{ "cells": [ { "cell_type": "code", "execution_count": 3, "id": "5aca2f6b-af4d-4e32-975e-fb93a637afff", "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 scipy.stats\n", "import statsmodels.api as sm" ] }, { "cell_type": "markdown", "id": "3fbac626-d91f-407f-bd96-7c48a48c516f", "metadata": {}, "source": [ "# Model Specification " ] }, { "cell_type": "markdown", "id": "c7c1736e-fe01-477c-ae7e-ee9aad9a2177", "metadata": {}, "source": [ "So far we have assumed correct **model specification**, if not correctly specified, we are encountering **model specification errors**. Here's the ideal criteria for specifying a model\n", "1. Be able to make logical prediction.\n", "2. Be consistent with theory.\n", "3. Independent variables are uncorrelated with disturbance terms.\n", "4. Parameter constancy.\n", "5. Residuals should be white noise.\n" ] }, { "cell_type": "markdown", "id": "aa6d4c29-23b1-4a80-bd42-e415e56a3e6a", "metadata": {}, "source": [ "However, in practice, model specification errors are almost unavoidable, here are the list of how we might encounter specification errors.\n", "\n", "1. Omission of a relevant variable(s).\n", "2. Inclusion of an unnecessary variable(s).\n", "3. Adoption of the wrong functional form.\n", "4. Errors of measurement in data collection process.\n", "5. Incorrect specification of the stochastic error term.\n", "6. Assumption that the error term is normally distributed." ] }, { "cell_type": "markdown", "id": "b8325720-6b6a-4500-a28a-51847e582f3b", "metadata": {}, "source": [ "This chapter is exceedingly difficult in multiple independent variable case without linear algebra, we will simply scratch the surface of the topic. " ] }, { "cell_type": "markdown", "id": "d8de37c0-3647-4903-ab8f-e38b6974831d", "metadata": {}, "source": [ "# Omission of A Relevant Variable " ] }, { "cell_type": "markdown", "id": "a2dfb3e2-1bc9-4f7f-a774-549955cc16be", "metadata": {}, "source": [ "Suppose the true relationship is\n", "$$\n", "Y_i = 3 + 4X_2 + 5X_3 + u_i\n", "$$" ] }, { "cell_type": "markdown", "id": "bc705db9-e9ce-4ecc-942d-b1221bdd1452", "metadata": {}, "source": [ "However we estimate only $X_3$, i.e. the model with omission of $X_2$\n", "$$\n", "\\hat{Y} = b_1+b_3X_3\n", "$$" ] }, { "cell_type": "markdown", "id": "716e9e14-a4f2-4638-a3df-f007e911556f", "metadata": {}, "source": [ "We will do $100000$ rounds of Monte Carlo simulation (might take some time, dial it down if necessary). " ] }, { "cell_type": "code", "execution_count": 155, "id": "1fc0abd4-ae83-4348-afc5-21a0ad6e29d6", "metadata": {}, "outputs": [], "source": [ "n = 100\n", "beta1, beta2, beta3 = 3, 4, 5\n", "X2 = np.random.rand(n)\n", "X3 = np.random.rand(n)\n", "beta3_hat_list = []\n", "for i in range(100000):\n", "\n", " u = np.random.randn(n)\n", " Y = beta1 + beta2 * X2 + beta3 * X3 + u\n", " df = pd.DataFrame(\n", " np.concatenate(\n", " (Y[:, np.newaxis], X2[:, np.newaxis], X3[:, np.newaxis]), axis=1\n", " ),\n", " columns=[\"Y\", \"X2\", \"X3\"],\n", " )\n", "\n", " X = df[\"X3\"]\n", " Y = df[\"Y\"]\n", "\n", " X = sm.add_constant(X) # adding a constant\n", "\n", " model = sm.OLS(Y, X).fit()\n", " beta3_hat_list.append(model.params[1])" ] }, { "cell_type": "markdown", "id": "7a71e466-df63-4358-8fd3-f2583481115c", "metadata": {}, "source": [ "Print the full reports of last round of simulation." ] }, { "cell_type": "code", "execution_count": 156, "id": "f84e2d87-b8e2-4cb1-ba61-bab7a5e03a63", "metadata": {}, "outputs": [], "source": [ "model = sm.OLS(Y, X).fit()\n", "print_model = model.summary()" ] }, { "cell_type": "code", "execution_count": 157, "id": "b28ba3d6-2412-41e1-9ea7-117417d4be94", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(figsize=(12, 7))\n", "ax.hist(beta3_hat_list, bins=150)\n", "ax.axvline(x=np.mean(beta3_hat_list), color=\"tomato\", label=\"mean of $b_3$\")\n", "ax.set_xlabel(\"$b_3$\")\n", "ax.set_title(\"Sampling Distribution of $b_3$\")\n", "ax.legend()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 97, "id": "cbad7b5f-d207-4b38-9448-003928e27c57", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The mean of b3 is 5.998913199703797.\n" ] } ], "source": [ "print(\"The mean of b3 is {}.\".format(np.mean(beta3_hat_list)))" ] }, { "cell_type": "markdown", "id": "08bc587a-9a2a-4b46-9853-4c5e9478e81f", "metadata": {}, "source": [ "With $100000$ rounds of simulation, we could easily notice the sample distribution is biased, and mean of $b_3$ is far from true value $5$. We will demonstrate why this is the case without linear algebra." ] }, { "cell_type": "markdown", "id": "4833e7b7-4787-41c1-b535-86a362f54686", "metadata": {}, "source": [ "We have shown in the second chapter, that estimator $b_3$ in a two-independent variable case is\n", "$$\n", "b_{3}=\\frac{\\operatorname{Cov}\\left(X_{3}, Y\\right) \\operatorname{Var}\\left(X_{2}\\right)-\\operatorname{Cov}\\left(X_{2}, Y\\right) \\operatorname{Cov}\\left(X_{3}, X_{2}\\right)}{\\operatorname{Var}\\left(X_{3}\\right) \\operatorname{Var}\\left(X_{2}\\right)-\\left[\\operatorname{Cov}\\left(X_{3}, X_{2}\\right)\\right]^{2}}\n", "$$" ] }, { "cell_type": "markdown", "id": "1b489b69-d358-4b96-8e2f-bbbcc068d208", "metadata": {}, "source": [ "However, without being aware the omission of $\\beta_2$, we end up using estimator\n", "$$\n", "b_{3}=\\frac{\\operatorname{Cov}\\left(X_{3}, Y\\right)}{\\operatorname{Var}\\left(X_{3}\\right)}\n", "$$\n", "Substitute $Y$ by its true relationship can share some insight\n", "$$\n", "\\begin{aligned}\n", "b_{3} &=\\frac{\\operatorname{Cov}\\left(X_{3}, Y\\right)}{\\operatorname{Var}\\left(X_{3}\\right)}=\\frac{\\operatorname{Cov}\\left(X_{3},\\left[\\beta_{1}+\\beta_{2} X_{2}+\\beta_{3} X_{3}+u\\right]\\right)}{\\operatorname{Var}\\left(X_{3}\\right)} \\\\\n", "&=\\frac{1}{\\operatorname{Var}\\left(X_{3}\\right)}\\left[\\operatorname{Cov}\\left(X_{3}, \\beta_{1}\\right)+\\operatorname{Cov}\\left(X_{3}, \\beta_{2} X_{2}\\right)+\\operatorname{Cov}\\left(X_{3}, \\beta_{3} X_{3}\\right)+\\operatorname{Cov}\\left(X_{3}, u\\right)\\right] \\\\\n", "&=\\frac{1}{\\operatorname{Var}\\left(X_{3}\\right)}\\left[0+\\beta_{2} \\operatorname{Cov}\\left(X_{2}, X_3\\right)+\\beta_{3} \\operatorname{Var}\\left(X_{3} \\right)+\\operatorname{Cov}\\left(X_{3}, u\\right)\\right] \\\\\n", "&=\\beta_{3}+\\beta_{2} \\frac{\\operatorname{Cov}\\left(X_{2}, X_{3}\\right)}{\\operatorname{Var}\\left(X_{3}\\right)}+\\frac{\\operatorname{Cov}\\left(X_{3}, u\\right)}{\\operatorname{Var}\\left(X_{3}\\right)}\n", "\\end{aligned}\n", "$$" ] }, { "cell_type": "markdown", "id": "33e6de26-1155-40c0-80f7-35651d80cf3b", "metadata": {}, "source": [ "This is the relationship of $b_3$ and $\\beta_3$ when $b_2$ is omitted in the model. Whether upward biased or downward biased, it depends on the sign of $\\beta_2$ and covariance of $X_2$ and $X_3$. And it's safe to assume $\\operatorname{Cov}(X_3, u )=0$." ] }, { "cell_type": "markdown", "id": "9437230f-02f3-4e53-809f-a95b82019704", "metadata": {}, "source": [ "As you can see now the biased term is \n", "$$\n", "\\beta_{2} \\frac{\\operatorname{Cov}\\left(X_{2}, X_{3}\\right)}{\\operatorname{Var}\\left(X_{3}\\right)}\n", "$$\n", "Therefore $b_3$ can be unbiased after all on condition that $\\operatorname{Cov}(X_2, X_3)=0$, however this is extremely unlikely. \n", "\n", "In general, omission of relevant variables also cause invalid hypotheses test, however we'll skip it entirely." ] }, { "cell_type": "markdown", "id": "c046b303-4a86-4139-be83-3ea52533ee04", "metadata": {}, "source": [ "# Inclusion Of An Unnecessary Variable " ] }, { "cell_type": "markdown", "id": "3482e7c0-7d97-4050-a6df-fb7318d79b7a", "metadata": {}, "source": [ "Now again suppose the true relationship is\n", "$$\n", "Y_i = 3 + 4X_2 + u_i\n", "$$" ] }, { "cell_type": "markdown", "id": "312fabf4-1a75-4686-b595-249f754b735b", "metadata": {}, "source": [ "However we include an unnecessary variable $X_3$, i.e.\n", "$$\n", "\\hat{Y} = b_1+b_2X_2 + b_3X_3\n", "$$" ] }, { "cell_type": "code", "execution_count": 149, "id": "5c438595-c432-4d1d-bf9b-c9dca403ec7e", "metadata": {}, "outputs": [], "source": [ "n = 100\n", "beta1, beta2 = 3, 4\n", "X2 = np.random.rand(n)\n", "X3 = np.random.rand(n)\n", "beta2_hat_list = []\n", "for i in range(100000):\n", "\n", " u = np.random.randn(n)\n", " Y = beta1 + beta2 * X2 + u\n", " df = pd.DataFrame(\n", " np.concatenate(\n", " (Y[:, np.newaxis], X2[:, np.newaxis], X3[:, np.newaxis]), axis=1\n", " ),\n", " columns=[\"Y\", \"X2\", \"X3\"],\n", " )\n", "\n", " X = df[[\"X2\", \"X3\"]]\n", " Y = df[\"Y\"]\n", "\n", " X = sm.add_constant(X) # adding a constant\n", "\n", " model = sm.OLS(Y, X).fit()\n", " beta2_hat_list.append(model.params[1])" ] }, { "cell_type": "markdown", "id": "cbf468f3-698f-4de0-8699-ada4b8dc2ba7", "metadata": {}, "source": [ "Print the full reports of last round of simulation." ] }, { "cell_type": "code", "execution_count": 150, "id": "3d45f7c9-f8cc-4a34-97dc-e4f444888e72", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " OLS Regression Results \n", "==============================================================================\n", "Dep. Variable: Y R-squared: 0.642\n", "Model: OLS Adj. R-squared: 0.635\n", "Method: Least Squares F-statistic: 87.05\n", "Date: Thu, 16 Sep 2021 Prob (F-statistic): 2.25e-22\n", "Time: 11:26:42 Log-Likelihood: -146.33\n", "No. Observations: 100 AIC: 298.7\n", "Df Residuals: 97 BIC: 306.5\n", "Df Model: 2 \n", "Covariance Type: nonrobust \n", "==============================================================================\n", " coef std err t P>|t| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "const 2.6164 0.271 9.646 0.000 2.078 3.155\n", "X2 4.3287 0.328 13.187 0.000 3.677 4.980\n", "X3 0.1170 0.377 0.311 0.757 -0.631 0.865\n", "==============================================================================\n", "Omnibus: 0.315 Durbin-Watson: 1.933\n", "Prob(Omnibus): 0.854 Jarque-Bera (JB): 0.457\n", "Skew: 0.113 Prob(JB): 0.796\n", "Kurtosis: 2.758 Cond. No. 5.20\n", "==============================================================================\n", "\n", "Notes:\n", "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n" ] } ], "source": [ "print_model = model.summary()\n", "print(print_model)" ] }, { "cell_type": "code", "execution_count": 151, "id": "cf5f5146-47d3-427c-9f1f-e6667ecd9b56", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(figsize=(12, 7))\n", "ax.hist(beta2_hat_list, bins=150)\n", "ax.axvline(x=np.mean(beta2_hat_list), color=\"tomato\", label=\"mean of $b_2$\")\n", "ax.set_xlabel(\"$b_2$\")\n", "ax.set_title(\"Sampling Distribution of $b_2$\")\n", "ax.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "69382d5a-1c33-4107-99d4-e07eb127a160", "metadata": {}, "source": [ "It turns out that the $b_2$ is not biased after all! However, let's compare with $b_2$'s sampling distribution which is from correct specification." ] }, { "cell_type": "code", "execution_count": 133, "id": "efe4a15b-e85b-499f-8baa-6472137037cc", "metadata": {}, "outputs": [], "source": [ "beta2_hat_list_correct = []\n", "for i in range(100000):\n", "\n", " u = np.random.randn(n)\n", " Y = beta1 + beta2 * X2 + u\n", " df = pd.DataFrame(\n", " np.concatenate((Y[:, np.newaxis], X2[:, np.newaxis]), axis=1),\n", " columns=[\"Y\", \"X2\"],\n", " )\n", "\n", " X = df[\"X2\"]\n", " Y = df[\"Y\"]\n", "\n", " X = sm.add_constant(X) # adding a constant\n", "\n", " model = sm.OLS(Y, X).fit()\n", " beta2_hat_list_correct.append(model.params[1])" ] }, { "cell_type": "code", "execution_count": 146, "id": "83e80796-d9cc-4acc-9136-f466aa7dd229", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(figsize=(12, 7))\n", "ax.hist(beta2_hat_list, bins=150, alpha=0.5, label=\"$b_2$ With An Unnecessary Variable\")\n", "ax.hist(\n", " beta2_hat_list_correct,\n", " bins=150,\n", " alpha=0.3,\n", " color=\"tomato\",\n", " label=\"$b_2$ With Correct Model\",\n", ")\n", "ax.set_xlabel(\"$b_2$\")\n", "ax.set_title(\"Sampling Distribution of $b_2$\")\n", "ax.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "abaa1ef7-cbb2-451f-ad78-3a1d15b4c655", "metadata": {}, "source": [ "If you take a closer look at the distribution, they are actually different, the $b_2$ estimated by correct model has a more concentrated distribution, i.e. the middle area is higher, in contrast the $b_2$ estimated by incorrectly specified model has fatter tails. We can check their standard deviation." ] }, { "cell_type": "code", "execution_count": 135, "id": "1749a2e0-2d28-43cf-aa2e-4c33cc1e4125", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.3638018676170749" ] }, "execution_count": 135, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.std(beta2_hat_list)" ] }, { "cell_type": "code", "execution_count": 136, "id": "03317d9c-8d83-4237-82e8-1c2c9e1b15db", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.35998511144681705" ] }, "execution_count": 136, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.std(beta2_hat_list_correct)" ] }, { "cell_type": "markdown", "id": "f8de9ce1-929f-4b28-98e4-864db433de07", "metadata": {}, "source": [ "To remind you why this is the case, here we reproduce the formula of standard deviation of $b_2$ in two independent variable regression\n", "$$\n", "\\sigma_{b_{2}}^{2}=\\frac{\\sigma_{u}^{2}}{n \\operatorname{Var}\\left(X_{2}\\right)} \\frac{1}{1-r_{X_{2} X_{3}}^{2}}\n", "$$\n", "$\\sigma_{b_{2}}^{2}$ is enlarged by a the correlation $r^2_{X_2X_3}$, which is why the blue color distribution has fatter tails." ] }, { "cell_type": "markdown", "id": "efe4ae12-7ca8-41cb-adb0-99ec5f96fa38", "metadata": {}, "source": [ "
\n", " Asymmetry of Specification Errors
\n", "The asymmetry of two types of specification errors: adding irrelevant variables still grant us unbiased and consistent estimate though standard errors are inflated, however excluding a relevant variable would cause bias and inconsistency.

\n", " \n", "Does it suggest that we should err on the side of adding irrelevant variables? In the field of econometrics, not really. The _best practice_ is to include only explanatory variables that, on theoretical grounds, directly influence the dependent variable and that are not accounted for by other\n", "included variables.\n", "
" ] }, { "cell_type": "markdown", "id": "9dcff64e-6d2d-4952-98a4-75d28131326f", "metadata": {}, "source": [ "# Tests of Specification Errors " ] }, { "cell_type": "markdown", "id": "3a2b4be5-9f33-466f-8800-e8d86f09f4dd", "metadata": {}, "source": [ "Truth to be told, we will never be sure how the observed data were generated, but we can make educated guess about specification errors with the help of statistical tests. " ] }, { "cell_type": "markdown", "id": "45e21a8a-667b-4277-b65a-82500cd19ad0", "metadata": {}, "source": [ "## Tests of Overfitted Models " ] }, { "cell_type": "markdown", "id": "be87926f-89d0-4ac4-8cc1-b9007e0109db", "metadata": {}, "source": [ "The most convenient methods of detecting unnecessary variables are $F$ and $t$-tests, take a look the regression results with unnecessary variable, the $X_3$ has a p-value of $ 0.757$, we can surely deem $X_3$ as unnecessary.\n", "\n", "The purist econometrician objects the method of adding independent variables iteratively by testing $t$, however data scientists have a more practical view that they believe the model should be driven by data, i.e. the model should learn the data and express the data." ] }, { "cell_type": "markdown", "id": "51065062-1ded-400f-bd38-ce7790361084", "metadata": {}, "source": [ "## Tests of Underfitted Models " ] }, { "cell_type": "markdown", "id": "50b23435-96ab-4926-bc64-92f8e9b74cbf", "metadata": {}, "source": [ "To detect an underfitting model is more than looking at $t$'s or $F$'s, we also need to investigate the broad features of results, such $R^2$, signs of estimates, residuals and other relevant tests. " ] }, { "cell_type": "markdown", "id": "fb0eafa5-fc95-4f59-bfce-3e2ae327d217", "metadata": {}, "source": [ "### Investigating Residuals " ] }, { "cell_type": "markdown", "id": "80565bf1-f14c-4112-993c-bb4e74007ab9", "metadata": {}, "source": [ "Here we reproduce the underfitted model example. The unfitted model obviously has larger dispersion of residual. " ] }, { "cell_type": "code", "execution_count": 185, "id": "401d8f8a-7541-46e1-aa41-553e5e98be1b", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n = 200\n", "beta1, beta2, beta3 = 3, 4, 5\n", "X2 = np.random.rand(n)\n", "X3 = np.random.rand(n)\n", "\n", "u = np.random.randn(n)\n", "Y = beta1 + beta2 * X2 + beta3 * X3 + u\n", "df = pd.DataFrame(\n", " np.concatenate((Y[:, np.newaxis], X2[:, np.newaxis], X3[:, np.newaxis]), axis=1),\n", " columns=[\"Y\", \"X2\", \"X3\"],\n", ")\n", "\n", "X_underfit = df[\"X2\"]\n", "Y = df[\"Y\"]\n", "X_underfit = sm.add_constant(X_underfit)\n", "model_underfit = sm.OLS(Y, X_underfit).fit()\n", "\n", "X_wellfit = df[[\"X2\", \"X3\"]]\n", "X_wellfit = sm.add_constant(X_wellfit)\n", "model_wellfit = sm.OLS(Y, X_wellfit).fit()\n", "\n", "\n", "fig, ax = plt.subplots(nrows=2, ncols=1, figsize=(14, 12))\n", "ax[0].scatter(np.arange(len(model_underfit.resid)), model_underfit.resid)\n", "ax[0].set_xlim(0, n)\n", "ax[0].set_ylim(-10, 10)\n", "ax[0].grid()\n", "ax[0].set_title(\"Residuals Plot of Underfitted Model\")\n", "\n", "ax[1].scatter(np.arange(len(model_wellfit.resid)), model_wellfit.resid)\n", "ax[1].set_xlim(0, n)\n", "ax[1].set_ylim(-10, 10)\n", "ax[1].grid()\n", "ax[1].set_title(\"Residuals Plot of Wellfitted Model\")\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "abd55090-624c-4a91-a7cc-c40b465c0e88", "metadata": {}, "source": [ "### Ramsey’s RESET Test" ] }, { "cell_type": "markdown", "id": "04e14dbd-6d2a-43d5-9256-f982dde479b5", "metadata": {}, "source": [ "Ramsey's **Regression Equation Specification Error Test** (RESET) is general test for specification error. " ] }, { "cell_type": "code", "execution_count": 216, "id": "57b68559-264e-41e1-b333-f19185f02043", "metadata": {}, "outputs": [], "source": [ "reset_results = sm.stats.diagnostic.linear_reset(model_underfit)" ] }, { "cell_type": "code", "execution_count": 217, "id": "14e69739-8320-4e81-b3e7-05c2e80e0c7a", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n" ] } ], "source": [ "print(reset_results.summary())" ] }, { "cell_type": "markdown", "id": "09583bbb-30aa-49d2-bf12-548c3f572855", "metadata": {}, "source": [ "### Lagrange Multiplier (LM) Test for Adding Variables" ] }, { "cell_type": "markdown", "id": "5b1a4aac-616c-4604-a383-5bcd85ca7d0a", "metadata": {}, "source": [ "# $AIC$ and $SIC$ " ] }, { "cell_type": "markdown", "id": "a5318e11-7797-416d-a0dd-445f5bdbcc6f", "metadata": {}, "source": [ "Besides $R^2$ and $\\bar{R}^2$ that were discussed in the first two chapters. Here are another two statistics for model selection: **Akaike’s Information Criterion** (AIC) and **Bayesian information criterion** (BIC).\n", "\n", "Both statistics are standard output printed in the estimation report, you can check the report above." ] }, { "cell_type": "markdown", "id": "53b8d210-2760-4b62-8108-4a482f571cc4", "metadata": {}, "source": [ "## **Akaike’s Information Criterion**" ] }, { "cell_type": "markdown", "id": "615f68cd-3e97-451a-8b1c-48f168c896f6", "metadata": {}, "source": [ "$AIC$ imposes stronger penalty than $\\bar{R}^2$. The formula is \n", "$$\n", "\\mathrm{AIC}=e^{2 k / n} \\frac{\\sum e_{i}^{2}}{n}=e^{2 k / n} \\frac{\\mathrm{RSS}}{\\mathrm{n}}\n", "$$" ] }, { "cell_type": "markdown", "id": "4c3cc855-f647-49e0-81a3-4030345aaba0", "metadata": {}, "source": [ "$AIC$ is commonly used in time series model to determine the lag length, where $n$ is number of observations and $k$ is the number of independent variables." ] }, { "cell_type": "markdown", "id": "2cf003f9-2901-4370-85db-33c6d7f595ba", "metadata": {}, "source": [ "## **Bayesian Information Criterion**" ] }, { "cell_type": "markdown", "id": "b5c61547-141d-4f2e-ad43-99d553ffb826", "metadata": {}, "source": [ "$BIC$ imposes even harsher penalty than $AIC$\n", "$$\n", "\\mathrm{SIC}=n^{k / n} \\frac{\\sum e^{2}_i}{n}=n^{k / n} \\frac{\\mathrm{RSS}}{n}\n", "$$" ] }, { "cell_type": "markdown", "id": "c7167eb7-a6d6-42c2-be06-dd60514b1c07", "metadata": {}, "source": [ "As you can see from their formula, both criteria prefer smaller result, because $RSS$ is smaller. \n", "\n", "We can plot the $AIC$ and $BIC$ as functions of number of variables, it is easy to see that $BIC$ has higher penalty, however it doesn't meant $BIC$ is superior than $AIC$." ] }, { "cell_type": "code", "execution_count": 232, "id": "ee946f87-452f-4173-8bb9-62ffb774f1ac", "metadata": {}, "outputs": [ { "data": { "image/png": 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S6toVfkMIxSTB9xcxxrtarF8NXAqcF2M02EqSpPQt/g389v2wZ2d2bepZcPkPoWx0amUpNxwy/IYQAnALsCTG+LUW6xcBHwPOjjHWdF+JkiRJ7bC3Gv78cXjmZ9m1UADn3ABnfgQKCtOrTTmjPTu/LwXeDCwIIczLrH0C+BbQH7gvycc8FmO8tjuKlCRJOqh1TyU3tW1fkV0rn5i0OUw+Pb26lHPaM+3hn0Bo49Ifu74cSZKkDmhsgIe/kRxR3FifXZ9zJbziqzBwSFqVKUd5m6MkSeqdKtbCXe9qfVNbv7Ik9B77Wght7d0p3xl+JUlS77PwLvj9B2FPRXZtwslJm8OwqamVpdxn+JUkSb3H3ir408dgXosJq6EAzvr3ZH6vs3t1CL5DJElS77D2SbjzHbBjZXZtyCS4/H9h0qnp1aVexfArSZJyW2MD/PNr8MAXITZk1499HVzyFRhQnl5t6nUMv5IkKXftXJPc1Lbmkexa/8Hwiq/BsVelV5d6LcOvJEnKTQt+Db//MOxtcVPbxFOTm9qGTk6vLvVqhl9JkpRb9lTCHz8K82/ProVCOOfjcMaHvalNh8V3jyRJyh0vPpHc1LZzdXZtyGS44ocw8eT06lKfYfiVJEnpa6iHh74K//hS65vajnsDXPxlGDA4vdrUpxh+JUlSunashruugRcfy671L4dLvwbHXJleXeqTDL+SJCk9838Jf/gI7K3Mrk06PbmpbcjE9OpSn2X4lSRJPW9PRRJ6F/wquxYK4dxPwBkfgoLC9GpTn2b4lSRJPWvNY3DnO6FiTXZt6FS44haYcGJ6dSkvGH4lSVLPaKiHB78MD34FYmN2/fg3wcU3Qf+y9GpT3jD8SpKk7rd9Jdz1Tlj7r+zagHJ45Tdh9mvSq0t5x/ArSZK6T4ww/w74w/VQW5Vdn3wGXP4DKJ+QXm3KS4ZfSZLUPXbvhD98GBbemV0rKIJzb4SXfsCb2pQKw68kSep6Kx+Ee94DFS9m14YdkZzUNv6E9OpS3jP8SpKkrlNbA/d/Fh7/fuv1E94CL/8i9C9Npy4pw/ArSZK6xov/gnuuhW3Ls2sDhyY3tR39qvTqklow/EqSpMNTvxf+fhM8/I3WI8yOfDlc9i0oG5NaadK+DL+SJKnzNsyHu6+FzYuya/3K4KIvwkveBCGkV5vUBsOvJEnquIZ6+OfX4R83QWN9dn3KmfDq/4Ehk9KrTToIw68kSeqYLc/B3e+C9U9n14oGwgWfhZPeCQUF6dUmHYLhV5IktU9jIzz+Pbj/P6F+T3Z9wknw6u/DiOnp1Sa1k+FXkiQd2o5Vydze1Q9n1wr7wbmfgNPf74EV6jUMv5Ik6cBihKd+DH+5Eep2ZdfHHAOv+QGMnp1aaVJnGH4lSVLbKtfDb66DFfdn10IhnPkROOujUNQvvdqkTjL8SpKk1mKE+b+EP30U9lRk10cclfT2Tjgxvdqkw2T4lSRJWdVb4A8fgiW/a7EY4LT3wss+CcUDUytN6gqGX0mSlFjyO/jdB6Fma3ZtyGR49fdgyktTK0vqSoZfSZLy3e4d8KePwfw7Wq/PfTtc8DnoX5pOXVI3MPxKkpTPlv8VfvM+qFqfXSsbB6/6Nkw/P726pG5i+JUkKR/trYJ7PwVP/aj1+rGvh4tvgoFD06lL6maGX0mS8s2qh+Ged8PO1dm1QSPg0q/D0ZelV5fUAwy/kiTli7rdcP/n4LH/AWJ2fealcOk3oHRkWpVJPeaQ4TeEMBH4KTAGaARujjF+M4QwDLgDmAKsAl4bY9zRfaVKkqROW/cU3H0tbH0uu9a/HC75Chz7WgghvdqkHlTQjsfUAx+JMc4CTgXeG0I4Gvg4cH+M8Ujg/szPJUlSLqmvhb99Hn54Qevge8R58J5H4bjXGXyVVw658xtj3ABsyPy4KoSwBBgPvAo4J/OwnwB/Bz7WLVVKkqSO27QI7n4XbFyQXSsugZf/F5z4VkOv8lKHen5DCFOAlwCPA6MzwZgY44YQwqgDPOca4BqASZMmHVaxkiSpHepr4eFvwD++DI112fVJp8Or/weGTU2tNClt7Q6/IYRS4E7ggzHGytDOvy3GGG8GbgaYO3duPMTDJUnS4Vj3NPz2fbBpYXatsD+c/2k45d1Q0J6OR6nvalf4DSEUkwTfX8QY78osbwohjM3s+o4FNndXkZIk6RDqdsMDX4BHvwOxMbs+/sTkeOKRM9KrTcoh7Zn2EIBbgCUxxq+1uPRb4Grgpszn33RLhZIk6eBW/TPZ7d3+QnataCC87JNw6ruhoDC92qQc056d35cCbwYWhBDmZdY+QRJ6fxlC+DdgDXBVt1QoSZLatqcS/vppePLW1utTzoTLvgXDpqVTl5TD2jPt4Z/AgRp8z+vaciRJUrs8dy/8/kNQuTa71n8wXPg5OOFqJzlIB+AJb5Ik9Sa7tsFfboD5d7ReP+oieMXXoHx8OnVJvYThV5Kk3iBGWHQ3/PGjULM1uz5oOFz8ZZhzhbu9UjsYfiVJynWVG+APH4Flf2i9fsxVcNFNUDIinbqkXsjwK0lSrooRnvkZ/OWTsLciu142Di79Gsy4OL3apF7K8CtJUi7avhJ+9wFY+Y/W6ye+DS74LAwoT6cuqZcz/EqSlEsaG+DxH8DfPgd1Ndn1oVOT8WVTz0qvNqkPMPxKkpQrNi+F314Ha/+VXQsFcOp74Nwbod+g9GqT+gjDryRJaauvhYe/AQ9+BRpqs+ujjobLvgMTTkytNKmvMfxKkpSmdU8nRxNvWphdKyiGs66HMz4MRf3Sq03qgwy/kiSloW43PPAFePQ7EBuz6+NPTHZ7Rx+dXm1SH2b4lSSpp616ONnt3b4iu1Y0EF72STj13VBQmF5tUh9n+JUkqafsqYS/fgaevKX1+pQzk0kOw6alUpaUTwy/kiT1hOfuhd9/CCrXZtf6D4YLPwcnXO3RxFIPMfxKktSddm2Dv9wA8+9ovX7URfCKr0H5+HTqkvKU4VeSpO4QIyy6G/74UajZml0fNBwu/jLMucLdXikFhl9JkrrajtXwx+vh+Xtbrx9zFVx0E5SMSKcuSYZfSZK6TEMdPPpd+PtNUL87u142Di79Gsy4OL3aJAGGX0mSusaax+H3H4TNi1uvn/g2uOCzMKA8lbIktWb4lSTpcNRsT8aXPf2T1uuj58ClX4eJJ6dSlqS2GX4lSeqMGGH+L+Evn2h9Q1vxIDjnhuSwisLi9OqT1CbDryRJHbV1Ofzhw7DyH63Xj7oYLvkyDJmUTl2SDsnwK0lSe9XtgYe/AQ99FRpqs+tl45LQO/NSx5dJOc7wK0lSe7zwj2S3d9vy7FoogFOuhXM/Af3L0qtNUrsZfiVJOpjqLXDvjfuf0DbuJXDpN2Dc8WlUJamTDL+SJLWlsRGe+Snc92nYszO73q8Mzv80zH07FBSmVp6kzjH8SpK0r02L4fcfghcfa70++zXw8i/C4LHp1CXpsBl+JUlqUlsD//gSPPodaKzPrg+ZDK/4Ghx5fnq1SeoShl9JkgCeuxf++BHYuSa7VlAEp78fzvoo9BuUXm2SuozhV5KU3yo3wJ8/Bot/03p90mnJCW2jZqVTl6RuYfiVJOWnxgb41w/h/s9BbVV2feBQuOA/4fg3QUFBevVJ6haGX0lS/ln/DPzug7BhXuv1494IF34OSkakUZWkHmD4lSTljz2V8MB/wRM3Q2zMrg8/MmlxmHpmerVJ6hGGX0lS3xcjLPkt/OljULUhu17YH866Hl76ASjqn159knqM4VeS1LftWA1//Cg8/5fW69POScaXDT8ilbIkpcPwK0nqmxrq4NHvJnN762qy6yUjk4MqjrkSQkivPkmpMPxKkvqeFQ8kLQ5bl7VeP/FtydHEA4emU5ek1Bl+JUl9x84X4d4b95/ZO2o2vPIbMPHkVMqSlDsMv5Kk3q9uDzzybXjoq1C/O7verxTO/hic+m4oLE6vPkk545DhN4RwK3ApsDnGOCezdjzwfWAAUA+8J8b4RDfWKUlS25b9OTmhbceq1uvHvi45rKJsTCplScpN7dn5/THwHeCnLda+DHw2xvinEMIlmZ+f0+XVSZJ0INtWwJ9v2H+Kw+hj4JKvwOTT0qlLUk47ZPiNMT4YQpiy7zIwOPPjcmB9F9clSVLbancl7Q2PfBsaarPrA8rhZZ9KbmortKtPUts6+93hg8BfQgj/DRQApx/ogSGEa4BrACZNmtTJX06SlPdihMX3wF9uhMp1LS4EOOEtcN5/eCyxpEPqbPh9N/ChGOOdIYTXArcA57f1wBjjzcDNAHPnzo2d/PUkSfls85LkoIpVD7VeHz83aXEYf0I6dUnqdTobfq8GPpD58a+AH3ZNOZIktbCnAv7+JXj8+xAbsuuDRsAFn4Xj3ggFBenVJ6nX6Wz4XQ+cDfwdeBnwfFcVJEkSjY0w/w647z9g1+bseiiEk6+Bcz4OA4ekVp6k3qs9o85uI5nkMCKEsBb4NPBO4JshhCJgD5meXkmSDtv6eUmLw9p9JmhOPgMu+TKMnp1KWZL6hvZMe3jDAS6d2MW1SJLyWc12+Nvn4MkfkQwVyigbBy//PMy+HEJIrTxJfYOzYCRJ6WpsgKd/Avf/J+zekV0vKIbTr4Mzr4f+penVJ6lPMfxKktLz4hPwx+thw7Ot16efDxd9CUZMT6cuSX2W4VeS1POqNsFfPwPP/l/r9SGT4aKbYMbFtjhI6haGX0lSz2mogyduhr/fBHsrs+tFA+CMD8NL3w/FA9OrT1KfZ/iVJPWMlQ8mUxy2LG29PuuVcOF/wdDJ6dQlKa8YfiVJ3atiLdz7SVh0d+v14UfCxV+C6eelU5ekvGT4lSR1j7o98Nh34cH/hrqa7Hq/Ujj7Y3DKtVDUL736JOUlw68kqWvFCIvvgfs+DTtXt752zGvhgv+EwWNTKU2SDL+SpK6z7in4y42w5tHW66PnwCVfgcmnp1OXJGUYfiVJh69iXXJIxfzbW68PHAbnfgJOfBsU+keOpPT5nUiS1Hm1u+Dhb8HD34T63dn1gmI45V1w1vUwcGh69UnSPgy/kqSOa2yE+XfA/Z+Fqg2tr828NOnrHX5EOrVJ0kEYfiVJHbP6EfjzDbBhXuv1McfAy78AU89KpSxJag/DrySpfbavhPv+A5b8tvV66Wh42afg+DdCQWE6tUlSOxl+JUkHt6cimdX7+PehoTa7XjQATn8fvPSD0L80tfIkqSMMv5KktjXUw9M/gQe+ADVbW1875io479MwZGI6tUlSJxl+JUn7W35/Mq93y5LW6xNOgpd/ESaelE5dknSYDL+SpKwty+DeT8Lz97ZeL58I538G5lwBIaRSmiR1BcOvJAl2bYO/fxGevBViQ3a9Xymc8SE47b1QPDC9+iSpixh+JSmf1dfCEzfDg19ObmxrFuAlb0qmOJSNTq08Sepqhl9JykcxwtI/wH2fgu0vtL425cxkXu/YY9OpTZK6keFXkvLNhvnwl0/Aqodarw+bBhd+HmZcYl+vpD7L8CtJ+aJqI/ztc/DML4CYXR9QDmd/DE56JxT1S608SeoJhl9J6uvqdsOj34GHvg51u7LroRBO+jc45wYYNCy9+iSpBxl+JamvihEW3gl//QxUvNj62pEXJi0OI2ekUpokpcXwK0l90coH4b7/gPXPtF4fdXQSeqefl05dkpQyw68k9SUbFyQ7vcv/2np90Ah42Y3wkrdAod/6JeUvvwNKUl+wYzU88AWYfwetbmYr7A+nXgtnfiS5sU2S8pzhV5J6s5rt8NBXk4MqGmqz66EAjnsjnHsDlE9Irz5JyjGGX0nqjWpr4PHvwT+/AXsrW1876mI47z9g9NGplCZJuczwK0m9SUM9zPsF/P2LULWh9bUJJ8H5n4UpL02nNknqBQy/ktQbxAjL/gh//SxsXdb62vDpcN6nYdYrPZlNkg7B8CtJuW7NY3Dfp+HFx1qvl46Gcz7uBAdJ6gC/W0pSrtqyLNnpXfaH1uv9yuCMD8Cp74F+JenUJkm9lOFXknJN5fqkp/eZn0NszK4XFMNJ74CzroeSEenVJ0m9mOFXknLFnopkesNj34P63a2vHXMVnHsjDJuaSmmS1FcYfiUpbfV74V8/hAe/Art3tL427Vw4/zMw7vg0KpOkPueQ4TeEcCtwKbA5xjinxfr7gOuAeuAPMcZ/77YqJakvamyEBb+Cv30eKta0vjbmWLjgs3DEy9KpTZL6qPbs/P4Y+A7w06aFEMK5wKuAY2OMe0MIo7qnPEnqg2KEFffDfZ+BTQtaXxsyOTmgYvblUFCQSnmS1JcdMvzGGB8MIUzZZ/ndwE0xxr2Zx2zuhtokqe9Z9zT89dOw8sHW64OGw1n/DnPfBkX906lNkvJAZ3t+jwLODCH8F7AHuD7G+K+2HhhCuAa4BmDSpEmd/OUkqZfbtiJpb1h0V+v14kFw2nvh9PfDgMHp1CZJeaSz4bcIGAqcCpwE/DKEMC3GGPd9YIzxZuBmgLlz5+53XZL6tOot8OCX4clbobE+ux4K4YS3JIdUlI1Jrz5JyjOdDb9rgbsyYfeJEEIjMALY0mWVSVJvtnsnPPY/8Oh3oba69bVZlyV9vSOOTKU0ScpnnQ2/9wAvA/4eQjgK6Ads7aqiJKnX2lsNT/wAHv4W7NnZ+tqk0+GC/4SJJ6VSmiSpfaPObgPOAUaEENYCnwZuBW4NISwEaoGr22p5kKS8Ubc7aW146GtQs89ewMhZyazeo14OIaRSniQp0Z5pD284wKU3dXEtktT71NfCMz+DB/8bqta3vjZ0Kpz7CZhzBRQUplOfJKkVT3iTpM5oqIf5d8A/boKd+xxQUT4Rzv53OO4NUFicTn2SpDYZfiWpIxobk3Flf/8ibFve+lrpaDjro8kUB2f1SlJOMvxKUnvECEv/AA98ATYvan1t4DA440Nw0jug36B06pMktYvhV5IOpuko4r99HtY/0/pa/3I4/X1w6rXQvyyd+iRJHWL4laQDWfVwEnrXPNJ6vbgETn03nH4dDByaTm2SpE4x/ErSvtY+mYTeFx5ovV40IGltOONDUDIindokSYfF8CtJTTbMT3p6n/tT6/WCYjjxajjzIzB4XDq1SZK6hOFXkrYsS0Lv4ntar4cCOO6NydiyoZNTKU2S1LUMv5Ly1/aV8PebYMEvITa2uBCSgynOuQFGTE+tPElS1zP8Sso/FWvhwa/AMz+HxvrW12ZempzKNnp2OrVJkrqV4VdS/qjaBP/8Gjx5KzTUtr42/YIk9I4/IZ3aJEk9wvArqe+r2Q4PfxOeuBnqalpfm3ImvOyTMOnUdGqTJPUow6+kvmv3Tnjse/Dod6G2qvW1CScloXfq2RBCKuVJknqe4VdS37N7Bzz2/ST47q1ofW3MMfCyT8GRFxp6JSkPGX4l9R0125Nd3sd/sP9O74gZ8LIbYeYroaAgnfokSakz/Erq/XZthUe+Df/6IdRWt742fDqc9e9wzJVQUJhOfZKknGH4ldR7VW+GR74F/7pl/xvZRsxIDqeY/RpDrySpmeFXUu9TtTGZ3vDkj6B+d+tro45OQu+sV9neIEkpqaip49m1O5n34k7OOmokx08cknZJzQy/knqPinVJ6H3qx9Cwt/W10cckoXfmpYZeSepBtfWNLNlQmYTdNUngfWHrrubrdQ2Nhl9J6pCdL8I/vw7P/Gz/wynGHgdnfwxmXOL0BknqZjFG1myvYd6LO5s/Fq2vpLa+8YDPmffizp4rsB0Mv5Jy147VyYlsz/wCGutaXxt3ApzzcUeWSVI32llT2xxyn31xJ8+urWD7rtpDPq+oIDBr7GCOnziEU6cN74FK28/wKyn3bH8BHvoqPHs7NNa3vjbhJDj74zD9PEOvJHWhvfUNLNlQxbw1O5Kwu7aClS3aFw5m0rBBHDdxCMdnPmaPG8yA4ty82djwKyl3bF2ehN75d0BsaH1t0mlJe8O0cwy9knSYYoys3pZtX3jmxZ0sWV9JbcOB2xealA8sToLuhHKOnzSE4yYMYXhp/x6oumsYfiWlb8syePC/YeGvIe7zjXfKmcmNbFPONPRKUift2FXLvBY3pD27dic7a+oO+bziwsDRmfaFpp3dqSNKCL34+7HhV1J6Ni+BB78CC+8CYutrU89OdnqnvDSV0iSpt9pT18Ci9ZXMX5u9KW31tppDPxGYPHxQc+vC8ROHMGts7rYvdJbhV1LP27gQHvwyLP7N/teOOC/Z6Z10as/XJUm9TENjZPnmap59cSfz1iY3pS3bWEV9Yzzkc4cMKua4CZmgO2kIx08YwtCSfj1QdboMv5J6zoZn4R9fhqW/3//akRcmO70T5vZ8XZLUC8QYWbtjN8+uzU5eWLiugprahkM+t19hAUePG9xqV3fy8EG9un2hswy/krrfuqfgH1+B5/60/7UZl8BZH4XxJ/R8XZKUw7ZV72X+2opWYbc9Y8YApo0s4fgJQzh2QjnHTxrKrLFl9C/qW+0LnWX4ldR91jye9PQuv2//a7NemYTescf1fF2SlGNqautZuK6yuX1h/tqdvLh996GfCIwZPIBjJ5Q335A2Z3w55QOLu7ni3svwK6lrxQjL/woPfQ3WPLLPxQBHvyoJvWPmpFKeJKWtrqGRZRureHbtTua/mOzsPrepina06VI2oIjjJgzhuInlHDshGTM2pnxA9xfdhxh+JXWNhnpYfA/88xuwacE+FwPMuQLOuh5GzUqhOElKR4yRVdtqmicvPJs5DnjvQY4DbtKvqIDZ4wY3h93jJgxhyvASCgryr0+3Kxl+JR2euj3w7P/Bw9+EHataXysogmNfBy/9IIw8Ko3qJKnHxBjZWLmHZ1+sYMG6ncxfW8H8tRVU7D70PN0Q4MhRpZmgm+zozhhTRr+igh6oPL8YfiV1zp5KePJWeOx/oHpT62vFg+CEq+H066B8Qjr1SVI3a7ohLfnYyfx1FWyp2tuu544fMpDjJw5p7tWdM76c0v7Gsp7gqyypY6q3wOPfgyd+CHsrWl8bMAROeRec/C4oGZ5KeZLUHSp217FwXYugu7aCdTvbd0Na0zzd5Ia0pFd3RC86DrivMfxKap8dq+GRb8MzP4P6Pa2vlY2F066DE98K/UtTKU+SukpNbT2L1ieTFxZkAu/Krbva9dySfoXMGV/OsROSkHvshHImDcvPebq5yvAr6eA2LYaHvwELfg1xn0Hqw46AMz6Y9PUWuYshqffZW9/A0g1Vzbu589dW8Pzm9k1eaHlD2jHjyzluYjlTR5RS6A1pOc3wK6ltLz6RjCtr62CKscfBGR9OZvUWODRdUu9Q39DI85urWwXdpRsrqWs4dNItKgjMGFPWvJt77IRyjhpdRnGhN6T1NocMvyGEW4FLgc0xxjn7XLse+AowMsa4tXtKlNRjYoTl98M/vwarH97/+pQz4cwPw7Rzk1uTJSlHNTZGXti6iwXrdmamL1SwaH0Fe+oOPWIsBJg+srRV0J01djADiv3Lfl/Qnp3fHwPfAX7acjGEMBG4AFjT9WVJ6lGNDZkZvV+HjfvO6AVmXgpnfAgmzO3x0iTpUBobI6u27WLBugoWrG0KupVU761v1/MnDx+UBN1Mr+6c8eWUOHmhzzrk/9kY44MhhCltXPo68O/Ab7q6KEk9pH4vzGua0buy9bWCIjjmtUlP78gZqZQnSfuKMbJ6W00SdDNhd+G6CqraGXTHlg9odTPaMePLGTKoXzdXrVzSqb/WhBAuA9bFGJ891N2LIYRrgGsAJk2a1JlfTlJX21MJT/0IHv0fqN7Y+lrRQDjx6mR6w5CJ6dQnSSRB98Xtu5m/bmeroFu5p31Bd0RpP44Z3yLoTihnVJlHAee7DoffEMIg4EbgwvY8PsZ4M3AzwNy5c9tx76SkbrNrKzz2PfjX/8KefWf0lifzeU95F5SMSKc+SXkrxsjaHbtb7eguWNe+09EAhpU0Bd3y5lFjYwYPcMSY9tOZnd8jgKlA067vBODpEMLJMcaNB32mpHTsXJPM6H36Z1C/z1D2srFw2nszM3rLUilPUn6JMbK+Yg8L1mbn6C5cV8GOmvYF3aGDipsD7jHjyzlmwhDGlRt01T4dDr8xxgXAqKafhxBWAXOd9iDloE2Lk37eBb9qY0bvNHjpB+G41zujV1K3iTGysXJPc8Cdn9nR3b6rtl3PLx9YnN3NHZ98njB0oEFXndaeUWe3AecAI0IIa4FPxxhv6e7CJHVSjLDib/Dod5LP+xpzbDKubNZlzuiV1KWadnQXrqto/liwroKt1e0LumUDiloE3eTgiInDDLrqWu2Z9vCGQ1yf0mXVSOq8+lpY+Gt45DuwedH+16ecmYwrO+JlzuiVdNha9ug2hdxF6yvbvaNb1r+ouXWh6bPHAKsnOMRO6u1qtieTGx6/ef/JDaEgmdF7+vth4knp1Cep12tsjKzeXpPd0V1fwcJ1le2+Ga20fxGzxw1uEXSHMHnYIAo8BlgpMPxKvdX2F5LJDc/8HOpqWl8rLoGXvAlOvTbp7ZWkdmpojKzcuqtV28Li9ZXtnqNb1r+I2eMHc0ymP3fO+HKmDi8x6CpnGH6l3mbN4/Dot2HJ74F9pgeWjklGlc19Gwwcmkp5knqP+oZGVmzZ1aJtIWldqKltOPSTSW5GO2Z8eTbsjktaFwy6ymWGX6k3aGyApb9P+nnXPrH/9VGz4fTrYM6VUORJRZL2V9fQyPObqpvbFhasq2DJhkr21DW26/nDSvolO7njsru6Tl1Qb2T4lXLZ3mqY9wt47H9gx6r9rx9xXhJ6p53rTWySmu2pa+C5TVUsWl+Z7Oiuq2DJxipq69sXdEeU9ueYzG7u7PHJLN2xztFVH2H4lXJR5QZ44gfw5I9gz87W1wqK4djXJgdTjJ6dSnmSckf13nqWbKhk0boKFq6vZOG6CpZvrqa+sX2Hqo4ZPCDTm5vd0R092COA1XcZfqVcsnEhPPrd5FCKxn3uoh4wBE76Nzj5Gigbk0p5ktK1Y1cti9ZXsjDTm7toXQUrt+0iti/nMn7IQOaMH8ycceXMmZD06I4s85Ab5RfDr5S2GGHF/Uk/7wsP7H996NRkl/f4N0K/kp6vT1KPizGyqXIvCzOzcxeuTyYurNu5+9BPzpgyfBCzx5VzdIse3WEl3hMgGX6ltNTvTXZ4H/0ubF68//WJpyb9vDMu8SQ2qQ+LMbJmew0L11WyaH3SurB4fftPRSssCEwfWcrs8YOZPS65IW3WuMEMHlDczZVLvZPhV+ppNdvhyVvhiZuhelPra6EgOXb49PfBhLnp1Cep29Q3NPJC8wzdJOx2ZIZuv6ICZo4pY/a4cmaPG8yc8eXMHFPGgGL/giy1l+FX6inbViSHUsz7RduHUpzwZjj13TB0SirlSepaLScuNLUvLNlQyd52Tlwo6VfY3LYwZ3wSdqePKqW4sKCbK5f6NsOv1J1ihBcfh0e+DUv/wH6HUpSNTQ6lOPGtHkoh9WIVNXUs2pDs4i5eX8mi9ZUs31JNQzsnLgwdVJwJuNkdXY//lbqH4VfqDvV7YdE98Pj3Yf3T+18ffUzSzzv7cg+lkHqRGCPrK/awaF0FizckIbejN6KNLR/A7HGZ/tzMjq4zdKWeY/iVulLVxmQ275O3wq7N+1+ffn7Szzv1bA+lkHJcU39uU1/uovWVLN5Qyc6aukM/OWPqiBKOHjuY2ZnxYrPHDWZ4qaPFpDQZfqWusPapZJd30d37z+ct7A/HXgWnXQejZqVTn6SDqqmtZ8mGKhZvSCYtLF5fydKNVe3uz+1XWMBRY0qZPTbp0Z09bjAzxw6mtL9/zEq5xt+VUmfV18Li3yShd92T+18vG5ccSnHiW6FkRI+XJ6lt26r3Nu/iLsqMFXtha/sPiigbUJTs5o7LBl1vRJN6D8Ov1FFVm+CpTGvDvqPKIJnPe8q7YNYrodA5m1JaGhsja3fsZlHmNLQk7FawqXJvu79GU3/u0WMHc3SmbWHC0IH250q9mOFXaq91T8HjP4CFd7XR2tAP5lwJp1wD416STn1SHttT18CyjVUs2ZCE3CUbKlmyoYrqds7PLQgwbWRpc9Bt2tX1RDSp7zH8SgdTXwtLfpu0Nqz91/7XS8fASe9IWhtKR/Z4eVI+2lK1tzngLs7Mzl2xpZp2ThWjf1EBM8cOzkxcSMLuzDGDGdjPgyKkfGD4ldpSvRme+jH86xao3rj/9QknJ60NR7/K1gapmzQ0RlZurc4cDtF0M1olW6vb37YwdFBxpi+3vDnoTh1RQpH9uVLeMvxKLa17Ojl2eOGd0FDb+lphP5hzBZx8DYw/IZ36pD6qem89SzMtC027uR2ZthACTB1ewqyxgzk6E3JnjR3M6MH97c+V1IrhV2qoy0xt+AGsfWL/66WjW7Q2jOrx8qS+pOmQiKaAu3h9JUs2VrJ6W82hn5wxsLiQmWPLkqCbCbszRpdR4lgxSe3gdwrlr+otSWvDk7dA1Yb9r084CU65FmZd5ilsUifsrW9g+ebqTNCtYvGGCpZsqKJid/sPiRg9uH9zwG0Ku5OHl1Dosb+SOsnwq/yzfl5masOv929tKCiGOZfDye+CCSemUp7U28QY2VK9lyUbkmkLSzOTFlZsqaa+nXehFRYEpo8sbdWyMGtsmaehSepyhl/lh4Y6WPK7JPS++Nj+10tGZQ6keBuUje75+qReora+keWbqzOjxJK+3CUbKtm2q/bQT85oOiSiZX/u9FGlDCh22oKk7mf4Vd9WvQWe/kkytaFq/f7Xx50Ap74bjn61rQ3SPrZU7d0v5C7f3P7dXICJwwYya0zrm9A8JEJSmgy/6ntihDWPJoF38W/2P5CioAhmvybp550wN50apRxSW9/Iii3VrULukg2VbK1u/27uoH6FzBiT3IQ2K/N5xpgyygY4ClBSbjH8qu/YUwnz70hC75Yl+18vGQlz3560Ngwe2/P1STlga3WL3dzM7NwVW6qpa+jYbu7MMYNbBd1JwwZR4E1oknoBw696vw3zk4kN838Fdbv2vz5+bjKqbM7lUOTNM8oPTbu5SzdmQ+6SDVUdOiBiYHF2N/fosWXMzOzmDnY3V1IvZvhV71S3BxbdnYTeto4dLh4Ex1yV3MQ29rier0/qITFGNlXuZUkm5C7dWMmyjVUd7s2dMDTZzW0KubPGDmayu7mS+iDDr3qXbSvgyVth3i9g9479r4+cCXP/DY57HQwo7/n6pG5UU1vPso1VLNtY1dybu3Rjx+bmDiguYEYm5M4aO5iZYwYzc6y7uZLyh+FXua+hHp77U9LL+8ID+18vKIajL0tC7+TTk3NOpV6ssTGyZnsNSzcmrQpNu7mrt9cQ27+Z27ybO7PpRrSxZR4QISnvGX6Vuyo3JGPKnvpJ22PKyifB3LfCS97sscPqtXbsqmXpxiqWbcxMWthYxXMbq9hd19Dur1HWv4iZY8uYMaYscyNaGUeNdtKCJLXF8Kvc0tgIK/+R9PIu/SPEfQNAgCMvTHp5p58PBQ7FV+9QW9/IC1urWbqhiiWZndylG6rYWLmn3V+jsCAwdURJ807uzDFJ4B0/xLm5ktRehl/lhprtMO//kn7e7Sv2v14yMtnhPfGtMHRyj5cntVeMkfUVe5p3cp/L9Od2dJzYiNL+zBpbxozRyQ1oM8eUeQqaJHUBw6/SEyOseyrp5V10F9S3sQM2+aXJbN5Zl3kCm3JORU1d0o+7qar5RrRlm6qo2lPf7q/Rr6iAo0aXturNnTGmjBGljuWTpO5g+FXPq90FC36VhN6N8/e/3n8wHPf6JPSOmtXz9Un72FPXwPLN1SzbWMVzm6oyPboda1mA7A1os1r0504ZPoiiwoJuqlyStC/Dr3rO5qVJL++zt8Peyv2vjzk26eU95iroV9Lz9SnvZacsJCE3GSlWyaptNTR0YGbu4AFFzByT7OAmh0R4A5ok5YpDht8Qwq3ApcDmGOOczNpXgFcCtcAK4G0xxp3dWKd6q/paWPLbpJd39cP7Xy8aALMvT0Lv+BMdU6Yes6Vqb3ObwrLMDWjPbaru0JSFfkUFHDmqlBmjy5qD7swxgxk9uL83oElSjmrPzu+Pge8AP22xdh9wQ4yxPoTwJeAG4GNdX556rS3L4OmfwrO3Qc22/a8POyJpazj+jTBoWM/Xp7yxa289z21q3a6wbGMV23bVtvtrhACThw1KAu7oMmZkdnVtWZCk3ueQ4TfG+GAIYco+a/e2+OljwJVdXJd6o9pdsOieJPS++Nj+10MhzHxFEnqnng0FhgZ1nb31DbywZVdzu0ISeKtZs72mQ19nRGl/Zo5J2hSaRokdObqUQf3sEpOkvqArvpu/HbjjQBdDCNcA1wBMmjSpC3455Zz1zySBd8Gv2+7lHTwBTngznHA1DB7b8/WpT2lojKze1hRyq5PPm6pYuXVXh/pyB/UrbA64LYPucKcsSFKfdljhN4RwI1AP/OJAj4kx3gzcDDB37twOHMypnLZ7ZzKx4emftj2xoaAIZlycBN4jXuZhFOqwGCPrdu5uFXKf21TF85urqa1vbPfXKSwITBtR0qJlIenLnTB0IAUe8ytJeafT4TeEcDXJjXDnxdiR0+bVa8UIax5NjhtefE/bc3mHT4cT3gLHvcEjh9UuMUa2Vte2aldYtqmK5zdVU723/fNyASYNG8RRo8uYMaY087mMqSNK6F/kX74kSYlOhd8QwkUkN7idHWPsWEOdep/qzcmNa0//FLYt3/960QA4+tVJ6J18uhMbdEAVNXU8tznTj5uZtPDcpmq2d+DmM4DRg/sn4XZ0GUdldnSnjyqlpL99uZKkg2vPqLPbgHOAESGEtcCnSaY79Afuy4zzeSzGeG031qme1tgAKx6Ap38Cy/4IjW3swI0+Bk68OpnLO3BIj5eo3LVrbz3Pb05aFZ7fVMWyTdU814lDIYYMKm5uVWjayT1qVBnlg5yXK0nqnPZMe3hDG8u3dEMtygU7X4R5v4Bnfg4VL+5/vV8ZHHtVsss79nh3efNcTW09yzdX89ymap7flJ2wsG7n7g59nZJ+hRy5z07uUWNKGVnqvFxJUtfy3wiVHETx3J+Stobl9wNttHBPPDUJvLNf7elreajpeN+mcPv8piqe21zF2h276UjHf7+iAqaPLOWo0aXZkDu6jPFDvPlMktQzDL/5bOvz2YModm3Z//rAYckhFCe8BUbO6Pn61OP21CWzcp/fXNUq6K7eXtOhkFtUEJgyooSjRpdy5KjMOLExZUwe5qEQkqR0GX7zTW1NctzwUz+BNY+0/Zhp5ya9vDMugSJnnvZFtfWNvLC1dbvC85uqWbVtFx0YlUthQWDy8EEcNaosCbqZndypI0roV2TIlSTlHsNvvtjwbLLLO/9XsLdi/+tl4+Alb4KX/D8YOqXHy1P3qK1vZFXmQIiWQXfVtpoOHQhREGDy8BKOHJWMEDtydPJ52kjHiEmSehfDb19Wsx0W3gnP/CwJv/sKhZmDKN4C08/3IIpebG99Ayu37uL5TdU8vzkJuc9vrmbV1l3UdyDkhgAThw5q3sWdkQm6R4wsZUCx7w9JUu9n+O1rGuphxf3JxIZlf4KGNuanDpuWOYjijVA2uudrVKftqWtgxZZqlm+uzgTdpF1h9faO7eQCTBg6MLuLOyppV5g+qpSB/Qy5kqS+y/DbV2xaDM/+H8z/JVRv2v96YX84+lVJ6J1yhiPKctzu2iTkNh3n+/ymapZvrmLN9poO9eQCjCsfkOzijilrblvwQAhJUr7yT7/erGY7LPh1ssu7YV7bjxl3QjKxYc4VMGhYj5anQ9u1N5mT27JV4flOjBADmDhsIEeOSgLukaOTz0eMKqXUkCtJUjP/VOxtGupg+V9h3v8lbQ2Ndfs/pnQMHPvaJPSOmtXzNWo/lXvqWL65muWZVoXnNiWtCx09DCIEmDxsENNHJe0KTTu500aWMKifv50lSToU/7TsLTYtSgLv/Dvanslb2A9mvgKO/3/JqLJC/9emYVv13uad3OWbq1mxJWlZ6Oixvk0jxI4clczJPXJ0KdNHeeOZJEmHy4SUy3ZtgwW/Snp525rWADB+bqat4XIYOLRn68tTMUY2VOxJdnIzQXdFpl1hR00bO/EHUVQQmDqiJBNum1oWSpk6whFikiR1B8Nvrmmog+fvS/p4n/tL220NZWPh2NclodeT17pNQ2NkzfaaFiG3ihWbq1mxZRfVe+s79LWKCwPTRpRmWhWyLQuTh3sYhCRJPcnwmys2Lsi0NfwSarbuf72wP8y6NAm80851Jm8X2lvfwKqtNTy/uao56C7fXM0LW3dRW9/Yoa81sLiQI0aVcOSoZKJC04fH+kqSlBsMv2mq3pJta9i4oO3HTDgZjn8DzL4cBg7p0fL6ml1765t7cJdvyYbcNZ2YkVs+sJjpo5Ld2+mZqQpHjiplXPlACgocIydJUq4y/Pa0+lp4/t5kl/f5v0BjG/98XjYOjnt9sss74sier7EXizGybVctKzYnAXfF5l2Zzx2frAAwqqx/8+5t0+iw6aNKGVnan+CsZEmSeh3Db0+IETbOTwLvgl9Bzbb9H1M0AGa9Mgm8U8+2reEQGhoj63bsZvmWqiTgNoXdLdXs7OBNZ5CcdnZkizaF6Zm2hfKBxd1QvSRJSovhtztVroeFd8Kzt8OmhW0/ZuKpmbaG18CA8p6trxfYU9fAC1uyu7dNn1du3cXeDvbjFmXGh03PjA9rCrrOyJUkKX/4J35X21MBS36XzONd+RDQRi/p4AlJW8Nxb4AR03u8xFy0Y1dtNuBm5uMu31LdqZPOmm46mz4ymYvbfNOZkxUkScp7ht+uUF8Ly+9LJjUs+xM07N3/MUUD4ejLkraGKWdBQf6FsMbGyLqdu5NgmxkZ1rSbu31XbYe/3ojS/hwxsqT58IemG8/GDh7gTWeSJKlNht/OamyEFx9PdngX3wO7d7TxoABTz0qOGp51GQwY3NNVpqKpVeGFrckNZ01h94Wt1eyp61irQkGAicMGMX1k9oSzI0aVcsTIEoYM6tdN/wWSJKmvMvx21OalSeBd8GuoWNP2Y8YckxxCMecKGDyuZ+vrITFGtlTvbQ63K7ZU88KW5Mfrdna8VWFAcQHTRpTus4tbwpThJR7nK0mSuozhtz0qN8DCXyeh90DzeMsnwTFXJru8o2b1bH3dqLa+kdXbdiUtCpmQu2LLLl7YUk3Vno6dcgYwrKRf0oub2b1tCrvjhzgfV5IkdT/D74E037j2S1j5IG3euDZgSDKl4djXwcRTenUf745dtfvt4K7YsqtTB0AUBJg0bBBHjEwmKUxr0bIwrMRWBUmSlB7Db0stb1x77s9Qv2f/xxT2hxkXJzu80y+Aot4T5uobGlm7Y3d2BzfTsvDC1l2duuGstH8RR4wsadWHO21kKZOHD6J/ka0KkiQp9xh+233j2pnJDu+sV+b8PN6dNbW8sHVXctNZi53cVdt2UdfQwWZcYPyQga3C7REjkzFiI8s85UySJPUu+Rt+Ny+FBb+E+b/qlTeu1TU0snpbTRJut2ZDbmd3cQcWFzJ1RElzyD0iMyN36ogSBvZzF1eSJPUN+RV+m29c+2Vy3HBbcujGtRgjW6tr2wy4nenFBRg9uH9zsJ3WomXB2biSJCkf9P3w26Eb116bHDfcwzeu7alrYNW2fdoUtnZ+okL/ooJkF7f5hrOS5l3csgHF3fBfIEmS1Dv07fC7+Ddw1zXtuHHtfCjq362lxBjZULGn+fCHpj7clVt3dWouLsC48gFMawq4I0qafzyu3LFhkiRJbenb4XfscfsE3+6/ca1idx2rtiYBd2WmReGFLbtYuXUXu+saOvz1SvsXMW1kCVNHlDBtRHYnd+qIEgb169v/+yRJkrpa305PQ6ck83frarr0xrW99Q2s2VbDC1uTUPtCZgd35dZdbK3u+M1mTUf4tty9nTYiufHMiQqSJEldp2+HX4A33Qn9yzr8tMbGyPqK3c2htmn39oWt1azbsZtO3GvGkEHFbQbcSc7FlSRJ6hF9P/weIvhu31XLykwPblPQbfrYW9/Y4V+uX1EBU4cnbQlTM724UzOB19PNJEmS0tX3wy+wuzY7TWHl1urmdoWVW3exs6auw18vBJgwdCBTR5S2CLfJZ282kyRJyl19OvzOe3En7/n5U6yvaGPaQzsML+nXHGqnjihtDrmThg1iQLFtCpIkSb1Nnw6/wwb1O2TwbTrZrGlc2NSRmaA7vITyQc7ElSRJ6kv6dPgdP3Qg/QoLaIiRScMGZcaFNQXc5Iaz0YOdpiBJkpQvDhl+Qwi3ApcCm2OMczJrw4A7gCnAKuC1McYd3Vdm5xQWBO7/yNmMKR9AcWHPntomSZKk3NOeRPhj4KJ91j4O3B9jPBK4P/PznDRx2CCDryRJkoB2hN8Y44PA9n2WXwX8JPPjnwCv7tqyJEmSpK7X2S3R0THGDQCZz6MO9MAQwjUhhCdDCE9u2bKlk7+cJEmSdPi6vR8gxnhzjHFujHHuyJEju/uXkyRJkg6os+F3UwhhLEDm8+auK0mSJEnqHp0Nv78Frs78+GrgN11TjiRJktR9Dhl+Qwi3AY8CM0IIa0MI/wbcBFwQQngeuCDzc0mSJCmnHXLOb4zxDQe4dF4X1yJJkiR1KwfgSpIkKW8YfiVJkpQ3DL+SJEnKG4ZfSZIk5Q3DryRJkvKG4VeSJEl5w/ArSZKkvBFijD33i4WwBVjdY79g3zEC2Jp2Eb2cr+Hh8fU7PL5+h8fX7/D4+h0eX7/Dk+brNznGOHLfxR4Nv+qcEMKTMca5adfRm/kaHh5fv8Pj63d4fP0Oj6/f4fH1Ozy5+PrZ9iBJkqS8YfiVJElS3jD89g43p11AH+BreHh8/Q6Pr9/h8fU7PL5+h8fX7/Dk3Otnz68kSZLyhju/kiRJyhuGX0mSJOUNw2+OCCFMDCE8EEJYEkJYFEL4QBuPOSeEUBFCmJf5+I80as1VIYRVIYQFmdfmyTauhxDCt0IIy0MI80MIJ6RRZy4KIcxo8b6aF0KoDCF8cJ/H+P7bRwjh1hDC5hDCwhZrw0II94UQns98HnqA514UQliWeT9+vOeqzh0HeP2+EkJYmvk9encIYcgBnnvQ3+/54ACv32dCCOta/D695ADP9f3X9ut3R4vXblUIYd4Bnuv77wC5pTd8D7TnN0eEEMYCY2OMT4cQyoCngFfHGBe3eMw5wPUxxkvTqTK3hRBWAXNjjG0O0878IfA+4BLgFOCbMcZTeq7C3iGEUAisA06JMa5usX4Ovv9aCSGcBVQDP40xzsmsfRnYHmO8KfMNfWiM8WP7PK8QeA64AFgL/At4Q8vf7/ngAK/fhcDfYoz1IYQvAez7+mUet4qD/H7PBwd4/T4DVMcY//sgz/P9R9uv3z7XvwpUxBj/s41rq/D912ZuAd5Kjn8PdOc3R8QYN8QYn878uApYAoxPt6o+51Uk3+RijPExYEjmN69aOw9Y0TL4qm0xxgeB7fssvwr4SebHPyH5w2BfJwPLY4wvxBhrgdszz8srbb1+McZ7Y4z1mZ8+Bkzo8cJ6iQO8/9rD9x8Hf/1CCAF4LXBbjxbVixwkt+T890DDbw4KIUwBXgI83sbl00IIz4YQ/hRCmN2zleW8CNwbQngqhHBNG9fHAy+2+Pla/AtGW17Pgb/h+/47tNExxg2Q/OEAjGrjMb4X2+ftwJ8OcO1Qv9/z2XWZtpFbD/BPzr7/Du1MYFOM8fkDXPf918I+uSXnvwcafnNMCKEUuBP4YIyxcp/LT5OcU30c8G3gnh4uL9e9NMZ4AnAx8N7MP2m1FNp4jn0/LYQQ+gGXAb9q47Lvv67je/EQQgg3AvXALw7wkEP9fs9X3wOOAI4HNgBfbeMxvv8O7Q0cfNfX91/GIXLLAZ/WxlqPvQcNvzkkhFBM8gb6RYzxrn2vxxgrY4zVmR//ESgOIYzo4TJzVoxxfebzZuBukn9WaWktMLHFzycA63umul7jYuDpGOOmfS/4/mu3TU3tNJnPm9t4jO/FgwghXA1cCvy/eIAbU9rx+z0vxRg3xRg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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n = 100\n", "k = np.arange(1, 21)\n", "RSS = 1000\n", "\n", "\n", "def aic(RSS):\n", " return np.exp(2 * k / n) * RSS / n\n", "\n", "\n", "def bic(RSS):\n", " return n ** (k / n) * RSS / n\n", "\n", "\n", "aic = aic(RSS)\n", "bic = bic(RSS)\n", "\n", "fig, ax = plt.subplots(figsize=(12, 7))\n", "ax.plot(k, aic, label=\"AIC\", lw=3)\n", "ax.plot(k, bic, label=\"BIC\", lw=3)\n", "ax.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "023a8460-0471-4f33-8787-562084c0c716", "metadata": {}, "source": [ "Also, both these criteria are commonly used in data science to compare **in-sample** and **out-of-sample** performance." ] }, { "cell_type": "markdown", "id": "4c75aad6-83b9-46e4-8dd1-fdf87cf9e7b6", "metadata": {}, "source": [ "# Measurement Error " ] }, { "cell_type": "markdown", "id": "dc5b3dea-9d97-4764-85f2-70017db5295b", "metadata": {}, "source": [ "Keep in mind that any data might have certain extent of measurement error, either due to mis-recording or mis-communication. Most of time we assume the data are correctly measured, but that's more precise, we will discuss what consequences measurement error can cause with examples of simple linear regression." ] }, { "cell_type": "markdown", "id": "44acf35d-207c-4826-9a18-74efd34c9bf3", "metadata": {}, "source": [ "## Measurement Error in Independent Variables " ] }, { "cell_type": "markdown", "id": "3e7b9ed0-d374-4fe5-89c1-fcae9d43b723", "metadata": {}, "source": [ "Assume the true relationship is\n", "$$\n", "Y_{i}=\\beta_{1}+\\beta_{2} Z_{i}+v_{i}\n", "$$\n", "However due to some technical reason, we are unable to precisely measure $Z_i$,what we can observe is $X_i$ which has a relationship with $Z_i$\n", "$$\n", "X_{i}=Z_{i}+w_{i}\n", "$$\n", "Combine them\n", "$$\n", "Y_{i}=\\beta_{1}+\\beta_{2}\\left(X_{i}-w_{i}\\right)+v_{i}=\\beta_{1}+\\beta_{2} X_{i}+v_{i}-\\beta_{2} w_{i}\n", "$$" ] }, { "cell_type": "markdown", "id": "ee579ba8-9d30-476b-8de7-469d96207000", "metadata": {}, "source": [ "The disturbance term $v_i-\\beta_2w_i$ is a composite term and also $X_i$ not independent from composite disturbance term, because of common part $w_i$.\n", "\n", "Recall the estimator of $b_2$ can be decomposed as\n", "$$\n", "b_{2}=\\frac{\\operatorname{Cov}(X, Y)}{\\operatorname{Var}(X)}=\\beta_{2}+\\frac{\\operatorname{Cov}(X, u)}{\\operatorname{Var}(X)}\n", "$$\n", "however in this case $X_i$ and $u_i$ are not independent, we can expand covariance expression.\n", "$$\n", "\\operatorname{Cov}(X, u)= \\operatorname{Cov}((Z+w),(v-\\beta_2w)) = \\operatorname{Cov}(Z,v)+\\operatorname{Cov}(w,v)+\\operatorname{Cov}(Z,-\\beta_2w)+\\operatorname{Cov}(w,-\\beta_2w) = -\\beta_2\\sigma_w^2\n", "$$\n", "Also expand variance at the denominator \n", "$$\n", "\\operatorname{Var}(X)=\\operatorname{Var}(Z+w)=\\operatorname{Var}(Z)+\\operatorname{Var}(w)+2\\operatorname{Cov}(Z,w)=\\sigma_Z^2+\\sigma_w^2\n", "$$\n", "Therefore\n", "$$\n", "b_{2}=\\frac{\\operatorname{Cov}(X, Y)}{\\operatorname{Var}(X)}=\\beta_{2}-\\frac{\\beta_2\\sigma_w^2}{\\sigma_Z^2+\\sigma_w^2}\n", "$$" ] }, { "cell_type": "markdown", "id": "94e633d0-12f7-4f83-abc8-d899c0d49dc7", "metadata": {}, "source": [ "This is how measurement error will affect estimates theoretically, the $b_2$ will be always biased downward." ] }, { "cell_type": "markdown", "id": "b82e7320-6f0e-496d-8272-0e4046b9540e", "metadata": {}, "source": [ "We can show this with Monet Carlo simulation" ] }, { "cell_type": "code", "execution_count": 261, "id": "8613ba60-57f3-4507-8e5f-fee69b1af5c9", "metadata": {}, "outputs": [ { "data": { "image/png": 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3Jjl/SdtlSa7v7nOSXD+9T1U9PclFSZ4xrXNFVZ0wrFoAgDl24koduvvDVbV1SfOFSV4wvb4qyQ1JXje1X93dDyS5s6r2Jjk3yR8Pqhf+3tbLPrDRJcCqLffneN+OCzagEmCEo51j9eTuvidJpufTpvYzkty1qN/+qe0RquqSqtpZVTsPHDhwlGUAAMyP0ZPXa5m2Xq5jd1/Z3du7e/uWLVsGlwEAsP6ONljdW1WnJ8n0fN/Uvj/JWYv6nZnk7qMvDwDg2HG0weraJBdPry9O8r5F7RdV1clVdXaSc5LcuLoSAQCODStOXq+q38jCRPVTq2p/kp9IsiPJNVX1yiSfS/KyJOnu3VV1TZLbkzyY5NLuPrhGtQMAzJVZzgp8+SEWvfAQ/S9PcvlqigLYzGY949XZgzB/XHkdAGAQwQoAYBDBCgBgEMEKAGAQwQoAYBDBCgBgEMEKAGAQwQoAYJAVLxAKwHxa7kKiLhoKG8uIFQDAIIIVAMAgghUAwCCCFQDAIIIVAMAgghUAwCCCFQDAIIIVAMAgghUAwCCCFQDAIIIVAMAg7hXIMWG5e6IBwLwxYgUAMIgRK+aO0SkAjlVGrAAABhGsAAAGEawAAAYRrAAABhGsAAAGcVYgwHFkubNq9+24YAMqgc3JiBUAwCCCFQDAIA4FsqFcDBSA44kRKwCAQQQrAIBBHApk3TjsB8DxzogVAMAgghUAwCCCFQDAIIIVAMAgghUAwCCCFQDAIC63AHCcc2NmWD+rClZVtS/Jl5IcTPJgd2+vqicl+c0kW5PsS/Ivu/uLqysTAGD+jTgU+B3dva27t0/vL0tyfXefk+T66T0AwHFvLeZYXZjkqun1VUlesgafAQAwd1YbrDrJh6rqpqq6ZGp7cnffkyTT82nLrVhVl1TVzqraeeDAgVWWAQCw8VY7ef353X13VZ2W5Lqq+tSsK3b3lUmuTJLt27f3KusAANhwqxqx6u67p+f7kvx2knOT3FtVpyfJ9HzfaosEADgWHHWwqqrHVtXjv/w6yXcluS3JtUkunrpdnOR9qy0SAOBYsJpDgU9O8ttV9eXt/Hp3f7CqPpHkmqp6ZZLPJXnZ6ssEAJh/Rx2suvszSZ61TPsXkrxwNUVx7FvugoTA/HDRUFgbrrwOwCEd6kuSEAbLc69AAIBBBCsAgEEEKwCAQQQrAIBBTF5n1ZwBCAALjFgBAAwiWAEADCJYAQAMIlgBAAwiWAEADOKsQACSOMMXRjBiBQAwiGAFADCIQ4EAHLHlDhvu23HBBlQC88WIFQDAIIIVAMAgDgVySIb6AeDICFYADOHLGAhWTGa9fo3r3ADAoQlWAKwZo1hsNiavAwAMIlgBAAwiWAEADGKOFQAbzlwsjhdGrAAABjFiBcC6ctkWjmdGrAAABjFiBcBcMu+KY5ERKwCAQYxYbULmNwDA2jBiBQAwiBErADYFc7ZYD0asAAAGMWJ1DPKtCwDmk2B1nDjUhHSBC+DQfFFlNIcCAQAGMWIFwDFj1hEml5VhowhWxzn/uADA+hGsADim+QLJPBGsAOAomfzOUoLVOvAXD+DY4d9sVmPNglVVnZ/kF5KckORt3b1jrT4LANbSkRxuFMw2tzUJVlV1QpJfSvLPkuxP8omqura7b1+LzztemCcAsLnN+v+AoDa/1mrE6twke7v7M0lSVVcnuTDJhgar9fgWMetfCiEKYPMY/W/+PI2KbeQFqufp9/Bl1d3jN1r1PUnO7+4fmN6/Islzu/vVi/pckuSS6e3XJ7ljeCHHl1OTfH6ji+Bh7JP5Yn/MH/tk/tgnY3xtd29ZbsFajVjVMm0PS3DdfWWSK9fo8487VbWzu7dvdB18hX0yX+yP+WOfzB/7ZO2t1S1t9ic5a9H7M5PcvUafBQAwF9YqWH0iyTlVdXZVfVWSi5Jcu0afBQAwF9bkUGB3P1hVr07yv7NwuYV3dPfutfisTcRh0/ljn8wX+2P+2Cfzxz5ZY2syeR0AYDNaq0OBAACbjmAFADCIYDVHquqsqvqDqtpTVbur6jWH6PeCqto19fk/613nZjLLPqmqJ1TV71TVJ6c+378RtW4GVfXoqrpx0e/6J5fpU1X1P6tqb1XdUlXP2YhaN4sZ98m/mfbFLVX10ap61kbUulnMsk8W9f0nVXVwuv4kA5hjNUeq6vQkp3f3zVX1+CQ3JXnJ4lsBVdUTk3w0Cxdg/VxVndbd921Mxce/GffJG5I8obtfV1VbsnCx23/U3X+7MVUfv6qqkjy2u++vqpOSfCTJa7r7Y4v6vDjJv0/y4iTPTfIL3f3cDSl4E5hxn3xrkj3d/cWq+u4kb7RP1s4s+2Tqd0KS65L8vyycZPZb61/t8ceI1Rzp7nu6++bp9ZeS7ElyxpJu/zrJe7r7c1M/oWoNzbhPOsnjp3/MHpfkz5M8uK6FbhK94P7p7UnTY+m3wwuT/MrU92NJnjgFZNbALPukuz/a3V+c3n4sC9c2ZI3M+PckWfgC8u4k/h8ZSLCaU1W1Ncmzk3x8yaKnJTmlqm6oqpuq6vvWvbhN6jD75BeTfGMWLoJ7axa+GT60vtVtHlV1QlXtysJ/Btd199L9cUaSuxa9359HhmEGmmGfLPbKJL+3LoVtYivtk6o6I8lLk7xlA8o7rglWc6iqHpeFbxGv7e6/WrL4xCTfnOSCJOcl+c9V9bR1LnHTWWGfnJdkV5KnJNmW5Ber6h+ua4GbSHcf7O5tWRj1OLeqnrmky4q31GKsGfZJkqSqviMLwep161jepjTDPvn5JK/r7oPrXdvxTrCaM9Px8HcneVd3v2eZLvuTfLC7/7q7P5/kw0lMBF1DM+yT78/C4dnu7r1J7kzyDetZ42bU3X+R5IYk5y9Z5JZaG+Qw+yRV9Y+TvC3Jhd39hfWtbPM6zD7ZnuTqqtqX5HuSXFFVL1nP2o5XgtUcmebovD0LkzzfdIhu70vy7VV1YlX9gyxMzt2zXjVuNjPuk88leeHU/8lJvj7JZ9anws2lqrZMJ3Ckqh6T5EVJPrWk27VJvm86O/B5Sf6yu+9Z30o3j1n2SVV9TZL3JHlFd//puhe5ycyyT7r77O7e2t1bk/xWkld193vXudTj0prc0oaj9vwkr0hy63RsPEnekORrkqS739Lde6rqg0luSfJQkrd1920bUewmseI+SfJTSd5ZVbdm4TDU66bRRMY7PclV09lMj0pyTXe/v6p+KPn7/fG7WTgjcG+Sv8nCiCJrZ5Z98l+SfHUWRkWS5MHu3r5RBW8Cs+wT1ojLLQAADOJQIADAIIIVAMAgghUAwCCCFQDAIIIVAMAgghUAwCCCFQDAIIIVbBJV9Z+qandV3VJVu6rquWv8efcfYf+DU127q+qTVfUjVfWoRcs/eph1n1hVr1ph+x+dnrdW1cwX1V1u24erZZSq+uGq2lNV71pm2Quq6lfXugbgyLnyOmwCVfUtSf55kud09wNVdWqSr9rgspb6v9NNY1NVpyX59SRPSPITSdLd33qYdZ+Y5FVJrli6YLotUa2w/uE8Ytur2NaReFWS7+7uO5dZti3Jn6xDDcARMmIFm8PpST7f3Q8kSXd/vrvvTpKqem9V3TSNFF0ytW2tqk9V1duq6raqeldVvaiq/qiqPl1V5y7pd9U0EvZb0z0sH6aqvreqbpxGpH55utXGIXX3fUkuSfLqKRj9/QhYVT22qj4wjWrdVlX/KsmOJE+dtv/TU117quqKJDcnOWvJCNqJS2teOpJVVT9WVW9cuu3FtUyvf2Sq47aqeu2i38ueqnrr9Hv90HTPtkc4xPpvSfJ1Sa6tqv+wzGrPSnJGVX28qj5TVS843O8TWEfd7eHhcZw/kjwuya4kf5qFkZd/umjZk6bnxyS5LQv3dNua5MEk35SFL2A3JXlHFu6FeGGS907rbE3SSZ4/vX9Hkh+bXt8/PX9jkt9JctL0/ook37dMjfcv0/bFJE9esr1/keSti/o8YarjtkVtW7NwL83nLd3+oWpeZhs/luSNS9uXbOubk9ya5LHT73h3kmcv+v1tm/pdk+R7l/n5ll1/WrYvyamH2J9/kuSN0+vvSvKHG/1nzMPDY+FhxAo2ge6+Pwv/iV+S5ECS36yqfzst/uGq+mSSjyU5K8k5U/ud3X1rdz+Uhf/wr+/uzkIQ2Lpo83d19x9Nr38tybct+fgXTp/9ielG1i/MwmjMLGqZtluTvKiq/kdVfXt3/+Uh1v1sd3/sEMtWqnlW35bkt7v7r6ff8XuSfPu07M7u3jW9vikP/53Nsv6yqurELITf/zY17UpyalW9ZBohe19VfddR/jzAKpljBZtEdx9MckOSG6rq1iQXV9W+JC9K8i3d/TdVdUOSR0+rPLBo9YcWvX8oD/+3Y+md3Je+ryRXdffrj6Teqvq6JAeT3Lfk5/jTqvrmJC9O8t+r6kNJfmWZTfz1YTa/XM0P5uHTIx6dlS0X/L5s8e/vYBZGBI9k/UN5epK93f230/vnJPlkd783yXur6pQkP5PkQ0exbWCVjFjBJlBVX19V5yxq2pbks1k4jPbFKVR9Q5LnHcXmv2aaHJ8kL0/ykSXLr0/yPdOE9FTVk6rqa1eod0uStyT5xWmUbPGypyT5m+7+tSwEiOck+VKSx6+y5nuTnFZVX11VJ2dhsn9W2PaHk7xkmqP12CQvTfKHR1DH0az/rCRnV9XJVfW4LEzu//lFy388yS8dQQ3AQEasYHN4XJL/VVVPzMLIzN4sHBb8UpIfqqpbktyRhcOBR2pPFka/fjnJp5O8efHC7r69qn48yYdq4fIJf5fk0iwEu8UeMx0qPGmq8VeTvGmZz/umJD9dVQ9N2/p33f2FaWL9bUl+LysHi0fU3N1/V1X/NcnHk9yZ5FNT/Q/bdnf/x0U/281V9c4kN05Nb+vuP6mqrSt8/mHXX2G1ZyV5V5KPZmEU7Ke6+2PTJP8dU403z/L5wHi15MsgwMymAPH+7n7mRtey2VXVDye5OMknkuzq7rdscEmwKQlWwFETrAAeTrACABjE5HUAgEEEKwCAQQQrAIBBBCsAgEEEKwCAQQQrAIBBBCsAgEH+P9m5iUIC375PAAAAAElFTkSuQmCC\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "Z = np.arange(1, 21)\n", "b2_array = []\n", "for i in range(10000):\n", " w = np.random.randn(20)\n", " v = np.random.randn(20)\n", " X = Z + w\n", " beta1, beta2 = 2, 3\n", " Y = beta1 + beta2 * Z + v\n", " b2_array.append(sp.stats.linregress(X, Y)[0])\n", "fig, ax = plt.subplots(figsize=(10, 7))\n", "n, bins, patches = ax.hist(b2_array, bins=100)\n", "ax.axvline(x=np.mean(b2_array), lw=3, color=\"tomato\")\n", "ax.set_xlabel(\"Sample Distribution of $b_2$\")\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "3d799dcf-37ee-4964-a9d5-a399e600ed07", "metadata": {}, "source": [ "With $10000$ times of simulations, the estimates are always less than $3$, which is biased downward." ] }, { "cell_type": "markdown", "id": "8d9dbca1-ba68-4097-962f-dae476df4538", "metadata": {}, "source": [ "## Measurement Error in Dependent Variables " ] }, { "cell_type": "markdown", "id": "019f25d3-1fbb-472a-96e6-cca34d0dd78c", "metadata": {}, "source": [ "If the true relationship is \n", "$$\n", "Q_i = \\beta_1+\\beta_2X_1+v_i\n", "$$\n", "However $Q_i$ cannot be precisely recorded, instead $Y_i$ is recorded, where\n", "$$\n", "Y_i = Q_r + r_i\n", "$$\n", "The true relationship can rewritten as\n", "$$\n", "Y_i= \\beta_1+\\beta_2X_i+v_i+r_i \n", "$$" ] }, { "cell_type": "markdown", "id": "297f404a-dd4c-4341-b965-f78ec511c75d", "metadata": {}, "source": [ "But note that $X_i$ is note affected, so OLS still provide unbiased and consistent estimates." ] }, { "cell_type": "markdown", "id": "069332dd-82a8-446e-9194-f2f681e5ed96", "metadata": {}, "source": [ "The composite disturbance term increase the the population variance of slope coefficient\n", "$$\n", "\\sigma_{b_{2}}^{2}=\\frac{\\sigma_{v}^{2}+\\sigma_{r}^{2}}{n \\operatorname{Var}(X)}\n", "$$" ] }, { "cell_type": "markdown", "id": "f08c5130-1a5c-4e64-87eb-927ddff1b110", "metadata": {}, "source": [ "# Instrumental Variables Regression " ] }, { "cell_type": "markdown", "id": "8c060bc2-04b8-4a47-baf5-8c0231bf6755", "metadata": {}, "source": [ "We have discussed how $X$ can be correlated with disturbance term $u$, either due to omitted variables or measurement errors in independent variables. In next chapter we will also see how simultaneous equations model also cause interdependence between $X$ and $u$.\n", "\n", "Here we will discuss a general method **Instrumental Variable** (IV) Regression to obtain consistent estimator when $X$ is correlated with $u$. The idea of this method is extract the part of $X$ that is not correlated with $u$ and extraction is called IV, which can be used in obtaining consistent estimators." ] }, { "cell_type": "markdown", "id": "782b2657-01c5-4712-ab91-a0c31c4b6684", "metadata": {}, "source": [ "Consider the model\n", "$$\n", "Y_{i}=\\beta_{1}+\\beta_{2} X_{i}+u_{i}\n", "$$\n", "where $X_i$ and $u_i$ are correlated, OLS estimators are inconsistent. IV method requires identify an instrument $Z_i$, which is correlated with $X_i$, but not with $u_i$. \n", "\n", "For the time being, we define that variables that correlated with disturbance term are called **endogenous variable**, otherwise called **exogenous variable**. In the context of simultaneous equation, we will come back to these terms again.\n", "\n", "To be a valid instrument, two conditions needs to be satisfied\n", "$$\n", "\\begin{aligned}\n", "&\\operatorname{Cov}\\left(Z_{i}, X_{i}\\right) \\neq 0 \\\\\n", "&\\operatorname{Cov}\\left(Z_{i}, u_{i}\\right)=0\n", "\\end{aligned}\n", "$$\n", "The philosophy of IV is to use $Z_i$ to capture the exogenous part of movements of $X_i$." ] }, { "cell_type": "markdown", "id": "17be0668-2240-4de0-b31c-d95b64415b9c", "metadata": {}, "source": [ "## Two Stage Least Squares " ] }, { "cell_type": "markdown", "id": "b2876b14-8148-4260-95ec-77ee6126749c", "metadata": {}, "source": [ "If both condition satisfied, the estimation process with IV is called **Two Stage Least Square** (2SLS).\n", "\n", "_$1$st Stage_: decomposes $X$ into two components: a problematic component that may be correlated with the regression error and another problem-free component that is uncorrelated with the disturbance term. For simple linear regression model, the first stage begins with a regression model that links $X$ and $Z$\n", "\n", "$$\n", "X_i = \\alpha_1 + \\alpha_2Z_i + v_i\n", "$$\n", "\n", "The problem-free component is the estimated values of $\\hat{X}_i= a_1 + a_2 Z_i$, which is uncorrelated with $u_i$.\n", "\n", "_$2$nd Stage_: uses the problem-free component to estimate $\\beta_2$.\n", "\n", "In the context of simple linear regression, regress $Y_i$ on $\\hat{X}_i$ using OLS. The resulting estimators are 2SLS estimators." ] }, { "cell_type": "markdown", "id": "5cdfb3e5-b97f-4c83-99bf-91e120d1e2b4", "metadata": {}, "source": [ "If you can derive the formula of IV estimator, you won't need to go through these two steps. We can demonstrate how IV estimator of $\\beta_2$ is derived. We start from the covariance of $Z_i$ and $Y_i$\n", "$$\n", "\\operatorname{Cov}(Z_i, Y_i) = \\operatorname{Cov}(Z_i, \\beta_{1}+\\beta_{2} X_{i}+u_{i}) = \\operatorname{Cov}(Z_i, \\beta_{0}) + \\operatorname{Cov}(Z_i, \\beta_{2} X_{i}) + \\operatorname{Cov}(Z_i, u_{i}) =\\beta_{2} \\operatorname{Cov}(Z_i, X_{i})\n", "$$" ] }, { "cell_type": "markdown", "id": "698c45cd-2874-461e-aa3a-756e6dafd3f0", "metadata": {}, "source": [ "Rearrange the result, denote estimator as $b_2$\n", "$$\n", "b_2^{\\mathrm{IV}} = \\frac{\\operatorname{Cov}(Z_i, Y_i)}{\\operatorname{Cov}(Z_i, X_{i})}\n", "$$" ] }, { "cell_type": "markdown", "id": "63e46475-d557-477b-97f7-fe54c6d54793", "metadata": {}, "source": [ "To compare OLS and IV estimator for simple linear regression, here we reproduce the $b_2$ estimator of OLS\n", "$$\n", "b_{2}^{\\mathrm{OLS}}=\\frac{\\operatorname{Cov}(X, Y)}{\\operatorname{Var}(X)}=\\frac{\\operatorname{Cov}(X, Y)}{\\operatorname{Cov}(X, X)}\n", "$$" ] }, { "cell_type": "markdown", "id": "2d77fe44-24ce-4da0-855f-e2a7818147f2", "metadata": {}, "source": [ "IV estimator is obtained by replacing an $X$ both in nominator and denominator.\n", "$$\n", "\\begin{aligned}\n", "b_{2}^{\\mathrm{IV}} &=\\frac{\\operatorname{Cov}(Z, Y)}{\\operatorname{Cov}(Z, X)}=\\frac{\\operatorname{Cov}\\left(Z,\\left[\\beta_{1}+\\beta_{2} X+u\\right]\\right)}{\\operatorname{Cov}(Z, X)} \\\\\n", "&=\\frac{\\operatorname{Cov}\\left(Z, \\beta_{1}\\right)+\\operatorname{Cov}\\left(Z, \\beta_{2} X\\right)+\\operatorname{Cov}(Z, u)}{\\operatorname{Cov}(Z, X)} \\\\\n", "&=\\beta_{2}+\\frac{\\operatorname{Cov}(Z, u)}{\\operatorname{Cov}(Z, X)}\n", "\\end{aligned}\n", "$$" ] }, { "cell_type": "markdown", "id": "a6f27b39-68a1-4843-bfb4-62ed89e89c1b", "metadata": {}, "source": [ "It tell that the accuracy of $b_{2}^{\\mathrm{IV}}$ depends on relative quantity of covariance. In large sample, we expect IV estimator to be consistent\n", "$$\n", "\\operatorname{plim} b_{2}^{\\mathrm{IV}}=\\beta_{2}+\\frac{\\operatorname{plim} \\operatorname{Cov}(Z, u)}{\\operatorname{plim} \\operatorname{Cov}(Z, X)}=\\beta_{2}+\\frac{0}{\\sigma_{z x}}=\\beta_{2}\n", "$$" ] }, { "cell_type": "markdown", "id": "4fbc875c-a36a-48d6-a58c-ce54822bc53d", "metadata": {}, "source": [ "We can also compare variance of OLS and IV from simple linear regression\n", "$$\n", "\\sigma_{b_{2}^{\\mathrm{IV}}}^{2}=\\frac{\\sigma_{u}^{2}}{n \\sigma_{X}^{2}} \\times \\frac{1}{r_{X Z}^{2}}\\\\\n", "\\sigma_{b_{2}^{\\mathrm{OLS}}}^{2}=\\frac{\\sigma_{u}^{2}}{n \\sigma_{X}^{2}}\n", "$$\n", "The greater the correlation between $X$ and $Z$, the smaller will be the variance of $\\sigma_{b_{2}^{\\mathrm{IV}}}^{2}$.\n", "\n", "We will walk through a numerical example in next chapter, after discussing the identification issue." ] }, { "cell_type": "markdown", "id": "9d938568-feed-4e5f-af2c-c129c630d851", "metadata": {}, "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": "3.8.8" } }, "nbformat": 4, "nbformat_minor": 5 }