Tom's Mar 15 edits

Author: thomassargent30

lectures/_static/quant-econ.bib

@@ -224,6 +224,14 @@ @book{Burns_2023
   address = {New York}
 }
 
+@book{lucas1981rational,
+  title={Rational expectations and econometric practice},
+  author={Lucas, Robert E and Sargent, Thomas J},
+  year={1981},
+  publisher={U of Minnesota Press},
+  address = {Minneapolis, Minnesota}
+}
+
 @article{Orcutt_Winokur_69,
   issn      = {00129682, 14680262},
   abstract  = {Monte Carlo techniques are used to study the first order autoregressive time series model with unknown level, slope, and error variance. The effect of lagged variables on inference, estimation, and prediction is described, using results from the classical normal linear regression model as a standard. In particular, use of the t and x^2 distributions as approximate sampling distributions is verified for inference concerning the level and residual error variance. Bias in the least squares estimate of the slope is measured, and two bias corrections are evaluated. Least squares chained prediction is studied, and attempts to measure the success of prediction and to improve on the least squares technique are discussed.},

lectures/_toc.yml

@@ -104,7 +104,8 @@ parts:
   - file: cross_product_trick
   - file: perm_income
   - file: perm_income_cons
-  - file: theil 
+  - file: theil_1 
+  - file: theil_2
   - file: lq_inventories
 - caption: Optimal Growth
   numbered: true

lectures/theil_1.md

@@ -0,0 +1,192 @@
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+
+(certainty_equiv_robustness)=
+```{raw} jupyter
+<div id="qe-notebook-header" align="right" style="text-align:right;">
+        <a href="https://quantecon.org/" title="quantecon.org">
+                <img style="width:250px;display:inline;" width="250px" src="https://assets.quantecon.org/img/qe-menubar-logo.svg" alt="QuantEcon">
+        </a>
+</div>
+```
+
+# Certainty Equivalence 
+
+```{index} single: Certainty Equivalence; Robustness
+```
+
+```{index} single: LQ Control; Permanent Income
+```
+
+```{contents} Contents
+:depth: 2
+```
+
+
+In addition to what's in Anaconda, this lecture will need the following libraries:
+
+```{code-cell} ipython3
+---
+tags: [hide-output]
+---
+!pip install quantecon
+```
+
+
+## The Central Problem of Empirical Economics
+
+The papers collected in {cite}`lucas1981rational` address a single overarching question: given observations on an agent's behavior in a particular economic environment, what can we infer about how that behavior **would have differed** had the environment been altered? This is the problem of policy-invariant structural inference.
+
+The difficulty is immediate. Observations arise under one environment; we wish to predict behavior under another. Unless we understand *why* the agent behaves as he does—that is, unless we recover the deep objectives that rationalize observed decisions—estimated behavioral relationships are silent on this question.
+
+---
+
+## A Formal Setup
+
+Consider a single decision maker whose situation at date $t$ is fully described by two state variables $(x_t, z_t)$.
+
+**The environment** $z_t \in S_1$ is selected by "nature" and evolves exogenously according to
+
+```{math}
+:label: eq:z_transition
+z_{t+1} = f(z_t,\, \epsilon_t),
+```
+
+where the innovations $\epsilon_t \in \mathcal{E}$ are i.i.d. draws from a fixed c.d.f. $\Phi(\cdot) : \mathcal{E} \to [0,1]$. The function $f : S_1 \times \mathcal{E} \to S_1$ is called the **decision maker's environment**.
+
+**The endogenous state** $x_t \in S_2$ is under partial control of the agent. Each period the agent selects an action $u_t \in U$. A fixed technology $g : S_1 \times S_2 \times U \to S_2$ governs the transition
+
+```{math}
+:label: eq:x_transition
+x_{t+1} = g(z_t,\, x_t,\, u_t).
+```
+
+**The decision rule** $h : S_1 \times S_2 \to U$ maps the agent's current situation into an action:
+
+```{math}
+:label: eq:decision_rule
+u_t = h(z_t,\, x_t).
+```
+
+The econometrician observes (some or all of) the process $\{z_t, x_t, u_t\}$, the joint motion of which is determined by {eq}`eq:z_transition`, {eq}`eq:x_transition`, and {eq}`eq:decision_rule`.
+
+---
+
+## The Lucas Critique: Why Estimated Rules Are Not Enough
+
+Suppose we have estimated $f$, $g$, and $h$ from a long time series generated under a fixed environment $f_0$. This gives us $h_0 = T(f_0)$, where $T$ is the (unknown) functional mapping environments into optimal decision rules. But this single estimate, however precise, **reveals nothing** about how $T(f)$ varies with $f$.
+
+Policy evaluation requires knowledge of the entire map $f \mapsto T(f)$. Under an environment change $f_0 \to f_1$, agents will in general revise their decision rules $h_0 \to h_1 = T(f_1)$, rendering the estimated rule $h_0$ invalid for forecasting behavior under $f_1$.
+
+The only nonexperimental path forward is to recover the **return function** $V$ from which $h$ is derived as the solution to an optimization problem, and then re-solve that problem under the counterfactual environment $f_1$.
+
+---
+
+##  An Optimization Problem
+
+Assume the agent selects $h$ to maximize the expected discounted sum of current-period returns $V : S_1 \times S_2 \times U \to \mathbb{R}$:
+
+```{math}
+:label: eq:objective
+E_0\!\left\{\sum_{t=0}^{\infty} \beta^t\, V(z_t,\, x_t,\, u_t)\right\}, \qquad 0 < \beta < 1,
+```
+
+given initial conditions $(z_0, x_0)$, the environment $f$, and the technology $g$. Here $E_0\{\cdot\}$ denotes expectation conditional on $(z_0, x_0)$ with respect to the distribution of $\{z_1, z_2, \ldots\}$ induced by {eq}`eq:z_transition`.
+
+In principle, knowledge of $V$ (together with $g$ and $f$) allows one to compute $h = T(f)$ theoretically and hence to trace out $T(f)$ for any counterfactual $f$. The empirical question is whether $V$ can itself be recovered from observations on $\{f, g, h\}$—a problem of structural identification that, at this level of generality, is formidably difficult.
+
+:::{note}
+The decision rule is in general a functional $h = T(f, g, V)$. The dependence on $g$ and $V$ is suppressed in the main text but made explicit when needed.
+:::
+
+---
+
+## A Linear-Quadratic Specialization and Certainty Equivalence
+
+Progress at the level of generality of Section 4 requires restricting the primitives. The most productive restriction, exploited in the bulk of the volume, imposes **quadratic** $V$ and **linear** $g$, which forces $h$ to be linear. Beyond computational tractability, this specialization delivers a striking structural result: the **certainty equivalence** theorem of Simon {cite}`simon1956dynamic`  and Theil {cite}`theil1957note`. 
+
+###  The Composite Decomposition of $h$
+
+Under quadratic $V$ and linear $g$, the optimal decision rule $h$ decomposes into two components applied in sequence.
+
+**Step 1 — Forecasting.** Define the infinite sequence of optimal point forecasts of all current and future states of nature:
+
+```{math}
+:label: eq:forecast_sequence
+\tilde{z}_t \;=\; \bigl(z_t,\;\; {}_{t+1}z_t^e,\;\; {}_{t+2}z_t^e,\;\ldots\bigr) \;\in\; S_1^\infty,
+```
+
+where ${}_{t+j}z_t^e$ denotes the least-mean-squared-error forecast of $z_{t+j}$ formed at time $t$. The optimal forecast sequence is a (generally nonlinear) function of the current state:
+
+```{math}
+:label: eq:forecast_rule
+\tilde{z}_t = h_2(z_t).
+```
+
+The function $h_2 : S_1 \to S_1^\infty$ depends entirely on the environment $(f, \Phi)$ and is obtained as the solution to a **pure forecasting problem**, with no reference to preferences or technology.
+
+**Step 2 — Optimization.** Given the forecast sequence $\tilde{z}_t$, the optimal action is a **linear** function of $\tilde{z}_t$ and $x_t$:
+
+```{math}
+:label: eq:optimization_rule
+u_t = h_1(\tilde{z}_t,\, x_t).
+```
+
+The function $h_1 : S_1^\infty \times S_2 \to U$ depends entirely on preferences $(V)$ and technology $(g)$ but **not** on the stochastic environment $(f, \Phi)$.
+
+The full decision rule is therefore the **composite**:
+
+```{math}
+:label: eq:composite_rule
+\boxed{h(z_t, x_t) \;=\; h_1\!\bigl[h_2(z_t),\; x_t\bigr].}
+```
+
+###  The Separation Principle
+
+{eq}`eq:composite_rule` embodies a clean **separation** of the two sources of dependence in $h$:
+
+| Component | Depends on | Independent of |
+|-----------|-----------|----------------|
+| $h_1$ (optimization) | $V$, $g$ | $f$, $\Phi$ |
+| $h_2$ (forecasting)  | $f$, $\Phi$ | $V$, $g$ |
+
+Since policy analysis concerns changes in $f$, and since $h_1$ is invariant to $f$, the policy analyst need only re-solve the forecasting problem $h_2 = S(f)$ under the new environment, keeping $h_1$ fixed. The relationship of original interest, $h = T(f)$, then follows directly from {eq}`eq:composite_rule`.
+
+###  Certainty Equivalence and Perfect Foresight
+
+The name "certainty equivalence" reflects a further implication of the LQ structure: the function $h_1$ can be derived as if the agent **knew the future path $z_{t+1}, z_{t+2}, \ldots$ with certainty** — i.e., by solving the deterministic problem in which $\tilde{z}_t$ is treated as the realized path rather than a forecast. The stochasticity of the environment affects actions only through the forecast $\tilde{z}_t$; conditional on $\tilde{z}_t$, the optimization problem is deterministic.
+
+This means the LQ problem decouples into:
+
+ *  **Dynamic optimization under perfect foresight** — solve for $h_1$ from $(V, g)$ by treating $\tilde{z}_t$ as known. This is a standard deterministic LQ regulator problem and is independent of the environment $(f, \Phi)$.
+
+ *  **Optimal linear prediction** — solve for $h_2 = S(f)$ from $(f, \Phi)$ using least-squares forecasting theory. If $f$ is itself linear, $h_2$ is also linear and reduces to a standard Kalman/Wiener prediction formula.
+
+###  Cross-Equation Restrictions
+
+A hallmark of the rational expectations hypothesis as it appears in this framework is that it ties together what would otherwise be free parameters in different equations. The requirement that $\tilde{z}_t = h_2(z_t) = S(f)(z_t)$ — i.e., that agents' forecasts be *optimal* with respect to the *actual* law of motion $f$ — imposes **cross-equation restrictions** between the parameters of the forecasting rule $h_2$ and the parameters of the environment $f$. These restrictions, rather than any conditions on distributed lags within a single equation, are the operative empirical content of rational expectations.
+
+---
+
+##  A Trouble with  Ad Hoc Expectations 
+
+Prior practice, exemplified by the adaptive expectations mechanisms of Friedman {cite}`Friedman1956` and Cagan {cite}`Cagan`, directly postulated a particular form of {eq}`eq:forecast_rule`:
+
+```{math}
+:label: eq:adaptive_expectations
+\theta_t^e = \lambda \sum_{i=0}^{\infty} (1-\lambda)^i\, \theta_{t-i}, \qquad 0 < \lambda < 1,
+```
+
+treating the coefficient $\lambda$ as a free parameter to be estimated from data, with no reference to the underlying environment $f$.
+
+The deficiency is not that {eq}`eq:adaptive_expectations` is a distributed lag — linear forecasting rules are perfectly acceptable simplifications. The deficiency is that the **coefficients** of the distributed lag are left unrestricted by theory. The mapping $h_2 = S(f)$ shows that optimal forecasting coefficients are *determined* by $f$: when $f$ changes, $h_2$ changes, and so does $h$. An estimated $\lambda$ calibrated under $f_0$ is therefore non-structural and will give incorrect predictions whenever $f$ is altered. This is the econometric content of the critique that Muth's paper delivers.
+
+Rational expectations equates the subjective distribution that agents use in forming $\tilde{z}_t$ to the objective distribution $f$ that actually generates the data, thereby closing the model and eliminating free parameters in $h_2$.