{"id":2522,"date":"2021-02-16T07:10:48","date_gmt":"2021-02-16T06:10:48","guid":{"rendered":"http:\/\/fee.carlarey.es\/?page_id=2522"},"modified":"2021-04-06T05:12:29","modified_gmt":"2021-04-06T04:12:29","slug":"propiedades-ajuste-mco","status":"publish","type":"page","link":"https:\/\/fee.carlarey.es\/index.php\/e-learning\/tecnicas-de-calculo\/propiedades-ajuste-mco\/","title":{"rendered":"Propiedades ajuste MCO"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"2522\" class=\"elementor elementor-2522\">\n\t\t\t\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-68b12dd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"68b12dd\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-00ba5fa\" data-id=\"00ba5fa\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6855c8e elementor-widget elementor-widget-heading\" data-id=\"6855c8e\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Propiedades del ajuste m\u00ednimo cuadr\u00e1tico ordinario (MCO)<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-81bb21d elementor-widget elementor-widget-text-editor\" data-id=\"81bb21d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p style=\"text-align: left;\">Las propiedades del m\u00e9todo de m\u00ednimos cuadrados ordinarios se deducen de las expresiones que definen al error\u00a0<span class=\"katex-eq\" data-katex-display=\"false\"> (e = Y &#8211; \\widehat Y) <\/span> y al vector de estimadores m\u00ednimo cuadr\u00e1ticos ordinarios: <span class=\"katex-eq\" data-katex-display=\"false\"> b = (X\u00b4X)^{-1} X\u00b4Y <\/span><\/p><p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> X^{\\prime}e =\\begin{pmatrix} 1 &amp; 1 &amp; \u00b7\u00b7\u00b7 &amp; 1 \\\\ x_{11} &amp; x_{12} &amp; \u00b7\u00b7\u00b7 &amp; x_{1T} \\\\ \u00b7\u00b7\u00b7 &amp; \u00b7\u00b7\u00b7 &amp; \u00b7\u00b7\u00b7 &amp; \u00b7\u00b7\u00b7 \\\\ x_{k1} &amp; x_{k2} &amp; \u00b7\u00b7\u00b7 &amp; x_{kT} \\end{pmatrix} \\begin{pmatrix} e_1\u00a0 \\\\ e_2 \\\\ \u00b7\u00b7\u00b7\u00a0 \\\\ e_{T}\u00a0 \\end{pmatrix} = \u00a0\\begin{pmatrix} \\sum_{t=1}^{T} e_{t } \\\\ \\sum_{t=1}^{T} e_{t }x_{1t} \\\\ \u00b7\u00b7\u00b7\u00a0 \\\\ \\sum_{t=1}^{T} e_{t }x_{kt} \\end{pmatrix} = \u00a0\\begin{pmatrix} 0\u00a0 \\\\ 0 \\\\ \u00b7\u00b7\u00b7\u00a0 \\\\ 0 \\end{pmatrix} <\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-31dd76d elementor-widget elementor-widget-image\" data-id=\"31dd76d\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t<figure class=\"wp-caption\">\n\t\t\t\t\t\t\t\t\t\t<img width=\"525\" height=\"315\" src=\"https:\/\/fee.carlarey.es\/wp-content\/uploads\/2021\/02\/MinimosCuadradosOrdinarios-768x461.jpg\" class=\"attachment-medium_large size-medium_large wp-image-4095\" alt=\"\" loading=\"lazy\" srcset=\"https:\/\/fee.carlarey.es\/wp-content\/uploads\/2021\/02\/MinimosCuadradosOrdinarios-768x461.jpg 768w, https:\/\/fee.carlarey.es\/wp-content\/uploads\/2021\/02\/MinimosCuadradosOrdinarios-300x180.jpg 300w, https:\/\/fee.carlarey.es\/wp-content\/uploads\/2021\/02\/MinimosCuadradosOrdinarios-1024x615.jpg 1024w, https:\/\/fee.carlarey.es\/wp-content\/uploads\/2021\/02\/MinimosCuadradosOrdinarios-1536x923.jpg 1536w, https:\/\/fee.carlarey.es\/wp-content\/uploads\/2021\/02\/MinimosCuadradosOrdinarios.jpg 1653w\" sizes=\"100vw\" \/>\t\t\t\t\t\t\t\t\t\t\t<figcaption class=\"widget-image-caption wp-caption-text\">Dados los pares de valores (x,y) la recta de regresi\u00f3n MCO es la que minimiza la suma de cuadrados de errores<\/figcaption>\n\t\t\t\t\t\t\t\t\t\t<\/figure>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c771870 elementor-widget elementor-widget-text-editor\" data-id=\"c771870\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Los errores de la estimaci\u00f3n son la diferencia entre los valores observados y los estimados de la variable explicada (<span class=\"katex-eq\" data-katex-display=\"false\"> e_t = y_t &#8211; \\widehat y_t <\/span>). Para cada punto del diagrama de dispersi\u00f3n hay un error, algunos son positivos y otros negativos de tal forma que se compensan.<\/p><ol><li style=\"list-style-type: none;\"><ol><li><strong>La suma<\/strong> <strong>de los errores de estimaci\u00f3n es cero<\/strong>:<\/li><\/ol><\/li><\/ol><p style=\"text-align: center;\"><span style=\"text-align: center; font-size: 1rem;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\sum_{t=1} ^T e_t = 0 \\rightarrow \\overline e = \\frac{ \\sum_{t=1} ^T e_t}{T} = 0 <\/span><\/span><\/p><p><strong>2. Entre los errores y las variables explicativas no existe correlaci\u00f3n muestral<\/strong>.<\/p><p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\sum_{t=1} ^T (x_{it} &#8211; \\overline {x}_i) e_t = \\sum_{t=1}^T x_{it} e_t &#8211; \\overline {x}_i \\sum_{t=1} ^T e_t = 0 \\thinspace \\forall i = 1, 2, \u00b7\u00b7\u00b7 , k <\/span><\/p><p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> Cov( x_{it}, e_t) = 0 \\rightarrow r_{x_{i}, e} = 0 <\/span><\/p><p>Como consecuencia de estas propiedades, se tiene que:<\/p><ul><li style=\"list-style-type: none;\"><ul><li><strong>Las sumas y las medias muestrales de los valores observados y estimados del regresando coinciden<\/strong>.<\/li><\/ul><\/li><\/ul><p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\sum_{t=1} ^T e_t = \\sum_{t=1} ^T (y_t &#8211; \\widehat y_t) = 0 \\rightarrow \u00a0\\sum_{t=1} ^T y_t = \\sum_{t=1} ^T\u00a0 \\widehat y_t\u00a0 \\rightarrow<\/span><\/p><p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\frac {\\sum_{t=1} ^T y_t}{T} = \\frac{\\sum_{t=1} ^T\u00a0 \\widehat y_t}{T} \\rightarrow \\large \\overline y =\\overline{\u00a0 \\widehat y}\u00a0 <\/span><\/p><ul><li style=\"list-style-type: none;\"><ul><li><strong>Entre los errores y los valores estimados de la variable explicada no existe correlaci\u00f3n muestral<\/strong>.<\/li><\/ul><\/li><\/ul><p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\sum_{t=1} ^T (\\widehat y_{t} &#8211; \\overline {\\widehat {y}}) e_t = \\sum_{t=1}^T \\widehat y_{t} e_t &#8211; \\overline {\\widehat y} \\sum_{t=1} ^T e_t = 0 <\/span><\/p><p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> Cov( \\widehat y_{t}, e_t) = 0 \\rightarrow r_{\\widehat y, e} = 0 <\/span><\/p><ul><li style=\"list-style-type: none;\"><ul><li><strong>La funci\u00f3n de regresi\u00f3n estimada pasa por el punto de coordenadas medias:<\/strong><\/li><\/ul><\/li><\/ul><p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\overline y = b_0 + b_1 \\overline {x_1} + b_2 \\overline x_2 + \u00b7\u00b7\u00b7 + b_k \\overline x_k <\/span><\/p><p>Si el modelo no se estima por m\u00ednimos cuadrados ordinarios y\/o no incluye ordenada en el origen, algunas de estas propiedades no se cumplen.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-14364b6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"14364b6\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4240aa5\" data-id=\"4240aa5\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-67d34aa elementor-widget elementor-widget-heading\" data-id=\"67d34aa\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Caso pr\u00e1ctico: modelo Ventas<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8dc8f1b elementor-widget elementor-widget-video\" data-id=\"8dc8f1b\" data-element_type=\"widget\" data-settings=\"{&quot;video_type&quot;:&quot;hosted&quot;,&quot;show_image_overlay&quot;:&quot;yes&quot;,&quot;image_overlay&quot;:{&quot;url&quot;:&quot;https:\\\/\\\/fee.carlarey.es\\\/wp-content\\\/uploads\\\/2021\\\/04\\\/PropAjusteMCO.jpg&quot;,&quot;id&quot;:18532,&quot;size&quot;:&quot;&quot;},&quot;controls&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"e-hosted-video elementor-wrapper elementor-open-inline\">\n\t\t\t\t\t<video class=\"elementor-video\" src=\"https:\/\/fee.carlarey.es\/wp-content\/uploads\/2021\/03\/PropiedadesAjusteMCOREvisado.mp4\" controls=\"\" preload=\"metadata\" controlsList=\"nodownload\"><\/video>\n\t\t\t\t\t\t<div class=\"elementor-custom-embed-image-overlay\" style=\"background-image: url(https:\/\/fee.carlarey.es\/wp-content\/uploads\/2021\/04\/PropAjusteMCO.jpg);\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"elementor-custom-embed-play\" role=\"button\" aria-label=\"Reproducir v\u00eddeo\" tabindex=\"0\">\n\t\t\t\t\t\t\t<i aria-hidden=\"true\" class=\"eicon-play\"><\/i>\t\t\t\t\t\t\t<span class=\"elementor-screen-only\">Reproducir v\u00eddeo<\/span>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Propiedades del ajuste m\u00ednimo cuadr\u00e1tico ordinario (MCO) Las propiedades del m\u00e9todo de m\u00ednimos cuadrados ordinarios se deducen de las expresiones que definen al error\u00a0 y al vector de estimadores m\u00ednimo cuadr\u00e1ticos ordinarios: Dados los pares de valores (x,y) la recta de regresi\u00f3n MCO es la que minimiza la suma de cuadrados de errores Los errores &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/fee.carlarey.es\/index.php\/e-learning\/tecnicas-de-calculo\/propiedades-ajuste-mco\/\" class=\"more-link\">Continuar leyendo<span class=\"screen-reader-text\"> \u00abPropiedades ajuste MCO\u00bb<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":2935,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"elementor_header_footer","meta":[],"_links":{"self":[{"href":"https:\/\/fee.carlarey.es\/index.php\/wp-json\/wp\/v2\/pages\/2522"}],"collection":[{"href":"https:\/\/fee.carlarey.es\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/fee.carlarey.es\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/fee.carlarey.es\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fee.carlarey.es\/index.php\/wp-json\/wp\/v2\/comments?post=2522"}],"version-history":[{"count":295,"href":"https:\/\/fee.carlarey.es\/index.php\/wp-json\/wp\/v2\/pages\/2522\/revisions"}],"predecessor-version":[{"id":18620,"href":"https:\/\/fee.carlarey.es\/index.php\/wp-json\/wp\/v2\/pages\/2522\/revisions\/18620"}],"up":[{"embeddable":true,"href":"https:\/\/fee.carlarey.es\/index.php\/wp-json\/wp\/v2\/pages\/2935"}],"wp:attachment":[{"href":"https:\/\/fee.carlarey.es\/index.php\/wp-json\/wp\/v2\/media?parent=2522"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}