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🍇Python宏观经济学矩估计
矩估计就是模拟模型数据S S S x ~ = { x ~ 1 , x ~ 2 , … x ~ s , … x ~ S } \tilde{x}=\left\{\tilde{x}_1, \tilde{x}_2, \ldots \tilde{x}_s, \ldots \tilde{x}_S\right\} x ~ = { x ~ 1 , x ~ 2 , … x ~ s , … x ~ S } S S S
m ^ ( x ~ ∣ θ ) = 1 S ∑ s = 1 S m ( x ~ s ∣ θ ) \hat{m}(\tilde{x} \mid \theta)=\frac{1}{S} \sum_{s=1}^S m\left(\tilde{x}_s \mid \theta\right) m ^ ( x ~ ∣ θ ) = S 1 s = 1 ∑ S m ( x ~ s ∣ θ ) 一旦我们从 S 模拟中估计出模型矩 m ^ ( x ~ ∣ θ ) \hat{m}(\tilde{x} \mid \theta) m ^ ( x ~ ∣ θ ) θ ^ S M M \hat{\theta}_{S M M} θ ^ SMM θ \theta θ m ( x ) m(x) m ( x ) m ^ ( x ~ ∣ θ ) \hat{m}(\tilde{x} \mid \theta) m ^ ( x ~ ∣ θ )
θ ^ S M M = θ : min θ ∣ ∣ m ^ ( x ~ ∣ θ ) − m ( x ) ∣ ∣ \hat{\theta}_{S M M}=\theta: \quad \min _\theta|| \hat{m}(\tilde{x} \mid \theta)-m(x)|| θ ^ SMM = θ : θ min ∣∣ m ^ ( x ~ ∣ θ ) − m ( x ) ∣∣ 在此，矩估计量如下：
θ ^ S M M = θ : min θ e ( x ~ , x ∣ θ ) T W e ( x ~ , x ∣ θ ) \hat{\theta}_{S M M}=\theta: \quad \min _\theta e(\tilde{x}, x \mid \theta)^T W e(\tilde{x}, x \mid \theta) θ ^ SMM = θ : θ min e ( x ~ , x ∣ θ ) T W e ( x ~ , x ∣ θ ) 其中 W 是准则函数中的 R × R R \times R R × R e ( x ~ , x ∣ θ ) T W e ( x ~ , x ∣ θ ) e(\tilde{x}, x \mid \theta)^T W e(\tilde{x}, x \mid \theta) e ( x ~ , x ∣ θ ) T W e ( x ~ , x ∣ θ ) R × R R \times R R × R W W W W W W R × R R \times R R × R
矩估计需要强调的最后一项是，为模型的 S S S θ ^ S M M \hat{ \theta}_{S M M} θ ^ SMM θ \theta θ θ \theta θ
💦正态分布拟合到中等宏观经济学测试分数
数据位于文本文件 tpts.txt 中。回想一下，这些测试分数在 0 到 450 之间。下图显示了数据的直方图，以及三个截断的正常概率密度函数。黑线是截断的正态概率密度函数的 μ \mu μ σ \sigma σ μ \mu μ σ \sigma σ
Copy import requests
from IPython . display import Image
url = ( 'https://raw.githubusercontent.com/Notebooks/' +
'master/DMM/images/Mplots.png' )
image_file = requests . get (url, allow_redirects = True )
open ( 'Mplots.png' , 'wb' ). write (image_file.content)
Image ( "Mplots.png" ) 让我们尝试根据矩估计的截断正态分布来估计参数 μ \mu μ σ \sigma σ
mean ( scores i ) = 1 N ∑ i = 1 N scores i var ( scores i ) = 1 N − 1 ∑ i = 1 N ( scores i − mean ( scores i ) ) 2 \begin{gathered} \text { mean }\left(\operatorname{scores}_i\right)=\frac{1}{N} \sum_{i=1}^N \operatorname{scores}_i \\ \operatorname{var}\left(\operatorname{scores}_i\right)=\frac{1}{N-1} \sum_{i=1}^N\left(\operatorname{scores}_i-\operatorname{mean}\left(\operatorname{scores}_i\right)\right)^2 \end{gathered} mean ( scores i ) = N 1 i = 1 ∑ N scores i var ( scores i ) = N − 1 1 i = 1 ∑ N ( scores i − mean ( scores i ) ) 2 因此，矩估计的数据矩向量 m ( x ) m(x) m ( x )
m ( scores i ) ≡ [ mean ( scores i ) var ( scores i ) ] m\left(\operatorname{scores}_i\right) \equiv\left[\begin{array}{c} \operatorname{mean}\left(\operatorname{scores}_i\right) \\ \operatorname{var}\left(\operatorname{scores}_i\right) \end{array}\right] m ( scores i ) ≡ [ mean ( scores i ) var ( scores i ) ] 测试分数的一次模拟（某次模拟）会是什么样子？数据文件 tpts.txt 中有 161 个测试分数观测值。因此，一次模拟（某次模拟）将是从参数 μ 、 σ \mu、\sigma μ 、 σ
Copy import numpy as np
import numpy . random as rnd
import numpy . linalg as lin
import scipy . stats as sts
import scipy . integrate as intgr
import scipy . optimize as opt
import matplotlib
import matplotlib . pyplot as plt
from mpl_toolkits . mplot3d import Axes3D
from matplotlib import cm
cmap1 = matplotlib . cm . get_cmap ( 'summer' )
% matplotlib notebook
Copy url = ( 'https://raw.githubusercontent.com/Notebooks/' +
'master/DMM/data/tpts.txt' )
data_file = requests . get (url, allow_redirects = True )
open ( 'tpts.txt' , 'wb' ). write (data_file.content)
pts = np . loadtxt ( 'tpts.txt' ) 令随机变量 y ∼ N ( μ , σ ) y \sim N(\mu, \sigma) y ∼ N ( μ , σ ) μ \mu μ σ \sigma σ ϕ ( y ∣ μ , σ ) \phi(y \mid \mu, \sigma) ϕ ( y ∣ μ , σ ) Φ ( y ∣ μ , σ ) \Phi(y \mid \mu, \sigma) Φ ( y ∣ μ , σ ) x ∈ ( a , b ) x \in(a, b) x ∈ ( a , b ) a ≥ − ∞ a \geq-\infty a ≥ − ∞ a < b ≤ ∞ a<b \leq \infty a < b ≤ ∞
f ( x ∣ μ , σ , a , b ) = { 0 if x ≤ a ϕ ( x ∣ μ , σ ) Φ ( b ∣ μ , σ ) − Φ ( a ∣ μ , σ ) 0 if x ≥ b if a < x < b f(x \mid \mu, \sigma, a, b)=\left\{\begin{array}{l} 0 \text { if } \quad x \leq a \\ \frac{\phi(x \mid \mu, \sigma)}{\Phi(b \mid \mu, \sigma)-\Phi(a \mid \mu, \sigma)} \\ 0 \quad \text { if } \quad x \geq b \end{array} \quad \text { if } a<x<b\right. f ( x ∣ μ , σ , a , b ) = ⎩ ⎨ ⎧ 0 if x ≤ a Φ ( b ∣ μ , σ ) − Φ ( a ∣ μ , σ ) ϕ ( x ∣ μ , σ ) 0 if x ≥ b if a < x < b 截断法线的累计密度函数可以表示为：
F ( x ∣ μ , σ , a , b ) = { 0 if x ≤ a Φ ( x ∣ μ , σ ) − Φ ( a ∣ μ , σ ) Φ ( b ∣ μ , σ ) − Φ ( a ∣ μ , σ ) 0 if x ≥ b if a < x < b F(x \mid \mu, \sigma, a, b)=\left\{\begin{array}{l} 0 \quad \text { if } \quad x \leq a \\ \frac{\Phi(x \mid \mu, \sigma)-\Phi(a \mid \mu, \sigma)}{\Phi(b \mid \mu, \sigma)-\Phi(a \mid \mu, \sigma)} \\ 0 \quad \text { if } \quad x \geq b \end{array} \quad \text { if } a<x<b\right. F ( x ∣ μ , σ , a , b ) = ⎩ ⎨ ⎧ 0 if x ≤ a Φ ( b ∣ μ , σ ) − Φ ( a ∣ μ , σ ) Φ ( x ∣ μ , σ ) − Φ ( a ∣ μ , σ ) 0 if x ≥ b if a < x < b 请注意，z 只是 p 的变换，使得z ∼ U ( Φ − 1 ( a ∣ μ , σ ) , Φ − 1 ( b ∣ μ , σ ) ) z \sim U\left(\Phi^{-1}(a \mid \mu, \sigma), \Phi^{-1}(b \mid \mu, \sigma)\right) z ∼ U ( Φ − 1 ( a ∣ μ , σ ) , Φ − 1 ( b ∣ μ , σ ) )
定义函数，根据截断正态分布给出概率密度函数值
Copy def trunc_norm_pdf ( xvals , mu , sigma , cut_lb , cut_ub ):
if (cut_lb == None ) & (cut_ub == None ) :
cut_ub_cdf = 1.0
cut_lb_cdf = 0.0
elif (cut_lb != None ) & (cut_ub == None ) :
cut_ub_cdf = 1.0
cut_lb_cdf = sts . norm . cdf (cut_lb, loc = mu, scale = sigma)
elif (cut_lb == None ) & (cut_ub != None ) :
cut_ub_cdf = sts . norm . cdf (cut_ub, loc = mu, scale = sigma)
cut_lb_cdf = 0.0
elif (cut_lb != None ) & (cut_ub != None ) :
cut_ub_cdf = sts . norm . cdf (cut_ub, loc = mu, scale = sigma)
cut_lb_cdf = sts . norm . cdf (cut_lb, loc = mu, scale = sigma)
pdf_vals = (sts . norm . pdf (xvals, loc = mu, scale = sigma) /
(cut_ub_cdf - cut_lb_cdf))
return pdf_vals 定义从截断的结果中提取 N x S 测试分数值的函数
Copy def trunc_norm_draws ( unif_vals , mu , sigma , cut_lb , cut_ub ):
if (cut_lb == None ) & (cut_ub == None ) :
cut_ub_cdf = 1.0
cut_lb_cdf = 0.0
elif (cut_lb != None ) & (cut_ub == None ) :
cut_ub_cdf = 1.0
cut_lb_cdf = sts . norm . cdf (cut_lb, loc = mu, scale = sigma)
elif (cut_lb == None ) & (cut_ub != None ) :
cut_ub_cdf = sts . norm . cdf (cut_ub, loc = mu, scale = sigma)
cut_lb_cdf = 0.0
elif (cut_lb != None ) & (cut_ub != None ) :
cut_ub_cdf = sts . norm . cdf (cut_ub, loc = mu, scale = sigma)
cut_lb_cdf = sts . norm . cdf (cut_lb, loc = mu, scale = sigma)
unif2_vals = unif_vals * (cut_ub_cdf - cut_lb_cdf) + cut_lb_cdf
tnorm_draws = sts . norm . ppf (unif2_vals, loc = mu, scale = sigma)
return tnorm_draws 从平均值 μ = 300 , σ = 30 \mu=300, \sigma=30 μ = 300 , σ = 30
Copy mu_1 = 300.0
sig_1 = 30.0
cut_lb_1 = 0.0
cut_ub_1 = 450.0
unif_vals_1 = sts . uniform . rvs ( 0 , 1 , size = 161 )
draws_1 = trunc_norm_draws (unif_vals_1, mu_1, sig_1,
cut_lb_1, cut_ub_1)
print ( 'Mean score =' , draws_1. mean ())
print ( 'Variance of scores =' , draws_1. var ())
print ( 'Standard deviation of scores =' , draws_1. std ())
count_d , bins_d , ignored_d = \
plt . hist (pts, 30 , density = True , color = 'b' , edgecolor = 'black' ,
linewidth = 0.8 , label = 'Data' )
count_m , bins_m , ignored_m = \
plt . hist (draws_1, 30 , density = True , color = 'r' , edgecolor = 'black' ,
linewidth = 0.8 , label = 'Simulated data' )
xvals = np . linspace ( 0 , 450 , 500 )
plt . plot (xvals, trunc_norm_pdf (xvals, mu_1, sig_1, cut_lb_1, cut_ub_1),
linewidth = 2 , color = 'k' , label = 'PDF' )
plt . title ( 'Econ 381 scores: 2011-2012' , fontsize = 20 )
plt . xlabel ( 'Total points' )
plt . ylabel ( 'Percent of scores' )
plt . xlim ([ 0 , 550 ])
plt . legend (loc = 'upper left' ) 通过该模拟，我们可以根据模拟数据计算矩，就像根据实际数据计算矩一样。
