{
"cells": [
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"E:\\Anaconda\\lib\\site-packages\\statsmodels\\compat\\pandas.py:65: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead.\n",
" from pandas import Int64Index as NumericIndex\n"
]
}
],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import matplotlib as mpl\n",
"import pandas as pd\n",
"import sympy as sy\n",
"import statsmodels.api as sm\n",
"\n",
"from sympy import init_printing\n",
"init_printing() \n",
"\n",
"plt.style.use('ggplot')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This chapter focuses on the features of sampling distribution of OLS estimator. We would like to know that if OLS can provide us with unbiased, consistent and efficient estimates. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Unbaisedness of OLS "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this chapter, we assume regression model has the form\n",
"\n",
"$$\n",
"\\boldsymbol{y} = \\boldsymbol{X\\beta} +\\boldsymbol{u}, \\quad \\boldsymbol{u}\\sim \\text{IID}(\\boldsymbol{0},\\sigma^2\\mathbf{I}) \n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The notation $\\text{IID}$ means **identically independent distributed**, $\\sigma^2\\mathbf{I}$ is the **covariance matrix**, being a diagonal matrix means regressors are independently distributed. We also assume all diagonal elements are the same, i.e. $\\sigma^2$, which means **homoscedasticity**. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If an estimator $\\theta$ is unbiased, it should always satisfy the condition\n",
"\n",
"$$\n",
"E(\\hat{\\theta}) -\\theta_0 = 0\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To show that the OLS estimator is unbiased, we denote $\\boldsymbol{\\beta}_0$ as the true parameter in the _data generating process_ (DGP), that is to say, we would like to see $E(\\boldsymbol{\\hat{\\beta}})=\\boldsymbol{\\beta}_0$.\n",
"\n",
"To show the conditions that makes OLS unbiased, we substitute DGP back in OLS formula\n",
"\n",
"\\begin{align} \n",
"\\hat{\\boldsymbol{\\beta}}&= (\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T\\boldsymbol{y}\\\\\n",
"&=(\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T(\\boldsymbol{X}\\boldsymbol{\\beta}_0+u)\\\\\n",
"& = \\boldsymbol{\\beta}_0 + (\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T\\boldsymbol{u}\n",
"\\end{align}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First, we assume $\\boldsymbol{X}$ to be _nonstochastic_, also with assumption $E(\\boldsymbol{u}) = 0$, we obtain"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{align}\n",
"E(\\boldsymbol{\\hat{\\beta}}) &= \\boldsymbol{\\beta}_0 + (\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T\\boldsymbol{u}\\\\\n",
" &= \\boldsymbol{\\beta}_0 + (\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^TE(\\boldsymbol{u})\\\\\n",
" &=\\boldsymbol{\\beta}_0\n",
"\\end{align}\n",
"However nonstochastic assumption is limited mostly in cross section data."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Second, we can assume $\\boldsymbol{X}$ **exogenous**, such that $E(\\boldsymbol{u}|\\boldsymbol{X}) = 0$, also proves $E(\\boldsymbol{\\hat{\\beta}})=\\boldsymbol{\\beta}_0$. Exogeneity simply means that the randomness of $\\boldsymbol{X}$ has nothing to do with $\\boldsymbol{u}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have even weaker version of exogeneity, $E(u_t|\\boldsymbol{X}_t) = 0$, which excludes the possibility that $u_t$ might depend on $X_{t-1}$ or $X_{t-2}$, etc. It is called **predeterminedness condition**, which is suitable for time series data."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"However, for time series data, OLS shall be used with cautions, it is widely known that OLS is not suitable for $\\text{ARMA}$ model, because $\\boldsymbol{y}$ has lagged dependent variables.\n",
"\n",
"For $\\text{VAR}$, OLS is a common practice, because it doesn't differentiate endogenous and exogenous variables."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"
Predetermined Variable vs Exogenous Variable
\n",
"In strictly speaking, predetermined variables has two categories: exogenous variables and predetermined endogenous variables.
\n",
" \n",
"The former one can either be current or lagged, as long as it doesn't evolve with model's dynamics. The latter one is common in autoregressive models.\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Consistency of OLS "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Consistency means the estimates tends to the quantity of true parameters as sample tends to infinity. Now let's show why OLS estimator is consistent. Start from \n",
"\n",
"$$\n",
"\\boldsymbol{\\hat{\\beta}} = \\boldsymbol{\\beta}_0 + (\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T\\boldsymbol{u} \\label{1}\\tag{1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We would to like to know if $\\text{plim}\\boldsymbol{\\hat{\\beta}} = \\boldsymbol{\\beta}_0$, which boils down to verify if\n",
"\n",
"$$\n",
"\\text{plim}(\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T\\boldsymbol{u} =0 \\label{2}\\tag{2}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"However $\\boldsymbol{X}^T\\boldsymbol{X}$ and $\\boldsymbol{X}^Tu$ do not have probability limit even if $n \\rightarrow \\infty$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Modify the probability limit we obtain\n",
"\n",
"$$\n",
"\\bigg(\\text{plim}\\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}\\bigg)^{-1}\\text{plim}\\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{u} =0 \n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We assume a weaker version of exogeneity $E(u_t|X_t) = 0$, so is $E(X_t^Tu_t|X_t)=0$, law of iterated expectation tells that $E(X_t^Tu_t)=0$. To summarize\n",
"\n",
"$$\n",
"\\text{plim}\\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{u}=\\text{plim}\\frac{1}{n}\\sum_{t=1}^nX_t^T u_t = 0 \n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"which proves the consistency of OLS."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Simulation of Consistency of OLS "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we can demonstrate a simple Monte Carlo simulation with true parameters $\\boldsymbol{\\beta}_0 = [2, 3, 4, 5]^T$, we increase sample size from $5$ to $100$ with a unit increment in each loop, and also with each sample size we redraw the disturbance term $10$ times, you can experiment with codes below, they might not be concise but considerably intuitive. Pay attention to the plots how they approach the true parameters as sample sizes increase."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"sampl_size = np.arange(5, 100)\n",
"re_draw = 10\n",
"\n",
"beta_array = np.array([2, 3, 4, 5])\n",
"beta_array = beta_array[np.newaxis, :].T\n",
"\n",
"beta_hat_mean = []\n",
"for i in sampl_size:\n",
" const = np.ones(i)\n",
" const = const[np.newaxis, :]\n",
"\n",
" X_inde = np.random.randn(3, i)\n",
" X = np.concatenate((const.T, X_inde.T), axis=1)\n",
" \n",
" beta_hat = []\n",
" for j in range(re_draw):\n",
" u = np.random.randn(i)\n",
" u = u[np.newaxis, :].T\n",
" y = X@beta_array + u\n",
" beta_hat.append(np.linalg.inv(X.T@X)@X.T@y)\n",
" beta_hat = np.array(beta_hat).T\n",
" beta_hat_mean.append(np.mean(beta_hat[0], axis = 1))\n",
"\n",
"beta0, beta1, beta2, beta3 = [], [], [], []\n",
"for i in beta_hat_mean:\n",
" beta0.append(i[0])\n",
" beta1.append(i[1])\n",
" beta2.append(i[2])\n",
" beta3.append(i[3])"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig, ax = plt.subplots(figsize =(18, 8), nrows = 2, ncols=2)\n",
"fig.suptitle('Simulation of OLS Consistency', size = 18)\n",
"ax[0,0].plot(beta0)\n",
"ax[0,0].axhline(2, color ='r')\n",
"ax[0,0].set_xlabel(r'$\\beta_1$ Sample Size')\n",
"\n",
"ax[0,1].plot(beta1)\n",
"ax[0,1].axhline(3, color ='r')\n",
"ax[0,1].set_xlabel(r'$\\beta_2$ Sample Size')\n",
"\n",
"ax[1,0].plot(beta2)\n",
"ax[1,0].axhline(4, color ='r')\n",
"ax[1,0].set_xlabel(r'$\\beta_3$ Sample Size')\n",
"\n",
"ax[1,1].plot(beta3)\n",
"ax[1,1].axhline(5, color ='r')\n",
"ax[1,1].set_xlabel(r'$\\beta_4$ Sample Size')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Efficiency of the OLS "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The **efficiency** of estimator represents the capacity of utilizing the information efficiently. For instance, OLS is more efficient than many other estimators, because with the same sample size, OLS yields the estimates with highest precision."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Positive Definite Matrices "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Before moving any further, we will refresh the definition of positive definite matrices. \n",
"\n",
"A **quadratic form** is a function with form $Q(\\mathbf{x})=\\mathbf{x}^TA\\mathbf{x}$, where $A$ is an $n\\times n$ symmetric matrix, which is called the **the matrix of the quadratic form**. For more details, check linear algebra material."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"Now some terms to need to be defined, a quadratic form $Q$ is: \n",
"\n",
"
positive definite if $Q(\\mathbf{x})>0$ for all $\\mathbf{x} \\neq \\mathbf{0}$
\n",
"
negative definite if $Q(\\mathbf{x})<0$ for all $\\mathbf{x} \\neq \\mathbf{0}$
\n",
"
positive semidefinite if $Q(\\mathbf{x})\\geq0$ for all $\\mathbf{x} \\neq \\mathbf{0}$
\n",
"
negative semidefinite if $Q(\\mathbf{x})\\leq0$ for all $\\mathbf{x} \\neq \\mathbf{0}$
\n",
"
indefinite if $Q(\\mathbf{x})$ assumes both positive and negative values.
\n",
"\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If $B$ is an $m\\times n$ matrix, both $B^TB$ and $BB^T$ yield symmetric matrices. Here is an example of $B_{3\\times 2}$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}1 & -1\\\\3 & 4\\\\0 & 2\\end{matrix}\\right]$"
],
"text/plain": [
"⎡1 -1⎤\n",
"⎢ ⎥\n",
"⎢3 4 ⎥\n",
"⎢ ⎥\n",
"⎣0 2 ⎦"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B = sy.Matrix([[1, -1], [3, 4], [0, 2]]); B"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$B^TB$ yields a $2\\times2$ matrix"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}10 & 11\\\\11 & 21\\end{matrix}\\right]$"
],
"text/plain": [
"⎡10 11⎤\n",
"⎢ ⎥\n",
"⎣11 21⎦"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B.T*B"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$BB^T$ yields a $3\\times3$ matrix"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"tags": []
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}2 & -1 & -2\\\\-1 & 25 & 8\\\\-2 & 8 & 4\\end{matrix}\\right]$"
],
"text/plain": [
"⎡2 -1 -2⎤\n",
"⎢ ⎥\n",
"⎢-1 25 8 ⎥\n",
"⎢ ⎥\n",
"⎣-2 8 4 ⎦"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B*B.T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Since $B^TB$ and $BB^T$ are both symmetric, $A$ can be replaced by either of them, technically the **Cholesky decomposition** can achieve $A = BB^T$ or $A = B^TB$.\n",
"\n",
"Actually any matrix of form $B^TB$ or $BB^T$ is positive semidefinite, to see this, observe that\n",
"$$\n",
"x^TB^TBx = (Bx)^T(Bx) = \\|Bx\\|^2 \\geq 0\\\\\n",
"xBB^Tx^T = xB (xB)^T = \\|xB\\|^2\\geq 0\n",
"$$\n",
"where $x\\neq0$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In practice, the second form is quite rare, so we focus the on the first form $x^TB^TBx$, since $x\\neq 0$, the only condition of $\\|Bx\\|^2 = 0$ is $B$ does not have **full rank**, in other words, if $B$ has full rank, then $B^TB$ is positive definite rather than positive semidefinite."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Covariance Matrix of the OLS Parameter Estimates "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The covariance matrix of error term is \n",
"\n",
"$$\n",
"\\text{Var}(\\boldsymbol{u}) = E(\\boldsymbol{u}\\boldsymbol{u}^T) = \\sigma^2_0\\mathbf{I}\n",
"$$\n",
"where $\\sigma^2_0\\mathbf{I}$ is a positive definite matrix\n",
"$$\n",
"\\boldsymbol{x}^{T} \\sigma_0^2\\mathbf{I} \\boldsymbol{x}=\\sigma_0^2\\sum_{i=1}^{k} x_{i}^{2}\n",
"$$\n",
"Now we can construct covariance matrix of $\\boldsymbol{\\hat{\\beta}}$, from discussions above we know $\\boldsymbol{\\hat{\\beta}}- \\boldsymbol{\\beta}_0 = (\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T\\boldsymbol{u}$, therefore"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{align} \n",
"\\text{Var}(\\boldsymbol{\\hat{\\beta}}) = E\\big[(\\boldsymbol{\\hat{{\\beta}}}-\\boldsymbol{\\beta_{0}})(\\boldsymbol{\\hat{{\\beta}}}-\\boldsymbol{{\\beta}}_{0})^{T}\\big]&=\n",
"(\\boldsymbol{X}^{\\top} \\boldsymbol{X})^{-1} \\boldsymbol{X}^{T}E( \\boldsymbol{u} \\boldsymbol{u}^{T}) \\boldsymbol{X}(\\boldsymbol{X}^{T} \\boldsymbol{X})^{-1}\\\\\n",
"&=(\\boldsymbol{X}^{\\top} \\boldsymbol{X})^{-1} \\boldsymbol{X}^{T}\\sigma_0^2\\mathbf{I} \\boldsymbol{X}(\\boldsymbol{X}^{T} \\boldsymbol{X})^{-1}\\\\\n",
"& = \\sigma_0^2 (\\boldsymbol{X}^{\\top} \\boldsymbol{X})^{-1}\n",
"\\end{align}\n",
"which is the standard result for the covariance matrix of $\\boldsymbol{\\hat{\\beta}}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It is illuminating if we rewrite the last equation, \n",
"\n",
"$$\n",
"\\text{Var}(\\boldsymbol{\\hat{\\beta}}) =\\frac{1}{n}\\sigma_0^2 \\left(\\frac{1}{n}\\boldsymbol{X}^{\\top} \\boldsymbol{X}\\right)^{-1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We need to make one extra assumption to interprete the equation properly, that is\n",
"\n",
"$$\n",
"\\text{plim}\\left(\\frac{1}{n}\\boldsymbol{X}^{T} \\boldsymbol{X}\\right)^{-1} = \\boldsymbol{S}_{\\boldsymbol{X}^{T} \\boldsymbol{X}}\n",
"$$\n",
"\n",
"where $\\boldsymbol{S}_{\\boldsymbol{X}^{T} \\boldsymbol{X}}$ is a nonstochastic matrix with full rank $k$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Thus we get a different form of variance of OLS estimator\n",
"\n",
"$$\n",
"\\text{Var}(\\boldsymbol{\\hat{\\beta}}) =\\frac{\\sigma_0^2}{n} \\boldsymbol{S}_{\\boldsymbol{X}^{T} \\boldsymbol{X}}\n",
"$$\n",
"\n",
"if we double the $\\sigma_0^2$ or double the sample size $n$, the variance of $\\boldsymbol{\\hat{\\beta}}$ will be doubled and halfed respectively."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And standard errors is\n",
"\n",
"$$\n",
"\\sigma_\\hat{\\beta}=\\frac{\\sigma_0}{\\sqrt{n}}\\sqrt{\\boldsymbol{S}_{\\boldsymbol{X}^{T} \\boldsymbol{X}}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For your reference, in basic statistics, we use formula for variance of sample means\n",
"\n",
"$$\n",
"\\sigma_{\\bar{x}}=\\frac{\\sigma_0}{\\sqrt{n}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Apart from $n$ and $\\sigma_0^2$, the third factor which influences $\\text{Var}(\\boldsymbol{\\hat{\\beta}})$ is $\\boldsymbol{X}$. To see this, we partition $\\boldsymbol{\\beta}$ into $[{\\beta}_1\\quad\\boldsymbol{\\beta}_2]$, where $\\boldsymbol{\\beta}_2$ denotes the $(k-1)$-vector. The linear regression model rewritten as\n",
"\n",
"$$\n",
"\\boldsymbol{y} = \\boldsymbol{x}_1\\beta_1+\\boldsymbol{X}_2\\boldsymbol{\\beta}_2+\\boldsymbol{u}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Use $\\text{FWL}$ regression\n",
"\n",
"$$\n",
"\\boldsymbol{M}_2\\boldsymbol{y}= \\boldsymbol{M}_2\\boldsymbol{x}_1\\beta_1 + \\boldsymbol{v}\n",
"$$\n",
"\n",
"where $\\boldsymbol{M}_2$ means projection off $\\boldsymbol{X}_2$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"According to OLS formula $\\boldsymbol{\\hat{\\beta}}= \\boldsymbol{(\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T\\boldsymbol{y}}$\n",
"\n",
"$$\n",
"\\hat{\\beta}_1 = (\\boldsymbol{x}_1^T\\boldsymbol{M}_2\\boldsymbol{x}_1)^{-1}\\boldsymbol{x}_1^T\\boldsymbol{M}_2\\boldsymbol{y}=\\frac{\\boldsymbol{x}_1^T\\boldsymbol{M}_2\\boldsymbol{y}}{\\boldsymbol{x}_1^T\\boldsymbol{M}_2\\boldsymbol{x}_1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And use variance formula $\\text{Var}(\\boldsymbol{\\hat{\\beta}}) = \\sigma_0^2 (\\boldsymbol{X}^{\\top} \\boldsymbol{X})^{-1}$, we get\n",
"\n",
"$$\n",
"\\text{Var}(\\boldsymbol{\\hat{\\beta}}_1) = \\sigma_0^2\\left(\\boldsymbol{x}_1^T\\boldsymbol{M}_2\\boldsymbol{x}_1\\right)^{-1}=\\frac{\\sigma_0^2}{\\boldsymbol{x}_1^T\\boldsymbol{M}_2\\boldsymbol{x}_1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Also note that denominator of $\\text{FWL}$ $\\text{Var}(\\boldsymbol{\\hat{\\beta}}_1)$ is actually $\\|\\boldsymbol{M}_2\\boldsymbol{x}_1\\|^2$. The geometric interpretation of precision (the inverse of variance) is provided in next section."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Multicollinearity and Visualization "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The formula of $\\text{FWL}$ $\\text{Var}(\\boldsymbol{\\hat{\\beta}})$ raises the problem of multicollinearity, which means that one or some of regressors have strong correlation with rest."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we visualize the subspace $\\text{span}({x}_1,\\ \\boldsymbol{X}_2)$ prepared for $\\text{FWL}$ regression above."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig, ax = plt.subplots(figsize = (10, 10))\n",
"\n",
"basis = np.array([[0, 0, 5, 7], [0, 0, 9, 0]])\n",
"X, Y, U, V = zip(*basis)\n",
"ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', scale=1) \n",
"\n",
"M2x1 = np.array([[0, 0, 0, 7]])\n",
"X, Y, U, V = zip(*M2x1)\n",
"ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', scale=1, color = 'red') \n",
"\n",
"x1_apos = np.array([[0, 0, 7, 5]])\n",
"X, Y, U, V = zip(*x1_apos)\n",
"ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', scale=1, color = 'gray') \n",
"\n",
"M2x1_apos = np.array([[0, 0, 0, 5]])\n",
"X, Y, U, V = zip(*M2x1_apos)\n",
"ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', scale=1, color = 'pink') \n",
"\n",
"X2c = np.array([[0, 0, 5, 0]])\n",
"X, Y, U, V = zip(*X2c)\n",
"ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', scale=1, color = 'aqua') \n",
"\n",
"point1 = [7, 5]\n",
"point2 = [0, 5]\n",
"line = np.array([point1, point2])\n",
"ax.plot(line[:,0], line[:,1], c = 'k', lw = 3.5,alpha =0.5, ls = '--')\n",
"\n",
"point1 = [5, 7]\n",
"point2 = [0, 7]\n",
"line = np.array([point1, point2])\n",
"ax.plot(line[:,0], line[:,1], c = 'k', lw = 3.5,alpha =0.5, ls = '--')\n",
"\n",
"point1 = [5, 0]\n",
"point2 = [5, 7]\n",
"line = np.array([point1, point2])\n",
"ax.plot(line[:,0], line[:,1], c = 'aqua', lw = 3.5,alpha =0.5, ls = '--')\n",
"\n",
"ax.set_xlim([-1, 10])\n",
"ax.set_ylim([-1, 10])\n",
"#plt.draw()\n",
"ax.grid()\n",
"\n",
"\n",
"ax.text(5, 7, r'$x_1$', size = 18)\n",
"ax.text(9, 0, r'$X_2$', size = 18)\n",
"ax.text(0, 7, r'$M_2 x_1$', size = 18)\n",
"ax.text(7, 5, r'$x_1^\\prime$', size = 18)\n",
"ax.text(0, 5, r'$M_2 x_1^\\prime$', size = 18)\n",
"ax.text(7, 7, r'$Var(\\hat{\\beta}_1)=\\frac{\\sigma_0^2}{\\|\\|M_2x_1\\|\\|^2}$', size = 18)\n",
"ax.text(.3, .1, r'$\\vartheta$', size = 22)\n",
"ax.text(5, 0.1, r'$X_2c$', size = 18)\n",
"ax.text(4.85, 3, r'$\\hat{u}$', size = 22)\n",
"\n",
"style = 'Simple, tail_width=0.5, head_width=8, head_length=16'\n",
"kw = dict(arrowstyle=style, color='orange', ls = '-.', alpha = .5, lw = 3)\n",
"arrow = mpl.patches.FancyArrowPatch((9,2),(2,9), connectionstyle='arc3,rad=.35', **kw)\n",
"ax.add_patch(arrow)\n",
"\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As we can see from the graph, if the length $\\|\\boldsymbol{M}_2{x_1}\\|$ is becoming longer means that the angle $\\vartheta$ is increasing on condition that $\\|x_1\\|$ remain the same.\n",
"\n",
"From linear algebra course, we know the formula\n",
"\n",
"$$\n",
"\\cos{\\vartheta} = \\frac{\\boldsymbol{u}\\cdot\\boldsymbol{v}}{\\|\\boldsymbol{u}\\|\\|\\boldsymbol{v}\\|}\n",
"$$\n",
"\n",
"which is exactly the correlation coefficient. The larger the angle, the lower the correlation, i.e. smaller $\\cos{\\vartheta}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this example, if $\\boldsymbol{x}_1$ is highly correlated with $\\boldsymbol{X}_2$, then $\\vartheta$ must be relatively small, therefore $\\text{Var}(\\hat{\\beta})$ is relatively high. This is what we call multicollinearity in practice, though it is not really collinear, rather, approximately pointing to the same direction.\n",
"\n",
"But remember this, multicollinearity always presents, but in a matter of degree, not yes or no."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can also interpret the graph by using a standard regression model\n",
"\n",
"$$\n",
"\\boldsymbol{x}_1 = \\boldsymbol{X}_2c + \\hat{u}\n",
"$$\n",
"\n",
"where $\\hat{u} = \\boldsymbol{M}_2\\boldsymbol{x}_1$. "
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"## Comparing Precision of Estimators "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If estimator $\\boldsymbol{\\hat{\\beta}}$ is more precise than $\\boldsymbol{\\tilde{\\beta}}$, the difference between them\n",
"$$\n",
"\\text{Var}(\\boldsymbol{\\tilde{\\beta}})- \\text{Var}(\\boldsymbol{\\hat{\\beta}})\n",
"$$\n",
"is a positive semidefinite matrix.\n",
"\n",
"Also any linear combination of $w^T\\boldsymbol{\\hat{\\beta}}$ is also more efficient than $w^T\\boldsymbol{\\tilde{\\beta}}$, because $\\text{Var}(w^T\\beta)=w^T\\text{Var}(\\beta)w$\n",
"$$\n",
"\\boldsymbol{w}^{\\top} \\operatorname{Var}(\\tilde{\\boldsymbol{\\beta}}) \\boldsymbol{w}-\\boldsymbol{w}^{\\top} \\operatorname{Var}(\\hat{\\boldsymbol{\\beta}}) \\boldsymbol{w}=\\boldsymbol{w}^{\\top}(\\operatorname{Var}(\\tilde{\\boldsymbol{\\beta}})-\\operatorname{Var}(\\hat{\\boldsymbol{\\beta}})) \\boldsymbol{w}\\geq0\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"## The Gauss-Markov Theorem "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The principal result on efficiency of OLS estimator is called the **Gauss-Markov Theorem**. The informal way of stating this theorem is to say that $\\hat{\\beta}$ is the **best linear unbiased estimator** (**BLUE**) ."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Formally, if these two conditions are satisfied\n",
"1. $E(\\boldsymbol{u}|\\boldsymbol{X})=\\boldsymbol{0}$\n",
"2. $E(\\boldsymbol{u}\\boldsymbol{u}^T|\\boldsymbol{X})=\\sigma^2\\mathbf{I}$ (homoskedasticity and uncorrelated $u_t$)\n",
"\n",
"OLS estimator is the BLUE."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The first condition holds because\n",
"$$\n",
"E\\left[\\begin{array}{c}\n",
"u_{1} \\mid \\boldsymbol{X} \\\\\n",
"u_{2} \\mid \\boldsymbol{X} \\\\\n",
"\\vdots \\\\\n",
"u_{n} \\mid \\boldsymbol{X}\n",
"\\end{array}\\right]=\\left[\\begin{array}{c}\n",
"E\\left(u_{1}\\right) \\\\\n",
"E\\left(u_{2}\\right) \\\\\n",
"\\vdots \\\\\n",
"E\\left(u_{n}\\right)\n",
"\\end{array}\\right]=\\left[\\begin{array}{c}\n",
"0 \\\\\n",
"0 \\\\\n",
"\\vdots \\\\\n",
"0\n",
"\\end{array}\\right]\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The second condition can be shown\n",
"\\begin{align}\n",
"E\\left(\\boldsymbol{u u}^T \\mid \\boldsymbol{X}\\right)&=E\\left[\\begin{array}{cccc}\n",
"u_{1}^{2} \\mid X & u_{1} u_{2} \\mid X & \\ldots & u_{1} u_{n} \\mid X \\\\\n",
"u_{2} u_{1} \\mid X & u_{2}^{2} \\mid X & \\ldots & u_{2} u_{n} \\mid X \\\\\n",
"\\vdots & \\vdots & \\vdots & \\vdots \\\\\n",
"u_{n} u_{1} \\mid X & u_{n} u_{2} \\mid X & \\ldots & u_{n}^{2} \\mid X\n",
"\\end{array}\\right]\\\\\n",
"&=\\left[\\begin{array}{cccc}\n",
"E\\left[u_{1}^{2} \\mid X\\right] & E\\left[u_{1} u_{2} \\mid X\\right] & \\ldots & E\\left[u_{1} u_{n} \\mid X\\right] \\\\\n",
"E\\left[u_{2} u_{1} \\mid X\\right] & E\\left[u_{2}^{2} \\mid X\\right] & \\ldots & E\\left[u_{2} u_{n} \\mid X\\right] \\\\\n",
"\\vdots & \\vdots & \\vdots & \\vdots \\\\\n",
"E\\left[u_{n} u_{1} \\mid X\\right] & E\\left[u_{n} u_{2} \\mid X\\right] & \\ldots & E\\left[u_{n}^{2} \\mid X\\right]\n",
"\\end{array}\\right]\\\\\n",
"&= \\left[\\begin{array}{cccc}\n",
"\\sigma^{2} & 0 & \\ldots & 0 \\\\\n",
"0 & \\sigma^{2} & \\ldots & 0 \\\\\n",
"\\vdots & \\vdots & \\vdots & \\vdots \\\\\n",
"0 & 0 & \\ldots & \\sigma^{2}\n",
"\\end{array}\\right]\n",
"\\end{align}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"in the linear regression model, then OLS estimator $\\boldsymbol{\\hat{\\beta}}$ is more efficient than any other $\\boldsymbol{\\tilde{\\beta}}$, i.e. $\\text{Var}(\\boldsymbol{\\hat{\\beta}}) - \\text{Var}(\\boldsymbol{\\tilde{\\beta}})$ is a positive semidefinite matrix. "
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"# Residuals and Disturbance Terms "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have discussed geometric and numerical properties of $\\hat{\\boldsymbol{u}}$ in last chapter, here we will discuss the properties of $\\hat{\\boldsymbol{u}}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"One important insight from geometric view of OLS is \n",
"$$\n",
"\\|\\boldsymbol{u}\\| \\geq \\|\\boldsymbol{\\hat{u}}\\| \n",
"$$\n",
"which is a direct deduction of $\\boldsymbol{\\hat{u}}=\\boldsymbol{M_Xu}$. "
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"## Expectation of Residuals "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Consider two matrices $\\boldsymbol{A}_{n\\times m}$ and $\\boldsymbol{B}_{m \\times k}$, then the $i^{th}$ row of $\\boldsymbol{AB}$ is the product of $i^{th}$ row of $\\boldsymbol{A}$ and entire $\\boldsymbol{B}$, if we want to represent the $t^{th}$ row of $\\boldsymbol{AB}$ we can use \n",
"\n",
"$$\n",
"\\boldsymbol{A}_t\\boldsymbol{B}\n",
"$$\n",
"\n",
"where $\\boldsymbol{A}_t$ is the $t^{th}$ row of $\\boldsymbol{A}$.\n",
"\n",
"In general we can represent $t^{th}$ row of a product of matrices by picking out the $t^{th}$ row of the leftmost matrix."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using this fact, we can denote the $t^{th}$ residual as\n",
"\n",
"$$\n",
"\\hat{u}_t = \\boldsymbol{M_X}\\boldsymbol{u} = (\\mathbf{I}-\\boldsymbol{P_X})\\boldsymbol{u} = u_t - \\boldsymbol{X}_t(\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T\\boldsymbol{u}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Although in Gauss-Markov Theorem, we assume the elements in error terms are independent, but this is not the case among residuals. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It is clear $\\boldsymbol{\\hat{u}}$ is a linear combination of $\\boldsymbol{u}$ i.e. $\\boldsymbol{M_Xu}$, so the $E(\\boldsymbol{\\hat{u}}) = \\boldsymbol{0}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Variance of Residuals "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Use the variance formula on $\\boldsymbol{\\hat{u}}$\n",
"\n",
"\\begin{align}\n",
"\\text{Var}(\\boldsymbol{\\hat{u}})&= \\text{Var}(\\boldsymbol{M_X}\\boldsymbol{{u}}) = E[(\\boldsymbol{M_X u}- E(\\boldsymbol{M_X u}))(\\boldsymbol{M_X u}- E(\\boldsymbol{M_X u})^T)]\\\\\n",
"&= E(\\boldsymbol{M_X}\\boldsymbol{u}\\boldsymbol{u}^T\\boldsymbol{M_X}) = \\boldsymbol{M_X}E(\\boldsymbol{u}\\boldsymbol{u}^T)\\boldsymbol{M_X} = \\boldsymbol{M_X}\\text{Var}(\\boldsymbol{u})\\boldsymbol{M_X}\\\\\n",
"& = \\boldsymbol{M_X}(\\sigma_0^2\\mathbf{I})\\boldsymbol{M_X}\\\\\n",
"& = \\sigma_0^2 \\boldsymbol{M_X}\n",
"\\end{align}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Obviously, $\\boldsymbol{M_X}$ is not an identity matrix, which means that variances of $\\hat{u}_t$ might not be homoscedasticity, also $E(u_tu_s)\\neq 0$, i.e. the residuals are correlated."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have denoted the $t^{th}$ diagonal element of $\\boldsymbol{P_X}$ as $h_t$, then the $t^{th}$ diagonal element of $\\boldsymbol{M_X}$ is $1-h_t$. In last chapter, we have shown that $0 \\leq h_t < 1$, hence $0 \\leq 1- h_t < 1$\n",
"\n",
"$$\n",
"\\text{Var}(\\hat{u}_t)<\\sigma_0^2\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The variance of any residual from OLS is always smaller than variance of disturbance term, this feature determined by the mechanism of OLS algorithm."
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"## Estimating the Variance of Error Terms "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We do not observe the disturbance terms $u_t$, neither does $\\sigma^2_0$. But we are able to estimate it by method of moments."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The simplest estimator of $\\sigma^2_0$ is \n",
"\n",
"$$\n",
"\\hat{\\sigma}^2 = \\frac{1}{n}\\sum_{t=1}^n\\hat{u}_t^2\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"However we have shown $\\text{Var}(\\hat{u}_t)<\\sigma_0^2$, so $\\hat{\\sigma}^2$ must be biased downward."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Before showing how to neutralize the bias, here is a side note about the trace of matrices."
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"### The Trace of Matrices "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The trace is the sum of the elements on the principal diagonal of a $n\\times n$ matrix, denoted $\\text{Tr}(\\boldsymbol{A})$. And it can be a product of any two matrices $\\boldsymbol{A}_{n\\times m} \\boldsymbol{B}_{m \\times n}$. But there is a convenient property called **cyclic permutation**.\n",
"\n",
"$$\n",
"\\text{Tr}(\\boldsymbol{ABC}) = \\text{Tr}(\\boldsymbol{CAB}) = \\text{Tr}(\\boldsymbol{BCA})\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Because $\\boldsymbol{P_X}$ is a square matrix, we can use the formula onto $\\boldsymbol{P_X}$ and we can show that\n",
"\n",
"$$\n",
"\\text{Tr}(\\boldsymbol{P_X})= \\text{Tr}(\\boldsymbol{X}(\\boldsymbol{X}^T\\boldsymbol{X})^{-1}\\boldsymbol{X}^T) = \\text{Tr}(\\boldsymbol{X}^T\\boldsymbol{X}(\\boldsymbol{X}^T\\boldsymbol{X})^{-1}) = \\text{Tr}(\\mathbf{I}_k) = k\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Unbiased Estimator of Variance of Error Term "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's process with $\\hat{\\sigma}^2$ by taking expectation\n",
"\n",
"$$\n",
"E(\\hat{\\sigma}^2) = \\frac{1}{n}\\sum_{t=1}^nE(\\hat{u}_t^2)\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Use the fact that $\\text{Var}(\\hat{u}_t) = E[(\\hat{u}_t- E(\\hat{u}_t))^2] = E[(\\hat{u}_t)^2] = (1-h_t)\\sigma_0^2$ and $\\text{Tr}(\\mathbf{I}_k)=k$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\frac{1}{n}\\sum_{t=1}^nE(\\hat{u}_t^2) = \\frac{1}{n}\\sum_{t=1}^n(1-h_t)\\sigma_0^2=\\frac{n-k}{n}\\sigma_0^2\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have the unbiased estimator of $\\sigma^2$\n",
"\n",
"$$\n",
"s^2 = \\frac{1}{n-k}\\sum_{t=1}^nE(\\hat{u}_t^2)= \\frac{1}{n-k}\\sum_{t=1}^n\\hat{u}_t^2=\\frac{\\hat{u}^T\\hat{u}}{n-k}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And now we can replace the $\\sigma_0^2$ in the $\\text{Var}(\\boldsymbol{\\hat{\\beta}}) = \\sigma_0^2(\\boldsymbol{X^TX})^{-1}$, the square root of principal diagonal holds the **standard error** of $\\boldsymbol{\\hat{\\beta}}$.\n",
"\n",
"$$\n",
"\\widehat{\\text{Var}}(\\boldsymbol{\\hat{\\beta}}) = s^2(\\boldsymbol{X^TX})^{-1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Numerical Example of Residuals "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Use the class or functions that I wrote in ```linregfunc.py```, we can carry out a quick simulation comparing the modulus of residuals and disturbance term. \n",
"\n",
"There are both class and functions in the ```linregfunc.py```, we will choose the whatever the most convenient.\n",
"\n",
"The moment when an object is instantiated, all the randomized data for OLS will be generated accordingly."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [],
"source": [
"import linregfunc as lr"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"ols_obj = lr.OLS_Simu(100, 3, [6, 7, 8])"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The moduduls of residual: 9.6886\n",
"The moduduls of disturbance term: 9.8263\n"
]
}
],
"source": [
"print('The moduduls of residual: {:.4f}'.format(np.linalg.norm(ols_obj.resid)))\n",
"print('The moduduls of disturbance term: {:.4f}'.format(np.linalg.norm(ols_obj.u)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It holds because theory guarantees\n",
"$$\n",
"\\text{Var}(\\hat{u}_t)<\\sigma_0^2\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"However simulate this in a manner of distribution?\n",
"\n",
"The function, which generates disturbance terms, follows a standard normal distribution, i.e. $\\sigma_0^2=1$. $\\hat{u}_t$ is a specific term of the residual vector, for instance to estimate the variance of $\\hat{u}_1$, we simulate the estimation process $10000$ times, then calculate the variance from those $10000$ $\\hat{u}_1$'s.\n",
"\n",
"Set sample size $n=100$, create projection matrix $\\boldsymbol{M}$."
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [],
"source": [
"proj_matM = ols_obj.proj_mat_M()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have create $\\boldsymbol{\\beta} = [6, 7, 8]^T$. Now add a placeholder array for residuals."
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [],
"source": [
"resid_array = np.zeros([100, 1])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Generate $10000$ vectors of residuals and stack them horizontally by specifying ```axis=1```. ```gen_u``` is a function for generating disturbance term which is identical to the one used in the class ```OLS_Simu```, for convenience of loops we use the function rather than method."
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [],
"source": [
"for i in range(10000):\n",
" u = lr.gen_u(100)\n",
" resid = proj_matM@u\n",
" resid_array = np.concatenate((resid_array, resid), axis = 1)\n",
"resid_array = resid_array[:, 1:]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Because this is a sequence of simulations, the variance will be a frequency distribution rather than a single number. Here is the histogram, obviously the mean is less than $1$ as we have expected."
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig, ax = plt.subplots(figsize=(10, 10))\n",
"ax.hist(np.var(resid_array, axis = 0), bins= 80)\n",
"ax.axvline(x = np.mean(np.var(resid_array, axis = 0)), color = 'blue')\n",
"ax.set_title('Variance Distribution of Sampled Residuals')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Standard Errors Simulation "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we simply show a numerical example of covariance matrix and the square root of its principal diagonal which holds the standard errors of $\\hat{\\boldsymbol{\\beta}}$."
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [],
"source": [
"ols_obj = lr.OLS_Simu(100, 4, [6, 7, 8, 9])\n",
"cov_betahat, cov_betahat_diag = ols_obj.cov_beta_hat()"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([0.10323776, 0.11105161, 0.11171944, 0.0987561 ])"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.sqrt(cov_betahat_diag)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Compare with ```statsmodels``` results. "
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[0.10323776 0.11105161 0.11171944 0.0987561 ]\n"
]
}
],
"source": [
"ols_obj_fit = sm.OLS(ols_obj.y, ols_obj.X).fit()\n",
"print(ols_obj_fit.bse)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Results are exactly the same."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.12"
},
"toc": {
"base_numbering": 1,
"nav_menu": {},
"number_sections": true,
"sideBar": true,
"skip_h1_title": false,
"title_cell": "Table of Contents",
"title_sidebar": "Contents",
"toc_cell": true,
"toc_position": {},
"toc_section_display": true,
"toc_window_display": false
},
"toc-autonumbering": false
},
"nbformat": 4,
"nbformat_minor": 4
}