochastic difference equation. As an example illustrating the use of this formula, return to the portfolio balance example (285), which leads to a solution for the log of the price level of the form

(301)#\[p_t = (1 - \lambda) \sum_{j=0}^\infty \lambda^j P_t m_{t+j}\]

where \(\lambda = -\alpha/(1-\alpha)\), and where \(m_t\) is the log of the money supply. Suppose that \(m_t\) is the first element of a vector \(x_t\) that evolves according to \(x_t = A x_{t-1} + \epsilon_t\) where \(\epsilon_t\) is a vector white noise. Let \(e\) be the unit vector that validates our writing \(m_t = e x_t\). Then substituting (300) into the above solution for \(p_t\) gives

\[ p_t = (1 - \lambda) e\left(\sum_{j=0}^\infty \lambda^j A^j\right)x_t. \]

If the eigenvalues of \(A\) are bounded by \(1/\lambda\) in modulus,[1] then we have that \(\sum_{j=0}^\infty \lambda^j A^j = (I -\lambda A)^{-1}\). Therefore our solution can be represented

(302)#\[\begin{split}\begin{aligned} p_t &= (1 - \lambda)e(I - \lambda A)^{-1}x_t \\ x_t &= A x_{t-1} + \epsilon_t \end{aligned}\end{split}\]

Two comments about this derivation are in order. First, in the special case in which only lagged \(m\)’s appear in the \(x_{t-1}\) vector, the above formula is equivalent with formula (282). In fact formula (282) could be derived from the above one simply by explicitly inverting \((I - \lambda A)\).

Second, we notice from (302) that not only lagged \(m\)’s but also any other variables that appear in the vector \(x_t\) also enter the equation (302) for \(p_t\). Thus, any variables that help predict future \(m\)’s end up in the equation (302) expressing \(p_t\) as a function of current and lagged variables.