{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Bayesian estimation of a univariate first-order autoregression\n", "\n", "#### David Evans, Chase Coleman, and Thomas Sargent" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook uses Bayesian methods offered by [pymc](https://github.com/pymc-devs/pymc) to estimate two parameters of a univariate first order VAR model. We'll let $\\{y_t\\}_{t=0}^T$ denote the sample. \n", "\n", "The model is a good laboratory for illustrating some\n", "important issues: \n", "\n", "* The consequences of alternative ways of modeling the distribution of the initial random variable $y_0$:\n", "\n", " * As a fixed number.\n", " \n", " * As a random variable drawn from the stationary distribution of the $\\{y_t\\}$ stochastic process\n", " " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Sun Jan 10 16:45:01 EST 2016\r\n" ] } ], "source": [ "!date" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For those who don't have pymc installed, the process is straightforward as long as you are using [Anaconda](https://www.continuum.io/why-anaconda). Assuming that you are, just execute the following command in a terminal." ] }, { "cell_type": "raw", "metadata": { "collapsed": false }, "source": [ "conda install -c https://conda.binstar.org/pymc pymc" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The model\n", "\n", "\n", "The model is\n", "\n", "$$y_{t+1} = \\rho y_t + \\sigma_x \\epsilon_{t+1}, \\quad t \\geq 0$$\n", "\n", "where the scalars $\\rho$ and $\\sigma_x$ satisfy $|\\rho| < 1$ and $\\sigma_x > 0$; and\n", "$\\{\\epsilon_{t+1}\\}$ is a sequence of i.i.d. normal random variables with mean $0$ and variance $1$;\n", "and $y_0 \\sim {\\cal N}(\\mu_0, \\sigma_0^2)$ \n", "\n", "Consider a sample $\\{y_t\\}_{t=0}^T$ governed by this statistical model. The first-order autoregression\n", "implies that the likelihood function of $\\{y_t\\}_{t=0}^T$ can be factored as follows:\n", "\n", "$$f(y_T, y_{T-1}, \\ldots, y_0) = f(y_T| y_{T-1}) f(y_{T-1}| y_{T-2}) \\cdots f(y_1 | y_0 ) f(y_0)$$\n", "\n", "where we use $f$ to denote a probability density. The statistical model above implies that\n", "\n", "\\eqalign{ f(y_t | y_{t-1}) & \\sim {\\mathcal N}(\\rho y_{t-1}, \\sigma_x^2) \\cr\n", " f(y_0) & \\sim {\\mathcal N}(\\mu_0, \\sigma_0^2) }\n", "\n", "A focus of the notebook is to study how inferences about the unknown parameters $(\\rho, \\sigma_x)$ depend on what is assumed, right or wrong, about the parameters $\\mu_0, \\sigma_0$ of the distribution of $y_0$.\n", "\n", "We pretend that the parameters $\\mu_0, \\sigma_0$ are known numbers that may or may not be functions of $\\rho, \\sigma_x$. Below, we compare two cases:\n", "\n", " * $y_0$ is drawn from the stationary distribution implied by $\\rho, \\sigma_x$. In this case, $\\mu_0,\\sigma_0$ are functions of $\\rho, \\sigma_x$.\n", " \n", " * $y_0$ is drawn from the distribution ${\\mathcal N}(y_0, 0)$. This amounts to conditioning on the initial value. \n", " \n", "Unknown parameters are $\\rho, \\sigma_x$. We know prior probability distributions for these two and want to compute a posterior probability distribution of them associated with a sample $\\{y_{t}\\}_{t=0}^T$. \n", "\n", "The notebook uses pymc to compute the posterior distribution of $\\rho, \\sigma_x$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Note:** We do not treat $\\mu_y,\\sigma_y$ as parameters to be estimated. Instead we treat them either as fixed\n", "parameters or as particular functions of $\\rho, \\sigma_x$. In particular, we explore consequences of two alternative assumptions about the distribution of $y_0$: \n", "\n", "\n", "* The first procedure is to condition on whatever value of $y_0$ is observed. This amounts to assuming that the probability distribution of the random variable $y_0$ is a Dirac delta function that puts probability one on the observed value of $y_0$. It is as if $\\mu_y = y_0, \\sigma_y =0$. \n", "\n", "* The second procedure is to assume that $y_0$ is drawn from the stationary distribution of \n", "$$y_{t+1} = \\rho y_t + \\sigma_x \\epsilon_{t+1}, ,\\quad t \\geq 0,$$ namely, $y_0 \\sim {\\cal N} \\left(0, {\\sigma_x^2\\over (1-\\rho)^2} \\right)$\n", "\n", "When the initial value $y_0$ is far out in the tails of the stationary distribution, conditioning on the initial value gives a posterior that is \"much more accurate\", whatever that might mean to a Bayesian. \n", "\n", "Here is what is going on. When you don't condition on $y_0$, the likelihood function for $\\{y_t\\}_{t=0}^T$ adjusts the parameter pair $\\rho, \\sigma_x$ to make the observed value of $y_0$ much more likely than it really is, thereby adversely twisting the posterior in short samples.\n", "\n", "The example below shows how not conditioning on $y_0$ adversely shifts the posterior probability distribution of $\\rho$ toward larger values." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline\n", "import pymc as pmc\n", "import numpy as np\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First simulate an AR(1) process.\n", "\n", "How we select the initial value $y_0$ matters.\n", "\n", " * If we think $y_0$ is drawn from the stationary distribution ${\\mathcal N}(0, \\frac{\\sigma_x}{1-\\rho^2})$, then it is a good idea to use this distribution as $f(y_0)$. Why? Because $y_0$ contains some information\n", " about $\\rho, \\sigma_x$. \n", " \n", " * If we suspect that $y_0$ is far in the tails of the stationary distribution -- so that variation in early observations in the sample have a significant transient component -- it is better to condition on $y_0$ by setting $f(y_0) = 1$.\n", " \n", "\n", "To illustrate the issue, we'll begin by choosing an initial $y_0$ that is far out in the stationary distribution." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def ar1_simulate(rho, sigma, y0, T):\n", "\n", " # Allocate space and draw epsilons\n", " y = np.empty(T)\n", " eps = np.random.normal(0.,sigma,T)\n", "\n", " # Initial condition and step forward\n", " y[0] = y0\n", " for t in range(1, T):\n", " y[t] = rho*y[t-1] + eps[t]\n", " \n", " return y\n", "\n", "sigma = 1.\n", "rho = 0.5\n", "T = 50\n", "\n", "np.random.seed(145353452)\n", "y = ar1_simulate(rho, sigma, 10, T)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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9Fn0NA7AQwHPgMzuEiJwMm5guV/T1rKreVbSPG59ZAiLSCcAgVe3O\nZxafiDQGMBP2e5kHYJqqjnH63LhQl4iIAolbvhMRUSAxQBERUSAxQBERUSAxQBERUSAxQBERUSAx\nQBERUSAxQBERUSAxQBERUSD9PwZPfO4Vak7OAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(y)\n", "plt.tight_layout()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we shall use Bayes law, conditioning on the initial value of $y_0$. (Later we'll assume that $y_0$ is drawn from the stationary distribution, but not now.)\n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def AR1_model():\n", " \n", " #start with priors\n", " rho = pmc.Uniform('rho',lower=-1.,upper=1.) #assume stable rho\n", " sigma = pmc.HalfNormal('sigma', tau = 0.1)\n", " \n", " # Expected value of y at the next period (rho * y)\n", " yhat = rho*y[:-1]\n", " \n", " # Likelihood of the actual realization.\n", " y_like = pmc.Normal('y_obs', mu = yhat, tau=1./sigma, observed=True,value=y[1:])\n", "\n", " return locals()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ "AR1_MCMC = pmc.MCMC(AR1_model())" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " [-----------------100%-----------------] 50000 of 50000 complete in 3.6 sec" ] } ], "source": [ "AR1_MCMC.sample(50000,burn = 10000)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Plotting rho\n", "Plotting sigma\n" ] }, { "data": { "image/png": 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y96Vm9nPgHWA1MMPdHzez7u6+IrXOcjPrltqkB/BcaBeNqWVrCT7D0paklouI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\nAGe6+67ArmY2tLqHIiLtjQIsEWmT3H25u89OPf4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wi4gU1ioCrGL07w+//33uY6ZfFHbcMfie7Zb+sOqT+XiadJnzaaXLdvH697+L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\n3cwGAGPdfUTi9YWApye6m9nC5CKwBfA58BN3n5Z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+Fjg9P0tyHnAU2YSeRVL3WFVedyFp\nIVmh/mO9bWbDKmNTbi3wQXhmJvsnSkO7BVWzL4nFpNrnSbmU4jJpX1KJi6QXlc52lvQ84O0894fe\nm45LTyYajYQn8ZN0PHAmcHc+PhvAJWRnSt0o6SPATuC0/rWybatIsy/nAT+QNJ3sTLzlZAXjyfQl\nIjZIuons7Lr9+d/VwPNJoB+SfggsAQ6T9ACwkuz19KPKtkfEZkk3kiWQ+4GPR35KTlHUOlZJOje7\nOVYD75P0MbI+/Bt4f/9aXFuN2AyT9yMifiHpFEn3kQ2xL+9faydXry+kE5NanydzSSwujfSFROIC\nvAS4RtIQ2fv+hjwOz7zvW4mLJxo1MzMz6zAXuZuZmZl1mBMsMzMzsw5zgmVmZmbWYU6wzMzMzDrM\nCZaZmZlZhznBMjMzM+swJ1hmZmZmHeYEy8zMzKzD/geGL4ePaHiLpQAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pmc.Matplot.plot(AR1_MCMC)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice how the posteriors are centered on the true values hard wired in above\n", "\n", "Here is more detail about the posterior" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "rho:\n", " \n", "\tMean SD MC Error 95% HPD interval\n", "\t------------------------------------------------------------------\n", "\t0.536 0.071 0.001 [ 0.39 0.671]\n", "\t\n", "\t\n", "\tPosterior quantiles:\n", "\t\n", "\t2.5 25 50 75 97.5\n", "\t |---------------|===============|===============|---------------|\n", "\t0.393 0.49 0.537 0.583 0.675\n", "\t\n", "\n", "sigma:\n", " \n", "\tMean SD MC Error 95% HPD interval\n", "\t------------------------------------------------------------------\n", "\t1.055 0.229 0.003 [ 0.664 1.517]\n", "\t\n", "\t\n", "\tPosterior quantiles:\n", "\t\n", "\t2.5 25 50 75 97.5\n", "\t |---------------|===============|===============|---------------|\n", "\t0.697 0.89 1.025 1.184 1.583\n", "\t\n" ] } ], "source": [ "AR1_MCMC.summary()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we shall use Bayes Law but shall assume that $y_0$ is drawn from the stationary distribution. This means that we expect the value of $y_0$ to come from\n", "\n", "$$y_0 \\sim N \\left(0, \\frac{\\sigma_x}{1 - \\rho} \\right)$$\n", "\n", "Note how the code has been altered to achieve this" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def AR1_model_y0():\n", " \n", " #start with priors\n", " rho = pmc.Uniform('rho',lower=-1.,upper=1.) #assume stable rho\n", " sigma = pmc.HalfNormal('sigma', tau = 0.1)\n", " \n", " #standard deviation of ergodic y\n", " @pmc.deterministic(trace=False)\n", " def y_sd(rho = rho, sigma = sigma):\n", " return sigma/np.sqrt(1-rho**2)\n", " \n", " #yhat\n", " yhat = rho*y[:-1]\n", " y_data = pmc.Normal('y_obs', mu = yhat, tau=1./sigma, observed=True,value=y[1:])\n", " y0_data = pmc.Normal('y0_obs',mu = 0., tau= 1./y_sd, observed = True, value = y[0])\n", " \n", " return locals()" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [], "source": [ "AR1_MCMC_y0 = pmc.MCMC(AR1_model_y0())" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " [-----------------100%-----------------] 50000 of 50000 complete in 4.1 sec" ] } ], "source": [ "AR1_MCMC_y0.sample(50000,burn = 10000,thin=10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Some nerdy comments about how pymc works\n", "\n", "The pymc package expects the model function to return all of its local variables by returning locals(). Remember that a model is simply a probability distribution over outcomes and for a Bayesian this probability distribution is determined by the posterior. The posterior is the product of the priors and the likelihood. The way pymc deals with this is by identifying which variables are stochastic and which are deterministic (one can also specify whether a variable is deterministic or stochastic by using the @pymc.deterministic or @pymc.stochastic decorators as explained in the pymc [documentation](https://pymc-devs.github.io/pymc/modelfitting.html#creating-models). The stochastic variables will be either priors (such as the distributions on $\\rho$ and $\\sigma$ in our case) or likelihoods (the probability of seeing a specific history of $y$ values). Thus pymc knows that the posterior probability will just be the product of all of the stochastic values.\n" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Plotting sigma\n", "Plotting rho\n" ] }, { "data": { "image/png": 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TJhqtbz17bs277x4FfCPjlWNSPydHLqtDhxNYunQx66yzTlLNEymrqkw0amajCG6CeB0w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\nsWoyVaafCVa/jNI/6n4b9VhHvX119RKXE6xx3Zxg9XdzgjU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vrnv+6U+n04dGjcgkCbbcK8+Sfuyx8eoqTjZPUxozrF91Venl1bZz5JHpTpXw\nj3+Uf+/FF9OZCV9EpB00PcCKo9n5QNFREYCFC2H33auv5//+b11dcW4FVEtyeJrl464XDXpKJZtH\n1y+U7dMH9tijdP1f/CKcemq8tuP2sbif5SYKbeQIVqW2Ntig/OhmnPUB7rwTLrusun51Z7PN0qtL\nRNanHKzsaMg8WOENVK9k3b0ILy1RJgdcAWwI/N3dxyfqWUrSTJIuKHefwko+/OHqyie9AKAa9Zwu\nYNCg9Ze99Vb59b/zndragfi/02i5TTaB118PnvfrF2/9WlXznVu1Kl65NOb7ysqpUZGeTvlX2VH3\nHCwz6wVMByYAuwBTzWynojIDgO8DR7r7rsDHEvUqRe184DArv3133llbna++Wnt/yoleiJDGfRDj\nvFdKqba/8Q0YMWLd60mT4Iknqqs3jnpe8fjUU9XVLSIi9RfnFOE4YJG7L3H3VcBMYHJRmROAW919\nOYC7v1xLZ8odfNMYpSmVfBu3vTTzvKpZp7t8o2OP7T7f6MgjS69XSakcmiwFqZVysGpZN6pXL9ht\nt+r6VKukt89p9KnzMcn+mRMR6VHiBFiDgaWR18vCZVE7AJub2e/MbI6ZxbgGKh3nnANf+UqjWquv\naq4I/MUvYO+9S79X6ynMLCoOIqJB0i67lC9XbllxPbUGDVkKOhthiy3guuua3QuR9qccrOzIyr0I\nNwDGAAcBmwB/MLM/uPuzaVTe3YHykEOCRz2VO5jut1+QJ5PWdAB/+UuQ/J304H3yyesv+9OfYNy4\nZPWmYcMN119W60jM5z4HU6YEB/9jjqlcPlomekXiV79aW/tJtVKQ9nJNY9IiUi3lYGVH0hysOAHW\ncmBo5PWQcFnUMuBld38LeMvMHgBGAesFWPl8PvIqh3uumv5WrXAQq8fEmT/6UfAzrRG0DTcMTlGt\nWZOsnuJt/elPg5GaagKZcp9X375dE9SrdfnlcPDBXZclubqxMOFrIY/qggvWzaheLM2RvXK3Jjrj\njNITdxa2sZWCqtp1hg/o8ucuItKDxAmw5gDDzWwY8AIwBZhaVOYO4Htm1hvoA+wFfLdUZdEAq3gU\ntNnTMZST1X7FVbj/Xb22o5p6q51/qtp50rbaqvJo1kYbdR3NqyXoOemkYASz2NVXly6fpdv61F8u\nfAQBlk53iEhPVDEHy93XANOAe4GngJnuvsDMTjOzU8MyC4F7gCeAh4Fr3X1+Wp1s9QCnlCRJ7s1U\nqk/VjAzJn5+EAAAgAElEQVQl/V2m8Zm8/TZMLf4XoUobbAAjR1a/XvGcYFn5bpf6XM86q/H9EOnp\nlIOVHQ3JwXL3u4Edi5ZdU/T628C3k3QmzsHmzDPLjxKUcswx8PzzML9EuHfRRcHyUu+VEndeokYq\nPjCeemq8fKS0vPlm8PhuyfHKeOLezDsrwUgShd/XPvvAhAnly9V7Pq56WrEiuM/no482uycirUc5\nWNnR4+5FWM2VdhDs6O+/v/zIy5Qp6y8vPpAPHBj8TPP+g+VE+1nLaM0113Sd1wnWbU+tN/WNziZe\n/Nn07VtdMFBqpve4sjiiV6v/9/9g331Lvzd6dLq3YqpVrZ/3lltqSgcRkZYIsGoZxTjuuHjlSl3V\nVqwetwdJK1jo7vOYO7fr61oP2pVOp/XtG397qp1OoVhxO602qtUOQeLGGwe5dN1ty/e+B2+80bg+\niYhkTdMDrGqTmJMeoIrzZs46C+bNS1ZnvZT6PH78YzjllHWv63F1ZMGTT1ZXvlZ77gnTp5d+LxoA\n1yuYatWgp97BZbn6FyyAxx/vft0NNmjt05wizaIcrOzIyjxYdVXLgWTXXUsvnzMHXnoJ3n03eN23\nL4waVXvfIN0DdGfnur6Vqvczn6m97mo/x8JnmOb2lepDnz5w+unrLy/VbisGQ9OnB0EkVH+XgO6k\n+VlUU9fQoZXLiEhtlIOVHY2YB6th0vqPfNWqdfMjFRQOIJtsAttt15h+dKfcAW3//Usv33jjrjOX\nt6osntKrd5+iwWMrBogiIlK9TJ0iTMsGG9R+0HzggWAUqRqNOHX18svwk59Ut34Wg5mkiqeEiLON\nlUb9mhn0jBxZ+r6R5WyzDaxcGTxP8/dbqq5aL4oQEZGMjWCVk8aBJO5BtNQI0vXXw+LFyetOolw+\ny0knlU/U32ij4Gfh86vlc/zxj2GHHapfr5ykv8vtt6/+887lsnMfveKgpVcvuOIKeOedeOubNe6i\ni69+FY4/Pv22RKS8Qv6VThU2Xz7/WKK7UWQ+wPr5z7sGPc0YbRgxYv2pD5JKazu23x6+/vXS7xUn\n9MedbyoqSc5XNe30FFdfDV/+ctdlw4fDPfc0pz/d2WSTrlN0iEj9KbDKjrbPwfroRxvfj54obsB3\n/fXJ7kWYJMBKcs/C7lx/fTAZbSMMHLhuXrUs6emBr4hI2jIVYNVT1pKLm9GfNHJqkt5iJu0DeZz6\nKpXZY4/gkXUf+ADsvnt96v7IR+DXv65P3SIiPVGsJHczm2hmC83sGTM7r8T7B5rZv8zssfDx1bgd\nqHYerJ4k7SCssxP++tfS7xV/9sVtDx5ceSb7++8PHt1pxu84yezxWbJwIcycWZ+6N9wQJk6sT90i\nEp/mwcqOus+DZWa9gOnAwcDfgDlmdkd4g+eoB9x9Uq0dueKKdXMF1UOWRrC++c1gjqmjj25su0OG\n1L7ub35TORH7oINqr7+exowJ7v3X6ooD3MGD4Yc/bE5fRKQ+lIOVHY3IwRoHLHL3JQBmNhOYDBQH\nWInGJs46K165z38+GJEoThRuJV+NOb63775w223J20tj1GjAgPhl+/Ytn6fVjFOE739/cO+/dtO7\nN/zHfzS7FyIiUkqckyeDgaWR18vCZcX2MbN5ZvYrM9s5ld6VsMkmcOih9aq9NvUaHfv5z+GTn0xe\nT6NH7155Bc48M/16SwVTcU7/6dSziIg0WlrZKX8Chrr7aILTibfHXXGLLVLqQQVZOkUYV69e689I\nX61f/apxn3FBv37l87XSzofq1w8WLUq3ThGRZlEOVnY04l6Ey4Ho3ceGhMvWcvfXIs9/bWY/MLPN\n3f2V4srykVm7fve7HJtvnquyy9kbkchaf6KOOKLZPegqyWc1pszp8OHD69dmo221FYwdG69s0uC7\nXjo7O+ms9nYIIgIoBytLGpGDNQcYbmbDgBeAKUCXi/XNbJC7rwifjwOsVHAVdDifqMO1qucIVpK6\nK12ZV+n9nqIVRyBrsWxZvIDw978vH3A2Wy6XI5fLrX2t/8ZFpCeqePh29zVmNg24l+CU4nXuvsDM\nTgve9muBj5rZ54BVwJuAbrARw5w5ledfuuwyOOWUxvSnYPfd6zcZZiuNJjVD3FGp/farbz9ERCSZ\nWOMj7n43sGPRsmsiz78PfD/drrW/ONNSDBwIe+1V/75EPfJI/QKhRueDiYi0Et2LMDva/l6Epey8\nM/zP/zS7F+2r3M2jqzFtGvzoR/Dvf3dd3r9/40/3adRMRFqFAqvsSJqD1ZJzXPfuDR//eHXr1POg\nftJJ8JWv1K/+VrTddvWZqkFERKQVtGSAVYt6Blg77AAXXVS/+iUZjWCJiEij9ZgAS0REJOs0D1Z2\nNGIeLJGaZGFqhTlzgslIRURagXKwsqNH5mCJxFXPG4iLiIiU02MCrCyMpvQ0yn0SEZGeqscEWNI9\nBUMiIs2nHKzsUA5WTJdcAitXNrsX2aURPmk1ZjYEuBEYBLwL/MjdrzazgcDNwDBgMXCcu68M17kA\nOBlYDXzB3e8Nl48BbgD6Ane5+1mN3RqRgHKwskM5WDF97nNw/vnN7kXPoqBN6mw18EV33wXYBzjd\nzHYCzgfuc/cdgd8CFwCY2c7AccBI4HDgB2Zrx25/CHzG3XcAdjCzCY3dFBFpN7ECLDObaGYLzewZ\nMzuvm3JjzWyVmR2TXhdFRNbn7i+6+7zw+WvAAmAIMBmYERabARwVPp8EzHT31e6+GFgEjDOzrYD+\n7j4nLHdjZB0RkZpUDLDMrBcwHZgA7AJMDf9LLFXuEuCetDspItIdM9sWGA08DAxy9xUQBGHAlmGx\nwcDSyGrLw2WDgWWR5cvCZSINpxys7GhEDtY4YJG7LwEws5kE/yEuLCp3BvALYGyiHklT1CPJXYnz\n0ghmtinBvucL7v6amRWfnNbJamkZysHKjqQ5WHECrOL/+pYRBF1rmdnWwFHuPt7MurwnIlIvZrYB\nQXD1U3e/I1y8wswGufuK8PTfS+Hy5cA2kdWHhMvKLS8pn8+vfZ7L5cjlcgm3QkSyozN8ADydqKa0\nriK8EojmZmnsQpTkLo1wPTDf3a+KLJsFfBq4FPgUcEdk+c/M7AqCfxyHA4+6u5vZyvCfwznAicDV\n5RqMBlgi0m5y4QPgTmBmzTXFCbCWA0Mjr0v9d7cnMDO8Iue9wOFmtsrdZxVXpv/+RNpbZ2cnnZ2d\ndW/HzD4EfBx40szmEpwK/E+CwOoWMzsZWEJw5SDuPt/MbgHmA6uAz7uv/TfgdLpO03B33TdApIRC\n/pVOFTZfPv8YSf6fMq8wzGBmvQnGyQ4GXgAeBaa6+4Iy5X8C3Onut5V4zyu1J41nBjffDMcdl269\nX/kKfOtbGsnq6cwMd2+LUW3tw0Sybfbs2Uye/HVWrpxd5ZqFXVT07/tOYFLN+6+KI1juvsbMpgH3\nElx1eJ27LzCz04K3/driVWrpiIiIiEi7iJWDFQ6X71i07JoyZU9OoV/SBrbfvtk9EBERaY4eM5O7\nNN7JJ8MbbzS7FyIirUPzYGWH7kUomWUG/fo1uxciIq1Dye3ZoXsRioiIiGSMAiwRERGRlCnAEhER\nyQjlYGWHcrBERETahHKwskM5WCIiIiIZowBLREREJGUKsERERDJCOVjZoRwsSYW1xZ3iRERam3Kw\nsqMhOVhmNtHMFprZM2Z2Xon3J5nZ42Y218weDe9yLy1E968VERFJT8URLDPrBUwHDgb+Bswxszvc\nfWGk2H3uPissvxtwCzCyDv0VERERybw4I1jjgEXuvsTdVwEzgcnRAu4evePcpsC76XVRGkGnCEVE\nmk85WNnRiByswcDSyOtlBEFXF2Z2FHAx8D7gw4l6JSIi0gMpBys7MjMPlrvf7u4jgaOAi9KqV0RE\nRKTVxBnBWg4MjbweEi4ryd0fNLPtzWxzd3+l+P18Pr/2eS6XI5fLxe6s1I+S3CUtnZ2ddHZ2Nrsb\nIiJNFSfAmgMMN7NhwAvAFGBqtICZfcDd/xI+HwNsVCq4gq4Bloi0n+J/nJRPIhJf4e9FpwqbL59/\njCQhS8UAy93XmNk04F6CU4rXufsCMzsteNuvBY41sxOBd4A3geNq75I0g5LcRUSaT4FVdiTNwYo1\n0ai73w3sWLTsmsjzy4DLEvVEmqZfP9htt2b3QkREpH1oJnfhjTcqlxEREZH4dC9CERGRjNA8WNmh\nexGKiIi0CeVgZUdm5sESERERkYACLBEREZGUKcASERHJCOVgZYdysERERNqEcrCyQzlYIiIiIhmj\nAEtEREQkZQqwREREMkI5WNnRkBwsM5sIXMm6exFeWvT+CcB54ct/A59z9ycT9UxERKSHUQ5WdtQ9\nB8vMegHTgQnALsBUM9upqNhzwAHuPgq4CPhRol7VSWdnZ49tvydve7Pb78nbLiLSU8U5RTgOWOTu\nS9x9FTATmBwt4O4Pu/vK8OXDwOB0u5mOZh9odJDvme335G0XEemp4gRYg4GlkdfL6D6AOgX4dZJO\niYiI9ETKwcqOTM2DZWbjgZOA/dKsV0R6BjM7HviFu69pdl9EmkE5WNmRNAfL3L37AmZ7A3l3nxi+\nPh/wEonuHwRuBSa6+1/K1NV9YyLSltzd4pQzs6OBY4EFwLXu/ve6dqxKZuaV9pki0jyzZ89m8uSv\ns3Ll7CrXLOyion/fdwKTYu+/isUZwZoDDDezYcALwBRgapdumQ0lCK4+WS64gvg7WRHpmdz9l2b2\nJPAdYKyZzXV3nS8RkZZTMQcrHKqfBtwLPAXMdPcFZnaamZ0aFvsasDnwAzOba2aP1q3HItK2zOwG\n4ATgVHc/Cni1uT0SaSzlYGVH0hysiqcIRUQaxcyGuvvz4fP3uvvLze5TlE4RimRblk4RNmwmdzOb\naGYLzewZMzuv8ho1tbHYzB6PjqKZ2UAzu9fMnjaze8xsQKT8BWa2yMwWmNlhNbR3nZmtMLMnIsuq\nbs/MxpjZE+Fnc2XC9jvMbJmZPRY+JtajfTMbYma/NbOnzOxJMzuzkdtfov0zGrX9ZtbHzB4Jv2dP\nmllHg7e9XPsN+d1H1u0VtjMrxe0/M9LEl6vpj4hIprh73R8EgdyzwDBgQ2AesFMd2nkOGFi07FLg\n3PD5ecAl4fOdgbkEeWjbhv2zKtvbDxgNPJGkPeARYGz4/C5gQoL2O4Avlig7Ms32ga2A0eHzTYGn\ngZ0atf3dtN+o7d84/NmbYO63cQ3+3ZdqvyHbHqn3bOB/gFkpfvfvidR/bZL9QT0eBBf4iEhGdXZ2\n+oABBzh4lQ/CR3TZLA//5mvaXzRqBKviZKUpMdYflZsMzAifzwCOCp9PIsgnW+3ui4FFYT9jc/cH\ngX8mac/MtgL6u/ucsNyNkXVqaR/WjXUW9yu19t39RXefFz5/jeCqryE0aPvLtF+Yn60R2/9G+LQP\nQeDgjdr2btqHBmw7BCOIwBHAj4vaSbr975jZrWb2c4ILZ0R6FOVgZUfSHKxGBVjVTlZaKwd+Y2Zz\nzOyUcNkgd18BwUEZ2LJMn5an1Kctq2xvMMHnUZDGZzPNzOaZ2Y8jp2nq1r6ZbUswkvYw1X/eabb/\nSLio7tsfnh6bC7wI/CYMEhq27WXah8b97q8AzqFrwkIa2+/A6QSnBxfE7ItI2+jo6NBcWBlR93sR\ntpgPufsYgv+sTzez/el6AKDE63prdHs/ALZ399EEB9/v1LMxM9sU+AXwhXAkqaGfd4n2G7L97v6u\nu+9OMGo3zsx2oYHbXqL9nWnQtpvZh4EV4Qhid8mftWz/aOBLwH+EDxGRlpTqTO7dWA4MjbweEi5L\nlbu/EP78u5ndTnDKb4WZDXL3FeEpiZcifdqmDn2qtr1U++FdJ2b8EcFlEHVp38w2IAhufurud4SL\nG7b9pdpv5PaH7b1qZp3ARJrwu4+27+7fjbxVz23/EDDJzI4A+gH9zeynwIspbP9Sdz8nRh9ERDKt\nUSNYaycrNbONCCYrnZVmA2a2cTiagZltAhwGPBm28+mw2KeAQiAwC5hiZhuZ2XbAcKCW+buMrv/F\nV9VeeCplpZmNMzMDToysU3X74YGt4Bjgz3Vs/3pgvrtfFVnWyO1fr/1GbL+Zvbdw+s3M+gGHEpzO\nasi2l2l/YaN+9+7+n+4+1N23J/hb/q27f5IgoEu6/f3N7PtmdpmZXVapLyLtRjlY2ZE0B6uRV99M\nJLjSaxFwfh3q347g6sS5BIHV+eHyzYH7wrbvBd4TWecCgiuaFgCH1dDmTcDfgLeB5wnuwziw2vaA\nPcI+LwKuStj+jcAT4WdxO0FeTOrtE4xirIl85o+Fv+OqP++U26/79gO7he3NC9v6Sq3ftRq3vVz7\nDfndF/XlQNZdRZh4+wmuNF77qND2dcAK1r+Kdln4+TxGMLIX7cOiEn0YE35uzwBXVmjTRSS7snQV\noSYaFZHMMLMvALu6+2fN7Gvu/s1uyu4HvAbc6O4fDJd1AP/2rqdLMbORBP+QjCU4FXkfMMLd3cwe\nAaa5+xwzu4sg0LynTJuufaZIdvXIiUZFRGL4AOuuOOzfXUFv4jQlIiKVKMASkSxxoJ+Z7QpsXWMd\nDZ2mRCRNysHKjqQ5WI26ilBEJI7vAJ8HPkmQM1WtHwDfCE/9XRTWd0qFdaqSz+fXPs/lcuRyuTSr\nlx5Oc2A1W2f4gHz+6UQ1KcASkSwZz7oJRscTnLKLzRswTUc0wBKRdpMLHxDsPmbWXJNOEYpIlrwY\nPv4N7B+jfDOnKRERKUsjWCKSGdGr98xsx+7KmtlNBP9qbmFmzxNM0TDezEYD7wKLgdPCeueb2S3A\nfGAV8PnI5YCnAzcAfYG73P3uFDdJpCqF/CudKmy+fP4xkgxYa5oGEcmM8CbPThAgPeHu32pyl7rQ\nNA0i2ZalaRo0giUimeHuH2t2H0RE0qAAS0Qyw8z+ALxFOF0Dwb0Jj2tur0REqqckdxHJkvvcfby7\nHwTcr+BKehrNg5UdmgdLRNrJcDMrXD24fVN7ItIESm7Pjnx+TKL1FWCJSJacCRxPcIrwzCb3RUSk\nZjpFKCJZchgwzN2/TxBoiYi0JAVYIpIl+xBMMgqwbRP7IdIUysHKDuVgiUg7WQ0Q3qR5qwplRdqO\ncrCyI2kOlkawRCRLbiC4jc1/A99tbldERGqnESwRyYTwXoAHuPuJze6LiEhSGsESkUwI70Ez1sym\nmtkRZnZEs/sk0mjKwcqOlsrBMjPdxEukB4pzLy8zmwTcB7wX2KjunRLJIOVgZUfL5WC5e1s8Ojo6\nmt4HbUt7bke7bUsVJrr7DGCku88In4uItCSdIhSRrBgWnhYcplOEItLqFGCJSFbcArwv8vN9ze2O\nSOMpBys7WioHq53kcrlmdyE17bIt7bId0F7bEpdOCYooBytLkuZgWZU5EskaM/NGticizWdmeIwk\n91agfZhIts2ePZvJk7/OypWzq1yzsIuK/n3fCUyqef+lU4QiIiIiKasYYJnZdWa2wsye6KbM1Wa2\nyMzmmdnodLsoIiLSMygHKzsakYP1E+B7wI2l3jSzw4EPuPsIM9uL4BYXeyfqlYiISA+kHKzsqPs8\nWO7+IPDPbopMJgy+3P0RYICZDUrUKxEREZEWlkYO1mBgaeT18nCZiIiISI+kJHcREZGMUA5WdmRh\nHqzlwDaR10PCZSXl8/m1z3O5XI+c70eknXV2dtLZ2dnsboi0JOVgZUdD5sEys22BO919txLvHQGc\n7u4fNrO9gSvdvWSSu+aQEel5NA+WiDRKlubBqjiCZWY3ATlgCzN7HugguNO9u/u17n5XeN+wZ4HX\ngZNq6YiIiIhIu6gYYLn7CTHKTEunOyIiIj1XIf9KpwqbL59/jEhWU9UyfS/CsWPHMmfOnGZ3Q0RE\npCEUWGVH3efBahZ3x6w5aRvFORbKuRAREZFqZCrAmjFjBlOnTmXy5MncfffdrFmzhjPOOIO9996b\nyy67DIDly5dz6KGHksvlOPPMM9er49vf/jbjx49nzz335P777wfgL3/5C4cccggHHXQQ55xzDgDf\n/e532XfffTnggAOYN28eAHvssQdnn302J554Ype+3HPPPQ36BERERKQtuHvDHkFz5d1www1+4okn\nrn29/fbb+9KlS33NmjW+6667urv7tGnT/N5773V391NOOcV///vfd6njzTffdHf3l156yQ888EB3\ndz/mmGN87ty5a8u8+OKLa99bvHixH3rooe7uvt122/lzzz1Xsi8iUpvw776h+5p6PSrtw0SSyufz\nns/nm92NltXZ2ekDBhzg4FU+CB/rluXz+UT7r8zlYI0dO3bt880335whQ4YA0K9fPwCeffZZ9txz\nTwD23HNPFi1axH777bd2nRkzZnDTTTfRq1cvXnzxRQCWLl3K6NHr7kG9ePFiRo0aBcCwYcNYuXIl\nAAMHDmS77bYr2RcREZF6Uw5WdrRdDlavXqW75GEe1IgRI3jkkUcAmDNnDiNGjOhSbvr06XR2dnLz\nzTevXWfo0KHMnTt3bT3bbrst8+bNw91ZvHgx73nPe0q2Xa4vIiIiIt3J3AhWVDTJvfD83HPP5VOf\n+hQXX3wxu+66a5fRK4D999+f/fbbj7322otNN90UgEsvvZTPfvazQJBndfnllzNp0iT23Xdfevfu\nzfTp09drT0RERKRWsWZyT60xzYIs0uNoJneR+DQPVjJpzuSez19IPp+v30zuIiIi0hgKrLKj7XKw\nRERERFqdAiwRERGRlCnAEhERyYgLL7xwbR6WNFc+/1ii9ZWDJSIikhHKwcoO5WCJiIiIZIwCLBER\nEZGUKcASERHJCOVgZYdysERERNqEcrCyoyE5WGY20cwWmtkzZnZeifc3M7NZZjbPzJ40s08n6pWI\niIhIC6sYYJlZL2A6MAHYBZhqZjsVFTsdeMrdRwPjge+YmUbHREREpEeKM4I1Dljk7kvcfRUwE5hc\nVMaB/uHz/sA/3H11et0UEenKzK4zsxVm9kRk2UAzu9fMnjaze8xsQOS9C8xskZktMLPDIsvHmNkT\n4Qj9lY3eDpEo5WBlR9IcrDgB1mBgaeT1snBZ1HRgZzP7G/A48IVEvRIRqewnBCPrUecD97n7jsBv\ngQsAzGxn4DhgJHA48AMzK9zA9YfAZ9x9B2AHMyuuU6RhOjo6lIeVEVmZB2sCMNfdtwZ2B75vZpum\nVLeIyHrc/UHgn0WLJwMzwuczgKPC55OAme6+2t0XA4uAcWa2FdDf3eeE5W6MrCMiUrM4eVLLgaGR\n10PCZVEnARcDuPtfzOyvwE7AH4sry+fza5/ncjlyuVxVHRaRbOvs7KSzs7NZzW/p7isA3P1FM9sy\nXD4Y+EOk3PJw2WqCUfmCUiP0IiJVixNgzQGGm9kw4AVgCjC1qMwS4BDgITMbBOwAPFeqsmiAJSLt\np/gfpybnk3gzGxepVuHvRacJmy+ff4wkIUvFAMvd15jZNOBeglOK17n7AjM7LXjbrwUuAm6IJJue\n6+6v1N4tEZGarDCzQe6+Ijz991K4fDmwTaRcYSS+3PKyNAov9aTAqtk6wwfk808nqinWVArufjew\nY9GyayLPX2D9ZFMRkXqz8FEwC/g0cCnwKeCOyPKfmdkVBKcAhwOPurub2UozG0cwWn8icHV3DWoU\nXqSd5cIHwJ0EEyfURnNViUhLMrObCPaEW5jZ80AHcAnwczM7mSB14TgAd59vZrcA84FVwOfdvXD6\n8HTgBqAvcFf4D6WISCIKsESkJbn7CWXeOqRM+YsJL8YpWv4nYLcUuyZSM+VgZUfdc7BERESkMRRY\nZUdW5sESERERkZACLBEREZGUKcASERHJCN2LMDuS3otQOVgiIiIZoRys7FAOloiIiEjGKMASERER\nSZkCLBERkYxQDlZ2KAdLRESkTSgHKzuUgyUiIiKSMQqwRERERFKmAEtERCQjlIOVHcrBEhERaRPK\nwcoO5WCJiIiIZEysAMvMJprZQjN7xszOK1MmZ2ZzzezPZva7dLspIiIi0joqBlhm1guYDkwAdgGm\nmtlORWUGAN8HjnT3XYGP1aGvIiIibU05WNnRiBysccAid18CYGYzgcnAwkiZE4Bb3X05gLu/nKhX\nIiIiPZBysLKjETlYg4GlkdfLwmVROwCbm9nvzGyOmX0yUa9EREREWlhaVxFuAIwBDgI2Af5gZn9w\n92dTql9ERESkZcQJsJYDQyOvh4TLopYBL7v7W8BbZvYAMApYL8DK5/Nrn+dyOXK5XHU9FpFM6+zs\npLOzs9ndEGlJhfwrnSpsvnz+MSIhS9XM3bsvYNYbeBo4GHgBeBSY6u4LImV2Ar4HTAT6AI8Ax7v7\n/KK6vFJ7ItJezAx3t2b3Iw3ah4lk2+zZs5k8+eusXDm7yjULu6jo3/edwKSa918VR7DcfY2ZTQPu\nJcjZus7dF5jZacHbfq27LzSze4AngDXAtcXBlYiIiEhPESsHy93vBnYsWnZN0etvA99Or2siIiIi\nrUkzuYuIiGSE5sHKDt2LUEREpE0ouT07dC9CERERkYxRgCUiIiKSMgVYIiIiGaEcrOxQDpaIiEib\nUA5WdigHS0RERCRjFGCJiIiIpEwBloiISEYoBys7lIMlIiLSJpSDlR3KwRIRERHJGAVYIiIiIilT\ngCUiIpIRysHKDuVgiYiItAnlYGWHcrBEREREMkYBloiIiEjKFGCJiIhkhHKwsiNpDlasAMvMJprZ\nQjN7xszO66bcWDNbZWbHVKozn89X0U0REZH219HRoTysjKh7DpaZ9QKmAxOAXYCpZrZTmXKXAPfE\naVgRuoiIiLSrOCNY44BF7r7E3VcBM4HJJcqdAfwCeCnF/omIiIi0nDgB1mBgaeT1snDZWma2NXCU\nu/8QsPS6JyIi0nMoBys7sjIP1pVANDerbJAVzb3q7Owkl8ul1AURyYLOzk46Ozub3Q2RlqT8q+xI\nmoNl7t59AbO9gby7Twxfnw+4u18aKfNc4SnwXuB14FR3n1VUlxfaMzMqtS0irS/8W2+Lke3oPkxE\nsmHG7toAABE/SURBVGf27NlMnvx1Vq6cXeWahV1U9O/7TmBSzfuvOCNYc4DhZjYMeAGYAkyNFnD3\n7dd20ewnwJ3FwZWIiIhIT1ExB8vd1wDTgHuBp4CZ7r7AzE4zs1NLrZJyH0VEqmJmi83scTOba2aP\nhssGmtm9Zva0md1jZgMi5S8ws0VmtsDMDmtez6WnUw5WdiTNwap4ijBNOkUo0vM04xRhmLawh7v/\nM7LsUuAf7n5ZOJ/fQHc/38x2Bn4GjAWGAPcBI0qdC9QpQpFsy9IpQs3kLiLtyFh//zYZmBE+nwEc\nFT6fRDAyv9rdFwOLCKanERGpmQIsEWlHDvzGzOaY2SnhskHuvgLA3V8EtgyXF09Fs5yiqWhERKql\nAEtE2tGH3H0McARwupntz/r5oTrXJ5mjHKzsyMo8WCIimeHuL4Q//25mtxOc8lthZoPcfYWZbcW6\nu04sB7aJrD4kXFZSdC6/XC6nufwkVZoHq9k6wwfk808nqkkBloi0FTPbGOjl7q+Z2SbAYcCFwCzg\n08ClwKeAO8JVZgE/M7MrCE4NDgceLVe/blQv0s5y4QOCJPeZNdekAEtE2s0g4Jdm5gT7uJ+5+71m\n9kfgFjM7GVgCHAfg7vPN7BZgPrAK+LwuFRSRpBRgiUhbcfe/AqNLLH8FOKTMOhcDF9e5ayIVFfKv\ndKqw+fL5x0gyYK0AS0REJCMUWGVH0nsR6ipCERERkZQpwBIRERFJmQIsERGRjNA8WNmhebBERETa\nhHKwskM5WCIiIiIZowBLREREJGUKsERERDJCOVjZoRwsERGRNqEcrOxImoMVK8Ays4nAlQQjXte5\n+6VF758AnBe+/DfwOXd/MlHPREREpEf485//zOuvv564nvnz56fQm3RUDLDMrBcwHTgY+Bswx8zu\ncPeFkWLPAQe4+8owGPsRsHc9OiwiIiLt4+mnn2bMmL3YeONdU6nvnXeOTqWepOKMYI0DFrn7EgAz\nmwlMBtYGWO7+cKT8wwR3pBcREZEq9MR7Eb755pv06zeClSsfaXZXukh6L8I4Se6DgaWR18voPoA6\nBfh13A7kk/ReRESkjXR0dPSo4CrLMjUPlpmNB05iXT5WRbpaQkRERNpNnFOEy4GhkddDwmVdmNkH\ngWuBie7+z3KVRUesOjs7Y3ZTRFpFZ2en/rZFpMczd+++gFlv4GmCJPcXgEeBqe6+IFJmKHA/8Mmi\nfKziurzQnpnh7mt/ikh7Cv/Grdn9SEN0HyZSDz0xB2vevHkceOCnefXVeU3sRWEXte7vO5+/kHw+\nX/P+q2KABWunabiKddM0XGJmpwHu7tea2Y+AY4AlYS9Xufu4EvUowBLpYRRgiUh3shpgwZ3ApJr3\nX7HmwXL3u4Edi5ZdE3n+WeCztXRAREREpN3oVjkiIiIiKVOAJSIikhG6F2F2JL0XYawcrLQoB0uk\n51EOloh0p11zsDSCJSIiIpIyBVgiIiIiKctUgKXb5oiISE+mHKzsaKscLOVjibQf5WCJtB935847\n7+Sf/yx745bYlixZwne+c1vb5WDFmgdLREREpGDJkiV89KMnsNFGx6ZS3zvvnJFKPVmiAEtERESq\n8u6779Knz5a89tqMZnclszKVg1WgXCwREemJlIOVHW2Zg6VcLJH2oRwskfbz3HPPMWrUIbz22nPN\n7kpKNA+WiIiISOYpwBIRERFJmZLcRUREMqKQf9XR0ZF63W+++SajRu3LSy+tSFzXu++uxn3b5J3K\nsHz+MZKkhCsHS0TqSjlYItnwj3/8g8GDh/P220+lVOMAYJOU6mq2HpiDpSsKpSfS915E6qMXsHVK\nj3YJruojVoBlZhPNbKGZPWNm55Upc7WZLTKzeWY2Oq0O6nJVyYJGBTyFdqLf+1YKtlqpryIi9VQx\nwDKzXsB0YAKwCzDVzHYqKnM48AF3HwGcBvx3HfoqLSx64G2lg3CpgKeeSrWTdttpf/7R+vQPkUgy\nmgcrO5LOg4W7d/sA9gZ+HXl9PnBeUZn/Bo6PvF4ADCpRlxcUnsddVkpHR0fZ99KUdjvR+hq1Dd0p\n1Ye4y+Iq9TvN2udQEO1Lqe9kkr5W+ly7+xuI226pz7VUG2mp9PcaLqu4r2mFR9qfnUgjvfzyy96n\nz+YOrsd6D8JHdNmsRPuvODuUY4FrI68/AVxdVOZOYN/I6/uAMSXqWvuLrjbAKnXQaNTOLq2AoFS/\nG7ENcQ/qcZeVqqfS55EkkK5FLcFIqb6kEfCUq7u770OlZcV1lGujnp913O+zAiyRZFauXOkvvPBC\n4sdTTz2lAKvsowcHWJWW1XOUKclBqpbRibhBS9y2uznoVexXzINnxc8j7YN+LUFjqbJpBTzd/a7K\nBalpfN+T9D+puN9nBVgitVu1apX377+59+s3KJXHpptOzEAwk8VH+gFWxWkazGxvIO/uE8PX54cN\nXhop89/A79z95vD1QuBAd19RVJdDR2RJLnyISPvoDB8FF+KapkEkluJ5sFatWkXfvhvz7rurmtmt\nHmD9aRry+QvJ5/O1778qRWBAb+BZYBiwETAPGFlU5gjgV+HzvYGHy9RVnxC/jEqnT1opt6rQ70r9\nr9RulnKdSqm0TWnnP9VT1j/rStLKw0MjWCI1e+edd7xXrw0yMMLT7o8mjGBBME0DcBXBVYfXufsl\nZnZa2PC1YZnpwETgdeAkd18v/T4L//216iSm+Xy+y0OkVWiiUemJHn/8cZYuXZq4ntWrV3PssR/T\nCFbdpT/RaNNmcm8WBSgijaUAS3qijTcewIYb7otZ78R1vfPOB3jzzatS6JWUpwBLRFqMAixpFZ/7\n3DnMmTM3lbrmzv097767Euhb1Xr5/IXhz44KJSVd6edgKcASkbpSgCWtYsCArXj11auBzVOobUvg\ngynUI42R/gjWBon7JCIi0jYOALZqdiekDSjAEhGRlnXIIUfz4IP/v73zD7KqLOP45xs/NWSBCgwJ\ngYiMGYt2EDBsqGxIIxeG+kNtCHD6MZOZI2OiTY77RzMpM6k1Os1oaWYZTUK65TAwiFTUEOQCG7gR\nhqlA0oihiQPjwtMf77twWe6y98K7770Hns/MmX3ve8893+c55+xzn/v+Ok8nOdaRI32Ac5Icy3E8\nwXIcx+HobOl7OTZb+q4ePuLUAdu2bePQobXA+ARH60etEywfg1U/NDe3cjpz4np82LNTnrVr19ba\nhGScKb6cKX7AmeVLEajkofa1ptb3REr9jo4ODhw4UNW2YsWKsvVmR4BBwOAEW3fJVTrfe6K5+Y4y\nyVU+/fLUUr922s3Njaf1eU+wTpFaB7uUnCm+nCl+wJnlS0GYAuwwsxfN7G1gKTC7xjYdR63viZT6\nM2d+noaGdzFkyPCKt1mzmsrW79//FjA0mW3lWdvLx3f9+tQ+PbyL0HEcBy4ASleF3EVIupxeYNeu\nPRw+/Efgkio+1UxHR/MJtR0dqaxynLR4guU4jlNHLFlyD4sXLyr7Xudz6mpFSv0BA+YxYMAHKt7/\n4MHtDBz4bDL9asipvWjRZADuvvuvNdEvx9lw7t94I/wdPPiqo3WLFk0+rTFY2dfByibmOE7dUO/r\nYFXyUPtY7zHMcc4yCrHQqOM4Tj2i8DyT7cDlwL+BDcA1ZtZeU8Mcxyks3kXoOM5Zj5kdlvQNYBXH\nlmnw5MpxnFPGW7Acx3Ecx3ESk22ZBklXSPq7pH9IWpxL93SRNErSGknbJP1N0jdj/VBJqyRtl7RS\nUkOtba0USe+Q1CqpJb4upC+SGiT9WlJ7vD5Ti+iLpJskbZXUJukXkvoXxQ9JP5G0V1JbSV23tku6\nTdKOeM1m1sbqk9NTrJJ0raQtcVsn6eLM+k1Re5OkDZKm59Qv2e8SSW9LmptLW9IMSftj/GqV9J1U\n2pXox30+Ec/9VknP5NSXdHPUbo3fRx2ShmTSHiypRdLmqL0ghW4V+kMkLY/3/npJExNqnxDHyuzz\nwxi7NkuaVNGBzazXN0Ii9zxwIWGp3M3ARTm0E9h+PjAplgcRxmlcBNwF3BLrFwN31trWKny6Cfg5\n0BJfF9IX4KfAwljuCzQUzRdgJLAT6B9f/wqYXxQ/gMuASUBbSV1Z24GJwKZ4rcbEmKBa+9DFnx5j\nFTANaIjlK4D1mfXPLSlfDLTn1C/Z72ngd8DcjL7P6IxbNbr2DcA24IL4+t25z33J/p8DVmf0/Tbg\ne51+A/uAvhn1lwC3x/IHU/kej3dCHOvy/pXAU7E8tdL/+VwtWHW/iF93mNkrZrY5lt8E2oFRBPsf\nibs9AsypjYXVIWkU8FngxyXVhfNF0mDg42b2MICZdZjZ6xTQF6AP8E5JfQlLSe+mIH6Y2Trgv12q\nu7O9CVgar9W/gB3U31pTPcYqM1sf7zWA9YQ1tHLqv1XychBwJKd+5AbgceA/NdDurRmplehfCywz\ns90AZvZqZv1SrgF+mVHbgPNi+Txgn5mlWoWsEv2JwBoAM9sOjJH0nhTi3cSxUmYDP4v7/gVokDSi\np+PmSrDKLeKXMihlQdIYQpa7HhhhZnshJGHA8NpZVhX3AN8i/LN0UkRfxgKvSno4Npc/IOlcCuaL\nme0Bvg+8REisXjez1RTMjy4M78b2rnFgN/UXB6qNVV8GVuTWlzRHUjvwW+C6nPqSRgJzzOxHpE12\nKj33l8ZumqdSdhNVqD8BGCbpGUkbJc3LrA+ApHMIrafLMmrfB0yUtAfYAtyYSLtS/S3AXABJU4DR\nhMaOHJxS7PJH5VSIpEGEX2w3xpasrrMD6n62gKRZwN7YIneywFj3vhC6mRqB+82sETgA3ErBrksc\nPzGb0DQ+ktCS9UUK5kcPFNn2bpH0SWAhoRs0K2b2hJl9iNA6+N3M8vdyvM851zh7FhhtZpMIX/hP\nZNSGY3HnSkKCc7ukFE+ZrpargHVmtj+j5meATWY2EvgocH/8XszFncBQSa3A9YThBocz6ldNrgRr\nNyHb7GRUrCsEsevmceBRM3syVu/tbCKUdD5pm8p7i+lAk6SdhKblT0l6FHilgL7sAl42s87ljpcR\nAl/RrsungZ1m9pqZHQZ+A3yM4vlRSne27wbeV7JfPcaBimKVpA8DDwBNZnayroVe0e8kdm2MkzQs\no/5kYKmkF4AvEL5om3Jom9mbnV2kZrYC6JfZ913ASjM7aGb7gD8AH8mo38nVpOserFR7IbAcwMz+\nCbxAGI+cRd/M/mdm15lZo5nNJ7SM70ykX4l9VceuXAnWRmC8pAsl9SfcHC2ZtFPwEPCcmf2gpK4F\nWBDL84Enu36o3jCzb5vZaDMbR7gGa8xsHqGbYUHcrSi+7AVeljQhVl1OGHxatOvyEjBN0kBJIvjx\nHMXyQxzfitGd7S3A1QqzJMcC4wkLetYTPcYqSaMJCf28+EWTW//9JeVGwgSJ13Lpm9m4uI0l/PD8\nupmliOeV+D6ipDyFMEkim++Ee/kySX3ikISphHG5ufRRmJU7g7QxoRLtFwk/CDuvwwTSJTiVXPsG\nSf1i+SvA72NvUiq6xrFSWoAvRe1pwP7OYRAnJdUo/J42QnPqdsLA1ltz6SawezqhGXIzoUmyNfoy\nDFgdfVoFDKm1rVX6dXQ2TlF9Ifxy3BivzXLCDJ/C+QLcQQjSbYRB4f2K4gfwGLAHOERIFhcCQ7uz\nnTAT6fno78xa29+NTyfEKuBrwFdj+UHCDKrWGBM2ZNa/Bdga9f8EXJpTv8u+D5FoFmGFvl8ffd8E\n/BmYmtt34GbCj7k24IYa6M8HHkupW+G5fy+wMvrdRnjSQU79afH9dkJi35BQu1wc63re74uxawvQ\nWMlxfaFRx3Ecx3GcxPggd8dxHMdxnMR4guU4juM4jpMYT7Acx3Ecx3ES4wmW4ziO4zhOYjzBchzH\ncRzHSYwnWI7jOI7jOInxBMtxHMdxHCcxnmA5juM4juMk5v/moSrwNGG2UQAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pmc.Matplot.plot(AR1_MCMC_y0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Look what happened to the posterior! It has moved far from the true values of the parameters used to generate the data because poor Bayes Law is struggling to fit the early \"explosive\" observations. The way in which Bayes Law is able to generate a reasonable likelihood for the first observation is by driving $\\rho \\rightarrow 1$ and $\\sigma \\uparrow$ in order to raise the variance of the stationary distribution.\n", "\n", "This small example highlights the importance of specifying your initial conditions in the proper way. If initial conditions are improperly specified then your estimates will be very far from their true values and may tell a different story than they would otherwise have told.\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.1" } }, "nbformat": 4, "nbformat_minor": 0 }