19. Prediction Formulas for Continuous Time Linear Rational Expectations Models

by Lars Peter Hansen and Thomas J. Sargent

Note

This chapter reprints a note by Lars Peter Hansen and Thomas J. Sargent. It keeps their first person and their section, equation, and footnote numbering. The sentences below that begin “Within this book” are editorial additions that tie the note to 12. Linear Least Squares Prediction and 16. Faster Methods for Solving Recursive Linear Models of Dynamic Economies; so is the reference list at the end.

In this note we derive optimal prediction formulas to be used in solving continuous time rational expectations models. In these derivations we employ Laplace transforms in a manner analogous to the use of \(z\) transforms for solving discrete time optimal prediction problems in Hansen and Sargent (1980a, Appendix A). The formulas are intended to play the same role for continuous time models that the discrete time formulas for optimal predictions of geometric distributed leads did in Hansen and Sargent (1980a). Within this book, they generalize the continuous time geometric-distributed-lead forecast of 12. Linear Least Squares Prediction from first-order Markov forcing to the full class of rational and nonstationary processes, making good the promise of equation (70) there. They supply the prediction calculus that 16. Faster Methods for Solving Recursive Linear Models of Dynamic Economies invokes to evaluate the optimal feedforward (117).

1. Convolutions and Prediction

Let \(L^1\) and \(L^2\) denote the spaces of all real-valued Borel measurable functions \(\phi\) on \(\mathbb{R}\) that are absolutely integrable and square integrable, respectively. Let \(W\) denote a random measure defined on \(\mathbb{R}\) with increments that are orthogonal and second-moment stationary. In other words,

(152)\[E\left[ W\{[t_2, t_1)\}^2 \right] = t_2 - t_1 \quad \text{for } t_2 > t_1,\]

and

(153)\[E\left[ W\{[t_4, t_3)\}\, W\{[t_2, t_1)\} \right] = 0 \quad \text{for } t_4 > t_3 > t_2 > t_1.\]

Here \(W\{[t_2, t_1)\}\) denotes the increment of \(W\) over the interval between \(t_1\) and \(t_2\); we follow the original in writing the larger endpoint first, so that (152) assigns that interval a variance equal to its length, and (153) says that increments over disjoint intervals are orthogonal.

Using functions in \(L^2\) and the random measure \(W\), we construct second-moment stationary processes as convolutions:

(154)\[x(t) = \int_{-\infty}^{+\infty} \phi(\tau)\, dW(t - \tau).\]

The stochastic integral in (154) can be interpreted as the limit point of a mean-square convergent sequence of random variables (e.g. see Rozanov 1967). Relation (154) gives a convenient mapping between the space \(L^2\) of functions and the space \(X\) of stochastic processes. It turns out that inner products on these two spaces coincide. More precisely, let \(\phi_1\) and \(\phi_2\) be any two functions in \(L^2\). An implication of (152) and (153), is

(155)\[\int_{-\infty}^{+\infty} \phi_1(\tau)\phi_2(\tau)\, d\tau = E\left[ x_1(t)\, x_2(t) \right]\]

where \(x_1\) and \(x_2\) are given by convolution (154) using \(\phi_1\) and \(\phi_2\) respectively.

Let \(L^2_+\) denote the subspace of \(L^2\) consisting of all functions that are zero on \((-\infty, 0)\) and let \(L^2_-\) denote the subspace of all functions that are zero on \([0, \infty)\). Clearly, \(L^2_+\) and \(L^2_-\) are orthogonal and \(L^2 = L^2_- \oplus L^2_+\). Any \(\phi\) in \(L^2\) can be decomposed uniquely into the sum of two functions \(\phi^+ \in L^2_+\) and \(\phi^- \in L^2_-\) via:[1]

(156)\[\begin{split}\begin{aligned} \phi^{+}(t) &\equiv \begin{cases} \phi(t) & t \geq 0 \\ 0 & t < 0 \end{cases} \\[4pt] \phi^{-}(t) &\equiv \begin{cases} 0 & t \geq 0 \\ \phi(t) & t < 0. \end{cases} \end{aligned}\end{split}\]

To formulate the prediction problems of interest, we use the random measure \(W\) to induce a family of information sets indexed by calendar time. Let \(H(t)\) denote the space of random variables \(x(t)\) given by (154) for \(\phi\)’s restricted to be in \(L^2_+\). It follows from (155) that since \(L^2_+\) is a Hilbert space, so is \(H(t)\). Furthermore, the family of Hilbert spaces \(\{H(t)\}\) is increasing in the sense that if \(t_2 > t_1\), then \(H(t_2) \supset H(t_1)\). Since \(H(t)\) is constructed using the random measure \(W\), the least squares projection operator \(P[\cdot \mid H(t)]\) onto the space \(H(t)\) is given by

(157)\[P\left[ \int_{-\infty}^{+\infty} \phi(\tau)\, dW(t-\tau) \;\Big|\; H(t) \right] = \int_{-\infty}^{+\infty} \phi^{+}(\tau)\, dW(t-\tau).\]

Hence the prediction process obtained by taking a process \(x \in X\) constructed as a convolution of \(\phi\) and \(dW\) and projecting it onto \(H(t)\) for each \(t\) is a convolution of \(\phi^+\) and \(dW\) for \(\phi^+\) given in (156).

2. Transforms

One convenient way to represent functions in \(L^2\) and characterize mapping (156) involves the use of transforms. For instance, Fourier transforms are valuable in characterizing the second moment properties of processes in \(X\). For any \(\phi\) in \(L^1 \cap L^2\), the Fourier transform of \(\phi\) is defined to be

(158)\[\mathcal{F}t(\phi)(\theta) \equiv \int_{-\infty}^{+\infty} \exp(-i\theta t)\, \phi(t)\, dt.\]

There is a well known extension of \(\mathcal{F}t\) from \(L^1 \cap L^2\) to \(L^2\). Using this extension, the spectral density function for \(x\) generated via (154) is just \(\lvert \mathcal{F}t(\phi) \rvert^2\).

To characterize the implied second moment properties of the solutions to prediction problems of the form (157), we use Laplace transforms. These transforms are defined as follows. For any \(\phi\) in \(L^2\) and any \(\rho\) in \(\mathbb{R}\) we construct a new function \(\exp(-\rho t)\phi\). This new function may or may not be in \(L^2\) depending on the value of \(\rho\). Whenever it is in \(L^2\), we define the Laplace transform to be:

(159)\[\mathcal{L}p(\phi)(\mathbf{c}) \equiv \mathcal{F}t\left[ \exp(-\rho t)\phi \right](\theta)\]

where \(\mathbf{c} \equiv \rho + i\theta\).

The question of interest is the following. Given the Laplace transform \(\mathcal{L}p(\phi)\) of a function \(\phi \in L^2\), how can we compute or characterize \(\mathcal{L}p(\phi^+)\) where \(\phi^+\) is defined in (156)? To answer this question, we first study Laplace transforms of functions \(\phi \in L^2_+\). For any such \(\phi\), \(\exp(-\rho t)\phi\) is also in \(L^2_+\) as long as \(\rho > 0\). Hence the Laplace transform \(\mathcal{L}p(\phi)(\mathbf{c})\) is well defined on the closed right plane \(\mathbf{C}^+_0\) where \(\mathbf{C}^+_\delta \equiv \{\mathbf{c} \in \mathbf{C} : \operatorname{real}(\mathbf{c}) \geq \delta\}\). Moreover, \(\mathcal{L}p(\phi)\) is analytic in the interior of \(\mathbf{C}^+_0\) (relative to \(\mathbf{C}\)). For \(\delta > 0\) and \(\mathbf{c} \in \mathbf{C}^+_\delta\),

(160)\[\begin{split}\begin{aligned} \lvert \mathcal{L}p(\phi)(\mathbf{c}) \rvert &\leq \int_0^\infty \lvert \phi(t) \rvert \exp(-\delta t)\, dt \\ &\leq \left[ \int_0^\infty \lvert \phi(t) \rvert^2\, dt \int_0^\infty \exp(-2\delta t)\, dt \right]^{1/2} \\ &\leq \left[ \int_0^\infty \lvert \phi(t) \rvert^2\, dt \,/\, 2\delta \right]^{1/2} \end{aligned}\end{split}\]

where the second inequality is an application of the familiar Cauchy–Schwarz inequality. The right side of (160) gives a uniform bound (in \(\mathbf{c}\)) on \(\mathcal{L}p(\phi)\) over the set \(\mathbf{C}_{\delta}^+\). This bound becomes arbitrarily small as \(\delta\) tends to plus infinity.

Consider next functions \(\phi \in L^2_-\). For any such \(\phi\), \(\mathcal{L}p(\phi)(\mathbf{c})\) is always well defined for \(\mathbf{c}\) in the left half plane \(\mathbf{C}_0^- \equiv \{\mathbf{c} \in \mathbf{C} : \operatorname{real}(\mathbf{c}) \leq 0\}\), and \(\mathcal{L}p(\phi)\) is analytic in the interior of that domain. Define \(\mathbf{C}_\delta^- \equiv \{\mathbf{c} \in \mathbf{C} : \operatorname{real}(\mathbf{c}) \leq \delta\}\). Mimicking the previous argument, it can be shown that for any \(\delta < 0\), \(\mathcal{L}p(\phi)\) is bounded on the domain \(\mathbf{C}_\delta^-\) and that the bound can be made arbitrarily small by driving \(\delta\) towards minus infinity.

For general functions \(\phi\) in \(L^2\), \(\mathcal{L}p(\phi)\) may only be defined on the imaginary axis, i.e. for \(\operatorname{real}(\mathbf{c}) = 0\). We are interested in a smaller class of functions, however. Let \(\Phi\) be the set of all functions \(\phi \in L^2\) such that \(\mathcal{L}p(\phi^-)\) is analytic in the interior of a region \(\mathbf{C}_{\delta}^-\) for some \(\delta > 0\). In this case \(\mathcal{L}p(\phi)\) is analytic in the interior of the strip \(\mathbf{C}_{\delta}^- \cap \mathbf{C}_0^+\). Furthermore, for any closed interval \(J \subset (0, \delta)\), \(\mathcal{L}p(\phi)\) is bounded on \(\{\mathbf{c} \in \mathbf{C} : \operatorname{real}(\mathbf{c}) \in J\}\). Define \(\mathcal{A}\) to be the collection of all Laplace transforms of functions \(\phi \in \Phi\).

The following result gives the decomposition for \(a \in \mathcal{A}\) corresponding to the decomposition \(\phi = \phi^+ + \phi^-\).

Lemma 1

For any \(a \in \mathcal{A}\) there is a unique decomposition \(a = a^+ + a^-\) where

(i) \(a^-\) is analytic in the interior of \(\mathbf{C}_\delta^-\), uniformly bounded on any closed half plane \(\mathbf{C}_\rho^-\) for \(\rho < \delta\) and

\[ \lim_{\rho \to -\infty} \max_{\mathbf{c} \in \mathbf{C}_{\rho}^{-}} \lvert a^{-}(\mathbf{c}) \rvert = 0\,; \]

(ii) \(a^+\) is analytic in the interior of \(\mathbf{C}_0^+\), uniformly bounded on any closed half plane \(\mathbf{C}_\rho^+\) for any \(\rho > 0\).

Proof. Functions \(a^-\) and \(a^+\) satisfying (i) and (ii) are obtained by letting \(a^- = \mathcal{L}p(\phi^-)\) and \(a^+ = \mathcal{L}p(\phi^+)\). To show that the decomposition is unique, we let \(a = b^+ + b^-\) be any other decomposition where \(b^-\) satisfies (i) and \(b^+\) satisfies (ii). Note that

\[ a^+ - b^+ = b^- - a^- \]

at least in the interior of the strip \(\mathbf{C}_{\delta}^- \cap \mathbf{C}_0^+\). Since \(a^+ - b^+\) is analytic in the interior of \(\mathbf{C}_0^+\) and \(b^- - a^-\) is analytic in the interior of \(\mathbf{C}_{\delta}^-\), \(b^- - a^-\) can be extended to be analytic on all of \(\mathbf{C}\). Furthermore, the uniform bounds on \(a^+ - b^+\) and \(b^- - a^-\) on overlapping half planes ensure that the extension of \(b^- - a^-\) is bounded as well. The only functions that are bounded and analytic on \(\mathbf{C}\) are constant. Since \(a^-\) and \(b^-\) satisfy (i) and the extension of \(b^- - a^-\) to \(\mathbf{C}\) is constant, \(a^+ - b^+\) must be identically zero.

Decompositions like that given in Lemma 1 apply to a much more general collection of analytic functions than the Laplace transforms of functions in \(L^2\). For instance, they also apply to Laplace transforms of generalized functions (e.g. see Beltrami and Wohlers 1966). However, these more general decompositions may not be unique. For instance, suppose we ignore the requirement

\[ \lim_{\rho \to -\infty} \max_{\mathbf{c} \in \mathbf{C}_{\rho}^{-}} \lvert a^{-}(\mathbf{c}) \rvert = 0 \]

in (i) of the Lemma. Then one can always add complex numbers to \(a^+\) and subtract the same numbers from \(a^-\) to obtain other decompositions of \(a\). If in addition, we ignore the bound restrictions in (i) and (ii) of the Lemma then one can add functions, such as polynomials, that are analytic in the entire complex plane to \(a^+\) and subtract them from \(a^-\) to obtain other decompositions of \(a\). Therefore in applying the Lemma to compute \(\mathcal{L}p(\phi^+)\), it is important to check whether the candidates for \(\mathcal{L}p(\phi^+)\) and \(\mathcal{L}p(\phi^-)\) satisfy the bounds restrictions in (i) and (ii).

3. Examples

We now apply the Lemma to obtain frequency domain characterizations of the solutions to prediction problems that occur in rational expectations models. These problems all have the following structure. Let \(\psi \in L^2_+\), and define \(y(t)\) by the convolution:

\[ y(t) = \int_0^{+\infty} \psi(\tau)\, dW(t - \tau). \]

Construct a new process by forming a forward-looking convolution using a function \(\gamma \in L^1\):[2]

(161)\[x(t) = \int_{-\infty}^{+\infty} \gamma(\tau)\, y(t-\tau)\, d\tau \equiv \int_{-\infty}^{+\infty} \phi(\tau)\, dW(t-\tau)\]

where \(\phi\) is given by the convolution:

\[ \phi(\tau) = \int_{-\infty}^{+\infty} \gamma(s)\, \psi(\tau - s)\, ds. \]

Applying the well known product representation for Fourier transforms of convolutions, we have that

\[ \mathcal{F}t(\phi) = \mathcal{F}t(\psi)\, \mathcal{F}t(\gamma). \]

This same result extends to Laplace transforms on the common domain of \(\mathcal{L}p(\phi)\) and \(\mathcal{L}p(\gamma)\). For the examples we consider, there will exist a \(\delta > 0\) such that \(\mathcal{L}p(\gamma)\) is defined on the interior of \(\mathbf{C}_{\delta}^-\). Hence on the interior of the strip \(\mathbf{C}_{\delta}^- \cap \mathbf{C}_{0}^+\),

(162)\[\mathcal{L}p(\phi) = \mathcal{L}p(\psi)\, \mathcal{L}p(\gamma).\]

We now investigate three related examples.

A geometric distributed lead

Example 1

Suppose that \(\gamma\) is given by

(163)\[\begin{split}\gamma(t) = \begin{cases} 0 & t \geq 0 \\ \exp(\delta t) & t < 0 \end{cases}.\end{split}\]

Then

\[ \mathcal{L}p(\gamma)(\mathbf{c}) = \int_{-\infty}^{0} \exp[(\delta - \mathbf{c})t]\, dt = 1/(\delta - \mathbf{c}) \]

for \(\operatorname{real}(\mathbf{c}) < \delta\). Thus

\[ \mathcal{L}p(\phi)(\mathbf{c}) = \mathcal{L}p(\psi)(\mathbf{c})/(\delta - \mathbf{c}). \]

Note that \(\mathcal{L}p(\phi)\) is analytic on \(\mathbf{C}\) except possibly at the point \(\delta\) where it may have a pole. If \(\mathcal{L}p(\psi)(\delta)\) is zero, the singularity at \(\delta\) is removable and \(\mathcal{L}p(\phi^+) = \mathcal{L}p(\phi)\). Usually \(\mathcal{L}p(\phi)\) will have a pole at \(\delta\), and to compute \(\mathcal{L}p(\phi^+)\) we must eliminate this pole. One candidate for \(\mathcal{L}p(\phi^+)\) is

\[ a^{+}(\mathbf{c}) = [\mathcal{L}p(\psi)(\delta) - \mathcal{L}p(\psi)(\mathbf{c})]/(\mathbf{c} - \delta). \]

Notice that the singularity of \(a^+\) at \(\delta\) is removable. The corresponding choice of \(a^-\) is

\[ a^{-}(\mathbf{c}) = a(\mathbf{c}) - a^{+}(\mathbf{c}) = \mathcal{L}p(\psi)(\delta)/(\delta - \mathbf{c}). \]

It is straightforward to show that \(a^+\) and \(a^-\) satisfy the requirements of the Lemma. Therefore,

(164)\[\mathcal{L}p(\phi^{+}) = [\mathcal{L}p(\psi)(\delta) - \mathcal{L}p(\psi)(\mathbf{c})]/(\mathbf{c} - \delta).\]

Formula (164) is the continuous time counterpart to formula (5) in Hansen and Sargent (1980a). It is the operator that evaluates the geometric distributed lead (66) of 12. Linear Least Squares Prediction and the optimal feedforward (117) of 16. Faster Methods for Solving Recursive Linear Models of Dynamic Economies. Indeed, the \(\gamma\) of (163) generates precisely the geometric distributed lead of 12. Linear Least Squares Prediction: substituting \(u = -\tau\) in (161) gives \(x(t) = \int_0^\infty e^{-\delta u} y(t+u)\, du\), so that \(\delta = -\rho\) in the notation there. With that substitution (164) reads \([-\tilde P(\mathbf{c}) + \tilde P(-\rho)]/(\mathbf{c} + \rho)\), which is exactly (70). 12. Linear Least Squares Prediction derived that formula only for a first-order \(\tilde P\) and then asserted it for general \(\tilde P\); the Lemma above is what establishes the general case, and in Example 3 below it is turned into a finite recursion.

A rational \(\gamma\)

Example 2

More generally, suppose

\[ \mathcal{L}p(\gamma)(\mathbf{c}) = p_n(\mathbf{c})/p_d(\mathbf{c}) \]

where \(p_n\) and \(p_d\) are finite-order polynomials with real coefficients. To ensure that \(p_n(\mathbf{c})/p_d(\mathbf{c})\) is the Laplace transform of a function in \(L^2_-\), we assume that the order of \(p_d\) exceeds the order of \(p_n\) and that the zeros of \(p_d\) are in the interior of \(\mathbf{C}_0^+\). In this case

\[ \mathcal{L}p(\phi)(\mathbf{c}) = \mathcal{L}p(\psi)(\mathbf{c})\, p_n(\mathbf{c})/p_d(\mathbf{c}), \]

which has poles in the interior of \(\mathbf{C}_0^+\) only at the zeros of \(p_d\). Let \(a_j\) denote the principal part of the Laurent series expansion of \(\mathcal{L}p(\phi)(\mathbf{c})\) at the \(j^{\text{th}}\) zero of \(p_d\). It follows from the partial fractions decomposition of a meromorphic function that

\[ a^{+} = \mathcal{L}p(\phi) - \sum_{j} a_{j} \]

is analytic in the interior of \(\mathbf{C}_0^+\). Furthermore, the principal parts, \(a_j\), are each sums of reciprocals of first and higher-order polynomials and hence satisfy

\[ \lim_{\rho \to -\infty} \max_{\mathbf{c} \in \mathbf{C}_{\rho}^{-}} \lvert a_{j}(\mathbf{c}) \rvert = 0 \]

for each \(j\). By construction,

\[ a^- = \sum_j a_j \]

satisfies (i) of the Lemma where \(\delta\) is the real part of the zero of \(p_d\) closest to the imaginary axis and is bounded on \(\mathbf{C}_{\rho}^-\) for any \(\rho < \delta\). Therefore, we have the following generalization of (164):

\[ \mathcal{L}p(\phi^{+}) = \mathcal{L}p(\phi) - \sum_{j} a_{j}. \]

A finite recursion for rational \(\psi\)

Example 3

Suppose that \(\gamma\) is given by (163), and \(\mathcal{L}p(\psi)\) is a rational function:

\[ \mathcal{L}p(\psi) = q_n/q_d \]

where \(q_n\) and \(q_d\) are polynomials with real coefficients. To guarantee that \(q_n/q_d\) is the Laplace transform of a function in \(L^2_+\), we assume that the order of \(q_d\) exceeds the order of \(q_n\) and that the zeros of \(q_d\) are in the interior of \(\mathbf{C}_0^-\). Solution (164) now becomes

\[ \mathcal{L}p(\phi)(\mathbf{c}) = q_n(\mathbf{c})/[q_d(\mathbf{c})(\delta - \mathbf{c})]. \]

From Example 1, we know that

(165)\[\begin{split}\begin{aligned} \mathcal{L}p(\phi^{+})(\mathbf{c}) &= [q_{n}(\delta)/q_{d}(\delta) - q_{n}(\mathbf{c})/q_{d}(\mathbf{c})]/(\mathbf{c} - \delta) \\ &= [q_{n}(\delta)q_{d}(\mathbf{c}) - q_{n}(\mathbf{c})q_{d}(\delta)]/[q_{d}(\mathbf{c})q_{d}(\delta)(\mathbf{c} - \delta)]. \end{aligned}\end{split}\]

The right side of (165) has a removable singularity at \(\delta\) by construction. This is evident because the polynomial \([q_n(\delta)q_d(\mathbf{c}) - q_n(\mathbf{c})q_d(\delta)]\) has a zero at \(\delta\). Canceling the common factor \((\mathbf{c} - \delta)\) in the numerator and denominator results in

\[ \mathcal{L}p(\phi^+)(\mathbf{c}) = q_n^+(\mathbf{c})/q_d^+(\mathbf{c}) \]

where

\[ q_d^+(\mathbf{c}) = q_d(\mathbf{c})q_d(\delta) \]

and \(q_n^+\) satisfies

(166)\[q_n^+(\mathbf{c})(\mathbf{c} - \delta) = [q_n(\delta)q_d(\mathbf{c}) - q_n(\mathbf{c})q_d(\delta)].\]

By equating coefficients of the polynomials on both sides of (166), one can construct a linear system of equations in the coefficients of \(q_n^+(\mathbf{c})\). In fact there is a recursive structure to this equation system that can be exploited as follows. Let \(\eta_j\) denote the coefficient on \(\mathbf{c}^j\) in \([q_n(\delta)q_d(\mathbf{c}) - q_n(\mathbf{c})q_d(\delta)]\) and let \(\epsilon_j\) denote the corresponding coefficient in \(q_n^+(\mathbf{c})\). Then

\[ -\delta\epsilon_0 = \eta_0 \]

and

\[ \epsilon_{j-1} - \delta \epsilon_j = \eta_j \quad \text{for } j \geq 1 \]

which can be solved recursively beginning with \(\epsilon_0\). The solution to this recursion gives a continuous time counterpart to formulas reported in Hansen and Sargent (1980a, 1981b) for autoregressive and autoregressive moving-average processes.

4. Vector Information Structures

Suppose that \(W\) is a \(k\)-dimensional vector random measure with second moment stationary increments. We now replace (152) and (153) with

\[ E\left[ W\{[t_2, t_1)\}\, W\{[t_2, t_1)\}' \right] = (t_2 - t_1)I_k \quad \text{for } t_2 > t_1, \]

and

\[ E\left[ W\{[t_4, t_3)\}\, W\{[t_2, t_1)\}' \right] = 0 \quad \text{for } t_4 > t_3 > t_2 > t_1 \]

where \(I_k\) is a \(k\)-dimensional identity matrix. Processes in \(X\) are now constructed using a \(k\)-dimensional vector \(\phi\) of functions in \(L^2\) via:

\[ x(t) = \int_{-\infty}^{+\infty} \phi(\tau) \cdot dW(t - \tau). \]

The analyses in Sections 2 and 3 extend by applying the decompositions to each of the \(k\) Laplace transforms of entries in \(\phi\). In Example 1 formula (164) still applies where \(\mathcal{L}p(\psi)\) is the vector of Laplace transforms of entries in \(\psi\). The recursions derived in Example 3 still apply where \(q_n\) is now a \(k\)-dimensional vector of polynomials, each with orders less than the scalar polynomial \(q_d\).

5. Nonstationarities

In Section 3, the assumption that \(\psi \in L_+^2\) guaranteed that process \(y\) is second moment stationary. Our analysis can be extended to a more general class of processes, however. To accommodate nonstationarities, it is most convenient to think of the underlying information process as starting at some initial time, say \(t = 0\). Hence we imagine (152) and (153) holding for nonnegative values of \(t_1\), \(t_2\), \(t_3\) and \(t_4\), and we assume that the random measure of any interval contained in \((-\infty, 0)\) is zero. This permits formula (161) to be well defined for a much larger class of functions \(\psi\). We might view the process \(y\) as being the deviation from a path that is perfectly predictable from time zero forward. We impose the weaker requirement that \(\exp(-\rho t)\psi\) be in \(L^2\) for strictly positive values of \(\rho\) which allows for polynomial growth in the second moment of \(y\).[3] The calculations in Example 1 through Example 3 still apply. In the case of Example 3, to accommodate polynomial growth we now allow \(q_d\) to have zeros on the imaginary axis of the complex plane \(\mathbf{C}\).

Notes

References

Beltrami, E. J., and M. R. Wohlers (1966). Distributions and the Boundary Values of Analytic Functions. New York: Academic Press.

Hansen, L. P., and T. J. Sargent (1980). Formulating and Estimating Dynamic Linear Rational Expectations Models. Journal of Economic Dynamics and Control, 2, 7–46.

Hansen, L. P., and T. J. Sargent (1981). Linear Rational Expectations Models for Dynamically Interrelated Variables, in R. E. Lucas, Jr., and T. J. Sargent, eds., Rational Expectations and Econometric Practice. Minneapolis: University of Minnesota Press.

Rozanov, Y. A. (1967). Stationary Random Processes. San Francisco: Holden-Day.

[1]

The uniqueness of this decomposition requires some qualification. Elements of \(L^2\) are only defined up to an equivalence of functions that are equal almost everywhere. Hence from the vantage point of \(L^2\), the construction of \(\phi^+\) and \(\phi^-\) at a particular point, say \(t = 0\), is inconsequential.

[2]

We take the right side of equation (161) as the definition of \(x(t)\). Alternatively, for particular classes of \(\gamma\) we could define \(x(t)\) using finite sum approximations for the middle integral.

[3]

In Example 1 and Example 3 it is also possible to allow for exponential growth in the second moments of \(y\) as long as \(\psi \exp(-\sigma t)\) is in \(L^2_+\) for some \(\sigma\) satisfying \(0 < \sigma < \delta\). The transform analysis now applies to the narrower strip \(\mathbf{C}^-_{\delta} \cap \mathbf{C}^+_{\sigma}\).