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Lecc08.html
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<title>Econometría Aplicada. Lección 8</title>
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<div id="content" class="content">
<h1 class="title">Econometría Aplicada. Lección 8</h1>
<div id="table-of-contents" role="doc-toc">
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#org71bafac">Carga de algunas librerías de R</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#orgc38f223">1. Identificación y diagnosis</a>
<ul>
<li><a href="#org3c7703c">1.1. Instrumentos de identificación</a></li>
<li><a href="#org6f5a589">1.2. Instrumentos de diagnosis</a></li>
</ul>
</li>
<li><a href="#orga6f0571">2. Raíces unitarias</a>
<ul>
<li><a href="#org602d3b3">2.1. Notación: operadores retardo y diferencia y modelos ARIMA</a>
<ul>
<li><a href="#org93b7704">2.1.1. Notación: ARIMA</a></li>
</ul>
</li>
<li><a href="#org76320fe">2.2. Raíces unitarias en los polinomios AR y MA</a></li>
<li><a href="#orgd761132">2.3. Paseos aleatorios</a></li>
</ul>
</li>
<li><a href="#org1c9df03">3. Modelos ARIMA estacionales (SARIMA)</a>
<ul>
<li><a href="#orgd6a2919">3.1. MA(1) estacional con raíz positiva</a></li>
<li><a href="#orgcbed32f">3.2. AR(1) estacional con raíz positiva</a></li>
<li><a href="#org3b20ddb">3.3. ARIMA\((0,0,1)\times(0,0,1)_{12}\)</a></li>
<li><a href="#orga3b07b1">3.4. ARIMA\((1,0,0)\times(0,0,1)_{12}\)</a></li>
<li><a href="#org15cf448">3.5. ARIMA\((1,0,0)\times(1,0,0)_{12}\)</a></li>
<li><a href="#org4a7af80">3.6. ARIMA\((0,0,1)\times(1,0,0)_{12}\)</a></li>
</ul>
</li>
<li><a href="#org5f8ebed">4. Resumen del análisis univariante de series temporales</a>
<ul>
<li><a href="#orgfc3840e">4.1. Ideas principales respecto a la modelización univariante</a></li>
<li><a href="#org2e5dceb">4.2. Metodología</a></div>
</div>
<div class="abstract" id="orgb13a59a">
<p>
En esta lección repasamos los instrumentos de identificación y
diagnosis del análisis univariante. Extendemos la notación para
incorporar modelos con raíces unitarias. Presentamos modelos
estacionales y finalmente resumimos las ideas principales del análisis
univariante.
</p>
</div>
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<h4 id="org71bafac">Carga de algunas librerías de R</h4>
<div class="outline-text-4" id="text-org71bafac">
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #8D8D84;"># </span><span style="color: #8D8D84; font-style: italic;">cargamos algunas librerías de R</span>
<span style="color: #D0372D;">library</span>(tfarima) <span style="color: #8D8D84;"># </span><span style="color: #8D8D84; font-style: italic;">librería de José Luis Gallego para Time Series</span>
<span style="color: #D0372D;">library</span>(readr) <span style="color: #8D8D84;"># </span><span style="color: #8D8D84; font-style: italic;">para leer ficheros CSV</span>
<span style="color: #D0372D;">library</span>(ggplot2) <span style="color: #8D8D84;"># </span><span style="color: #8D8D84; font-style: italic;">para el scatterplot (alternaticamente library(tidyverse))</span>
<span style="color: #D0372D;">library</span>(ggfortify) <span style="color: #8D8D84;"># </span><span style="color: #8D8D84; font-style: italic;">para pintar series temporales</span>
<span style="color: #D0372D;">library</span>(jtools) <span style="color: #8D8D84;"># </span><span style="color: #8D8D84; font-style: italic;">para representación resultados estimación</span>
<span style="color: #D0372D;">library</span>(zoo) <span style="color: #8D8D84;"># </span><span style="color: #8D8D84; font-style: italic;">para generar objetos ts (time series)</span>
<span style="color: #8D8D84;"># </span><span style="color: #8D8D84; font-style: italic;">y fijamos el tamaño de las figuras que se generan en el notebook</span>
<span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 4, repr.plot.res = 200)
</pre>
</div>
</div>
</div>
<div id="outline-container-orgc38f223" class="outline-2">
<h2 id="orgc38f223"><span class="section-number-2">1.</span> Identificación y diagnosis</h2>
<div class="outline-text-2" id="text-1">
<ul class="org-ul">
<li>Combinando las herramientas gráficas y estadísticas que hemos visto,
se puede inferir el modelo subyacente a los datos.</li>
<li>Este proceso de especificación empírica del modelo es conocido como
"<i>identificación</i>"</li>
</ul>
<p>
El proceso de identificación puede estructurarse como una secuencia de
preguntas:
</p>
<ol class="org-ol">
<li>¿Es estacionaria la serie?</li>
<li>¿Tiene una media significativa?</li>
<li>¿Es persistente la ACF? ¿sigue alguna pauta reconocible?</li>
<li>¿Es persistente la PACF? ¿sigue alguna pauta reconocible?</li>
</ol>
<ul class="org-ul">
<li>La identificación se basa en estadísticos, como la media muestral o
las autocorrelaciones, cuya representatividad depende de la
estacionariedad de las series</li>
<li>Tras inducir la estacionariedad, especificamos un modelo tentativo
decidiendo cuál de las funciones ACF o PACF es finita y cuál es
persistente</li>
</ul>
<table border="2" cellspacing="0" cellpadding="6" rules="all" frame="border">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"> </th>
<th scope="col" class="org-left"><b>ACF finita</b></th>
<th scope="col" class="org-left"><b>ACF persistente</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"><b>PACF finita</b></td>
<td class="org-left"><span class="underline">Ruido blanco</span>: retardos conjuntamente NO significativos</td>
<td class="org-left"><span class="underline">AR</span>: orden indicado por la PACF</td>
</tr>
<tr>
<td class="org-left"><b>PACF persistente</b></td>
<td class="org-left"><span class="underline">MA</span>: orden indicado por la ACF</td>
<td class="org-left"><span class="underline">ARMA</span></td>
</tr>
</tbody>
</table>
<p>
La parametrización de mayor orden en modelos ARMA con series
económicas suele ser ARMA(\(2,1\))
</p>
</div>
<div id="outline-container-org3c7703c" class="outline-3">
<h3 id="org3c7703c"><span class="section-number-3">1.1.</span> Instrumentos de identificación</h3>
<div class="outline-text-3" id="text-1-1">
<table border="2" cellspacing="0" cellpadding="6" rules="all" frame="border">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"> </th>
<th scope="col" class="org-left">Instrumento</th>
<th scope="col" class="org-left">Objetivo y observaciones</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Transf. logarítmica</td>
<td class="org-left">Gráficos rango-media y serie temporal</td>
<td class="org-left">Conseguir independizar la variabilidad de los datos de su nivel. Las series económicas necesitan esta transformación frecuentemente</td>
</tr>
<tr>
<td class="org-left">\(d\), orden de diferenciación</td>
<td class="org-left">Gráfico de la serie temporal. ACF (caída lenta y lineal). Contrastes de raíz unitaria (DF o ADF y KPSS)</td>
<td class="org-left">Conseguir que los datos fluctúen en torno a una media estable. En series económicas, \(d\) suele ser 0, 1 ó 2</td>
</tr>
<tr>
<td class="org-left">Constante</td>
<td class="org-left">Media de la serie transformada. Desviación típica de la media</td>
<td class="org-left">Si la media de la serie transformada es significativa, el modelo debe incluir un término constante</td>
</tr>
<tr>
<td class="org-left">\(p\), orden AR</td>
<td class="org-left">Si PACF de orden \(p\) y ACF infinita</td>
<td class="org-left">En series económicas \(p\) suele ser \(\leq2\)</td>
</tr>
<tr>
<td class="org-left">\(q\), orden MA</td>
<td class="org-left">Si ACF de orden \(q\) y PACF infinita</td>
<td class="org-left">En series económicas q suele ser \(\leq1\)</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org6f5a589" class="outline-3">
<h3 id="org6f5a589"><span class="section-number-3">1.2.</span> Instrumentos de diagnosis</h3>
<div class="outline-text-3" id="text-1-2">
<table border="2" cellspacing="0" cellpadding="6" rules="all" frame="border">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"> </th>
<th scope="col" class="org-left">Instrumento</th>
<th scope="col" class="org-left">Posible diagnóstico</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(d\), orden de diferenciación</td>
<td class="org-left"><i>Proximidad a 1</i> de alguna raíz de los polinomios AR o MA</td>
<td class="org-left">Conviene diferenciar si la raíz es AR; o quitar una diferencia si es MA (<i>salvo si hay tendencia determinista</i>)</td>
</tr>
<tr>
<td class="org-left">\(d\), orden de diferenciación</td>
<td class="org-left">Gráfico de los residuos</td>
<td class="org-left">Si muestra rachas largas de residuos positivos o negativos, puede ser necesaria una diferencia adicional.</td>
</tr>
<tr>
<td class="org-left">Constante</td>
<td class="org-left">Media de los residuos</td>
<td class="org-left">Si es significativa: añadir una constante</td>
</tr>
<tr>
<td class="org-left">Constante</td>
<td class="org-left">Constante estimada</td>
<td class="org-left">Si NO es significativa: el modelo mejorará quitando el término constante</td>
</tr>
<tr>
<td class="org-left">\(p\) y \(q\),</td>
<td class="org-left">Contrastes de significación de los parámetros estimados</td>
<td class="org-left">Pueden sugerir eliminar parámetros irrelevantes</td>
</tr>
<tr>
<td class="org-left">\(p\) y \(q\),</td>
<td class="org-left">ACF/PACF residuos. Test Q de Ljung-Box</td>
<td class="org-left">Indican posibles pautas de autocorrelación no modelizadas</td>
</tr>
<tr>
<td class="org-left">\(p\) y \(q\),</td>
<td class="org-left">Correlaciones elevadas entre los parámetros estimados</td>
<td class="org-left">Puede ser síntoma de sobreparametrización</td>
</tr>
</tbody>
</table>
<p>
Aún, una vez superadas las pruebas de diagnostico, se puede aplicar un
análisis exploratorio consistente en añadir parámetros AR y/o MA, para
comprobar si resultan significativos y mejoran el modelo
</p>
</div>
</div>
</div>
<div id="outline-container-orga6f0571" class="outline-2">
<h2 id="orga6f0571"><span class="section-number-2">2.</span> Raíces unitarias</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org602d3b3" class="outline-3">
<h3 id="org602d3b3"><span class="section-number-3">2.1.</span> Notación: operadores retardo y diferencia y modelos ARIMA</h3>
<div class="outline-text-3" id="text-2-1">
<p>
El operador diferencia \(\nabla\) se define a partir del operador retardo como \(\nabla=(1 - \mathsf{B})\):
\[\nabla Y_t = (1 - \mathsf{B})Y_t = Y_t - Y_{t-1}\]
El operador diferencia estacional es
\({\nabla}_{_S} = (1 - \mathsf{B}^S)\):
\[\nabla_{_S}Y_t = (1 - \mathsf{B}^S)Y_t = Y_t - Y_{t-S}\]
</p>
</div>
<div id="outline-container-org93b7704" class="outline-4">
<h4 id="org93b7704"><span class="section-number-4">2.1.1.</span> Notación: ARIMA</h4>
<div class="outline-text-4" id="text-2-1-1">
<p>
Extendemos la notación a procesos con raíces autorregresivas unitarias
Con ``ARIMA(\(p,d,q\))''; donde \(d\) indica el número de diferencias que
la serie necesita para ser \(I(0)\),
\[\boldsymbol{\phi}_p*\nabla^d*\boldsymbol{Y} = \boldsymbol{\theta}_q*
\boldsymbol{U}\] es decir \[\boldsymbol{\phi}_p(\mathsf{B})\nabla^d
Y_t = \boldsymbol{\theta}_q(\mathsf{B}) U_t; \quad t\in\mathbb{Z}\]
</p>
</div>
</div>
</div>
<div id="outline-container-org76320fe" class="outline-3">
<h3 id="org76320fe"><span class="section-number-3">2.2.</span> Raíces unitarias en los polinomios AR y MA</h3>
<div class="outline-text-3" id="text-2-2">
<p>
Cuando un polinomio tiene alguna raíz igual a uno se dice que tiene
“raíces unitarias”.
</p>
<p>
Si el polinomio AR estimado tiene alguna raíz "próxima a uno" es
síntoma de infradiferenciación.
</p>
<p>
Si el polinomio MA estimado tiene alguna raíz "próxima a uno" es
síntoma de sobrediferenciación.
</p>
<p>
Ejemplos:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="all" frame="border">
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Modelo expresado con raíces unitarias en \(\boldsymbol{\phi}\) o \(\boldsymbol{\theta}\)</th>
<th scope="col" class="org-left">Modelo equivalente sin raíces unitarias en \(\boldsymbol{\phi}\) o \(\boldsymbol{\theta}\)</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\((1-1.5\mathsf{B}+.5\mathsf{B}^2) Y_t = U_t\)</td>
<td class="org-left">\({\color{blue}{(1-0.5\mathsf{B})\nabla Y_t=U_t}}\)</td>
</tr>
<tr>
<td class="org-left">\((1-.5\mathsf{B}+0.7\mathsf{B}^2)\nabla^2Y_t=(1-\mathsf{B})U_t\)</td>
<td class="org-left">\({\color{blue}{(1-.5\mathsf{B}+0.7\mathsf{B}^2)\nabla Y_t = U_t}}\)</td>
</tr>
<tr>
<td class="org-left">\(\nabla Y_t = \beta+ (1-\mathsf{B}) U_t\)</td>
<td class="org-left">\({\color{blue}{Y_t = \beta t + U_t}}\quad\) (¡no estacionario!)</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orgd761132" class="outline-3">
<h3 id="orgd761132"><span class="section-number-3">2.3.</span> Paseos aleatorios</h3>
<div class="outline-text-3" id="text-2-3">
<p>
Un paseo aleatorio representa una variable cuyos cambios son ruido
blanco: \[Y_t = \mu + Y_{t-1} + U_t\]
</p>
<p>
Cuando \(\mu\ne0\) se denomina <i>paseo aleatorio con deriva</i>: \(\;\nabla Y_t = \mu + U_t\)
</p>
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 4, repr.plot.res = 200)
rwcd <span style="color: #D0372D;"><-</span> um(i = <span style="color: #008000;">"(1 - B)"</span>,
mu=.25)
ide(sim(rwcd, n = 500),
lag.max = 20,
graphs = c(<span style="color: #008000;">"plot"</span>, <span style="color: #008000;">"acf"</span>, <span style="color: #008000;">"pacf"</span>, <span style="color: #008000;">"pgram"</span>),
main = <span style="color: #008000;">"Paseo aleatorio con deriva (mu=0.25)"</span>)
</pre>
</div>
<div id="orgaea7bec" class="figure">
<p><img src="./img/lecc08/ACF-RWcd.png" alt="ACF-RWcd.png" width="900px" />
</p>
</div>
<p>
El proceso tiene mayor inercia cuanto mayor es \(|\mu|\). El signo de
\(\mu\) determina el signo de la pendiente global.
</p>
<p>
Cuando \(\mu=0\) se denomina sencillamente <i>paseo aleatorio</i>: \(\;\nabla Y_t = U_t\)
</p>
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 4, repr.plot.res = 200)
rw <span style="color: #D0372D;"><-</span> um(i = <span style="color: #008000;">"(1 - B)"</span>)
ide(sim(rw, n = 500), lag.max = 20, graphs = c(<span style="color: #008000;">"plot"</span>, <span style="color: #008000;">"acf"</span>, <span style="color: #008000;">"pacf"</span>, <span style="color: #008000;">"pgram"</span>), main = <span style="color: #008000;">"Paseo aleatorio"</span>)
</pre>
</div>
<div id="org211f318" class="figure">
<p><img src="./img/lecc08/ACF-RW.png" alt="ACF-RW.png" width="900px" />
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org1c9df03" class="outline-2">
<h2 id="org1c9df03"><span class="section-number-2">3.</span> Modelos ARIMA estacionales (SARIMA)</h2>
<div class="outline-text-2" id="text-3">
<p>
El período estacional \(S\) es el número mínimo de observaciones
necesarias para recorrer un ciclo estacional completo. Por ejemplo,
\(S=12\) para datos mensuales, \(S=4\) para datos trimestrales, \(S=24\)
para datos horarios, etc.
</p>
<p>
Describiremos comportamientos estacionales con modelos
ARIMA\((p,d,q)\times(P,D,Q)_S\)
</p>
<p>
\[\boldsymbol{\phi}_p(\mathsf{B})\boldsymbol{\Phi}_P(\mathsf{B}^S)\nabla^d\nabla_{_S}^D
Y_t =
\boldsymbol{\theta}_q(\mathsf{B})\boldsymbol{\Theta}_q(\mathsf{B}^S)
U_t; \quad t\in\mathbb{Z}\] donde
</p>
\begin{align*}
\boldsymbol{\Phi}_P(\mathsf{B}^S) = & 1-\Phi_1\mathsf{B}^{1\cdot S}-\Phi_2\mathsf{B}^{2\cdot S}-\cdots-\Phi_P\mathsf{B}^{P\cdot S}\\
\boldsymbol{\Theta}_Q(\mathsf{B}^S) = & 1-\Theta_1\mathsf{B}^{1\cdot S}-\Theta_2\mathsf{B}^{2\cdot S}-\cdots-\Theta_Q\mathsf{B}^{Q\cdot S}\\
{\nabla}_{_S}^D = & (1 - \mathsf{B}^S)^D
\end{align*}
<p>
Es decir, el modelo consta de polinomios autorregresivos y de media
móvil tanto regulares (en minúsculas) como estacionales (en
mayúsculas).
</p>
<p>
Veamos un ejemplo de un modelo MA(\(1\)) estacional y otro de un modelo
AR(\(1\)) estacional…
</p>
</div>
<div id="outline-container-orgd6a2919" class="outline-3">
<h3 id="orgd6a2919"><span class="section-number-3">3.1.</span> MA(1) estacional con raíz positiva</h3>
<div class="outline-text-3" id="text-3-1">
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 2, repr.plot.res = 200)
SMA1 <span style="color: #D0372D;"><-</span> um(ma = <span style="color: #008000;">"(1 - 0.9B12)"</span>)
display(list(SMA1), lag.max = 50, byrow = <span style="color: #6434A3;">TRUE</span>)
</pre>
</div>
<p>
MA(\(1\)) estacional:
\(\quad\boldsymbol{\Theta}=1-0.9z^{12}\quad\Rightarrow\quad X_t= (1-0.9
\mathsf{B}^{12})U_t\)
</p>
<div id="org35cdda6" class="figure">
<p><img src="./img/lecc08/ACF-SMA1p.png" alt="ACF-SMA1p.png" width="900px" />
</p>
</div>
<div class="org-src-container">
<pre class="src src-jupyter-R">roots(SMA1)
</pre>
</div>
<ol class="org-ol">
<li><table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">Real</th>
<th scope="col" class="org-right">Imaginary</th>
<th scope="col" class="org-right">Modulus</th>
<th scope="col" class="org-right">Frequency</th>
<th scope="col" class="org-right">Period</th>
<th scope="col" class="org-right">Mult.</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">1.008819e+00</td>
<td class="org-right">1.082287e-14</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.00000000</td>
<td class="org-right">Inf</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">8.736626e-01</td>
<td class="org-right">5.044094e-01</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.08333333</td>
<td class="org-right">12.0</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">8.736626e-01</td>
<td class="org-right">-5.044094e-01</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.08333333</td>
<td class="org-right">12.0</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">5.044094e-01</td>
<td class="org-right">-8.736626e-01</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.16666667</td>
<td class="org-right">6.0</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">5.044094e-01</td>
<td class="org-right">8.736626e-01</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.16666667</td>
<td class="org-right">6.0</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">1.288336e-14</td>
<td class="org-right">-1.008819e+00</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.25000000</td>
<td class="org-right">4.0</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">-2.057493e-17</td>
<td class="org-right">1.008819e+00</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.25000000</td>
<td class="org-right">4.0</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">-5.044094e-01</td>
<td class="org-right">-8.736626e-01</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.33333333</td>
<td class="org-right">3.0</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">-5.044094e-01</td>
<td class="org-right">8.736626e-01</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.33333333</td>
<td class="org-right">3.0</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">-8.736626e-01</td>
<td class="org-right">-5.044094e-01</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.41666667</td>
<td class="org-right">2.4</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">-8.736626e-01</td>
<td class="org-right">5.044094e-01</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.41666667</td>
<td class="org-right">2.4</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-right">-1.008819e+00</td>
<td class="org-right">-1.257046e-14</td>
<td class="org-right">1.008819</td>
<td class="org-right">0.50000000</td>
<td class="org-right">2.0</td>
<td class="org-right">1</td>
</tr>
</tbody>
</table></li>
</ol>
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 5, repr.plot.res = 200)
ide(sim(SMA1, n = 500),
lag.max = 50,
lags.at = 12,
graphs = c(<span style="color: #008000;">"plot"</span>, <span style="color: #008000;">"acf"</span>, <span style="color: #008000;">"pacf"</span>, <span style="color: #008000;">"pgram"</span>))
</pre>
</div>
<div id="org55a1d40" class="figure">
<p><img src="./img/lecc08/Sim-SMA1p.png" alt="Sim-SMA1p.png" width="900px" />
</p>
</div>
</div>
</div>
<div id="outline-container-orgcbed32f" class="outline-3">
<h3 id="orgcbed32f"><span class="section-number-3">3.2.</span> AR(1) estacional con raíz positiva</h3>
<div class="outline-text-3" id="text-3-2">
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 2, repr.plot.res = 200)
SAR1 <span style="color: #D0372D;"><-</span> um(ar = <span style="color: #008000;">"(1 - 0.9B12)"</span>)
display(list(SAR1), lag.max = 50, byrow = <span style="color: #6434A3;">TRUE</span>)
</pre>
</div>
<p>
AR(\(1\)) estacional:
\(\quad\boldsymbol{\Phi}=1-0.9z^{12}\quad\Rightarrow\quad (1-0.9
\mathsf{B}^{12})X_t= U_t\)
</p>
<div id="org39411f0" class="figure">
<p><img src="./img/lecc08/ACF-SAR1p.png" alt="ACF-SAR1p.png" width="900px" />
</p>
</div>
<div class="org-src-container">
<pre class="src src-jupyter-R">roots(SAR1)
</pre>
</div>
<p>
Evidentemente las raíces son iguales a las del caso anterior (aunque ahora corresponden al polinomio autorregresivo).
</p>
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 5, repr.plot.res = 200)
ide(sim(SAR1, n = 500),
lag.max = 50,
lags.at = 12,
graphs = c(<span style="color: #008000;">"plot"</span>, <span style="color: #008000;">"acf"</span>, <span style="color: #008000;">"pacf"</span>, <span style="color: #008000;">"pgram"</span>))
</pre>
</div>
<div id="orgf8d8818" class="figure">
<p><img src="./img/lecc08/Sim-SAR1p.png" alt="Sim-SAR1p.png" width="900px" />
</p>
</div>
<p>
Con estos dos ejemplos hemos podido apreciar que:
</p>
<ul class="org-ul">
<li>las pautas de autocorrelación son análogas a las de los MA(1) y
AR(2), pero ahora los retardos significativos corresponden a los
retardos estacionales, es decir, a múltiplos del período estacional
\(S\).</li>
<li>En estos ejemplos, en los que \(S=12\), los retardos estacionales son:
12, 24, 36, 48, 60,…</li>
<li>las correlaciones correspondientes a los “retardos regulares” (es
decir, todos menos menos los estacionales) son no significativas en
general.</li>
</ul>
<p>
Veamos ahora un par de ejemplos de modelos estacionales
multiplicativos (i.e., con parte regular y parte estacional).
</p>
</div>
</div>
<div id="outline-container-org3b20ddb" class="outline-3">
<h3 id="org3b20ddb"><span class="section-number-3">3.3.</span> ARIMA\((0,0,1)\times(0,0,1)_{12}\)</h3>
<div class="outline-text-3" id="text-3-3">
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 2, repr.plot.res = 200)
MA1SMA1 <span style="color: #D0372D;"><-</span> um(ma = <span style="color: #008000;">"(1 - 0.9B)(1 - 0.9B12)"</span>)
display(list(MA1SMA1), lag.max = 50, byrow = <span style="color: #6434A3;">TRUE</span>)
</pre>
</div>
<p>
ARIMA\((0,0,1)\times(0,0,1)_{12}\):
\(\quad X_t= (1-0.9 \mathsf{B})(1-0.9 \mathsf{B}^{12})U_t\)
</p>
<div id="org645a202" class="figure">
<p><img src="./img/lecc08/ACF-MA1SMA1.png" alt="ACF-MA1SMA1.png" width="900px" />
</p>
</div>
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 5, repr.plot.res = 200)
ide(sim(MA1SMA1, n = 500),
lag.max = 50,
lags.at = 12,
graphs = c(<span style="color: #008000;">"plot"</span>, <span style="color: #008000;">"acf"</span>, <span style="color: #008000;">"pacf"</span>, <span style="color: #008000;">"pgram"</span>))
</pre>
</div>
<div id="orgbee9fcf" class="figure">
<p><img src="./img/lecc08/Sim-MA1SMA1.png" alt="Sim-MA1SMA1.png" width="900px" />
</p>
</div>
</div>
</div>
<div id="outline-container-orga3b07b1" class="outline-3">
<h3 id="orga3b07b1"><span class="section-number-3">3.4.</span> ARIMA\((1,0,0)\times(0,0,1)_{12}\)</h3>
<div class="outline-text-3" id="text-3-4">
<div class="org-src-container">
<pre class="src src-jupyter-R"><span style="color: #D0372D;">options</span>(repr.plot.width = 12, repr.plot.height = 2, repr.plot.res = 200)
AR1SMA1 <span style="color: #D0372D;"><-</span> um(ar = <span style="color: #008000;">"(1 - 0.9B)"</span>, ma = <span style="color: #008000;">"(1 - 0.9B12)"</span>)
display(list(AR1SMA1), lag.max = 50, byrow = <span style="color: #6434A3;">TRUE</span>)
</pre>
</div>
<p>
ARIMA\((1,0,0)\times(0,0,1)_{12}\):
\(\quad (1-0.9 \mathsf{B})X_t= (1-0.9 \mathsf{B}^{12})U_t\)
</p>
<div id="org8b7eb12" class="figure">
<p><img src="./img/lecc08/ACF-AR1SMA1.png" alt="ACF-AR1SMA1.png" width="900px" />
</p>
</div>
<div class="org-src-container">