Predicting Geometric Distributed Leads

It is important to know the solution of the following problem in order to use a variety of linear rational expectations models. Let \(x_t\) be a covariance stationary stochastic process with Wold moving average representation

(274)\[x_t = c(L)\epsilon_t, \quad c_0 = 1\]

where \(\epsilon_t\) is a fundamental white noise for \(x\) and \(c(L) = \sum_{j=0}^\infty c_j L^j\) is square summable. We further assume that \(c(L)\) has an inverse \(a(L) = c(L)^{-1}\) which is one-sided in nonnegative powers of \(L\) and square summable. Thus, \(x_t\) has the autoregressive representation

(275)\[a(L)x_t = \epsilon_t\]

where \(a(L) = 1 - a_1 L - a_2 L^2 - \cdots\).

We want to calculate the following linear projection

(276)\[y_t = P\left[\sum_{j=0}^\infty \lambda^j x_{t+j}\Big|x_t, x_{t-1},\ldots\right] \equiv P_t\sum_{j=0}^\infty \lambda^j x_{t+j}\]

where \(|\lambda|<1\). Projections of such geometric distributed leads occur in a variety of linear rational expectations models. We begin by noting that \(y_t\) defined by (276) satisfies the stochastic difference equation

(277)\[y_t = \lambda P_t y_{t+1} + x_t.\]

That is, \(y_t\) is the stationary solution of the difference equation (277), as can be verified by repeated substitution in (277). We seek expressions for \(y_t\) of the forms

(278)\[y_t = g(L)x_t\]

and

(279)\[y_t = d(L)\epsilon_t\]

where

\[ d(L) = \sum_{j=0}^\infty d_j L^j,\quad g(L) = \sum_{j=0}^\infty g_j L^j,\quad \sum_{j=0}^\infty g_j^2 < +\infty,\quad \sum_{j=0}^\infty d_j^2 < + \infty \]

We know that representation (278) exists by definition, and therefore that \(d(L) = g(L) c(L)\) also exists. That is, a representation of the form (279) exists because \(\{x_t, x_{t-1}, \ldots\}\) and \(\{\epsilon_t, \epsilon_{t-1},\ldots\}\) span the same space.

We shall solve for \(d(L)\) using (277) and prediction theory. Using (279), we have that

\[ P_t y_{t+1} = \left[\frac{d(L)}{L}\right]_{+}\epsilon_t \]

or

\[ P_t y_{t+1} = \left[\frac{d(L)}{L} - \frac{d_0}{L}\right]\epsilon_t. \]

Substituting this and (274) and (279) into (277) gives

\[ d(L)\epsilon_t = \lambda\left[\frac{d(L)}{L} - \frac{d_0}{L}\right]\epsilon_t + c(L)\epsilon_t. \]

Since this equation holds for all \(\epsilon_t\) realizations, it implies, after rearranging, that

\[ (1 - \lambda L^{-1})d(L) = c(L) - \lambda d_0 L^{-1} \]

an equation that we desire to solve for \(d(L)\) as a function of \(c(L)\). We determine \(d_0\) by evaluating the above equation at \(L = \lambda\), to get \(c(\lambda) = d_0\). Using this value for \(d_0\) gives

(280)\[d(L) = \frac{c(L) - \lambda c(\lambda) L^{-1}}{1 - \lambda L^{-1}}\]

Using \(g(L) = d(L)c(L)^{-1}\) and \(c(L)^{-1} = a(L)\), we get

(281)\[g(L) = \frac{1 - \lambda a(\lambda)^{-1} a(L) L^{-1}}{1 - \lambda L^{-1}}.\]

For the case in which \(a(L)\) is an \(r\)th order polynomial \(a(L) = 1 - \sum_{j=1}^r a_j L^j\), Hansen and Sargent (1980) show using polynomial long division that (281) can be expressed

(282)\[g(L) = a(\lambda)^{-1}\left[1 + \sum_{j=1}^{r-1}\left(\sum_{k=j+1}^r \lambda^{k-j} a_k\right)L^j\right]\]

so that

\[ g(L) = \sum_{j=0}^{r-1} g_j L^j, \]

with

\[ g_0 = a(\lambda)^{-1},\quad g_j = a(\lambda)^{-1} \sum_{k=j+1}^r \lambda^{k-j} a_k \]

for \(j = 1,\ldots, r-1\). Evidently, the coefficients \(g_j\) can be computed recursively via the following formulas:

(283)\[\begin{split}\begin{aligned} g_0 &= a(\lambda)^{-1} \\ g_r &= 0 \\ g_{j-1} &= \lambda g_j + \lambda g_0 a_j \quad j=r,r-1, \ldots, 2. \end{aligned}\end{split}\]

Various versions of formulas (280), (281), and (282) were originally derived in papers by Saracoglu and Sargent (1978), Hansen and Sargent (1980), and Futia (1981).

See also

Formula (282) is used repeatedly in what follows. Some Applications to Rational Expectations Models uses it to solve Cagan’s model and a supply and demand model with inventories, and to derive their cross-equation restrictions. Optimal Prediction: Compact Notation recovers it in compact state space notation. Seasonal Adjustment and Forecasting Geometric Distributed Leads applies it to seasonally adjusted data and shows how the implied restrictions are distorted, and Seasonality and Approximation Errors asks whether the distortion matters. Bubbles uses it to isolate the bubble term. Exact Linear Rational Expectations Models embeds it in an exact linear rational expectations model, and Money Demand in Hyperinflations: A Misspecified Regression and Sims’s Approximation Formula uses it to derive an equilibrium money rule. Chapter XIV uses it to build a rational expectations equilibrium of investment under uncertainty.