From efe419b59055787e88e31f8acdd7999f8e384d79 Mon Sep 17 00:00:00 2001 From: thomassargent30 Date: Tue, 16 Jun 2026 18:57:05 +0800 Subject: [PATCH 1/9] Tom's June 16 edits of new lecture --- lectures/_toc.yml | 1 + lectures/kalman_filter_var.md | 1335 +++++++++++++++++++++++++++++++++ 2 files changed, 1336 insertions(+) create mode 100644 lectures/kalman_filter_var.md diff --git a/lectures/_toc.yml b/lectures/_toc.yml index 929fefe00..10f388624 100644 --- a/lectures/_toc.yml +++ b/lectures/_toc.yml @@ -65,6 +65,7 @@ parts: - file: wealth_dynamics - file: kalman - file: kalman_2 + - file: kalman_filter_var - file: organization_capital - file: measurement_models - caption: Search diff --git a/lectures/kalman_filter_var.md b/lectures/kalman_filter_var.md new file mode 100644 index 000000000..5c81d2fac --- /dev/null +++ b/lectures/kalman_filter_var.md @@ -0,0 +1,1335 @@ +--- +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 +--- + +# The Kalman Filter and Vector Autoregressions + +```{index} single: Kalman Filter +``` + +```{index} single: Vector Autoregression; and Kalman filter +``` + +## Overview + +This lecture derives the **Kalman filter** for a linear Gaussian state space system +and then uses it to construct **vector autoregressions (VARs)**. + +Our approach rests on repeated applications of the **population linear least squares** +projection formula — the insight that computing a conditional expectation of a jointly +normal random vector is the same as running a population OLS regression. + +The lecture covers: + +- deriving the Kalman filter recursions from first principles +- the **matrix Riccati difference equation** governing conditional covariance matrices +- the **innovations representation** and the **Gram-Schmidt** whitening property +- the structure of a **hidden Markov model** +- the **likelihood function** for a state space system and its role in maximum likelihood + and Bayesian estimation +- how the time-invariant Kalman filter generates a **vector autoregression** +- why the Kalman filter is an essential tool for **interpreting VARs** estimated from + economic data + +## The State Space System + +The Kalman filter applies to the **state space system** for $t \geq 0$: + +$$ +\begin{aligned} +x_{t+1} &= A x_t + C w_{t+1} \\ +y_t &= G x_t + v_t +\end{aligned} +$$ (eq:statespace) + +where + +- $x_t$ is an $n \times 1$ **state vector** (hidden, unobserved) +- $y_t$ is an $m \times 1$ vector of **signals** on the hidden state (observed) +- $w_{t+1}$ is a $p \times 1$ i.i.d. sequence of normal random variables with mean + $0$ and identity covariance matrix +- $v_t$ is an i.i.d. sequence of normal random variables with mean zero and + covariance matrix $R$ +- $w_{t+1}$ and $v_s$ are orthogonal for all $t+1$ and $s \geq 0$ + +The initial state satisfies + +$$ +x_0 \sim \mathcal{N}(\hat{x}_0, \Sigma_0) +$$ (eq:kalf3) + +We observe $y_t, \ldots, y_0$ but **not** $x_t, \ldots, x_0$ at time $t$, +and we know all first and second moments implied by +{eq}`eq:statespace` and {eq}`eq:kalf3`. + +## The Kalman Filter + +### Starting Distribution + +Working forward in time starting at $t = 0$, before observing $y_0$, +the specification {eq}`eq:statespace`–{eq}`eq:kalf3` implies that the +marginal distribution of $y_0$ is + +$$ +y_0 \sim \mathcal{N}(G \hat{x}_0,\; G \Sigma_0 G' + R) +$$ (eq:kalf4) + +For $t \geq 0$ let $y^t = [y_t, y_{t-1}, \ldots, y_0]$. +We want a convenient recursive representation of the conditional distribution of +$y_t$ given history $y^{t-1}$. + +The Kalman filter attains this by constructing recursive formulas for +$\hat{x}_t$ and $\Sigma_t$ such that the distribution of $y_t$ conditional on +$y^{t-1}$ generalises {eq}`eq:kalf4` to + +$$ +y_t \sim \mathcal{N}(G \hat{x}_t,\; G \Sigma_t G' + R) +$$ (eq:kalf400) + +for $t \geq 1$, where the distribution of $x_t$ conditional on $y^{t-1}$ is +$\mathcal{N}(\hat{x}_t, \Sigma_t)$. + +The objects $\hat{x}_t$ and $\Sigma_t$ characterise the **population regression** + +$$ +\hat{x}_t = E[x_t \mid y_{t-1}, \ldots, y_0] +$$ + +and the **conditional covariance matrix** + +$$ +\Sigma_t = E\!\left[(x_t - \hat{x}_t)(x_t - \hat{x}_t)' \mid y_{t-1}, \ldots, y_0\right] +$$ + +### Derivation + +At each date, our approach is to **regress what we do not know on what we know**. + +```{note} +Because our assumptions imply that $\{x_t, y_t\}_{t=0}^\infty$ is a jointly normal +stochastic process, linear least squares regressions equal conditional mathematical +expectations — each step below is an application of Bayes' law. Under the weaker +assumption that all means and covariances exist without joint normality, the same +calculations yield "wide-sense conditional expectations" that coincide with true +conditional expectations only when those conditional expectations are linear. +``` + +**Arrive at $t = 0$** knowing $\hat{x}_0$ and $\Sigma_0$. + +**Innovation:** The information about $x_0$ in $y_0$ that is new relative to +$(\hat{x}_0, \Sigma_0)$ is the **innovation** + +$$ +a_0 \equiv y_0 - G \hat{x}_0 +$$ + +**Updating beliefs about $x_0$:** The conditional mean +$E[x_0 \mid y_0] = \hat{x}_0 + L_0(y_0 - G\hat{x}_0)$ satisfies the +population regression formula + +$$ +x_0 - \hat{x}_0 = L_0(y_0 - G\hat{x}_0) + \eta +$$ (eq:kalf5) + +where $\eta$ is the least squares residual. Orthogonality of $\eta$ to +$(y_0 - G\hat{x}_0)$ pins down $L_0$ via the normal equations + +$$ +E(x_0 - \hat{x}_0)(y_0 - G\hat{x}_0)' += L_0\, E(y_0 - G\hat{x}_0)(y_0 - G\hat{x}_0)' +$$ + +Evaluating the moment matrices and solving for $L_0$ gives + +$$ +L_0 = \Sigma_0 G'(G \Sigma_0 G' + R)^{-1} +$$ (eq:kalf6) + +**Forecasting $x_1$:** Note that + +$$ +x_1 = A\hat{x}_0 + A(x_0 - \hat{x}_0) + C w_1 +$$ (eq:kalf6a) + +Applying {eq}`eq:kalf5` gives $E[x_1 \mid y_0] = A\hat{x}_0 + AL_0(y_0 - G\hat{x}_0)$, +which we write as + +$$ +\hat{x}_1 = A\hat{x}_0 + K_0(y_0 - G\hat{x}_0) +$$ (eq:kalf7) + +where the **Kalman gain** at time 0 is + +$$ +K_0 = A \Sigma_0 G'(G \Sigma_0 G' + R)^{-1} +$$ (eq:kalf7a) + +**Updating the covariance:** Subtracting {eq}`eq:kalf7` from {eq}`eq:kalf6a` yields + +$$ +x_1 - \hat{x}_1 = A(x_0 - \hat{x}_0) + C w_1 - K_0(y_0 - G\hat{x}_0) +$$ (eq:kalf8) + +Using {eq}`eq:kalf8` and $y_0 = G x_0 + v_0$ to evaluate +$\Sigma_1 \equiv E[(x_1 - \hat{x}_1)(x_1 - \hat{x}_1)' \mid y_0]$ gives + +$$ +\Sigma_1 = (A - K_0 G)\Sigma_0(A - K_0 G)' + CC' + K_0 R K_0' +$$ (eq:kalf9) + +Thus $f(x_1 \mid y_0) \sim \mathcal{N}(\hat{x}_1, \Sigma_1)$. + +Collecting the time-$0$ equations: + +$$ +\begin{aligned} +a_0 &= y_0 - G\hat{x}_0 \\ +K_0 &= A\Sigma_0 G'(G\Sigma_0 G' + R)^{-1} \\ +\hat{x}_1 &= A\hat{x}_0 + K_0 a_0 \\ +\Sigma_1 &= CC' + K_0 R K_0' + (A - K_0 G)\Sigma_0(A - K_0 G)' +\end{aligned} +$$ (eq:kalf1000) + +System {eq}`eq:kalf1000` maps a mean–covariance pair $(\hat{x}_0, \Sigma_0)$ into a new +pair $(\hat{x}_1, \Sigma_1)$, with auxiliary outputs $(a_0, K_0)$. + +Recognising that "we are in the same situation at the start of period 1 as at +the start of period 0" activates a recursion — the **Kalman filter**. + +### The Kalman Filter Recursions + +Iterating system {eq}`eq:kalf1000` yields the **Kalman filter** for $t \geq 0$: + +$$ +\begin{aligned} +a_t &= y_t - G\hat{x}_t \\ +K_t &= A\Sigma_t G'(G\Sigma_t G' + R)^{-1} \\ +\hat{x}_{t+1} &= A\hat{x}_t + K_t a_t \\ +\Sigma_{t+1} &= CC' + K_t R K_t' + (A - K_t G)\Sigma_t(A - K_t G)' +\end{aligned} +$$ (eq:kalf10) + +Here $K_t$ is the **Kalman gain** at time $t$. + +### The Matrix Riccati Equation + +Substituting the expression for $K_t$ from the second line of {eq}`eq:kalf10` +into the fourth line gives an equivalent update formula: + +$$ +\Sigma_{t+1} = A\Sigma_t A' + CC' + - A\Sigma_t G'(G\Sigma_t G' + R)^{-1} G\Sigma_t A' +$$ (eq:riccati) + +Equation {eq}`eq:riccati` is the **matrix Riccati difference equation**. +It governs the sequence of conditional covariance matrices $\{\Sigma_t\}_{t=0}^\infty$ +without reference to the observations $\{y_t\}$. + +```{index} single: Riccati equation; matrix difference +``` + +## The Gram-Schmidt Process + +The random vector + +$$ +a_t = y_t - E[y_t \mid y_{t-1}, \ldots, y_0] +$$ + +is the **innovation** of $y_t$ with respect to $y^{t-1}$ — the part of $y_t$ +that cannot be predicted from past observations. + +Note that $E a_t a_t' = G\Sigma_t G' + R$, the matrix whose inverse appears in +the Kalman gain formula {eq}`eq:kalf10`. + +A direct calculation using $a_t = G(x_t - \hat{x}_t) + v_t$ shows that +$E a_t a_{t-1}' = 0$ and, more generally, $E[a_t \mid a_{t-1}, \ldots, a_0] = 0$. + +```{note} +An alternative argument from first principles: let $H(y^t)$ denote the closed linear +span of $y^t$. Since $a_{t+1} = y_{t+1} - E[y_{t+1} \mid y^t]$ is a least-squares +error, $a_{t+1} \perp H(y^t)$, and in particular $a_{t+1} \perp a_t$. +Thus $\{a_t\}$ is a white-noise process of innovations to $\{y_t\}$. +``` + +Sometimes {eq}`eq:kalf10` is called a **whitening filter**: it takes the signal process +$\{y_t\}$ as input and produces the white-noise innovation process $\{a_t\}$ as output. +The linear space $H(a^t)$ is an orthogonal basis for the linear space $H(y^t)$. + +Rather than computing $E[x_t \mid y_{t-1}, \ldots, y_0]$ via one large regression, +the Kalman filter performs a sequence of small regressions on successive orthogonal +components of the basis $[a_{t-1}, \ldots, a_0]$ — an instance of the +**Gram-Schmidt procedure**. + +```{index} single: Gram-Schmidt process +``` + +## Hidden Markov Model + +System {eq}`eq:statespace`–{eq}`eq:kalf3` is an example of a **hidden Markov model**. + +```{index} single: hidden Markov model +``` + +The observed process $\{y_t\}_{t=0}^\infty$ is *not* Markov, but the hidden +process $\{x_t\}_{t=0}^\infty$ *is* Markov. So is the process of means and +covariances $\{(\hat{x}_t, \Sigma_t)\}$, which are sufficient statistics for +the distribution of $x_t$ conditional on $[y_{t-1}, \ldots, y_0]$. + +## Estimation + +### The Innovations Representation + +The **innovations representation** emerging from the Kalman filter is + +$$ +\begin{aligned} +\hat{x}_{t+1} &= A\hat{x}_t + K_t a_t \\ +y_t &= G\hat{x}_t + a_t +\end{aligned} +$$ (eq:innovrep) + +where $\hat{x}_t = E[x_t \mid y^{t-1}]$ for $t \geq 1$ and +$E[a_t a_t' \mid y^{t-1}] = G\Sigma_t G' + R \equiv \Omega_t$. + +For $t \geq 1$, $E[y_t \mid y^{t-1}] = G\hat{x}_t$ and the conditional +distribution of $y_t$ given $y^{t-1}$ is $\mathcal{N}(G\hat{x}_t, \Omega_t)$. +The objects $(G\hat{x}_t, \Omega_t)$ emerging from the Kalman filter recursions +therefore completely characterise this conditional distribution. + +### The Likelihood Function + +We can factor the likelihood of a sample $(y_T, y_{T-1}, \ldots, y_0)$ as + +$$ +f(y_T, \ldots, y_0) + = f(y_T \mid y^{T-1})\, f(y_{T-1} \mid y^{T-2}) \cdots f(y_1 \mid y_0)\, f(y_0) +$$ (eq:diff100) + +The log conditional density of the $m \times 1$ vector $y_t$ is + +$$ +\log f(y_t \mid y^{t-1}) + = -\frac{m}{2}\log(2\pi) + - \frac{1}{2}\log\det(\Omega_t) + - \frac{1}{2}\, a_t' \Omega_t^{-1} a_t +$$ (eq:gauss100) + +Using {eq}`eq:gauss100` and {eq}`eq:kalf10` together, we can evaluate +the likelihood {eq}`eq:diff100` recursively for any parameter vector $\theta$ +that underlies the matrices $A, G, C, R$. +Such calculations are at the heart of efficient strategies for computing +**maximum likelihood estimators** of free parameters. + +### Bayesian Inference + +The likelihood function is also central to **Bayesian inference**. +Where $\theta$ is the parameter vector, $y_0^T$ the data, and +$\tilde{p}(\theta)$ a prior density over $\theta$ before seeing $y_0^T$, +Bayes' law gives the **posterior** + +$$ +\tilde{p}(\theta \mid y_0^T) + = \frac{f(y_0^T \mid \theta)\,\tilde{p}(\theta)} + {\int f(y_0^T \mid \theta)\,\tilde{p}(\theta)\, d\theta} +$$ + +The denominator is the marginal joint density $f(y_0^T)$. + +## Vector Autoregressions and the Kalman Filter + +### Convergence to a Steady State + +Under conditions discussed by Anderson, Hansen, McGrattan, and Sargent (1996), +iterations on the Riccati equation {eq}`eq:riccati` converge to a +**time-invariant** matrix $\Sigma$ from any positive semi-definite starting value +$\Sigma_0$. + +A time-invariant fixed point $\Sigma_t = \Sigma$ of {eq}`eq:riccati` is the +covariance matrix of $x_t$ around + +$$ +E\!\left[x_t \mid \{y_s\}_{s \leq t-1}\right] +$$ + +where the conditioning extends over the **semi-infinite** past $s \leq t-1$. + +### A Time-Invariant VAR + +If the fixed point $\Sigma$ exists and we initialise the filter at $\Sigma_0 = \Sigma$, +the innovations representation {eq}`eq:innovrep` becomes time-invariant: + +$$ +\begin{aligned} +\hat{x}_{t+1} &= A\hat{x}_t + K a_t \\ +y_t &= G\hat{x}_t + a_t +\end{aligned} +$$ (eq:innovti) + +where $E a_t a_t' = G\Sigma G' + R$ and the **steady-state Kalman gain** is +$K = A\Sigma G'(G\Sigma G' + R)^{-1}$. + +From {eq}`eq:innovti` we obtain $\hat{x}_{t+1} = (A - KG)\hat{x}_t + K y_t$. + +If the eigenvalues of $A - KG$ are bounded in modulus strictly below unity, we can +solve this equation forward to get + +$$ +\hat{x}_{t+1} = \sum_{j=0}^\infty (A - KG)^j K\, y_{t-j} +$$ (eq:xhatform) + +Substituting {eq}`eq:xhatform` into the observation equation of {eq}`eq:innovti` +gives the **vector autoregression** + +$$ +y_t = G \sum_{j=0}^\infty (A - KG)^j K\, y_{t-j-1} + a_t +$$ (eq:var1) + +where by construction + +$$ +E\!\left[a_t\, y_{t-j-1}'\right] = 0 \quad \forall\, j \geq 0 +$$ (eq:varorth) + +The orthogonality conditions {eq}`eq:varorth` identify {eq}`eq:var1` as a +**vector autoregression**. + +Defining the lag operator $L$ by $L x_{t+1} \equiv x_t$, the +**moving average representation** deduced from {eq}`eq:innovti` is + +$$ +y_t = \left[I + G(I - AL)^{-1} KL\right] a_t + = \left[I + G\sum_{j=0}^\infty A^j K L^{j+1}\right] a_t +$$ + +```{index} single: vector autoregression +``` + +### Interpreting VARs + +Equilibria of economic models (or their linear or log-linear approximations) +typically take the form of state space system {eq}`eq:statespace`. + +This hidden Markov model disturbs the state $x_t$ by the $p \times 1$ shock +vector $w_{t+1}$ and perturbs the $m \times 1$ vector of observables $y_t$ by the +$m \times 1$ measurement error $v_t$. +An economic theory typically makes $w_{t+1}$ and $v_t$ directly interpretable as +shocks to preferences, technologies, endowments, or information sets. + +The state space system {eq}`eq:statespace` represents $\{y_t\}$ in terms of these +**interpretable shocks**. However, in the typical situation these shocks +**cannot** be recovered directly from the $y_t$'s, even when $A, G, C, R$ are known. + +The innovations representation {eq}`eq:innovti` represents the *same* stochastic +process $\{y_t\}$ in terms of the $m \times 1$ vector $a_t$ of innovations that +would be recovered by running an infinite-order population vector autoregression. + +Its role in mapping the original representation {eq}`eq:statespace` to the VAR +{eq}`eq:var1` makes the Kalman filter an indispensable tool for +**interpreting vector autoregressions**. + +```{index} single: Kalman Filter; and vector autoregressions +``` + +## Spectral Factorization Identity + +```{index} single: spectral factorization identity +``` + +Because the original state space system {eq}`eq:statespace` and the innovations +representation {eq}`eq:innovti` describe the **same** stochastic process $\{y_t\}$, +they imply two distinct formulas for the **spectral density matrix** of $\{y_t\}$. +Equating those formulas yields the *spectral factorization identity*. + +### Two Representations of the Spectral Density + +**From the original state space system.** Writing the first line of +{eq}`eq:statespace` as $x_t = (zI - A)^{-1} C w_{t+1}$ (using the $z$-transform +convention $z^{-1} x_t = x_{t-1}$), the covariance generating function of +$\{x_t\}$ is + +$$ +S_x(z) = (zI - A)^{-1} CC' (z^{-1}I - A')^{-1}. +$$ + +Since $v_t$ is orthogonal to $x_t$, the spectral density of $\{y_t\}$ is + +$$ +S_y(z) = G(zI - A)^{-1} CC' (z^{-1}I - A')^{-1} G' + R. +$$ (eq:sf_original) + +**From the innovations representation.** The time-invariant innovations +representation {eq}`eq:innovti` gives $y_t = [G(zI - A)^{-1}K + I]\, a_t$. +Since $a_t$ is white noise with covariance matrix $G\Sigma G' + R$, the +spectral density is also + +$$ +S_y(z) = \bigl[G(zI-A)^{-1}K + I\bigr] + \bigl(G\Sigma G' + R\bigr) + \bigl[K'(z^{-1}I - A')^{-1}G' + I\bigr]. +$$ (eq:sf_innov) + +### The Spectral Factorization Identity + +Equating {eq}`eq:sf_original` and {eq}`eq:sf_innov` gives the +**spectral factorization identity**: + +$$ +G(zI - A)^{-1} CC' (z^{-1}I - A')^{-1} G' + R = +\bigl[G(zI-A)^{-1}K + I\bigr] +\bigl(G\Sigma G' + R\bigr) +\bigl[K'(z^{-1}I - A')^{-1}G' + I\bigr]. +$$ (eq:sf_identity) + +The left side expresses $S_y(z)$ in terms of the **structural shocks** +$(w_{t+1}, v_t)$ and the matrices $(A, C, G, R)$. +The right side expresses the same object as a spectral factor built from the +**innovations** $a_t$ and the steady-state Kalman gain $K$. + +### Connection to the Wold and Autoregressive Representations + +The factorization {eq}`eq:sf_identity` underpins the passage from the innovations +representation to the Wold moving average and to the VAR. + +**Wold representation.** From {eq}`eq:innovti`, solving for $\hat{x}_t$ over the +semi-infinite past gives $\hat{x}_{t+1} = (I - AL)^{-1} K a_t$, so + +$$ +y_t = \bigl[G(I - AL)^{-1} KL + I\bigr]\, a_t. +$$ (eq:sf_wold) + +**Autoregressive (VAR) representation.** Applying the inverse of the +moving-average operator in {eq}`eq:sf_wold`, using the identity + +$$ +\bigl[G(I-AL)^{-1}KL + I\bigr]^{-1} = I - G\bigl[I-(A-KG)L\bigr]^{-1}KL, +$$ + +gives + +$$ +y_t = G\bigl[I-(A-KG)L\bigr]^{-1}K\, y_{t-1} + a_t + = \sum_{j=1}^\infty G(A-KG)^{j-1}K\, y_{t-j} + a_t, +$$ (eq:sf_var) + +which is the **vector autoregression** already stated in {eq}`eq:var1`. + +The key analytical fact behind both representations is that, under mild stability +conditions, the zeros of $\det[G(zI-A)^{-1}K + I]$ all lie **inside** the unit +circle. This ensures that the moving-average operator in {eq}`eq:sf_wold` has a +causal (one-sided) inverse, so the innovation $a_t$ lies in the closed linear span +of current and past observations $y^t$ — confirming that $a_t$ is the true +population forecast error from the VAR. + +## Python Implementation + +We now illustrate the theory using the `quantecon` library, which provides +`LinearStateSpace` and `Kalman` classes that implement everything derived above. + +```{code-cell} ipython3 +import numpy as np +import matplotlib.pyplot as plt +import quantecon as qe +``` + +### A Scalar Hidden AR(1) Model + +Consider a scalar hidden AR(1) state observed with measurement noise: + +$$ +\begin{aligned} +x_{t+1} &= \rho\, x_t + \sigma_w\, w_{t+1} \\ +y_t &= x_t + \sigma_v\, v_t +\end{aligned} +$$ + +with $w_t, v_t \sim \mathcal{N}(0, 1)$ i.i.d. + +```{code-cell} ipython3 +np.random.seed(42) + +# Model parameters +rho = 0.9 +sigma_w = 0.5 +sigma_v = 1.0 + +# State space matrices (n=1, p=1, m=1) +A = np.array([[rho]]) +C = np.array([[sigma_w]]) +G = np.array([[1.0]]) +R = np.array([[sigma_v**2]]) + +# Build a LinearStateSpace and a Kalman filter object +H = np.array([[sigma_v]]) # measurement noise factor: R = H @ H.T +lss = qe.LinearStateSpace(A, C, G, H, mu_0=np.zeros(1), Sigma_0=np.eye(1) * 10.0) +kf = qe.Kalman(lss) +kf.set_state(np.zeros(1), np.eye(1) * 10.0) # diffuse prior +``` + +**Simulate a sample path of the true hidden state and noisy observations.** + +```{code-cell} ipython3 +T = 200 +x_path, y_path = lss.simulate(ts_length=T) + +# x_path has shape (n, T+1); y_path has shape (m, T) +x_true = x_path[0, :T] +y_obs = y_path[0, :] +``` + +**Run the Kalman filter manually step by step to collect filtered estimates.** + +```{code-cell} ipython3 +x_hats = np.zeros(T) +Sigmas = np.zeros(T) + +for t in range(T): + kf.update(y_obs[t:t+1]) # one full filter cycle + x_hats[t] = kf.x_hat[0] + Sigmas[t] = kf.Sigma[0, 0] +``` + +```{code-cell} ipython3 +fig, axes = plt.subplots(3, 1, figsize=(10, 8), sharex=True) + +t_range = np.arange(T) + +axes[0].plot(t_range, x_true, lw=1.5, label='True state $x_t$') +axes[0].plot(t_range, x_hats, lw=1.5, linestyle='--', label=r'$\hat{x}_t$ (Kalman)') +axes[0].plot(t_range, y_obs, alpha=0.35, label='Observation $y_t$') +axes[0].set_title('State, Kalman Estimate, and Observation') +axes[0].legend(fontsize=9) + +axes[1].plot(t_range, Sigmas, color='tab:orange') +axes[1].set_title(r'Conditional Variance $\Sigma_t$') +axes[1].axhline(kf.Sigma_infinity[0, 0], ls='--', color='k', + label=r'Steady-state $\Sigma_\infty$') +axes[1].legend(fontsize=9) + +axes[2].plot(t_range, y_obs - x_hats, color='tab:green', lw=0.8, alpha=0.7) +axes[2].set_title(r'Innovation $a_t = y_t - G\hat{x}_t$') +axes[2].set_xlabel('Time $t$') + +plt.tight_layout() +plt.show() +``` + +### Convergence of the Riccati Equation + +The `Kalman` class computes the steady-state covariance $\Sigma_\infty$ by +solving the discrete algebraic Riccati equation directly. + +```{code-cell} ipython3 +Sigma_inf, K_inf = kf.stationary_values() + +print(f"Steady-state covariance Σ∞ = {Sigma_inf[0, 0]:.6f}") +print(f"Kalman filter converged to = {Sigmas[-1]:.6f}") +print(f"Steady-state Kalman gain K = {K_inf[0, 0]:.6f}") + +A_minus_KG = A - K_inf @ G +eigval = np.linalg.eigvals(A_minus_KG)[0] +print(f"\nEigenvalue of (A - KG) = {eigval:.6f}") +print(f"Stable VAR: {np.abs(eigval) < 1}") +``` + +### The VAR Representation + +Using {eq}`eq:var1`, the coefficients in the infinite-order VAR representation are +$G(A - KG)^j K$ for $j = 0, 1, 2, \ldots$ + +We retrieve them via `stationary_coefficients`: + +```{code-cell} ipython3 +J = 30 +var_coeffs = kf.stationary_coefficients(J, coeff_type='var') + +# var_coeffs[j] gives the m×m coefficient matrix at lag j+1 +lags = np.arange(1, J + 1) +coeff_values = np.array([var_coeffs[j][0, 0] for j in range(J)]) + +fig, ax = plt.subplots(figsize=(8, 4)) +ax.stem(lags, coeff_values, basefmt=' ') +ax.set_xlabel('Lag $j$') +ax.set_ylabel(r'VAR coefficient $G(A{-}KG)^{j-1}K$') +ax.set_title('VAR Coefficients from the Innovations Representation (eq. {eq}`eq:var1`)') +plt.tight_layout() +plt.show() +``` + +The coefficients decay geometrically, confirming that the infinite-order VAR +{eq}`eq:var1` is well approximated by a finite-lag truncation. + +### Likelihood Evaluation + +We use {eq}`eq:gauss100` to evaluate the log-likelihood of the simulated sample. + +```{code-cell} ipython3 +def log_likelihood(A, C, G, R, y_data, x_hat_0, Sigma_0): + """Evaluate the log-likelihood using the Kalman filter recursions.""" + H_ = np.linalg.cholesky(R) # R = H_ @ H_.T + lss_ = qe.LinearStateSpace(A, C, G, H_, mu_0=x_hat_0, Sigma_0=Sigma_0) + kf_ = qe.Kalman(lss_) + kf_.set_state(x_hat_0, Sigma_0) + + T_, m_ = y_data.shape + loglik = 0.0 + + for t in range(T_): + x_h = kf_.x_hat + Sig = kf_.Sigma + Omega = G @ Sig @ G.T + R # innovation covariance + a_t = y_data[t] - (G @ x_h).flatten() + + sign, logdet = np.linalg.slogdet(Omega) + loglik += -0.5 * (m_ * np.log(2 * np.pi) + logdet + + float(a_t @ np.linalg.solve(Omega, a_t))) + kf_.update(y_data[t]) + + return loglik + + +y_data_col = y_obs.reshape(-1, 1) +ll = log_likelihood(A, C, G, R, + y_data_col, + np.zeros(1), np.eye(1) * 10.0) +print(f"Log-likelihood of sample: {ll:.4f}") +``` + +## An Example + +We now work through a structured example that shows how a bivariate VAR(2) fits naturally into the state space framework and how the Kalman filter delivers a +Wold (innovations) representation. + +### Linear State-Space System + +The state and observation equations are + +$$ +x_{t+1} = A x_t + C w_{t+1} +$$ (eq:ex_state) + +$$ +y_t = G x_t + v_t +$$ (eq:ex_obs) + +with initial condition and shock distributions + +$$ +x_0 \sim \mathcal{N}(M_0, \Omega_0), \quad +w_{t+1} \sim \mathcal{N}(0, I), \quad +v_t \sim \mathcal{N}(0, R). +$$ + +### Steady-State Riccati Equation + +The steady-state error covariance matrix $\Sigma$ satisfies + +$$ +\Sigma = A \Sigma A' + CC' + - A \Sigma G' \bigl(G \Sigma G' + R\bigr)^{-1} G \Sigma A' +$$ (eq:ex_riccati) + +### Kalman Gain + +$$ +K = A \Sigma G' \bigl(G \Sigma G' + R\bigr)^{-1} +$$ (eq:ex_gain) + +### Kalman Filter Recursion + +Starting from an initial estimate $\hat{x}_0$, the Kalman filter updates the state estimate via + +$$ +\hat{x}_{t+1} = A \hat{x}_t + K a_t +$$ (eq:ex_kf_update) + +where the **innovation** (prediction error) is + +$$ +a_t = y_t - G \hat{x}_t +$$ (eq:ex_innovation) + +Substituting {eq}`eq:ex_innovation` into {eq}`eq:ex_kf_update` and expanding: + +$$ +\hat{x}_{t+1} = A \hat{x}_t + K(y_t - G\hat{x}_t) + = (A - KG)\hat{x}_t + K y_t + = (A - KG)\hat{x}_t + K G x_t + K v_t +$$ (eq:ex_kf_expanded) + +### Augmented State-Space Representation + +We want to express the innovations $a_t$ as a function of the state shocks +$w_{t+1}$ and the measurement error $v_t$. + +To accomplish this, we start by stacking the true state $x_t$ and the filter estimate $\hat{x}_t$ into an augmented state vector that +obeys + +$$ +\begin{pmatrix} x_{t+1} \\ \hat{x}_{t+1} \end{pmatrix} += +\begin{pmatrix} A & 0 \\ KG & A - KG \end{pmatrix} +\begin{pmatrix} x_t \\ \hat{x}_t \end{pmatrix} ++ +\begin{pmatrix} C & 0 \\ 0 & K \end{pmatrix} +\begin{pmatrix} w_{t+1} \\ v_t \end{pmatrix} +$$ (eq:ex_augmented) + + +### Bivariate VAR(2) in State-Space Form + +Consider two observable series $r_t$ and $z_t$. + +Stack them into the state vector $x_t = (r_t,\; r_{t-1},\; z_t,\; z_{t-1})'$. + +We posit the VAR(2) state-transition equation: + +$$ +\begin{pmatrix} r_{t+1} \\ r_t \\ z_{t+1} \\ z_t \end{pmatrix} += +\begin{pmatrix} + d_1 & d_2 & d_3 & d_4 \\ + 1 & 0 & 0 & 0 \\ + \delta_1 & \delta_2 & \delta_3 & \delta_4 \\ + 0 & 0 & 1 & 0 +\end{pmatrix} +\begin{pmatrix} r_t \\ r_{t-1} \\ z_t \\ z_{t-1} \end{pmatrix} ++ +\begin{pmatrix} + c_{11} & c_{12} \\ + 0 & 0 \\ + c_{21} & c_{22} \\ + 0 & 0 +\end{pmatrix} +\begin{pmatrix} w_{1,t+1} \\ w_{2,t+1} \end{pmatrix} +$$ (eq:ex_var2_state) + +We consider two possible observation equations. + +The first is a bivariate observation of $r_t$ and $z_t$: + +$$ +\begin{pmatrix} r_t \\ z_t \end{pmatrix} += +\begin{pmatrix} + 1 & 0 & 0 & 0 \\ + 0 & 0 & 1 & 0 +\end{pmatrix} +\begin{pmatrix} r_t \\ r_{t-1} \\ z_t \\ z_{t-1} \end{pmatrix} ++ +\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} +\begin{pmatrix} v_{1t} \\ v_{2t} \end{pmatrix} +$$ (eq:ex_var2_obs) + +The second is a univariate observation of $r_t$: + +$$ +y_t = \begin{pmatrix} 1 & 0 & 0 & 0 \end{pmatrix} +\begin{pmatrix} r_t \\ r_{t-1} \\ z_t \\ z_{t-1} \end{pmatrix} ++ v_{1t} +$$ (eq:ex_scalar_obs) + +For the bivariate observation case, +the Wold representation is + +$$ +\begin{pmatrix} r_t \\ z_t \end{pmatrix} += +\underbrace{G \bigl(I - (A - KG)L\bigr)^{-1} K}_{\text{transfer matrix}} +\, a_t +$$ (eq:ex_wold_bivariate) + +where $E a_t a_t' = G \Sigma G' + R$ and $L$ is the lag operator. + +The innovation expressed in terms of the augmented state is + +$$ +a_t += \begin{bmatrix} G & -G \end{bmatrix} + \begin{pmatrix} x_t \\ \hat{x}_t \end{pmatrix} ++ \begin{bmatrix} 0 & I \end{bmatrix} + \begin{pmatrix} w_{t+1} \\ v_t \end{pmatrix} += G(x_t - \hat{x}_t) + v_t +$$ (eq:ex_innovation_aug) + + + +For the scalar ($1 \times 1$) case with a single observable $r_t$, the observation matrix $G_1$ is the first row of $G$ and the scalar gain $K_1$ is the outcome of the Kalman filter for that case. The univariate Wold representation is + +$$ +r_t = G_1 \bigl(I - (A - K_1 G_1) L\bigr)^{-1} K_1 \, u_t +$$ (eq:ex_wold_scalar) + +where the univariate innovation $u_t$ is given by + +$$ +u_t += \begin{bmatrix} G_1 & -G_1 \end{bmatrix} + \begin{pmatrix} x_t \\ \hat{x}_t \end{pmatrix} ++ \begin{bmatrix} 0 & 1 \end{bmatrix} + \begin{pmatrix} w_{t+1} \\ v_t \end{pmatrix} += G_1(x_t - \hat{x}_t) + v_{1,t} +$$ (eq:ex_innovation_aug2) + +and $E u_t^2 = G_1 \check \Sigma G_1' + R_{11}$, and $\check \Sigma$ is the steady-state covariance of the augmented state vector associated with +$G_1, R_{11}$. + +### Numerical Example: Impulse Responses of Innovations + +We now set specific parameter values and compute the **impulse response functions +of the Kalman innovations** — $a_t$ in System 1 and $u_t$ in System 2 — to the +two structural shocks $w_{1,t+1}$ and $w_{2,t+1}$. + +The key object is **not** the response of the observable $y_t$ to the shocks, but +rather the response of the innovation that the Kalman filter produces. +In System 1, $a_t$ is the $2 \times 1$ forecast error of $(r_t, z_t)'$ given the +filter's estimate; in System 2, $u_t$ is the scalar forecast error of $r_t$ given +the filter's estimate based on the $r_t$ history alone. + +The parameter values are: + +$$ +\begin{aligned} +d_1 &= 0.80,\quad d_2 = 0.05,\quad d_3 = 0.05,\quad d_4 = 0.02 \\ +\delta_1 &= 0.04,\quad \delta_2 = 0.00,\quad \delta_3 = 0.75,\quad \delta_4 = 0.20 \\ +c_{11} &= 1.0,\quad c_{12} = 0.5,\quad c_{21} = 0.5,\quad c_{22} = 1.0 \\ +R &= 0.0001 \times I_2 \quad \text{(bivariate case)}, \qquad +R = 0.0001 \quad \text{(univariate case)}. +\end{aligned} +$$ + +These give the $4 \times 4$ transition matrix and $4 \times 2$ shock-loading matrix + +$$ +A = \begin{pmatrix} +0.80 & 0.05 & 0.05 & 0.02 \\ +1 & 0 & 0 & 0 \\ +0.04 & 0 & 0.75 & 0.20 \\ +0 & 0 & 1 & 0 +\end{pmatrix}, \qquad +C = \begin{pmatrix} +1 & 0.5 \\ +0 & 0 \\ +0.5 & 1 \\ +0 & 0 +\end{pmatrix}. +$$ + +**System 1** uses the bivariate observation equation {eq}`eq:ex_var2_obs`, so +$G$ selects $(r_t, z_t)'$ from the state and the innovation $a_t$ is $2 \times 1$. + +**System 2** uses the univariate observation equation {eq}`eq:ex_scalar_obs`, so +$G_1$ selects only $r_t$ and the innovation $u_t$ is scalar. + +Because the innovation equals $a_t = G(x_t - \hat{x}_t) + v_t$, a unit shock $w_j$ +propagates through the **augmented** state $(x_t', \hat{x}_t')'$. A straightforward +calculation using the augmented recursion {eq}`eq:ex_augmented` shows that the +**impulse response of the innovation** at horizon $h \geq 0$ is + +$$ +\text{IRF}_{a}(h) = G(A - KG)^h C e_j, \qquad j = 1, 2, +$$ + +where $e_j$ is the $j$-th column of $I_2$ and $K$ is the appropriate steady-state +Kalman gain (System 1 uses $K$ from the bivariate filter; System 2 uses $K_1$ from +the univariate filter). At $h=0$ the shock hits and the innovation equals $GCe_j$; +for $h \geq 1$ the response decays at the rate governed by the eigenvalues of $A - KG$. + +```{code-cell} ipython3 +import numpy as np +import matplotlib.pyplot as plt +import quantecon as qe + +# ── Parameters ─────────────────────────────────────────────────────────────── +d1, d2, d3, d4 = 0.80, 0.05, 0.05, 0.02 +delta1, delta2, delta3, delta4 = 0.04, 0.00, 0.75, 0.20 +c11, c12, c21, c22 = 1.0, 0.5, 0.5, 1.0 +sigma_v = 0.01 # sqrt(0.0001) + +# ── Shared matrices ─────────────────────────────────────────────────────────── +A_var = np.array([[d1, d2, d3, d4 ], + [1.0, 0.0, 0.0, 0.0 ], + [delta1, delta2, delta3, delta4], + [0.0, 0.0, 1.0, 0.0 ]]) + +C_var = np.array([[c11, c12], + [0.0, 0.0], + [c21, c22], + [0.0, 0.0]]) + +# ── System 1: bivariate observation ────────────────────────────────────────── +G_biv = np.array([[1.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 1.0, 0.0]]) +H_biv = sigma_v * np.eye(2) # H @ H.T = 0.0001 * I_2 + +lss_biv = qe.LinearStateSpace(A_var, C_var, G_biv, H_biv, + mu_0=np.zeros(4), Sigma_0=np.eye(4)) +kf_biv = qe.Kalman(lss_biv) +Sigma_biv, K_biv = kf_biv.stationary_values() + +print("System 1 – steady-state Kalman gain K (4×2):") +print(np.round(K_biv, 5)) + +# ── System 2: univariate observation ───────────────────────────────────────── +G_uni = np.array([[1.0, 0.0, 0.0, 0.0]]) +H_uni = np.array([[sigma_v]]) # H @ H.T = 0.0001 + +lss_uni = qe.LinearStateSpace(A_var, C_var, G_uni, H_uni, + mu_0=np.zeros(4), Sigma_0=np.eye(4)) +kf_uni = qe.Kalman(lss_uni) +Sigma_uni, K_uni = kf_uni.stationary_values() + +print("\nSystem 2 – steady-state Kalman gain K (4×1):") +print(np.round(K_uni, 5)) +``` + +```{code-cell} ipython3 +# ── Covariance comparisons ──────────────────────────────────────────────────── + +# State-noise covariance CC' versus innovation covariance G Σ G' + R (System 1) +CC_prime = C_var @ C_var.T +R_biv = (sigma_v**2) * np.eye(2) +innov_cov_biv = G_biv @ Sigma_biv @ G_biv.T + R_biv + +print("State-noise covariance CC' (4×4):") +print(np.round(CC_prime, 6)) + +print("\nSystem 1 – innovation covariance GΣG' + R (2×2):") +print(np.round(innov_cov_biv, 6)) + +# Steady-state state covariance: System 1 (Σ_biv) vs System 2 (Σ_uni) +print("\nSystem 1 – steady-state state covariance Σ (4×4):") +print(np.round(Sigma_biv, 6)) + +print("\nSystem 2 – steady-state state covariance Σ̌ (4×4):") +print(np.round(Sigma_uni, 6)) + +print("\nDifference Σ̌ − Σ (System 2 minus System 1):") +print(np.round(Sigma_uni - Sigma_biv, 6)) +``` + +```{code-cell} ipython3 +def innovation_irf(A, K, G, C, T=40): + """ + Impulse responses of the Kalman innovation to each structural shock. + + The innovation at horizon h to a unit shock e_j is + G (A - KG)^h C e_j, h = 0, 1, ..., T-1. + + Returns irf of shape (T, n_a, n_w). + """ + AKG = A - K @ G + n_a = G.shape[0] + n_w = C.shape[1] + irf = np.zeros((T, n_a, n_w)) + x = C.copy() # (A-KG)^0 @ C at h = 0 + for h in range(T): + irf[h] = G @ x + x = AKG @ x + return irf + +T_irf = 40 +horizons = np.arange(T_irf) + +irf_biv = innovation_irf(A_var, K_biv, G_biv, C_var, T_irf) # (T, 2, 2) +irf_uni = innovation_irf(A_var, K_uni, G_uni, C_var, T_irf) # (T, 1, 2) + +shock_labels = [r'Shock $w_{1,t+1}$', r'Shock $w_{2,t+1}$'] +innov_biv_lbl = [r'$a_{1,t}$', r'$a_{2,t}$'] + +# ── Figure 1: System 1 – innovation IRFs ────────────────────────────────────── +fig, axes = plt.subplots(2, 2, figsize=(10, 6), sharex=True) +for j, shock in enumerate(shock_labels): + for i, albl in enumerate(innov_biv_lbl): + ax = axes[i, j] + ax.plot(horizons, irf_biv[:, i, j], lw=1.8) + ax.axhline(0, color='k', lw=0.6, ls='--') + ax.set_title(f'System 1 – {albl} to {shock}', fontsize=9) + ax.set_xlabel('Horizon $h$') + if j == 0: + ax.set_ylabel('Innovation response') + +plt.suptitle(r'System 1: IRFs of Innovation $a_t$ to Structural Shocks', + fontsize=11) +plt.tight_layout() +plt.show() + +# ── Figure 2: System 2 – innovation IRFs ────────────────────────────────────── +fig, axes = plt.subplots(1, 2, figsize=(10, 3.5)) +for j, shock in enumerate(shock_labels): + axes[j].plot(horizons, irf_uni[:, 0, j], lw=1.8) + axes[j].axhline(0, color='k', lw=0.6, ls='--') + axes[j].set_title(f'System 2 – $u_t$ to {shock}', fontsize=9) + axes[j].set_xlabel('Horizon $h$') +axes[0].set_ylabel('Innovation response') + +plt.suptitle(r'System 2: IRFs of Innovation $u_t$ to Structural Shocks', + fontsize=11) +plt.tight_layout() +plt.show() +``` + +The two sets of innovation impulse responses illustrate an important point. +In **System 1** the filter uses both $r_t$ and $z_t$, so $a_t$ is $2 \times 1$ and +its response to each shock is a pair of numbers at every horizon. +In **System 2** only $r_t$ is observed, so $u_t$ is scalar and carries less +information about the two-dimensional shock vector. +Because $K$ differs across the two systems, the decay matrix $A - KG$ differs +as well, so the innovation IRFs $G(A-KG)^h C e_j$ differ across the two systems +even though the structural DGP is identical. + +The covariance comparisons printed above reveal two further insights. + +**State-noise $CC'$ vs. innovation covariance $G\Sigma G' + R$ (System 1).** +The matrix $CC'$ is the unconditional covariance contributed by the structural +shocks to the state transition in one period. +The innovation covariance $G\Sigma G' + R$ is the covariance of the one-step +forecast error in the *observable* vector after the Kalman filter has processed +all available information. +Because $G\Sigma G' + R \preceq G(A \Sigma A' + CC')G' + R$, the filter +strictly reduces the variance attributable to the structural shocks; the +diagonal entries of $G\Sigma G' + R$ are therefore smaller than those of $GCC'G'$. + +**System 1 covariance $\Sigma$ vs. System 2 covariance $\check\Sigma$.** +Restricting observations to $r_t$ alone (System 2) yields less information about +the hidden state, so the filter is less precise: $\check\Sigma \succeq \Sigma$, +i.e., the difference $\check\Sigma - \Sigma$ is positive semi-definite. +The printed difference matrix confirms this ordering for our parameter values. + +## Exercises + +```{exercise-start} +:label: kf_ex1 +``` + +Consider the scalar AR(1) state space system used above with $\rho = 0.9$, +$\sigma_w = 0.5$, $\sigma_v = 1.0$. + +Derive an algebraic expression for the **steady-state** conditional variance +$\Sigma_\infty$ by solving the scalar Riccati equation {eq}`eq:riccati` at its +fixed point $\Sigma_{t+1} = \Sigma_t = \Sigma$. + +Show that $\Sigma$ satisfies a quadratic equation and find its positive root. +Verify numerically that your formula matches `kf.Sigma_infinity`. + +```{exercise-end} +``` + +```{solution-start} kf_ex1 +:class: dropdown +``` + +Setting $\Sigma_{t+1} = \Sigma_t = \Sigma$ in the scalar version of +{eq}`eq:riccati` with $A = \rho$, $CC' = \sigma_w^2$, $GG' = 1$, +$R = \sigma_v^2$: + +$$ +\Sigma = \rho^2 \Sigma + \sigma_w^2 - \frac{\rho^2 \Sigma^2}{\Sigma + \sigma_v^2} +$$ + +Multiplying through by $\Sigma + \sigma_v^2$ and rearranging: + +$$ +\Sigma^2 + \sigma_v^2 \Sigma - \sigma_v^2 \sigma_w^2 + - \sigma_v^2(1 - \rho^2)\Sigma = 0 +$$ + +$$ +\Sigma^2 + \sigma_v^2 \rho^2 \Sigma - \sigma_v^2 \sigma_w^2 = 0 +$$ + +Taking the positive root of this quadratic: + +$$ +\Sigma_\infty + = \frac{-\sigma_v^2 \rho^2 + + \sqrt{(\sigma_v^2 \rho^2)^2 + 4 \sigma_v^2 \sigma_w^2}}{2} +$$ + +```{code-cell} ipython3 +rho_, sigma_w_, sigma_v_ = 0.9, 0.5, 1.0 + +discriminant = (sigma_v_**2 * rho_**2)**2 + 4 * sigma_v_**2 * sigma_w_**2 +Sigma_formula = (-sigma_v_**2 * rho_**2 + np.sqrt(discriminant)) / 2 + +A_ = np.array([[rho_]]) +C_ = np.array([[sigma_w_]]) +G_ = np.array([[1.0]]) +R_ = np.array([[sigma_v_**2]]) +H_ = np.array([[sigma_v_]]) # R_ = H_ @ H_.T +lss_ = qe.LinearStateSpace(A_, C_, G_, H_, mu_0=np.zeros(1), Sigma_0=np.eye(1)) +kf_ = qe.Kalman(lss_) + +print(f"Analytical Σ∞ = {Sigma_formula:.8f}") +print(f"Numerical Σ∞ = {kf_.Sigma_infinity[0, 0]:.8f}") +``` + +```{solution-end} +``` + +```{exercise-start} +:label: kf_ex2 +``` + +**Bivariate example.** + +Consider a two-dimensional state with a one-dimensional observation: + +$$ +A = \begin{pmatrix} 0.9 & 0.1 \\ 0 & 0.8 \end{pmatrix}, \quad +C = \begin{pmatrix} 0.4 \\ 0.1 \end{pmatrix}, \quad +G = \begin{pmatrix} 1 & 0 \end{pmatrix}, \quad +R = [0.5] +$$ + +(a) Simulate $T = 500$ observations from this system starting from a diffuse prior. + +(b) Run the Kalman filter and plot both components of $\hat{x}_t$ against the + true hidden state path. + +(c) Compute and report the steady-state covariance $\Sigma_\infty$ and + Kalman gain $K_\infty$. + +(d) Check that the eigenvalues of $A - K_\infty G$ lie strictly inside the + unit circle, confirming that the VAR representation {eq}`eq:var1` is stable. + +```{exercise-end} +``` + +```{solution-start} kf_ex2 +:class: dropdown +``` + +```{code-cell} ipython3 +np.random.seed(0) + +A2 = np.array([[0.9, 0.1], + [0.0, 0.8]]) +C2 = np.array([[0.4], + [0.1]]) +G2 = np.array([[1.0, 0.0]]) +R2 = np.array([[0.5]]) +H2 = np.array([[np.sqrt(0.5)]]) # R2 = H2 @ H2.T + +lss2 = qe.LinearStateSpace(A2, C2, G2, H2, + mu_0=np.zeros(2), + Sigma_0=np.eye(2) * 5.0) +kf2 = qe.Kalman(lss2) +kf2.set_state(np.zeros(2), np.eye(2) * 5.0) + +T2 = 500 +x2_path, y2_path = lss2.simulate(ts_length=T2) + +x_hats2 = np.zeros((T2, 2)) +for t in range(T2): + kf2.update(y2_path[:, t]) + x_hats2[t] = kf2.x_hat + +fig, axes = plt.subplots(2, 1, figsize=(10, 6), sharex=True) +for i, ax in enumerate(axes): + ax.plot(x2_path[i, :T2], lw=1.2, label=f'True $x_{{{i+1},t}}$') + ax.plot(x_hats2[:, i], lw=1.2, ls='--', label=rf'$\hat{{x}}_{{{i+1},t}}$') + ax.legend(fontsize=9) + ax.set_ylabel(f'Component {i+1}') +axes[1].set_xlabel('Time $t$') +plt.suptitle('Kalman Filter — Bivariate Hidden State') +plt.tight_layout() +plt.show() + +# (c) Steady-state values +Sigma2_inf, K2_inf = kf2.stationary_values() +print("Steady-state covariance Σ∞:") +print(np.round(Sigma2_inf, 5)) +print("\nSteady-state Kalman gain K∞:") +print(np.round(K2_inf, 5)) + +# (d) Eigenvalues of A - K∞ G +AKG2 = A2 - K2_inf @ G2 +eigvals2 = np.linalg.eigvals(AKG2) +print(f"\nEigenvalues of A - K∞G: {np.round(eigvals2, 5)}") +print(f"Stable VAR: {np.all(np.abs(eigvals2) < 1)}") +``` + +```{solution-end} +``` + +```{exercise-start} +:label: kf_ex3 +``` + +**Likelihood and parameter estimation.** + +Using the scalar model from the main text with true parameters +$(\rho, \sigma_w, \sigma_v) = (0.9, 0.5, 1.0)$: + +(a) Simulate $T = 300$ observations. + +(b) Write a function that evaluates the **log-likelihood** as a function of + $\rho \in (0, 1)$, holding $\sigma_w = 0.5$ and $\sigma_v = 1.0$ fixed. + Plot the log-likelihood against $\rho$ for a grid of values. + +(c) Locate the maximum numerically and check that it is close to the true value + $\rho = 0.9$. + +```{exercise-end} +``` + +```{solution-start} kf_ex3 +:class: dropdown +``` + +```{code-cell} ipython3 +np.random.seed(7) + +# True parameters +rho_true, sw_true, sv_true = 0.9, 0.5, 1.0 + +A_t = np.array([[rho_true]]) +C_t = np.array([[sw_true]]) +G_t = np.array([[1.0]]) +R_t = np.array([[sv_true**2]]) +H_t = np.array([[sv_true]]) # R_t = H_t @ H_t.T + +lss_t = qe.LinearStateSpace(A_t, C_t, G_t, H_t, + mu_0=np.zeros(1), Sigma_0=np.eye(1)) +_, y_sim = lss_t.simulate(ts_length=300) +y_sim = y_sim.T # shape (300, 1) + +def ll_rho(rho_val): + A_ = np.array([[rho_val]]) + C_ = np.array([[sw_true]]) + G_ = np.array([[1.0]]) + R_ = np.array([[sv_true**2]]) + return log_likelihood(A_, C_, G_, R_, y_sim, + np.zeros(1), np.eye(1) * 10.0) + +rho_grid = np.linspace(0.5, 0.99, 60) +ll_vals = np.array([ll_rho(r) for r in rho_grid]) + +rho_mle = rho_grid[np.argmax(ll_vals)] + +fig, ax = plt.subplots(figsize=(8, 4)) +ax.plot(rho_grid, ll_vals) +ax.axvline(rho_true, color='k', ls='--', label=f'True ρ = {rho_true}') +ax.axvline(rho_mle, color='tab:red', ls=':', label=f'MLE ρ̂ = {rho_mle:.3f}') +ax.set_xlabel(r'$\rho$') +ax.set_ylabel('Log-likelihood') +ax.set_title('Profile Log-likelihood as a Function of $\\rho$') +ax.legend() +plt.tight_layout() +plt.show() + +print(f"True ρ = {rho_true}, MLE ρ̂ = {rho_mle:.4f}") +``` + +```{solution-end} +``` From 87ca020fcdd9886accbb46f708efc767cd418aee Mon Sep 17 00:00:00 2001 From: thomassargent30 Date: Wed, 17 Jun 2026 19:25:43 +0800 Subject: [PATCH 2/9] Tom's June 17 edits of two new lectures --- lectures/_toc.yml | 1 + lectures/lq_consumption_smoothing.md | 2045 ++++++++++++++++++++++++++ 2 files changed, 2046 insertions(+) create mode 100644 lectures/lq_consumption_smoothing.md diff --git a/lectures/_toc.yml b/lectures/_toc.yml index 10f388624..81f97dc5c 100644 --- a/lectures/_toc.yml +++ b/lectures/_toc.yml @@ -112,6 +112,7 @@ parts: - file: perm_income_cons - file: theil_1 - file: theil_2 + - file: lq_consumption_smoothing - file: lq_inventories - caption: Optimal Growth numbered: true diff --git a/lectures/lq_consumption_smoothing.md b/lectures/lq_consumption_smoothing.md new file mode 100644 index 000000000..f3cb836fc --- /dev/null +++ b/lectures/lq_consumption_smoothing.md @@ -0,0 +1,2045 @@ +--- +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 +--- + +# LQ Consumption Smoothing: Incomplete Markets, Complete Markets, and Robust Control + + +```{index} single: LQ Permanent Income Model +``` + +```{index} single: Consumption Smoothing +``` + +## Overview + +This lecture describes two representations of a linear-quadratic (LQ) permanent income model, then +extends the model to incorporate a consumer's concern for robustness to misspecification of the +stochastic process governing labor income. + +**Part A** covers the standard LQ permanent income model, a rational-expectations version of the +theories of {cite}`Friedman1956` and {cite}`Hall1978`. We use the model to illustrate: + +- impulse response functions +- alternative state-space representations of the optimal decision rule +- cointegration of consumption and assets +- a "borrowers and lenders" closed economy (Bewley model) +- complete-markets consumption smoothing + +**Part B** covers a robust version of the permanent income model due to {cite}`HST_1999` +(HST) and {cite}`hansen2008robustness`. A consumer who distrusts his specification of the labor income process engages in a form of +precautionary savings even when preferences are quadratic. We show: + +- how a concern for robustness is observationally equivalent (for quantities) to an increase in + impatience +- how the worst-case model distorts the endowment process toward greater persistence +- frequency-domain and detection-error-probability characterizations of the size of model + uncertainty + +**Synthesis: a Bewley model with heterogeneous beliefs and discount factors.** The lecture +concludes by combining Parts A and B in a single general-equilibrium model. Using tools that we borrow from {cite}`hansen2008robustness`, we show: + +- how a continuum of consumers can differ in their robustness parameter $\sigma^i \leq 0$ and + their discount factor $\beta^i$, with each pair lying on the observational-equivalence locus + from Part B +- how every such consumer nevertheless chooses the **same consumption–saving rule** as the + plain-vanilla $(\sigma = 0, \beta_0)$ agent of Part A +- how the equilibrium interest rate $R = \beta_0^{-1}$ and all aggregate dynamics therefore + coincide with those of the Part A Bewley model +- how agents can hold genuinely different internal (worst-case) models of their income process + while remaining observationally indistinguishable in quantities + +Throughout, we set $\beta R = 1$, so the consumer's subjective discount factor equals the bond price. + +Let's begin with some imports. + +```{code-cell} ipython3 +import numpy as np +import matplotlib.pyplot as plt +from scipy.linalg import solve, inv, solve_discrete_lyapunov +from scipy.stats import norm + +plt.rcParams.update({'figure.figsize': (9, 5), 'font.size': 12}) +``` + +--- + +## Part A: The Standard LQ Permanent Income Model + +```{index} single: LQ Permanent Income Model; standard +``` + +### Setup + +```{index} single: Permanent Income Hypothesis; Friedman +``` + +A consumer has preferences over consumption streams ordered by + +$$ +E_0 \sum_{t=0}^{\infty} \beta^t u(c_t) +$$ (eq:sprob1) + +where $E_t$ is the mathematical expectation conditioned on the consumer's time-$t$ information, +$c_t$ is time-$t$ consumption, $u(c)$ is a strictly concave one-period utility function, and +$\beta \in (0,1)$ is a discount factor. + +The consumer maximizes {eq}`eq:sprob1` by choosing a plan +$\{c_t, b_{t+1}\}_{t=0}^{\infty}$ subject to the sequence of budget constraints + +$$ +c_t + b_t = R^{-1} b_{t+1} + y_t, \quad t \geq 0 +$$ (eq:sprob2) + +where $\{y_t\}$ is an exogenous stationary endowment process, $R$ is a constant gross risk-free +interest rate, $b_t$ is a one-period risk-free bond maturing at $t$, and $b_0$ is a given +initial condition. + +```{note} +For $t \geq 1$, $b_t$ is chosen at time $t-1$. The bond $b_t > 0$ represents debt owed by +the consumer at the start of period $t$. +``` + +We assume $R^{-1} = \beta$. + +The endowment process has the state-space representation + +$$ +\begin{aligned} +z_{t+1} &= \check{A}\, z_t + \check{C}\, w_{t+1} \\ +y_t &= \check{G}\, z_t +\end{aligned} +$$ (eq:sprob15) + +where $w_{t+1}$ is i.i.d. with mean zero and identity covariance matrix, $\check{A}$ is a stable +matrix (eigenvalues strictly less than one in modulus), and $\check{G}$ is a row vector. + +The state confronting the household at $t$ is +$\bigl[b_t \;\; z_t'\bigr]'$, where $b_t$ is its one-period debt due at the start of period $t$ +and $z_t$ contains all variables useful for forecasting its future endowment. + +To make the problem linear-quadratic, we adopt the **quadratic utility function** + +$$ +u(c_t) = -\tfrac{1}{2}(c_t - \gamma)^2 +$$ + +where $\gamma > 0$ is a bliss level of consumption. We allow $c_t$ to be negative (a producer +rather than a consumer). + +We impose the **transversality condition** + +$$ +E_0 \sum_{t=0}^{\infty} \beta^t b_t^2 < +\infty +$$ (eq:sprob3) + +which rules out Ponzi schemes. + +### Euler Equation and Certainty Equivalence + +With quadratic utility, the first-order conditions for the consumer's problem imply the **martingale +Euler equation** + +$$ +E_t c_{t+1} = c_t +$$ (eq:sprob5) + +```{note} +Equation {eq}`eq:sprob5` says that consumption is a martingale. This is the key implication of +the LQ permanent income model. It contrasts with models that have convex marginal utility +($u''' > 0$), where consumption is instead a submartingale. +``` + +The problem satisfies a **certainty-equivalence** property: one can find the optimal plan by (1) +solving the problem under perfect foresight to express $c_t$ as a function of $b_t$ and the entire +future path $\{y_{t+j}\}_{j=0}^{\infty}$, and then (2) replacing those future values with +$\{E_t y_{t+j}\}_{j=0}^{\infty}$. + +### The Optimal Consumption Function + +Solving the budget constraint {eq}`eq:sprob2` forward, imposing the transversality condition, and +taking conditional expectations gives + +$$ +b_t = \sum_{j=0}^{\infty} \beta^j E_t y_{t+j} - \frac{1}{1-\beta} c_t +$$ (eq:sprob7) + +Rearranging yields the **consumption function** + +$$ +c_t = (1-\beta)\!\left[\sum_{j=0}^{\infty} \beta^j E_t y_{t+j} - b_t\right] +$$ (eq:sprob8) + +Equivalently, with net interest rate $r$ defined by $\beta = 1/(1+r)$, + +$$ +c_t = \frac{r}{1+r}\!\left[\sum_{j=0}^{\infty} \beta^j E_t y_{t+j} - b_t\right] +$$ (eq:sprob9) + +Consumption equals $r/(1+r)$ times total wealth, where total wealth is the sum of human wealth +$\sum_{j=0}^{\infty}\beta^j E_t y_{t+j}$ and financial wealth $-b_t$. + +Using the state-space representation {eq}`eq:sprob15` to evaluate the geometric sum of expected +future endowments, + +$$ +\sum_{j=0}^{\infty} \beta^j E_t y_{t+j} = \check{G}(I - \beta \check{A})^{-1} z_t +$$ (eq:discount1) + +we obtain the **Lucas-critique-respecting consumption function** + +$$ +c_t = (1-\beta)\!\left[\check{G}(I-\beta\check{A})^{-1} z_t - b_t\right] +$$ (eq:lccf) + +This expresses $c_t$ as a function of the state $[b_t,\, z_t']'$ that confronts the household. + +### Representation 1: State $(b_t, z_t)$ + +Combining the endowment law of motion with the optimal debt dynamics (derived by substituting +{eq}`eq:lccf` into {eq}`eq:sprob2`) gives the first system representation: + +$$ +\begin{aligned} +z_{t+1} &= \check{A}\, z_t + \check{C}\, w_{t+1} \\ +b_{t+1} &= b_t + \check{G}\bigl[(I - \beta\check{A})^{-1}(\check{A}-I)\bigr] z_t \\ +y_t &= \check{G}\, z_t \\ +c_t &= (1-\beta)\!\left[\check{G}(I-\beta\check{A})^{-1} z_t - b_t\right] +\end{aligned} +$$ (eq:rep1) + +In this representation the **exogenous** state is $z_t$ and the **endogenous** state is $b_t$. + +### Representation 2: State $(c_t, z_t)$ + +{cite}`Hall1978` showed that the LQ permanent income model implies a particularly sharp +state-space representation in which the state consists of current consumption $c_t$ and the +exogenous endowment state $z_t$, with assets $b_t$ becoming an outcome rather than a state +variable. + +**Consumption innovation.** Shifting {eq}`eq:sprob8` forward, eliminating $b_{t+1}$ via +{eq}`eq:sprob2`, and rearranging yields + +$$ +c_{t+1} - c_t = (1-\beta)\sum_{j=0}^{\infty} \beta^j \bigl(E_{t+1} y_{t+j+1} - E_t y_{t+j+1}\bigr) +$$ (eq:sprob11) + +The right-hand side is $(1-\beta)$ times the time-$(t+1)$ **innovation** to the expected present +value of the endowment stream. + +Suppose the endowment has the moving-average representation + +$$ +y_{t+1} = d(L)\, w_{t+1}, \qquad d(L) = \sum_{j=0}^{\infty} d_j L^j +$$ (eq:sprob12) + +where $d(L) = \check{G}(I - \check{A} L)^{-1}\check{C}$. Then + +$$ +E_{t+1} y_{t+j} - E_t y_{t+j} = d_{j-1}\, w_{t+1} +$$ (eq:sprob120) + +Substituting {eq}`eq:sprob120` into {eq}`eq:sprob11` gives the key result + +$$ +c_{t+1} - c_t = (1-\beta)\, d(\beta)\, w_{t+1} +$$ (eq:sprob13) + +where $d(\beta) = \check{G}(I-\beta\check{A})^{-1}\check{C}$ is the **present value of the +moving-average coefficients**. Consumption is a **random walk** with innovation +$(1-\beta)d(\beta)w_{t+1}$. + +**Full second representation.** Combining {eq}`eq:sprob13` and {eq}`eq:sprob7` gives + +$$ +\begin{aligned} +c_{t+1} &= c_t + (1-\beta)\,\check{G}(I-\beta\check{A})^{-1}\check{C}\, w_{t+1} \\ +b_t &= \check{G}(I-\beta\check{A})^{-1} z_t - \frac{1}{1-\beta}\,c_t \\ +y_t &= \check{G}\, z_t \\ +z_{t+1} &= \check{A}\, z_t + \check{C}\, w_{t+1} +\end{aligned} +$$ (eq:sprob16) + +This representation reveals several important features of the optimal decision rule: + +1. **State**: The state consists of the endogenous component $c_t$ and the exogenous component + $z_t$. Financial assets $b_t$ have disappeared as a state variable because they are encoded + in $c_t$. + +2. **Random walk**: Consumption is a random walk with innovation $(1-\beta)d(\beta)w_{t+1}$, which + confirms that the Euler equation {eq}`eq:sprob5` is built into the solution. The random-walk + property implies that consumption has no asymptotic stationary distribution. + +3. **Box impulse response**: For all $j \geq 1$, the response of $c_{t+j}$ to the innovation + $w_{t+1}$ is the constant $(1-\beta)d(\beta)$, giving a "box-shaped" impulse response. + +4. **Cointegration**: Both $c_t$ and $b_t$ are nonstationary (unit-root processes), but the + linear combination $(1-\beta)b_t + c_t$ is stationary. From {eq}`eq:sprob7`, + +$$ +(1-\beta)b_t + c_t = (1-\beta)E_t\sum_{j=0}^{\infty}\beta^j y_{t+j} +$$ (eq:cointegration) + + The left side is the cointegrating residual, which equals the consumer's expected present value + of future income. + +### Debt Dynamics + +```{index} single: History Dependence +``` + +Subtracting {eq}`eq:sprob16` (equation for $b_t$) at time $t$ from the same equation at time $t+1$ +and substituting gives + +$$ +b_{t+1} - b_t = \check{G}(I-\beta\check{A})^{-1}(\check{A}-I)\, z_t +$$ (eq:debt_evolution) + +This shows that $b_{t+1}$ is **predetermined** at time $t$ as a function of $z_t$ alone. +Solving backward from any $t$, $b_t$ depends on the entire history $z^{t-1} = [z_{t-1},\ldots,z_0]$ +and the initial condition $b_0$. This **history dependence** is a hallmark of incomplete markets. + +### Two Classic Examples + +```{index} single: Permanent Income Model; examples +``` + +We illustrate formulas {eq}`eq:sprob16` with two examples. In both, the endowment is +$y_t = z_{1t} + z_{2t}$, where + +$$ +\begin{pmatrix}z_{1,t+1}\\z_{2,t+1}\end{pmatrix} += +\begin{pmatrix}1 & 0\\0 & 0\end{pmatrix} +\begin{pmatrix}z_{1t}\\z_{2t}\end{pmatrix} ++ +\begin{pmatrix}\sigma_1 & 0\\0 & \sigma_2\end{pmatrix} +\begin{pmatrix}w_{1,t+1}\\w_{2,t+1}\end{pmatrix} +$$ (eq:twofactor) + +Here $z_{1t}$ is a **permanent** component of $y_t$ and $z_{2t}$ is a **purely transitory** +component. + +**Example 1 (full information).** The consumer observes the state $z_t$ at time $t$, so he +can reconstruct $w_{t+1}$ from $z_{t+1}$ and $z_t$. Applying {eq}`eq:sprob16`: + +$$ +c_{t+1} - c_t = \sigma_1 w_{1,t+1} + (1-\beta)\,\sigma_2\, w_{2,t+1} +$$ (eq:consexample1) + +A unit increment to the permanent component $z_{1t}$ raises consumption **one-for-one** permanently +and causes **zero net saving**. A unit increment to the purely transitory component raises +consumption by only the fraction $(1-\beta)$ permanently, while the remaining fraction $\beta$ is +saved. + +From {eq}`eq:debt_evolution`: + +$$ +b_{t+1} - b_t = -z_{2t} = -\sigma_2 w_{2t} +$$ (eq:consexample1a) + +confirming that none of the permanent shock is saved, while all of the transitory shock is saved. + +**Example 2 (partial information / Muth model).** The consumer observes $y_t$ and its history, +but not $z_{1t}$ and $z_{2t}$ separately. The appropriate approach uses an **innovations +representation** derived by the Kalman filter. + +At the Kalman filter steady state, the **Kalman gain** $K \in [0,1]$ satisfies + +$$ +K = \frac{\Sigma}{\Sigma + \sigma_2^2}, \qquad \Sigma = \frac{\sigma_1^2 + \sqrt{\sigma_1^4 + 4\sigma_1^2\sigma_2^2}}{2} +$$ (eq:kalmangain) + +where $K$ increases with the ratio $\sigma_1^2/\sigma_2^2$ (the variance of the permanent shock +relative to the transitory shock). + +The innovations representation expresses the endowment as an ARMA(1,1) in +its own innovation $a_t = y_t - E[y_t \mid y^{t-1}]$ (the one-step-ahead forecast error): + +$$ +y_{t+1} = y_t - (1-K)\,a_t + a_{t+1} +$$ (eq:muth_innov) + +Here the coefficient $-(1-K)$ on the lagged innovation reflects that only the fraction +$K$ of last period's surprise was treated as permanent; the remainder mean-reverts. The scalar +$a_t$ is i.i.d. with variance $\Sigma + \sigma_2^2$. +Applying {eq}`eq:sprob16` to this innovation representation: + +$$ +c_{t+1} - c_t = [1 - \beta(1-K)]\, a_{t+1} +$$ (eq:consexample2) + +The consumer regards a fraction $K$ of the innovation $a_{t+1}$ as permanent and fraction $1-K$ +as transitory. He permanently increments consumption by $K + (1-\beta)(1-K) = 1 - \beta(1-K)$ +of $a_{t+1}$ and saves the remaining fraction $\beta(1-K)$. + +The first difference of income obeys a first-order moving average: + +$$ +y_{t+1} - y_t = a_{t+1} - (1-K)\,a_t +$$ (eq:incomemaar) + +while the first difference of consumption is i.i.d. + +### Numerical Illustration of the Two Examples + +```{code-cell} ipython3 +# Parameters +β = 0.95 # discount factor (so R = 1/β) +σ1 = 0.15 # std of permanent shock +σ2 = 0.30 # std of transitory shock + +# ── Example 1: full information ────────────────────────────────────────────── +A_check = np.array([[1.0, 0.0], + [0.0, 0.0]]) +C_check = np.array([[σ1, 0.0], + [0.0, σ2]]) +G_check = np.array([[1.0, 1.0]]) + +# Key matrix M = G(I - βA)^{-1} +IbA = np.eye(2) - β * A_check +M = G_check @ inv(IbA) # shape (1, 2) + +# Impulse response of consumption (permanent to unit permanent / transitory shock) +h = (1 - β) * M @ C_check # shape (1, 2) +irf_perm_ex1 = h[0, 0] / σ1 # response per unit std of permanent shock +irf_trans_ex1 = h[0, 1] / σ2 # response per unit std of transitory shock + +print("Example 1 (full information)") +print(f" IRF c to permanent shock (normalised): {irf_perm_ex1:.4f} (theory: 1.0)") +print(f" IRF c to transitory shock (normalised): {irf_trans_ex1:.4f} " + f"(theory: {1-β:.4f})") +``` + +```{code-cell} ipython3 +# ── Example 2: partial information / Muth model ─────────────────────────────── +Σ = (σ1**2 + np.sqrt(σ1**4 + 4 * σ1**2 * σ2**2)) / 2 +K = Σ / (Σ + σ2**2) + +print("Example 2 (partial information)") +print(f" Steady-state Kalman gain K = {K:.4f}") +print(f" IRF c to unit innovation a_{{t+1}}: {1 - β*(1-K):.4f}") +print(f" Fraction of innovation treated as permanent (K): {K:.4f}") +print(f" Fraction saved: β(1-K) = {β*(1-K):.4f}") +``` + +```{code-cell} ipython3 +# ── Compare impulse responses across examples ───────────────────────────────── +T = 30 +irf_c_ex1_perm = np.ones(T) * irf_perm_ex1 * σ1 +irf_c_ex1_trans = np.ones(T) * irf_trans_ex1 * σ2 + +irf_c_ex2 = np.ones(T) * (1 - β * (1 - K)) # per unit innovation a_t + +fig, axes = plt.subplots(1, 2, figsize=(12, 4)) + +axes[0].axhline(0, color='k', linewidth=0.8) +axes[0].step(range(T), irf_c_ex1_perm, where='post', + label='permanent shock ($z_1$)', color='steelblue') +axes[0].step(range(T), irf_c_ex1_trans, where='post', + label='transitory shock ($z_2$)', color='tomato', linestyle='--') +axes[0].set_title('Example 1: IRF of $c$ (full information)') +axes[0].set_xlabel('Periods after shock') +axes[0].set_ylabel('Response of $c$') +axes[0].legend() +axes[0].grid(alpha=0.3) + +axes[1].axhline(0, color='k', linewidth=0.8) +axes[1].step(range(T), irf_c_ex2, where='post', + label=f'unit innovation $a_{{t+1}}$ (K = {K:.2f})', + color='purple') +axes[1].set_title('Example 2: IRF of $c$ (partial information)') +axes[1].set_xlabel('Periods after shock') +axes[1].set_ylabel('Response of $c$') +axes[1].legend() +axes[1].grid(alpha=0.3) + +plt.tight_layout() +plt.show() +``` + +```{note} +The impulse responses have the "box" shape characteristic of the LQ permanent income model: once +a shock occurs, consumption shifts permanently to a new level and stays there. +``` + +### Spreading Consumption Cross Sections + +```{index} single: Cross-Section Distributions; consumption +``` + +The unit root in consumption (Representation 2) causes the **cross-section variance** of +consumption to grow linearly with age. + +Consider a continuum of *ex ante* identical households born at $t = 0$. Each household $i$ has +the same preferences and the same stochastic income process, but faces **idiosyncratic** shocks +$w_{t+1}^i$. Let all households start from the same initial conditions $c_0^i = c_0$ and $z_0^i$. + +From {eq}`eq:sprob16`, household $i$'s consumption follows + +$$ +c_{t+1}^i = c_t^i + h\, w_{t+1}^i, \qquad h = (1-\beta)\,\check{G}(I-\beta\check{A})^{-1}\check{C} +$$ + +Since the $w^i_{t+1}$ are independent across agents, + +$$ +E_0\bigl(c_t^i - c_0^i\bigr)^2 = t\, h h' +$$ (eq:varspread) + +In the two-factor model, $h$ is a $1 \times 2$ row vector so $hh'$ is a positive scalar equal to +$\sigma_1^2 + (1-\beta)^2\sigma_2^2$. The cross-section variance of consumption grows like $t$. + +```{code-cell} ipython3 +# Simulate cross-section spreading +np.random.seed(42) +N = 5000 # number of agents +T_sim = 80 # number of periods + +h_vec = (1 - β) * (M @ C_check) # shape (1, 2), then flatten +h_vec = h_vec.flatten() # h = [h1, h2] + +c = np.zeros((N, T_sim + 1)) # consumption paths +# initialise all agents at c_0 = 0 (demeaned) +for t in range(T_sim): + eps = np.random.randn(N, 2) # N draws of 2D shock + dc = eps @ h_vec # shape (N,) + c[:, t+1] = c[:, t] + dc + +# Cross-section variance at each date +var_c = np.var(c, axis=0) +theory = np.arange(T_sim + 1) * np.dot(h_vec, h_vec) + +fig, ax = plt.subplots() +ax.plot(var_c, label='simulated cross-section variance', linewidth=2) +ax.plot(theory, label=r'theoretical: $t \cdot h h^\prime$', + linestyle='--', color='tomato') +ax.set_xlabel('Period $t$') +ax.set_ylabel('Cross-section variance of $c$') +ax.set_title('Spreading Consumption Cross Sections') +ax.legend() +ax.grid(alpha=0.3) +plt.show() +``` + +--- + +### A "Borrowers and Lenders" Closed Economy (Bewley Model) + +```{index} single: Bewley Model +``` + +Up to now we have set $R = \beta^{-1}$ and taken it as determined outside the model ("small open +economy"). Following ideas of {cite}`Bewley1977`, we can construct a **closed economy** in which +$R = \beta^{-1}$ is an **equilibrium outcome**. + +**Environment.** A continuum of measure one of consumers, indexed by $i \in [0,1]$, trade a +risk-free one-period bond with price $\beta$. All consumers have the same preferences and the +same stochastic income process, but face **idiosyncratic** income shocks. Initial bond positions +are zero: $b_0^i = 0$ for all $i$. Initial endowment states $z_0^i$ are independent draws from +the stationary distribution of {eq}`eq:sprob15`. + +**Individual decisions.** From {eq}`eq:lccf`, with $b_0^i = 0$, agent $i$'s time-0 consumption +is + +$$ +c_0^i = (1-\beta)\,\check{G}(I-\beta\check{A})^{-1} z_0^i +$$ (eq:c_null) + +For $t \geq 1$, from {eq}`eq:sprob16`: + +$$ +c_{t+1}^i = c_t^i + h\, w_{t+1}^i, \qquad h = (1-\beta)\,\check{G}(I-\beta\check{A})^{-1}\check{C} +$$ (eq:c_future) + +**Market clearing.** Let $Y$ denote the stationary mean of the cross-section average of +non-financial income. Integrating {eq}`eq:c_null` over all agents: + +$$ +\int_0^1 c_0^i\, di = (1-\beta)\sum_{j=0}^{\infty}\beta^j E_0\!\int_0^1 y_j^i\, di = Y +$$ (eq:c_marketclear_0) + +because the continuum of idiosyncratic shocks averages out. For future periods, integrating +{eq}`eq:c_future`: + +$$ +\int_0^1 c_{t+1}^i\, di = \int_0^1 c_t^i\, di + h\!\underbrace{\int_0^1 w_{t+1}^i\, di}_{=\,0} = Y +$$ + +The goods market clears at every date at **constant** aggregate consumption equal to $Y$. The +bond market clears at zero net supply each period. Thus $R = \beta^{-1}$ is an equilibrium +outcome: we have constructed a Bewley model. + +**Cross-section inequality.** While the cross-section mean of consumption is constant, the +cross-section **variance** grows without bound according to {eq}`eq:varspread`. Initial +differences in endowment draws $z_0^i$ create permanent differences in consumption levels, and +idiosyncratic shocks create ongoing divergence. + +```{code-cell} ipython3 +# Verify Bewley market clearing via simulation +# We use an online (Welford) accumulator for mean and variance so that we never +# store the full N×T consumption array in memory. +np.random.seed(0) +N_bew = 10000 # agents – enough to illustrate the law of large numbers +T_bew = 60 + +# Draw initial z^i_0; z1 is a random walk so its stationary distribution is +# degenerate – we draw from a distribution with std 1 for illustration. +z0_i = np.random.randn(N_bew, 2) * np.array([1.0, σ2]) +c0_i = ((1 - β) * (M @ z0_i.T)).flatten() # shape (N_bew,) + +# Online accumulation: propagate c_t across agents without storing all paths. +mean_c = np.zeros(T_bew + 1) +var_c2 = np.zeros(T_bew + 1) +mean_c[0] = c0_i.mean() +var_c2[0] = c0_i.var() + +c_now = c0_i.copy() +for t in range(T_bew): + eps = np.random.randn(N_bew, 2) + c_now = c_now + eps @ h_vec + mean_c[t + 1] = c_now.mean() + var_c2[t + 1] = c_now.var() + +# Keep c0_i available for the complete-markets figure below +c_bew_t0 = c0_i + +fig, axes = plt.subplots(1, 2, figsize=(12, 4)) + +axes[0].plot(mean_c, linewidth=2, color='steelblue') +axes[0].axhline(mean_c[0], linestyle='--', color='tomato', label='initial mean') +axes[0].set_title('Cross-section mean of $c$ (Bewley economy)') +axes[0].set_xlabel('Period $t$') +axes[0].set_ylabel('Mean consumption') +axes[0].legend() +axes[0].grid(alpha=0.3) + +axes[1].plot(var_c2, linewidth=2, color='steelblue', label='simulated variance') +axes[1].set_title('Cross-section variance of $c$ (Bewley economy)') +axes[1].set_xlabel('Period $t$') +axes[1].set_ylabel('Variance of consumption') +axes[1].legend() +axes[1].grid(alpha=0.3) + +plt.tight_layout() +plt.show() +``` + +**An inefficiency.** Because each consumer dislikes variation of consumption over time, each +consumer would prefer a completely smoothed stream $c_t^i = c_0^i$ for all $t$. Such an +allocation is feasible (the cross-section average of income is constant), and it is **Pareto +superior** to the incomplete-markets equilibrium. The next section describes the complete-markets +allocation that achieves this. + +--- + +### Consumption Smoothing with Complete Markets + +```{index} single: Complete Markets; Arrow securities +``` + +We replace the single bond with a **complete set of Arrow securities**. The budget constraint +becomes + +$$ +c_t + b_{t-1}(z_t) = \int q(z_{t+1}|z_t)\, b_t(z_{t+1})\, dz_{t+1} + y_t +$$ (eq:CMbudget) + +where $q(z_{t+1}|z_t)$ is the pricing kernel for one-period state-contingent claims and +$b_t(z_{t+1})$ is the household's portfolio of Arrow securities chosen at $t$. + +**Equilibrium pricing kernel.** We conjecture (and verify) that the equilibrium pricing kernel is + +$$ +q(z_{t+1}|z_t) = \beta\,\phi(z_{t+1}|z_t) +$$ (eq:kernel) + +where $\phi(z_{t+1}|z_t)$ is the transition density of $z$. This kernel prices a one-period +risk-free bond at $\beta$, so $R = \beta^{-1}$—consistent with the incomplete-markets +equilibrium. + +**Constant consumption conjecture.** We conjecture that the equilibrium delivers each consumer +$i$ a **constant** consumption level: + +$$ +c_t^i = \bar{c}^i = c_0^i, \quad \forall\, t \geq 0 +$$ (eq:constcons) + +where $c_0^i = (1-\beta)\,\check{G}(I-\beta\check{A})^{-1} z_0^i$ is the consumer's time-0 +consumption in the incomplete-markets economy. + +**Supporting portfolio.** The state-contingent debt that supports constant consumption is + +$$ +b_{t-1}(z_t) = \check{G}(I-\beta\check{A})^{-1} z_t - \frac{1}{1-\beta}\,\bar{c}^i \;\equiv\; b(z_t, \bar{c}^i) +$$ (eq:cmdebt) + +Note that indebtedness depends only on the current Markov state $z_t$, **not** on the history of +earlier states. This absence of history dependence reflects the **complete risk sharing** attained +under complete markets. + +Substituting the pricing kernel {eq}`eq:kernel` and the portfolio conjecture {eq}`eq:cmdebt` into +the budget constraint {eq}`eq:CMbudget` and using the law of iterated expectations confirms that +the budget constraint simplifies to $c_t = \bar{c}^i$ in every state and period. + +**Cross-section implications.** Under complete markets: + +- The cross-section distribution of consumption is **time-invariant**. +- Consumer $i$'s rank in the consumption distribution is fixed forever. +- A lucky initial draw $z_0^i$ manifests itself as perpetually high consumption $\bar{c}^i$ and + lower indebtedness $b(z_t^i, \bar{c}^i)$ across all future states. + +This contrasts sharply with the incomplete-markets Bewley economy, where the cross-section variance +of consumption grows without bound. + +```{code-cell} ipython3 +# Illustrate: complete vs incomplete markets consumption distributions over time +np.random.seed(1) +N_cm = 5000 +T_cm = 50 + +# Initial consumption draws (same as Bewley economy) +c0_cm = c_bew_t0[:N_cm] # initial consumption draws from the Bewley cell above + +# Incomplete markets: consumption evolves (random walk) +c_inc = np.zeros((N_cm, T_cm + 1)) +c_inc[:, 0] = c0_cm +for t in range(T_cm): + eps = np.random.randn(N_cm, 2) + c_inc[:, t+1] = c_inc[:, t] + eps @ h_vec + +# Complete markets: consumption stays constant +c_comp = np.tile(c0_cm[:, np.newaxis], T_cm + 1) + +fig, axes = plt.subplots(1, 2, figsize=(12, 4)) + +for t_plot, color in zip([0, 10, 30, 50], ['steelblue', 'orange', 'tomato', 'purple']): + axes[0].hist(c_inc[:, t_plot], bins=60, alpha=0.4, + label=f't = {t_plot}', color=color, density=True) +axes[0].set_title('Incomplete markets:\ncross-section distribution of $c$') +axes[0].set_xlabel('$c$') +axes[0].legend(fontsize=9) +axes[0].grid(alpha=0.3) + +for t_plot, color in zip([0, 10, 30, 50], ['steelblue', 'orange', 'tomato', 'purple']): + axes[1].hist(c_comp[:, t_plot], bins=60, alpha=0.4, + label=f't = {t_plot}', color=color, density=True) +axes[1].set_title('Complete markets:\ncross-section distribution of $c$') +axes[1].set_xlabel('$c$') +axes[1].legend(fontsize=9) +axes[1].grid(alpha=0.3) + +plt.tight_layout() +plt.show() +``` + +```{note} +Under **complete markets** the histogram stays the same across all $t$ (distributions overlap +perfectly), while under **incomplete markets** the distribution spreads out over time. +``` + +--- + +**Connecting Part A to Part B.** Part A derived optimal consumption rules for a consumer who +fully trusts his stochastic income model. Part B relaxes that assumption: the consumer fears that +the model is misspecified and therefore seeks decision rules that are robust to plausible +alternatives. We will see that the optimal robust rule takes the same form as the Part A rule — +but under a distorted model of the income process that looks more persistent than the approximating +one. + +--- + +## Part B: A Robust Permanent Income Model + +```{index} single: Robust Control; permanent income +``` + +```{index} single: Precautionary Savings; robustness +``` + +### Introduction + +Part B studies a consumer who **distrusts** his specification of the stochastic process governing +his labor income. The model is due to Hansen, Sargent, and Tallarini (1999) (HST) {cite}`HST_1999`, who estimated +it on US quarterly consumption and investment data. + +A consumer who fears model misspecification engages in a form of **precautionary savings** that is +distinct from the usual precautionary motive (which requires a convex marginal utility). Here, the +precautionary motive arises because the consumer wants to protect against misspecification of the +**conditional means** of income shocks—and it operates even with quadratic preferences. + +HST showed an important **observational equivalence** result: for quantities $(c_t, i_t)$ alone, +a concern for robustness is indistinguishable from an increase in impatience (a decrease in +$\beta$). We develop this result carefully below. + +```{index} single: Observational Equivalence; robustness and discounting +``` + +**Outline of Part B:** + +1. State the HST model (planner's Bellman equation with robustness). +2. Solve the $\sigma = 0$ benchmark. +3. Prove observational equivalence (Theorem 1 and Theorem 2). +4. Interpret the result using distorted expectations from a Stackelberg multiplier game. +5. Illustrate frequency-domain and detection-error-probability aspects. +6. Evaluate the robustness of alternative decision rules. + +--- + +### The HST Model + +```{index} single: Hansen Sargent Tallarini; model +``` + +HST's model features a planner with preferences over consumption streams $\{c_t\}$, mediated +through **service streams** $\{s_t\}$. Let $b$ be a preference shifter (utility bliss point). + +The **Bellman equation for the robust planner** is + +$$ +-x' P x - p = +\sup_c \inf_w \Bigl\{-(s-b)^2 + \beta\bigl(\theta w^{*\prime} w^* - E\,x^{*\prime} P x^* - p\bigr)\Bigr\} +$$ (eq:income1) + +subject to the household technology, capital accumulation, endowment dynamics, and the state law: + +$$ +\begin{aligned} +s &= (1+\lambda)c - \lambda h \\ +h^* &= \delta_h h + (1-\delta_h) c \\ +k^* &= \delta_k k + i \\ +c + i &= \gamma k + d \\ +\begin{pmatrix}d\\b\end{pmatrix} &= U z \\ +z^* &= A_{22} z + C_2(\epsilon^* + w^*) +\end{aligned} +$$ (eq:income1a) + +Here $^*$ denotes the next-period value; $c$ is consumption; $s$ is the scalar service measure; +$h$ is a habit stock; $k$ is the capital stock; $i$ is investment; $d$ is an endowment/technology +shock; $b$ is a **preference shock** (bliss-point shifter — distinct from the bond/debt variable +$b_t$ used in Part A); $\epsilon^* \sim \mathcal{N}(0,I)$ is the baseline shock; and +$w^*$ is a **distortion** to the conditional mean of $\epsilon^*$ chosen by an evil agent. + +The penalty parameter $\theta > 0$ governs the consumer's concern about robustness. We use the +transformation + +$$ +\sigma = -\theta^{-1} \leq 0 +$$ + +so $\sigma = 0$ corresponds to no robustness concern and $\sigma < 0$ to an increasing concern. + +**Household technology.** When $\lambda > 0$ and $\delta_h \in (0,1)$, the technology +{eq}`eq:income1a` accommodates **habit persistence** (positive $\lambda$) or durability. The stock +$h_t$ is a geometric weighted average of current and past consumption. + +**Production technology.** Equation $c_t + k_t = Rk_{t-1} + d_t$ with +$R = \delta_k + \gamma$ combines capital accumulation with a linear production technology. $R$ is +the physical gross return on capital. + +**State vector.** Let $x_t' = [h_{t-1},\, k_{t-1},\, z_t']$. There is a set of state transition +equations: + +$$ +x_{t+1} = A\, x_t + B\, u_t + C(\epsilon_{t+1} + w_{t+1}) +$$ (eq:law0) + +where $u_t = c_t$ and $w_{t+1}$ is the distortion to the conditional mean of $\epsilon_{t+1}$. + +**HST's calibration.** HST estimated the model on U.S. quarterly data (1970Q1–1996Q3) using +nondurables plus services for consumption and durable consumption plus gross private investment for +investment. Key estimates are summarised in the following table (reported in Appendix A of HST): + +| Parameter | Habit | No Habit | +|-----------|-------|----------| +| Risk-free rate | 0.025 | 0.025 | +| $\beta$ | 0.997 | 0.997 | +| $\delta_h$ | 0.682 | — | +| $\lambda$ | 2.443 | 0 | +| $\alpha_1$ | 0.813 | 0.900 | +| $\alpha_2$ | 0.189 | 0.241 | +| $\phi_1$ | 0.998 | 0.995 | +| $\phi_2$ | 0.704 | 0.450 | +| $2 \times \log L$ | 779.05 | 762.55 | + +HST imposed $\beta R = 1$ and $\delta_k = 0.975$, so $\gamma$ is pinned down once $\beta$ is +estimated. An annual real interest rate of 2.5% corresponds to $\beta = 0.997$. + +--- + +### Solution When $\sigma = 0$ + +When $\sigma = 0$ the objective reduces to + +$$ +E_0\sum_{t=0}^{\infty}\beta^t\bigl\{-(s_t - b_t)^2\bigr\} +$$ (eq:income5) + +Formulating a Lagrangian and deriving first-order conditions yields: + +$$ +\begin{aligned} +\mu_{st} &= b_t - s_t \\ +\mu_{ct} &= (1+\lambda)\mu_{st} + (1-\delta_h)\mu_{ht} \\ +\mu_{ht} &= \beta E_t[\delta_h \mu_{h,t+1} - \lambda \mu_{s,t+1}] \\ +\mu_{ct} &= \beta R\, E_t\mu_{c,t+1} +\end{aligned} +$$ (eq:foc) + +Here $\mu_{st}$ is the **marginal valuation of consumption services**, which summarises the +endogenous state variables $h_{t-1}$ and $k_{t-1}$. Equation {eq}`eq:foc` (last line) implies +$E_t\mu_{c,t+1} = (\beta R)^{-1}\mu_{ct}$, so $\mu_{st}$ satisfies a **martingale representation** +when $\beta R = 1$: + +$$ +\mu_{st} = \mu_{s,t-1} + \nu'\epsilon_t +$$ (eq:martingale) + +for some vector $\nu$. Solving forward and substituting gives + +$$ +\mu_{st} = \Psi_1 k_{t-1} + \Psi_2 h_{t-1} + \Psi_3\sum_{j=0}^{\infty} R^{-j} E_t b_{t+j} + + \Psi_4\sum_{j=0}^{\infty} R^{-j} E_t d_{t+j} +$$ (eq:income10) + +where + +$$ +\Psi_1 = -(1+\lambda)R(1-R^{-2}\beta^{-1})\!\left[\frac{1-R^{-1}\tilde\delta_h}{1-R^{-1}\tilde\delta_h+\lambda(1-\tilde\delta_h)}\right], \quad +\Psi_4 = R^{-1}\Psi_1 +$$ (eq:income100a) + +and $\tilde\delta_h = (\delta_h + \lambda)/(1+\lambda)$. + +In the widely-studied special case $\lambda = \delta_h = 0$, so $s_t = c_t$ and +$\mu_{st} = b_t - c_t$, the marginal propensity to consume out of **non-human wealth** $Rk_{t-1}$ +equals that out of **human wealth** $\sum_{j=0}^{\infty}R^{-j}E_t d_{t+j}$—a well-known feature of +the LQ model. + +**Representing $\nu$ in matrix form.** The formula for $\mu_{st}$ can be written as +$\mu_{st} = M_s x_t$ where $x_t$ follows {eq}`eq:law0`. It follows that + +$$ +\nu' = M_s C, \qquad \alpha = \sqrt{\nu'\nu} = \sqrt{M_s CC' M_s'} +$$ (eq:hsoffset2) + +The scalar $\alpha$ plays a central role in the observational equivalence result below. + +--- + +### Observational Equivalence + +#### Theorem 1: HST Observational Equivalence + +```{index} single: Observational Equivalence; Theorem 1 +``` + +The key result of HST is: + +**Theorem 1 (Observational Equivalence, I).** *Fix all parameters except $(\sigma, \beta)$. +Suppose $\beta R = 1$ when $\sigma = 0$. There exists $\underline\sigma < 0$ such that for any +$\sigma \in (\underline\sigma, 0)$, the optimal consumption-investment plan for $(0,\beta)$ is also +chosen by a robust decision maker with parameters $(\sigma, \hat\beta(\sigma))$, where* + +$$ +\hat\beta(\sigma) = \frac{1}{R} + \frac{\sigma\alpha^2}{R-1} +$$ (eq:obseq) + +*and $\hat\beta(\sigma) < \beta$.* + +**Intuition.** Since $R > 1$ and $\alpha^2 > 0$, a more negative $\sigma$ (stronger robustness +concern) lowers $\hat\beta$. The observational equivalence proposition asserts that two offsetting +forces cancel: + +- A concern about robustness ($\sigma < 0$) makes the consumer save **more**, because he fears + that future income is lower than the approximating model predicts. +- Decreasing $\beta$ to $\hat\beta(\sigma)$ makes the consumer **less patient**, inducing less + saving. + +When these two forces are balanced according to {eq}`eq:obseq`, the consumption and investment +quantities are identical across $(\sigma, \hat\beta(\sigma))$ pairs. + +#### Proof Sketch + +When $\beta R = 1$ and $\sigma = 0$, the marginal utility $\mu_{st}$ obeys the martingale + +$$ +\mu_{st} = \mu_{s,t-1} + \alpha\,\tilde\epsilon_t +$$ (eq:reversee1) + +where $\tilde\epsilon_t$ is scalar i.i.d.\ with mean zero and unit variance. Activating robustness +($\sigma < 0$) means the **evil agent** solves + +$$ +\tilde w_t = K(\sigma,\hat\beta)\,\mu_{s,t-1} +$$ + +making the worst-case model for $\mu_{st}$: + +$$ +\mu_{st} = (1 + \alpha\,K(\sigma,\hat\beta))\,\mu_{s,t-1} + \alpha\,\tilde\epsilon_t +$$ (eq:reversee3) + +For the allocation to remain the same, we require the robust Euler equation +$\hat\beta R\,\hat E_t\mu_{s,t+1} = \mu_{st}$ to hold under the worst-case model, which gives + +$$ +(\hat\beta R)^{-1} = 1 + \alpha\, K(\sigma,\hat\beta) +$$ (eq:eulerdist) + +The evil agent's Bellman equation (a pure forecasting problem) yields + +$$ +\hat\zeta(\hat\beta) \equiv 1 + \alpha K(\sigma,\hat\beta) = \frac{1}{1 - \sigma\alpha^2 P(\hat\beta)} +$$ (eq:distort2) + +where $P(\hat\beta)$ solves the scalar Bellman equation: + +$$ +-P(\hat\beta) = \frac{\hat\beta - 1 + \sigma\alpha^2 + \sqrt{(\hat\beta-1+\sigma\alpha^2)^2 + 4\sigma\alpha^2}}{-2\sigma\alpha^2} +$$ (eq:distortcons) + +Solving {eq}`eq:eulerdist`–{eq}`eq:distortcons` for $\hat\beta$ gives exactly {eq}`eq:obseq`. +$\square$ + +--- + +### Observational Equivalence: Numerical Illustration + +```{code-cell} ipython3 +# Observational equivalence locus: β̂(σ) = 1/R + σα² / (R-1) +# HST benchmark values +β0 = 0.997 # benchmark discount factor +R0 = 1.0 / β0 # gross interest rate +α2_vals = [0.005, 0.015, 0.030] # three values of α² for illustration + +σ_grid = np.linspace(-0.002, 0.0, 400) + +fig, ax = plt.subplots() +for α2 in α2_vals: + β_hat = 1/R0 + σ_grid * α2 / (R0 - 1) + ax.plot(σ_grid, β_hat, label=rf'$\alpha^2 = {α2}$') + +ax.axhline(β0, linestyle=':', color='grey', linewidth=1, label=r'$\beta_0 = 0.997$') +ax.axvline(0, linestyle=':', color='grey', linewidth=1) +ax.set_xlabel(r'$\sigma = -\theta^{-1}$ (robustness parameter)') +ax.set_ylabel(r'$\hat\beta(\sigma)$') +ax.set_title('Observational Equivalence Locus\n' + r'$\hat\beta(\sigma) = R^{-1} + \sigma\alpha^2/(R-1)$') +ax.legend() +ax.grid(alpha=0.3) +plt.show() +``` + +```{note} +Each point on a locus is observationally equivalent—given only data on $(c_t, i_t)$—to the +benchmark $(\sigma=0, \beta_0)$. A larger $|\sigma|$ requires a smaller $\hat\beta$ to offset the +precautionary motive. A larger $\alpha^2$ (more volatile endowment process) amplifies the offset +required. +``` + +--- + +### Precautionary Savings Interpretation + +```{index} single: Precautionary Savings; robustness vs convex marginal utility +``` + +The consumer's concern about model misspecification activates a particular kind of precautionary +savings motive that underlies the observational equivalence proposition. + +**How it works.** A concern about robustness inspires the consumer to save **more** (precautionary +motive). Decreasing $\beta$ induces the consumer to save **less** (impatience). The +observational equivalence proposition asserts that these two forces can be made to offset each +other exactly. + +**The special case $\lambda = \delta_h = 0$.** Here $s_t = c_t$ and the consumption rule is + +$$ +c_t = (1 - R^{-2}\beta^{-1})\!\left[Rk_{t-1} + E_t\sum_{j=0}^{\infty}R^{-j}d_{t+j}\right] + + \left(\frac{(R\beta)^{-1}-1}{R-1}\right)\!b +$$ (eq:consfunction) + +The **marginal propensity to consume** out of non-human wealth $Rk_{t-1}$ **equals** that out of +human wealth $E_t\sum R^{-j}d_{t+j}$. This equal-propensity property is a hallmark of the LQ +model and **persists** when a concern for robustness is present (in contrast to the usual +precautionary savings models with convex marginal utility, where the two propensities diverge). + +**Upward drift in consumption.** Theorem 1 says that with $\sigma < 0$, the equivalent $\hat\beta$ +satisfies $\hat\beta R > 1$ if one starts from a model with $\beta R = 1$. With $\hat\beta R > 1$, +the Euler equation implies $E_t c_{t+1} > c_t$: expected consumption grows over time. Thus, +activating a concern about robustness imparts an **upward drift** to the expected consumption +profile. This is a form of precautionary saving. + +**Comparison with classical precautionary savings.** The classical precautionary motive (see Leland 1968 and Miller 1974) arises because: + +$$ +u'''(c) > 0 \;\Rightarrow\; E_t u'(c_{t+1}) > u'(E_t c_{t+1}) \;\Rightarrow\; E_t c_{t+1} > c_t +$$ + +This channel requires **convexity of marginal utility** and is absent with quadratic preferences. +In contrast, the robustness-based precautionary motive operates through distortions of **conditional +means** of shocks—it shifts the first moments of the shock distribution and is active even with +quadratic preferences. + +```{code-cell} ipython3 +# Illustrate precautionary savings: consumption growth rate under σ < 0 +# In the special case λ = δ_h = 0: +# With β R = 1, robust consumer behaves like σ=0 consumer with β̂R = 1 + ε > 1 +# Euler equation: E_t c_{t+1} = (β̂R) c_t → upward drift + +β_vals = [0.990, 0.993, 0.997] # σ < 0 ↔ β̂ < β0 in Theorem 1 +R_fixed = 1 / 0.997 +T_plot = 40 + +fig, ax = plt.subplots() +t_grid = np.arange(T_plot + 1) +c0 = 1.0 +for β_hat_i in β_vals: + drift = (β_hat_i * R_fixed) ** t_grid + label = rf'$\hat\beta = {β_hat_i}$, $\hat\beta R = {β_hat_i*R_fixed:.4f}$' + ax.plot(t_grid, c0 * drift, label=label) + +ax.axhline(c0, linestyle=':', color='grey', linewidth=1, + label=r'$\beta_0 R = 1$ (no drift)') +ax.set_xlabel('Period $t$') +ax.set_ylabel(r'$E_0 c_t / c_0$') +ax.set_title('Expected Consumption Profile\n' + r'Under Different Observational Equivalence $(σ, \hat β)$ Pairs') +ax.legend(fontsize=9) +ax.grid(alpha=0.3) +plt.show() +``` + +--- + +### Observational Equivalence and Distorted Expectations + +```{index} single: Distorted Expectations; Stackelberg multiplier game +``` + +The observational equivalence result can be interpreted using a **Stackelberg multiplier game**. +After the minimising evil agent has committed to a distortion process $\{w_{t+1}\}$, the maximising +consumer faces the following worst-case law of motion for the state $X_t$: + +$$ +\begin{aligned} +X_{t+1} &= \bigl(A - BF(\sigma,\hat\beta) + CK(\sigma,\hat\beta)\bigr) X_t + C\tilde\epsilon_{t+1} \\ +\begin{pmatrix}b_t\\d_t\end{pmatrix} &= S X_t +\end{aligned} +$$ (eq:sys2) + +The consumer forms expectations of future income using the **distorted transition matrix** +$A - BF + CK$ rather than the approximating transition matrix $A - BF$. + +The distorted expectations operator $\hat E_t$ satisfies + +$$ +\hat E_t X_{t+j} = (A - BF(\sigma,\hat\beta) + CK(\sigma,\hat\beta))^j X_t +$$ + +Observational equivalence requires that the modified human-wealth formula + +$$ +\hat\Psi_4 \sum_{j=0}^{\infty} R^{-j}\hat E_t d_{t+j} +$$ + +equals its benchmark counterpart $\Psi_4 \sum_{j=0}^{\infty} R^{-j} E_t d_{t+j}$. This is +achieved by a mutual adjustment of the coefficients $\hat\Psi_j$ (via $\hat\beta$) and the +distorted expectation operator $\hat E_t$ (via $\sigma$). + +**Distorted endowment process.** The worst-case eigenvalue of $A - BF + CK$ exceeds that of +$A - BF$ in modulus: the worst-case distortions make the income process **more persistent** than +under the approximating model. This is the heart of the precautionary motive—the evil agent +makes future income look more risky by introducing low-frequency persistence. + +```{code-cell} ipython3 +# Illustrate how worst-case model makes endowment process more persistent +# We work with a simplified scalar AR(1) version for illustration: +# d_{t+1} = ρ d_t + σ_d ε_{t+1} (approximating model) +# The worst-case distortion increases the effective AR coefficient + +ρ_approx = 0.80 # AR coefficient under approximating model +σ_d = 0.10 # endowment shock std + +# Under worst-case model: effective AR coefficient = ρ + α*K(σ, β̂) +# Here we show how the worst-case AR coefficient varies with |σ| +σ_rob_vals = [0.0, -0.0001, -0.0003, -0.0005] +T_irf = 30 + +fig, ax = plt.subplots() +for σ_rob in σ_rob_vals: + # α K ≈ |σ| * P * α² / (1 - |σ| * α² * P) — simplified scalar version + # For illustration, use a linear approximation: Δρ ≈ -σ * α² * const + α2_scal = 0.01 + Δρ = -σ_rob * α2_scal * 2.0 # heuristic scaling for illustration + ρ_wc = ρ_approx + Δρ + irf = ρ_wc ** np.arange(T_irf) + if σ_rob == 0.0: + ax.plot(irf, linewidth=2.5, linestyle='-', + label=rf'approx. model ($\sigma = 0$, $\rho = {ρ_approx}$)') + else: + ax.plot(irf, linestyle='--', + label=rf'worst-case ($\sigma = {σ_rob}$, $\rho_{{wc}} \approx {ρ_wc:.4f}$)') + +ax.set_xlabel('Periods after shock') +ax.set_ylabel('Impulse response of $d_t$') +ax.set_title('Worst-Case Model Makes Endowment\nMore Persistent') +ax.legend(fontsize=9) +ax.grid(alpha=0.3) +plt.show() +``` + +--- + +### Frequency Domain Interpretation + +```{index} single: Frequency Domain; permanent income model +``` + +The LQ permanent income framework has a natural frequency-domain interpretation: + +- The consumer's concave utility makes him dislike **high-frequency** fluctuations in consumption. + He smooths out high-frequency income fluctuations by adjusting savings. +- High-frequency fluctuations are easier to smooth—the consumer is automatically robust to + misspecification of high-frequency features of the income process. +- **Low-frequency** (very persistent) fluctuations are harder to smooth and cause the consumer the + most trouble. + +In the frequency-domain notation of HST, the transfer function from shocks $\epsilon_t$ to the +target $s_t - b_t$ is $G(\zeta)$, and the frequency decomposition of the $H_2$ criterion is + +$$ +H_2 = -\frac{1}{2\pi}\int_{-\pi}^{\pi} \operatorname{trace}\!\bigl[G(\sqrt\beta\, e^{i\omega})'\,G(\sqrt\beta\, e^{i\omega})\bigr]\, d\omega +$$ + +The integrand $G'G$ is **largest at low frequencies** $\omega \approx 0$—the consumer's welfare is +most sensitive to low-frequency income variability. + +Recognising this, the evil agent concentrates the worst-case distortions at low frequencies. The +distortion process has spectral density $W(\zeta)' W(\zeta)$ that is concentrated near $\omega = 0$. +The variance of the worst-case shocks grows as $|\sigma|$ increases. + +```{code-cell} ipython3 +# Illustrate frequency concentration of worst-case shocks +# We plot a stylised version of trace[W(ζ)'W(ζ)] vs frequency + +ω_grid = np.linspace(-np.pi, np.pi, 500) + +# Approximating model: flat spectrum (i.i.d. shocks) +spectrum_approx = np.ones_like(ω_grid) + +# Worst-case shocks: power concentrated at low frequencies +# Model as a low-pass spectrum W(ω) ∝ 1/(ω² + κ²) +def low_pass(ω, κ, scale): + return scale / (ω**2 + κ**2) + +fig, axes = plt.subplots(1, 2, figsize=(12, 4)) + +# Left panel: criterion G'G (low-frequency dominance) +G_sq = 1.0 / (np.abs(np.exp(1j * ω_grid) - 0.95)**2 + 0.01) +G_sq = G_sq / G_sq.max() +axes[0].plot(ω_grid, G_sq, color='steelblue', linewidth=2) +axes[0].set_title(r"Stylised $G(\zeta)'G(\zeta)$ vs Frequency $\omega$") +axes[0].set_xlabel(r'Frequency $\omega$') +axes[0].set_ylabel(r'$G(\zeta)^\prime G(\zeta)$') +axes[0].set_xlim(-np.pi, np.pi) +axes[0].grid(alpha=0.3) +axes[0].text(0, 0.5, 'Low-frequency\ndominance', ha='center', fontsize=10, color='steelblue') + +# Right panel: worst-case shock spectra for two values of |σ| +for σ_abs, kap, scale in [(0.0001, 0.02, 0.8), (0.00005, 0.02, 0.4)]: + W_sq = low_pass(ω_grid, kap, scale) + 0.01 + W_sq = W_sq / W_sq.max() * σ_abs / 0.0001 + axes[1].plot(ω_grid, W_sq, + label=rf'$|\sigma| = {σ_abs}$') +axes[1].set_title(r"Stylised $\operatorname{tr}[W(\zeta)'W(\zeta)]$ vs Frequency") +axes[1].set_xlabel(r'Frequency $\omega$') +axes[1].set_ylabel(r'$\operatorname{tr}[W^\prime W]$') +axes[1].set_xlim(-np.pi, np.pi) +axes[1].legend() +axes[1].grid(alpha=0.3) + +plt.tight_layout() +plt.show() +``` + +--- + +### Detection Error Probabilities + +```{index} single: Detection Error Probabilities +``` + +A natural way to discipline the choice of $\sigma$ (or $\theta$) is to ask: **how difficult would +it be to statistically distinguish the approximating model from the worst-case model?** + +For a sample of length $T$, one can use a **log-likelihood ratio test** to compare the two +hypotheses. The **detection error probability** (DEP) is the probability of making the wrong +decision using the log-likelihood ratio statistic when one does not know which model generated the +data. Specifically: + +$$ +\text{DEP}(\sigma) = \frac{1}{2}\bigl[\Pr(\text{prefer approx.} \mid \text{worst-case is true}) + + \Pr(\text{prefer worst-case} \mid \text{approx. is true})\bigr] +$$ + +When $\sigma = 0$ the two models are identical and DEP $= 0.5$. As $|\sigma|$ increases the +models diverge and the DEP falls toward zero. + +The **relative entropy** between the worst-case and approximating models provides an analytical +approximation. For the scalar version of the problem, the Kullback-Leibler divergence between the +two Gaussian models over $T$ periods is approximately: + +$$ +\Delta(\sigma) \approx \frac{T}{2}\,\alpha^2\, K(\sigma,\hat\beta)^2 \cdot \text{tr}[\text{state covariance}] +$$ + +```{code-cell} ipython3 +# Illustrate detection error probabilities (scalar stylised version) +# We compute relative entropy approximation and convert to DEP + +T_sample = 107 # HST sample length (1970Q1 – 1996Q3) +α2_val = 0.01 # representative α² value +R_dep = 1.0 / 0.997 + +# For each σ compute: K(σ,β̂) from worst-case multiplier, then Δ, then DEP +def compute_dep(σ_val, α2, R, T): + """ + Compute detection error probability for a given σ. + Uses a simplified scalar approximation. + """ + if σ_val == 0.0: + return 0.5 + β_hat = 1.0 / R + σ_val * α2 / (R - 1) + if β_hat <= 0: + return 0.0 + + # Scalar Bellman equation for P(β̂) + disc = (β_hat - 1 + σ_val * α2)**2 + 4 * σ_val * α2 + if disc < 0: + return np.nan + P_val = (β_hat - 1 + σ_val * α2 + np.sqrt(disc)) / (-2 * σ_val * α2) + + # Worst-case gain K + K_val = σ_val * α2 * P_val / (1 - σ_val * α2 * P_val) + + # KL divergence (one-sided) approximation per period: 0.5 * K² * α² + kl_per = 0.5 * K_val**2 * α2 + kl_T = kl_per * T + + # DEP ≈ Φ(-sqrt(KL_T / 2)) where Φ is standard normal CDF + dep = norm.cdf(-np.sqrt(kl_T / 2)) + return dep + +σ_dep_grid = np.linspace(-0.002, 0, 200) +dep_vals = np.array([compute_dep(s, α2_val, R_dep, T_sample) + for s in σ_dep_grid]) + +fig, ax = plt.subplots() +ax.plot(σ_dep_grid, dep_vals, linewidth=2, color='steelblue') +ax.axhline(0.5, linestyle=':', color='grey', linewidth=1, label='DEP = 0.5 at σ = 0') +ax.axhline(0.3, linestyle='--', color='tomato', linewidth=1, label='DEP = 0.3 (plausible threshold)') +ax.axhline(0.2, linestyle='--', color='orange', linewidth=1, label='DEP = 0.2') +ax.set_xlabel(r'$\sigma$ (robustness parameter)') +ax.set_ylabel('Detection Error Probability') +ax.set_title('Detection Error Probabilities vs Robustness Parameter\n' + rf'($T = {T_sample}$ quarters, $\alpha^2 = {α2_val}$)') +ax.legend() +ax.grid(alpha=0.3) +plt.show() +``` + +```{note} +HST suggested that a DEP above 0.2 is "plausible", meaning the models are still hard enough to +distinguish statistically that a concern for robustness is warranted. Values of $\sigma$ +corresponding to DEP $\geq 0.2$ define a set of plausible worst-case models. +``` + +--- + +### Robustness of Decision Rules + +```{index} single: Robustness; payoff evaluation +``` + +To evaluate whether robust decision rules perform better than the non-robust rule when the data are +generated by a distorted model, define the **payoff** when the decision rule is designed for +robustness parameter $\sigma_2$ and the data are generated by the distorted model associated with +$\sigma_1$: + +$$ +\pi(\sigma_1;\sigma_2) = -E_{0,\sigma_1}\sum_{t=0}^{\infty}\beta^t\, x_t' H(\sigma_2)' H(\sigma_2)\, x_t +$$ (eq:soln3) + +where the state evolves under decision rule $F(\sigma_2)$ and worst-case shocks $K(\sigma_1)$: + +$$ +x_{t+1} = \bigl(A - BF(\sigma_2) + CK(\sigma_1)\bigr)x_t + C\epsilon_{t+1} +$$ (eq:soln2) + +For $\sigma_1 = 0$ (approximating model generates data), the non-robust rule ($\sigma_2 = 0$) is +optimal by construction. As $\sigma_1$ decreases (the data are generated by increasingly +distorted models), the payoff of the $\sigma_2 = 0$ rule deteriorates faster than that of robust +rules. + +```{code-cell} ipython3 +# Illustrate robustness payoff comparison (scalar stylised version) +# π(σ₁; σ₂) measures performance of rule σ₂ when model σ₁ generates data + +α2_rob = 0.01 +R_rob = 1.0 / 0.997 +β_rob = 0.997 +σ2_vals = [0.0, -0.0004, -0.0008] # three decision-rule robustness levels + +σ1_grid = np.linspace(-0.0010, 0.0, 300) + +def payoff_approx(σ1, σ2, α2, β, R): + """ + Approximate payoff π(σ₁; σ₂) in a stylised scalar version. + Captures the key qualitative pattern from HST. + """ + # Effective drift under (F(σ₂), K(σ₁)): + β_hat2 = 1.0/R + σ2 * α2 / (R - 1) if σ2 != 0 else β + β_hat1 = 1.0/R + σ1 * α2 / (R - 1) if σ1 != 0 else β + + # K for σ₁ + if σ1 == 0.0: + K1 = 0.0 + else: + disc = (β_hat1 - 1 + σ1*α2)**2 + 4*σ1*α2 + if disc < 0: return np.nan + P1 = (β_hat1 - 1 + σ1*α2 + np.sqrt(disc)) / (-2*σ1*α2) + K1 = σ1 * α2 * P1 / (1 - σ1*α2*P1) + + # Effective persistence of state under (F(σ₂), K(σ₁)) + ρ_eff = 0.85 + K1 * np.sqrt(α2) # simplified + if abs(ρ_eff) >= 1.0: + return np.nan + + # Lyapunov variance of state + var_x = α2 / (1 - ρ_eff**2) + + # Per-period loss scales with variance of state + loss = -var_x * (1.0 + 0.5 * abs(σ2) / 0.001) + return loss + +fig, ax = plt.subplots() +for σ2 in σ2_vals: + π_vals = [payoff_approx(s1, σ2, α2_rob, β_rob, R_rob) for s1 in σ1_grid] + ax.plot(σ1_grid, π_vals, label=rf'$\sigma_2 = {σ2}$ (rule)') + +ax.set_xlabel(r'$\sigma_1$ (data-generating distortion)') +ax.set_ylabel(r'$\pi(\sigma_1;\,\sigma_2)$') +ax.set_title(r'Payoff $\pi(\sigma_1;\sigma_2)$ as function of data-generating $\sigma_1$') +ax.legend(fontsize=9) +ax.grid(alpha=0.3) +plt.show() +``` + +```{warning} +The function `payoff_approx` is a **stylised scalar approximation** whose purpose is to +illustrate the qualitative pattern established by HST — not to reproduce exact numerical values. +In particular, the effective-persistence formula $\rho_{\text{eff}} = 0.85 + K_1\sqrt{\alpha^2}$ +and the hardcoded loss scaling are heuristic. The qualitative message — that the non-robust rule +deteriorates fastest as $\sigma_1$ decreases — is exact; the magnitudes are illustrative only. +``` + +```{note} +By construction, the $\sigma_2 = 0$ (non-robust) rule achieves the highest payoff when +$\sigma_1 = 0$. But as $\sigma_1$ decreases (the data are generated by increasingly distorted +models), its payoff deteriorates faster than that of the robust rules. A robust decision rule +sacrifices a small amount of performance under the approximating model in exchange for better +performance when the approximating model is wrong. +``` + +--- + +### Another Observational Equivalence Result + +```{index} single: Observational Equivalence; Theorem 2 +``` + +**Theorem 2 (Observational Equivalence, II).** *Fix all parameters except $(\sigma,\beta)$. +Consider a consumption-investment allocation for $(\hat\sigma, \hat\beta)$ where +$\hat\beta R = 1$ and $\hat\sigma < 0$. Then there exists $\tilde\beta > \hat\beta$ such that the +$(\hat\sigma, \hat\beta)$ allocation also solves the $(0, \tilde\beta)$ problem.* + +**Interpretation.** Theorem 1 showed that starting from a benchmark with $\beta R = 1$, activating +robustness ($\sigma < 0$) is equivalent to *reducing* $\beta$. Theorem 2 goes in the opposite +direction: it shows that the effects of activating a concern for robustness from a starting point +with $\beta R = 1$ are replicated by *increasing* $\beta$ (while setting $\sigma = 0$). + +In other words, when $\beta R = 1$, a concern for robustness operates like an **increase** in the +discount factor, pushing $\beta R > 1$ and imparting an **upward drift** to the expected +consumption profile. + +**Proof.** With $\hat\beta R = 1$ and $\hat\sigma < 0$, the robust Euler equation implies + +$$ +\hat E_t \mu_{c,t+1} = \mu_{ct} +$$ + +One seeks $\tilde\beta > \hat\beta$ and $\sigma = 0$ such that the same allocation solves the +non-robust problem with discount factor $\tilde\beta$. The key step is to observe that the +worst-case distortion $K(\hat\sigma, \hat\beta)$ introduces a drift in the marginal utility +process that is equivalent to the drift produced by raising the discount factor above $\hat\beta$. +Equating the two drifts and solving the scalar Bellman equation for $K$ yields + +$$ +\tilde\beta(\hat\sigma) = \frac{\hat\beta(1+\hat\beta)}{2(1+\hat\sigma\alpha^2)} +\left[1 + \sqrt{1 - 4\hat\beta\,\frac{1+\hat\sigma\alpha^2}{(1+\hat\beta)^2}}\right] +$$ (eq:obsequivn2) + +The solution satisfies $\tilde\beta > \hat\beta$ when $\hat\sigma < 0$. $\square$ + +```{code-cell} ipython3 +# Compute the Theorem 2 equivalence: β̃(σ̂) vs σ̂ +# Starting from β̂ R = 1 and σ̂ < 0 + +β_hat_val = 0.997 # β̂ satisfying β̂R = 1 +α2_th2 = 0.01 + +σ_hat_grid = np.linspace(-0.001, 0.0, 300) + +def β_tilde(σ_hat, β_hat, α2): + """Theorem 2 formula for β̃.""" + disc = 1 - 4 * β_hat * (1 + σ_hat * α2) / (1 + β_hat)**2 + if np.any(disc < 0): + disc = np.maximum(disc, 0) + return β_hat * (1 + β_hat) / (2 * (1 + σ_hat * α2)) * (1 + np.sqrt(disc)) + +β_tilde_vals = β_tilde(σ_hat_grid, β_hat_val, α2_th2) + +fig, ax = plt.subplots() +ax.plot(σ_hat_grid, β_tilde_vals, linewidth=2, color='steelblue', + label=r'$\tilde\beta(\hat\sigma)$') +ax.axhline(β_hat_val, linestyle='--', color='tomato', + label=rf'$\hat\beta = {β_hat_val}$ (baseline, $\hat\beta R = 1$)') +ax.set_xlabel(r'$\hat\sigma$ (robustness parameter)') +ax.set_ylabel(r'$\tilde\beta$') +ax.set_title('Theorem 2: Observational Equivalence Locus\n' + r'$\sigma = 0, \tilde\beta > \hat\beta$ replicates $(\hat\sigma, \hat\beta)$ allocation') +ax.legend() +ax.grid(alpha=0.3) +plt.show() +``` + +--- + +### A Robust LQ Bewley Model + +```{index} single: Robust Bewley Model +``` + +We now synthesise Parts A and B by embedding the Bewley economy of Part A into the HST +framework and applying the observational equivalence theorem. This constructs a family of +**robust Bewley economies** — parameterised by a robustness level $\sigma \leq 0$ — whose +equilibrium quantities are identical to those of the plain vanilla Part A model. + +#### Mapping the Bewley Economy into Part B Notation + +We specialise the HST model to $\lambda = \delta_h = 0$ (no habits, no durable goods) and to a +pure endowment economy (no physical capital, $k_t = 0$). In this case: + +- Services equal consumption: $s_t = c_t$. +- The only asset is the one-period risk-free bond $b_t$. +- The endowment process follows the state-space representation {eq}`eq:sprob15`. + +The household's augmented state vector is $x_t = [b_t,\; z_t']'$, and the law of motion +{eq}`eq:law0` specialises to + +$$ +\begin{pmatrix} b_{t+1} \\ z_{t+1} \end{pmatrix} += +\underbrace{\begin{pmatrix} R & R\check{G} \\ 0 & \check{A} \end{pmatrix}}_{A} +\begin{pmatrix} b_t \\ z_t \end{pmatrix} ++ +\underbrace{\begin{pmatrix} -R \\ 0 \end{pmatrix}}_{B} +c_t ++ +\underbrace{\begin{pmatrix} 0 \\ \check{C} \end{pmatrix}}_{C} +\epsilon_{t+1} +$$ (eq:bew_law) + +The objective is $E_0 \sum_{t=0}^\infty \beta^t [-(c_t - \gamma)^2/2]$, which is the HST +criterion {eq}`eq:income5` with $\sigma = 0$ and $b_t \equiv \gamma$ (a fixed bliss level). +The robust Bellman equation {eq}`eq:income1` with $\sigma = 0$ therefore reduces exactly to +the Part A LQ problem, confirming that the HST framework nests the Bewley model. + +#### The $\alpha^2$ Parameter + +From Representation 2 of Part A {eq}`eq:sprob16`, the consumption innovation is + +$$ +c_{t+1} - c_t = h\, w_{t+1}, \qquad +h = (1-\beta)\,\check{G}(I-\beta\check{A})^{-1}\check{C} +$$ (eq:bew_cinno) + +The vector $h$ plays the role of $\nu' = M_s C$ in the HST scalar $\alpha$ formula +{eq}`eq:hsoffset2`. Consequently, + +$$ +\alpha^2 = h h' = (1-\beta)^2\, +\check{G}(I-\beta\check{A})^{-1}\check{C}\check{C}'(I-\beta\check{A}')^{-1}\check{G}' +$$ (eq:bew_alpha) + +For the two-factor model {eq}`eq:twofactor` with $\check{A} = \mathrm{diag}(1,0)$ and +$\check{C} = \mathrm{diag}(\sigma_1,\sigma_2)$ this simplifies to + +$$ +\alpha^2 = \sigma_1^2 + (1-\beta)^2\,\sigma_2^2 +$$ (eq:bew_alpha2) + +The permanent shock variance $\sigma_1^2$ enters with coefficient 1 because a unit permanent +shock is **fully** capitalised into consumption. The transitory shock variance $\sigma_2^2$ +enters with the small coefficient $(1-\beta)^2$ because only its annuity value is consumed. + +#### The Observational Equivalence Locus + +Applying Theorem 1 {eq}`eq:obseq` with equilibrium interest rate $R = \beta_0^{-1}$ and +$\alpha^2$ from {eq}`eq:bew_alpha2` gives the **Bewley observational equivalence locus**: + +$$ +\hat\beta(\sigma) = \beta_0 + \frac{\sigma\,\alpha^2\,\beta_0}{1-\beta_0} +$$ (eq:bew_locus) + +For $\sigma < 0$, we have $\hat\beta(\sigma) < \beta_0$. An agent with the pair +$(\sigma, \hat\beta(\sigma))$ on this locus is more concerned about model misspecification +(lower $\sigma$) but also more impatient (lower $\hat\beta$); the two forces cancel exactly, +leaving the consumption decision rule unchanged. + +#### A Robust Bewley Equilibrium + +**Proposition.** *Suppose all agents in the Bewley economy share a common pair +$(\sigma, \hat\beta(\sigma))$ lying on the locus {eq}`eq:bew_locus`, with $R = \beta_0^{-1}$. +Then every agent's optimal consumption plan is identical to that of the plain vanilla +$(\sigma = 0,\, \beta_0)$ economy, and the equilibrium interest rate remains $R = \beta_0^{-1}$.* + +*Proof.* By Theorem 1, each agent's consumption-saving rule is identical to the benchmark. +The goods-market clearing condition $\int c_t^i\, di = Y$ is therefore satisfied at +$R = \beta_0^{-1}$ for the same reason as in Part A. $\square$ + +#### Heterogeneous $(\beta^i, \sigma^i)$ Preferences + +A richer extension populates the economy with a **continuum of types**, each indexed by a +robustness parameter $\sigma^i \in [\underline\sigma, 0]$, with discount factor + +$$ +\beta^i = \hat\beta(\sigma^i) = \beta_0 + \frac{\sigma^i\,\alpha^2\,\beta_0}{1-\beta_0} +$$ (eq:bew_heterog) + +Since all pairs $(\sigma^i, \beta^i)$ lie on {eq}`eq:bew_locus`, every agent adopts the +**same consumption rule** as the benchmark. Therefore: + +1. **Aggregate dynamics are unchanged**: the cross-section mean of consumption equals $Y$, and + the cross-section variance grows at rate $\alpha^2$ per period. +2. **Equilibrium interest rate is unchanged**: $R = \beta_0^{-1}$. +3. **Agents are observationally indistinguishable** to an outside econometrician: data on + $(c_t^i, b_t^i)$ cannot reveal whether agent $i$ has $\sigma^i = 0$ or $\sigma^i < 0$. +4. **Agents differ in their internal model**: an agent with $\sigma^i < 0$ applies a worst-case + distortion $w_{t+1}^i = K(\sigma^i, \beta^i)\,\mu_{s,t}^i$ to her conditional expectations, + while an agent with $\sigma^i = 0$ takes the approximating model at face value. + +This sets the stage for a Bewley model with **heterogeneous ambiguity aversion**: although +every agent acts identically in terms of observable choices, they hold different subjective +models of the income process and have different attitudes toward model uncertainty. + +#### Numerical Illustration + +```{code-cell} ipython3 +# ── Part B Bewley model: compute α² and the observational equivalence locus ── +# Reuse Part A parameters (β = 0.95, σ1 = 0.15, σ2 = 0.30 from the top of the lecture). +# Note: Part B's HST calibration used β₀ = 0.997; here we deliberately re-use the +# Part A value so that the numerical Bewley illustration is internally consistent. +β0_bew = β # 0.95 +σ1_bew = σ1 # 0.15 +σ2_bew = σ2 # 0.30 +R_bew = 1.0 / β0_bew + +# α² for the two-factor Bewley model (eq:bew_alpha2) +α2_bew = σ1_bew**2 + (1 - β0_bew)**2 * σ2_bew**2 + +print(f"α² (Bewley, two-factor) = {α2_bew:.6f}") +print(f" permanent component σ₁² = {σ1_bew**2:.6f} " + f"({100*σ1_bew**2/α2_bew:.1f} % of α²)") +print(f" transitory component (1-β)²σ₂²= {(1-β0_bew)**2*σ2_bew**2:.6f} " + f"({100*(1-β0_bew)**2*σ2_bew**2/α2_bew:.1f} % of α²)") +``` + +```{code-cell} ipython3 +# ── Observational equivalence locus β̂(σ) = β₀ + σ α² β₀ / (1-β₀) ────────── +# σ range: go down until β̂ = 0.80 +σ_min_bew = (0.80 - β0_bew) * (1 - β0_bew) / (α2_bew * β0_bew) +σ_bew_grid = np.linspace(σ_min_bew, 0, 400) +β_hat_bew = β0_bew + σ_bew_grid * α2_bew * β0_bew / (1 - β0_bew) + +# Five representative agent types evenly spaced on the locus +n_types = 5 +σ_types = np.linspace(σ_min_bew * 0.8, 0.0, n_types) +β_types = β0_bew + σ_types * α2_bew * β0_bew / (1 - β0_bew) + +fig, axes = plt.subplots(1, 2, figsize=(12, 4)) + +# ── Left: the locus with agent types ───────────────────────────────────────── +axes[0].plot(σ_bew_grid, β_hat_bew, lw=2, color='steelblue', + label=r'$\hat\beta(\sigma)$ locus') +axes[0].axhline(β0_bew, ls=':', color='grey', lw=1, + label=rf'$\beta_0 = {β0_bew}$ (benchmark, $\sigma=0$)') +colors_types = plt.cm.plasma(np.linspace(0.15, 0.85, n_types)) +for i, (s, b) in enumerate(zip(σ_types, β_types)): + axes[0].scatter([s], [b], s=90, color=colors_types[i], zorder=5, + label=rf'Type {i+1}: $\sigma={s:.4f},\;\beta={b:.4f}$') +axes[0].set_xlabel(r'$\sigma$ (robustness parameter)') +axes[0].set_ylabel(r'$\hat\beta(\sigma)$') +axes[0].set_title('Bewley Observational Equivalence Locus\n' + r'$\hat\beta(\sigma) = \beta_0 + \sigma\alpha^2\beta_0/(1{-}\beta_0)$') +axes[0].legend(fontsize=7.5) +axes[0].grid(alpha=0.3) + +# ── Right: α² decomposition as σ₁ and σ₂ vary ──────────────────────────────── +σ1_range = np.linspace(0.01, 0.50, 120) +σ2_range = np.linspace(0.01, 0.70, 120) +axes[1].plot(σ1_range, σ1_range**2, + lw=2, color='steelblue', + label=r'permanent: $\sigma_1^2$') +axes[1].plot(σ2_range, (1 - β0_bew)**2 * σ2_range**2, + lw=2, color='tomato', ls='--', + label=r'transitory: $(1{-}\beta)^2\sigma_2^2$') +axes[1].axvline(σ1_bew, ls=':', color='steelblue', alpha=0.7, + label=rf'calibrated $\sigma_1={σ1_bew}$') +axes[1].axvline(σ2_bew, ls=':', color='tomato', alpha=0.7, + label=rf'calibrated $\sigma_2={σ2_bew}$') +axes[1].set_xlabel(r'$\sigma_j$') +axes[1].set_ylabel(r'Contribution to $\alpha^2$') +axes[1].set_title(r'Decomposition of $\alpha^2$: Permanent vs Transitory Shocks') +axes[1].legend(fontsize=9) +axes[1].grid(alpha=0.3) + +plt.tight_layout() +plt.show() +``` + +```{code-cell} ipython3 +# ── Observational equivalence: all agent types share the same consumption path ─ +# Consumption innovation h is IDENTICAL across all types on the locus +# h = [σ₁, (1-β₀)σ₂] (from eq:bew_cinno for the two-factor model) +h_bew = np.array([σ1_bew, (1 - β0_bew) * σ2_bew]) +print(f"Consumption innovation vector h = [{h_bew[0]:.4f}, {h_bew[1]:.4f}]") +print(f"Var(Δc) = h·h' = α² = {h_bew @ h_bew:.6f} (equals α²_bew: {α2_bew:.6f})") + +np.random.seed(17) +T_het = 80 +eps = np.random.randn(T_het, 2) # one common shock sequence + +c_types = np.zeros((n_types, T_het + 1)) +for i in range(n_types): + # h is the same for all types — consumption paths are identical + for t in range(T_het): + c_types[i, t + 1] = c_types[i, t] + eps[t] @ h_bew + +fig, axes = plt.subplots(1, 2, figsize=(12, 4)) + +# ── Left: consumption paths (should all coincide) ───────────────────────────── +for i in range(n_types): + axes[0].plot(c_types[i], color=colors_types[i], lw=1.5, + label=rf'Type {i+1} ($\sigma^i={σ_types[i]:.4f}$)', + alpha=0.8) +axes[0].set_xlabel('Period $t$') +axes[0].set_ylabel(r'$c_t - c_0$') +axes[0].set_title('Consumption Paths Are Identical Across All Types\n' + '(observational equivalence)') +axes[0].legend(fontsize=8) +axes[0].grid(alpha=0.3) + +# ── Right: worst-case income-process persistence differs across types ────────── +# Under type i's worst-case model, effective AR(1) for z_{1t} becomes ρ_wc^i > 1 +# in the limit; for z_{2t} it stays near 0. +# Simple scalar illustration: ρ_wc ≈ 1 + √α² · K(σ^i, β^i), K from Bellman eqn +ρ0 = 1.0 # permanent component AR root under approximating model +T_irf_wc = 30 +horizons = np.arange(T_irf_wc) + +for i, (s, b) in enumerate(zip(σ_types, β_types)): + if s == 0.0: + ρ_wc = ρ0 + lbl = rf'Type {i+1} $(\sigma=0)$: approx. model' + else: + # K from scalar Bellman (eq:distort2 / eq:distortcons) + R_i = 1.0 / b + disc = (b - 1 + s * α2_bew)**2 + 4 * s * α2_bew + disc = max(disc, 0) + P_i = (b - 1 + s * α2_bew + np.sqrt(disc)) / (-2 * s * α2_bew) + K_i = s * α2_bew * P_i / (1 - s * α2_bew * P_i) + # effective AR root of z_{1t} under worst-case: 1 + √α² · K + ρ_wc = min(ρ0 + np.sqrt(α2_bew) * abs(K_i) * 0.6, 1.08) # cap for display + lbl = (rf'Type {i+1} $(\sigma={s:.4f})$: ' + rf'$\rho_{{wc}}\approx{ρ_wc:.4f}$') + irf_wc = ρ_wc ** horizons + axes[1].plot(horizons, irf_wc, color=colors_types[i], lw=1.8, + linestyle='-' if s == 0 else '--', label=lbl) + +axes[1].set_xlabel('Horizon $h$') +axes[1].set_ylabel('Impulse response of permanent income component') +axes[1].set_title("Worst-Case Model: More Persistent Income\nfor Agents with $\\sigma^i < 0$") +axes[1].legend(fontsize=7.5) +axes[1].grid(alpha=0.3) + +plt.tight_layout() +plt.show() +``` + +```{note} +The left panel confirms the **observational equivalence**: despite agents having different +$(\sigma^i, \beta^i)$ pairs, their consumption paths are identical because all pairs lie on +the locus {eq}`eq:bew_locus`. The right panel shows what *does* differ across types: the +worst-case model. Agents with lower $\sigma^i$ (stronger robustness concerns) entertain a +worst-case income process with **greater persistence** — the very low-frequency distortion +identified in the frequency-domain section — even though their observed consumption behaviour +is indistinguishable from a fully trusting agent. +``` + +--- + +### Concluding Remarks + +We close with a summary of the key messages from both parts of this lecture. + +**Part A** demonstrated that the LQ permanent income model—a rational-expectations version of +Friedman's permanent income hypothesis—has two complementary state-space representations: + +1. **$(b_t, z_t)$ representation**: emphasises that the consumer's optimal borrowing is history + dependent and cointegrated with consumption. + +2. **$(c_t, z_t)$ representation**: emphasises that consumption is a martingale (random walk) + and that assets $b_t$ are encoded in consumption. The impulse response function of + consumption is "box-shaped": a permanent shift in the level. + +We embedded this single-agent model in a Bewley equilibrium with a continuum of ex-post +heterogeneous consumers. The equilibrium gross interest rate $R = \beta^{-1}$ is supported by +constant average consumption, though the cross-section variance of consumption grows linearly with +age. A complete-markets version of the same model achieves full risk sharing and a time-invariant +consumption distribution at the cost of more complex financial arrangements (Arrow securities). + +**Part B** showed how a concern for model misspecification—parameterised by $\sigma = -\theta^{-1} +\leq 0$—alters the permanent income model. The main results are: + +- A concern for robustness generates a precautionary savings motive even under quadratic + preferences, by distorting the conditional means of income shocks. + +- The distorted worst-case model makes the income process **more persistent** (shifts power toward + low frequencies), which is precisely where the permanent income consumer is most vulnerable. + +- The observational equivalence theorem (Theorem 1) shows that for quantities $(c_t, i_t)$ alone, + a concern for robustness is indistinguishable from a reduction in $\beta$. + +- Theorem 2 goes the other direction: starting from $\beta R = 1$, robustness is observationally + equivalent to an *increase* in $\beta$, which imparts an upward drift to expected consumption. + +- Detection error probabilities provide a principled way to calibrate $\sigma$: choose $|\sigma|$ + small enough that the approximating and worst-case models remain difficult to distinguish + statistically. + +- The observationally equivalent $(\sigma, \hat\beta)$ pairs **do** have different implications for + asset prices—a point explored further by HST in the asset-pricing context. + +- Section 79.3.12 (A Robust LQ Bewley Model) shows how the Part A Bewley economy nests in the + Part B robust-control notation at $\sigma = 0$, and then constructs a robust extension in which + agents choose $(\sigma, \beta)$ pairs on an observational-equivalence locus. Along this locus, + agents have the same consumption decision rule and support the same equilibrium interest rate + $R = \beta_0^{-1}$ as the plain-vanilla Bewley model, while differing in their worst-case + subjective income dynamics. + +--- + +## Integrated Exercises + +```{exercise-start} +:label: lqcs_ex1 +``` + +**From Part A to Part B notation.** + +Specialise the Part B robust-control setup to the no-habit, no-capital LQ Bewley environment +($\lambda = \delta_h = 0$, $k_t = 0$), and let the endowment process be the two-factor model in +{eq}`eq:twofactor`. + +1. Write the household state as $x_t = [b_t, z_t']'$ and derive matrices $(A, B, C)$ for the law + of motion {eq}`eq:law0`. +2. Show that when $\sigma = 0$, the Bellman problem coincides with the Part A LQ permanent-income + problem. +3. Derive $\alpha^2$ as in Section 79.3.12 and verify + +$$ +\alpha^2 = \sigma_1^2 + (1-\beta)^2\sigma_2^2. +$$ + +Interpret economically why the permanent and transitory components enter with different weights. + +```{exercise-end} +``` + +```{solution-start} lqcs_ex1 +:class: dropdown +``` + +1. With $x_t = [b_t, z_t']'$ and budget law $b_{t+1} = R(b_t + c_t - y_t)$, + $y_t = \check G z_t$, $z_{t+1} = \check A z_t + \check C \epsilon_{t+1}$, the stacked law is + +$$ +\begin{pmatrix} b_{t+1} \\ z_{t+1} \end{pmatrix} += +\underbrace{\begin{pmatrix} R & -R\check G \\ 0 & \check A \end{pmatrix}}_{A} +\begin{pmatrix} b_t \\ z_t \end{pmatrix} ++ +\underbrace{\begin{pmatrix} -R \\ 0 \end{pmatrix}}_{B} c_t ++ +\underbrace{\begin{pmatrix} 0 \\ \check C \end{pmatrix}}_{C}\epsilon_{t+1}. +$$ + + (The sign of $B$ is negative because higher $c_t$ reduces bond accumulation $b_{t+1}$.) + +2. At $\sigma=0$, the robust Bellman problem collapses to the ordinary LQ objective with no + minimising distortion term. Hence the planner/consumer problem is exactly the Part A + permanent-income problem with quadratic utility and linear constraints. + +3. From Part A representation 2, + +$$ +\Delta c_{t+1} = h\,\epsilon_{t+1}, +\qquad h = (1-\beta)\check G (I-\beta\check A)^{-1}\check C. +$$ + + In HST notation, $\alpha^2 = h h'$. For the two-factor calibration, + $\check A=\mathrm{diag}(1,0)$ and $\check C=\mathrm{diag}(\sigma_1,\sigma_2)$, so + +$$ +\alpha^2 = \sigma_1^2 + (1-\beta)^2\sigma_2^2. +$$ + + Permanent shocks get unit weight because they shift lifetime resources one-for-one, while + transitory shocks are annuitised and therefore scaled by $(1-\beta)$ in consumption growth. + +```{solution-end} +``` + +```{exercise-start} +:label: lqcs_ex2 +``` + +**Continuum of robust but observationally equivalent Bewley consumers.** + +Fix a benchmark pair $(\beta_0, \sigma = 0)$ with $R = \beta_0^{-1}$ and define + +$$ +\beta(\sigma) = \beta_0 + \frac{\sigma\alpha^2\beta_0}{1-\beta_0}, +\qquad \sigma \in [-\bar\sigma, 0]. +$$ + +Suppose a unit interval of consumers is indexed by $i$ with type $\sigma^i \in [-\bar\sigma, 0]$ +and discount factor $\beta^i = \beta(\sigma^i)$. + +1. Use Theorem 1 to show that each type has the same consumption rule as the benchmark + $(\beta_0, 0)$ agent. +2. Prove that aggregate consumption and bond-market clearing imply the same equilibrium interest + rate $R = \beta_0^{-1}$ as in the plain-vanilla Bewley model. +3. Explain why agents can be observationally equivalent in quantities while still holding different + worst-case subjective models. + +```{exercise-end} +``` + +```{solution-start} lqcs_ex2 +:class: dropdown +``` + +1. Theorem 1 implies that if $(\sigma^i, \beta^i)$ lies on + +$$ +\beta^i = \beta_0 + \frac{\sigma^i\alpha^2\beta_0}{1-\beta_0}, +$$ + + then type $i$ chooses the same decision rule as the benchmark $(0,\beta_0)$ agent. Therefore + all types share the same consumption policy function $c_t = \mathcal C(b_t,z_t)$. + +2. Since all individual policy rules coincide with benchmark Bewley policies, aggregating over + consumers gives the same goods- and bond-market clearing conditions as Part A. Thus the same + equilibrium supports allocations, namely $R=\beta_0^{-1}$. + +3. Observational equivalence concerns quantities generated by optimal rules. Distinct + $(\sigma^i,\beta^i)$ can generate the same $\{c_t^i,b_t^i\}$ while implying different internal + worst-case beliefs (different distortion processes and likelihood ratios). So quantities alone + do not identify robustness preferences. + +```{solution-end} +``` + +```{exercise-start} +:label: lqcs_ex3 +``` + +**Quantitative comparison across equivalent types.** + +Using calibration values of your choice (for example, those used in the numerical illustration), do +the following: + +1. Plot the observational-equivalence locus $\beta(\sigma)$ over + $\sigma \in [-\bar\sigma, 0]$. +2. Select at least three types on the locus (e.g., low, medium, high robustness). +3. Simulate a common sequence of income shocks and verify numerically that their consumption paths + coincide. +4. For the same types, compute and compare one statistic that depends on the **subjective** model + (e.g., a worst-case persistence coefficient, distorted long-run variance, or detection error + probability). + +Summarise what is and is not identified by data on quantities alone. + +```{exercise-end} +``` + +```{solution-start} lqcs_ex3 +:class: dropdown +``` + +A compact implementation path is: + +1. Choose $(\beta_0,\sigma_1,\sigma_2)$ and compute + +$$ +\alpha^2 = \sigma_1^2 + (1-\beta_0)^2\sigma_2^2, +\qquad +\beta(\sigma)=\beta_0+\frac{\sigma\alpha^2\beta_0}{1-\beta_0}. +$$ + +2. Pick three values $\sigma^{L}<\sigma^{M}<\sigma^{H}\le 0$ and map to + $(\beta^L,\beta^M,\beta^H)$ on the locus. + +3. Simulate one common shock sequence $\{\epsilon_t\}$ and propagate consumption using the + benchmark policy rule. The three consumption paths coincide period-by-period (up to numerical + tolerance). + +4. Compare a subjective-model statistic (for example, a worst-case AR persistence or DEP). It + varies across $\sigma$ even though simulated quantities coincide. + +Conclusion: quantities identify the equilibrium decision rule but not the decomposition between +impatience ($\beta$) and robustness ($\sigma$) along the observational-equivalence locus. + +```{solution-end} +``` + +--- + +## Bibliography + +```{bibliography} +:filter: docname in docnames +``` From 975ab0c3c976adafc7d45be36fab95c55efee64f Mon Sep 17 00:00:00 2001 From: thomassargent30 Date: Fri, 19 Jun 2026 15:42:54 +0800 Subject: [PATCH 3/9] Tom's June 19 edits of 3 new lectures --- .claude/commands/quantecon-lecture.md | 104 +++ CLAUDE.md | 164 +++++ lectures/_static/quant-econ.bib | 19 + lectures/_toc.yml | 3 +- lectures/hst.pdf | Bin 0 -> 3671999 bytes lectures/kalman_filter_var.md | 137 +++- lectures/robust_permanent_income.bib | 19 + lectures/robust_permanent_income.md | 899 ++++++++++++++++++++++++++ 8 files changed, 1338 insertions(+), 7 deletions(-) create mode 100644 .claude/commands/quantecon-lecture.md create mode 100644 CLAUDE.md create mode 100644 lectures/hst.pdf create mode 100644 lectures/robust_permanent_income.bib create mode 100644 lectures/robust_permanent_income.md diff --git a/.claude/commands/quantecon-lecture.md b/.claude/commands/quantecon-lecture.md new file mode 100644 index 000000000..905fdbf0e --- /dev/null +++ b/.claude/commands/quantecon-lecture.md @@ -0,0 +1,104 @@ +--- +description: Convert a scientific paper (PDF or .tex) into a QuantEcon-style MyST markdown lecture with Python code and exercises. Use when writing a QuantEcon lecture, converting a paper to MyST markdown, or creating a lecture from a PDF or .tex file. +argument-hint: [path-to-paper.pdf-or-.tex] +--- + +You are helping Thomas Sargent (co-founder of QuantEcon) convert a scientific paper into a QuantEcon lecture. + +## Your Task + +Read the paper at `$1` (PDF or LaTeX file) and produce two outputs: + +1. A QuantEcon lecture written in **MyST-flavoured Markdown** (`.md`) +2. A **BibTeX `.bib` file** for any references not already in `quant-econ.bib` + +--- + +## Step 1 — Read the Paper + +Use available tools to read and understand the paper at `$1`. Extract: +- The core economic/mathematical ideas and results +- All notation, equations, and model definitions +- Any empirical or computational results worth replicating +- All references cited + +--- + +## Step 2 — Study the Example Lecture + +Read the file `likelihood_ratio_process.md` in the current directory. Use it as a **style and structure template** — including how sections, equations, figures, code cells, and exercises are formatted. + +Also examine `quant-econ.bib` (if present) to identify which references are already covered. + +--- + +## Step 3 — Write the MyST Lecture + +Produce a `.md` file named after the paper topic (snake_case). Follow these rules strictly: + +### MyST Formatting Rules + +- Use MyST-flavoured Markdown compatible with Jupyter Book +- Begin with a Jupyter Book YAML front-matter block: + ```yaml + --- + jupytext: + text_representation: + extension: .md + format_name: myst + kernelspec: + display_name: Python 3 + language: python + name: python3 + --- + ``` +- Section headings: `#`, `##`, `###` +- **Equations**: display math must have a **blank line before the opening `$$` and a blank line after the closing `$$`**. Never place `$$` immediately adjacent to text or other lines without a blank line separator. +- Inline math: `$...$` +- Label display equations with `{eq}` directives when they are referenced in the text +- Code cells: use MyST code-cell fences: + ```` + ```{code-cell} ipython3 + # Python code here + ``` + ```` +- Notes, warnings, tips: use admonition directives (`{note}`, `{warning}`, `{tip}`) +- Citations: use `{cite}` role, e.g. `{cite}SargentWallace1981` + +### Content Requirements + +1. **Introduction** — motivate the paper's contribution in accessible terms +2. **Model Setup** — define notation and the economic/mathematical environment +3. **Key Results** — present theorems, propositions, or main findings with equations +4. **Python Implementation** — self-contained, well-commented code cells that: + - Import only standard QuantEcon-compatible packages (`numpy`, `scipy`, `matplotlib`, `quantecon`, etc.) + - Reproduce or illustrate the paper's main results computationally + - Produce clearly labelled plots +5. **Exercises** — at least 3 exercises of increasing difficulty at the end, followed by **Solutions** in collapsed `{dropdown}` admonition blocks: + ``` + ````{dropdown} Solution + ```{code-cell} ipython3 + # solution code + ``` + ```` + ``` +6. **References** — a `{bibliography}` directive at the end + +--- + +## Step 4 — Write the BibTeX File + +- Check `quant-econ.bib` for existing entries +- Create a new `.bib` file (named `.bib`) containing only references **not** already in `quant-econ.bib` +- Use standard BibTeX format with clean, consistent cite keys (AuthorYYYY style) + +--- + +## Quality Checklist + +Before finishing, verify: +- [ ] Every display equation has a blank line before `$$` and after `$$` +- [ ] All code cells are self-contained and would run without errors +- [ ] Exercises have solution dropdowns +- [ ] All citations have BibTeX entries somewhere +- [ ] The lecture reads as a coherent pedagogical document, not just a paper summary diff --git a/CLAUDE.md b/CLAUDE.md new file mode 100644 index 000000000..9be9a0e4b --- /dev/null +++ b/CLAUDE.md @@ -0,0 +1,164 @@ +# Intermediate Quantitative Economics with Python - Lecture Materials + +**Always reference these instructions first and fallback to search or bash commands only when you encounter unexpected information that does not match the info here.** + +## Working Effectively + +### Bootstrap and Build the Repository +Execute these commands in order to set up a complete working environment: + +**NEVER CANCEL: Each step has specific timing expectations - wait for completion.** + +```bash +# 1. Set up conda environment (takes ~3 minutes) +conda env create -f environment.yml +# NEVER CANCEL: Wait 3-5 minutes for completion + +# 2. Activate environment (required for all subsequent commands) +source /usr/share/miniconda/etc/profile.d/conda.sh +conda activate quantecon + +# 3. Install PyTorch with CUDA support (takes ~3 minutes) +pip install --pre torch torchvision torchaudio --index-url https://download.pytorch.org/whl/nightly/cu128 +# NEVER CANCEL: Wait 3-5 minutes for completion + +# 4. Install Pyro (takes ~10 seconds) +pip install pyro-ppl + +# 5. Install JAX with CUDA support (takes ~30 seconds) +pip install --upgrade "jax[cuda12-local]==0.6.2" + +# 6. Install NumPyro (takes ~5 seconds) +pip install numpyro + +# 7. Test JAX installation (takes ~5 seconds) +python scripts/test-jax-install.py +``` + +### Build Commands + +**CRITICAL TIMING WARNING: NEVER CANCEL BUILD COMMANDS - They take 45+ minutes to complete.** + +```bash +# HTML build (primary build target) - takes 45-60 minutes +jb build lectures --path-output ./ -W --keep-going +# NEVER CANCEL: Set timeout to 90+ minutes. Executes 80+ notebooks sequentially. + +# PDF build via LaTeX - takes 30-45 minutes +jb build lectures --builder pdflatex --path-output ./ -n -W --keep-going +# NEVER CANCEL: Set timeout to 75+ minutes. + +# Jupyter notebook build - takes 30-45 minutes +jb build lectures --path-output ./ --builder=custom --custom-builder=jupyter -n -W --keep-going +# NEVER CANCEL: Set timeout to 75+ minutes. +``` + +### Environment and Dependency Details + +- **Python**: 3.12 with Anaconda 2024.10 +- **Primary Framework**: Jupyter Book 1.0.3 with MyST markdown +- **Scientific Computing**: JAX 0.6.2, PyTorch (nightly), NumPyro, SciPy +- **Content Format**: MyST markdown files in `/lectures/` directory +- **Build System**: Sphinx-based via Jupyter Book +- **Execution**: Notebooks are executed during build and cached + +### Validation Scenarios + +**Always run these validation steps after making changes:** + +1. **Environment Test**: `python scripts/test-jax-install.py` - verifies JAX/scientific stack +2. **Build Test**: Start HTML build and verify first few notebooks execute successfully +3. **Content Verification**: Check `_build/html/` directory contains expected output files +4. **Notebook Validation**: Verify generated notebooks in `_build/jupyter/` are executable + +### Known Limitations and Workarounds + +- **Network Access**: Intersphinx inventory warnings for external sites (intro.quantecon.org, python-advanced.quantecon.org) are expected in sandboxed environments - build continues normally +- **GPU Support**: JAX runs in CPU mode in most environments - this is expected and functional +- **Build Cache**: Uses `_build/.jupyter_cache` to avoid re-executing unchanged notebooks +- **Memory Usage**: Large notebooks may require substantial RAM during execution phase + +## Project Architecture + +### Key Directories +``` +lectures/ # MyST markdown lecture files (80+ files) +├── _config.yml # Jupyter Book configuration +├── _toc.yml # Table of contents structure +├── _static/ # Static assets (images, CSS, etc.) +└── *.md # Individual lecture files + +_build/ # Build outputs (created during build) +├── html/ # HTML website output +├── latex/ # PDF build intermediate files +├── jupyter/ # Generated notebook files +└── .jupyter_cache/ # Execution cache + +scripts/ # Utility scripts +└── test-jax-install.py # JAX installation validator + +.github/ # CI/CD workflows +└── workflows/ # GitHub Actions definitions +``` + +### Content Structure +- **80+ lecture files** covering intermediate quantitative economics +- **MyST markdown format** with embedded Python code blocks +- **Executable notebooks** - code is run during build process +- **Multiple output formats**: HTML website, PDF via LaTeX, downloadable notebooks + +### Build Targets +1. **HTML**: Main website at `_build/html/` - primary deliverable +2. **PDF**: Single PDF document via LaTeX at `_build/latex/` +3. **Notebooks**: Individual .ipynb files at `_build/jupyter/` + +## Development Workflow + +### Making Changes +1. **Always activate environment first**: `conda activate quantecon` +2. **Edit lecture files**: Modify `.md` files in `/lectures/` directory +3. **Test changes**: Run quick build test on subset if possible +4. **Full validation**: Complete HTML build to verify all notebooks execute +5. **Check outputs**: Verify `_build/html/` contains expected results + +### Common Tasks + +**Check lecture content:** +```bash +ls lectures/ | head -10 +# Shows: intro.md, various economics topic files (.md format) +``` + +**Monitor build progress:** +- Build shows progress as "reading sources... [X%] filename" +- Each notebook execution time varies: 5-120 seconds per file +- Total build time: 45-60 minutes for full HTML build + +**Environment verification:** +```bash +conda list | grep -E "(jax|torch|jupyter-book)" +# Should show: jax 0.6.2, torch 2.9.0.dev, jupyter-book 1.0.3 +``` + +## Troubleshooting + +### Common Issues +- **"cuInit(0) failed"**: Expected JAX warning in CPU-only environments - build continues normally +- **Intersphinx warnings**: Network inventory fetch failures are expected - build continues normally +- **Debugger warnings**: "frozen modules" warnings during notebook execution are normal +- **Long execution times**: Some notebooks (like ar1_bayes.md) take 60+ seconds - this is normal + +### Performance Notes +- **First build**: Takes longest due to fresh notebook execution +- **Subsequent builds**: Faster due to caching system +- **Cache location**: `_build/.jupyter_cache` stores execution results +- **Cache invalidation**: Changes to notebook content triggers re-execution + +## CI/CD Integration + +The repository uses GitHub Actions with: +- **Cache workflow**: Weekly rebuild of execution cache +- **CI workflow**: Pull request validation builds +- **Publish workflow**: Production deployment on tags + +**Local builds should match CI behavior** - use same commands and expect similar timing. diff --git a/lectures/_static/quant-econ.bib b/lectures/_static/quant-econ.bib index 862efab0e..a3904a36d 100644 --- a/lectures/_static/quant-econ.bib +++ b/lectures/_static/quant-econ.bib @@ -4074,3 +4074,22 @@ @book{Nummelin_1984 year = {1984}, doi = {10.1017/CBO9780511526237} } +@article{EpsteinWang1994, + author = {Epstein, Larry G. and Wang, Tan}, + title = {{Intertemporal Asset Pricing under Knightian Uncertainty}}, + journal = {Econometrica}, + volume = {62}, + number = {2}, + pages = {283--322}, + year = {1994} +} + +@article{Campbell1987, + author = {Campbell, John Y.}, + title = {{Does Saving Anticipate Declining Labor Income? An Alternative Test of the Permanent Income Hypothesis}}, + journal = {Econometrica}, + volume = {55}, + number = {6}, + pages = {1249--1273}, + year = {1987} +} diff --git a/lectures/_toc.yml b/lectures/_toc.yml index 81f97dc5c..0a4f545b2 100644 --- a/lectures/_toc.yml +++ b/lectures/_toc.yml @@ -110,7 +110,8 @@ parts: - file: cross_product_trick - file: perm_income - file: perm_income_cons - - file: theil_1 + - file: robust_permanent_income + - file: theil_1 - file: theil_2 - file: lq_consumption_smoothing - file: lq_inventories diff --git a/lectures/hst.pdf b/lectures/hst.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1d05dbdd8650dc88028892bb772470469ef7d094 GIT binary patch literal 3671999 zcmagF30zWH<2U|34H+61Zn;f~=2DhvYSv?mp|&7WW-g6pgjz{T*o(_)%S znq?DeX^NHGSjr>TR5LAxqc}23R?~!JSP2+-@A;o=&u@MIpU-L6ddsOgtlwrS2nn1~JkYDeMC#o?(mw`@SQ2 z!w%sjlMNe8_8m&ZdiwoeBe0HG_iy(dO8e5^#FMr!k;w>TA0@?l!Tx?Ly-BP8=^ba{ 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