{"id":238571,"date":"2025-06-18T08:36:37","date_gmt":"2025-06-18T08:36:37","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=238571"},"modified":"2025-06-18T08:36:40","modified_gmt":"2025-06-18T08:36:40","slug":"deriving-the-simple-linear-regression-estimator-for-the-population-model-described-by-y-beta_0-beta_1x-mu-obtain-the-ols-ordinary-least-squares-estimator-of-the-intercept-hatbeta_","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/18\/deriving-the-simple-linear-regression-estimator-for-the-population-model-described-by-y-beta_0-beta_1x-mu-obtain-the-ols-ordinary-least-squares-estimator-of-the-intercept-hatbeta_\/","title":{"rendered":"Deriving the Simple Linear Regression Estimator For the population model described by y = \\beta_0 + \\beta_1x + \\mu obtain the OLS (Ordinary Least Squares) estimator of the intercept, \\hat{\\beta}_0, and the slope, \\hat{\\beta}_1"},"content":{"rendered":"\n

Deriving the Simple Linear Regression Estimator For the population model described by y = \\beta_0 + \\beta_1x + \\mu obtain the OLS (Ordinary Least Squares) estimator of the intercept, \\hat{\\beta}_0, and the slope, \\hat{\\beta}_1, 1. For the Least Squares Method, explain in detail the idea behind the method, and show clearly and precisely all of the steps of the derivations. 2. For the Method of Moments, explain in detail the idea behind the method, and show clearly and precisely all of the steps of the derivations.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Deriving the Simple Linear Regression Estimator<\/strong><\/h3>\n\n\n\n

Population Model:<\/strong><\/h4>\n\n\n\n

We begin with the simple linear regression population model:y=\u03b20+\u03b21x+\u03bcy = \\beta_0 + \\beta_1 x + \\muy=\u03b20\u200b+\u03b21\u200bx+\u03bc<\/p>\n\n\n\n

where:<\/p>\n\n\n\n

    \n
  • yyy is the dependent variable<\/li>\n\n\n\n
  • xxx is the independent variable<\/li>\n\n\n\n
  • \u03b20\\beta_0\u03b20\u200b is the intercept<\/li>\n\n\n\n
  • \u03b21\\beta_1\u03b21\u200b is the slope<\/li>\n\n\n\n
  • \u03bc\\mu\u03bc is the error term<\/li>\n<\/ul>\n\n\n\n

    We observe nnn data points: (xi,yi)(x_i, y_i)(xi\u200b,yi\u200b) for i=1,2,\u2026,ni = 1, 2, \\dots, ni=1,2,\u2026,n<\/p>\n\n\n\n

    \n\n\n\n

    1. Ordinary Least Squares (OLS) Estimation<\/strong><\/h3>\n\n\n\n

    Idea Behind the Method:<\/strong><\/h4>\n\n\n\n

    The Least Squares Method minimizes the sum of squared residuals<\/strong> (SSR). A residual is the difference between the observed value yiy_iyi\u200b and the predicted value y^i=\u03b2^0+\u03b2^1xi\\hat{y}_i = \\hat{\\beta}_0 + \\hat{\\beta}_1 x_iy^\u200bi\u200b=\u03b2^\u200b0\u200b+\u03b2^\u200b1\u200bxi\u200b.SSR=\u2211i=1n(yi\u2212\u03b2^0\u2212\u03b2^1xi)2\\text{SSR} = \\sum_{i=1}^n (y_i – \\hat{\\beta}_0 – \\hat{\\beta}_1 x_i)^2SSR=i=1\u2211n\u200b(yi\u200b\u2212\u03b2^\u200b0\u200b\u2212\u03b2^\u200b1\u200bxi\u200b)2<\/p>\n\n\n\n

    Steps of Derivation:<\/strong><\/h4>\n\n\n\n

    We minimize SSR with respect to \u03b2^0\\hat{\\beta}_0\u03b2^\u200b0\u200b and \u03b2^1\\hat{\\beta}_1\u03b2^\u200b1\u200b:<\/p>\n\n\n\n

    Step 1: Take partial derivatives<\/strong>\u2202SSR\u2202\u03b2^0=\u22122\u2211(yi\u2212\u03b2^0\u2212\u03b2^1xi)\\frac{\\partial \\text{SSR}}{\\partial \\hat{\\beta}_0} = -2 \\sum (y_i – \\hat{\\beta}_0 – \\hat{\\beta}_1 x_i)\u2202\u03b2^\u200b0\u200b\u2202SSR\u200b=\u22122\u2211(yi\u200b\u2212\u03b2^\u200b0\u200b\u2212\u03b2^\u200b1\u200bxi\u200b)\u2202SSR\u2202\u03b2^1=\u22122\u2211xi(yi\u2212\u03b2^0\u2212\u03b2^1xi)\\frac{\\partial \\text{SSR}}{\\partial \\hat{\\beta}_1} = -2 \\sum x_i (y_i – \\hat{\\beta}_0 – \\hat{\\beta}_1 x_i)\u2202\u03b2^\u200b1\u200b\u2202SSR\u200b=\u22122\u2211xi\u200b(yi\u200b\u2212\u03b2^\u200b0\u200b\u2212\u03b2^\u200b1\u200bxi\u200b)<\/p>\n\n\n\n

    Step 2: Set derivatives equal to zero (first-order conditions):<\/strong>\u2211(yi\u2212\u03b2^0\u2212\u03b2^1xi)=0(1)\\sum (y_i – \\hat{\\beta}_0 – \\hat{\\beta}_1 x_i) = 0 \\tag{1}\u2211(yi\u200b\u2212\u03b2^\u200b0\u200b\u2212\u03b2^\u200b1\u200bxi\u200b)=0(1)\u2211xi(yi\u2212\u03b2^0\u2212\u03b2^1xi)=0(2)\\sum x_i (y_i – \\hat{\\beta}_0 – \\hat{\\beta}_1 x_i) = 0 \\tag{2}\u2211xi\u200b(yi\u200b\u2212\u03b2^\u200b0\u200b\u2212\u03b2^\u200b1\u200bxi\u200b)=0(2)<\/p>\n\n\n\n

    Step 3: Solve the system of equations<\/strong><\/p>\n\n\n\n

    From (1):n\u03b2^0+\u03b2^1\u2211xi=\u2211yi\u21d2\u03b2^0=y\u02c9\u2212\u03b2^1x\u02c9n\\hat{\\beta}_0 + \\hat{\\beta}_1 \\sum x_i = \\sum y_i \\Rightarrow \\hat{\\beta}_0 = \\bar{y} – \\hat{\\beta}_1 \\bar{x}n\u03b2^\u200b0\u200b+\u03b2^\u200b1\u200b\u2211xi\u200b=\u2211yi\u200b\u21d2\u03b2^\u200b0\u200b=y\u02c9\u200b\u2212\u03b2^\u200b1\u200bx\u02c9<\/p>\n\n\n\n

    Substitute into (2) and solve for \u03b2^1\\hat{\\beta}_1\u03b2^\u200b1\u200b:\u03b2^1=\u2211(xi\u2212x\u02c9)(yi\u2212y\u02c9)\u2211(xi\u2212x\u02c9)2\\hat{\\beta}_1 = \\frac{\\sum (x_i – \\bar{x})(y_i – \\bar{y})}{\\sum (x_i – \\bar{x})^2}\u03b2^\u200b1\u200b=\u2211(xi\u200b\u2212x\u02c9)2\u2211(xi\u200b\u2212x\u02c9)(yi\u200b\u2212y\u02c9\u200b)\u200b<\/p>\n\n\n\n

    Then plug back to get \u03b2^0=y\u02c9\u2212\u03b2^1x\u02c9\\hat{\\beta}_0 = \\bar{y} – \\hat{\\beta}_1 \\bar{x}\u03b2^\u200b0\u200b=y\u02c9\u200b\u2212\u03b2^\u200b1\u200bx\u02c9<\/p>\n\n\n\n

    \n\n\n\n

    2. Method of Moments Estimation<\/strong><\/h3>\n\n\n\n

    Idea Behind the Method:<\/strong><\/h4>\n\n\n\n

    The Method of Moments sets the population moments equal to their sample counterparts. For simple linear regression, we focus on matching the population expectations and covariances.<\/p>\n\n\n\n

    From the model:y=\u03b20+\u03b21x+\u03bc\u21d2E[y]=\u03b20+\u03b21E[x]y = \\beta_0 + \\beta_1 x + \\mu \\Rightarrow E[y] = \\beta_0 + \\beta_1 E[x]y=\u03b20\u200b+\u03b21\u200bx+\u03bc\u21d2E[y]=\u03b20\u200b+\u03b21\u200bE[x]Cov(x,y)=Cov(x,\u03b20+\u03b21x+\u03bc)=\u03b21Var(x)Cov(x, y) = Cov(x, \\beta_0 + \\beta_1 x + \\mu) = \\beta_1 Var(x)Cov(x,y)=Cov(x,\u03b20\u200b+\u03b21\u200bx+\u03bc)=\u03b21\u200bVar(x)<\/p>\n\n\n\n

    Thus:\u03b21=Cov(x,y)Var(x),\u03b20=E[y]\u2212\u03b21E[x]\\beta_1 = \\frac{Cov(x, y)}{Var(x)}, \\quad \\beta_0 = E[y] – \\beta_1 E[x]\u03b21\u200b=Var(x)Cov(x,y)\u200b,\u03b20\u200b=E[y]\u2212\u03b21\u200bE[x]<\/p>\n\n\n\n

    Replace expectations and variances with sample versions:\u03b2^1=\u2211(xi\u2212x\u02c9)(yi\u2212y\u02c9)\u2211(xi\u2212x\u02c9)2,\u03b2^0=y\u02c9\u2212\u03b2^1x\u02c9\\hat{\\beta}_1 = \\frac{\\sum (x_i – \\bar{x})(y_i – \\bar{y})}{\\sum (x_i – \\bar{x})^2}, \\quad \\hat{\\beta}_0 = \\bar{y} – \\hat{\\beta}_1 \\bar{x}\u03b2^\u200b1\u200b=\u2211(xi\u200b\u2212x\u02c9)2\u2211(xi\u200b\u2212x\u02c9)(yi\u200b\u2212y\u02c9\u200b)\u200b,\u03b2^\u200b0\u200b=y\u02c9\u200b\u2212\u03b2^\u200b1\u200bx\u02c9<\/p>\n\n\n\n

    This produces the same result as OLS because both rely on matching theoretical and empirical moments.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Deriving the Simple Linear Regression Estimator For the population model described by y = \\beta_0 + \\beta_1x + \\mu obtain the OLS (Ordinary Least Squares) estimator of the intercept, \\hat{\\beta}_0, and the slope, \\hat{\\beta}_1, 1. For the Least Squares Method, explain in detail the idea behind the method, and show clearly and precisely all of […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-238571","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238571","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=238571"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238571\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=238571"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=238571"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=238571"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}