# A Difficulty in Interpreting Vector Autoregressions

```{note}
This section is based on Lars Peter Hansen and Thomas J. Sargent, "Two Difficulties in
Interpreting Vector Autoregressions," Chapter 4 of *Rational Expectations Econometrics*
(Westview Press, 1991). That chapter describes *two* difficulties — one for discrete-time
models and one arising from time aggregation of continuous-time models. We treat only the
first, discrete-time difficulty (the paper's Introduction and Section 1), and we set the
sampling interval to $\Delta = 1$ throughout, so that one period is one unit of time.
```

The equilibrium of a dynamic rational expectations model is a covariance stationary
$(n\times 1)$ vector process $z_t$. *Surprises* — random shocks to the agents' information
sets — prompt revisions in their plans and so move equilibrium prices and quantities. Since
every covariance stationary process has a vector autoregression (Wold's theorem of
{doc}`13_representation_theory`), it is tempting to summarize such an equilibrium by its
vector autoregression and to read the **innovation accounting** of {cite:t}`sims1980macroeconomics` — variance
decompositions and impulse responses to the autoregression's innovations — as though those
innovations *were* the shocks hitting agents. This section describes a class of models in
which that reading is wrong: the white noise a vector autoregression recovers is generally
**not** the white noise that is fundamental for the agents, and innovation accounting, taken
at face value, gives a distorted picture of how the economy responds to the agents' surprises.

## The vector autoregression and its innovations

Let $z_t$ be an $(n\times 1)$ covariance stationary process, observed at the integer dates
$t = 0, \pm 1, \pm 2, \dots$ (this is the paper's setup with $\Delta = 1$). Its **vector
autoregression** is the projection equation

```{math}
:label: eq-vd-1
z_t = \sum_{j=1}^{\infty} A_j\, z_{t-j} + a_t ,
```

where $a_t$ is the $(n\times 1)$ vector of population regression residuals, with
$E a_t a_t^T = V$, determined by the orthogonality (normal-equation) conditions

```{math}
:label: eq-vd-2
E\, z_{t-j}\, a_t^T = 0, \qquad j \geq 1 ,
```

and where the $A_j$ are square summable, $\sum_{j=1}^{\infty}\operatorname{tr}(A_j A_j^T) <
\infty$. Conditions {eq}`eq-vd-1`–{eq}`eq-vd-2` make $a_t$ a vector white noise, $E a_t
a_{t-j}^T = 0$ for $j\neq 0$, that lies in the closed linear space spanned by
$\{z_t, z_{t-1}, \dots\}$. Eliminating the lagged $z$'s from {eq}`eq-vd-1` gives the **vector
moving-average (Wold) representation**

```{math}
:label: eq-vd-3
z_t = \sum_{j=0}^{\infty} C_j\, a_{t-j}, \qquad C_0 = I ,
```

whose coefficients satisfy $A(L)\,C(L) = I$ with $A(L) = I - \sum_{j\geq 1} A_j L^j$ and
$C(L) = \sum_{j\geq 0} C_j L^j$, and are square summable. Because $a_t$ belongs to the space
spanned by current and lagged $z$'s, it is a **fundamental** white noise for $z_t$: current
and lagged $z$'s reveal it. There are also non-fundamental representations $z_t = \sum_j
\tilde C_j\, \tilde a_{t-j}$ in which $\{\tilde a_t, \tilde a_{t-1},\dots\}$ spans a *strictly
larger* space than $\{z_t, z_{t-1},\dots\}$; current and lagged $z$'s fail to be "fully
revealing" about such an $\tilde a_t$.

Representation {eq}`eq-vd-3` induces the decomposition of the $j$-step-ahead prediction-error
covariance that underlies innovation accounting,

```{math}
:label: eq-vd-4
E\big(z_t - \hat E_{t-j} z_t\big)\big(z_t - \hat E_{t-j} z_t\big)^T
 = \sum_{k=0}^{j-1} C_k\, V\, C_k^T ,
```

where $\hat E$ is the linear least squares projection operator. Sims's methods estimate the
autoregression {eq}`eq-vd-1`, form the moving average {eq}`eq-vd-3`, and decompose
{eq}`eq-vd-4` to attribute prediction-error variance to innovations in particular components
of $z_t$.

Now suppose the equilibrium of an economic model has its own moving-average representation in
terms of the shocks to agents' information sets,

```{math}
:label: eq-vd-5
z_t = \sum_{j=0}^{\infty} D_j\, \epsilon_{t-j},
```

where $\epsilon_t$ is the white noise that is fundamental for the agents. The interpretive
question is whether the vector-autoregression innovations $a_t$ equal the agents' shocks
$\epsilon_t$ — and whether the response coefficients $C_j$ equal the economic responses
$D_j$. If they do, innovation accounting reads off the economics directly. The next subsection
exhibits a class of models in which they do **not**.

##  Unrevealing models

Consider models whose equilibrium solves the pair of stochastic difference equations

```{math}
:label: eq-vd-6
H(L)\, y_t = E_t\, J(L^{-1})^{-1}\, p\, x_t , \qquad x_t = K(L)\,\epsilon_t ,
```

where $y_t$ is $n_1\times 1$, $x_t$ is $n_2\times 1$, $H(L) = H_0 + \dots + H_{m_1}L^{m_1}$
and $J(L) = J_0 + \dots + J_{m_2}L^{m_2}$ are matrix polynomials, $K(L) = \sum_{j\geq 0} K_j
L^j$ with $K_0 = I$, and $\epsilon_t = x_t - E(x_t\mid x_{t-1},x_{t-2},\dots)$ is the
fundamental white noise of the forcing process $x_t$. We assume the zeros of $\det H(z)$ lie
outside the unit circle, those of $\det J(z)$ inside, and those of $\det K(z)$ not outside.
Interrelated factor-demand models of the Lucas–Prescott type are special cases with $J(L^{-1})
= H(L^{-1})^T$; dominant-player and Kennan–Sargent market models give other cases.

Equation {eq}`eq-vd-6` is solved with the prediction technology of {doc}`14_linear_prediction`
and the partial-fraction calculus of {doc}`18a_partial_fractions`. Factor $\det J(z^{-1}) =
\lambda_0\prod_{j=1}^{k}(1-\lambda_j z^{-1})$, with $k = m_2 n_1$ and the distinct $\lambda_j$
inside the unit circle, and expand

```{math}
:label: eq-vd-7
J(z^{-1})^{-1} = \sum_{j=1}^{k} \frac{M_j}{1 - \lambda_j z^{-1}},
\qquad
M_j = \lim_{z\to\lambda_j} J(z^{-1})^{-1}\,(1 - \lambda_j z^{-1}) .
```

Each term is a geometric sum of *expected future* $x$'s, evaluated with the Hansen–Sargent
formula (the seasonal-lead calculation of {doc}`20_geometric_leads`),

```{math}
:label: eq-vd-8
E_t\, \frac{M_j}{1 - \lambda_j L^{-1}}\, p\, x_t
 = M_j\, p\left(\frac{L K(L) - \lambda_j K(\lambda_j)}{L - \lambda_j}\right)\epsilon_t .
```

Summing over $j$ and writing $M(K(L)) = \sum_{j=1}^{k} M_j\, p\big(\frac{L K(L) - \lambda_j
K(\lambda_j)}{L - \lambda_j}\big)$, the equilibrium becomes

```{math}
:label: eq-vd-9
H(L)\, y_t = M(K(L))\,\epsilon_t , \qquad x_t = K(L)\,\epsilon_t .
```

The full vector $(y_t^T, x_t^T)^T$ — $n_1 + n_2$ variables — is driven by only the $n_2$ white
noises $\epsilon_t$, so it has a *singular* spectral density: the model fits an internal subset
of equations with $R^2 = 1$. To avoid this, suppose the econometrician observes only a subset
$(y_t, x_{2t})$ of the variables, where $x_t = (x_{1t}, x_{2t})$ and $K(L) =
\operatorname{diag}(K_1(L), K_2(L))$. Writing $z_t = (y_t^T, x_{2t}^T)^T$ and
$\epsilon_t = (\epsilon_{1t}^T, \epsilon_{2t}^T)^T$, the observed system is

```{math}
:label: eq-vd-10
S(L)\, z_t = R(L)\, \epsilon_t ,
\qquad
S(L) = \begin{pmatrix} H(L) & 0 \\ 0 & I \end{pmatrix},
\quad
R(L) = \begin{pmatrix} M(K_1(L)) & M(K_2(L)) \\ 0 & K_2(L) \end{pmatrix}.
```

Equation {eq}`eq-vd-10` is a moving average expressing $z_t$ in terms of the agents' shocks
$\epsilon_t$. The shocks $\epsilon_t$ are fundamental for the agents' information set
$(x_{1t}, x_{2t})$; the question is whether they are also fundamental for the
*econometrician's* data $z_t = (y_t, x_{2t})$. By construction the econometrician's space is
contained in the agents'; the issue is whether it is as large. It is **not**, in general:
$\epsilon_t$ is fundamental for $z_t$ if and only if the zeros of $\det R(z)$ do not lie inside
the unit circle, i.e.

```{math}
:label: eq-vd-11
\det M(K_1(z)) = 0 \;\Longrightarrow\; |z| \geq 1 ,
```

and {cite:t}`hansensargent1980formulating` exhibit a class of models — not thin in any natural sense — for
which {eq}`eq-vd-11` fails.

When {eq}`eq-vd-11` fails, the Wold representation that a vector autoregression recovers is
**not** {eq}`eq-vd-10`. Instead there is a matrix polynomial $G(L)^T = \sum_{j\geq 0} G_j^T
L^j$, one-sided in nonnegative powers of $L$, with $G(L^{-1})G(L)^T = I$, that "flips" the
inside-the-circle zeros of $\det R(z)$ to the outside (a *Blaschke factorization*). Setting
$R^*(L) = R(L)\,G(L^{-1})$ and $\epsilon_t^* = G(L)^T\epsilon_t$, the equilibrium can be
re-expressed as

```{math}
:label: eq-vd-12
S(L)\, z_t = R^*(L)\,\epsilon_t^* ,
```

and now $\epsilon_t^*$ *is* fundamental for $z_t$: {eq}`eq-vd-12` is the Wold representation,
and $\epsilon_t^* = R^*(L)^{-1} S(L) z_t$ is recovered by the vector autoregression. The Wold
innovation $\epsilon_t^*$ is a *one-sided distributed lag of current and past* $\epsilon_t$'s,
$\epsilon_t^* = G(L)^T\epsilon_t$ — it mixes the agents' current surprise with *old news*. Only
when no zeros of $\det R(z)$ lie inside the unit circle can $G(L)^T$ be taken to be the
identity and $\epsilon_t^* = \epsilon_t$. Otherwise the two white noises differ, and their
contemporaneous covariances obey the strict inequality

```{math}
:label: eq-vd-13
E\big(R_0^*\epsilon_t^*\big)\big(R_0^*\epsilon_t^*\big)^T
 \;>\; E\big(R_0\,\epsilon_t\big)\big(R_0\,\epsilon_t\big)^T :
```

the contemporaneous innovation the econometrician sees carries *more* variance than the
agents' contemporaneous surprise, because the econometrician's innovation has folded in past
shocks that the agents already knew.

## A numerical example: price and quantity in a single market

A useful illustration is a single market in which the econometrician observes only the
$2\times 1$ vector $y_t = (q_t, p_t)$ of quantity and price — there are no separately observed
forcing variables $x_t$, so {eq}`eq-vd-10` specializes to $H(L)y_t = M(C_1(L))\epsilon_{1t}$.

A representative supplier and a representative demander each solve a linear-quadratic dynamic
problem, and their first-order (Euler) conditions are

```{math}
:label: eq-vd-14
-E_t\Big\{\big[h_s + g_s(1-\beta L^{-1})(1-L)\big]q_t\Big\} + p_t = s_t ,
```

```{math}
:label: eq-vd-15
-E_t\Big\{h_d + g_d\big[a(\beta L^{-1})\,a(L)\big]\Big\}q_t - p_t = d_t ,
```

where {eq}`eq-vd-14` is the supply Euler equation of a competitive firm with quadratic costs
of adjusting output (in $(1-L)q_t$), {eq}`eq-vd-15` is the demand Euler equation with
$a(L) = a_0 + a_1 L + a_2 L^2 + a_3 L^3 + a_4 L^4$ describing a durable-services technology,
$\beta$ is the discount factor, and $s_t, d_t$ are serially correlated supply and demand
shocks,

```{math}
:label: eq-vd-16
s_t = B_s(L)\, w_{st} , \qquad d_t = B_d(L)\, w_{dt} ,
```

with $B_s(z), B_d(z)$ having zeros outside the unit circle and $w_{st}, w_{dt}$ mutually
uncorrelated white noises that agents observe. Setting $y_t = (q_t, p_t)$, $\epsilon_t =
(w_{st}, w_{dt})$, $K_2(L) = 0$, and $K_1(L) = \operatorname{diag}(B_s(L), B_d(L))$, the model
is a member of the class {eq}`eq-vd-6`. Eliminating $p_t$ between {eq}`eq-vd-14` and
{eq}`eq-vd-15` gives a single Euler equation in $q_t$ whose two-sided characteristic operator
factors as $J(L^{-1})H(L) = E(L)$, where

$$
E(L) = \begin{pmatrix}
 -\big(h_s + g_s(1-L)(1-\beta L^{-1})\big) & 1 \\[2pt]
 -\big(h_d + g_d\, a(L)\,a(\beta L^{-1})\big) & -1
\end{pmatrix},
$$

with the zeros of $\det J(z)$ inside and those of $\det H(z)$ outside the unit circle. The
equilibrium then has the two representations

```{math}
:label: eq-vd-17
\underbrace{S(L)\begin{pmatrix} q_t \\ p_t \end{pmatrix} = R(L)\begin{pmatrix} w_{dt} \\ w_{st} \end{pmatrix}}_{\text{structural shocks (agents' surprises)}}
\qquad\text{and}\qquad
\underbrace{S(L)\begin{pmatrix} q_t \\ p_t \end{pmatrix} = R^*(L)\,\epsilon_t^*}_{\text{Wold innovations (what a VAR recovers)}},
```

with $S(L) = H(L)$ and $R(L) = M(C_1(L))$. For this market the zeros of $\det R(z)$ lie
*inside* the unit circle, so {eq}`eq-vd-11` fails and the two white noises differ.

We compute the example with the parameters

$$
h_s = h_d = 1, \quad g_s = 10, \quad g_d = 0.1, \quad \beta = 1/1.05,
$$
$$
a(L) = 1 + .8L + .6L^2 + .4L^3 + .2L^4,
$$
$$
B_d(L) = (1+.6L)(1+.4L)(1+.2L), \qquad B_s(L) = (1-.8L)(1+.4L)(1+.2L),
$$
$$
E w_{st}^2 = .5, \qquad E w_{dt}^2 = 4, \qquad E w_{st} w_{dt} = 0 .
$$

To solve the model we map it into the class of linear-quadratic economies of Hansen and
Sargent (2013) — the construction in the paper's appendix — and use QuantEcon's `DLE`
(dynamic linear economy) class, which solves the optimal resource-allocation problem by
dynamic programming and returns the state-space equilibrium $x_{t+1} = A^o x_t + C w_{t+1}$,
$z_t = G x_t$. Impulse responses to the agents' structural shocks are read directly off this
system; the Wold representation {eq}`eq-vd-12` and its innovations $\epsilon_t^*$ are obtained
by passing the equilibrium through the Kalman filter (the innovations representation), and the
filter that maps $\epsilon_t \mapsto \epsilon_t^*$ is the *whitener*. The figure below
collects the results.

```{figure} ../figures/ch36a_two_difficulties.png
:name: fig-vd
:width: 100%
:align: center

**Figure.** The single-market example of Hansen and Sargent (1991). *Row 1* — the response
of quantity and price to the agents' **structural shocks** (representation {eq}`eq-vd-17`,
left): a demand surprise sends price up sharply on impact with little quantity response, while
a supply surprise sends quantity and price off in opposite directions. *Row 2* — the response
to the **Wold innovations** a vector autoregression recovers (representation {eq}`eq-vd-17`,
right, with $q$ ordered first in the Gram–Schmidt orthogonalization). *Row 3* — the response
of the **Wold innovations $\epsilon_t^*$ to the structural shocks** ($\epsilon_t^* =
G(L)^T\epsilon_t$): the demand surprise is impounded almost contemporaneously (mostly in the
price innovation), but the supply surprise enters the recovered innovations as a *distributed
lag*. The recovered innovations are therefore mixtures of current and past structural shocks,
not the structural shocks themselves. Computed with QuantEcon's `DLE` class (a port of the
authors' MATLAB `twodiff1.m`); see
[`code/ch36a_two_difficulties.py`](https://github.com/thomassargent30/sargent-time-series/blob/main/code/ch36a_two_difficulties.py).
```

The economic reading of the rows is the substance of the example. The supply innovation
$w_{st}$ shows up in the recovered innovations only as a *distributed lag* (Row 3, right),
because the slow adjustment of quantity means that a supply surprise is revealed to the
econometrician gradually, through the path of $(q_t, p_t)$, rather than all at once. The demand
innovation $w_{dt}$, by contrast, hits price almost contemporaneously (Row 3, left), so the
price innovation in the autoregression is a fairly timely indicator of the demand surprise —
but the quantity innovation is a long distributed lag, mainly of the *supply* surprise. An
econometrician who interpreted the autoregression's innovations as the agents' shocks would
therefore misattribute the timing and the sources of the market's response.

The covariance inequality {eq}`eq-vd-13` is the quantitative signature of the discrepancy. For
this parameterization the contemporaneous covariance of the recovered (Wold) innovations
exceeds that of the agents' structural innovations,

$$
E\big(R_0^*\epsilon_t^*\big)\big(R_0^*\epsilon_t^*\big)^T
 - E\big(R_0\,\epsilon_t\big)\big(R_0\,\epsilon_t\big)^T \;\succeq\; 0
 \quad(\text{positive semidefinite, and nonzero}),
$$

confirming that the innovation the econometrician sees carries *more* contemporaneous variance
than the agents' surprise — it has folded in shocks that the agents already knew. None of this
can be detected from the autoregression alone; it takes the cross-equation restrictions of the
economic theory, estimated as in {cite:t}`hansensargent1980formulating`, to recover $R(L)$ — and hence the
agents' shocks $\epsilon_t$ — from a record on $(q_t, p_t)$, even when some zeros of $\det R(z)$
lie inside the unit circle.

```{admonition} The moral for innovation accounting
:class: tip

A vector autoregression always recovers *some* fundamental white noise $a_t$ for the observed
$z_t$, and Sims's innovation accounting always produces a tidy variance decomposition. But the
fundamental noise for the data need not be the fundamental noise for the agents. When it is
not — as in this market, where quantity adjusts sluggishly and so reveals supply surprises only
with a lag — the impulse responses and variance decompositions describe the data's own internal
forecasting structure, not the economy's response to the surprises that actually move agents.
Reading economic meaning into them requires the restrictions of a model, not just the
autoregression.
```

## Dynamic supply and demand curves

It is illuminating to solve the supplier's and the demander's Euler equations
{eq}`eq-vd-14`–{eq}`eq-vd-15` *separately*, before imposing market clearing. Each is an
expectational difference equation in $q_t$, and each is solved by the now-standard device of
{doc}`ch09_difference_equations` and {doc}`20_geometric_leads`: **factor the characteristic
operator into a stable root and an unstable root, solve the stable root backwards into a
feedback on lagged quantities, and solve the unstable root forwards into a geometric sum of
expected future variables.** The two solutions are *dynamic supply and demand curves*.

**The dynamic supply curve.** Write the supplier's Euler equation {eq}`eq-vd-14` as
$E_t\,\phi_s(L)\,q_t = p_t - s_t$, with characteristic operator $\phi_s(L) = h_s +
g_s(1-\beta L^{-1})(1-L)$. Because $\phi_s$ is symmetric under $L \mapsto \beta L^{-1}$, the
roots of its symbol come in a reciprocal pair $(\delta_s,\ \beta/\delta_s)$, the two solutions
of the supplier's characteristic equation

```{math}
:label: eq-vd-18
z^2 - \Big[(1+\beta) + \tfrac{h_s}{g_s}\Big]\,z + \beta = 0 .
```

Let $\delta_s$ be the smaller root, $|\delta_s| < \sqrt{\beta} < 1$ (for the chapter's
parameters $\delta_s \approx 0.709$). Then $\phi_s$ factors as

```{math}
:label: eq-vd-19
\phi_s(L) = \frac{g_s\beta}{\delta_s}\,\big(1 - \delta_s L^{-1}\big)\,\big(1 - \tfrac{\delta_s}{\beta}L\big),
```

an unstable **forward** factor $(1-\delta_s L^{-1})$, with $|\delta_s| < 1$, and a stable
**backward** factor $(1-\tfrac{\delta_s}{\beta}L)$, with $|\delta_s/\beta| < 1$. Solving the
forward root forward — a geometric sum of expected future variables, exactly as in
{doc}`20_geometric_leads` — and reading the backward root as a feedback on the lag of $q$, the
supplier's Euler equation becomes the **dynamic supply curve**

```{math}
:label: eq-vd-20
q_t = \frac{\delta_s}{\beta}\,q_{t-1}
 \;+\; \frac{\delta_s}{g_s\beta}\,E_t\sum_{j=0}^{\infty}\delta_s^{\,j}\,\big(p_{t+j} - s_{t+j}\big).
```

Current quantity supplied is a geometrically declining feedback on its own lag $q_{t-1}$ plus
the conditional expectation of a *discounted geometric sum of future prices* $p_{t+j}$ and
*future supply shocks* $s_{t+j}$, discounted at the rate $\delta_s$. Higher expected future
prices raise current supply (the curve slopes up in the price path); a higher expected cost
shock $s_{t+j}$ lowers it. Only the single lag $q_{t-1}$ appears, because the supplier's
adjustment cost penalizes $(1-L)q_t$ one period at a time.

**The dynamic demand curve.** The demander's Euler equation {eq}`eq-vd-15` is
$E_t\,\phi_d(L)\,q_t = -(p_t + d_t)$, with characteristic operator $\phi_d(L) = h_d +
g_d\,a(\beta L^{-1})\,a(L)$. Since $a(L)$ has degree four, $\phi_d$ is again symmetric under
$L \mapsto \beta L^{-1}$, but now its symbol has eight roots in four reciprocal pairs
$(\delta_{d,i},\ \beta/\delta_{d,i})$, $i = 1,\dots,4$ (for the chapter's parameters the four
stable $\delta_{d,i}$ are two complex-conjugate pairs of modulus $\approx 0.30$ and
$\approx 0.39$). Collecting the four stable roots $|\delta_{d,i}| < \sqrt{\beta}$, the
factorization is

```{math}
:label: eq-vd-21
\phi_d(L) = \nu_d\, c_d(L)\, c_d(\beta L^{-1}),
\qquad
c_d(L) = \prod_{i=1}^{4}\Big(1 - \tfrac{\delta_{d,i}}{\beta}\,L\Big) = 1 - \sum_{k=1}^{4}\gamma_{d,k}\,L^{k},
```

with $\nu_d > 0$ a normalizing constant and $c_d(\beta L^{-1}) = \prod_i(1 - \delta_{d,i}L^{-1})$
the forward factor. Inverting the forward factor with a partial-fraction expansion
({doc}`18a_partial_fractions`),

$$
\big[c_d(\beta L^{-1})\big]^{-1} = \sum_{i=1}^{4}\frac{A_{d,i}}{1 - \delta_{d,i}L^{-1}},
\qquad
A_{d,i} = \Big[\textstyle\prod_{k\neq i}\big(1 - \delta_{d,k}/\delta_{d,i}\big)\Big]^{-1},
$$

and reading the backward factor $c_d(L)$ as a feedback on lags, the demander's Euler equation
becomes the **dynamic demand curve**

```{math}
:label: eq-vd-22
q_t = \sum_{k=1}^{4}\gamma_{d,k}\,q_{t-k}
 \;-\; \frac{1}{\nu_d}\sum_{i=1}^{4} A_{d,i}\,E_t\sum_{j=0}^{\infty}\delta_{d,i}^{\,j}\,\big(p_{t+j} + d_{t+j}\big).
```

Now current quantity demanded depends on *four* lags $q_{t-1},\dots,q_{t-4}$ — the demander's
durable-services technology $a(L)$ spreads adjustment over four periods — and on a sum of
geometric feedforward terms, one per stable root $\delta_{d,i}$, each a discounted sum of
future prices and future demand shocks. (The $\delta_{d,i}$ come in conjugate pairs, so the
weights combine into real, damped-oscillatory coefficients.) The price terms enter with a
*negative* sign: higher expected future prices lower current demand, so the demand curve slopes
down in the price path.

**Both shocks shift both curves.** Taken at face value, the dynamic supply curve
{eq}`eq-vd-20` seems to involve only supply shocks $s_{t+j}$, and the dynamic demand curve
{eq}`eq-vd-22` only demand shocks $d_{t+j}$. But each curve also contains the conditional
expectations $E_t\,p_{t+j}$ of *future prices*, and in equilibrium the price process is driven
by **both** shocks. A demand surprise that moves expected future prices therefore shifts the
dynamic *supply* curve through its term $E_t\sum_j \delta_s^{\,j} p_{t+j}$, and a supply
surprise that moves expected future prices shifts the dynamic *demand* curve through its terms
$E_t\sum_j \delta_{d,i}^{\,j} p_{t+j}$. It is exactly this dependence on forecasts of future
prices — absent from static supply and demand curves — that couples the two sides of the market
and makes the equilibrium dynamics richer than a sequence of momentary intersections.

**Equilibrium.** The rational expectations equilibrium equates quantity demanded to quantity
supplied period by period. Imposing $q_t^{\text{supply}} = q_t^{\text{demand}} = q_t$ in
{eq}`eq-vd-20` and {eq}`eq-vd-22`, and requiring that the price forecasts $E_t\,p_{t+j}$ that
appear in both curves be the ones generated by the equilibrium price process itself, pins down
the joint $(q_t, p_t)$ process. That equilibrium is precisely the structural representation
$S(L)\,(q_t, p_t)^T = R(L)\,(w_{dt}, w_{st})^T$ of {eq}`eq-vd-17` — the one whose innovations a
vector autoregression generally fails to recover.

## Each side of the market as a price-taking linear regulator

The dynamic supply and demand curves are the first-order conditions of two *distinct*
optimization problems — one solved by the representative supplier, one by the representative
demander — and in the rational expectations equilibrium **both agents are price takers**. We
now make those problems explicit and cast each as a discounted **optimal linear regulator** in
which the agent faces the equilibrium price as an exogenous stochastic process. This is an
instance of the "Big $X$, little $x$" (or "Big $K$, little $k$") device used throughout modern
macroeconomics, and the recursive set-up is the one followed in
[Section 50.7.1, "Recursive formulation of a follower's problem," of the QuantEcon dynamic
Stackelberg lecture](https://python-advanced.quantecon.org/dyn_stack.html): append the
*exogenous* aggregate law of motion to the agent's own state, and solve an ordinary linear
regulator.

**The two optimization problems.** The representative supplier chooses $\{q_t\}$ to maximize

```{math}
:label: eq-vd-23
E_0\sum_{t=0}^{\infty}\beta^t\Big\{\, p_t q_t - \tfrac{h_s}{2}q_t^2 - \tfrac{g_s}{2}\big(q_t - q_{t-1}\big)^2 - s_t q_t \,\Big\},
```

taking the market price $\{p_t\}$ as given (and observing its cost shock $s_t$): it earns
revenue $p_t q_t$, pays a quadratic cost of *adjusting* output, and is buffeted by $s_t$. The
representative demander chooses $\{q_t\}$ to maximize

```{math}
:label: eq-vd-24
E_0\sum_{t=0}^{\infty}\beta^t\Big\{\, -p_t q_t - \tfrac{h_d}{2}q_t^2 - \tfrac{g_d}{2}\big(a(L)q_t\big)^2 - d_t q_t \,\Big\},
```

again taking $\{p_t\}$ as given: it pays $p_t q_t$ for the good, values the service flow
$a(L)q_t$ generated by current and past purchases, and is shifted by $d_t$. Differentiating
{eq}`eq-vd-23` and {eq}`eq-vd-24` with respect to $q_t$ returns exactly the supplier's and
demander's Euler equations {eq}`eq-vd-14` and {eq}`eq-vd-15`.

**The exogenous price process (Big $X$).** In a rational expectations equilibrium each agent's
forecasts of future prices must be model-consistent — they must be the forecasts implied by the
*equilibrium* price process, which is the one the Hansen–Sargent `DLE` computed above. Write
that equilibrium in state-space form,

```{math}
:label: eq-vd-25
X_{t+1} = A\,X_t + C\,w_{t+1}, \qquad p_t = G_p\,X_t ,
```

with $A = A^o$, $C$, and the price selector $G_p = M_c$ taken directly from the `DLE` solution
(the supply and demand shocks are linear in the same state, $s_t = G_s X_t$ and $d_t = G_d
X_t$). A price-taking agent treats $X_t$ as an exogenous Markov state it cannot influence. The
key consequence is that, because $X_t$ is Markov, every conditional expectation of a future
price is a *linear function of the current state*,

```{math}
:label: eq-vd-26
E_t\,p_{t+j} = G_p\,A^{\,j}\,X_t ,
```

so the geometric feed-forward sums $E_t\sum_j \delta^{\,j} p_{t+j}$ in the dynamic supply and
demand curves collapse into linear functions of $X_t$.

**The recursive formulation.** Following Section 50.7.1 of the dynamic Stackelberg lecture, we
append the exogenous law of motion {eq}`eq-vd-25` to the agent's own state and solve an optimal
linear regulator. For the supplier, the composite state stacks the aggregate price-process
state $X_t$ ("Big $X$") on top of the supplier's own lagged quantity $q_{t-1}$ ("little $x$"),
$\widehat X_t = (X_t^{\,\prime}, q_{t-1})'$, and evolves as

```{math}
:label: eq-vd-27
\widehat X_{t+1} =
\begin{pmatrix} A & 0 \\ 0 & 0 \end{pmatrix}\widehat X_t
+ \begin{pmatrix} 0 \\ 1 \end{pmatrix} q_t
+ \begin{pmatrix} C \\ 0 \end{pmatrix} w_{t+1} .
```

The block-triangular transition makes $X_t$ exogenous — the supplier's choice $q_t$ cannot move
it — while $q_t$ becomes next period's lag. Writing the one-period return {eq}`eq-vd-23` as a
quadratic form in $(\widehat X_t, q_t)$ (using $p_t = G_p X_t$, $s_t = G_s X_t$), the supplier
solves a standard discounted linear regulator, and its optimal policy is the feedback rule

$$
q_t = -F\,\widehat X_t = \frac{\delta_s}{\beta}\,q_{t-1} \;+\; (\text{a linear feed-forward in } X_t).
$$

The coefficient on the supplier's own lag is the stable root $\delta_s/\beta$ of the supply
curve {eq}`eq-vd-20`, and — by {eq}`eq-vd-26` — the feed-forward $-F_X X_t$ is exactly the
geometric sum $\frac{\delta_s}{g_s\beta}E_t\sum_j \delta_s^{\,j}(p_{t+j}-s_{t+j})$ written as a
linear function of the state. The demander's regulator has the same shape with a richer own
state $(q_{t-1},\dots,q_{t-4})$ — four lags, because the services technology $a(L)$ spreads
adjustment over four periods — and its feedback on those four lags reproduces the coefficients
$\gamma_{d,k}$ of the demand curve {eq}`eq-vd-22`.

Solving the two regulators with QuantEcon's `LQ` routine, facing the `DLE` equilibrium price
process $(A, C, G_p)$, returns precisely the dynamic-curve coefficients: the supplier's feedback
on $q_{t-1}$ is $\delta_s/\beta = 0.744$, and the demander's feedback on $(q_{t-1},\dots,q_{t-4})$
is $(\gamma_{d,1},\dots,\gamma_{d,4}) = (-0.117,\,-0.079,\,-0.045,\,-0.017)$ — exactly the
coefficients read off the factored Euler equations. See
[`code/ch36a_price_taker_lq.py`](https://github.com/thomassargent30/sargent-time-series/blob/main/code/ch36a_price_taker_lq.py).

**Big $X$ equals little $x$: the equilibrium fixed point.** Each agent's rule is a best response
to the price process $X_t$, and *only* a best response: the agent takes $X_t$ as given, and its
own little-$x$ choice does not move Big $X$. What closes the model is the requirement — the heart
of the "Big $X$, little $x$" trick — that the aggregate the agents respond to be consistent with
the aggregate their choices produce. Here that means the market clears, $q_t^{\text{supply}} =
q_t^{\text{demand}}$, at the common price, and the price process $X_t$ that both agents forecast
is the very process those market-clearing quantities generate. The Hansen–Sargent `DLE` computes
exactly this fixed point: its equilibrium price process is the one for which the suppliers' and
demanders' price-taking best responses clear the market period by period. Feeding that process
back into either agent's regulator — as the verification does — returns a quantity rule
consistent with it.

**Open-loop versus closed-loop decision rules.** The construction has produced two distinct
pairs of decision rules. We keep them apart. The dynamic supply curve
{eq}`eq-vd-20` and the dynamic demand curve {eq}`eq-vd-22` give each agent's optimal quantity as
a function of its own past quantities and of its forecasts $E_t\,p_{t+j}$ of an **arbitrary**
price process $\{p_t\}_t$ that it takes as exogenously given. They are best responses to
*whatever* price process the agent happens to face, and they assume nothing about how that price
is generated — in particular, they do not assume it is produced by the market the agent trades
in. In this sense {eq}`eq-vd-20` and {eq}`eq-vd-22` are an **open-loop** pair: the loop between
the market's quantity and its price is left open, and the rules are valid **outside** any
particular rational expectations equilibrium. The regulator feedback rules $q_t = -F\,\widehat
X_t$ are a different pair. They are the *same* optimizing behavior with the *equilibrium* price
process {eq}`eq-vd-25` substituted in: evaluating the feed-forward sums with $E_t\,p_{t+j} = G_p
A^{\,j} X_t$ collapses each open-loop curve into a feedback on the equilibrium state $X_t$
(together with the agent's own lags). These are a **closed-loop** pair — the price each agent
responds to is now the very one the equilibrium system produces, so the market-clearing loop is
closed — and they are the supply and demand decision rules that obtain **inside** the rational
expectations equilibrium. Equivalently, {eq}`eq-vd-20`–{eq}`eq-vd-22` are the *outside-an-REE*
pair of rules and $q_t=-F\widehat X_t$ the *inside-an-REE* pair. The two coincide only after one
fixes the price process to be the equilibrium one; passing from the open-loop curve to the
closed-loop rule — from outside the REE to inside it — is exactly the act of imposing rational
expectations on the agents' price forecasts.

**The key lesson.** Whichever representation one adopts — the open-loop supply and demand curves
{eq}`eq-vd-20`–{eq}`eq-vd-22` or their closed-loop counterparts $q_t = -F\,\widehat X_t$ — both
deliver the same message: each decision maker's quantity *today* depends not on today's price
alone but on the entire prospective **continuation path** of the product price,
$\{p_{t+j}\}_{j\ge 0}$, forecast from today out into the indefinite future. In the open-loop
curves the dependence is explicit, through the discounted geometric sums
$E_t\sum_{j\ge 0}\delta^{\,j}\,p_{t+j}$ of expected future prices; in the closed-loop rules it is
encoded in the feedback on the equilibrium state $X_t$, which carries exactly those forecasts via
$E_t\,p_{t+j} = G_p A^{\,j} X_t$. Either way, supply and demand are inescapably
**forward-looking**: current decisions are governed by the whole expected future path of prices,
not by the current price in isolation. And because that equilibrium price path is itself driven
by *both* disturbances, both pairs of rules make each side's quantity depend on both shock
processes: the supply shock $s_t$ and the demand shock $d_t$ each appear in the dynamic supply
curve and in the dynamic demand curve alike. Today's suppliers and today's demanders both react
to supply *and* demand shocks — through their common effect on the prospective path of prices.

## References

```{bibliography}
:labelprefix: IV
:filter: key in {"hansensargent1980formulating", "hansensargent1991twodifficulties", "hansensargent2013recursive", "sargentstachurski_dynstack", "sims1980macroeconomics"}
```