quant-econ Solutions: LLN and CLT

Solutions for http://quant-econ.net/lln_clt.html



In [1]:

    
%matplotlib inline

Standard imports



In [2]:

    
import numpy as np
import matplotlib.pyplot as plt

Exercise 1

Here is one solution

You might have to modify or delete the lines starting with rc, depending on your configuration



In [4]:

    
"""
Illustrates the delta method, a consequence of the central limit theorem.
"""

from scipy.stats import uniform, norm
from matplotlib import rc

# == Specifying font, needs LaTeX integration == #
rc('font',**{'family':'serif','serif':['Palatino']})
rc('text', usetex=True)

# == Set parameters == #
n = 250
replications = 100000
distribution = uniform(loc=0, scale=(np.pi / 2))
mu, s = distribution.mean(), distribution.std()

g = np.sin
g_prime = np.cos

# == Generate obs of sqrt{n} (g(\bar X_n) - g(\mu)) == #
data = distribution.rvs((replications, n)) 
sample_means = data.mean(axis=1)  # Compute mean of each row
error_obs = np.sqrt(n) * (g(sample_means) - g(mu))

# == Plot == #
asymptotic_sd = g_prime(mu) * s
fig, ax = plt.subplots(figsize=(10, 6))
xmin = -3 * g_prime(mu) * s
xmax = -xmin
ax.set_xlim(xmin, xmax)
ax.hist(error_obs, bins=60, alpha=0.5, normed=True)
xgrid = np.linspace(xmin, xmax, 200)
lb = r"$N(0, g'(\mu)^2  \sigma^2)$"
ax.plot(xgrid, norm.pdf(xgrid, scale=asymptotic_sd), 'k-', lw=2, label=lb)
ax.legend()
plt.show()

What happens when you replace $[0, \pi / 2]$ with $[0, \pi]$?

In this case, the mean $\mu$ of this distribution is $\pi/2$, and since $g' = \cos$, we have $g'(\mu) = 0$

Hence the conditions of the delta theorem are not satisfied

Exercise 2

First we want to verify the claim that

$$ \sqrt{n} \mathbf Q ( \bar{\mathbf X}_n - \boldsymbol \mu ) \stackrel{d}{\to} N(\mathbf 0, \mathbf I) $$

This is straightforward given the facts presented in the exercise

Let

$$ \mathbf Y_n := \sqrt{n} ( \bar{\mathbf X}_n - \boldsymbol \mu ) \quad \text{and} \quad \mathbf Y \sim N(\mathbf 0, \Sigma) $$

By the multivariate CLT and the continuous mapping theorem, we have

$$ \mathbf Q \mathbf Y_n \stackrel{d}{\to} \mathbf Q \mathbf Y $$

Since linear combinations of normal random variables are normal, the vector $\mathbf Q \mathbf Y$ is also normal

Its mean is clearly $\mathbf 0$, and its variance covariance matrix is

$$ \mathrm{Var}[\mathbf Q \mathbf Y] = \mathbf Q \mathrm{Var}[\mathbf Y] \mathbf Q' = \mathbf Q \Sigma \mathbf Q' = \mathbf I $$

In conclusion, $\mathbf Q \mathbf Y_n \stackrel{d}{\to} \mathbf Q \mathbf Y \sim N(\mathbf 0, \mathbf I)$, which is what we aimed to show

Now we turn to the simulation exercise

Our solution is as follows



In [5]:

    
from scipy.stats import uniform, chi2
from scipy.linalg import inv, sqrtm

# == Set parameters == #
n = 250
replications = 50000
dw = uniform(loc=-1, scale=2)  # Uniform(-1, 1)
du = uniform(loc=-2, scale=4)  # Uniform(-2, 2)
sw, su = dw.std(), du.std()
vw, vu = sw**2, su**2
Sigma = ((vw, vw), (vw, vw + vu))
Sigma = np.array(Sigma)

# == Compute Sigma^{-1/2} == #
Q = inv(sqrtm(Sigma))  

# == Generate observations of the normalized sample mean == #
error_obs = np.empty((2, replications))
for i in range(replications):
    # == Generate one sequence of bivariate shocks == #
    X = np.empty((2, n))
    W = dw.rvs(n)
    U = du.rvs(n)
    # == Construct the n observations of the random vector == #
    X[0, :] = W
    X[1, :] = W + U
    # == Construct the i-th observation of Y_n == #
    error_obs[:, i] = np.sqrt(n) * X.mean(axis=1)

# == Premultiply by Q and then take the squared norm == #
temp = np.dot(Q, error_obs)
chisq_obs = np.sum(temp**2, axis=0)

# == Plot == #
fig, ax = plt.subplots(figsize=(10, 6))
xmax = 8
ax.set_xlim(0, xmax)
xgrid = np.linspace(0, xmax, 200)
lb = "Chi-squared with 2 degrees of freedom"
ax.plot(xgrid, chi2.pdf(xgrid, 2), 'k-', lw=2, label=lb)
ax.legend()
ax.hist(chisq_obs, bins=50, normed=True)
plt.show()



In [ ]: