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   "source": "# HW4\n\n## 2.3 Question 1: Unbiasedness\n\n### a. $\\frac{1}{N}\\sum_{i=1}^N{Y_i}$\n    \n $E[\\bar{Y}]$ = $E[\\frac{1}{N}\\sum_{i=1}^N{Y_i}]$\n \n $E[\\bar{Y}]$ = $\\frac{1}{N}\\sum_{i=1}^NE[{Y_i}]$\n \n $E[\\bar{Y}]$ = $\\frac{1}{N}\\sum_{i=1}^N\\mu$\n \n $E[\\bar{Y}]$ = $\\frac{N\\mu}{N}$\n \n $E[\\bar{Y}]$ = ${\\mu}$\n\n It is the sample mean estimator which is Linear and Unbiased.\n \n### b. $\\sum_{i=1}^N{Y_i}$\n    \n $E[\\bar{Y}]$ = $E[\\sum_{i=1}^N{Y_i}]$\n \n $E[\\bar{Y}]$ = $\\sum_{i=1}^NE[{Y_i}]$\n \n $E[\\bar{Y}]$ = $\\sum_{i=1}^N\\mu$\n \n $E[\\bar{Y}]$ = $N\\mu$\n\n It is not the sample mean estimator which is unbiased.\n\n### c. ${Y_3}$\n\n $E[\\bar{Y}]$ = $E[{Y_3}]$\n \n $E[\\bar{Y}]$ = ${Y_3}$\n \n It is not the sample mean estimator which is unbiased."
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   "source": "## 2.4 Question 2: More Unbiasedness\n\n### a. \n\n$E[b_1]$ = $E[\\sum_{i=1}^N{w_iY_i}]$\n\n$E[b_1]$ = $E[\\sum_{i=1}^N{w_i(\\beta_0+\\beta_1X_i+\\epsilon_i)}]$\n\n### b.\n\n$\\sum_{i=1}^N{E[w_i\\epsilon_i]} = 0$, as X is uncorrelated\nwith $\\epsilon$ in the population model.\n\n### c. \n\nExogeneity Gauss-Markov assumption is used i.e. the regressors aren’t correlated with the error term.\n\n### d.\n\nExpected value of $\\frac{1}{N}\\sum_{i=1}^N{Y_i}$ is $\\mu$.\n\n### e.\n\n$\\mu_x$ should be equal to $\\mu$ so that estimator is unbiased."
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   "source": "\n## 2.5 Question 3: Variance of an Estimator\n\n### a.\n\n$V[\\bar{Y}]$ = $V[\\frac{1}{N}\\sum_{i=1}^N{Y_i}]$\n\n$V[\\bar{Y}]$ = $\\frac{1}{N^2}\\sum_{i=1}^NV[{Y_i}]$ since $Cov(\\epsilon_i,\\epsilon_j) = 0$\n\n$V[\\bar{Y}]$ = $\\frac{1}{N^2}\\sum_{i=1}^N\\sigma^2$ since \n$Var(\\epsilon_i) = \\sigma^2$\n\n$V[\\bar{Y}]$ = $\\frac{1}{N^2}N\\sigma^2$\n\n$V[\\bar{Y}]$ = $\\frac{\\sigma^2}{N}$\n\n### b.\n\n$V[\\bar{Y}]$ = $V[\\frac{1}{N}\\sum_{i=1}^N{Y_i}]$\n\n$V[\\bar{Y}]$ = $\\frac{1}{N^2}\\sum_{i=1}^NV[{Y_i}]$ since $Cov(\\epsilon_i,\\epsilon_j) = 0$\n\n$V[\\bar{Y}]$ = $\\frac{1}{N^2}\\sum_{i=1}^N\\sigma_i^2$ since \n$Var(\\epsilon_i) = \\sigma_i^2$\n\n\n### c.\n\n$V[\\bar{Y}]$ = $V[\\frac{V_1+V_2}{2}]$\n\n$V[\\bar{Y}]$ = $\\frac{1}{2^2}V[{Y_1+Y_2}]$ \n\n$V[\\bar{Y}]$ = $\\frac{1}{4}(V[Y_1]+V[Y_2] + 2Cov(Y_1, Y_2))$ \n\n$V[\\bar{Y}]$ = $\\frac{1}{4}(\\sigma^2+\\sigma^2 + 2Cov(Y_1, Y_2))$ \n\nsince $Var(\\epsilon_i) = \\sigma^2$\n\n$V[\\bar{Y}]$ = $\\frac{1}{2}(\\sigma^2 + Cov(Y_1, Y_2))$\n\n since $Cov(\\epsilon_i,\\epsilon_j) \\ne 0$\n\n### d.\n\n$Cov(\\epsilon_i,\\epsilon_j) = 0$ i.e. \n\nThe errors should be uncorrelated across observations. \n"
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   "source": "## 2.6 Question 4: Showing Unbiasedness with Data\n\n### a. Before running the code, look at the code.                What Gauss-Markov assumption is responsible for the             fact that $ε_i$ comes from a normal distribution?\n\nAnswer: Randomness i.e. data is randomly sampled from the population.\n\n### b. Before running the code, What Gauss-Markov assumption is responsible for the fact that $ε_i$ comes from the fact that $ε_i$ comes from a distribution whose variance (or in this case standard deviation) that doesn’t change?\n\nAnswer: Homoscedasticity assumption i.e. independent of the regressors coefficient, the error of the variance is constant."
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   "source": "### c. Run the regression of Y on X. What is the estimate? What is the normalized z-score for this estimate? Put your anwer in a print statement f-string (Put this print statement before the summary() call so both show up).\n"
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HW4 Homework 4 2.1 Name: 2.2 SSID: Please put your name and SSI...