From 9b4397de33cd8f58352ac65e491a3f8c105ed224 Mon Sep 17 00:00:00 2001 From: Humphrey Yang Date: Wed, 24 Jan 2024 11:31:40 +1100 Subject: [PATCH] clean up _toc and redundant files --- lectures/_toc.yml | 1 - lectures/ar1_processes.md | 605 -------------------------------------- lectures/intro.mda | 24 -- 3 files changed, 630 deletions(-) delete mode 100644 lectures/ar1_processes.md delete mode 100644 lectures/intro.mda diff --git a/lectures/_toc.yml b/lectures/_toc.yml index c72bb3d..23e40f3 100644 --- a/lectures/_toc.yml +++ b/lectures/_toc.yml @@ -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 deleted file mode 100644 index 92076bc..0000000 --- a/lectures/ar1_processes.md +++ /dev/null @@ -1,605 +0,0 @@ ---- -jupytext: - text_representation: - extension: .md - format_name: myst -kernelspec: - display_name: Python 3 - language: python - name: python3 ---- - -(ar1)= -```{raw} html -
- - QuantEcon - -
-``` - -# 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 index 7d0ac4c..0000000 --- a/lectures/intro.mda +++ /dev/null @@ -1,24 +0,0 @@ ---- -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