---
jupytext:
  text_representation:
    extension: .md
    format_name: myst
    format_version: 0.13
    jupytext_version: 1.11.1
kernelspec:
  display_name: Python 3
  language: python
  name: python3
---

# 20. Aggregation Over Time and the Inverse Optimal Predictor Problem for Adaptive Expectations in Continuous Time

```{eval-rst}
.. index::
   single: inverse optimal predictor problem
   single: adaptive expectations; optimality of
   single: Cagan model; aggregation over time
   single: Granger causality; induced by aggregation
```

```{note}
This chapter reports and paraphrases the analysis of Lars Peter Hansen and Thomas J. Sargent,
"Aggregation Over Time and the Inverse Optimal Predictor Problem for Adaptive Expectations in
Continuous Time," *International Economic Review* **24** (1983), no. 1, pp. 1–20. We retell the
argument in the third person, writing "Hansen and Sargent show" and "the authors now assume,"
and we preserve the original section, equation, and table numbering. We recompute the four
tables of the original paper in Python, and the code adds figures that are not in the original.
The recomputation reproduces the published tables to all printed digits, except for three
entries of Table 2 that are typographical errors in the 1983 original. {ref}`sec-20-5` flags
them.
```

## 1. Introduction

In 1956 Milton Friedman (1957) and Phillip Cagan (1956) formulated and applied the adaptive
expectations hypothesis. Shortly thereafter, John F. Muth (1960) solved the following
"inverse optimal predictor" problem: for what discrete-time, *univariate* stochastic process
is the discrete-time version of the adaptive expectations mechanism optimal in the sense of
delivering linear least squares forecasts? Much later Sargent (1977) solved the following
extended inverse optimal predictor problem: in the context of a discrete-time version of
Cagan's model of portfolio balance, for what *bivariate* money creation, inflation
stochastic process does a discrete-time version of adaptive expectations deliver linear
least squares forecasts for inflation?

Hansen and Sargent solve the continuous-time version of both of these inverse optimal predictor
problems. The continuous-time adaptive expectations scheme itself was derived as an *optimal*
forecast in {doc}`14_nonstationary_examples` (equation {eq}`eq-14-2` there); this chapter takes
that scheme as its starting point and asks what aggregation over time does to it. In the context
of a continuous-time version of Cagan's portfolio schedule, they find
the continuous-time, generalized stochastic process for the money supply and price level that
makes the adaptive expectations mechanism yield linear least squares forecasts for inflation.
The problem is of interest if only because Cagan himself formulated his model in continuous
time, as have many others, even though he ultimately estimated an approximating discrete-time
model. To pin down the "optimal" discrete-time approximating model, the authors deduce the
discrete-time process for point-in-time observations on the money supply and price level
implied by the continuous-time model that makes adaptive expectations rational. This makes
precise a sense in which the discrete-time adaptive expectations scheme approximates a model in
which agents form adaptive expectations optimally in continuous time. Hansen and Sargent also
derive an exact formula linking the discrete-time adaptive expectations decay parameter
$\lambda$ to the continuous-time decay parameter $\beta$, and compare it to the approximation
$\lambda = \exp(-\beta)$ used by Cagan.

The continuous-time process for inflation and money creation that makes adaptive expectations
optimal has, *ipso facto*, the property that money creation fails to Granger (1969) cause
inflation in continuous time. Yet for discrete-time samples drawn from that same process, money
creation *does* Granger cause inflation. This is an instance of how aggregation over time
interrupts Granger non-causality patterns that hold in continuous time. Sims (1971, 1973)
studied that phenomenon, {doc}`18_time_aggregation_var` takes it up systematically, and the
spectral folding of {doc}`17_discrete_sampling_folding` is its root. The model is simple enough that
Hansen and Sargent can analyze the effect quite completely.

The calculations are interesting for their own sake, and also because they illustrate a method
for analyzing the effects of aggregation over time that applies to a wide variety of linear
rational expectations models.

```{eval-rst}
.. index::
   single: inverse optimal predictor problem; continuous time
   single: Cagan's adaptive expectations; continuous time version
   single: Muth, J. F.
   single: money creation and inflation
   single: exponentially weighted forecast
```

## 2. The continuous-time inverse optimal predictor problem

Hansen and Sargent begin with Cagan's portfolio balance schedule in continuous time

```{math}
:label: eq-20-1
m(t) - p(t) = \alpha\, D^{+} \hat E_t\, p(t) + a(t), \qquad \alpha < 0
```

where $p(t)$ is the logarithm of the price level, $m(t)$ is the logarithm of the money supply,
$a(t)$ is a random disturbance to the portfolio balance schedule, $D^{+}$ is the mean square
right time derivative operator, and $\hat E_t$ is the linear least squares projection operator
onto an information set that includes at least current and past observations on $p$, $m$, and
$a$. The authors do not require that the $p$ process be mean square differentiable, but only
that $\{\hat E_t\, p(t+v): v \geq 0\}$ be mean square differentiable for $v > 0$ and have a mean
square right derivative at $v = 0$. This right derivative is denoted $D^{+}\hat E_t\, p_t$.

To solve stochastic differential equation {eq}`eq-20-1`, Hansen and Sargent write it shifted
ahead $v$ time units as

```{math}
:label: eq-20-2
m(t+v) - p(t+v) = \alpha\, D^{+}\hat E_{t+v}\, p(t+v) + a(t+v).
```

Projecting both sides of {eq}`eq-20-2` onto the $t$ period information set gives

```{math}
:label: eq-20-3
\hat E_t\, m(t+v) - \hat E_t\, p(t+v) = \alpha D\, \hat E_t\, p(t+v) + \hat E_t\, a(t+v)
```

where $D$ is the time derivative operator. The realizable, time invariant solution to
{eq}`eq-20-3` is just

$$
\hat E_t\, p(t+v) = \rho\, \hat E_t \int_0^\infty e^{\rho u}[a(t+v+u) - m(t+v+u)]\, du
$$

where $\rho = 1/\alpha < 0$. Taking limits as $v$ declines to zero, and noting that
$\lim_{v \downarrow 0}\hat E_t\, p(t+v) = \hat E_t\, p(t) = p(t)$, one obtains

```{math}
:label: eq-20-4
p(t) = \rho\, \hat E_t \int_0^\infty e^{\rho u}[a(t+u) - m(t+u)]\, du
```

as the solution to {eq}`eq-20-1`. (Dividing {eq}`eq-20-3` by $\alpha$ puts it in the form
$(D + \rho)\hat E_t p(t+v) = \rho\,[\hat E_t m(t+v) - \hat E_t a(t+v)]$; since $\rho < 0$ the
root $-\rho$ lies in the right half plane, so the bounded solution is the forward one displayed
above. As a check, if $m$ and $a$ are constant there is no expected inflation and
{eq}`eq-20-1` requires $p = m - a$, which is what {eq}`eq-20-4` delivers, since
$\rho\int_0^\infty e^{\rho u}\, du = -1$.)

Hansen and Sargent now specialize the assumptions so that Cagan's adaptive expectations
mechanism is optimal: they seek specifications for $a$ and $m$ that, together with
{eq}`eq-20-4`, imply that

```{math}
:label: eq-20-5
D\hat E_t\, p(t+v) = \beta \int_{-\infty}^t e^{-\beta(t-u)} Dp(u)\, du, \qquad \beta > 0,\ v > 0.
```

In expression {eq}`eq-20-5` $Dp$ is not necessarily required to be an ordinary stochastic
process but rather can be a generalized stochastic process so long as the integral on the
right-hand side is well defined. On the other hand, $\{D\hat E_t\, p(t+v): v > 0\}$ is assumed
to be an ordinary stochastic process. Thus, even though inflation itself may not be physically
realizable, the authors assume that anticipated inflation is. Equation {eq}`eq-20-5` also
implies that at each instant inflation is expected to be constant over the entire future, since
its right-hand side does not depend on $v$.

The authors assume that the joint process $x$ given by

$$
x(t) = \begin{bmatrix} p(t) \\ m(t) \\ a(t) \end{bmatrix}
$$

has a time invariant Wold representation

```{math}
:label: eq-20-6
x(t) = c(D)\, w(t).
```

In equation {eq}`eq-20-6` $c(D)$ is a one-sided matrix convolution operator and $w$ is a
continuous-time white noise vector with $Ew(t) = 0$ and

$$
Ew(t) w(t-v)' = I\, \delta(t-v)
$$

where $\delta$ is the Dirac delta generalized function. They assume that {eq}`eq-20-6` holds for
some $t$ greater than or equal to a start-up time $T$, with $w(t) = 0$ for $t < T$. That
{eq}`eq-20-6` is a Wold representation means that instantaneous forecast errors in predicting an
element of $x(t)$ from past $x$'s are linear combinations of the elements of $w(t)$.

Hansen and Sargent write the first row of {eq}`eq-20-6` as

```{math}
:label: eq-20-7
p(t) = c_1(D)\, w(t).
```

The operator that shifts a time subscript $v$ units ahead can be represented as $e^{vD}$.
Shifting {eq}`eq-20-7` forward $v$ time units and taking expectations, they find

$$
\hat E_t\, p(t+v) = [c_1(D)\, e^{vD}]_+\, w(t)
$$

where $[\ ]_+$ is the annihilation operator that instructs one to ignore portions of the
convolution operator that are concentrated on the negative numbers. Equation {eq}`eq-20-5`
implies that

$$
[Dc_1(D)\, e^{vD}]_+ = \frac{\beta D c_1(D)}{\beta + D}
$$

for all $v > 0$. It is verified in the appendix that the solution to this operator equation is

$$
c_1(D) = \frac{D + \beta}{D^2}\, k_0
$$

where $k_0$ is an arbitrary row vector of constants. The authors are free to normalize $c$ and
$w$ so that

```{math}
:label: eq-20-8
p(t) = \frac{(D + \beta)}{D^2}\, k_1\, w_1(t)
```

where $w_1$ is the first element of $w$ and $k_1$ is an arbitrary scalar constant.
Specification {eq}`eq-20-8` implies that

```{math}
:label: eq-20-9
D^{+}\hat E_t\, p(t+v) = \frac{\beta}{D}\, k_1\, w_1(t),
```

as one verifies by evaluating the annihilation operator of the appendix: with
$c_1(D) = (D+\beta)k_1/D^2$,

$$
[\,D c_1(D)\, e^{vD}\,]_+ = \frac{\beta D\, c_1(D)}{\beta + D}
= \frac{\beta D (D + \beta) k_1}{(\beta + D) D^2} = \frac{\beta k_1}{D}.
$$

Substituting {eq}`eq-20-9` into equation {eq}`eq-20-1` shows that

```{math}
:label: eq-20-10
m(t) - p(t) = \frac{\alpha \beta D}{(\beta + D)}\, p(t) + a(t).
```

Equation {eq}`eq-20-10` is a version of Cagan's model in continuous time, since

$$
\frac{\beta D}{\beta + D}\, p(t) = \beta \int_{-\infty}^t e^{-\beta(t-u)} Dp(u)\, du,
$$

which is the right-hand side of the adaptive expectations scheme {eq}`eq-20-5`.

Equation {eq}`eq-20-10` reveals an exact relationship among $m$, $p$, and $a$. This singularity
means that the white noise vector $w$ can have at most two elements. The authors assume that the
$x$ process has maximal rank, so that $w$ has exactly two elements. Partitioning $w$ and $c$,
they write

```{math}
:label: eq-20-11
\begin{bmatrix} p(t) \\ m(t) \\ a(t) \end{bmatrix}
= \begin{bmatrix} c_{11}(D) & c_{12}(D) \\ c_{21}(D) & c_{22}(D) \\ c_{31}(D) & c_{32}(D) \end{bmatrix}
\begin{bmatrix} w_1(t) \\ w_2(t) \end{bmatrix}
```

where

$$
c_{11}(D) = \frac{(D + \beta) k_1}{D^2}, \qquad c_{12}(D) = 0.
$$

Substituting {eq}`eq-20-11` into equation {eq}`eq-20-1` and equating coefficients on $w_1(t)$
and $w_2(t)$ shows that

```{math}
:label: eq-20-12
\begin{aligned}
c_{22}(D) &= c_{32}(D) \\
c_{21}(D) - \frac{(D+\beta) k_1}{D^2} &= \frac{\alpha \beta k_1}{D} + c_{31}(D).
\end{aligned}
```

The stochastic process $a$ is assumed not to be observed by the econometrician. To give
equation {eq}`eq-20-1` empirical content, something must be said about the dynamic correlation
between $a$ and $m$. Hansen and Sargent require that, for $v > 0$,

```{math}
:label: eq-20-13
\hat E[a(t+v)\mid a(u): u \leq t] = \hat E_t\, a(t+v).
```

Assumption {eq}`eq-20-13` says that no other variables observed by private agents Granger cause
(help predict) $a$. It implies that

```{math}
:label: eq-20-14
c_{31}(D) = k_2\, c_{32}(D)
```

for some scalar constant $k_2$. Combining restrictions {eq}`eq-20-14` and {eq}`eq-20-12`
determines that

```{math}
:label: eq-20-15
c_{21}(D) = \frac{\alpha \beta k_1}{D} + \frac{(D+\beta) k_1}{D^2} + k_2\, c_{22}(D).
```

Restriction {eq}`eq-20-15` is a restriction on the bivariate moving average representation for
the observable process

$$
\begin{bmatrix} p(t) \\ m(t) \end{bmatrix}
= \begin{bmatrix} c_{11}(D) & c_{12}(D) \\ c_{21}(D) & c_{22}(D) \end{bmatrix}
\begin{bmatrix} w_1(t) \\ w_2(t) \end{bmatrix}
```

where the authors have already imposed the restrictions that

```{math}
:label: eq-20-16
c_{11}(D) = \frac{(D+\beta) k_1}{D^2}, \qquad c_{12}(D) = 0.
```

An identification question for this model is whether parameters $\alpha$, $\beta$, $k_1$, and
$k_2$ can be identified from the continuous-time "reduced form" convolutions $c_{11}$, $c_{12}$,
$c_{21}$, $c_{22}$. It is clear that $\beta$ and $k_1$ can be identified from {eq}`eq-20-16`. In
general, $\alpha$ and $k_2$ can be identified from equation {eq}`eq-20-15`. However, for a
special and convenient parameterization of $c_{22}(D)$, they are not identified. Suppose that
the derivative of $a$ is a white noise. Thus

```{math}
:label: eq-20-17
c_{32}(D) = \frac{k_3}{D} = c_{22}(D).
```

When $a$ is a Gaussian process, {eq}`eq-20-17` implies that $a$ is in fact a Brownian motion.
Substituting {eq}`eq-20-17` into {eq}`eq-20-15` yields

```{math}
:label: eq-20-18
c_{21}(D) = \frac{\alpha \beta k_1 + k_1 + k_2 k_3}{D} + \frac{\beta k_1}{D^2}.
```

The parameters $\alpha$ and $k_2$ are not identifiable in {eq}`eq-20-18`. It remains true,
however, even in this case that the model imposes testable cross equation restrictions in that
$\beta k_1 / D^2$ enters both $c_{11}$ and $c_{21}$.

For the remainder of the analysis, Hansen and Sargent adopt {eq}`eq-20-17` as a simplifying
assumption and require that

$$
\alpha \beta k_1 + k_1 + k_2 k_3 = 0.
$$

In this case

$$
c_{21}(D) = \frac{\beta k_1}{D^2},
$$

and the moving average representation for $p$ and $m$ is

```{math}
:label: eq-20-19
\begin{bmatrix} p(t) \\ m(t) \end{bmatrix}
= \begin{bmatrix} \dfrac{(D+\beta) k_1}{D^2} & 0 \\[2ex] \dfrac{\beta k_1}{D^2} & \dfrac{k_3}{D} \end{bmatrix}
\begin{bmatrix} w_1(t) \\ w_2(t) \end{bmatrix}.
```

In the continuous-time system {eq}`eq-20-19`, $m$ fails to Granger cause $p$ because
$c_{12}(D) = 0$, while $p$ does Granger cause $m$. That these features characterize the system
is no surprise: {eq}`eq-20-19` was constructed precisely to guarantee that Cagan's adaptive
expectations mechanism {eq}`eq-20-5` is optimal. In light of {eq}`eq-20-4`, if Cagan's mechanism
is rational there must be extensive feedback from $p$ to $m$.[^fn20-1]

```{eval-rst}
.. index::
   single: aggregation over time; effects on a bivariate model
   single: matrix covariogram
   single: bivariate Wold representation
   single: temporal aggregation; and adaptive expectations
```

## 3. Effects of aggregation over time

Hansen and Sargent now deduce the implications of the continuous-time version of Cagan's model
for point-in-time sampled, discrete-time observations on $(p, m)$, assuming that such
observations are available at the integers $t = 0, 1, 2, \ldots$.

The presence of $D$ and $D^2$ in the denominator of the "moving average" polynomials on the
right side of {eq}`eq-20-19` indicates that $(p, m)$ is a nonstationary process. It turns out
that the second differences of $(p, m)$ form a stationary process with a very simple
representation.

Consider now the discrete-time process formed by taking second differences of the point-in-time
observations on $(p, m)$ at the integers. Note first that the lag operator $L$ can be
represented as $L = e^{-D}$. Then the first difference operator is $(1 - L) = (1 - e^{-D})$,
while the second difference operator is $(1-L)^2 = (1 - e^{-D})^2$. Applying this operator to
{eq}`eq-20-19` gives

$$
\begin{bmatrix} (1-L)^2 p(t) \\ (1-L)^2 m(t) \end{bmatrix}
= \begin{bmatrix} \dfrac{(1-e^{-D})^2}{D^2}(\beta + D) k_1 & 0 \\[2ex] \dfrac{(1-e^{-D})^2}{D^2}\beta k_1 & \dfrac{(1-e^{-D})^2}{D} k_3 \end{bmatrix}
\begin{bmatrix} w_1(t) \\ w_2(t) \end{bmatrix}.
$$

Now recall the following Laplace transform pairs:

```{math}
:label: eq-20-20
\frac{(1-e^{-s})^2}{s} \leftrightarrow \begin{cases} 1 & t \in [0, 1] \\ -1 & t \in [1, 2] \\ 0 & t > 2 \end{cases}
```

```{math}
:label: eq-20-21
\frac{(1-e^{-s})^2}{s^2} \leftrightarrow \begin{cases} t & t \in [0, 1] \\ 2 - t & t \in [1, 2] \\ 0 & t > 2. \end{cases}
```

Using the Laplace transforms {eq}`eq-20-21` and {eq}`eq-20-20` gives the desired representation:

```{math}
:label: eq-20-22
\begin{aligned}
(1-L)^2 p(t) &= k_1 \int_0^1 (\beta \tau + 1) w_1(t-\tau)\, d\tau + k_1 \int_1^2 [\beta(2-\tau) - 1] w_1(t-\tau)\, d\tau \\
(1-L)^2 m(t) &= k_1 \int_0^1 \beta \tau\, w_1(t-\tau)\, d\tau + k_1 \int_1^2 \beta(2-\tau) w_1(t-\tau)\, d\tau \\
&\quad + k_3 \int_0^1 w_2(t-\tau)\, d\tau - k_3 \int_1^2 w_2(t-\tau)\, d\tau.
\end{aligned}
```

To represent things compactly, define

$$
y(t) = \begin{bmatrix} (1-L)^2 p(t) \\ (1-L)^2 m(t) \end{bmatrix}.
$$

Then {eq}`eq-20-22` can be written

```{math}
:label: eq-20-23
\begin{aligned}
y(t) &= \int_0^1 \begin{bmatrix} (\beta\tau + 1) k_1 & 0 \\ \beta\tau k_1 & k_3 \end{bmatrix} \begin{bmatrix} w_1(t-\tau) \\ w_2(t-\tau) \end{bmatrix} d\tau \\
&\quad + \int_1^2 \begin{bmatrix} [\beta(2-\tau) - 1] k_1 & 0 \\ \beta(2-\tau) k_1 & -k_3 \end{bmatrix} \begin{bmatrix} w_1(t-\tau) \\ w_2(t-\tau) \end{bmatrix} d\tau.
\end{aligned}
```

Evidently, by virtue of the white noise property of $w$, $y$ sampled at the integers is a
first-order, bivariate moving average process with unconditional mean $Ey(t) = 0$. The
autocovariogram of the $y$ process is readily computed from[^fn20-2]

$$
\Gamma_0 = Ey(t) y(t)', \qquad \Gamma_1 = Ey(t) y(t-1)', \qquad \Gamma_{-1} = Ey(t) y(t+1)' = \Gamma_1', \qquad \Gamma_j = \Gamma_{-j} = 0 \text{ for } j > 1.
$$

Carrying out the integrations (with $v_{11} = (k_1)^2$ and $v_{22} = (k_3)^2$) gives

```{math}
:label: eq-20-24
\begin{aligned}
\Gamma_0 &= \begin{bmatrix} 2v_{11}\left(\tfrac{1}{3}\beta^2 + 1\right) & 2v_{11}\tfrac{1}{3}\beta^2 \\[1ex] 2v_{11}\tfrac{1}{3}\beta^2 & 2v_{11}\left[\tfrac{1}{3}\beta^2 + \tfrac{v_{22}}{v_{11}}\right] \end{bmatrix} \\[2ex]
\Gamma_1 &= \begin{bmatrix} v_{11}\left(\tfrac{1}{6}\beta^2 - 1\right) & v_{11}\tfrac{\beta}{2}\left(\tfrac{1}{3}\beta - 1\right) \\[1ex] v_{11}\tfrac{\beta}{2}\left(\tfrac{1}{3}\beta + 1\right) & v_{11}\left(\tfrac{1}{6}\beta^2 - \tfrac{v_{22}}{v_{11}}\right) \end{bmatrix}.
\end{aligned}
```

```{note}
The $(2,1)$ element of $\Gamma_1$ in {eq}`eq-20-24` is $v_{11}\tfrac{\beta}{2}\left(\tfrac{1}{3}\beta + 1\right)$.
The original article prints this entry as $v_{11}\tfrac{\beta}{2}\left(\tfrac{1}{3}\beta + 1\right) - v_{22}$.
The $-v_{22}$ term cannot be present: $(1-L)^2 m(t)$ and $(1-L)^2 p(t-1)$ share only the white
noise $w_1$, so $\Gamma_1(2,1)$ depends on $v_{11}$ alone. The value above is what one obtains by
integrating {eq}`eq-20-23` directly, and it is the value used in the recomputations below; it
also makes $\det[\Gamma_1' + \Gamma_0 + \Gamma_1]$ vanish, the property that the original paper
relies on (see {ref}`sec-20-5`).
```

The matrix covariogram {eq}`eq-20-24` of the discrete-time process $y_t$ contains all of the
information required to compute the discrete-time Wold moving average representations. By
studying the univariate and bivariate discrete-time Wold representations for $y_t$, one can
characterize the effects of aggregation over time, as Hansen and Sargent do in the next two
sections.

We set up the computational tools used below.

```{code-cell} ipython3
import numpy as np
import matplotlib.pyplot as plt

def Gamma(beta, v11=1.0, v22=1.0):
    """Autocovariances Gamma_0, Gamma_1 of y_t = [(1-L)^2 p, (1-L)^2 m], eq (24)."""
    G0 = np.array([[2*v11*(beta**2/3 + 1),  2*v11*beta**2/3],
                   [2*v11*beta**2/3,        2*v11*(beta**2/3 + v22/v11)]])
    G1 = np.array([[v11*(beta**2/6 - 1),            v11*(beta/2)*(beta/3 - 1)],
                   [v11*(beta/2)*(beta/3 + 1),      v11*(beta**2/6 - v22/v11)]])
    return G0, G1

def ma1_factor(c0, c1):
    """Scalar MA(1) spectral factorization: (1-L)^2 x = (1 - lam L) eps,
    covariance generating function c1/z + c0 + c1 z.  Returns lam (|lam|<1) and sigma^2."""
    r = c0 / c1
    disc = (r/2)**2 - 1
    lam = min(-r/2 + np.sqrt(disc), -r/2 - np.sqrt(disc), key=abs)
    return lam, c0 / (1 + lam**2)

def bivariate_factor(G0, G1, iters=20000):
    """Fundamental Wold factorization y_t = u_t + F u_{t-1}, E u u' = Vbar,
    via the monotone (innovations) Riccati  Vbar = G0 - G1 Vbar^{-1} G1'."""
    V = G0.copy()
    for _ in range(iters):
        V = G0 - G1 @ np.linalg.solve(V, G1.T)
    F = G1 @ np.linalg.inv(V)
    return F, V
```

```{eval-rst}
.. index::
   single: univariate prediction; of inflation
   single: approximation error; of adaptive expectations
```

## 4. Predicting inflation using information on lagged inflation only

Hansen and Sargent first consider the univariate Wold representation for the $(1-L)^2 p$
process. From {eq}`eq-20-24`, $(1-L)^2 p$ is a first-order moving average with covariance
generating function

```{math}
:label: eq-20-25
g(z) = c(1) z^{-1} + c(0) + c(1) z
```

where from {eq}`eq-20-19`, $c(0) = 2v_{11}\left(\tfrac{1}{3}\beta^2 + 1\right)$ and
$c(1) = v_{11}\left(\tfrac{1}{6}\beta^2 - 1\right)$. They seek the Wold moving average
representation for $(1-L)^2 p$, of the form

```{math}
:label: eq-20-26
(1-L)^2 p(t) = (1 - \lambda_p L)\, \varepsilon_{pt}, \qquad |\lambda_p| < 1
```

with $\varepsilon_p$ a discrete-time white noise that is fundamental for $(1-L)^2 p$; the
variance of the one-step-ahead prediction error $\varepsilon_p$ is $\sigma_{\varepsilon p}^2$.
A routine application of the spectral factorization theorem gives the following formulas
for $\lambda_p$ and $\sigma_{\varepsilon p}^2$:

```{math}
:label: eq-20-27
\lambda_p = -\frac{1}{2}\frac{c(0)}{c(1)} \pm \sqrt{\frac{c(0)^2}{4 c(1)^2} - 1}
\qquad \text{subject to } |\lambda_p| < 1, \qquad
\sigma_{\varepsilon p}^2 = \frac{c(0)}{1 + \lambda_p^2}.
```

Using the preceding formulas for $c(0)$ and $c(1)$,

```{math}
:label: eq-20-28
\lambda_p = -\frac{\tfrac{1}{3}\beta^2 + 1}{\tfrac{1}{6}\beta^2 - 1} \pm \sqrt{\left(\frac{\tfrac{1}{3}\beta^2 + 1}{\tfrac{1}{6}\beta^2 - 1}\right)^2 - 1}
\qquad \text{subject to } |\lambda_p| < 1.
```

Now consider the discrete-time inflation rate $X(t) = p(t) - p(t-1)$, defined for $t$ at the
integers. Representation {eq}`eq-20-26` can then be written

```{math}
:label: eq-20-29
(1 - L) X(t) = (1 - \lambda_p L)\, \varepsilon_{pt}.
```

As shown by John F. Muth (1960), the optimal $j$-step-ahead forecast of $X$ governed by process
{eq}`eq-20-29`, given current and lagged values of $X$ alone, is the discrete-time version of
Cagan's adaptive expectation schemes

```{math}
:label: eq-20-30
\hat E[X(t+j)\mid X(t), X(t-1), \ldots] = (1 - \lambda_p) \sum_{i=0}^\infty \lambda_p^i\, X(t-i), \qquad j \geq 1.
```

Now equation {eq}`eq-20-30` is precisely the discrete-time representation which Cagan used for
approximating the continuous-time adaptive expectations scheme

$$
\hat E_t\, \tilde x(t+\tau) = \beta \int_0^\infty e^{-\beta s}\, \tilde x(t-s)\, ds, \qquad \tau > 0.
$$

Cagan took $\lambda_p$ to be related to $\beta$ via the equation

```{math}
:label: eq-20-31
\lambda_p = e^{-\beta}.
```

For various values of $\beta$, Table 1 reports the values of $\lambda_p$ given by formula
{eq}`eq-20-28` and Cagan's formula {eq}`eq-20-31`. For $\beta$ close to zero, exp$(-\beta)$ is a
close approximation to {eq}`eq-20-28`. However, for large values of $\beta$, $\exp(-\beta)$ is
approximately zero, while equation {eq}`eq-20-28` implies a $\lambda_p$ of approximately
$-0.25$. (As $\beta \to \infty$, $\lambda_p \to -2 + \sqrt{3} \approx -0.267949$.)

This comparison is of interest in the following context. Suppose the continuous-time model is
correct, and that an analyst possesses discrete-time observations on $p$ at integer points in
time. A procedure recommended by Nerlove (1967) and Nerlove, Grether, and Carvalho (1979)
would be to determine the optimal predictors for the univariate process for $p$, and then
attribute them to the private agents in the model. This procedure is motivated by an appeal to
the rational expectations hypothesis and is termed the method of "quasi-rational expectations."
In an infinitely large sample, the analyst could recover the parameter $\lambda_p$ given by
formula {eq}`eq-20-27`, if he followed Nerlove, Grether, and Carvalho's method. Using formula
{eq}`eq-20-28` or Table 1, the analyst could then infer the value of $\beta$. Table 1 provides a
fairly complete characterization of Cagan's approximation {eq}`eq-20-31` as a vehicle for
inferring $\beta$ from $\lambda$.

We now recompute Table 1.

```{code-cell} ipython3
def lambda_p(beta):
    c0 = 2*(beta**2/3 + 1)      # v11 cancels in the ratio
    c1 = (beta**2/6 - 1)
    lam, _ = ma1_factor(c0, c1)
    return lam

betas_tab = [0.0, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00, 2.25, 2.50,
             3.00, 4.00, 5.00, 7.00, 10.00]
print(f"{'beta':>6} {'lambda_p':>12} {'exp(-beta)':>12}")
for b in betas_tab:
    lp = lambda_p(b) if b > 0 else 1.0
    print(f"{b:6.2f} {lp:12.6f} {np.exp(-b):12.6f}")
print(f"{'+inf':>6} {-2+np.sqrt(3):12.6f} {0.0:12.6f}")
```

The recomputed column $\lambda_p$ reproduces Table 1 of the original paper to six decimal
places. The figure below plots the exact decay parameter $\lambda_p$ against Cagan's
approximation $\exp(-\beta)$.

```{code-cell} ipython3
bb = np.linspace(0.001, 10, 400)
lp = np.array([lambda_p(b) for b in bb])

fig, ax = plt.subplots(figsize=(9, 5))
ax.plot(bb, lp, lw=2, label=r'exact $\lambda_p$  (eq. 28)')
ax.plot(bb, np.exp(-bb), lw=2, ls='--', label=r"Cagan's approximation $e^{-\beta}$")
ax.axhline(-2 + np.sqrt(3), color='0.5', ls=':', lw=1,
           label=r'$\lambda_p \to -2+\sqrt{3}$ as $\beta\to\infty$')
ax.axhline(0, color='k', lw=0.6)
ax.set_xlabel(r'continuous-time decay parameter $\beta$')
ax.set_ylabel(r'discrete-time decay parameter')
ax.set_title("Exact adaptive-expectations decay $\\lambda_p$ vs. Cagan's $e^{-\\beta}$")
ax.legend()
plt.show()
```

The two curves coincide for small $\beta$ but diverge sharply for large $\beta$: Cagan's
$e^{-\beta}$ collapses to $0$, whereas the exact $\lambda_p$ turns *negative* and approaches
$-2 + \sqrt{3}$. For values of $\lambda_p$ in the range estimated by Cagan (1956) and Sargent
(1977), the two are close, so aggregation over time imparts at most a very small asymptotic bias
to Cagan's estimator of $\beta$.

(sec-20-5)=
```{eval-rst}
.. index::
   single: bivariate prediction; of inflation
   single: Granger causality; spurious
   single: information; discarded by aggregation
```

## 5. Predicting inflation using information on lagged inflation and lagged money creation

Hansen and Sargent now turn to the bivariate moving average of the discrete-time process for
inflation and money creation. A Wold moving average representation for
$((1-L)^2 p(t), (1-L)^2 m(t))' = y(t)$ is

```{math}
:label: eq-20-32
y(t) = u_t + F u_{t-1}
```

where $u_t$ is a $(2 \times 1)$ vector discrete-time white noise with $Eu_t u_t' = \bar V$, where
$\bar V$ is a positive semidefinite matrix; $u_t = y(t) - \hat E y(t)\mid y(t-1), y(t-2), \ldots$;
and the eigenvalues of $F$ are less than or equal to unity in absolute value. Given $\Gamma_0$
and $\Gamma_1$ from {eq}`eq-20-24`, $F$ and $\bar V$ are determined by solving

```{math}
:label: eq-20-33
\Gamma_0 = \bar V + F \bar V F', \qquad \Gamma_1 = F \bar V.
```

The spectral factorization theorem implies that these equations have a unique solution with the
properties indicated above.

Letting $X_t = p(t) - p(t-1)$, $M_t = m(t) - m(t-1)$, one can write {eq}`eq-20-32` as

```{math}
:label: eq-20-34
\begin{bmatrix} (1-L) X_t \\ (1-L) M_t \end{bmatrix} = u_t + F u_{t-1}.
```

By carrying out a series of calculations paralleling those of Muth (1960), it is straightforward
to verify that {eq}`eq-20-34` admits the alternative representation

```{math}
:label: eq-20-35
\begin{bmatrix} X_{t+1} \\ M_{t+1} \end{bmatrix} = (I + F) \sum_{i=0}^\infty (-F)^i \begin{bmatrix} X_{t-i} \\ M_{t-i} \end{bmatrix} + u_{t+1}.
```

From the fact that $u$ is fundamental for $(X, M)$, it can be readily verified that there obtains
the following bivariate generalization of Cagan's adaptive expectations scheme:

```{math}
:label: eq-20-36
\hat E\begin{bmatrix} X_{t+j} \\ M_{t+j} \end{bmatrix}\,\Big|\, X_t, M_t, X_{t-1}, M_{t-1}, \ldots
= (I + F) \sum_{i=0}^\infty (-F)^i \begin{bmatrix} X_{t-i} \\ M_{t-i} \end{bmatrix}, \qquad j \geq 1.
```

The one-step-ahead prediction error vector is $u_{t+1}$, which has covariance matrix $\bar V$.

Representation {eq}`eq-20-34` is usefully compared to the one constructed by Sargent (1977). He
posited a discrete-time, bivariate, first-order moving average

```{math}
:label: eq-20-37
\begin{bmatrix} (1-L) X_t \\ (1-L) M_t \end{bmatrix}
= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \varepsilon_{1t} \\ \varepsilon_{2t} \end{bmatrix}
+ \begin{bmatrix} -\lambda & 0 \\ (1-\lambda) & -1 \end{bmatrix} \begin{bmatrix} \varepsilon_{1t-1} \\ \varepsilon_{2t-1} \end{bmatrix}
```

where $(\varepsilon_1, \varepsilon_2)' = \varepsilon$ is a discrete-time vector white noise with
arbitrary contemporaneous covariance $E\varepsilon_t \varepsilon_t' = W$; $\varepsilon$ is
fundamental for $((1-L) X, (1-L) M)$; and $|\lambda| < 1$. It is evident from the first equation
of {eq}`eq-20-37` that Cagan's discrete-time adaptive expectations formulation for inflation is
rational, given {eq}`eq-20-37`. In form, {eq}`eq-20-37` matches {eq}`eq-20-34`. A central task is
now to study the relation between the $(2 \times 2)$ matrix $F$ in {eq}`eq-20-34` and the
corresponding matrix

$$
E = \begin{bmatrix} -\lambda & 0 \\ (1-\lambda) & -1 \end{bmatrix}
$$

in {eq}`eq-20-37`. Notice that the eigenvalues of the matrix $E$ are $-1$ and $-\lambda$. It can
be proved that one of the eigenvalues of $F$ in {eq}`eq-20-34` is $-1$.[^fn20-3] A comparison
between the value of $-\lambda_p$ given by equation {eq}`eq-20-28` and the nonunit eigenvalue of
$F$ is one interesting measure of the effects of time aggregation.

For various values of $\beta$ and $V = \begin{bmatrix} v_{11} & 0 \\ 0 & v_{22} \end{bmatrix}$,
Hansen and Sargent calculate $F$ and $\bar V$, together with $\lambda_p$ and
$\sigma_{\varepsilon p}^2$ in the univariate Wold representation for $(1-L)X$ and the analogous
$\lambda_m$ and $\sigma_{\varepsilon m}^2$ for $(1-L) M$. Writing $\bar v_{11}, \bar v_{22}$ for
the diagonal
elements of $\bar V$ (the discrete-time one-step-ahead prediction error variances for $X$ and $M$
*given lagged $X$'s and lagged $M$'s*), the quantities

$$
\text{percentage gain for } p = \frac{\sigma_{\varepsilon p}^2 - \bar v_{11}}{\sigma_{\varepsilon p}^2}, \qquad
\text{percentage gain for } m = \frac{\sigma_{\varepsilon m}^2 - \bar v_{22}}{\sigma_{\varepsilon m}^2}
$$

measure, respectively, the marginal assistance of lagged $M$'s in predicting $X$, and of lagged
$X$'s in predicting $M$. These "percentage gains" are measures of the strength of the Granger
causality that occurs between the discrete-time $X$ and $M$ processes. In the continuous-time
system {eq}`eq-20-19`, $m$ fails to Granger cause $X$; however, in the discrete-time model, $M$
will in general Granger cause $X$ due to the effects of aggregation over time.[^fn20-4]

The following code recomputes Tables 2–4 of the paper (for $V = I$, $V = \text{diag}(10, 1)$ and
$V = \text{diag}(1, 10)$ respectively).

```{code-cell} ipython3
def table_row(beta, v11, v22):
    G0, G1 = Gamma(beta, v11, v22)
    F, V = bivariate_factor(G0, G1)
    lam_p, s2p = ma1_factor(G0[0, 0], G1[0, 0])     # univariate (1-L)X
    lam_m, s2m = ma1_factor(G0[1, 1], G1[1, 1])     # univariate (1-L)M
    gain_p = (s2p - V[0, 0]) / s2p * 100
    gain_m = (s2m - V[1, 1]) / s2m * 100
    ev = sorted(np.linalg.eigvals(F).real, key=lambda e: abs(e + 1))
    return gain_p, gain_m, lam_p, ev[1]             # ev[0] ~ -1, ev[1] = nonunit

def print_table(v11, v22, betas):
    print(f"V = diag({v11}, {v22})")
    print(f"{'beta':>6} {'%gain p':>9} {'%gain m':>9} {'lambda_p':>10} {'eig F':>10}")
    for b in betas:
        gp, gm, lp, ef = table_row(b, v11, v22)
        print(f"{b:6.2f} {gp:9.3f} {gm:9.3f} {lp:10.6f} {ef:10.6f}")

betas = [0.05, 0.25, 0.55, 1.00, 2.00, 3.00, 5.00, 6.00, 7.00, 10.00, 20.00]
print_table(1.0, 1.0, betas)
```

```{code-cell} ipython3
print_table(10.0, 1.0, betas)   # Table 3
print()
print_table(1.0, 10.0, betas)   # Table 4
```

These reproduce the published Tables 2–4 to all printed digits, with three exceptions in Table 2
($V = I$) that are evidently typographical errors in the 1983 original: the nonunit eigenvalue of
$F$ at $\beta = 1.00$ is printed as $-0.127017$, which is a duplicate of the $\beta = 2.00$ entry
(the correct value is $-0.367138$); the percentage gain for $p$ at $\beta = 10.00$ is printed as
$6.289$ (a digit transposition of $6.989$); and the percentage gain for $m$ at $\beta = 6.00$ is
printed as $28.205$ (the recomputation gives $28.251$).

One outstanding characteristic that emerges is that for small values of $\beta$, $-\exp(-\beta)$,
$-\lambda_p$, and the nonunit eigenvalue of $F$ are very close to each other; further, for small
$\beta$, money creation only very weakly Granger-causes inflation in the discrete-time data. On
the other hand, for large values of $\beta$, $\exp(-\beta)$ fails to approximate $\lambda_p$, and
substantial Granger causality can extend from money creation to inflation in discrete time. The
figure below plots the two percentage gains against $\beta$ for the symmetric case $V = I$.

```{code-cell} ipython3
bb = np.array([0.05, 0.1, 0.2, 0.35, 0.5, 0.75, 1, 1.5, 2, 3, 4, 5, 7, 10, 14, 20])
gp = np.array([table_row(b, 1.0, 1.0)[0] for b in bb])
gm = np.array([table_row(b, 1.0, 1.0)[1] for b in bb])

fig, ax = plt.subplots(figsize=(9, 5))
ax.plot(bb, gm, 'o-', lw=2, label=r'gain for $m$: lagged $X$ helps predict $M$')
ax.plot(bb, gp, 's-', lw=2, label=r'gain for $p$: lagged $M$ helps predict $X$')
ax.set_xscale('log')
ax.set_xlabel(r'continuous-time decay parameter $\beta$  (log scale)')
ax.set_ylabel('percentage gain  (Granger-causality strength)')
ax.set_title('Aggregation-over-time induced Granger causality ($V=I$)')
ax.legend()
plt.show()
```

In continuous time money creation does *not* Granger cause inflation, yet in the point-in-time
sampled discrete data it does (the "gain for $p$" curve is strictly positive and rises with
$\beta$). The much larger "gain for $m$" reflects the strong feedback from $p$ to $m$ that
{eq}`eq-20-19` builds in to make Cagan's mechanism optimal. For values of $\lambda_p$ in the
range estimated by Cagan and Sargent, the gain for $p$ is small, which is comforting; for large
$\beta$, however, the effects of aggregation over time can be considerable.

## 6. Conclusions

Hansen and Sargent produce a continuous-time model that solves the inverse optimal predictor
problem for a continuous-time version of Cagan's model of hyperinflation with adaptive
expectations. They then deduce the restrictions this continuous-time model places on
discrete-time data, obtaining exact formulas that link the parameters of the discrete-time
representation to those of the continuous-time model. These formulas make it possible to
evaluate the quality of the approximations that Cagan and others have used in connecting the
discrete-time and continuous-time parameterizations.

The computational techniques are useful for studying the effects of aggregation over time in a
variety of dynamic models under rational expectations, and in later work the authors applied
them in substantially richer dynamic settings.

## Appendix

This appendix shows that the only choice of $c_1(D)$ that has a rational Laplace transform and
satisfies

```{math}
:label: eq-20-a1
[D c_1(D) e^{vD}]_+ = \frac{\beta D c_1(D)}{\beta + D}
```

is

```{math}
:label: eq-20-a2
c_1(D) = \left(\frac{D + \beta}{D^2}\right) k_0
```

where $k_0$ is an arbitrary row vector constant.

The fact that no other variables Granger cause $p$ implies that $c_1(D)$ must take the form
$c_1(D) = c_1(D)^* k_0$ where $c_1(D)^*$ is a scalar operator and $k_0$ is a row vector constant
that satisfies $k_0 k_0' = 1$. Thus the fundamental representation for $p$ can be written

```{math}
:label: eq-20-a3
p(t) = c_1(D)^*\, w(t)^*
```

where $w(t)^* = k_0 w(t)$. The function $c_1(s)^*$ is assumed to be rational, written as

$$
c_1(s)^* = \frac{\mu(s)}{\gamma(s)}
$$

where

$$
\gamma(s) = (s - \gamma_1)(s - \gamma_2) \cdots (s - \gamma_n), \qquad
\mu(s) = \mu_0 (s - \mu_1)(s - \mu_2) \cdots (s - \mu_m).
$$

Consistent with the requirement that anticipated inflation be physically realizable, the
restriction $\operatorname{Real}(\gamma_j) \leq 0$ is imposed for $j = 1, 2, \ldots, n$. This
guarantees that the vector function $c_1^*(s)$ is analytic in the open right half plane. The
assumption that $p$ is physically realizable implies that $m < n$.

The function $G(s, v) = s c_1(s)^* e^{vs}$ is analytic in its first argument everywhere in the
complex plane except possibly at $\gamma_1, \gamma_2, \ldots, \gamma_n$. Let
$H_1(s, v), H_2(s, v), \ldots, H_q(s, v)$ denote the principal parts of $G$ at the corresponding
$q$ distinct zeroes of $\gamma(s)$. Using a result from Hansen and Sargent (1982), it follows
that $[G(D, v)]_+ = H_1(D, v) + H_2(D, v) + \ldots + H_q(D, v)$.

However from {eq}`eq-20-a1` it is clear that $H_j(s, v)$ cannot depend on $v$. It follows that
$n = 2$ and $\gamma_1 = \gamma_2 = 0$. Therefore

$$
c_1(s)^* = \frac{\mu_0 (s - \mu_1)}{s^2}.
$$

Computing $[D c_1(s)^* e^{vD}]_+$ gives

$$
\left[\mu_0 \frac{(D - \mu_1) e^{vD}}{D}\right]_+ = -\frac{\mu_0 \mu_1}{D}.
$$

By equation {eq}`eq-20-a1`,

```{math}
:label: eq-20-a4
-\frac{\mu_0 \mu_1}{D} = \frac{\beta \mu_0 (D - \mu_1)}{(\beta + D) D}.
```

Equation {eq}`eq-20-a4` implies that $\mu_1 = -\beta$. Therefore
$c_1(D)^* = \mu_0 \dfrac{(D + \beta)}{D^2}$, which proves the desired result.

## Exercises

```{exercise-start}
:label: ex-20-1
```

The text states that as $\beta \to \infty$ the exact decay parameter $\lambda_p$ of {eq}`eq-20-28`
approaches $-2 + \sqrt{3}$, while Cagan's approximation $e^{-\beta}$ approaches $0$.

(a) Using {eq}`eq-20-27` with $c(0) = 2v_{11}(\tfrac{1}{3}\beta^2 + 1)$ and
$c(1) = v_{11}(\tfrac{1}{6}\beta^2 - 1)$, show analytically that $\lim_{\beta \to \infty}\lambda_p
= -2 + \sqrt{3}$. (Hint: the ratio $c(0)/c(1) \to 4$.)

(b) Plot the *absolute approximation error* $|e^{-\beta} - \lambda_p|$ as a function of $\beta$,
and confirm numerically that the error is negligible for the small-$\beta$ range relevant to
Cagan's and Sargent's estimates but grows toward $|{-2+\sqrt 3}| \approx 0.27$ as $\beta$ grows.

```{exercise-end}
```

```{solution-start} ex-20-1
:class: dropdown
```

For (a): as $\beta \to \infty$, $c(0)/c(1) = 2(\tfrac{1}{3}\beta^2 + 1)/(\tfrac{1}{6}\beta^2 - 1)
\to 2 \cdot \tfrac{1/3}{1/6} = 4$. With $r = c(0)/c(1) \to 4$, {eq}`eq-20-27` gives
$\lambda_p = -r/2 \pm \sqrt{r^2/4 - 1} \to -2 \pm \sqrt{4 - 1} = -2 \pm \sqrt 3$. The root with
$|\lambda_p| < 1$ is $-2 + \sqrt 3 \approx -0.267949$.

```{code-cell} ipython3
bb = np.linspace(0.01, 12, 500)
err = np.abs(np.exp(-bb) - np.array([lambda_p(b) for b in bb]))

fig, ax = plt.subplots(figsize=(9, 4.5))
ax.plot(bb, err, lw=2)
ax.axhline(abs(-2 + np.sqrt(3)), color='0.5', ls=':',
           label=r'$|{-2+\sqrt{3}}|\approx 0.268$')
ax.set_xlabel(r'$\beta$'); ax.set_ylabel(r'$|e^{-\beta} - \lambda_p|$')
ax.set_title("Error of Cagan's approximation $\\lambda \\approx e^{-\\beta}$")
ax.legend()
plt.show()

print(f"limit  -2+sqrt(3)      = {-2 + np.sqrt(3):.6f}")
print(f"lambda_p at beta=50    = {lambda_p(50.0):.6f}")
print(f"error at beta=0.5      = {abs(np.exp(-0.5) - lambda_p(0.5)):.4f}")
print(f"error at beta=10       = {abs(np.exp(-10) - lambda_p(10.0)):.4f}")
```

```{solution-end}
```

```{exercise-start}
:label: ex-20-2
```

In continuous time, money creation fails to Granger cause inflation, yet aggregation over time
makes money creation Granger cause inflation in the point-in-time sampled discrete data. The
"percentage gain for $p$," $(\sigma_{\varepsilon p}^2 - \bar v_{11})/\sigma_{\varepsilon p}^2$,
measures the strength of this induced causality.

(a) For the symmetric case $V = I$, verify numerically the claim in {ref}`sec-20-5` that **one
eigenvalue of $F$ is exactly $-1$** for several values of $\beta$.

(b) Compute the percentage gain for $p$ at $\beta = 0.35$ (a value implying $\lambda_p$ in
Cagan's estimated range) and at $\beta = 10$, and comment on how the strength of
aggregation-induced Granger causality depends on $\beta$.

```{exercise-end}
```

```{solution-start} ex-20-2
:class: dropdown
```

```{code-cell} ipython3
# (a) one eigenvalue of F equals -1
print("beta     eigenvalues of F")
for b in [0.35, 1.0, 5.0, 10.0]:
    G0, G1 = Gamma(b, 1.0, 1.0)
    F, V = bivariate_factor(G0, G1)
    ev = np.sort(np.linalg.eigvals(F).real)
    print(f"{b:5.2f}   {np.round(ev, 6)}")
```

```{code-cell} ipython3
# (b) percentage gain for p
for b in [0.35, 10.0]:
    gp, gm, lp, ef = table_row(b, 1.0, 1.0)
    print(f"beta={b:5.2f}:  gain for p = {gp:6.3f}%   (lambda_p={lp:.4f})")
```

One eigenvalue of $F$ sits at $-1$ for every $\beta$, confirming the text's claim (it reflects
the over-differencing implicit in applying $(1-L)^2$). At $\beta = 0.35$ the percentage gain for
$p$ is a small fraction of a percent. Money creation barely improves the inflation forecast, so
Cagan's univariate procedure loses almost nothing. At $\beta = 10$ the gain is several percent:
when the continuous-time decay is fast relative to the unit sampling interval, aggregation over
time induces economically meaningful Granger causality from money creation to inflation that is
entirely absent in continuous time.

```{solution-end}
```

## References

Cagan, P. (1956). The Monetary Dynamics of Hyperinflation, in M. Friedman, ed., *Studies in the
Quantity Theory of Money*. Chicago: University of Chicago.

Christiano, L. (1980). Rational Expectations, Hyperinflation, and the Demand for Money.
Manuscript.

Friedman, B. M. (1978). Stability and Rationality in Models of Hyperinflation. *International
Economic Review*, **19** (February), 45–64.

Friedman, M. (1957). *A Theory of the Consumption Function*. Princeton: Princeton University
Press.

Granger, C. W. J. (1969). Investigating Causal Relations by Econometric Models and Cross
Spectral Methods. *Econometrica*, **37** (July), 424–438.

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

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.

Hansen, L. P., and T. J. Sargent (1982). Continuous Time Models of Control, Prediction, and
Rational Expectations.

Khan, M. S. (1980). Dynamic Stability in the Cagan Model of Hyperinflation. *International
Economic Review*, **21** (October), 577–582.

Muth, J. F. (1960). Optimal Properties of Exponentially Weighted Forecasts. *Journal of the
American Statistical Association*, **55** (June), 299–306.

Nerlove, M. (1967). Distributed Lags and Unobserved Components in Economic Time Series, in
W. Fellner, et al., *Ten Economic Studies in the Tradition of Irving Fisher*. New York: Wiley.

Nerlove, M., D. M. Grether, and J. L. Carvalho (1979). *Analysis of Economic Time Series: A
Synthesis*. New York: Academic Press.

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

Sargent, T. J. (1977). The Demand for Money during Hyperinflations under Rational Expectations:
I. *International Economic Review*, **18** (February), 59–82.

Sims, C. A. (1971). Discrete Approximations to Continuous Time Distributed Lags in Econometrics.
*Econometrica*, **39** (May), 545–564.

Sims, C. A. (1972). Money, Income and Causality. *American Economic Review*, **62** (September),
540–552.

[^fn20-1]: This is a continuous-time version of the assumption about the disturbance to the
    portfolio balance schedule considered in Sargent (1977).

[^fn20-2]: Hansen and Sargent use the rules for taking expected values of products of integrals
    of white noises described by Kwakernaak and Sivan (1972, pp. 97–99).

[^fn20-3]: First, note that $(I + Fz)\bar V(I + F'z^{-1}) = \Gamma_{-1} z^{-1} + \Gamma_0 +
    \Gamma_1 z$ and therefore the zeroes of $\det(\Gamma_{-1} z^{-1} + \Gamma_0 + \Gamma_1 z)$
    are comprised of the zeroes of $\det(I + Fz)$ and the reciprocals of the zeroes of
    $\det(I + Fz)$. The zeroes of $\det(I + Fz)$ are minus the reciprocals of the eigenvalues of
    $F$. Using formulas (18) it can be proved that unity is a zero of
    $\det(\Gamma_{-1} z^{-1} + \Gamma_0 + \Gamma_1 z)$, which implies that $-1$ is an eigenvalue
    of $F$.

[^fn20-4]: In a more general context, Sims (1971, 1972) has emphasized that $\tilde y$'s failure
    to Granger-cause $\tilde x$ in continuous time does not imply that $\tilde y$ fails to
    Granger-cause $\tilde x$ in discrete time.