{"id":3256,"date":"2022-07-02T15:17:35","date_gmt":"2022-07-02T15:17:35","guid":{"rendered":"https:\/\/ricovictor.com\/?p=3256"},"modified":"2022-07-02T15:55:41","modified_gmt":"2022-07-02T15:55:41","slug":"entiendo-el-modelo-arima-parte-2","status":"publish","type":"post","link":"https:\/\/ricovictor.com\/index.php\/2022\/07\/02\/entiendo-el-modelo-arima-parte-2\/","title":{"rendered":"Entiendo el MODELO ARIMA (Parte 2)"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"3256\" class=\"elementor elementor-3256\">\n\t\t\t\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5daa402 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5daa402\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-wide\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-50c644bb\" data-id=\"50c644bb\" 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-2a78f3bf elementor-widget elementor-widget-text-editor\" data-id=\"2a78f3bf\" 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>El <strong>Modelo ARIMA<\/strong> (Autoregresivo Integrado de Media M\u00f3vil) es un tipo de modelo econom\u00e9trico muy extendido para la predicci\u00f3n en series de tiempo. En este tutorial vamos a estudiar como se interpretan los resultados del modelo as\u00ed como se relacionan con la formula te\u00f3rica ayud\u00e1ndonos de Rstudio. Esta ser\u00e1 la segunda parte de la explicaci\u00f3n.<\/p><p>La primera clase puedes encontrarla aqu\u00ed :<\/p><p><blockquote class=\"wp-embedded-content\" data-secret=\"sSEXWytWUz\"><a href=\"https:\/\/ricovictor.com\/index.php\/2022\/07\/01\/entendiendo-el-modelo-arima-parte-1\/\">Entendiendo el MODELO ARIMA | Parte 1<\/a><\/blockquote><iframe class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; clip: rect(1px, 1px, 1px, 1px);\" title=\"\u00abEntendiendo el MODELO ARIMA | Parte 1\u00bb \u2014 Finanzas Cuantitativas en Espa\u00f1ol\" src=\"https:\/\/ricovictor.com\/index.php\/2022\/07\/01\/entendiendo-el-modelo-arima-parte-1\/embed\/#?secret=XRDR8mJ7OA#?secret=sSEXWytWUz\" data-secret=\"sSEXWytWUz\" width=\"500\" height=\"282\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe><\/p><p>Tambi\u00e9n m\u00e1s abajo est\u00e1 el c\u00f3digo en R completo:<\/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<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-4f96f7d8\" data-id=\"4f96f7d8\" 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-5e8dfa28 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"5e8dfa28\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-74b52843 elementor-widget elementor-widget-video\" data-id=\"74b52843\" data-element_type=\"widget\" data-settings=\"{&quot;youtube_url&quot;:&quot;https:\\\/\\\/youtu.be\\\/LOQ3G12wCIM&quot;,&quot;video_type&quot;:&quot;youtube&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=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<div class=\"elementor-video\"><\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3fc6c7f1 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"3fc6c7f1\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-745ed16c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"745ed16c\" 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-39dd739\" data-id=\"39dd739\" 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-262defc0 elementor-view-stacked elementor-shape-circle elementor-widget elementor-widget-icon\" data-id=\"262defc0\" data-element_type=\"widget\" data-widget_type=\"icon.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-icon-wrapper\">\n\t\t\t<div class=\"elementor-icon\">\n\t\t\t<i aria-hidden=\"true\" class=\"fas fa-arrow-circle-down\"><\/i>\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6ddbe935 elementor-widget elementor-widget-text-editor\" data-id=\"6ddbe935\" 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<h2>Codigo R completo<\/h2><p><br \/>#INTERPRETACION ARIMA<\/p><p>#video Proceso Autoregresivo (AR)<\/p><p><br \/>#install.packages(\u00abquantmod\u00bb)<br \/>library(quantmod)<br \/>#install.packages(\u00abtseries\u00bb)<br \/>library(tseries)<br \/>#install.packages(\u00abforecast\u00bb)<br \/>library(forecast)<br \/>#install.packages(\u00abTSA\u00bb)<br \/>library(TSA)<br \/>#install.packages(\u00abQuandl\u00bb)<br \/>library(Quandl)<br \/>library(stats)<br \/>#install.packages(\u00ablmtest\u00bb)<br \/>library(lmtest)<\/p><p><br \/># INTERPRETACI\u00d3N ARIMA :<\/p><p><br \/># 3 COMPONENTES : AR &#8211; I- MA<\/p><p>#AR = Componente Autoregresivo<br \/>#I = Componente de Diferenciaci\u00f3n. N\u00famero de veces que la serie debe ser diferenciada para hacerla estacionaria<br \/>#MA = Componente de Media M\u00f3vil<\/p><p>#(Termino AR) ( TERMINO MA)<br \/>#Y[t] = constante + b1*Y[t-1] + e[t] + theta * e[t-1]<br \/><br \/><br \/># AR (AUTOREGRESIVO)<br \/><br \/>#Implica relaci\u00f3n de un valor de una serie en un punto del tiempo <br \/>#con sus propios valores previos.Tal relaci\u00f3n puede existir con cualquier orden de retraso(lag en ingl\u00e9s)<br \/><br \/># Mirar ejemplo en la tabla antes de la ecuaci\u00f3n<\/p><p># INTERPRETACION DE LA ECUACION<\/p><p># AR(1)<\/p><p># Y(t)= c + b1*Y(t-1) + e(t) donde e(t) ruido blanco ~ N(0,s^2)<\/p><p># AR(2) (como vimos en la tabla)<\/p><p># Y(t)= c + b1*Y(t-1) + B2*Y(t-2) + e(t)<\/p><p><br \/>#IMPORTANTE: \u00abc\u00bb dentro del Modelo<\/p><p># c = mu * (1 &#8211; b1 ) donde mu es la media de Y<\/p><p>\u00a0<\/p><p># Y(t)= mu * (1 &#8211; b1 ) + b1*Y(t-1) + e(t)<br \/># \u00abconstante\u00bb<\/p><p><br \/># de forma que al sustituir nos queda as\u00ed<\/p><p>\u00a0<\/p><p># y^[t] &#8211; mu = B1(Y[t-1]-mu) + e[t]<br \/><br \/><br \/>#y[t] = mu + (1 &#8211; B1) + b1*y[t-1] + e[t]<\/p><p>#Podremos ver el modelo escrito de 2 formas, pero interpretaci\u00f3n es la misma<\/p><p># 1. y^[t] &#8211; mu = B1(Y[t-1]-mu) + e[t]<\/p><p># 2. Y(t)= c + b1*Y(t-1) + e(t) <br \/><br \/>#########################################<br \/><br \/>#install.packages(\u00abquantmod\u00bb)<br \/>library(quantmod)<br \/>#install.packages(\u00abtseries\u00bb)<br \/>library(tseries)<br \/>#install.packages(\u00abforecast\u00bb)<br \/>library(forecast)<br \/>#install.packages(\u00abTSA\u00bb)<br \/>library(TSA)<br \/>#install.packages(\u00abQuandl\u00bb)<br \/>#library(Quandl)<br \/>library(stats)<br \/>#install.packages(\u00ablmtest\u00bb)<br \/>library(lmtest)<br \/><br \/><br \/>## HACEMOS MODELO AR(1)<br \/><br \/><br \/>## PRECIOS DE LAS ACCIONES<br \/><br \/>acciones &lt;- c(235,320,115,355,190,320,275,205,395,240,355,175,285,<br \/>200,290,220,400,275,185,370,255,285,250,300,225,285,<br \/>250,225,125,295,250,355,280,370,250,290,225,270,180,<br \/>270,240,275,225,285,250,310,220,320,215,260,190,295,<br \/>275,205,265,245,170,175,270,225,340,190,250,300,195)<br \/><br \/>plot(acciones,type=\u00bbl\u00bb)<br \/><br \/><br \/>adf.test(acciones) # test de raiz unitaria<br \/><br \/>#EL MODELO ARIMA PIDE QUE LAS SERIES SEAN ESTACIONARIAS<br \/><br \/># REALIZAMOS UN MODELO AR(1) arima(1,0,0)<br \/><br \/>arima_acc &lt;- arima(acciones,order=c(1,0,0))<br \/>arima_acc<br \/><br \/>#? \u00bfC\u00f3mo saca el ARIMA los coeficientes que vemos en el SUMARIO?<br \/>#method = c(\u00abCSS-ML\u00bb, \u00abML\u00bb, \u00abCSS\u00bb)<br \/><br \/><br \/>?arima<br \/><br \/>coeftest(arima_acc)<br \/><br \/>z_value1&lt;- &#8211; 0.40472\/ 0.11329<br \/>z_value1<br \/>zvalue&lt;- 259.31594\/4.88598<br \/><br \/>summary(arima_acc)<br \/><br \/><br \/>residuos&lt;- arima_acc$residuals<br \/>betas = arima_acc$coef<br \/><br \/><br \/>coef_ar &lt;-betas[1]<br \/>intercepto &lt;- betas[2]<br \/><br \/>mean(acciones)<br \/><br \/>predict(arima_acc) # arima(1,0,0) serie acciones 285.3459<br \/><br \/>forecast(arima_acc)<br \/><br \/><br \/>#Calculo de los valores fijados reescribiendo ecuaci\u00f3n<br \/><br \/><br \/># Y(t)= mu * (1 &#8211; b1 ) + b1*Y(t-1) + e(t)<br \/><br \/>y_fijados = intercepto*(1- coef_ar) + coef_ar*lag(acciones,default=0)<br \/>y_fijados[1:64]<br \/><br \/><br \/><br \/>#Me falta poner el valor 1<br \/>fitted.values(arima_acc)<br \/><br \/># Hacemos una Predicci\u00f3n para el periodo 66 con predict<br \/><br \/>predict(arima_acc)<br \/><br \/># hacemos la predicci\u00f3n con la ecuaci\u00f3n escrita <br \/>#t-1<br \/>Predicci\u00f3n_t66 &lt;- intercepto*(1- coef_ar) + coef_ar*acciones[65]<br \/>Predicci\u00f3n_t66<br \/><br \/><br \/>### RECAPITULANDO<br \/><br \/># Hemos visto la ecuaci\u00f3n del Modelo AR, su interpretaci\u00f3n<br \/># y luego hemos comparado los resultados y predicci\u00f3n hechos por R<br \/># con los resultados que a nosotros nos da en la ecuaci\u00f3n escrita<br \/><br \/># as\u00ed como hemos apuntado los m\u00e9todos por los que Arima de R busca los <br \/>#coeficientes<br \/><br \/>#############################<br \/><br \/>## COMPONENTE MA <br \/><br \/># El promedio m\u00f3vil aqu\u00ed tienes caracter hist\u00f3rico y no hay que confundir<br \/>#con la media m\u00f3vil como m\u00e9todo de predicci\u00f3n. Es decir, aqu\u00ed MA se refiere<br \/># al hecho de que la desviaci\u00f3n de la respuesta , (Yt &#8211; mu), es una combinaci\u00f3n<br \/># lineal de los errores actuales y pasados y que conforme avanaza el tiempo, los errores<br \/># tambi\u00e9n se mueven hacia adelante<br \/><br \/># EN CONCLUSI\u00d3N: Implica que la desviaci\u00f3n actual de la media depende de desviaciones previas<br \/><br \/><br \/># MODELOS MA (MOVING AVERAGES)<br \/><br \/>#y[t] = mu + e[t] + theta * e[t-1]<br \/><br \/><br \/>#LUEGO LO CONVERTIREMOS EN :<br \/><br \/>#y^[t] = mu + (theta * residuos )<br \/><br \/><br \/>#Modelo MA(1) con DIferencia<br \/><br \/>#y^[t] &#8211; y[t-1] = mu + e[t] + theta * e[t-1]<br \/><br \/><br \/>## PRECIOS DE LAS ACCIONES<br \/><br \/>#y[t] = mu + e[t] + theta * e[t-1]<br \/><br \/>adf.test(acciones)<br \/><br \/># modelo ARIMA(0,0,1) O MA(1)<br \/><br \/>arima_acc &lt;- arima(acciones,order=c(0,0,1))<br \/>arima_acc<br \/><br \/>coeftest(arima_acc)<br \/><br \/># arima(0,0,1)<br \/><br \/>residuos&lt;- arima_acc$residuals<br \/>plot(residuos,type=\u00bbl\u00bb)<br \/><br \/>#aqu\u00ed dejaremos a un lado el estudio de los residuos y dem\u00e1s&#8230;<br \/><br \/>betas = arima_acc$coef<br \/><br \/>coef_ma &lt;-betas[1]<br \/>intercepto &lt;- betas[2]<br \/><br \/>#e = rnorm(65,sd=sqrt(arima_acc$sigma2))<br \/>#plot(e,type=\u00bbl\u00bb)<br \/><br \/>error_arima &lt;- acciones-intercepto <br \/>error_arima<br \/><br \/><br \/># ECUACION 1 valores fijados<br \/><br \/>y_ma= NULL<br \/><br \/>for( i in 1:65) # e[t] &#8211; residuos[i] (Yt- pred)<br \/>y_ma[i] = intercepto + error_arima[i] -(residuos[i])<br \/><br \/>#y_ma[1] = intercept+error[1]<br \/><br \/>fitted.values(arima_acc)<br \/><br \/>y_ma[1:65]<br \/><br \/>tail(y_ma)<br \/><br \/>comp_valores_fija &lt;- data.frame(fitted.values(arima_acc),y_ma)<br \/><br \/>#t65 = intercept + error_arima[65] &#8211; ( residuos[65])<br \/><br \/>#PREDICCI\u00d3N CON MA(1)<br \/><br \/>predict(arima_acc)<br \/><br \/># fORMULA PARA LA PREDICCI\u00d3N CON LA ECUACI\u00d3N<br \/><br \/># res[t-1]<br \/><br \/>t66 = intercepto + (coef_ma*residuos[65])<br \/>t66<br \/><br \/>#y_pred= NULL<br \/><br \/>#for( i in 1:65) #e[t-1]<br \/>y_pred = intercepto + coef_ma*lag(residuos,default=0)<br \/>y_pred[1:64] <br \/>fitted.values(arima_acc) <br \/><br \/><br \/>#y[t] = mu + b1* residuos[t-1]<br \/><br \/>#y[t] = mu + b1* (serie acciones &#8211; fijados)<br \/><br \/><br \/><br \/><br \/><br \/><br \/>## ARIMA (1,0,1)<br \/># ( parte AR) (parte ma)<br \/>Y[t] &lt;- c + coef_ar* Y[t-1] + e[t] + coef_ma * e[t-1]<br \/><br \/>##<br \/>arima_acc &lt;- arima(acciones,order=c(1,0,1))<br \/>arima_acc<br \/><br \/>coeftest(arima_acc)<br \/><br \/><br \/>residuos&lt;- arima_acc$residuals<br \/>betas = arima_acc$coef<br \/><br \/>coef_ar &lt;-betas[1]<br \/>coef_ma &lt;- betas[2]<br \/>intercepto &lt;- betas[3]<br \/><br \/>mean(acciones)<br \/><br \/>predict(arima_acc)<br \/>forecast(arima_acc)<br \/><br \/>#y^[t] = intercepto+ coef_ar(acciones[t-1])+e[t]+coef_ma*e[t-1]<br \/><br \/><br \/># RECORDAR COMO SE HALLABA mu EN EL MODELO AR<br \/><br \/># c = mu * (1 &#8211; b1 ) donde mu es la media de Y<br \/><br \/># ENTONCES <br \/><br \/>Y[t] &lt;- intercepto*(1-coef_ar) + coef_ar*acciones[t-1] + e[t]+coef_ma*e[t-1]<br \/><br \/>predict(arima_acc)<br \/><br \/><br \/><br \/>#formula para la predicci\u00f3n ARIMA (1,0,1)<br \/><br \/>Predicci\u00f3n_66 &lt;- intercepto*(1-coef_ar) + coef_ar*acciones[65] + coef_ma*residuos[65]<br \/>Predicci\u00f3n_66<br \/><br \/><br \/>fija_1_1&lt;- intercepto*(1-coef_ar) + coef_ar*lag(acciones,default=0) + coef_ma*lag(residuos,default=0)<br \/>fija_1_1<br \/><br \/>fija_1_1[1:64]<br \/><br \/><br \/>fijados_arima&lt;-fitted.values(arima_acc)<br \/>fijados_arima<br \/>(fijados_arima)[62:65]<br \/>fija_1_1[61:64] #quitamos el ultimo ya que es ser\u00eda prediccion <br \/><br \/>#ARIMA 1,1,1<br \/><br \/>##<br \/><br \/>arima_acc &lt;- arima(acciones,order=c(1,1,1))<br \/>arima_acc # \u00bf Por qu\u00e9 elimina el intercepto? Porque le establecemos diff<br \/><br \/>coeftest(arima_acc)<br \/><br \/><br \/>residuos&lt;- arima_acc$residuals<br \/>betas = arima_acc$coef<br \/><br \/>coef_ar &lt;-betas[1]<br \/>coef_ma &lt;- betas[2]<br \/>#intercepto &lt;- betas[3]<br \/><br \/>mean(acciones)<br \/><br \/>predict(arima_acc)<br \/>forecast(arima_acc)<br \/><br \/>fitted.values(arima_acc)<br \/><br \/><br \/>#formula para la predicci\u00f3n ARIMA (1,1,1)<br \/><br \/><br \/><br \/>#Y^[t]-Y[t-1]= coef_ar*(Y[t]-Y[t-1]) + e[t]+coef_ma*e[t-1]<br \/><br \/>acciones[66]-acciones[65]=coef_ar*(acciones[66]-acciones[65])+coef_ma*residuos[65]<br \/><br \/>acciones[66]=acciones[65]+ coef_ar*(acciones[66]-acciones[65])+coef_ma*residuos[65]<br \/><br \/>Prediccion_66= acciones[65]+ coef_ar*(acciones[65]-acciones[64])+coef_ma*residuos[65]<br \/>Prediccion_66<br \/><br \/>#Predicci\u00f3n &lt;- coef_ar*acciones[65] + coef_ma*residuos[65]<br \/><br \/>fija_1_1_1 = NULL<br \/><br \/>for(i in 1:65)<br \/>fija_1_1_1[i]&lt;- acciones[i-1]+ coef_ar*(acciones[i-1]-acciones[i-2])+coef_ma*residuos[i-1]<br \/><br \/>fija_1_1_1[1:65]<\/p><p>\u00a0<\/p><p>\u00a0<\/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\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>El Modelo ARIMA (Autoregresivo Integrado de Media M\u00f3vil) es un tipo de modelo econom\u00e9trico muy extendido para la predicci\u00f3n en series de tiempo. En este tutorial vamos a estudiar como se interpretan los resultados del modelo as\u00ed como se relacionan con la formula te\u00f3rica ayud\u00e1ndonos de Rstudio. Esta ser\u00e1 la segunda parte de la explicaci\u00f3n. &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/ricovictor.com\/index.php\/2022\/07\/02\/entiendo-el-modelo-arima-parte-2\/\"> <span class=\"screen-reader-text\">Entiendo el MODELO ARIMA (Parte 2)<\/span> Leer m\u00e1s &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":3259,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","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":"","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":"","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","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-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-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-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-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-gradient":""}},"footnotes":""},"categories":[2],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Entiendo el MODELO ARIMA (Parte 2) - V\u00edctor A. 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