diff --git a/lectures/_toc.yml b/lectures/_toc.yml
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@@ -15,7 +15,6 @@ parts:
chapters:
- file: prob_meaning
- file: bayes_nonconj
- - file: ar1_processes
- file: ar1_bayes
- file: ar1_turningpts
- file: likelihood_ratio_process
diff --git a/lectures/ar1_processes.md b/lectures/ar1_processes.md
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----
-jupytext:
- text_representation:
- extension: .md
- format_name: myst
-kernelspec:
- display_name: Python 3
- language: python
- name: python3
----
-
-(ar1)=
-```{raw} html
-
-```
-
-# AR1 Processes
-
-```{index} single: Autoregressive processes
-```
-
-```{contents} Contents
-:depth: 2
-```
-
-## Overview
-
-In this lecture we are going to study a very simple class of stochastic
-models called AR(1) processes.
-
-These simple models are used again and again in economic research to represent the dynamics of series such as
-
-* labor income
-* dividends
-* productivity, etc.
-
-AR(1) processes can take negative values but are easily converted into positive processes when necessary by a transformation such as exponentiation.
-
-We are going to study AR(1) processes partly because they are useful and
-partly because they help us understand important concepts.
-
-Let's start with some imports:
-
-```{code-cell} ipython
-import numpy as np
-%matplotlib inline
-import matplotlib.pyplot as plt
-plt.rcParams["figure.figsize"] = (11, 5) #set default figure size
-```
-
-## The AR(1) Model
-
-The **AR(1) model** (autoregressive model of order 1) takes the form
-
-```{math}
-:label: can_ar1
-
-X_{t+1} = a X_t + b + c W_{t+1}
-```
-
-where $a, b, c$ are scalar-valued parameters.
-
-This law of motion generates a time series $\{ X_t\}$ as soon as we
-specify an initial condition $X_0$.
-
-This is called the **state process** and the state space is $\mathbb R$.
-
-To make things even simpler, we will assume that
-
-* the process $\{ W_t \}$ is IID and standard normal,
-* the initial condition $X_0$ is drawn from the normal distribution $N(\mu_0, v_0)$ and
-* the initial condition $X_0$ is independent of $\{ W_t \}$.
-
-### Moving Average Representation
-
-Iterating backwards from time $t$, we obtain
-
-$$
-X_t = a X_{t-1} + b + c W_t
- = a^2 X_{t-2} + a b + a c W_{t-1} + b + c W_t
- = \cdots
-$$
-
-If we work all the way back to time zero, we get
-
-```{math}
-:label: ar1_ma
-
-X_t = a^t X_0 + b \sum_{j=0}^{t-1} a^j +
- c \sum_{j=0}^{t-1} a^j W_{t-j}
-```
-
-Equation {eq}`ar1_ma` shows that $X_t$ is a well defined random variable, the value of which depends on
-
-* the parameters,
-* the initial condition $X_0$ and
-* the shocks $W_1, \ldots W_t$ from time $t=1$ to the present.
-
-Throughout, the symbol $\psi_t$ will be used to refer to the
-density of this random variable $X_t$.
-
-### Distribution Dynamics
-
-One of the nice things about this model is that it's so easy to trace out the sequence of distributions $\{ \psi_t \}$ corresponding to the time
-series $\{ X_t\}$.
-
-To see this, we first note that $X_t$ is normally distributed for each $t$.
-
-This is immediate from {eq}`ar1_ma`, since linear combinations of independent
-normal random variables are normal.
-
-Given that $X_t$ is normally distributed, we will know the full distribution
-$\psi_t$ if we can pin down its first two moments.
-
-Let $\mu_t$ and $v_t$ denote the mean and variance
-of $X_t$ respectively.
-
-We can pin down these values from {eq}`ar1_ma` or we can use the following
-recursive expressions:
-
-```{math}
-:label: dyn_tm
-
-\mu_{t+1} = a \mu_t + b
-\quad \text{and} \quad
-v_{t+1} = a^2 v_t + c^2
-```
-
-These expressions are obtained from {eq}`can_ar1` by taking, respectively, the expectation and variance of both sides of the equality.
-
-In calculating the second expression, we are using the fact that $X_t$
-and $W_{t+1}$ are independent.
-
-(This follows from our assumptions and {eq}`ar1_ma`.)
-
-Given the dynamics in {eq}`ar1_ma` and initial conditions $\mu_0,
-v_0$, we obtain $\mu_t, v_t$ and hence
-
-$$
-\psi_t = N(\mu_t, v_t)
-$$
-
-The following code uses these facts to track the sequence of marginal
-distributions $\{ \psi_t \}$.
-
-The parameters are
-
-```{code-cell} python3
-a, b, c = 0.9, 0.1, 0.5
-
-mu, v = -3.0, 0.6 # initial conditions mu_0, v_0
-```
-
-Here's the sequence of distributions:
-
-```{code-cell} python3
-from scipy.stats import norm
-
-sim_length = 10
-grid = np.linspace(-5, 7, 120)
-
-fig, ax = plt.subplots()
-
-for t in range(sim_length):
- mu = a * mu + b
- v = a**2 * v + c**2
- ax.plot(grid, norm.pdf(grid, loc=mu, scale=np.sqrt(v)),
- label=f"$\psi_{t}$",
- alpha=0.7)
-
-ax.legend(bbox_to_anchor=[1.05,1],loc=2,borderaxespad=1)
-
-plt.show()
-```
-
-## Stationarity and Asymptotic Stability
-
-Notice that, in the figure above, the sequence $\{ \psi_t \}$ seems to be converging to a limiting distribution.
-
-This is even clearer if we project forward further into the future:
-
-```{code-cell} python3
-def plot_density_seq(ax, mu_0=-3.0, v_0=0.6, sim_length=60):
- mu, v = mu_0, v_0
- for t in range(sim_length):
- mu = a * mu + b
- v = a**2 * v + c**2
- ax.plot(grid,
- norm.pdf(grid, loc=mu, scale=np.sqrt(v)),
- alpha=0.5)
-
-fig, ax = plt.subplots()
-plot_density_seq(ax)
-plt.show()
-```
-
-Moreover, the limit does not depend on the initial condition.
-
-For example, this alternative density sequence also converges to the same limit.
-
-```{code-cell} python3
-fig, ax = plt.subplots()
-plot_density_seq(ax, mu_0=3.0)
-plt.show()
-```
-
-In fact it's easy to show that such convergence will occur, regardless of the initial condition, whenever $|a| < 1$.
-
-To see this, we just have to look at the dynamics of the first two moments, as
-given in {eq}`dyn_tm`.
-
-When $|a| < 1$, these sequences converge to the respective limits
-
-```{math}
-:label: mu_sig_star
-
-\mu^* := \frac{b}{1-a}
-\quad \text{and} \quad
-v^* = \frac{c^2}{1 - a^2}
-```
-
-(See our {doc}`lecture on one dimensional dynamics ` for background on deterministic convergence.)
-
-Hence
-
-```{math}
-:label: ar1_psi_star
-
-\psi_t \to \psi^* = N(\mu^*, v^*)
-\quad \text{as }
-t \to \infty
-```
-
-We can confirm this is valid for the sequence above using the following code.
-
-```{code-cell} python3
-fig, ax = plt.subplots()
-plot_density_seq(ax, mu_0=3.0)
-
-mu_star = b / (1 - a)
-std_star = np.sqrt(c**2 / (1 - a**2)) # square root of v_star
-psi_star = norm.pdf(grid, loc=mu_star, scale=std_star)
-ax.plot(grid, psi_star, 'k-', lw=2, label="$\psi^*$")
-ax.legend()
-
-plt.show()
-```
-
-As claimed, the sequence $\{ \psi_t \}$ converges to $\psi^*$.
-
-### Stationary Distributions
-
-A stationary distribution is a distribution that is a fixed
-point of the update rule for distributions.
-
-In other words, if $\psi_t$ is stationary, then $\psi_{t+j} =
-\psi_t$ for all $j$ in $\mathbb N$.
-
-A different way to put this, specialized to the current setting, is as follows: a
-density $\psi$ on $\mathbb R$ is **stationary** for the AR(1) process if
-
-$$
-X_t \sim \psi
-\quad \implies \quad
-a X_t + b + c W_{t+1} \sim \psi
-$$
-
-The distribution $\psi^*$ in {eq}`ar1_psi_star` has this property ---
-checking this is an exercise.
-
-(Of course, we are assuming that $|a| < 1$ so that $\psi^*$ is
-well defined.)
-
-In fact, it can be shown that no other distribution on $\mathbb R$ has this property.
-
-Thus, when $|a| < 1$, the AR(1) model has exactly one stationary density and that density is given by $\psi^*$.
-
-## Ergodicity
-
-The concept of ergodicity is used in different ways by different authors.
-
-One way to understand it in the present setting is that a version of the Law
-of Large Numbers is valid for $\{X_t\}$, even though it is not IID.
-
-In particular, averages over time series converge to expectations under the
-stationary distribution.
-
-Indeed, it can be proved that, whenever $|a| < 1$, we have
-
-```{math}
-:label: ar1_ergo
-
-\frac{1}{m} \sum_{t = 1}^m h(X_t) \to
-\int h(x) \psi^*(x) dx
- \quad \text{as } m \to \infty
-```
-
-whenever the integral on the right hand side is finite and well defined.
-
-Notes:
-
-* In {eq}`ar1_ergo`, convergence holds with probability one.
-* The textbook by {cite}`MeynTweedie2009` is a classic reference on ergodicity.
-
-For example, if we consider the identity function $h(x) = x$, we get
-
-$$
-\frac{1}{m} \sum_{t = 1}^m X_t \to
-\int x \psi^*(x) dx
- \quad \text{as } m \to \infty
-$$
-
-In other words, the time series sample mean converges to the mean of the
-stationary distribution.
-
-As will become clear over the next few lectures, ergodicity is a very
-important concept for statistics and simulation.
-
-## Exercises
-
-```{exercise}
-:label: ar1p_ex1
-
-Let $k$ be a natural number.
-
-The $k$-th central moment of a random variable is defined as
-
-$$
-M_k := \mathbb E [ (X - \mathbb E X )^k ]
-$$
-
-When that random variable is $N(\mu, \sigma^2)$, it is known that
-
-$$
-M_k =
-\begin{cases}
- 0 & \text{ if } k \text{ is odd} \\
- \sigma^k (k-1)!! & \text{ if } k \text{ is even}
-\end{cases}
-$$
-
-Here $n!!$ is the double factorial.
-
-According to {eq}`ar1_ergo`, we should have, for any $k \in \mathbb N$,
-
-$$
-\frac{1}{m} \sum_{t = 1}^m
- (X_t - \mu^* )^k
- \approx M_k
-$$
-
-when $m$ is large.
-
-Confirm this by simulation at a range of $k$ using the default parameters from the lecture.
-```
-
-
-```{solution-start} ar1p_ex1
-:class: dropdown
-```
-
-Here is one solution:
-
-```{code-cell} python3
-from numba import njit
-from scipy.special import factorial2
-
-@njit
-def sample_moments_ar1(k, m=100_000, mu_0=0.0, sigma_0=1.0, seed=1234):
- np.random.seed(seed)
- sample_sum = 0.0
- x = mu_0 + sigma_0 * np.random.randn()
- for t in range(m):
- sample_sum += (x - mu_star)**k
- x = a * x + b + c * np.random.randn()
- return sample_sum / m
-
-def true_moments_ar1(k):
- if k % 2 == 0:
- return std_star**k * factorial2(k - 1)
- else:
- return 0
-
-k_vals = np.arange(6) + 1
-sample_moments = np.empty_like(k_vals)
-true_moments = np.empty_like(k_vals)
-
-for k_idx, k in enumerate(k_vals):
- sample_moments[k_idx] = sample_moments_ar1(k)
- true_moments[k_idx] = true_moments_ar1(k)
-
-fig, ax = plt.subplots()
-ax.plot(k_vals, true_moments, label="true moments")
-ax.plot(k_vals, sample_moments, label="sample moments")
-ax.legend()
-
-plt.show()
-```
-
-```{solution-end}
-```
-
-
-```{exercise}
-:label: ar1p_ex2
-
-Write your own version of a one dimensional [kernel density
-estimator](https://en.wikipedia.org/wiki/Kernel_density_estimation),
-which estimates a density from a sample.
-
-Write it as a class that takes the data $X$ and bandwidth
-$h$ when initialized and provides a method $f$ such that
-
-$$
-f(x) = \frac{1}{hn} \sum_{i=1}^n
-K \left( \frac{x-X_i}{h} \right)
-$$
-
-For $K$ use the Gaussian kernel ($K$ is the standard normal
-density).
-
-Write the class so that the bandwidth defaults to Silverman’s rule (see
-the “rule of thumb” discussion on [this
-page](https://en.wikipedia.org/wiki/Kernel_density_estimation)). Test
-the class you have written by going through the steps
-
-1. simulate data $X_1, \ldots, X_n$ from distribution $\phi$
-1. plot the kernel density estimate over a suitable range
-1. plot the density of $\phi$ on the same figure
-
-for distributions $\phi$ of the following types
-
-- [beta
- distribution](https://en.wikipedia.org/wiki/Beta_distribution)
- with $\alpha = \beta = 2$
-- [beta
- distribution](https://en.wikipedia.org/wiki/Beta_distribution)
- with $\alpha = 2$ and $\beta = 5$
-- [beta
- distribution](https://en.wikipedia.org/wiki/Beta_distribution)
- with $\alpha = \beta = 0.5$
-
-Use $n=500$.
-
-Make a comment on your results. (Do you think this is a good estimator
-of these distributions?)
-```
-
-
-```{solution-start} ar1p_ex2
-:class: dropdown
-```
-
-Here is one solution:
-
-```{code-cell} ipython3
-K = norm.pdf
-
-class KDE:
-
- def __init__(self, x_data, h=None):
-
- if h is None:
- c = x_data.std()
- n = len(x_data)
- h = 1.06 * c * n**(-1/5)
- self.h = h
- self.x_data = x_data
-
- def f(self, x):
- if np.isscalar(x):
- return K((x - self.x_data) / self.h).mean() * (1/self.h)
- else:
- y = np.empty_like(x)
- for i, x_val in enumerate(x):
- y[i] = K((x_val - self.x_data) / self.h).mean() * (1/self.h)
- return y
-```
-
-```{code-cell} ipython3
-def plot_kde(ϕ, x_min=-0.2, x_max=1.2):
- x_data = ϕ.rvs(n)
- kde = KDE(x_data)
-
- x_grid = np.linspace(-0.2, 1.2, 100)
- fig, ax = plt.subplots()
- ax.plot(x_grid, kde.f(x_grid), label="estimate")
- ax.plot(x_grid, ϕ.pdf(x_grid), label="true density")
- ax.legend()
- plt.show()
-```
-
-```{code-cell} ipython3
-from scipy.stats import beta
-
-n = 500
-parameter_pairs= (2, 2), (2, 5), (0.5, 0.5)
-for α, β in parameter_pairs:
- plot_kde(beta(α, β))
-```
-
-We see that the kernel density estimator is effective when the underlying
-distribution is smooth but less so otherwise.
-
-```{solution-end}
-```
-
-
-```{exercise}
-:label: ar1p_ex3
-
-In the lecture we discussed the following fact: for the $AR(1)$ process
-
-$$
-X_{t+1} = a X_t + b + c W_{t+1}
-$$
-
-with $\{ W_t \}$ iid and standard normal,
-
-$$
-\psi_t = N(\mu, s^2) \implies \psi_{t+1}
-= N(a \mu + b, a^2 s^2 + c^2)
-$$
-
-Confirm this, at least approximately, by simulation. Let
-
-- $a = 0.9$
-- $b = 0.0$
-- $c = 0.1$
-- $\mu = -3$
-- $s = 0.2$
-
-First, plot $\psi_t$ and $\psi_{t+1}$ using the true
-distributions described above.
-
-Second, plot $\psi_{t+1}$ on the same figure (in a different
-color) as follows:
-
-1. Generate $n$ draws of $X_t$ from the $N(\mu, s^2)$
- distribution
-1. Update them all using the rule
- $X_{t+1} = a X_t + b + c W_{t+1}$
-1. Use the resulting sample of $X_{t+1}$ values to produce a
- density estimate via kernel density estimation.
-
-Try this for $n=2000$ and confirm that the
-simulation based estimate of $\psi_{t+1}$ does converge to the
-theoretical distribution.
-```
-
-```{solution-start} ar1p_ex3
-:class: dropdown
-```
-
-Here is our solution
-
-```{code-cell} ipython3
-a = 0.9
-b = 0.0
-c = 0.1
-μ = -3
-s = 0.2
-```
-
-```{code-cell} ipython3
-μ_next = a * μ + b
-s_next = np.sqrt(a**2 * s**2 + c**2)
-```
-
-```{code-cell} ipython3
-ψ = lambda x: K((x - μ) / s)
-ψ_next = lambda x: K((x - μ_next) / s_next)
-```
-
-```{code-cell} ipython3
-ψ = norm(μ, s)
-ψ_next = norm(μ_next, s_next)
-```
-
-```{code-cell} ipython3
-n = 2000
-x_draws = ψ.rvs(n)
-x_draws_next = a * x_draws + b + c * np.random.randn(n)
-kde = KDE(x_draws_next)
-
-x_grid = np.linspace(μ - 1, μ + 1, 100)
-fig, ax = plt.subplots()
-
-ax.plot(x_grid, ψ.pdf(x_grid), label="$\psi_t$")
-ax.plot(x_grid, ψ_next.pdf(x_grid), label="$\psi_{t+1}$")
-ax.plot(x_grid, kde.f(x_grid), label="estimate of $\psi_{t+1}$")
-
-ax.legend()
-plt.show()
-```
-
-The simulated distribution approximately coincides with the theoretical
-distribution, as predicted.
-
-```{solution-end}
-```
diff --git a/lectures/intro.mda b/lectures/intro.mda
deleted file mode 100644
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--- a/lectures/intro.mda
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----
-jupytext:
- text_representation:
- extension: .md
- format_name: myst
-kernelspec:
- display_name: Python 3
- language: python
- name: python3
----
-
-# Statistics for Computational Economics
-
-This website presents a set of statistical methods, designed and written by
-[Thomas J. Sargent](http://www.tomsargent.com/) and [John Stachurski](http://johnstachurski.net/).
-
-For an overview of the series, see [this page](https://quantecon.org/python-lectures/)
-
-```{tableofcontents}
-```
-
-```{admonition} Previous website
-While this new site will receive all future updates, you may still view the [old site here](http://rst-python-advanced.quantecon.org) for the next month.
-```
\ No newline at end of file
-
-