{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import pandas as pd\n",
"import scipy as sp\n",
"import statsmodels.api as sm\n",
"import statsmodels.formula.api as smf\n",
"\n",
"import seaborn as sns\n",
"from scipy.optimize import minimize"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Simple Linear Regression"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is a simple linear regression model as in every econometrics textbooks\n",
"$$\n",
"Y_i=\\beta_1+\\beta_2X_i+u_i\n",
"$$\n",
"where $Y$ is **dependent variable**, $X$ is **independent variable** and $u$ is **disturbance term**. $\\beta_1$ and $\\beta_2$ are unknown parameters that we are aiming to estimate by feeding the data in the model. Without disturbance term, the model is simple a function of a straight line in $\\mathbb{R}^2$, such as\n",
"$$\n",
"Y = 2 + 3X\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the context of _machine learning_ (ML), the $X$ is usually called **feature variable** and $Y$ called **target variable**. And linear regression is the main tool in **supervised learning**, meaning that $Y$ is supervising $X$./\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"X = np.linspace(1, 10, 10)\n",
"Y = 2 + 3 * X\n",
"fig, ax = plt.subplots(figsize=(7, 7))\n",
"ax.plot(X, Y)\n",
"ax.scatter(X, Y, c=\"r\")\n",
"ax.grid()\n",
"ax.set_title(\"$Y=2+3x$\")\n",
"ax.set_xlim(0, 10)\n",
"ax.set_ylim(0, 40)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are five reasons justified that we need a disturbance term:\n",
"\n",
"1. omission of independent variables \n",
"2. aggregation of variables \n",
"3. model misspecification \n",
"4. function misspecification, eg. should be nonlinear rather than linear \n",
"5. measurement error\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The second one means that if we intend to aggregate the variable to a macro level, for instance every family has a consumption function, but aggregation on a national level causes discrepancies which contribute to the disturbance term.\n",
"\n",
"The third and forth one will be discussed in details in later chapter.\n",
"\n",
"The fifth one includes all types of error, man-made or natural. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Odinary Least Squares"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Odinary Least Squares** is the most common estimation technique used in ML or econometrics, it is popular due to its _simplicity_ and _transparency_. You'll be able to derive the whole estimation process by hand-calculation, all steps will have _closed-form expression_.\n",
"\n",
"We'll demonstrate OLS with our first plot. Every time you run this script, the result will be different than mine, because no random seeds are set."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"beta1, beta2 = 2, 3\n",
"\n",
"\n",
"def gen_linreg_data(beta1, beta2, samp_size, disturb_scale):\n",
"\n",
" X = np.linspace(1, 10, samp_size)\n",
" u = disturb_scale * np.random.randn(samp_size)\n",
" Y = beta1 + beta2 * X + u\n",
" Y_hat = beta1 + beta2 * X\n",
" return X, Y, Y_hat\n",
"\n",
"\n",
"def plot_lin_reg(X, Y, Y_hat):\n",
" fig, ax = plt.subplots(figsize=(7, 7))\n",
"\n",
" for i in range(len(Y)):\n",
" dot_fit_values = [X[i], X[i]]\n",
" dot_org_values = [Y[i], Y_hat[i]]\n",
" ax.plot(\n",
" dot_fit_values,\n",
" dot_org_values,\n",
" linestyle=\"--\",\n",
" color=\"red\",\n",
" label=\"residual\",\n",
" )\n",
"\n",
" ax.plot(X, Y_hat)\n",
" ax.scatter(X, Y_hat, c=\"k\")\n",
" ax.scatter(X, Y, c=\"r\")\n",
" ax.grid()\n",
" ax.set_title(\"$\\hat Y ={}+{}X$\".format(beta1, beta2))\n",
" plt.show()\n",
"\n",
"\n",
"if __name__ == \"__main__\":\n",
" X, Y, Y_hat = gen_linreg_data(\n",
" beta1=beta1, beta2=beta2, samp_size=10, disturb_scale=5\n",
" )\n",
" plot_lin_reg(X, Y, Y_hat)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have plotted a fitted line onto $10$ observations (red dots) which was generated by $Y_i = 2+3X_i+5u_i$, where $u_i \\sim N(0, 1)$. For easy demonstration, say we have a 'perfect' estimator that provides\n",
"$$\n",
"\\hat{\\beta}_1 = 2\\\\\n",
"\\hat{\\beta}_2 = 3\n",
"$$\n",
"where $\\hat{\\beta}_1 $ and $\\hat{\\beta}_2$ are estimates, in contrast $\\beta_1$ and $\\beta_2$ are model parameters.\n",
"\n",
"Therefore we can plot a fitted line (blue line) $\\hat{Y} = 2+3X$. The red dashed line is the difference of $Y_i$ and $\\hat{Y}_i$, we officially call it **residual**, denoted as $\\varepsilon_i$.\n",
"$$\n",
"\\varepsilon_i = Y_i - \\hat{Y}_i = Y_i - \\hat{\\beta}_1-\\hat{\\beta}_2X_i\n",
"$$\n",
"The OLS algorithm is aiming find the estimates of $\\beta_1$ and $\\beta_2$ such that\n",
"$$\n",
"\\text{min}\\text{ RSS}=\\sum_{i=1}^n \\varepsilon^2_i \n",
"$$\n",
"where $RSS$ is the **residual sum of squares** $(\\text{RSS})$. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In ML context, notation is slightly different, but the ideas are the same, a **cost function** is defined with $\\text{RSS}$\n",
"$$\n",
"J(\\hat{\\beta_1}, \\hat{\\beta}_2) = \\frac{1}{2m} \\sum_{i=1}^n \\varepsilon^2_i \n",
"$$\n",
"Then minimize the cost function\n",
"$$\n",
"\\text{min }J(\\hat{\\beta_1}, \\hat{\\beta}_2)\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To minimise the $RSS$, we simply take _partial derivatives_ w.r.t. $b_2$ and $b_1$ respectively. The results are the OLS estimators of simple linear regression, which are \n",
"\\begin{align}\n",
"\\hat{\\beta}_2 &=\\frac{\\sum_{i=1}^n(X_i-\\bar{X})(Y_i-\\bar{Y})}{\\sum^n_{i=1}(X_i-\\bar{X})^2}=\\frac{\\text{Cov}(X, Y)}{\\text{Var}(X)}\\\\\n",
"\\hat{\\beta}_1 &= \\bar{Y}-\\hat{\\beta}_2\\bar{X}\n",
"\\end{align}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"With these formulae in mind, let's perform a serious OLS estimation. Considering possible repetitive use of OLS in this tutorial, we will write a class for OLS. "
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"class S_OLS:\n",
" \"\"\"Create instances with S_OLS(X, Y), where X and Y are data array.\"\"\"\n",
"\n",
" def __init__(self, X, Y):\n",
" self.X = X\n",
" self.Y = Y\n",
"\n",
" def ols(self):\n",
" \"\"\"Estimate the data with OLS method, and return b1 and b2.\"\"\"\n",
" cov_mat = np.cov(self.X, self.Y)\n",
" self.b2 = cov_mat[0, 1] / cov_mat[0, 0]\n",
" self.b1 = np.mean(self.Y) - self.b2 * np.mean(self.X)\n",
" self.Y_hat = self.b1 + self.b2 * self.X\n",
" print(\"b1 estimate: {:.4f}\".format(self.b1))\n",
" print(\"b2 estimate: {:.4f}\".format(self.b2))\n",
" return self.Y_hat, self.b2, self.b1\n",
"\n",
" def simul_plot(self, beta1, beta2):\n",
" \"\"\"Plot scatter plot and fitted line with ols_plot(self, beta1, beta2),\n",
" beta1 and beta2 are parameters of data generation process.\"\"\"\n",
" fig, ax = plt.subplots(figsize=(7, 7))\n",
" for i in range(len(Y)):\n",
" dot_fit_values = [self.X[i], self.X[i]]\n",
" dot_org_values = [self.Y[i], self.Y_hat[i]]\n",
" ax.plot(dot_fit_values, dot_org_values, linestyle=\"--\", color=\"red\")\n",
" ax.scatter(self.X, self.Y_hat, c=\"k\")\n",
" ax.scatter(self.X, self.Y, c=\"r\")\n",
" ax.plot(self.X, self.Y_hat, label=\"$b_1$= {:.2f}, $b_2$={:.2f}\".format(b1, b2))\n",
" ax.grid()\n",
" ax.set_title(\"$\\hat Y ={:.2f}+{:.2f}X$\".format(b1, b2))\n",
" Y_hat_perfect = beta1 + beta2 * X\n",
" ax.plot(X, Y_hat_perfect, label=r\"$\\beta_1=2, \\beta_2=3$\")\n",
" ax.legend()\n",
" plt.show()\n",
"\n",
" def ols_plot(self, xlabel, ylabel):\n",
" self.xlabel = xlabel\n",
" self.ylabel = ylabel\n",
" fig, ax = plt.subplots(figsize=(7, 7))\n",
" ax.scatter(self.X, self.Y_hat, c=\"k\")\n",
" ax.scatter(self.X, self.Y, c=\"r\")\n",
" ax.plot(\n",
" self.X,\n",
" self.Y_hat,\n",
" label=\"$b_1$= {:.2f}, $b_2$={:.2f}\".format(self.b1, self.b2),\n",
" )\n",
" ax.grid()\n",
" ax.set_title(\"$\\hat Y ={:.2f}+{:.2f}X$\".format(self.b1, self.b2))\n",
" ax.set_xlabel(self.xlabel)\n",
" ax.set_ylabel(self.ylabel)\n",
"\n",
" def r_sq(self):\n",
" \"\"\"Calculate coefficient of determination and correlation of Y and Yhat\"\"\"\n",
" self.ESS = np.var(self.Y_hat)\n",
" self.RSS = np.var(self.Y - self.Y_hat)\n",
" self.R_sq = self.ESS / self.RSS\n",
" return self.ESS, self.RSS, self.R_sq"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"X, Y, Y_hat = gen_linreg_data(beta1=4, beta2=2, samp_size=15, disturb_scale=3)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1. , 0.88496253],\n",
" [0.88496253, 1. ]])"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.corrcoef(X, Y)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"b1 estimate: 4.8314\n",
"b2 estimate: 1.9802\n"
]
}
],
"source": [
"s_ols = S_OLS(X, Y)\n",
"Y_hat, betahat2, betahat1 = s_ols.ols()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A convenient function ```np.polyfit``` of curve fitting could verify our results."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([1.98022248, 4.83137211])"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.polyfit(X, Y, 1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The plot the fitted line $b_1+b_2X$, original line $\\beta_1+\\beta_2X$ and observations."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"s_ols.ols_plot(\"X\", \"Y\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"From the estimation result and graph above, we can notice $\\hat{\\beta}_1$ and $\\hat{\\beta}_2$ are close to true parameters $\\beta_1$ and $\\beta_2$ if scalar of disturbance term is small."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Interpretation of Estimated Coefficients"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will check on a real data set to understand basic principles of interpreting estimates. The data set collected average apartment price ¥/$m^2$ and average annual disposable income from $25$ Chinese cities in 2020. \n",
"\n",
"Load the data with Pandas."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"s_ols_house_income.ols_plot(\"Disposable Income\", \"House Price\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\\hat{\\beta}_2$ can be interpreted literally as the graph shows, as disposable income increases $1$ yuan (Chinese currency unit), the house price increases $1.1$ yuan. \n",
"\n",
"As for $\\hat{\\beta}_1$, it is tricky to interpret the face value that if the disposable income is zero, i.e. the house price is $-29181$ yuan if disposable income is $0$, it doesn't make any sense. The basic principle of interpreting constant term is to check if it has a plausible meaning when all independent/feature variables equals zero. If no sensible meaning, you don't need to interpret it, this is a well known defect of linear regression model."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Important Results of OLS"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Some features of OLS could provide us insights of mechanism of the algorithm. \n",
"\n",
"The _first_ one is \n",
"$$\n",
"\\bar{\\varepsilon}=0\n",
"$$\n",
"It is true because\n",
"$$\n",
"\\bar{\\varepsilon}=\\bar{Y}-\\hat{\\beta}_1-\\hat{\\beta}_2\\bar{X}=\\bar{Y}-(\\bar{Y}-\\hat{\\beta}_2\\bar{X})-\\hat{\\beta}_2\\bar{X}=0\n",
"$$\n",
"holds. We can demonstrate numerically with the variables that we have defined in house price example."
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.473825588822365e-12"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"epsilon = df[\"house_price\"] - Y_hat\n",
"np.mean(epsilon)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It is not theoretically zero due to some numerical round-off errors, but we treat it as zero."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The _second_ feature is \n",
"$$\n",
"\\bar{\\hat{Y}}=\\bar{Y}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Mean of Y hat: 30218.879999999994\n",
"Mean of Y: 30218.88\n"
]
}
],
"source": [
"print(\"Mean of Y hat: {}\".format(np.mean(Y_hat)))\n",
"print(\"Mean of Y: {}\".format(np.mean(df[\"house_price\"])))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The _third_ and _forth_ features are\n",
"$$\n",
"\\sum_i^n X_i\\varepsilon_i=0\\\\\n",
"\\sum_i^n \\hat{Y}_i\\varepsilon_i=0\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This can be shown by using a _dot product_ function. "
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"3.814697265625e-06\n",
"1.9073486328125e-06\n"
]
}
],
"source": [
"print(np.dot(df[\"salary\"], epsilon))\n",
"print(np.dot(Y_hat, epsilon))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Actually, lots of econometric theory can be conveniently derived by linear algebra. \n",
"\n",
"For instance, in linear algebra, covariance has a geometric interpretation\n",
"$$\n",
"\\text{Cov}(X, Y)=x\\cdot y= ||x||||y||\\cos{\\theta}\n",
"$$\n",
"where $x$ and $y$ are vectors in $\\mathbb{R}^n$. If dot product equals zero, geometrically these two vectors are perpendicular, denote as $x\\perp y$. Therefore the third and forth features are equivalent to\n",
"$$\n",
"\\text{Cov}(X, e)=0\\\\\n",
"\\text{Cov}(\\hat{Y}, e)=0\n",
"$$\n",
"i.e. $x\\perp e$ and $\\hat{y} \\perp e$. Traditionally, the vectors are denoted as lower case letters."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Variance of Decomposition"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Variance of decomposition** is based on analysis of variance (ANOVA), if you don't know what ANOVA is, check here. We know that any observation can be decomposed as a fitted value and a residual\n",
"$$\n",
"Y_i = \\hat{Y}_i+\\varepsilon_i\n",
"$$\n",
"Take variance on both sides\n",
"$$\n",
"\\text{Var}(Y)=\\text{Var}(\\hat{Y}+\\varepsilon)=\\operatorname{Var}(\\hat{Y})+\\operatorname{Var}(\\varepsilon)+ \\underbrace{2 \\operatorname{Cov}(\\hat{Y}, \\varepsilon)}_{=0}\n",
"$$\n",
"Or in the explicit form\n",
"$$\n",
"\\frac{1}{n} \\sum_{i=1}^{n}\\left(Y_{i}-\\bar{Y}\\right)^{2}=\\frac{1}{n} \\sum_{i=1}^{n}\\left(\\hat{Y}_{i}-\\overline{\\hat{Y}}\\right)^{2}+\\frac{1}{n} \\sum_{i=1}^{n}\\left(\\varepsilon_{i}-\\bar{\\varepsilon}\\right)^{2}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Use the OLS features, i.e. $\\bar{\\hat{Y}}=\\bar{Y}$ and $\\bar{\\varepsilon}=0$, the equation simplifies into\n",
"$$\n",
"\\underbrace{\\sum_{i=1}^{n}\\left(Y_{i}-\\bar{Y}\\right)^{2}}_{TSS}=\\underbrace{\\sum_{i=1}^{n}\\left(\\hat{Y}_{i}-\\bar{Y}\\right)^{2}}_{ESS}+\\underbrace{\\sum_{i=1}^{n} \\varepsilon_{i}^{2}}_{RSS}\n",
"$$\n",
"where $TSS$ means **total sum of squares**, $ESS$ means **explained sum of squares**."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Coefficient of Determination"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Though $ESS$ is called 'explained' part, it might be entirely wrong due to misspecification of model. That being said, we still need a quantitative indicator that tells us how much the model is able to 'explain' the behaviour of the dependent variables. \n",
"\n",
"The **coefficient of determination** is most intuitive indicator\n",
"$$\n",
"R^2 = \\frac{ESS}{TSS}=\\frac{\\sum_{i=1}^{n}\\left(\\hat{Y}_{i}-\\bar{Y}\\right)^{2}}{\\sum_{i=1}^{n}\\left(Y_{i}-\\bar{Y}\\right)^{2}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have written a ```r_sq()``` method in the ```S_OLS``` class."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0.5744603125471681\n"
]
}
],
"source": [
"ess, rss, r_sq = s_ols_house_income.r_sq()\n",
"print(r_sq)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It means the disposable income can explain $57\\%$ of house price variation. Furthermore, \n",
"$$\n",
"R^2 = \\frac{TSS - RSS}{TSS}=1-\\frac{RSS}{TSS}\n",
"$$\n",
"it is clear that minimise $RSS$ is equivalent to maximise $R^2$.\n",
"\n",
"Alternatively, the $R^2$ can be shown its relationship with correlation coefficient $r_{Y, \\hat{Y}}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\begin{aligned}\n",
"r_{Y, \\hat{Y}} &=\\frac{\\operatorname{Cov}(Y, \\hat{Y})}{\\sqrt{\\operatorname{Var}(Y) \\operatorname{Var}(\\hat{Y})}}=\\frac{\\operatorname{Cov}([\\hat{Y}+e], \\hat{Y})}{\\sqrt{\\operatorname{Var}(Y) \\operatorname{Var}(\\hat{Y})}}=\\frac{\\operatorname{Cov}(\\hat{Y}, \\hat{Y})+\\operatorname{Cov}(e, \\hat{Y})}{\\sqrt{\\operatorname{Var}(Y) \\operatorname{Var}(\\hat{Y})}}=\\frac{\\operatorname{Var}(\\hat{Y})}{\\sqrt{\\operatorname{Var}(Y) \\operatorname{Var}(\\hat{Y})}} \\\\\n",
"&=\\frac{\\sqrt{\\operatorname{Var}(\\hat{Y}) \\operatorname{Var}(\\hat{Y})}}{\\sqrt{\\operatorname{Var}(Y) \\operatorname{Var}(\\hat{Y})}}=\\sqrt{\\frac{\\operatorname{Var}(\\hat{Y})}{\\operatorname{Var}(Y)}}=\\sqrt{R^{2}}\n",
"\\end{aligned}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Gauss-Markov Conditions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In order to achieve the best estimation by OLS, the disturbance term ideally has to satisfy four conditions which are called **Gauss-Markov Conditions**. Provided that all G-M conditions satisfied, OLS is the preferred over all other estimators, because mathematically it is proved to be the **Best Linear Unbiased Estimator** (BLUE). This conclusion is called **Gauss-Markov Theorem**."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"1. $E(u_i|X_i)=0$\n",
"2. $E(u_i^2|X_i)= \\sigma^2$ for all $i$, **homoscedasticity**\n",
"3. $\\text{Cov}(u_i, u_j)=0, \\quad i\\neq j$, no **autocorrelation**.\n",
"4. $\\text{Cov}(X_i, u_i)=0$, assuming $X_i$ is non-stochastic."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In addition to G-M conditions, we also assume normality of disturbance term, i.e. $u_i\\sim N(0, \\sigma^2_u)$, which is guaranteed by _Central Limit Theorem_.\n",
"\n",
"In practice, almost impossible to have all conditions satisfied simultaneously or even one perfectly, but we must be aware of severity of violations, because any violation of G-M condition will compromise the quality of estimation results. And identifying which condition is violated could also lead us to corresponding remedies."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Random Components of Regression Coefficients"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"According to the OLS formula of $\\beta_2$\n",
"$$\n",
"\\hat{\\beta}_{2}=\\frac{\\operatorname{Cov}(X, Y)}{\\operatorname{Var}(X)}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plug in $Y=\\beta_1+\\beta_2X+u$:\n",
"$$\n",
"\\hat{\\beta}_{2}=\\frac{\\operatorname{Cov}(X, Y)}{\\operatorname{Var}(X)}=\\frac{\\operatorname{Cov}(X, \\beta_1+\\beta_2X+u)}{\\operatorname{Var}(X)}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The covariance operation rules come in handy\n",
"$$\n",
"\\operatorname{Cov}(X, \\beta_1+\\beta_2X+u)=\\operatorname{Cov}\\left(X, \\beta_{1}\\right)+\\operatorname{Cov}\\left(X, \\beta_{2} X\\right)+\\operatorname{Cov}(X, u)\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"where $\\operatorname{Cov}\\left(X, \\beta_{1}\\right)=0$, and $\\operatorname{Cov}\\left(X, \\beta_{2} X\\right)=\\beta_2 \\operatorname{Var}(X)$, therefore\n",
"$$\n",
"\\hat{\\beta}_{2}=\\frac{\\operatorname{Cov}(X, Y)}{\\operatorname{Var}(X)}=\\beta_{2}+\\frac{\\operatorname{Cov}(X, u)}{\\operatorname{Var}(X)}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If $u$ perfectly uncorrelated with $X$ as in G-M condition, the second term should be $0$. However that rarely happens, so the $\\hat{\\beta}_2$ and $\\beta_2$ will always have certain level of discrepancy. And note that we can't decompose $\\hat{\\beta}_2$ in practice, because we don't know the true value of $\\beta_2$.\n",
"\n",
"And also note that there are two ways to improve the accuracy of $\\hat{\\beta}_2$, either lower the correlation of $X$ and $u$ or increase the variance of $X$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Monte Carlo Sampling Distribution"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We want to perform a Monte Carlo simulation to show the sampling distribution of $\\hat{\\beta}_1$ and $\\hat{\\beta}_2$. First, we write a simple class for OLS Monte Carlo experiment. \n",
"\n",
"We can set $\\beta_1$, $\\beta_2$, $N$ and $a_u$ for initialisation. The model is\n",
"$$\n",
"Y_i=\\beta_1+\\beta_2X_i +a_uu_i, \\qquad i\\in(1, n),\\qquad u_i\\sim N(0,1)\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [],
"source": [
"class OLS_Monte_Carlo:\n",
" def __init__(self, beta1, beta2, N, u_scaler):\n",
" \"\"\"Input beta1, beta2, sample size, scaler of disturbance\"\"\"\n",
" self.beta1 = beta1\n",
" self.beta2 = beta2\n",
" self.u_scaler = u_scaler\n",
" self.N = N\n",
" self.X = self.N * np.random.rand(\n",
" self.N\n",
" ) # generate N random X's in the range of (0, N)\n",
"\n",
" def ols(self):\n",
" \"\"\"Estimate the data with OLS method, and return b1 and b2.\"\"\"\n",
" self.u = self.u_scaler * np.random.randn(self.N)\n",
" self.Y = self.beta1 + self.beta2 * self.X + self.u\n",
" cov_mat = np.cov(self.X, self.Y)\n",
" self.b2 = cov_mat[0, 1] / cov_mat[0, 0]\n",
" self.b1 = np.mean(self.Y) - self.b2 * np.mean(self.X)\n",
" return self.b2, self.b1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Instantiate the OLS Monte Carlo object with $\\beta_1=2$, $\\beta_2=3$, $n=10$ and $a_u=1$, then run $10000$ times of simulations, each time generated a new $u$. All estimated $\\hat{\\beta}_1$ and $\\hat{\\beta}_2$ are collected in their arrays."
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [],
"source": [
"ols_mt = OLS_Monte_Carlo(beta1=2, beta2=3, N=10, u_scaler=1)\n",
"b2_array, b1_array = [], []\n",
"for i in range(10000):\n",
" b2, b1 = ols_mt.ols()\n",
" b2_array.append(b2)\n",
" b1_array.append(b1)\n",
"b2_mean = np.mean(b2_array)\n",
"b1_mean = np.mean(b1_array)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plot the histogram and the mean of estimates. Not difficult to notice that the mean of the $\\hat{\\beta}_1$ and $\\hat{\\beta}_2$ are very close to 'true values'."
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
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",
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"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"fig, ax = plt.subplots(nrows=2, ncols=1, figsize=(9, 9))\n",
"ax[0].hist(b1_array, bins=100)\n",
"ax[0].axvline(b1_mean, color=\"salmon\", label=r\"$\\bar{\\beta}_1$=%f\" % b1_mean)\n",
"ax[0].legend()\n",
"\n",
"ax[1].hist(b2_array, bins=100)\n",
"ax[1].axvline(b2_mean, color=\"salmon\", label=r\"$\\bar{\\beta}_2$=%f\" % b2_mean)\n",
"ax[1].legend()\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we try again with a larger disturbance scaler $a_u = 2$ and keep rest parameters unchanged."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"ols_mt = OLS_Monte_Carlo(beta1=2, beta2=3, N=10, u_scaler=2)\n",
"b2_array, b1_array = [], []\n",
"for i in range(10000):\n",
" b2, b1 = ols_mt.ols()\n",
" b2_array.append(b2)\n",
" b1_array.append(b1)\n",
"b2_mean = np.mean(b2_array)\n",
"b1_mean = np.mean(b1_array)\n",
"\n",
"fig, ax = plt.subplots(nrows=2, ncols=1, figsize=(9, 9))\n",
"ax[0].hist(b1_array, bins=50)\n",
"ax[0].axvline(b1_mean, color=\"salmon\", label=r\"$\\bar{b}_1$=%f\" % b1_mean)\n",
"ax[0].legend()\n",
"\n",
"ax[1].hist(b2_array, bins=50)\n",
"ax[1].axvline(b2_mean, color=\"salmon\", label=r\"$\\bar{b}_2$=%f\" % b2_mean)\n",
"ax[1].legend()\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Though the histogram has been scaled as seemingly identical as above, but pay attention to the $x$-axis and vertical line. Apparently the variance of $u$ affects the accuracy of estimates, i.e. the variance of sampling distribution of $b_1$ and $b_2$.\n",
"\n",
"It's straightforward to see why this happens from the formula\n",
"$$\n",
"\\hat{\\beta}_{2}=\\beta_{2}+\\frac{\\operatorname{Cov}(X, a_uu)}{\\operatorname{Var}(X)}=\\beta_{2}+a_u\\frac{\\operatorname{Cov}(X, u)}{\\operatorname{Var}(X)}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We know from statistics course the increase sample is always doing good for quality of estimates, thus we dial it up to $N=100$, rest are unchanged."
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
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",
"text/plain": [
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"ols_mt = OLS_Monte_Carlo(beta1=2, beta2=3, N=100, u_scaler=2)\n",
"b2_array, b1_array = [], []\n",
"for i in range(10000):\n",
" b2, b1 = ols_mt.ols()\n",
" b2_array.append(b2)\n",
" b1_array.append(b1)\n",
"b2_mean = np.mean(b2)\n",
"b1_mean = np.mean(b1)\n",
"\n",
"fig, ax = plt.subplots(nrows=2, ncols=1, figsize=(9, 9))\n",
"ax[0].hist(b1_array, bins=50)\n",
"ax[0].axvline(b1_mean, color=\"salmon\", label=r\"$\\bar{b}_1$=%f\" % b1_mean)\n",
"ax[0].legend()\n",
"\n",
"ax[1].hist(b2_array, bins=50)\n",
"ax[1].axvline(b2_mean, color=\"salmon\", label=r\"$\\bar{b}_2$=%f\" % b2_mean)\n",
"ax[1].legend()\n",
"plt.show()"
]
},
{
"cell_type": "raw",
"metadata": {},
"source": [
"Notice how we have improved the accuracy of estimates, pay attention to $x$-axis that the sample distributions are tremendously concentrated than previous runs."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Statistical Features of Estimates"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"After a visial exam of sample distributions, now we can formally discuss the statistical features of estimator."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Unbiased of Estimator"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If an estimator is biased, we rarely perform any estimation with it, unless you have no choices which is also rare. But how to prove unbiasness of an estimator? \n",
"\n",
"The _rule of thumb_ is to take expectation on both sides of estimator. Here is the OLS example.\n",
"$$\n",
"E\\left(b_{2}\\right)=E\\left[\\beta_{2}+\\frac{\\operatorname{Cov}(X, u)}{\\operatorname{Var}(X)}\\right]=\\beta_{2}+E\\left[\\frac{\\operatorname{Cov}(X, u)}{\\operatorname{Var}(X)}\\right]=\\beta_2+\\frac{E[\\operatorname{Cov}(X, u)]}{E[\\operatorname{Var}(X)]}\n",
"$$\n",
"To show $E[\\operatorname{Cov}(X, u)]=0$, rewrite covariance in explicit form\n",
"$$\n",
"E[\\operatorname{Cov}(X, u)]=\\frac{1}{n} \\sum_{i=1}^{n}\\left(X_{i}-\\bar{X}\\right) E\\left(u_{i}-\\bar{u}\\right)=0\n",
"$$\n",
"Therefore\n",
"$$\n",
"E\\left(\\hat{\\beta}_{2}\\right)=\\beta_{2}\n",
"$$\n",
"Again take expectation on $b_1$, immediately we get\n",
"$$\n",
"E(\\hat{\\beta}_1) = E(\\bar{Y})-E(\\hat{\\beta}_2)E(\\bar{X})= E(\\bar{Y})-\\beta_2\\bar{X}=\\beta_1\n",
"$$\n",
"where we used $E(\\bar{Y})=\\beta_{1}+\\beta_{2} \\bar{X}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Precision of Estimator"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If we know the variance of disturbance term, the population variance of $b_1$ and $b_2$ can be derived\n",
"$$\n",
"\\sigma_{\\hat{\\beta}_1}^{2}=\\frac{\\sigma_{u}^{2}}{n}\\left[1+\\frac{\\bar{X}^{2}}{\\operatorname{Var}(X)}\\right]\\\\\\sigma_{\\hat{\\beta}_2}^{2}=\\frac{\\sigma_{u}^{2}}{n \\operatorname{Var}(X)}\n",
"$$\n",
"Though we will never really know $\\sigma_{u}$, the formulae provide the intuition how the variance of coefficients are determined. \n",
"\n",
"In the visual example of last section, we have seen that the larger $\\sigma_{u}$ causes larger $\\sigma_{\\hat{\\beta}_1}^{2}$ and $\\sigma_{\\hat{\\beta}_2}^{2}$, here the formulae also present the relation. \n",
"\n",
"And there are two ways to contract the variance of $b_1$ and $b_2$, one is increasing sample size $n$, the other is increasing $\\operatorname{Var}(X)$.\n",
"\n",
"In practice, we substitute $\\sigma_{u}^{2}$ by its unbiased estimator\n",
"$$\n",
"s_{u}^{2}=\\frac{n}{n-2} \\operatorname{Var}(e)\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"where $\\operatorname{Var}(\\epsilon)$ is the sample variance of residuals. The term $\\frac{n}{n-2}$ is for upscaling variance, because generally residuals are smaller than disturbance term, this is determined by its mathematical nature. You will get a very clear linear algebraic view in advanced econometric theory. \n",
"\n",
"After plug-in we get the _standard error_ of $\\hat{\\beta}_1$ and $\\hat{\\beta}_2$\n",
"$$\n",
"\\text { s.e. }\\left(\\hat{\\beta}_1\\right)=\\sqrt{\\frac{s_{u}^{2}}{n}\\left[1+\\frac{\\bar{X}^{2}}{\\operatorname{Var}(X)}\\right]}\\\\\n",
"\\text { s.e. }\\left(\\hat{\\beta}_2\\right)=\\sqrt{\\frac{s_{u}^{2}}{n \\operatorname{Var}(X)}}\n",
"$$\n",
"The standard error is used when we are referring to the _standard deviation of sampling distribution of estimates_, specifically in econometric term, regression coefficients."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Hypothesis Testing and $t$-Test"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Essentially, hypothesis testing in econometrics is the same as statistics. We propose a theory and test against data collected by experiment or sampling. \n",
"\n",
"That being said, in econometrics, we mainly investigate if the linear relation between independent and dependent variables is plausible, so we rarely test a specific null hypothesis such that $\\beta_2 = 2$, but rather $\\beta_2 =0$. Therefore\n",
"$$\n",
"H_0: \\beta_1 = 0, \\beta_2 =0\\\\\n",
"H_1: \\beta_1 \\neq 0, \\beta_2 \\neq 0 \n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's reproduce the house price example. We have the estimates, but we would like to investigate how reliable the results are. Reliability hinges on relativity, that means even if the absolute value of estimates are small, such as $.1$, as long as the standard error are smaller enough, such as $.01$, we can safely conclude a rejection of null hyphothesis without hesitation. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$t$ statistic of $\\hat{\\beta}_1$ and $\\hat{\\beta}_2$ are \n",
"$$\n",
"\\frac{\\hat{\\beta}_1-\\beta_1^0}{s.e.(\\hat{\\beta}_1)}\\qquad\\frac{\\hat{\\beta}_2-\\beta_2^0}{s.e.(\\hat{\\beta}_2)}\n",
"$$\n",
"where $\\beta^0$ is null hypothesis. \n",
"\n",
"Generally, statistics are intuitive to interpret, it measures how many standard deviations (with known $\\sigma$) or standard errors (with unknown $\\sigma$) away from the null hypothesis. The further way, the stronger evidence supporting the rejection of null hypothesis. "
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"b1 estimate: -29181.1698\n",
"b2 estimate: 1.1010\n"
]
},
{
"data": {
"image/png": 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",
"text/plain": [
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"df = pd.read_excel(\n",
" \"Basic_Econometrics_practice_data.xlsx\", sheet_name=\"CN_Cities_house_price\"\n",
")\n",
"s_ols_house_income = S_OLS(df[\"salary\"], df[\"house_price\"])\n",
"Y_hat, b2, b1 = s_ols_house_income.ols()\n",
"s_ols_house_income.ols_plot(\"Disposable Income\", \"House Price\")\n",
"resid = df[\"house_price\"] - Y_hat"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Compute $t$-statistic as in formulae"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"t_b1:-1.7580\n",
"t_b2:3.6349\n"
]
}
],
"source": [
"n = len(df[\"house_price\"])\n",
"s_u_sqr = n / (n - 2) * np.var(resid)\n",
"\n",
"std_err_b1 = np.sqrt(\n",
" s_u_sqr / n * (1 + np.mean(df[\"salary\"]) ** 2 / np.var(df[\"salary\"]))\n",
")\n",
"std_err_b2 = np.sqrt(s_u_sqr / (n * np.var(df[\"salary\"])))\n",
"\n",
"t_b1 = b1 / std_err_b1\n",
"t_b2 = b2 / std_err_b2\n",
"print(\"t_b1:{:.4f}\".format(t_b1))\n",
"print(\"t_b2:{:.4f}\".format(t_b2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Seems both $t$'s are far away from null hypothesis $\\beta^0=0$, but how far is real far? Unless we have a quantitative criterion, the interpretations won't sound objective. \n",
"\n",
"That's why we have **significant level** (denoted as $\\alpha$) as the decisive criterion. The common levels is $5\\%$ and $1\\%$. If $t$ falls on the right of **critical value** of $t$ at $5\\%$, we can conclude a rejection of null hypothesis.\n",
"$$\n",
"t > t_{.05}\n",
"$$\n",
"However in econometrics two-side test is more common, then rejection rules of $\\alpha=.05$ are\n",
"$$\n",
"t t_{.025}\n",
"$$\n",
"Critical value of $t$-distribution is obtained by ```sp.stats.t.ppf```, and the shape of $t$-distribution approximate to normal distribution as degree of freedom raise.\n",
"\n",
"The two-side test $t$-statistic is"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.0638985616280205"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t_crit = sp.stats.t.ppf(0.975, df=n - 1)\n",
"t_crit"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Therefore we conclude a rejections of $\\beta_2=0$ ($t_{b_2}>t_{crit}$) but fail to reject $\\beta_1=0$ ($t_{\\hat{\\beta}_1}\n",
"Side note of $t$-distribution \n",
"\n",
"
Here below is a figure for refreshing $t$-statistics, we set $\\alpha=.05$. The first axes demonstrates how $t$-statistic changes as degree of freedom raises, the second shows that $t$-distribution approximates normal distribution indefinitely with rising degree of freedom.
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"degree_frd = np.arange(1, 50, 1)\n",
"t_array = sp.stats.t.ppf(0.975, df=degree_frd)\n",
"fig, ax = plt.subplots(nrows=2, ncols=1, figsize=(14, 12))\n",
"ax[0].plot(degree_frd, t_array)\n",
"ax[0].set_xlim([0, 20])\n",
"ax[0].set_xlabel(\"degree of freedom\")\n",
"ax[0].set_ylabel(\"critical value of $t$-statistic\")\n",
"ax[0].set_title(\"The Change of $t$-statistic As d.o.f. Increases\")\n",
"x = np.linspace(-3, 3, len(degree_frd))\n",
"for i in range(len(degree_frd)):\n",
" t_pdf = sp.stats.t.pdf(\n",
" x, degree_frd[i], loc=0, scale=1\n",
" ) # pdf(x, df, loc=0, scale=1)\n",
" if i % 8 == 0:\n",
" ax[1].plot(x, t_pdf, label=\"t-Distribution with d.o.f. = {}\".format(i))\n",
" else:\n",
" ax[1].plot(x, t_pdf)\n",
"ax[1].set_title(\n",
" \"The Shape of $t$-Distribution Approximate Normal Distribution As d.o.f. Increases\"\n",
")\n",
"ax[1].legend()\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# $p$-Values"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Personally, I myself prefer to $p$-values. It is more informative than $t$-statistic, $p$-value gives the probability of obtaining corresponding $t$-statistic if null hypothesis is true, which is exactly the probability of **type I error**. \n",
"\n",
"With proper use of ```sp.stats.t.cdf```, we can access $p$-value conveniently. "
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.0006284669089314798"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 - sp.stats.t.cdf(\n",
" t_b2, df=n\n",
") # because t_b2 is positive, so p-value should be deducted from 1, if negative then without"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$p$-value tells that if null hypothesis is true, the probability of obtaining $t_{b_2}=3.6349$ or even higher is merely $0.0006$. That means, very unlikely the null hypothesis is true, we can safely reject null hypothesis with a tiny probability of Type I error.\n",
"\n",
"Medical researches conventionally uses $p$-value, econometrics tend to use estimates with standard error bracketed below. But they just different ways of expressing the same ideas, pick based on your preference unless you don't are writing an economic paper. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"Side note of Type I and II Error \n",
"The blue shaded area are genuinely generated by null distribution, however they are too distant (i.e. $2\\sigma$ away) from the mean ($0$ in this example), and they are mistakenly rejected, this is what we call Type I Error.\n",
" \n",
" \n",
"The orange shaded area are actually generated by alternative distribution, however they are in the adjacent area of mean of null hypothesis, so we failed to reject they, but wrongly. And this is called Type II Error.\n",
" \n",
" \n",
"As you can see from the chart, if null distribution and alternative are far away from each other, the probability of both type of errors diminish to trivial. \n",
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"from plot_material import type12_error\n",
"\n",
"type12_error()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Confidence Interval "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Why bother with confidence interval if we have hypothesis testing?\n",
"\n",
"If you have a theory that house price has a certain linear relationship with disposable income, you test theory with model, this is called _hypothesis testing_. But what if you don't have a theory yet, and runs the regression, you are wondering how confident these estimates can represent the true parameters, the range that you feel confident is called **confidence interval**.\n",
"\n",
"These two procedures complementing each other, that's why we see them often reported together."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Recall the rejection rules are\n",
"$$\n",
"\\frac{\\hat{\\beta}-\\beta}{\\text { s.e. }\\left(\\hat{\\beta}\\right)}>t_{\\text {crit }} \\quad \\text { or } \\quad \\frac{\\hat{\\beta}-\\beta}{\\text { s.e. }\\left(\\hat{\\beta}\\right)}<-t_{\\text {crit }}\n",
"$$\n",
"If we slight rearrange and join them, we get the confidence interval\n",
"$$\n",
"\\hat{\\beta}-\\text { s.e. }\\left(b\\right) \\times t_{\\text {crit }} \\leq \\beta\\leq \\hat{\\beta}+\\text { s.e. }\\left(\\hat{\\beta}\\right) \\times t_{\\text {crit }}\n",
"$$\n",
"The higher significance level, the smaller $\\alpha$ is, the larger confidence interval (because of larger $t_{crit}$). For example, if the significance level is $\\alpha=.05$, then confidence level is $0.95$.\n",
"\n",
"Here's the confidence interval ($95\\%$) of our house price example."
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"C.I. of b1: [-63439.54567466995, 5077.206045321891]\n",
"C.I. of b2: [0.4758472405690616, 1.7261233292171854]\n"
]
}
],
"source": [
"t_crit = sp.stats.t.ppf(0.975, df=n - 1)\n",
"print(\"C.I. of b1: [{}, {}]\".format(b1 - t_crit * std_err_b1, b1 + t_crit * std_err_b1))\n",
"print(\"C.I. of b2: [{}, {}]\".format(b2 - t_crit * std_err_b2, b2 + t_crit * std_err_b2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are $95\\%$ chances the true parameter $\\beta$ will 'fall' in this confidence interval."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# One-Tailed vs Two-Tailed Test "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So far we have been discussing about two-tailed test, but there are scenarios that one-tailed test make more sense. In our house price example, some practitioners would prefer to test the theory or common sense: _disposable income would not have negative effects on house price_. The alternative would be that _disposable income would have either no effect or positive effects on house price_.\n",
"\n",
"Thus the one-tailed test hypotheses are\n",
"$$\n",
"H_0: \\beta_2<0\\\\\n",
"H_1: \\beta_2\\geq 0\n",
"$$\n",
"In one-tailed test, we don't split $\\alpha$ anymore since there is only one side, that means the critical value will be smaller, easier to reject null hypothesis. \n",
"\n",
"Here is $t_{crit}$ of $\\alpha=5\\%$. However, these are conventional rules, if you still prefer $2.5\\%$ on one-side, feel free to do so. Especially you have a very significant $t$-statistic, such as $10$, one-tailed or two-tailed won't really matter. "
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.7108820799094275"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t_crit_oneside = sp.stats.t.ppf(0.95, df=n - 1)\n",
"t_crit_oneside"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So here the rule of thumb for one-tailed test."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"Rules of Thumb for One-Tailed Test \n",
"1. If the theory or common sense supports one side test, e.g. household consumption increases as disposable incomes increase. \n",
"2. If two-tailed test failed to reject, but one-tailed reject, you can report one-tailed test results if the first rule satisfied too.\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# $F$-test"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$F$-test is based on _Analysis of Variance_ (ANOVA), the goal is to test **multiple restrictions** on the regression model. In simple linear regression model, the **joint hypothesis** is usually\n",
"\\begin{align}\n",
"H_0&: \\beta_1 = 0,\\qquad\\beta_2=0\\\\\n",
"H_1&: \\text{One or more restrictions does not hold}\n",
"\\end{align}\n",
"\n",
"once you have ANOVA done, $F$-statistic is an natural byproduct.\n",
"$$\n",
"F=\\frac{E S S /(k-1)}{R S S /(n-k)}\n",
"$$\n",
"where $k$ is the number of number of parameters in the regression model, here in simple regression model $k=2$ and $n$ is the sample size.\n",
"\n",
"You might have doubt now: why aren't we using same old $t$-tests such that\n",
"$$\n",
"H_0: \\hat{\\beta}_1=0 \\qquad H_0: \\hat{\\beta}_2=0 \\qquad \\\\\n",
"H_1: \\hat{\\beta}_1\\neq0 \\qquad H_1: \\hat{\\beta}_2\\neq0\\qquad\\\\\n",
"$$\n",
"\n",
"Apparently, the number of $t$-tests will be as large as ${k \\choose 2} $ where $k$ is the number of parameters. If there are $5$ parameters, then we have to test ${5 \\choose 2}=10$ pairs. With $95\\%$ confidence level, $10$ $t$-tests would cut back confidence level dramatically to $95\\%^{10}=59.8\\%$, which also means the probability of _type I_ error would be around $40\\%$.\n",
"\n",
"We have user-defined functions written in the OLS class, so $F$-statistic is "
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"F-statistic is 13.2126.\n"
]
}
],
"source": [
"f_stat = (ess / (2 - 1)) / (rss / (len(df[\"salary\"]) - 2))\n",
"print(\"F-statistic is {:.4f}.\".format(f_stat))"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"p-value is 0.0014.\n"
]
}
],
"source": [
"p_value = 1 - sp.stats.f.cdf(\n",
" f_stat, 1, len(df[\"salary\"]) - 2\n",
") # sp.stats.f.cdf(df on nominator, df on denom)\n",
"print(\"p-value is {:.4f}.\".format(p_value))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To explore further, we can even prove that in simple linear regression $F$ are the just the square of $t$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"$F$-statistic and $t$-statistic \n",
"Here's the proof that $F$ and $t$ are connected\n",
"$$\n",
"F=\\frac{R^{2}}{\\left(1-R^{2}\\right) /(n-2)}=\\frac{\\frac{\\operatorname{Var}(\\hat{Y})}{\\operatorname{Var}(Y)}}{\\left\\{1-\\frac{\\operatorname{Var}(\\hat{Y})}{\\operatorname{Var}(Y)}\\right\\} /(n-2)}\\\\\n",
"=\\frac{\\frac{\\operatorname{Var}(\\hat{Y})}{\\operatorname{Var}(Y)}}{\\left\\{\\frac{\\operatorname{Var}(Y)-\\operatorname{Var}(\\hat{Y})}{\\operatorname{Var}(Y)}\\right\\} /(n-2)}=\\frac{\\operatorname{Var}(\\hat{Y})}{\\operatorname{Var}(\\varepsilon) /(n-2)}\\\\\n",
"=\\frac{\\operatorname{Var}\\left(\\hat{\\beta}_{1}+\\hat{\\beta}_{2} X\\right)}{\\left\\{\\frac{1}{n} \\sum_{i=1}^{n} \\varepsilon_{i}^{2}\\right\\} /(n-2)}=\\frac{\\hat{\\beta}_{2}^{2} \\operatorname{Var}(X)}{\\frac{1}{n} s_{u}^{2}}=\\frac{\\hat{\\beta}_{2}^{2}}{\\frac{s_{u}^{2}}{n \\operatorname{Var}(X)}}=\\frac{\\hat{\\beta}_{2}^{2}}{\\left[\\operatorname{s.e.} \\left(\\hat{\\beta}_{2}\\right)\\right]^{2}}=t^{2}\n",
"$$\n",
"
\n",
"$F$-statistic and $R^2$ \n",
"$F$-statistic is a different angle of evaluating the goodness of fit, it can be shown that $F$ and $R^2$ are closely connected, divide both nominator and denominator by $TSS$:\n",
"$$\n",
"F=\\frac{(E S S / T S S) /(k-1)}{(R S S / T S S) /(n-k)}=\\frac{R^{2} /(k-1)}{\\left(1-R^{2}\\right) /(n-k)}\n",
"$$\n",
"We prefer to $F$ for hypothesis test, it's because critical value of $F$ is straightforward, and critical value of $R^2$ has to be calculated based on $F_{crit}$:\n",
"$$\n",
"R_{\\mathrm{crit}}^{2}=\\frac{(k-1) F_{\\mathrm{crit}}}{(k-1) F_{\\text {crit }}+(n-k)}\n",
"$$\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Regression vs Correlation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here is the message for all beginner that misinterpret regression relationship as causation."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"Does Regression Imply Causation \n",
"\n",
"It's tempting to interpret regression result as causality, but it's not. Regression only implies a statistical relationship, the independent variables may or may not be the cause of dependent variables, sometimes we know thanks to theories, sometimes we don't.\n",
" \n",
"For instance, researches found that parents with higher education tend to have healthier children, but this is hardly a causality. Perhaps higher education parents are in general wealthier, they can afford decent medical packages. Or they spend time with their kids on sports and dining. We can form some hypothesis, but not a definite causality based on one regression.\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"But regressions do resemble the correlation to some extent"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"Does Regression Imply Correlation \n",
"From formula of $\\hat{\\beta}_2$\n",
"$$\n",
"\\hat{\\beta}_2 =\\frac{\\text{Cov}(X, Y)}{\\text{Var}(X)}\n",
"$$\n",
"We can see the regression indeed has a component of correlation (covariance in the formula), but it's normalized by variance (i.e. $\\sigma_X\\sigma_X$) rather than $\\sigma_X\\sigma_Y$. To compare with correlation coefficient of $X$ and $Y$ \n",
"$$\n",
"\\rho_{XY}=\\frac{\\text{Cov}(X, Y)}{\\sigma_X\\sigma_Y}\n",
"$$\n",
"We can see one important difference is that regression coefficient does not treat both variables symmetrically, but correlation coefficient does. Joining two formulae, we have a different view of the coefficient.
\n",
"$$\n",
"\\hat{\\beta}_2=\\frac{\\rho_{XY}\\sigma_X\\sigma_Y}{\\text{Var}(X)}= \\frac{\\rho_{XY}\\sigma_X\\sigma_Y}{\\sigma_X\\sigma_X}=\\rho_{XY}\\frac{\\sigma_Y}{\\sigma_X}\n",
"$$\n",
" \n",
"Besides that, the purpose of these two techniques are different, regression are mainly predicting dependent variables behaviors, but correlation are mainly summarizing the direction and strength among two or more variables. \n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Maybe a chart can share insight of their relationship. All data are simulated by $u\\sim N(0, 1)$. It's easy to notice the smaller correlation implies a smaller slope coefficient in terms of absolute value. And larger disturbance term also implies lower correlation coefficient. "
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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QrdBK3jBxyeptah7Yq2tbH5Yk7RsYoW++cara7vuT5n5gQtmuG5lVQsyNOnDggFasWKHFixfneTdAcS3f0JF2PJmqknXJ6m0FGXeOb2lURxbttwr5mSi8UsXddGPTTPmCVPGYOA2UVk1Xyvb19enpp5/WscceK0nq7OzUoYceqhEjRsTOyWYWa8KECQlVADt27Ei5pPXCCy/UU089pYcfflijR4/WMccck9fxCRMmaMKECbEKrk9/+tN66qmnCvjTASqD69qeNiEbFb9BwrqFsxikVig/xN2olpYWffzjH4/1M3zggQd09NFHq7W1VcFgUPPmzdP//u//Dvk8wI/SJRokKbhvl64d/XDsuW++caokf/buriWVFHMladWqVfrABz6gww47rNC3CgxbNlWwqWJeoWLhgtmThqz+SoX4W5lKGXfTjU0z5QtSxWPidPUoZqsVFE5NV8pu2bJFb775pv785z/rqKOO0hVXXKF/+Id/SDgnm1msv/qrv9Lzzz+vF198UW1tbbrtttv061//2vPc3bt3a8yYMXrllVe0bNkyPfroo3kdHzt2rA4//HBt27ZNkyZN0po1a3TccccN58cBVBzXtV2hJwb/LqVKyMI/Kj3udnV1KRgMqqWlRT09PXrggQd0+eWXS5KOOOIIPfbYY9q/f78aGxu1Zs0azZgxY5g/EaAypEs0uK7tuqb1cfVGVl7Gb7Do097dCcysQdLDkg5SeGz8W+fcd8xstKTbJR0l6SVJn3XO7Ym85gpJF0oKSfqac251GS49o0qKuZJ06623siQWFSubKthUlayFioVeq7/2H+jXnv19RftMFFYp4266sWmmfEGqeEycrg6ZVkChctR0UnbDhg363Oc+p/nz5+vtt9/WvHnzdPHFF+f8PvX19frpT3+q2bNnKxQK6Utf+pKmTJkiSTrzzDP1i1/8Ijar9alPfUpvvPGGgsGgfvazn2nUqFF5HZekn/zkJ/rc5z6nAwcO6D3veY/+4z/+Y7g/EqBiuN0vKPTHW2OPSchWh0qPu08//bQuuOAChUIhDQwM6LOf/azOPvtsSdJJJ52kT3/60/rABz6g+vp6TZ8+Pa9rB0op01LcqFSJhjnvfkOhJ36tiWOatLVzry7aPSf2nF97d3t4R9Is59w+MwtK+oOZrZI0T9Ia51y7mS2UtFDS5WZ2nKTzJU2RNF7SA2Z2bCVuslhJMXf//v26//77tXTp0oT3nj9/vh566CG9/vrrmjBhgq655prYxjTpngMKLZsq2AWzJyUkOqTCx8K509sS4nRycqUYn4nCKWXczTQ2TZUvSBWPUx2H/xS71QoKx7x6lZTajBkz3Pr160v+uZdddpk+/OEP67zzziv5ZwNIbWD38xr4422xx6VKyJrZk865mih9JO4C1S/VF/nF86YOGZB7nTun6VV9e+KrGtvcIEm6e/yXCt67u9LirpmNlPQHSX8v6T8lfdw512lm4yQ95JybFKmSlXNuceQ1qyVd7Zx7NNX7EnOByjezfa3n5FRbS6PWLZwVe5ztZFchFeozKy3mFku5Yq5E3EVlOHrhSnll+kzSi+1nlfpyalqmuFvTPWU3btyoadOmlfsyAMQpV0IWpUHcBUonU5/YeHOnt2nxvKlqa2mUSbp8zDMJCdlqj8VmFjCzjZJ2S7rfOfe4pMOcc52SFPl9TOT0Nkmvxr18R+RY8ntebGbrzWx9V1dXUa8/FWIukD2vfq6m8LLf+H6M5djHIP4z/3DR0Tp7500a2PZg0T8XuSPuohKkam9C25PKU9PtCx566KFyXwIADc7+H7r/Zf3fd2/UxDFNGtvcUPVJgFpE3AVKJ9cNaaJLZkOPLJXb2ydpMCFb7b3JIq0HpplZi6Tfm9nxaU43j2NDClKcczdKulEKV20V4jpzRcwFshffz7Wju0emwb/YlRLzBl55UgPP3BN+MHJU+pNRFsRdVIJStFpBYdR0pSyA8ot+0T90/0v6yiFPqrc/pK2de3X3+C+V+9IAwNfyqZLoX7lIbu/u2OPo5FguVbd+5pzrlvSQpDmSXou0LVDk9+gPZoekw+NeNkHSztJdJYBiiVaktrU0DplpKXfM61+5KJaQrfvgZ1R3+LSyXQuAypa8AqqtpdGzfRXKr6YrZQGU35LV2zS17lVd2Lwpduyi3XPURhNyABiWXKsk+lcuSngcv1oh16pbPzGzVkl9zrluM2uU9DeSvidphaQLJLVHfr8z8pIVkn5tZj9UeKOvYyQ9UfILB1A0lRbz4uNz4K//j+yQsWW5DgD+kbxpoN+Vo593KZCUBVBWh/e8oC82Px17fEnXGZKq44s+AJRT/FLcTAPYdAlZKVxd67UBTpX0Jhsn6RYzCyi8iuwO59zdZvaopDvM7EJJr0j6jCQ55zab2R2Stkjql3RppP0BgCpRKTHPDYQUWvXPsceBj14saz6spNcAAOVWzW20SMoCKJuBjqf196M3q7c//DiakJWq5os+AJRVNlUSmRKyUnX3JnPOPS1pusfxNySdmuI110m6rsiXBqBMKiHmud63FFpzfexx4PQFsmBDyT4fACpFujZaxU7KFrtCl6QsgLIY2LFJA5tWaOKYJm3t3KuLds+JPVctX/QBoNLFJ2R37e3VeS+fqp2PrBwy6Myl6hYA/K7cMc+98ZJCj/1X7HHgzG/LzGuPQQCofuVqKVOKCl2SsgCKLnl2qf1DTh9++38lSWObG7R+8lfUxhd9ACip5ITsaVs+qp6+8ODWa9BZbb3JACCdTDGvWNVToT/eJrf7+dhjr9ULAFBLytVSphQVuiRlARTV8g0dWvCbTeobCO9he1Tv8+p58lntGt+ssc0Nqj/rSs2V/3vBAICfJLcsOO/lU2MJ2ahSLQsDAL8pVvVUNu1kAKDWlKulTCkqdOsK9k4A4OHqFZtjCdmZDTv0/73rGTk5/WnXWww0AaCAlm/o0Mz2tTp64UrNbF+r5Rs6PM/z+tJfaTuNA0AlS1c9lS8SsgDgbe70Ni2eN1VtLY0ySW0tjVo8b2rRCwdSVeIWskKXSlkABRW/lOuQxqC6e/okSX/d8Kr+7l3Pxs770muz9VKZrhEAqk22VVupvvSnWhbmJM1sX0tbGQCIU+iJrPjYbEdMV2Dq2Xm9DwBUq3K00SpFhS5JWQAFk5wUiCZkP3vwczp15Eux8y7pOqMclwcAVSubnlcJX/qbDlXgY38fe+w16IwqxqYGAOBnhepv6AZCCq3659jjuuPPVN2RHxz29QGA3xSrT/dw3rcUmz6SlAVQMF5JgfMO3qJZI1+OPY4mZOvYQBYACiZT1VZCQnb8cQpM/1TCefGDTq9EA/1lAdSCbL+8F6J6yr3VpdDD/x57HPj4pbKDRw/vBnzAzBokPSzpIIXzEb91zn3HzEZLul3SUZJekvRZ59yeyGuukHShpJCkrznnVpfh0gHkKNuYWqw+3YV432JX6NJTFkDBJCcFvtL8lGdCVpIibWYBAAWQrudVfEK27piPDknIRs2d3qZ1C2cp1ZwZ/WUBVLPol/eO7h45DX559+rPPdz+hgN/fiwxIXvGt2oiIRvxjqRZzrkTJE2TNMfMPiRpoaQ1zrljJK2JPJaZHSfpfElTJM2RdIOZBcpx4QCyl0tMLUaf7mK+byFRKQugYOKXcn21+UlNPWh37LnklgVtBWyODQC1LlXV1u1HrpHUIEmqO+Ec1U04IeV7RKsZUs2ZFXJTAwCoNNm0gYmXb/VU/0M/ld7eE3tcaxt6OeecpH2Rh8HILyfpXEkfjxy/RdJDki6PHL/NOfeOpBfN7AVJJ0p6tHRXDSBXucTUYm0464eNbKmUBVAwC2ZPUmMwoH84ZH3ahGyhm2MDQK1avqFDM9vX6hu3b1RDsE4tjcFI1VaD7j/uEY1tjiRkP/CpjAnZaDWDF+I2gGpXii/v/SsX1XRCNsrMAma2UdJuSfc75x6XdJhzrlOSIr+PiZzeJunVuJfviBxLfs+LzWy9ma3v6uoq6vUDyCyXmJpuxddwFOt9C4mkLIBhi08KfK35j5re+EbsueSEbMAsp+VdAABvycvC9uzv05s9farTgP65cUXsvMBHvqi6ccelfS+vaoaoXJflAkCpRceiRy9cqZntaz2Xx2ZS7C/v8a1kpNpNyEqScy7knJsmaYKkE83s+DSne3XVGbKowzl3o3NuhnNuRmtra4GuFEC+comp0eKueIUoCCjW+xYSSVkAw7J8Q4cW/GaTOrp79PVD/qj31L2m/pBTW0ujLus+O+HcxmBAP/jsCXyxB4AC8EqkNlqfbmhdrd7+kLZ27tV9rZ+RjZqQ8b1SVTOYpHULZxG3AVSsXPoWppPtl/d8EsAJCdnG5ppOyMZzznUr3KZgjqTXzGycJEV+jy672yHp8LiXTZC0s3RXCSAfuSREh9unO5VCvK87sF/9Kxep/4EfKtx9pbDoKQsgL9Heg9Glrpe3PKr3BLslSU5On99xmhbPm5LVbosAgNwlJ1LHBvbpmtGPxB5/tetUvXvtTp154uSM7xXfEzz5OABUslx7waYSPTfd2DXXnbzdQEihVf8ce1x31F+pbsqc7G+uCplZq6Q+51y3mTVK+htJ35O0QtIFktojv98ZeckKSb82sx9KGi/pGElPlPzCAeQkm5iafH70uWiu4Ru3bxx2HiHf/t+SNLBjkwY2RVafuQGZpdoON38kZQHkLHlAGp+QlaItC/qGFQABAIOD0p3dPWoZGZRz0ps9fRrf0qiWkUHt2d8nSTp+RJf+4ZD1sdf9fddsDagu616IqTYKq6TlXQDgpZC9YDONXXNJALu/vKLQo7dIknbt7dX/e3W6Hn8kpPEta2u9UGGcpFvMLKDwyt07nHN3m9mjku4wswslvSLpM5LknNtsZndI2iKpX9KlzjnvfjsAfC/Xya9icAMhhdb+WHrnbUlS3cSPqG7yqUX5LJKyAHIWPyC9ouV/dVTwzdhzyT1kAQD5SR6URhOwUniAGqwzBQOmk0e8qM82PRd7Lj4Ot4wMamb72owVCrlWMwBApShlpX+2CeDQ+tvlXvuTpHBC9owtH9FbkRBejgRDJXHOPS1pusfxNyR5Zj2cc9dJuq7IlwaggPJNrhZq9UO+3J4dCv3vf8QeBz72FVnTu4v2eSRlAeQsOvC8dtTDOqz+7djx+ETAwSMCQ14HAMheus23JKlvwOmy0Zs0/aDX1NsfPhYfh4MB077e/lgyN9NgmNUNAPyolJX+2SSAkzf0Ou/lU/VWX+JrSplgAIByyDe5WsjVD7mKj982aoLqPvyForQsiMdGXwBysnxDh+rM9M+jH0qZkJWkYIDwAgDDkWnw+YN3r9H7Ajs1872H6tTJh6ln1hUJGxkcPKJefQOJGxJEB8MAUC2KtUGMl0wb1yQnZOvPurKsCQYAKJd8Y1+qVQ7F3OfAde9MiN91k2cp8JEvFj0hK1EpCyAH0SUI141aq1GB3thxr5YFb/b0DTkGAMheqoosSVraukqS1FAfTg7Un3Wl5iqxAvbohSs9X0siAEC1KVWlf7pWL14JWYmNFAHUpnxjX6n3Oehf+2OpZ2/sceC0b8pGjCzKZ3khKQsgo+hGMx3dPfre6AfVkiEhKzHQBIBsxG/kldzH1WtQKg0mZANmmjimKfbFPxmJAAAoPK8EcKqErMRGigBqU76xr1T7HLi+XoXuWzJ4YMRI1Z/2zdjDdGP0QsqYlDWzwyX9p6SxkgYk3eic+7GZXS3p/0jqipz6LefcPZHXXCHpQkkhSV9zzq0u+JUDKIn4Bt1L3r1WzXXvxJ5LlZBloAkAmWXaACF5UNoyMqgfvOtu9Q2EK2QnjmnShPmp9z0hEQAAxeWcU+ie7w4eaHiX6k+9LOEcNlIEUIuGE/uKvfph4LkHNPDnR2OPAx/5omzUhNjjfDcpy0c2lbL9kr7pnHvKzN4l6Ukzuz/y3I+cc/8Sf7KZHSfpfElTJI2X9ICZHeucS71TBYCKE18dK4V7FzbVHYg9H5+QHTUyqJEj6hloAkAOstkAIX5QGq7Eao2dm6pCNopEAAAUj/vLKwo9ekvscd20c1XX9n7Pc9lIEUAlK1ZVaLFiX77XO2QiTd7j6Xw3KctHxqSsc65TUmfkz2+Z2XOS0l3FuZJuc869I+lFM3tB0omSHk3zGgAVYvmGDl29YrO643rCRpfKRsUnZBuDAX3nE1MYaAJAjnLZACHd0th0SAQAQOGF1t0k190Rexw4fYEs2JBwTqmWvgLAcJSyKrQQ8r3egS33aeDFx2OP6973N6p7z4c9zy3lBo05bY9uZkdJmi4peidfNbOnzewmMxsVOdYm6dW4l+1Q+iQugAoRDXDZJmSLubstAFS7bHeXzTchCwAovP6VixISsvVnXemZkF3w203q6O6RUzhpsOC3m7R8Q4cAoJKkqwqtRPlcb//KRQkJ2cDpC1ImZKXsx+iFkHVS1syaJP1O0mXOub2S/k3SREnTFK6k/UH0VI+XO4/3u9jM1pvZ+q6uLo+XACi15ACXKiHbGAzo+vOmad3CWSRkASBPp0xuzXichCwAVI7+lYu0a2+v1r3wutZsfU0fe2amZ6L1mrs2qy+U+BW4L+R0zV2bS3WpAJCVUlaFprN8Q4dmtq/V0QtXamb72pSTWLlcr+ve6TmWTp5IS7Zg9iQ1BgMJx4q1L0NWSVkzCyqckP1v59wySXLOveacCznnBiT9XOEWBVK4MvbwuJdPkLQz+T2dczc652Y452a0tnp/KQFQWvG7dKdKyFIdW1xmdriZPWhmz5nZZjP7euT41WbWYWYbI7/OjHvNFWb2gpltM7PZ5bt6ALl4cKv3pHT0OAlZAKgc0YTs1s696u0P6ZKuM2LLZpOTB3v293m+R6rjAFAupawKTSW6Yjd+dYFXbE13XV4rzULrfhl7XDfjszm1/1o8b6raWhplKm4OJGNPWTMzSb+U9Jxz7odxx8dF+s1K0iclPRv58wpJvzazHyq80dcxkp4o6FUDKIqAmULOeSZkG4MBkrGlweaKQI1IN9NPQhYAKkc0Jm/fvU8h5xLaeRVr8xcAKIUFsycl9GiVilcVmkouG2tlul4X6lfo3sUJr8lnHF2qfRkyJmUlzZT0d5KeMbONkWPfkjTfzKYp3JrgJUmXSJJzbrOZ3SFpi8LJhUtJDgD+kCohK4mEbImwuSJQO1pGBj2rpv69dZXWvRDQxDFNGtvcQEIWAMokeafuaIVssuRJtpbGYMIeDfHHAaCSRL/jZ9qYsJibF+bSkiDd9YYe/5Xc6y/GzrXWiQqc+LcFucZiyZiUdc79Qd59Yu9J85rrJF03jOsCUAa3jLtPvf2Dj+NbFpCQLb2kzRVnKry54v8nab3C1bR7FE7YPhb3spSbK5rZxZIulqQjjjiieBcOICtuSMf9wdYxvf0hbe3cq/WTv6K5pb0sAIAkt/sFhf54a+yxjT5C1/bNlDQ0SZC8bPbqc6ZowW82qW9gMNAH60xXnzOlaNcLAPnKVBUabS8QrU6NtheIvna4xrc0JrRSjD+e7fUmrzILnPEtWV1iX9hKlPVGXwCq1/INHfqfH39dvf2DRe3xm3qVcukCwgq9uaJEL2+g0ryZVEWVvFLhot1zKnbnWwCoZv0rFyUkZAMnf1mBD1+QcoPG7v0HEnofzp3epiWfOSGhH+GSz5xAkQMAX0rXXqAQhrOx1sCurd6befkgIStl174AQBVbvqFDTQ+2qzeuZOvLcRWyhVyWgOyk2lwx7vmfS7o78jCrzRUBVI7o8q/42ZNUrWNKvfMtANS6dD29U23Q+PaB0JCqsVL1IwQAL4VsN5BLe4F8ZNtCIdmQ6ti/vlB2yPiCXFOpkJQFatyoh7+fkJCNb1mwbuGscl1WzWJzRaC6JS//klInZKXS7nxbq8zscEn/KWmspAFJNzrnfmxmV0v6P5KiWZhvOefuibzmCkkXSgpJ+ppzbnXJLxxAwWXaZDFdAoINvwBUikK3G8i1vUA+cpnIcgf2K3T/DxKO+XUPBpKyQA3rX7nIs2WBRHVWGbG5IlDFkpd/pUvINgYDOmVyq2a2ry3KpgqI6Ve4T/dTZvYuSU+a2f2R537knPuX+JPN7DhJ50uaovBk2ANmdiyxF6hc2VSMZUrISqkTE1GMnwFUgnTtBvIZRy6YPWlIUUG52hz237dE6uuNPa57z4dU977TEs7JpUq4mBuYZYOkLFCjogPPhvqA506yVGeVB5srAtUt/st8ckK2Z9YVaosbFJ4yuVW/e7KjaJsqICyyCqEz8ue3zOw5pdgwMeJcSbc5596R9KKZvSDpREmPFv1igRpSqC/K2VSMZZOQlbwTE/EYPwOoBIVuN5Bve4FCG9Ku4MxvK7zQdFAuVcLF3sAsGyRlgRqQPKi9/cg1GtvcIEmaOKZJp235qMIrMMPY3AsAspNr0iBgppBznhWybau3Jbx+ZvvaglY5IDMzO0rSdEmPK7xy4atm9v9JWq9wNe0ehRO2j8W9bIc8krhmdrGkiyXpiCOOKO6FA1WmkF+UM1WMZZuQjf/sq1dsVnfSZo2MnwFUimK0Gyhnn+yBPz+qgeceSDiWKlbnUiVc6IrifNSV5FMAlE10UNvR3SMn6argMm3t3Ktde8Ml/xPmX6fF86Ym7A67eN5UvvADQAbJ8TWaNIjfgTtZckJ2d//BsZUKya8v9qYKSGRmTQpvsniZc26vpH+TNFHSNIUraaPNy7xWM7ghB5y70Tk3wzk3o7XVe8d2AN4KudN3qph50L7OnBKyUXOnt2njd07X9edNY/wMoCItmD1JjcFAwrFSTRwt39Chme1rdfTClZrZvjbtuDgb/SsXJSRkA7O+njZW5zJ+roSxNpWyQJWLH9RGEwEh57R99z5NmB9e8c7usACQu1xn150b0C3j7lNvf/jxup4J+s99U1O+vhSbKiDMzIIKJ2T/2zm3TJKcc6/FPf9zSXdHHu6QdHjcyydI2lmiSwVqQiG/KHvF0h+9+wGNHjEg6VBJUuCkz8kOfU/O7w0Alahc7QYKucrBvf0XhR76WcKxbCbOchk/V8JYm0pZoMpFB6/JS2W/0Hl6OS4HAKpGLkkDd2C/Qvdcp4ljmhQw0817pw5JyCa/vpxVDrXEws3IfinpOefcD+OOj4s77ZOSno38eYWk883sIDM7WtIxkp4o1fUCtSDVF+J8vignx9Klrav0rkC/Jo5pkhTpSZhjQjaflRIAUEpzp7dp3cJZerH9LK1bOKskRViFWuXQv3JRQkK2bupZWSVkJe/xsykcp5MrdythrE2lLFDlxrc06qrgsoRjl3SdoTYqrQBgWLKdXXdv7lToD7+UJI1tbtCWIz+tVx7eI73jndSNvr5SNlWoATMl/Z2kZ8xsY+TYtyTNN7NpCrcmeEnSJZLknNtsZndI2iKpX9KlzjnvXX8A5KWQO33Hx9KrgsvUUB/QxDFNGtvckPWX/GSV0IcQACpNrqschuzNcPoxOrvz5oRzco3T8TG/o7tHpsEeU8mVu5Uw1iYpC1S5249co62d4Y1lpHBClkorABi+Uya36lePveJ5PGpgxyYNbFoRexw47Zs6fcRInf6RoUu8pKFJB9rLFJ9z7g/y7hN7T5rXXCfpuqJdFFDjCv1Fee70Np298yZJh8WO5ZuQlSqjDyEAVJpc2gEkj4OvCi5T4CHTrnHNsU3J843T0fHzzPa1Q64neQKt3GNtkrJAFetfuSgW0Lbv3qcvdJ6uNiqtAKAgHtzalfZ46Om75F7dGDseOPOfZDbYOaoSZucBoFIV8otyPht6pVMJfQgBoNLkssoh3d43bZ/+tiw4/Hjqhwk0krJAFYkv/7953H2xpVljmxs0Yf51erHcFwgAVSTdQK9/zfVS71uxY6kSAOWenQeAalfohKxU2PYKAFAtcik42Nndo481vKK/fdfmhONf6DxdLxYgISv5YwKNpCxQJeLL/5e2rlJvv7S1c68kacJ8VlgCQKGlGujdPO4+qffQ2ONCJAAAALlxb3Up9PC/JxwrVDxmpQMAeMu24ODmcfept39wYmv528dq1f6JBd37xg8TaCRlgSoRLf+Plv5L4fL/814+VevKeF0AUG2iqxKSNw+QpF+MuVcTxzTHHmdKAAzZ4IAv9QAwbP0P/lTavyf2uO6Ec1Q34YSCfgYrHQAgd66vV6H7lmjimCZt7dyrkHO6pOsMSYMJ00KNj/0wgUZSFqgSO7t7EhKyUnhTL1Pl9EsBAL9L3pTASbHE7C3j7tPEMdlvTpD8Xsk7wgJAtSnFRFRyu4Lkft4AUG38MskfH5/HNjeoe/8BnfvCKZKcAmb61AfD11zI8XGlT6DxrxNQJW4ed1/C4+hsUyX1SwEAv4vflCAqmpCd+d5Dc9ot1uu9ojvCAkC1iU5EdXT3yGnwi/byDR0F+wyv/rEkZAFUs1LE1kJIjs8rx16gz7x0qkIuvOYs5Jx+92SHrrlrc02Nj6mUBXwqeVOvdzeN0K43ez3L/wEAheG1uVe0j3dUtj0L/bAjLAAUSrqJqEJUMRVjQy8AqHTFjq3DFXrsv+TeeCnhWP1ZV+r77Ws9rzv5WFSpx8elqj4mKQv4kNemXrve7NXYQxpiLQsqedkCAPhV8uZe0bYxDfUBSbklAfywIywAFEoxJ6JIyAKoVZU8yZ8cm+umf1J144+XlPv1lXJ8XMoWY6zlAHwo1aZel3SdoXULZ+nF9rO0buEsErIAUGALZk9SYzCcgI3G4ICZJo5pyjkJEP9eUaxwAFCtUn2hHu4XbRKyAGpZsWLrcLjuDs/YHE3ISqmvr6UxWPbxcSlbjJGUBXxk+YYOzWxfq44Um3pVwmwYAPhJNK4evXClZravzdh/a+70Ni2eN1W3RPp4N9QHNHlcsybMvy7nz46+V1tLo0xSW0ujFs+byoQagKpU6Iko9/ZfSMgCqHmVNsnfv3KRQutuSjjmFZtTXffV50wp+/i4lNXHtC8AfGD5hnDD6z37+yTJMyErseQVAHKR79Kks3feJL330Njj4SQBKn1HWAAolGisK0SPvv6Hfiq9vSf2uO6ov1LdlDm+2YEcQGXyYwwpZGwdDuecQvd8N+FY4Mxvy8w8z8903eX8uZeyxRhJWaDCJScNUiVkg3XGklcAyEE+GyNQlQUA+SvERFRyHA7MWSgLBEvaAxBA9fFzDCn3JH9yXJayGyOX+7pTWTB7UsL/C1Lxqo9pXwBUuPikQaqErCTJewIKAJBCrkuTSMgCQHl5xWELBCWVtgcggOpDDMnPkImyj3zR92PkUrYYo1IWqHDR5EDahKykvpBLW90FAEiUy9IkErIAUFi5LhPOFIcreQdyAJWvnDHEj20TBl58XANb7ks4Vk3j41JV8VIpC1S48S2NGROyUQw6ASB72W6MQEIWAAoruky4o7tHToPLhFNttphNHK7EHcgB+Ee5Ykiu8bAS9K9clJiQrQ8yPs4TSVmgwt1+5BoF4ppjp0rISgw6ASAX8UuTJClgFlumFh0Ik5AFgMLLZZlwtnG40nYgB+Av5YohubZNWL6hQzPb1+rohSs1s31tSZO3rv+AZ0yun72wZNeQSjl/LsNB+wKggvWvXKSxzQ2SpD+99pa+tGt2ynMZdAJA7qLLkhb8ZpP6BpykcIXCgt9s0oytN8RisERCFgAKJZtlwu6dtxV64IcJz6eLw5WyAzkAfypXDMmlbUI5NyPLdzOvUvDzJm0kZYEKFR/0xjY3aML863R9XK+ZQxqDMpO69/cx6ASAYbh6xeZYQjbqp+++R3/aFV5QtH33Pl3QeboCf7hHIefURswFgGHJ1NM79Nh/yr3xcuy4HXq0Aid9PuP7VupO3gD8oRwxJJc9DtJV1Rbzuods5nXKP8hGthTt83JVrp9LIZCUBSpQqmVaDDQBoPC6e/oSHkf7ePcNDGjzzjdjbWNCbrCS1i+z7wBQiRbMnpRQ1SQNrvoa8uX/tG/KRows9SUCQEmki4fJSr0ZWf/aH0s9exOOVUp1bDw/b/RIT1mgwtC/EADKJ9uNFdP1+gIApBff09sktbU0avG8qTp7500J59WfdSUJWQBVLVU89Jr4L+VmZP0rFyUkZG30ERWbm/DzRo9UygIVhIQsAJTeqJFB7dnfl3VCNsoPs+8AUC7L49puebXaSl4B1r9ykXbt7dX23fvU2x/StX3ztGB8BysSAFS9bFfE5lJVmy/35i6F/vDzhGOVnpcoxc+lWEjKAhWChCwAlMd3PjFFI9e2K9pVdu/AQVrwxqyMr/PD7DsAlEOum65EE7JbO/cq5FxkUoxWMQAQL5vNyDJNiKVTyZt5pePnjR5JygIVgIQsAJRO/GD1iJYR+u8jH5LGN2v77n1au2+cbnlrasb38MvsOwCUQy6brkTHwdt374tLyA6+5rLbN2rJ6m2++YINAMWUrqo21wmxeEP6ec9ZKAsEC3DFpeHX/XdIygJlRkIWAEonfrDaWve2vhVcpa2dpsnjmvXRT1+kPbtH646k5U8myUkKmCnknNp8NPsOAOWQzaYrrv+AQqu/F3vc2x9K2TaGDRYBILNcJsSi/FodWy1IygJlREIWAHI3nGVZ0cHqCSNe01cOeUqSFHJOX39lhn7XNlVz2wbP89vyJwCoFONbGtXhkZg9pDGome1r9YmB/9WHD+7SxDFNGtvcIAUbdG3fPEmpe3X39IX0zTs2SRqamB3OvwsA/IG/55llMyEWLzkfYYdPU+D9nyj4dSG1jElZMztc0n9KGitpQNKNzrkfm9loSbdLOkrSS5I+65zbE3nNFZIulBSS9DXn3OqiXD3gYyRkASB3w1mWJYUHpZ8+eKtOG/li7NjXX/8b9bpwooCBPgAMn9emK8E609sH+vW9phWSpN5+aWvnXj1z9N/qjA9N0YLWjiGvSRZybkjMH+6/CwAqH3/Ps5NqQix5H4SB7es0sHVtwjHyEeVRl8U5/ZK+6Zx7n6QPSbrUzI6TtFDSGufcMZLWRB4r8tz5kqZImiPpBjMLFOPiAb8iIQsA+Um3LCtq+YYOzWxfq6MXrtTM9rVavqEj9tyPxv4hISF7Sdcc9bqgTOEBvtPgQD/+dQCA7M2d3qbF86aqraVRJqmtpVFNDfX66eh7Es67aPccffeh14a8Jp3kmJ/NvwsA/I2/59lZMHuSGoOJ6bfkfRD6Vy4iIVtBMlbKOuc6JXVG/vyWmT0nqU3SuZI+HjntFkkPSbo8cvw259w7kl40sxcknSjp0UJfPOBHJGQBIH+ZlmWlq6Q4e+dNmnFYnbZ2WsJmMtGesfEy9d8CAKSXvOnK6h98NeH5aAyOj+vR1yTH8mTxr8l1uS4A/+HveXaiMderzYNzTqF7vptwPrmI8supp6yZHSVpuqTHJR0WSdjKOddpZmMip7VJeizuZTsix4CaR0IWmdAyBkgv07KsVJUUox7+vvTeQ8O9CyWd9/KpMvWkfD+JgT4AFEr/ykVqqA+otz8cn+M39EpeVisNJha+eccmhVzytFnia7JdrgvAv/h7nr3kCTHJf5t51VL/4GzaF0iSzKxJ0u8kXeac25vuVI9jQ/4lNbOLzWy9ma3v6urK9jIA3yIhiyzRMgZII9OyLK9E6tLWVbFEgCRNmH+d1i2cpRfbz9K6hbNSLpVloA+gGqRr6VIK0THwxDFNCpglJGSTl9XGmzu9TT/47AkZl+Jms1wXgL/x9zx/yXmIug98qqJzEdGVErXSViyrSlkzCyqckP1v59yyyOHXzGxcpEp2nKTdkeM7JB0e9/IJknYmv6dz7kZJN0rSjBkzhk5/AlXEKyFbS7M/yB4tY4D00i3LkoZWUixtXSVJaqgPD+STB6HLN3Ro/4H+IZ/DQB9ANSjn5jhuIKTQqn+OPR7b3KD1k7+ithzGv5lifrbnAPA3/p7nrv/+H0gH9iccq5RkbLpcSLr+wdX43ztjUtbMTNIvJT3nnPth3FMrJF0gqT3y+51xx39tZj+UNF7SMZKeKORFA36SKiHL7pHIpNAtY8zsYkkXS9IRRxxRpKsGis9rWVZU/I7f0YRswEwTxzR5JmS9eha2NAZ19TlTiMc1grYxqGbF+nKbqbhgYOtaDWxfl/Ca+rOu1FzlPtZNF/NzOQeVgZiLfPH3PHuV3K4gUy6k1voHZ1MpO1PS30l6xsw2Ro59S+Fk7B1mdqGkVyR9RpKcc5vN7A5JWxRehnupc867QztQ5VK1LKi12R/kLrllTHh+zPtUj2Oeqw9YoYBaEI2hox7+vnr7wxWyE8c0acL864ac6xWLJengg+qJxbUl2jbmKTN7l6Qnzex+SV9QuG1Mu5ktVLhtzOVJbWPGS3rAzI5lvItKVOgvt8s3dOjqFZvV3dMXO5b8hTp5/Bv4yBdloybk9XmoSsRcoEhc71sKrbk+4VilJGOjMuVCaq1/cMakrHPuD/L+0i9Jp6Z4zXWShn77AWpIuh6ytTb7g9wUo2UMUEvO3nmT9N5DY49TDUaJxZBoG4PqVsgvt6lWF0iDX6jP3nlTwvFKSwag/Ii5QHFUcnVsvEzj7/hVb1HV3FYs642+AGQv06ZeqQbC1Tr7g+xl0TJGGtoy5nwzO8jMjhYtY1DjctlUkViMZOnaxkiKbxvzatzLPNvGsKktKkEhN8dJtbog6qrgsoTHlZgMQGUpZMwFatmQFQof/0rFxuBM4++509u0eN5UtbU0yiS1tTRq8bypVbuKLauNvgBkL5uEQK3N/iAntIwB8pRLQlYiFiNRodvG0DIGlaCQm+OkW0WwtHVVbENFiYQsMit0zGXvBNQiv1THxstm/F1L/YNJygIFlG1CgN0jkQotY4DMvDaYyWfJLLEYUbSNQTUr1JfbVK0Qlrauim2oKFV+QgDlV4yYy0QYao0fE7IS4+9kJGWBAsm1QquWZn8AoFC8dmxterBdu8Y1a2xzg6TcBqTEYmTRNqZdQ9vG/NrMfqjwpjO0jUFNGFrd5LS09V4FA3U69rB3aWxzgy8SAigvYi4wPAOvbtTA03clHPNb7GX8PYikLFAAuSZkAQD5Se5puLR1lUJO2r57HwkB5Iu2MUAW4qub3vfOc/q7lj9p4phD8poQQ00j5gJ58mt1LFIjKQvkKbp89qrgMjXUBzRxTBMJAQAosviehktbV8X+3NsfIv4iL7SNAbI3d3pbXLuYQyVJdR/8jOrGTi7fRcFXiLlAfoZs5nXGt2R1gRRnwy9IygJ5iC6fvb7lbknhZMDWzr1aP/krmlveSwOAqhbtaRifkJWka/vmaXaZrgkAagWrwwCgtKiOrW4kZYEcRKtjvRICF+2eo7bV2+iNAgDD9O3lz+jWx19VyA3u09EW2QRgwexJanqwXaG4LTwu6z5bi+dN8ngnAEChkJAFgNJKjrs2aoICH/lima4GxUBSFshS/OYyyQnZS7rOkJS4rBYAMCg6qZVpl9VvL39Gv3rslSHHO7p7dMWyZ3T/cY9I45q1ffc+9faHdG3fPC2eV7s7tgJAKZCQBYDSCa37pVz3zoRjxN3qRFIWyFJ0c5lUCVkpvKwWAJAoflJLGkywShqSTL318VdTvs/1LXdr807F+nhPmH8dLQsAoMjKlZDNdjIPAKoJ7QpqC0lZIEs7PVoWxCdkG4MBLZjN8lkASBad1IrX0xfSEo+WL/EtC+Ilb+p12paPavGGDr6gA0ARlTMhm+1kHgBUAxfqV+jexQnHqjUZy6TbIJKyQJZuHnefevsHH8cnZNtqPJAAQDqpWrt4HQ+YDUnMek+IeSd1AQC58fpyfM5hezSw4fcJ55UyOZDLZB4A+F0tVccy6ZaIpCyQhf6VizRxTJO2du5VyLlYQrYxGNDieVNrMngAQLbGtzSqwyMB69XyZf5Jhyf0lE23QoE+3gCQWbqKJK8vx00PtmvnuGaNbW6QJNUd+zHVHXNySa85l8k8APCz5IRs3fvPVt3h0zO+zq/Vpky6JSIpC2QQDZLRgel5L58qk78CHwCU04LZkxK+9EupW758d+5USdKtj7+iGw4dTMi+1n+wrtqTmBSgjzcApJepIin5y/HS1lUKOWn77n0a29ygwJnflpmV/LpzmcwDAD8aTnWsn6tNmXRLRFIWiBOdbero7lHATDccek9sQ5mxzQ2aMP86rSv3RQKAz0QHh9nO5i86c6KuDi6XdJgkaVNgki5b1ywpc1IXADAoU0VS/Jfg5N7d5Vw6m8tkHgD4zXDbFfi52pRJt0QkZYGI5NmmGw69R1J4ULq1c6/WT/6K5pbx+gDAz+ZOb8tqkOi6OxRad1Pscd2M8/XBw47R4sP8uUQLAMopVeVRR3ePZravVbSDd3KrmGv75ml2ka8tnVwn8wDAD9ybnQr94RcJx/KZAPNztSmTbolIygIR8bNNyQPTi3bPUZsPZp0AwM8GXl6vgWcH42/glH+QjWyR5J3U9WsvLQAolVQVSZJix5PHvZd1n63F87L/clysWJztZB4A+EEhN/Pyc7Upk26JSMoCETtTDEyjm8r4YdYJAPwq9MR/y3X9OfY4MOcKWSD1MMXPvbQAoFS8KpLieVXILp6X/ZdjYjEAZJackI0vPMiH36tNmXQbRFIWiBjf0qirgssSjsXv8n1IY7DUlwQANSF5oJpN1YCfe2kBQKl4VSR1dPdocvANfaPliYRzL+k6Qy+1z8rqfeP3YUhGLAaAsEJWx8aj2rR6kJQFIm4/co22dppCLtxdKz4hK0l9oYFyXBYAVLV8ErKSv3tpAUApJVckrf7BVxOef+qdsVq6d3rW75dcHeuFWAyg1hUrIRtFtWl1ICmLmvft5c/o5O03ysW2OhiakJWktw+kHngCQK3Lp6dgvglZyd+9tACgXPpXLlIwUBcrNri063T1KyBJGjUyu1VhXisVkhGLAVSaUuxFsHxDhzauXa4P6zk11Ac0cUyTxjY3FDQZi+pCUhY17dvLn9FHty+NS8d6J2QBAKnl01MwOSF79/gvaUn72qwHyn7vpQUAyYqdMIjG3WMPe5e27Nyri7vmxJ4LBkzf+cSUrN4nUxUssRhApSlF/+vlGzrU9GC7PhzJLvT2h7S1c6/WT/6K5hbkE7K/Dtoa+EdduS8AKKeTt9+Y8DhdQraFnrIA4Cldf1cvXgnZK5Y9o47uHjkNDpSXb+hI+Zlzp7dp8bypamtplElqa2nU4nlTGXQC8KVowiCXOJiL+Lg7trlB+2ctTIifSz59QtbxM10VLLEYQCXKdayaj1EPfz/WClGSLumao4t2zynoZ2RS7H9LUHhUyqJm9a9clLFlQVSwznT1OdlVDwBArcmlv6tXy4Il7Wvz2rSLXloAqkWumxfmUgnlFXfnKv/qsFQrFUjGAqhUxdyLIBpje/sHY2J8bqGUPbbZCNd/SMqipkQHsFcFl6mhPhA7npyQrTNp3CGNlPwDQBay7e+aqocsm3YBqHW5xMFcluHm2rs7m2Qvu34D8Jti7UUQH2Mb6gNat2+MbnxrWkE/IxeMqf2HpCxqRnQAe33L3ZIGZ7K8KmT/9qQj9N25U0t6fQDgV6mqpk6Z3KqZkT6xN4+7L7bZgZSYGGDTLgC1Lpc4mE0llNv7mkKPJLbpyiYhm22yl5UKAEqhUP1RC70XQf+97VKoL+HYnpP/Uf+17BlJ5dvvgDG1/9BTFjVjyeptsYRs1CVdZ2hksE4BM0lSwEyf/xAJWQDIhVd/1099sE2/e7JDHd09+vfWVbHNDnbt7R2SGFgwe5Iag4GEY2wUA6CW5BIHM1VC9a9clJiQPejgrHb+LkXPRQDI1vINHVrwm00J/VEX/GZTXv1RC7kXQf/KRUMSsvVnXVkR+x0wpvYfKmVRM64KLkt4HK2Q3d83oOvPm8ZsPwAMQ3LV1MxIn9ilratix0LO6byXT9U6j9dKLIUFULtyiYPpKqGS2xUETvumbMTIrK6BZa8AKsnVKzarb8AlHOsbcLp6xea8xojDrfB3fe8odN/3E44lT3iVexUBY2r/ISmLqrd8Q4dGPZwYPJNbFqRamgUASBS/jKxlZFDOSW/29A0Z9O3s7klIyErh2Gvy/nJf7kEsAJRbtnEw1TLc249cI6khdiyb6th4LHsFUEm6e/pyOl5MyRNeUu4xtlQYU/sL7QtQ1ZZv6FDTg+0pd0KMYmkWAGQW7TcYXUa2Z3+funv6YkvKrlj2TGxJ2c3j7kt4bTT28uUeAIbHa4ns/cc9EuvZLeWXLGDZKwAMFZ+Q3bW3VxdvnqhjHvmAZravzaqVwvINHZrZvlZHL1yZ9WtQO6iURVUb9fD31esGlzx4JWSjWJoFAOl59RuMF53gOnvnTZo4pklbO/cq5Fws9vLlHgAKI74SKpwwGF5CNvqeEsteAVSGUSOD2rN/aFXsqJHBon1m/Iqw5E1qd+3t1WlbPprVZojx75ftBoqoTSRlUbX6Vy7KWCEbj+otAEgvm8mrcP/uw2ID2PNePlUmvtwD8I9C7fZdCslLaoe7nJZlrwAqxXc+MUULfrtJfaHBIqtgwPSdT0wpyufFJ1CXtq5Sb7+0tXOvJGlsc4POe/lU9fQljoWjBQmp4ma6DRSJtZBIyqLKRAfRVwWXqaE+oGBdnfoGBoYkZE1SfMtwqrcAILNU/QajlrauUkP94NLXCfOvG7KpFwBUMr9UNbnevQqt+XHCsUrtbwgA+Sh19f6S1dt0lF7TZa1/jB2LbVK7cJZ2PrLS83XpihbYQBGZkJSFL3lVMEjhDbuub7lbktTbH5LJ9NU3zlR8CrYxGNCnPtimB7d2+aICAgAqhdfmMlFLW1cpYKaJY5okkRwA4E9+qGry04YzADAcpazevyq4TGpJPBa/SW0+myGme42fVmWgeEjKwndSVTA0BOtiCdmoi7vmaNTIeo0cUU+wA4BhSq5YaBkZlHPS95pWqKE+EOu7RXIAgF9VelVTckI28PFLZQePLtPVAEB16F+5SA31gVj7w2++Pkv73EGSBpOuXsUJmVbcpnrNKZNbfbEqA8VHUha+k6qCITkhG21ZsGd/nzZcdXrJrg8A/CKfGfrkioVwguCw2GMSsgD8LJ9KqELJFJML3T8WAPxuuNWm8XE1ukntRbvnxI7FJ13zaaeQ6jV+WJWB0siYlDWzmySdLWm3c+74yLGrJf0fSV2R077lnLsn8twVki6UFJL0Nefc6iJcN2qYV6XC0tZVCY/je8gGzIp+TQDgN6lWHax/+S+x9i6HNAZlJnXv7yNBgKrFWBfx8qmEKoRMvWyJtwCQaLg9wKNxddfeXm3fvU+9/SEtfPsctTRKb/Z4j33zaafg9Zpv3L7R89xKWZWB0smmUvZmST+V9J9Jx3/knPuX+ANmdpyk8yVNkTRe0gNmdqxzbmjzOSBPyRUM6RKyUrg5N+AnJAhQCqlm6P/7sVdiXbi7e/piz5EgQBW7WYx1EVGqjWWSq7vefqc/ZdXUjK03xBIGDfUB7Tn5HzW3oFcDAP6Tb7Vp6Om75F7dKCmckE2sju1TYzCgH503ragVq+VclYHKkjEp65x72MyOyvL9zpV0m3PuHUkvmtkLkk6U9Gj+lwiERQevHd09MoW37oomZANmqjPTl16bPeR1bQQ2+M/NIkGAIks1E59uGis60D17502SBisLvtB5usY/s5ae3fAlxrpIVuyNZbyqu1K5KrhMWzstVmRwQefpaqTvIADk1QM8uahg++59Ce0KpNK0ESjXqoxcsRlZ8dUN47VfNbOnzewmMxsVOdYm6dW4c3ZEjgHDEh28Rget8QnZhvqAJo9r1t6PX67GYCDhdZUY2IBMnHMPS/pLlqfHEgTOuRclRRMEQFr5zMQHNBDemVbhhOyKV4O6oPN0OQ1W0i7f0FHgKwXKZlhjXTO72MzWm9n6rq4ur1NQo7yqu5KNUL+Wtq6SaTAhG10NFk0YAEAtSzWW9TrunPNc5fWFTu+9Z4rdRmDu9DYtnjdVbS2NMoULyRbPm1pRCc/4HAxj/eLJNyn7b5ImSpomqVPSDyLHvZp3ehbdMFBFLpIHr/EJ2ZnvPVQT5l/ni8AGDBOTYSiYBbMnDZnISteBe3Rdj25oXa2G+vBr/rXjKH3/L3+VcA6JAlSRYY91nXM3OudmOOdmtLa2FuUi4U+ZvuxfM+oR/aT1fgXM5JSYkM32PQCg2nmNZb2KsvpXLlLonu8mHIu23colsVtoc6e3ad3CWXqx/SytWzir4vIW6dpDoHCy6Sk7hHPuteifzeznkqLb3u+QdHjcqRMk7UzxHjdKulGSZsyYQdNPpLUzRQ/Z3v5QQh/DYi83A8ro3yQtUvjL/yKFEwRfUo6TYZIulqQjjjiiOFcJ3/Dqm3jK5Fb97smOIQOw94/YrUsPeVIBM00c06TAR76o3zyyyfN9SRSgGhRirAukkqqX4KiRQV1/yMpY79iJY5q04NXpevQvQ5MD9B0EUOuy6QGeXB1bN3Gm6ibPij32SxuBcsinPQRyl1dS1szGOec6Iw8/KenZyJ9XSPq1mf1Q4d6Gx0h6YthXiZoXHbwmb+p1bd88De0iC1QfJsNQKJl6Q804cnTs+UMag5rbsFkfrt8eSxC0feqfZCNGanzLn9igAFWLsS6KKVUS4K5j/kdjmw+NHas/60qdt6FDG0kYAICnVEVZyclYyXtT2mJu7uj3fqxsRlYaGZOyZnarpI9LOtTMdkj6jqSPm9k0hauxXpJ0iSQ55zab2R2Stkjql3Qpm82gEBbMnqSmB9sViksjXdZ9thbPY0CK2kCCAIXgtbnMFUkbxsQPbvsf+KH0ztuSDpMkBc78tszCxdkLZk/Sgt9sUt/AYGAO1hmJAvgOY10UQ7ov415JgNuPXKOxzQ2x10eTB8VMGABANco2IRtVjNW22Yy5Kx1VxKWRMSnrnJvvcfiXac6/TtJ1w7koINnZO2/SrnHN2r57n3r7Q7q2b54Wz2NAiupEggDFkq43VHI89doMYYjk5hnpmtICFYqxLgot5wmwlYskDU3IRtGeCwAyc+/sU+iBHyUcS5eMLaZcxtyViknB0sirfQFQStHEwNjmBo1tblD9WVfSsgBVjQQBiiXb3lBeCdnkqq/9B/rVF0rsgtEXcr4abAJAMRR8AsyD35fFAkAh5VodW2zV0o+VScHiIymLipbvQBUAMFQ2vaFSJWSTq75S8dtgEwAKLZsv484NKHRP4nxqLglZvy+LBYBCSR67Bv76Qt35Z6cl7WvLNnFFP1Zkq67cFwCkQkIWAAprwexJagwGEo7F94ZKFXe9qr5SYbAJoNalioPR46ENy/JOyErpK3EBlNfyDR2a2b5WRy9cqZnta7V8Q0e5L6lq9a9c5Dl2vfPPTlcse0Yd3T1yGpy4KuV/i0xjbiCKpCwqEglZACi8udPbtHjeVLW1NMoktbU0avG8qZo7vS1t3M22+pXBJgCk/zLev3KR3M7NseN10+flPM6tlmWxQLWJVrGXMxlYK9K1K6iEiat0Y24gHu0LUHH6Vy7Srr29CZt6LRjfQQADgALw6g2VaSIs1RKslsagDj6onp6GAGqCVx9XKfUmKMnHz955U8L75Vt0wLJYoDJVw+ZOlW5g9/Ma+ONtCceSY2mpJq4y9famHyuyQVIWFWP5hg6Nevj76u0f/Ifskq4zJNEnCwDSGc6GL9msTFgwe1JC/0IpXPV19TlTiMsAaoJXH9cFv90kOalvwMWOxY9Z4+NjIYsOUsVkVioA5VUrVezl2mgw2828Uk1cHdIY1MwC9ZmltzcKhaQsKsLyDR1qerBdvW5wJ+9wQjaMGUYA8DacQWG2rWJSVX0RkwH4XbbVr14VcH0hN+T9vMas0YTs1s69Cjk37KIDYjJQmWqhir1cycghm3nNWSgLBD3P9Zq4CtaZ3j7Qr+6ePknDv26/VUWXK5GOzEjKoiKMevj7KROyUdU2wwgAhZDvoDDX3t0swQJQbTyrX3+zSbLBhGv0i3u2mx1KiWPWaKzdvntfXEI2bDhf4InJQOWphSr2Uicjs62Ojec1cbX/QL/27O9LOG841+2nqmiqeisbSVmUXf/KRR4tC4aqphlGACiUVIO/ju4ezWxf6zkTzmaKAOCdXIi2IojX0xdSwEwhN/Q5L9Exa3ys7e0PlbTogKoooPQKUcVe6X93S5mMTB6v2ruPUuBDf5fVa5Mnro5euNLzvHyv209V0X6r6q01JGVRVtFA21AfSDlYlapvhhEACiXVoFDyngknIQsAYbl8GQ85p8ZgIHE5bMASespK4THrDz+4d0isvbZvnqTSfIGnKgoon+FUsfvh724xk5HRhPRpocd1StMuTRzTpLHNDZKGP14t9HX7qSraT1W9taiu3BeA2hUdrO7a2ztkOZckWeT3tpZGLZ43tWL+IQKASrJg9qRwYiCFnr6Qrrlrs2a2r9XqH3xV6154Xbv29koiIQugtuXyZTw6Hm1raZRFHi/59Ala8pkTEo7df9wj+uCBZ2Kvq5s8S/VnXakFsyepMRhIeM90X+CXb+jQzPa1OnrhSs1sX6vlGzqyvtZ0VVEAKpcf/u7mGsuyFU1IXxVcpg83dKi3P6StnXu1a29vQcarhb7uudPbhvybUOqcRbb/TqT6t64Sq3prEZWyKIv4hOzWzr26aPechOdbGoPs6g0A2cqwonbP/j61H7xCkmKD3PWTv6K5xb8yAKhYqTaDie8pKw1+cU9VAZe4EqEhdjw+kZDLsubhVstRFQX4kx/+7qaKZZI0s31t3m0X/mX1Vl3fcnfCsYt2z1HbgUatK+J1DyffUM7e3rn8O+Gnqt5aRFIWJRe/nGv77n1DErKSdPBB9SRkASAiXX+xJau3efZAjLe0dVXC44t2z1EbfaQA1Li509u0/uW/6NbHX1XIOQXMdN6Jh2vGkaNz/uKeTWuYbL/AD7f/n596HQIY5Je/u8mxbLgTSf0rF+nK4GsJx6KraAuZkK6mDRJz+XeiGAlpFA5JWZTM8g0dGvXw99XbH1JDfUATxzTpC52ne55bSbOBAFAO0URsR3ePTIPFsMkD3UzxMjkhW4xBLgD40fINHfrdkx2xDbxCzul3T3ZoxpGjtW7hrKzfp9C9uodbLUdVFOBPfv27O5yJpOQ9Zv7zrala1zsh9nylJaQrRa7/TlRTQrra0FMWRbd8Q4emXXOfGtcuVm9/OFj39od02paP6pDGoOdrCL4Aalm04iBaLZFcBxvfXyxdvEyVkM30OgCoBYXo31iMzRPz7f8X7S/4jds3qiFYp5bGYNl6HQLIXSX0Kc1HqkRgR3dPyl6n/SsXJcTPiWOadFn32QkJWT8kpMuFPrHVg0pZFFU0sZDcHyacGAipIVg3ZCdbgi+AWueVKEgWHQCnqqr42eh71DcweH58QpY4C6DU0rVhKZfhVqQWIyEr5Vctl7x8eM/+PjUGA/rRedPK/nMGKl0lxSc/VjSmarsgebcySI6dkjRh/nVaHPff4ZDGoMykb9y+UUtWb6uIfzMqiV+rqjEUlbIoqiWrt6VIyIZ17+/z5WwgABRTNgmB6Ey4V1XF/cc9omPHvksBM0mJcXfUyCBxFkBJxVf/Ow1+SU+1U3Sp5FtpNNDxbNESslJ+1XJ+2LUdqESVGp/8ZMHsSWoMBlI+H41F7u2/eMbOaPycO71N6xbO0o/Om6Z3+ge0Z38f/01S8GtVNYaiUhZFEZ1tvCq4LOF4fGJACg96/TgbCADFlK7iQBo6Ex4fR5N3/z7v5VNlKn/lB4DaNdyNq4oln0qj5ISCHT5dgfefXfBry3V87Idd2wFJMrObJJ0tabdz7vjIsdGSbpd0lKSXJH3WObcn8twVki6UFJL0Nefc6kJeT6XGJz+J30gq1fj1quAyhR5al3As1WRWsf6bVFJFdCGQR6kOJGVRcOlbFgyivB4AvHklCqKbfbWlGUQmJwsmzL9O64acBQClVakJw1x3pE6OsYEzvy2LrEgoN7/s2g5IulnSTyX9Z9yxhZLWOOfazWxh5PHlZnacpPMlTZE0XtIDZnascy59j6ccVGp88ptognBm+9ohsWhp6yo11A9W0gZO+aps5KiU75XLf5NsE63JLV682ipUmmpLIsMbSVkUXKaWBVJ4+ex3PjGFoAIAHnJNFDjnFLrnuwnH4qsPGNQBKKdKThhmW2lUzHYFhUB/QfiFc+5hMzsq6fC5kj4e+fMtkh6SdHnk+G3OuXckvWhmL0g6UdKjhbqeSo5PfhQfi6IbzgbMNHFMk6TsYme2/01ySbT6pSI6Ombv6O6JFWRI/kgiIz8kZVFw6VoWpKvwAgAMyjZR4PrfUWj192OPrW2qAtPmxh77sTIAQHXxe8Kw0hOyUu6TeUCFOcw51ylJzrlOMxsTOd4m6bG483ZEjg1hZhdLuliSjjjiiKw/2O/xqdJEY86oh7+v3n6poT6giWOaNLZlpOrP+Kes3iPb/ya5JFr9UBGdPGZ3Sc9XYhIZw0dSFgXVv3KRGuoD6u0PB5LkhOy6hbPKdWkAUHXc3tcUeuTG2OO6D3xKdeOOSzjHL5UBAKqXnxOGqRKylbgCgf6CqEJe/UGSc1Xhg87dKOlGSZoxY4bnOV78HJ8q0UDnFp2983fSew+NHct1Iivb/ya5JFr9UBHtNWZPVklJZBQGSVkUTHTQOnFMk7Z27tVFu+fEnmO2EQAKa6DjaQ1svDP2OPCxr8ia3p1wzvINHSk3XGBQB6CU/JgwTE7I3j3+S1oS6ZfIslKgoF4zs3GRKtlxknZHju+QdHjceRMk7Sz0h/sxPlWi5Jgp5b+yIJv/JrkkWotZEV2oSbp0m/xGVVISGYVBUhYFER+AxzY3aP3kr6iN2UYAyFk2A7vQxuVyHc/EHgfmLJQFgkPeJ5ok8MKgDkAlKHfFqdfnn3vkgEKP3pxw3t3jv8SyUqB4Vki6QFJ75Pc7447/2sx+qPBGX8dIeqIsV4i0hm6E+E8yqyvqZ+aSaC1WRXQh24QFzBRyqYu8KXSrTiRlMWxey7rmikoBAMhVNgO7/nu+K8UN2FJVIKRbAsWgDkAlKHfPa6/Pb3qwXR3jmjW2uUGSZIcercBJn9eS9rUsKwUKwMxuVXhTr0PNbIek7yicjL3DzC6U9Iqkz0iSc26zmd0haYukfkmXOufS/0VESRWyOjZXuSZai1ERXcg2YekSsuzNU71IymJY/LDxAQD4RaaBXS4xN11yYPG8qQzqAJRduXteJ3/+0tZVCjlp++59GtvckLAKIZuEKysQgMycc/NTPHVqivOvk3Rd8a4I+Uoel9YdfaLqjptd0msod+uJQm4g1paiHQN781S34taTo6qRkAWAwko3sMs15qZKDrS1NJKQBVARyr0bdvznLG1dFftzb39I9WddmdAWJlPClRUIAGpF6Knfeo5LS52QrQSp/m3IZ5JuwexJagwGEo7xb0v1IymLvOy49Z+07oXXtWbra1r3wuu6e/yXyn1JAOB7qQZwN4+7L+FxNpNgDOwAVLpCfpkdzufHJ2Ql6ZKuMzSzfa2Wb+iIHfOKqdFt4dtaGlmBAKAm9K9cJNf5XMKxWi7OKuR4e+70Ni2eN1VtLY0y8W9LraB9AXK249Z/0tbOvbGeJxd0nq5GdpwFgGHz2rDgF2Pu1cQxzbHH2Q58i7WhAQAUSjF3w87285sebFcoro3fJV1nSBra35aYCqDQyr3RYS5cqE+he9sTjtVyMjaq0P82lLsdA0qPpCxy0r9ykbbv3hdLyEYHruw4CwDDlzywu3ncfZo4ZnDDmVwHvwzsAFSycic6z955k3aNa9b23fvU2x+KjWujkse3xFQAheK10eBlt2/U1Ss26+pzpuQda4qR6C3nZl7xKjWJne+/DZV6PygtkrLIWjQY9/aH/+FIHriy4ywADF90YBeOuYfGjlONAKAalSPR6fZ3K/TgTyRJY5sbNLa5Qcc88gHPcxnfAigGr40OJam7py+hSj8XXonefN8rKjkhGzjpc7JD35PXew1HMe6tnKrtfpA/esoiK/HBuKE+MCQhK7HjLAAUChspAqVjZjeZ2W4zezbu2Ggzu9/Mno/8PiruuSvM7AUz22Zmtberic/1r1wUS8hKkr2rVfVnXVn2/rYAaku6CZ9olX6uvBK9+b5X/8pFnuPRciRkpcLeWyWotvtB/kjKwtPyDR2a2b5WRy9cqf/58de1a29v7Lk9J/8jm8cAQJGQkAVK7mZJc5KOLZS0xjl3jKQ1kccys+MknS9pSuQ1N5hZQPCFIRVfp31TgZO/LInNEQFkJ/57cvKGgLnINOGTT5V+qtfk+l6V0q4gXqHurVJU2/0gf7QvwBDLN3RowW82qW/AaWnrKvX2S1t27pUkTZh/neZGzqP/CQBkJ9ueUSRkgdJzzj1sZkclHT5X0scjf75F0kOSLo8cv805946kF83sBUknSnq0JBeLvGWKr+Xubwug8hVyybnXRofx8qnSH9/SqA6PpF627+X271HowZ8mHKuUsehw763SVNv9IH8kZTHE1Ss2xxKyUU5OZ//pY9oYecxGBwCQnVQD+PUv/0UPbu2Kffm//cg1sQ29pMoZBAM16jDnXKckOec6zWxM5HibpMfiztsROTaEmV0s6WJJOuKII4p4qf5Qzg1Nsp3wYnwLIJ10S85zjR3R86+5a7P27O9LeC7fKn2vRG+271WJ1bHxTpncql899orncT8azn8rVBeSshiiu6cvISErRTf16vN+AQAgpVQD+P9+7BW5yOOrgsu0tdMkhTedqaRBMIAE5nHMeRyTc+5GSTdK0owZMzzPqRWl2tDEK/F79s6bEs4hvgLIV6GXnEcnggo1aZVvxf+Q1i6nL5AFG1KcXR4Pbu3K6XilY3UGojImZc3sJklnS9rtnDs+cmy0pNslHSXpJUmfdc7tiTx3haQLJYUkfc05t7ooV46i8U7IAigV4m51STVQj2ZoojE35Jy2796nCfOvK9GVAUjjNTMbF6mSHSdpd+T4DkmHx503QdLOkl+dzxSyuiwVr8Rv49rFejhQp2MPexcTXgCGrVhLzgtZpZ/Le1V6dWy8auzByuoMSNlt9HWz2PygZvSvXKSADRaBxCdkDx7Bf0qgRG4WcbfiZdroIfp8uvK45EmwL3SeXoQrBZCHFZIuiPz5Akl3xh0/38wOMrOjJR0j6YkyXJ+vFPrLtFf8jU/8HmwHYvG1LzSgrZ17dff4L+X1vgAQVU0bAiYnZO2wSRWbkJVSJ77pwQq/y1gpy+YHtSMamOvMFHJuSIVsMJBNDh/AcBF3K1+mpbjJzyczSf/usSqhjYElUHJmdqvC8fVQM9sh6TuS2iXdYWYXSnpF0mckyTm32czukLRFUr+kS51z3n/REVPI6rJU8Tf6+OpRj2hc/b6E11y0e47aUlTlRhO6Hd09Mg2uYihWiwUA/lUNS84HXt2ogafvSjhWf9aV4VjYvrZi7yubHqzl7F0O5CvfnrLD3vwAlSV+pqxvYMCzZcGbPfSUBcqITWcqSKaluF7PR7W1NGpp6yrtejM8ASaFE7J+rbQA/M45Nz/FU6emOP86SfQZydLyDR3af6B/yPF8Y16q+Bsw0w2H3pNw/NtvnKyugYMleVflJid4k1c2FLrFAgD/8/OS81TtCgrd97sYydFMCfFS9S4HCq3QG31lvfkByYHKkRycr+2bJ6nwvXIAFAWbzpRBpqW4qZ43Sf8zdZ2kZrWMHKHtu/fpC52nq43ZfABVKNWqgZbGoK4+Z0peMS9VfL3h0HsUsMTJrnhe49h0E2iZPg8A/GTIZl5nflsWaVtYyL7fxUyOpkuIl6J3OVAM+SZlh735AcmByrDj1n/S9t371NsfUkN9QHtO/kctGK+MSwMAlBybzlSQTEtxUz1/87j7JB0qSRrb3KAJ86/TixqsKPjG7RtZbgWgaqRKeh58UH3eMc4rvi5tXaWG+oAmjmnSn3a9pS+9Njvh+VTj2GwSrhQlAPCzbDbzKmTf73IlR6txIzDUhnybhLL5gQ8lb17w7C8XaGvnXvX2h4PmBZ2nx2axFs+bqraWRpnCS20Xz5tKggAoL+JuBcm00YPX878Yc68mjmmKPY4OiKMVBR3dPXIarChggxkAfleML8nJ8XVp6yoFzDRxTJPGNjdo1jf+VdefNy2rcewhjcG0n0VRAgA/S07I1p1wjudmXoXcRMurKCHd8VRy3XiRjcDgVxkrZdn8oDokLyO4KrhMHd2Dz0eXeEVnsdYtnEUSFigT4m7ly9TXKvn5m8fdp4ljmjW2uUFSYoUCy60AVKtCbvAVFR9frwoui1XIjm1uiMXWbHs+mlcDoAjaygDwq9CmO+V2PJ1wzCsZG5XNJlrZim8jk3w8W/m0QCjkPQCllDEpy+YH1SH+S/9Sjx2/41HiD5QXcdcfMn3pjz4frlI4NHa8mEvGAKCSFOtL8rnvP0xn77xJ0mGxY+kSDql07/fexNYkrVs4K8+rA4DyyaZdQbJMxQa58ErIpjvuJZ+ChULeA1BKhd7oCxXAa7fD6Jf7TAlZiRJ/AEgll91knRtQ6J7BXLkddqwCM84bcl4xKskAoBIU40ty6Om75F7dmHAsn4SsRPwFUD1c/wGFVn8v4Vh8bMw0hs12hUEmbSnialsOcTXfgoVC3QNQSiRlq0yqUv9DGoP6XtOKhHMv6TpDpsRt2inxBwBvyzd0aMFvN6kvFI6aHd09WvDbTZISl1It39ChG1Y/rW8GV8aW1Y7/8FzVHfVXnu/LcisA1ayQX5KH7B7+1xfqzj87LWlfm1fSl/gLoBpkqo7Npx1AvgoRV5kwQy3Jd6MvVKhUpf4/POTuhD4ul3SdocZgQJ/70BFs6AUAWbjmrs2xhGxUX8jpmrs2xx4v39Chf7/zEX0zuFKS1Nsf0v/ZdrxW7Bmf8n3nTm9jc0UAyCA56VB/1pW6889uWBslEn8B+E3yBlg7bv2nhOcDJ395yOqBdO0ACq0QcTXThrpANaFStsp4lfQvbV2lvpA0Zfwh2r57n77QeTqbFwBAjvak6D0Yf/yhB1brG+96Kvb4m6/P0j53UMZNu1huBQCpeSVkpcJslEj8BeAX8RWv33/3Wh1S9462doYLr+I3O0yWatl/R3ePZua50iCd4cZV+sOilpCUrTINwTr19A3EHkd7yAbMNLa5QRPmX6cXy3VxAFDFQhuW6Yy6wYTsl7vmyCk8UGbTLgC1JJf+25mkSshKbJQIoLZEJ6Li94kJOac/vub0yfmpe2unagdgUux4MVsa5IMJM9QKkrJV5p3+oQlZKRys890EAQAgtTQG1d0ztFq2pTGo/tXtUn+fGuoD6u0PDdlEkR5YAPzOK9EqDa1kklSw3oXpErISfQcB1JbQW69raevDCcei+8R8Ms3rvPq8Ju8tI+W+0gDA8NFTtsoMRCJrfEJW0pAEAQAgN1efM2XIP5p1ku4+9n+k/nCyduKYJl3WfXbCOfTAAuB30SWz8b1bF/x2kxb8ZtOQfq5Xr9g87N6FzrmMCVmJvoMAakf/ykX63ph1Ccei3/EzTUR59XlNTshGsdIAKC0qZX0uuWpB8k7ImteLAQA5CQRMA3Gbff17672SmmOPJ8y/TosLuGwXACqBV+/W5I0PpXDyNfm8qGy/6A88/4gG/vRQwrFUq73oOwigFkQnqSaOadLWzr36+92nqU/hCalsJ6KS2wHMbF/LSgOgApCU9bH4Rt9SuEIhVYXsyBGBIa8HAGRvyeptCUmIpa2r5CRt3vmmtu/epz0n/6Pmih5YAKpPISqnsvmin1wdWzfjPNUddmza1xBzAVSSQvbUDj30M7m3/xJ7PLa5Qesnf0VjCvD+Xi0NWGkAlB5JWR9LrlpI17Jg/wHvqgUAQHbikxLJ8faCztPVWEGbIwBAIaXq3epl1MigevsGcv6in027gmwVMikCANnyKpoqVE/tuhPOUd2EE2IFAMPFSgOgMpCU9bF0CQI2mQGAwoomJVLF256+kC67faOWrN7GoBZAVfGqqAoGTHJS38DgCoLGYEDf+cQUSbl90S90QrZQSREASMVr8ser1Uuum2e5v7yi0KO3JBwr1obd5VppwMQZMIikrI+lShB8OSkhyzIEALWo0AO+BbMnqenBdsW3UfTaRJEEAIBqk6qiyutY9Nxs418hE7LR6xluUgQA0vGa/PnG7RuHvXlWcjyUipeQLRcmzoBEJGUrXLqkgleC4LLus/W5D7Xpwa1dzDwBqFnFGPCdvfMm7RrXrO2796m3P+SZkI0iAQCg2qSqqBpOnCt0QlZKnfxgR3EAheI1+ZMqIStlXrXqnFPonu8mHKu2ZGwUE2dAIpKyFSzVDNxlt29UW0ujbj9yjRSXILi2b54WzyMBCwCFGPDFT4rdPO4+TRzTpLHNDRrb3KC7x39JjUlLeZORAACA1IqRkJVS97+llReAQslljJdp1Wr//T+QDuxPOFatCVmJiTMgGUnZCpZuBu6q4DJt7TRNHtesme89VPVnXanZpb9EAKhI+Q74oonYju4emcIxd2nrKvX2S1s790qSJsy/TnMj50fP9UICAACGGti5WQMbliUcyzUBkWklGTuKAyimbDc/bMuwajV5cirw8UtlB48uyDVWKibOgEQkZStYquRBtIdsyDlt371PE+ZfV8rLAoCKl8+AL3l1QjQhGxVyTue9fKrWRR5Hl/Imv04iAQDA34q1CcuQ3cSnnqW6Iz6Q87Wla0/DjuIAis1r8idZW0uj1i2c5fmc69qu0BO/TjhWzdWx8Zg4AxKRlK1gXkmF5E29Lug8XS+V8JoAwA/yGfAlr05IjreXdJ0h09BELwkAANUkl57cuSRvC9WuIJv2NOXaURyAP+U6ERU/9otfXRWVbsw5ZHLq2JNVd8zHhnsLRVPoSTrGzUAikrIVLDmp4JUgCJiV49IAoKLlM+CLX53gFW+l1JW2JAAAVItse3LnkrwtZP9Y+hECKKR8N4eNH/tlk7h0oT6F7m1POBaNhcVanTBcxdg4N/raSrg/oBKQlK1g8UmFq4KJvbeiCYKQS7fPIwDUrlwHfNHVCakSsiytAlALUiU3O7p7tHxDR8L4NJvkbaE39KIfIYBCKsTmsJnGnMlxUEpMyBYj8VkIhfjZAEiPpGwFSp4pu/3INfrTrjr1DQxIGkwQSFJLY7BclwkAVWXB7ElqerBdobi5ri9H4q3XRg2VWtUAAMORbgOb+ERBNhWrhU7ISvQjBFBYxa6+H7KZ1+zLZfUjYo8rOfHJygSg+EjKVpjkmbKrgsu0tXOwRUF8QlaS6F4AAMO3fEOHRj38ffU6J5PJyenavnn60XneidZKrmoAgOFIt4FNfKIgU8VqMRKyEv0IARRWsarvBzqe1sDGOxOOecXBSk58sjIBKD6SshVk+YYOffOOTbGWBNEltNHHyQlZSere31e6CwQAH8m2knX5hg41Pdiu3kisdXK6rPtsLZ6X+kt+JVc1AMBwRGPYZbdv9Hw+mihIVbF61cdGFy0hG3+NxFoAhVCM6vshm3l94FOqG3ec57mVnPhkZQJQfHXlvgCERauukhOyUV4JWakygjUAVJpoTO3o7pHTYCXr8g0dQ84d9fD3E/pzX9J1RizBmkolVzUAwHDNnd6mthRjzOjYc+70Ni2eN1VtLY0yhdu8rJ7ymE79y12xc+umnlnwhCwAFJJXLFs8b2peEz/uwH7PSalUCVkpnPhsDAYSjlVK4rOQPxsA3qiUrRDxVVdeCdlRI4Pq7RtglgoAspCqkvWauzYP6dnd2z94XvwEWKqeilJlVzUAQLbSrSjIpkIqvmI1nIgIJxZ27e3V+S/PUscjuzS+ZS3tBQBUtEJU3ycnY+3QoxU46fNZfbZUuS1ZWJkAFBdJ2QoRra7ySsg2BgP6ziemSKrcYA0AlSRVxeqe/X3aE2n7clVwmbbsDDfmfqnvEC3u/kjCuYE0TbtZzgXA7zL1xk6XKPDalHZsc4OkcEL2tC0fVU9fr+f7AkC1GbKZ15nfluWw+QuJT6B2kZStEC0jg2o/eEXCsUu6zlCdKWGJAMEaADJLt3u4NDgB5uS0dv+Ruv3tocvK4lsaJKv0qgYAhWFmL0l6S1JIUr9zboaZjZZ0u6SjJL0k6bPOuT3lusZ8ZdMb2ytRkG5T2rHNDTrv5VPV09eT9n0BoBoM/PkxDTx3f8KxUrZsyXb/hFzPBVA6JGUrxJKmuxSK+/4fXULbUF9HsASAHKXaPTygAd3Qujr2+D/2vl+PveMdY9NVykpUNQA15BTn3OtxjxdKWuOcazezhZHHl5fn0vKXb29sr5ZbIee0ffc+TZh/nXY+sjKv941H8gBApRtSHXvyl2Xvai3Z52da7ZDvuQBKi42+KkD/ykVDNpmJ2t83UI5LAgBf89qYoMkOJCRk2/d8OGVCVkpfKQugpp0r6ZbIn2+RNLd8l5K/VD2wM/XGTtVy6wudp0sKr/7ykup4slw2agSAUnO9+zw38yplQlZKv9phOOcCKC0qZcssOaDHJ2QBAPmLr2R1+17Xg7+4VtHx6D++cYreHAj3P6wzacAj/5pq53EANcVJus/MnKSlzrkbJR3mnOuUJOdcp5mN8XqhmV0s6WJJOuKII0p1vVnLtzf2pBbTN4L3JBy7pOuMWMxMNZ+V7TxXNm0VAKDYvCr2P7Hvbrm9u2Pn1H3g06ob976yXF8uqx3yPcDd0wAAG6RJREFUXRkBoPhIypZRfEI2WFenL702e8g5LY3ZVRUAQK3Jdnmr2/2CQn+8Vcce9i5t2blXl3adpr7IDuHBgOm8vzpcv3uyg027AHiZ6ZzbGUm83m9mW7N9YSSBe6MkzZgxoyil98NZ5p9Pb+zQ03fpF0c+pq2dppBz+q+3jtcfeg9PiJlv9vR5vra7p08z29dm/CySBwDKbehy//1qerBdneOaY5salrJ3rJdU+yd4rXbI5VwApUVStgyWb+jQqIe/r97+kBrqA5o4pkl7P365gr/ZpL64cq1gnenqc6aU8UoBoDJl2xtr4MXHNLAlvAHD2OYGrZ/09xpz35+GJAVmHDma/oUAhnDO7Yz8vtvMfi/pREmvmdm4SJXsOEm7075JkRSiR2AuvbGjxQTRhMTfvnyKXu19R21JMTPVl3+LXGOmayV5AKDc4iv2ZzW+pPOanlPISdt379O4Y6cp8MHPlPkKc1vtkO/KCADFR1K2xJZv6FDTg+3qjazh6u0P6bQtH9XiydKSz5xAUgAAspDN8taBbQ9q4IU/xJ6vP+tKzZU09wMThrwfm3YBSGZmB0uqc869Ffnz6ZKulbRC0gWS2iO/31mO6yvlMv/kdlsT5l+nh1Oc6/Xl3xTuA5HNtZI8AFBuqXpnX7JrlrZ+8NxyXNIQuax2yGdlBIDSIClbYqMe/n4sIStFe8iGB6XrFs4iMAJAFjItbw098Wu5ru2x4+VeYgbAlw6T9Hszk8Jj5l875+41sz9KusPMLpT0iqSylEwVepl/fCuElpFBORduRXDzuPs0cUxT1kt2vb78e1W+prpWkgcAym3yKOmy+qSEbFzv7ELLtxVNLkUFFCAAlYmkbJHFB9ibx92n3v7BWf/4Tb3okwUA2Uu3vLX/viVSX2/sWP1ZVw6r7yKA2uSc+7OkEzyOvyHp1NJfUaJCLvNPboWwZ3+4L+zS1lXq7Ze2du6VFK6QzUbyl/+Z7WtzulaSBwDKZeClP+rnR/xvrHf2VX85Wa+FDi5axX4hWtEA8K+6cl9ANYsG2I7uHv1766qUCVmJPlkAsmNmL5nZM2a20czWR46NNrP7zez5yO+jyn2dxbZg9iQ1BgMJxxqDAd1+5BrPhGw0FjsNDnaXb+go8VUDQGbLN3RoZvtaHb1wpWa2r00Zq1LFwXySBl6tEOKX7Yac03kv55+HLuS1AkAxOOcUeugGDWy+V2ObGzR5XLOu7Zun3aGD1dbSqMXzphYlSZquFQ2A6kelbBFEK7I6UvSi+XJSQpZBKYAcneKcez3u8UJJa5xz7Wa2MPL48vJcWml4LW+9/cg1seW10uAS21L2XQSA4cilYqqQy/zjV2wFFdJPW+9LeP6SrjNkyn9VFy0JAFQy55xC93w39jjw0Ys1ofkwrSvBZ6drRTPclV6sFAMq37CSsmb2kqS3JIUk9TvnZpjZaEm3SzpK0kuSPuuc2zO8y/SP5MH0kObgkYRsW0sjwRFAoZwr6eORP98i6SFVeVJWSlzeGt6EZmhCVip830UAKJZcJ5EKtcw/2gph2ojX9PeHPBU7vvTN6XrqwNjYOcNBSwIAw2FmcyT9WFJA0i+cc+0Fe/OBkOzg0VJdQHUfvUSRXuIlkaoVzSGNwWG1NaAtAuAPhaiUremKrfiq2ICF+85EpUvIrls4q6TXCaBqOEn3mZmTtNQ5d6Okw5xznZLknOs0szFeLzSziyVdLElHHHFEqa636JJ3BU/ehKaQfRcBoJjKNYm0YPYk9a75V42yt2PHvtp1uvoUbjnAqi4A5WRmAUk/k3SapB2S/mhmK5xzWwry/oF6BT5+aSHeKmcLZk9KSJ5K4ZhrpmGt9GKlGOAPxegpe67ClVqK/D63CJ9REeL7FErKKiHLoBbAMM10zn1A0hmSLjWzk7N9oXPuRufcDOfcjNbW1uJdYQllSshK9DIE4B+pJouKPYl09s6b9Ndt9WqoD8fKhW+fo4MbG2RSUXspAkCWTpT0gnPuz865A5JuUzjv4Htzp7dp8bypamtpTIi53ZENF5NlO0nHSjHAH4ZbKVvTFVtes09S+gpZWhUAGA7n3M7I77vN7PcKD1JfM7NxkZg7TtLusl5kiWSTkJXoZQjAP1JVTBVzEikaS8c2N2hsc4Pqz7pSs4v2aQCQlzZJr8Y93iHppPgTCp1fKGU/Vq/2LvF71MTLdpKOlWKAPww3KTvTObczkni938y2ZvvCSAL3RkmaMWOGy3B6RfKaZfJKyDYGA1QYABg2MztYUp1z7q3In0+XdK2kFZIukNQe+f3O8l1laWSbkI2ilyEAPyj1JFKqWJptMoJNZACUiFeT14QcQiHzC5XQj3W4k3SnTG7Vrx57xfM4gMoxrKRsrVdsJc8+eSVkqY4FUECHSfp9ZPOBekm/ds7da2Z/lHSHmV0o6RVJnynjNRZdrglZAPCTVJNIhU6ApkvIZpOMqISkBYCasUPS4XGPJ0jaWawPq4R+rPlO0sXveePlwa1dBb9WAPnLOylLxVbi7JV3hWwdG3oBKBjn3J8lneBx/A1Jp5b+ikqPhCyAWlTIBKgbCCm06p9jj23MexX4q/mxx9kmIyohaQGgZvxR0jFmdrSkDknnS/rbYn1YpfRjzXWlV/K/FV7oKQtUluFUytZ8xdbc6W1a//Jf9NHtSxOOR3vI9vQNlOOyAKAqkZAFUKsKlQB1b+5S6A8/jz0OnPQ52aHvSTgn22REpSQtAFQ/51y/mX1V0mpJAUk3Oec2F+vz/NqPNdWeN/Eq/R6AWpN3UraWK7bilwSk2tQLAFA4JGQB1LJCJEAHNt+rgZf+GHscmH25rH7EkPOyTUb4NWkBwJ+cc/dIuqcUn1WOTRcLIdO/CX64B6DW1JX7AvwmuiQgm4TsyCA/XgAYLhKyAGpdqkRntgnQ/pWLEhKy9Wdd6ZmQlcLJiMZgIOGY1xf5bM8DAL+ZO71Ni+dNVVtLo0xSW0ujLzbuTvdvgl/uAag1w9roq5YkN8zOpkL2oKSBKgAgNyRkAWB4VVu5xtFsN5fJdxMaAPCDXPu5VoJU/1aQjAUqF0nZLCQ3zM62ZUH3/r6iXxsAVCsSsgAQlm8CNN84mm0ywo9JCwCoVkyWAf5DUjYL8Q2zc+khS08tAMhPfCLBDhmrwF//nzJeDQCUX64JUCa2AKD2MFkG+AtJ2SzszKFlQRQ9tQBgqGgrmHSz9/GJhLqjT9KKd6ZoSftaZvwBIAvOOYXu+W7ssbWMV2DmhZ7nZhOTAQAAUBwkZbMwvqVRVwWXJRyLJmTbIgNYiWUCAJBOciuYju4eXbHsGUmDy60SErJTz9SKN8ZmfA0AIMzte0Oh/7kh9rjuA59S3bjjPM/NJiYDAACgeEjKZuH2I9doa6cp5JykcELWq2E2A1gASC2+FUxUT19IS1Zv07nTxidUdgVO/FtZ60Qt+fXalK8h5gLAoIE/PaSB5x+JPQ6cvkAWbEh5frqYTHwFAAAoPpKyHuKXct087j5NHNOkyeOatX33Pn2h8/RYdSwDVgDIXrQVTLLXut9OTMh+9GJZ82FpX5PqOADUonz6x5YqvtIiAQAAwBtJ2YjogLGju0cmySncQ7a3X9rauVeTxzXrY1//sV4s94UCgE+Nb2lUR9KX/Ubr0/WHPqB1LwQ0cUyT2j55hayhKe1roscBAPlv6FWK+EqLBAAAgNTqyn0BlSA6YIwOTKMJ2aiQczrv5VPLdHUAUB0WzJ6kxmAg9viQul5df+gDkqTe/pDO2PIR3fncm2lfI7GRIgBE5ZuQlUoTX9O1SAAAAKh1VMpq6IAxPiErhXvImlgqCwDDEa2KWrJ6mwbe2q2rR/8h9twlXXNiz3n16mbpKwAkGk5CVipNfKUFDQAAQGo1mZRN7m0Vv3TLKyErsVQWAAph7vQ2nXv4Ad13y7LYsWiclby/qM+d3kYSFgAinHMJfbjtyA8qcPyZeb1XseMrLWgAAABSq7n2BfGtCpyUVUKWpbIAUBgDrz2v0OO/UkN9eMlsfEJW4os6AKTjDuxP3BjxpM/nnZAtBVrQAAAApFZzlbJeva2koQnZL0cSBW0slQWAghh45UkNPHOPJGnimCadtuWjkgbjMV/UASA19/YbCj10Q+xx4PQFsmBDGa8oM1rQAAAApFZzSVmvpbHJCdlr++bpR+cxYASAQhnY9qAGXhjsITth/nVanNRKhi/qAOBtoPM5DTz129jjXPvHlhMtaAAAALzVXFI2mx6yL7XPKvVlAUDVSk7IRpMJfFEHgMwGtqzWwItPxB7HJ2ST90mIrjZgwgsAAKDy1VxSdsHsSbpi2TPq6Qt5JmQDZmW6MgCoPqFNd8rteDr22E/VXQBQbgMvPjaYkG1sVv2sr8eei+6TEG3L1dHdowW/3SQ5qW/AxY5dsewZSSIxCwAAUGFqLikbHZA2PdiukBs8Ht1sZv5Jh5fjsgCg6kQTsrv29mr77n36QufpGv/MWqq2ACALoY3L5TrCCdW6E85R3YQTEp732iehL35wG9HTF9KS1duIuwAAABWm5pKyknT2zpukSWO0ddde7ezu1cVdcxQw0/yTDtd3504t9+UBgO+FnvyN3K6t2rW3V1s79+qi3XMkUbUFANkYeP5/YgnZwMlflr2rdcg5XvskpJLLuQAAACiNmkvK9q9cFPvz5LHNOv7CJXqpfJcDAFUntO4mue4OSdL23ftiCdkoqrYAID07pE0yU+Bv/q9sxEjPc5L3SUhnfEtjIS8PAFBhvHqMM9YGKl9duS+glKIJ2V17e7Xuhdd1zCMf0Mz2tVq+oaPMVwYA1aH/gR/KdXfE4uwFnad7nhdftbV8Q4dmtq/V0QtXEpMB+IKZzTGzbWb2gpktLPj7j3mv6s/8dsqErBTeJ6ExGEg4FgyYgnWJ+yM0BgOxDcAAANUn2mO8o7tHToMr0xhTA5Wvqitl42eLbh53nyaOaZIkltICQBHET3zFx1kv0aotr41qojFZYgdxAJXHzAKSfibpNEk7JP3RzFY457aU8jqi8TA5TnodI3YCQPXy6jHOyjTAH6omKRufgG0ZGVRvX0g9fQOSpKWtq9TbL23ZuVf1dcZSWgAosPjWME+/1qevpEnIxldtpRpEXnPXZvX2DXgma4nVAMrsREkvOOf+LElmdpukcyUVNSmbammqV0wkTgJA7UjVN5x+4kDlq4r2Bcs3dGjBbzfFyvX37O9LSMhGOTl96bXZnu9BwAKA/PSv/XHsz9YyXpfu+ljKc9taGrV43tRYwiBV7A3Hce8ZfwAoszZJr8Y93hE5lsDMLjaz9Wa2vqura1gfyNJUAEAqqfqG008cqHxVkZS95q7N6gu5IcfjE7KSdEnXGSnfg4AFALnrv+e7Us9eSVLd5FkKzLwwZTxta2nUuoWzEiq4co29TKABqADmcWzIQNQ5d6NzboZzbkZra+uwPjDd0lQAQG3z6jFOP3HAH6oiKbtnf9+QY+kSsgQsABi+0KO3SC6ch6h7/9mqmzhTUm4Dw1TntjQGPT+TCTQAFWCHpMPjHk+QtLOYH8jSVABAKnOnt2nxvKlqa2mUaejKNACVq2p6ysZLl5BtaQzq6nOmsAECAAxD6Om75P7yiiSp7gOfVt2498WeS7X5TLq+h14b1cRvACYxgQagYvxR0jFmdrSkDknnS/rbYn7g+JZGdXgkYJmoAgBIStljHEBlq4qkbEtjUN094WrZdAnZYJ3p6nOmELAAYBhCT/1WrvM5SVLglK/KRo4ack4ucTbduUygAag0zrl+M/uqpNWSApJucs5tLuZnLpg9yXOi6pTJrZrZvpY4CQAA4ENVkZS9+pwp+r+3b9S/eSRkA2YKOac2BqoAMGyhx38l9/qLkqTAqV+XNTQX7bOYQANQqZxz90i6p1Sf57Wq4JTJrfrdkx2xRG1086/48wEAAFC5qiIpK0nfaHki4fFX3jhT1593AoNSACiQ0B9+LvfmLklS4G/+r+ygg8t8RQBQO5Inqma2r025+RfjXwAAgMpXFRt97V7zCx0bfEOStPnAobqk6wyFBpyuuauoK8kAoGaEnvj1YEL29AUkZAGgzNj8CwAAwN98WSm7fENHbPnWV1r/pPdblyTp3v3v0e/fHtwEZs/+vnJdIgBUlxEjJUmB2ZfL6kckxOFDGoMyk7r399HTEABKhM2/AAAA/M13lbLLN3ToimXPqKO7Rx8+aIfer+2ShiZkAaCWmNkcM9tmZi+Y2cJCv39g2lzVn3VlLCEbjcNOUndPn/bs75PTYE/D5Rs6Cn0JAIA4C2ZPUmMwkHCsMRjQgtmMhwEAAPzAd0nZJau3qacvpI8ctEMXNIc3M7i++688E7ItjcFSXx4AlJyZBST9TNIZko6TNN/MjivW50XjcCrRnoYAgOKZO71Ni+dNVVtLo0xSW0ujFs+bykoFAAAAn/Bd+4Kd3T0aof6EhOxzfYcOOS9YZ7r6nCmlvjwAKIcTJb3gnPuzJJnZbZLOlbSlEG8e36og1XLZZNmcAwBILzn+JreHSd78CwBQeJliMQDky3dJ2fEtjXqte5/u33+0njnQqm197449Z5KcwpUCBEoANaRN0qtxj3dIOqkQbxxtVRCtjO3o7onF2nQCZoX4eACoWV7x94pl4aIExrgAUBrEYgDFVLT2BcXqb7hg9iQFgyP027cnJyRkpcGE7LqFswiQAGqJVwZ0SN7UzC42s/Vmtr6rqyurN/ZqVeBSfGC8kMuUtgUApOMVf2kPAwClRSwGUExFScoWs79htH9WKjtZMgug9uyQdHjc4wmSdiaf5Jy70Tk3wzk3o7W1Nas3ThVTo5NgqaR7DgCQWar4y1gXAEqHWAygmIpVKRvrb+icOyAp2t+wIOZOb0v5hX88iQAAteePko4xs6PNbISk8yWtKMQbp4qp0VUJ1583jd2/AaAIUsVfxroAUDrEYgDFVKykrFd/w4R+Avkso423YPYkEgEAIMk51y/pq5JWS3pO0h3Ouc2FeO9MsZbdvwGgOBjrAkD5EYsBFFOxNvrK2N/QOXejpBslacaMGTk3H4x+4WcXRACQnHP3SLqn0O+bTaxl928AKDzGugBQfsRiAMVUrKRsVv0Nh4tEAAAUH7EWAMqD+AsA5UcsBlAsxWpfULT+hgAAAAAAAADgZ0WplHXO9ZtZtL9hQNJNhepvCAAAAAAAAAB+Vqz2BUXrbwgAAAAAAAAAflas9gUAAAAAAAAAAA8kZQEAAAAAAACghEjKAgAAAAAAAEAJkZQFAAAAAAAAgBIiKQsAAAAAAAAAJURSFgAAAAAAAABKyJxz5b4GmVmXpJezPP1QSa8X8XLKhfvyF+7LX7K9ryOdc63FvphKkEPcrfX/J/yG+/IX7qtG4i5jXUncl99wX/7CWDcOMVdS9d6XxL35UbXel5T53tLG3YpIyubCzNY752aU+zoKjfvyF+7LX6r1vkqhWn923Je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"text/plain": [
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"from plot_material import reg_corr_plot\n",
"\n",
"reg_corr_plot()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Joint Confidence Region"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Joint confidence region** are the joint distribution of regression coefficients, it is theoretically an ellipse. It shows the distributed location of the coefficient pair. \n",
"\n",
"Here is a Monte Carlo simulation, we set $\\beta_1 = 3$, $\\beta_2 = 4$ and $u\\sim N(0, 10)$, run $1000$ times then plot the estimates."
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"tags": []
},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"beta1, beta2 = 3, 4\n",
"beta1_array, beta2_array = [], []\n",
"for i in range(1000):\n",
" u = 10 * np.random.randn(30)\n",
" X2 = np.linspace(10, 100, 30)\n",
" Y = beta1 + beta2 * X2 + u\n",
"\n",
" df = pd.DataFrame([Y, X2]).transpose()\n",
" df.columns = [\"Y\", \"X2\"]\n",
"\n",
" X_inde = df[\"X2\"]\n",
" Y = df[\"Y\"]\n",
"\n",
" X_inde = sm.add_constant(X_inde)\n",
"\n",
" model = sm.OLS(Y, X_inde).fit()\n",
" beta1_array.append(model.params[0])\n",
" beta2_array.append(model.params[1])\n",
"\n",
"fig, ax = plt.subplots(figsize=(10, 10))\n",
"ax.grid()\n",
"for i in range(1000):\n",
" ax.scatter(\n",
" beta1_array[i], beta2_array[i]\n",
" ) # no need for a loop, i just want different colors\n",
"ax.set_xlabel(r\"$\\beta_1$\")\n",
"ax.set_ylabel(r\"$\\beta_2$\")\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"But why the joint distribution of coefficient has an elliptic shape? If you take a look at any linear regression plot, it wouldn't be difficult to notice that the high slope coefficient $\\beta_2$ would cause low the intercept coefficient $\\beta_1$, this is a geometric feature of linear regression model.\n",
"\n",
"And from the plot, we can see the range of $\\beta_1$ is much larger than $\\beta_2$ and even include $0$, especial the data points are far away from $0$, $\\beta_1$ can have erratic results, that's also the reason we don't expect to interpret $\\beta_1$ most of time."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Stochastic Regressors"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"One of G-M condition is $\\text{Cov}(X_i, u_i)=0$, assuming $X_i$ is non-stochastic. But we commonly encounters that $X_i$ is stochastic, for instance you sample $10$ family's annual income, you have no clue on how much they are earning eventually, therefore assuming they are stochastic would be more appropriate. \n",
"\n",
"If $X_i$ is stochastic and distributed independently of $u_i$, it guarantees that $\\text{Cov}(X_i, u_i)=0$, but not vice versa."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Maximum Likelihood Estimation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In more advanced level of econometric research, **maximum likelihood estimation**(MLE) is used more often than OLS. The reason is that MLE is more flexible on assumption of disturbance term, i.e. not assuming the normality of the disturbance term. But it requires you to have an assumption of certain distribution, it can be exponential or gamma distribution or whatever. \n",
"\n",
"We will provide two examples to illustrate the philosophy of MLE, one is to estimate simple linear regression with MLE and the other is to estimate a mean of an exponential distribution."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## MLE for Simple Linear Regression "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Once you have a dataset, you should know the data are just observations of random variables. For instance, you are about to collect $100$ family's annual income data for year 2021, each family's income is essentially a random variable, hence it follows some distribution, it can be normal distribution or skewed gamma distribution. \n",
"\n",
"Once the data is collected, the observances are done. Each data point, i.e. $Y_1, Y_2, ..., Y_n$ is just single realization of its distribution, the joint distribution of all random variables is\n",
"$$\n",
"f\\left(Y_{1}, Y_{2}, \\ldots, Y_{n}\\right)\n",
"$$\n",
"We assume a simple linear regression $Y_{i}=\\beta_{1}+\\beta_{2} X_{i}+u_{i}$ model to explain $Y$, then the joint distribution is conditional\n",
"$$\n",
"f\\left(Y_{1}, Y_{2}, \\ldots, Y_{n} \\mid \\beta_{1}+\\beta_{2} X_{i}, \\sigma^{2}\\right)\n",
"$$\n",
"We also assume each family's income is independent of rest, then joint distribution equals the product of all distributions.\n",
"$$\n",
"\\begin{aligned}\n",
"&f\\left(Y_{1}, Y_{2}, \\ldots, Y_{n} \\mid \\beta_{1}+\\beta_{2} X_{i}, \\sigma^{2}\\right)=f\\left(Y_{1} \\mid \\beta_{1}+\\beta_{2} X_{i}, \\sigma^{2}\\right) f\\left(Y_{2} \\mid \\beta_{1}+\\beta_{2} X_{i}, \\sigma^{2}\\right) \\cdots f\\left(Y_{n} \\mid \\beta_{1}+\\beta_{2} X_{i}, \\sigma^{2}\\right) =\\prod_{i=0}^nf\\left(Y_{i} \\mid \\beta_{1}+\\beta_{2} X_{i}, \\sigma^{2}\\right)\n",
"\\end{aligned}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we need to ask ourselves, what distribution does $Y$ follow? It's the same as asking what's the distribution of $u$? It's reasonable to assume $u$ follow normal distribution, so are $Y$'s.\n",
"$$\n",
"f(Y_i)= \\frac{1}{\\sigma \\sqrt{2\\pi}}e^{-\\frac{1}{2}\\frac{[Y_i-E(Y_i)]^2}{\\sigma^2}}=\\frac{1}{\\sigma \\sqrt{2\\pi}}e^{-\\frac{1}{2}\\frac{[Y_i-\\beta_1-\\beta_2X_i]^2}{\\sigma^2}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then the joint distribution is the product of this PDF function\n",
"$$\n",
"\\prod_{i=0}^nf\\left(Y_{i} \\mid \\beta_{1}+\\beta_{2} X_{i}, \\sigma^{2}\\right)=\\frac{1}{\\sigma^{n}(\\sqrt{2 \\pi})^{n}} e^{-\\frac{1}{2} \\sum \\frac{\\left(Y_{i}-\\beta_{1}-\\beta_{2} X_{i}\\right)^{2}}{\\sigma^{2}}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Once we have joint distribution function, in a frequentist view, the observed data is generated by this distribution with certain parameters, in this case $\\beta_1, \\beta_2$ and $\\sigma^2$. So the fundamental the questions are what those parameters are? How to find them?\n",
"\n",
"We give a name to the join distribution function above - **likelihood function**, which means how likely a set of parameters can generate a set of data. We denote likelihood function as $LF(\\beta_1, \\beta_2, \\sigma^2)$.\n",
"\n",
"The MLE, as its name indicates, is to estimate the parameters that in such manner that the probability of generating $Y$'s is the highest. \n",
"\n",
"To derive a analytical solution, usually we use log form likelihood function\n",
"$$\n",
"\\ln{LF}=\\ln{\\prod_{i=0}^nf\\left(Y_{i} \\mid \\beta_{1}+\\beta_{2} X_{i}, \\sigma^{2}\\right)} =-\\frac{n}{2} \\ln \\sigma^{2}-\\frac{n}{2} \\ln (2 \\pi)-\\frac{1}{2} \\sum \\frac{\\left(Y_{i}-\\beta_{1}-\\beta_{2} X_{i}\\right)^{2}}{\\sigma^{2}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Take derivative with respect to $\\beta_1$, $\\beta_2$ and $\\sigma^2$ and equal them to $0$ will yield the **maximum likelihood estimators** for simple linear regression with assumption of normally distributed disturbance term\n",
"\\begin{align}\n",
"\\hat{\\beta}_2 &= \\frac{\\sum_{i=1}^n (X_i - \\bar{X})(Y_i-\\bar{Y})}{\\sum_{i=1}^n(X_i-\\bar{X})^2}=\\frac{\\text{Cov}(X, Y)}{\\text{Var}(X)}\\\\\n",
"\\hat{\\beta}_1 &= \\bar{Y}-\\beta_2\\bar{X}\\\\\n",
"s^2 &= \\frac{1}{n} \\sum_{i=1}^{n}\\left(Y_{i}-\\left(\\hat{\\beta}_{1}+\\hat{\\beta}_{2} X_{i}\\right)\\right)^{2} = \\frac{1}{n}\\sum_{i=1}^n e_i^2\n",
"\\end{align}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"MLE $\\hat{\\beta}_1$ and $\\hat{\\beta}_2$ are exactly the same as OLS estimators, only the $s^2$ differs from OLS estimator $s^2 = \\frac{\\sum e^2_i}{n-2}$. The MLE $s^2$ is biased in small sample, but _consistent_ as sample size increases."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We'll experiment with MLE with simulated data. Generate a data for OLS estimation and print the estimated coefficients."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"```statsmodel``` library doesn't have a direct api for maximum likelihood estimation, but we can construct it with ```Scipy```'s ```minimize``` function, so we define a negative log likelihood. The reason of negative function is because ```Scipy``` has only ```minimize``` function."
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"tags": []
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"outputs": [
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