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.pull-left[

# Índice de precios al productor (IPP)
### Oferta Interna: Agricultura, Ganaderia, Pesca y Silvicultura
#### Por: Santiago Bula M.

### [`#Help_Ukraine 🇺🇦`](https://helpua.bank.gov.ua)

]

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# Introducción
Objetivos de la presentación, metodología y justificación

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<h1 align="left">Librerías</h1>

Para el desarrollo de este proyecto se hizo uso de la siguiente lista de paquetes:

.pull-left[

```r
library(rmarkdown) #Unnecessary 
library(bslib) #References 
library(DT) #Data Tables
library(plotly) #Ggplot Pro
library(xaringan) #Presentation Gen.
library(plyr) #Data Filtration
library(readxl) #Import Excel Files
library(kableExtra) #Interactive Tables
library(revealjs) #Scrolling
library(knitr) #Render
library(forecast) #ARIMA Command

# How to see the fonts? 
names(xaringan:::list_css()) 
```
]

---

# Introducción
El IPP un indicador de la evolución de los precios de venta del productor, correspondientes al primer canal de comercialización o distribución de los bienes transados en la economía, su diferencia con el IPC son los intercambios entre los intermediarios de mercado.

La metodología usada para la realización del IPP usada por el DANE tiene como objetivo proporcionar una medición de la variación mensual promedio de los precios de una canasta de bienes representativa de la producción nacional en la cual el diseño muestral es no probabilístico tomando fuentes que se relacionan con la canasta de bienes entre ellas empresas productoras, comercializadoras y manufactureras, entre otros.

---
### Variables de Clasificación:
#### 1. Según Actividad Económica: Sección, División, Grupo y Clase.
- 1.1.1.     Agricultura, ganadería, caza, silvicultura y pesca
- 1.1.2.    Explotación de minas y canteras 
- 1.1.3.    Industrias manufactureras

---
# Falta por complementar
1. Año base 
1. Formula de indice Paasche o Laspayres??
1. Ficha técnica
1. Historia e importancia del del indice 
1. Medicion economia, tipo de calculo del indice

???

Cual es la diferencia entre Paasche o Laspayres

---

--- 
## Objetivos

### Objetivo general:

1.  Analizar el IPP por oferta interna en el período 2001-2019, mediante la metodología de Box-Jenkins.

### Objetivos especīficos:

1.  Transformar la serie de IPP por oferta interna del sector agropecuario para garantizar el cumplimiento de estacionariedad.

1.  Identificar los órdenes p,d,q para la estimación y pronóstico del modelo ARIMA.  
--

1.  Verificar el cumplimiento en los residules de los supuestos básicos del modelo.

1. Comprender la programación por comandos directos y manuales  para desarrollar la metodología de Box-Jenkins.

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## Metodología
Método empleado para el desarrollo del ejercicio econométrico

---
## Justificación

Para la realización de este proyecto se tendrá en cuenta el análisis de una serie de datos llamada el índice de precios al productor, la cual a términos generales es un indicador de la evolución de los precios de venta del productor, este proyecto se realizará para ver qué tan eficaz será el uso de esta serie y por ende realizar la verificación de ciertos supuestos básicos para poder determinar la verificación del modelo para luego realizar el uso de esta y poder generar pronósticos y simular.

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# Identificación
Recolección, tratamiento y manipulación de datos

---

## Identificación

### Recolección de datos.

Para el ejericio econometrico importamos los datos de la base de datos Totoro del Banco de la República de Colombia. 
- El periodo de estudio del ejercicio es: **01/01/2001 - 31/12/2019**
- Se agrupa la serie en series anuales y se quita n=0 observaciones de 2019

<center> **N−n**= 228 &nbsp; **H**= 19 &nbsp; **R**= 12

---
## Recolección de datos.

```
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   52.01   68.64   86.04   86.91  101.69  133.36
```

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class: center 
## Análisis de la Serie

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00:00:00.000000","2009-08-01 00:00:00.000000","2009-09-01 00:00:00.000000","2009-10-01 00:00:00.000000","2009-11-01 00:00:00.000000","2009-12-01 00:00:00.000000","2010-01-01 00:00:00.000000","2010-02-01 00:00:00.000000","2010-03-01 00:00:00.000000","2010-04-01 00:00:00.000000","2010-05-01 00:00:00.000000","2010-06-01 00:00:00.000000","2010-07-01 00:00:00.000000","2010-08-01 00:00:00.000000","2010-09-01 00:00:00.000000","2010-10-01 00:00:00.000000","2010-11-01 00:00:00.000000","2010-12-01 00:00:00.000000","2011-01-01 00:00:00.000000","2011-02-01 00:00:00.000000","2011-03-01 00:00:00.000000","2011-04-01 00:00:00.000000","2011-05-01 00:00:00.000000","2011-06-01 00:00:00.000000","2011-07-01 00:00:00.000000","2011-08-01 00:00:00.000000","2011-09-01 00:00:00.000000","2011-10-01 00:00:00.000000","2011-11-01 00:00:00.000000","2011-12-01 00:00:00.000000","2012-01-01 00:00:00.000000","2012-02-01 00:00:00.000000","2012-03-01 00:00:00.000000","2012-04-01 00:00:00.000000","2012-05-01 00:00:00.000000","2012-06-01 00:00:00.000000","2012-07-01 00:00:00.000000","2012-08-01 00:00:00.000000","2012-09-01 00:00:00.000000","2012-10-01 00:00:00.000000","2012-11-01 00:00:00.000000","2012-12-01 00:00:00.000000","2013-01-01 00:00:00.000000","2013-02-01 00:00:00.000000","2013-03-01 00:00:00.000000","2013-04-01 00:00:00.000000","2013-05-01 00:00:00.000000","2013-06-01 00:00:00.000000","2013-07-01 00:00:00.000000","2013-08-01 00:00:00.000000","2013-09-01 00:00:00.000000","2013-10-01 00:00:00.000000","2013-11-01 00:00:00.000000","2013-12-01 00:00:00.000000","2014-01-01 00:00:00.000000","2014-02-01 00:00:00.000000","2014-03-01 00:00:00.000000","2014-04-01 00:00:00.000000","2014-05-01 00:00:00.000000","2014-06-01 00:00:00.000000","2014-07-01 00:00:00.000000","2014-08-01 00:00:00.000000","2014-09-01 00:00:00.000000","2014-10-01 00:00:00.000000","2014-11-01 00:00:00.000000","2014-12-01 00:00:00.000000","2015-01-01 00:00:00.000000","2015-02-01 00:00:00.000000","2015-03-01 00:00:00.000000","2015-04-01 00:00:00.000000","2015-05-01 00:00:00.000000","2015-06-01 00:00:00.000000","2015-07-01 00:00:00.000000","2015-08-01 00:00:00.000000","2015-09-01 00:00:00.000000","2015-10-01 00:00:00.000000","2015-11-01 00:00:00.000000","2015-12-01 00:00:00.000000","2016-01-01 00:00:00.000000","2016-02-01 00:00:00.000000","2016-03-01 00:00:00.000000","2016-04-01 00:00:00.000000","2016-05-01 00:00:00.000000","2016-06-01 00:00:00.000000","2016-07-01 00:00:00.000000","2016-08-01 00:00:00.000000","2016-09-01 00:00:00.000000","2016-10-01 00:00:00.000000","2016-11-01 00:00:00.000000","2016-12-01 00:00:00.000000","2017-01-01 00:00:00.000000","2017-02-01 00:00:00.000000","2017-03-01 00:00:00.000000","2017-04-01 00:00:00.000000","2017-05-01 00:00:00.000000","2017-06-01 00:00:00.000000","2017-07-01 00:00:00.000000","2017-08-01 00:00:00.000000","2017-09-01 00:00:00.000000","2017-10-01 00:00:00.000000","2017-11-01 00:00:00.000000","2017-12-01 00:00:00.000000","2018-01-01 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---
background-image: url(Immagini/Desarrollo.jpg)
background-size: cover
background-position: middle
class: inverse center middle

# Desarrollo práctico
Programación en R

---

## Estabilización de Varianza

El sentido metodologico de este ejercicio es poder identificar el proceso generador de los datos de la serie, motivo por ello es relavente explorar la existencia de estacionariedad sobre la serie.

> “Si la variabilidad tiende a cambiar con la media de las observaciones,  la varianza
  solo se estabilizará con un cambio apropiado de escala”  
-**Misas, 2018**

Para hallar la varianza es necesario conocer la media `\(\bar{Z_h}\)` y la desviación estándar `\(S_h\)`. Las cuales fueron calculadas de la siguiente manera:

.pull-left[
.pull-right[
<img src="https://latex.codecogs.com/svg.latex?\Large&space;\bar{Z}_h=\frac{\sum^R_{r=1}{Z_{h,r}}}{R}" 
title="Media Zh"/>
]
]

.pull-right[ 
.pull-left[
<img src="https://latex.codecogs.com/svg.latex?\Large&space;S_h=\sqrt{\frac{ \sum_{r=1}^R{(Z_{h,r}-\bar{Z}_h)^2}}{R-1}}" 
title="Desviación Estándard"/>
]
]

---

## Media y desviación estándar por grupo
.left-column[
**Media y desviación estándar**

Potencias Lambda

Diferenciación

ACF

PACF

]

```r
## Media 
Zh <- matrix(data=NA, nrow = H, ncol = 1)
Eich <-matrix(data=seq(from=12, to=N-n , by=R), nrow = H, ncol = 1) 
#Eich
for(z in 2:H ){
 Zh[1] <- mean(IPP_Estudio[1:12])
 Zh[z] <- mean(IPP_Estudio[Eich[z-1]:Eich[z]]) } 
# Zh 
## Desv. Standard 
Sh <- matrix(data=NA, nrow = H, ncol = 1)
#Eich
for(s in 2:H ){
 Sh[1] <- sd(IPP_Estudio[1:12]) #<<
 Sh[s] <- sqrt(sum((IPP_Estudio[,s]-mean(IPP_Estudio[,s]))^2)/(R-1))} 
```

---
## Media y desviación estándar por lambda
.left-column[
Media y desviación estándar

**Potencias Lambda**

Diferenciación

ACF

PACF

]
.right-column[
Los candidatos a lambdas son: `\(\lambda_1 = -1\)`, `\(\lambda_2=-0.5\)` , `\(\lambda_3=0\)`, `\(\lambda_4=0.5\)` `\(\lambda_5=1\)`

<div id="htmlwidget-8c2b3d45f3fcc8f1b8dc" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-8c2b3d45f3fcc8f1b8dc">{"x":{"filter":"none","vertical":false,"caption":"<caption>Tabla de potencias lambda<\/caption>","fillContainer":false,"data":[[2019,2018,2017,2016,2015,2014,2013,2012,2011,2010,2009,2008,2007,2006,2005,2004,2003,2002,2001],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[0.00037,0.00055,0.00033,0.00021,0.00022,0.00022,0.0002,0.00036,0.00047,0.00024,0.0002,0.00031,0.0003,0.00051,0.00039,0.00028,7e-05,0.00013,0.00022],[0.00277,0.00416,0.00253,0.00165,0.00184,0.00182,0.00169,0.00315,0.00429,0.00223,0.00191,0.00302,0.00284,0.00491,0.004,0.0031,0.00073,0.00144,0.00253],[0.02042,0.03139,0.01966,0.01312,0.01515,0.0153,0.01441,0.02773,0.03891,0.02064,0.01858,0.02921,0.02658,0.04773,0.04094,0.03383,0.00782,0.01556,0.0286],[0.1508,0.23679,0.15286,0.10443,0.1247,0.12833,0.12288,0.24433,0.35259,0.1911,0.18079,0.28207,0.24878,0.46348,0.41906,0.36938,0.08423,0.16875,0.32294],[1.11362,1.78636,1.1888,0.83107,1.02637,1.07648,1.04771,2.15277,3.19475,1.76958,1.759,2.72395,2.32836,4.50103,4.28964,4.03356,0.90777,1.82999,3.64697]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Year<\/th>\n <th>Group (H)<\/th>\n <th>Lambda -1<\/th>\n <th>Lambda -0.5<\/th>\n <th>Lambda 0<\/th>\n <th>Lambda 0.5<\/th>\n <th>Lambda 1<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"scrollX":true,"columnDefs":[{"className":"dt-right","targets":[0,1,2,3,4,5,6]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>
]

---
# Transformación potencia
.left-column[
Media y desviación estándar

**Potencias Lambda**

Diferenciación

ACF

PACF

]

.right-column[
<table class=" lightable-paper" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; width: auto !important; margin-left: auto; margin-right: auto;'>
 <thead>
 <tr>
 <th style="text-align:left;"> Paso </th>
 <th style="text-align:left;"> Fórmula </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> 1. Media </td>
 <td style="text-align:left;"> <html><body><img src="Immagini/M_Kable.png" width="384" height="128"></body></html>
</td>
 </tr>
 <tr>
 <td style="text-align:left;"> 2. Desviación estándar </td>
 <td style="text-align:left;"> <html><body><img src="Immagini/SD_Kable.png" width="384" height="128"></body></html>
</td>
 </tr>
 <tr>
 <td style="text-align:left;"> 3. Coeficiente de variación </td>
 <td style="text-align:left;"> <html><body><img src="Immagini/CV_Kable.png" width="384" height="128"></body></html>
</td>
 </tr>
</tbody>
</table>
]

---
## Selección del Lambda

.left-column[
Media y Std.Desv

**Potencias Lambda**

Diferenciación

ACF

PACF

]

.right-column[
<table class=" lightable-paper lightable-hover" style='font-family: "Arial Narrow", arial, helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
<caption>Coeficientes de variación para lambda</caption>
 <thead>
 <tr>
 <th style="text-align:left;"> </th>
 <th style="text-align:right;"> Coeficiente de variación </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Lambda -1 </td>
 <td style="text-align:right;"> 0.4287876 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Lambda -0.5 </td>
 <td style="text-align:right;"> 0.4122601 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Lambda 0 </td>
 <td style="text-align:right;"> 0.4362937 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Lambda 0.5 </td>
 <td style="text-align:right;"> 0.4901177 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Lambda 1 </td>
 <td style="text-align:right;"> 0.5624301 </td>
 </tr>
</tbody>
</table>

```
## El valor mínimo escogido va ser  0.4122601 , en este caso es  Lambda -0.5
```
]

---
## Transformar la serie

.left-column[
Media y Std.Desv

Potencias Lambda

**Diferenciación**

ACF

PACF

]
Teniendo en cuenta los siguientes casos:

```r
G.Zh. <- matrix(data = NA, ncol = 1, nrow = N)
for(h in 1:N){
 G.Zh.[h] <- IPP_Estudio[h]^(-0.5) # This shape changes according the lambda value
}
#G.Zh.

Diff1G.Zh. <- matrix(data = NA, ncol = 1, nrow = N)
for(h in 2:N){
 Diff1G.Zh.[h] <- G.Zh.[h]-G.Zh.[h-1] 
}
#Diff1G.Zh.

Diff2G.Zh.<- matrix(data = NA, ncol = 1, nrow = N)
for(h in 3:N){
 Diff2G.Zh.[h] <- Diff1G.Zh.[h]-Diff1G.Zh.[h-1]
} # In case that we would wish differentiate twice we apply the code again
```

---
class: center 
### ACF - Serie transformada. 
.pull-left[
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![](Slides_files/figure-html/ACF Comando 1-1.png)

.footpage[
Nótese que la ACF decrece lentamente, por tanto no presenta estacionariedad y requiere ser diferenciada.]

---
class: center 
### ACF - Comparación de correlogramas. 
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![](Slides_files/figure-html/ACF Comando-1.png)

.footpage[
Nótese que la ACF decrece rápidamente presentando estacionariedad, gracias a que fue diferenciada 1 vez.
]

---
class: center 
### PACF - Comparación de correlogramas. 
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.pull-right[
![](Slides_files/figure-html/PACF Comando-1.png)

]

---
class: center 
## Estimación ARIMA(2,1,3)

![](Slides_files/figure-html/ARIMA-1.png)

---
class: center 
## Pronóstico ARIMA(2,1,3)

![](Slides_files/figure-html/p values-1.png)

---
class: inverse middle center
background-image: url(Immagini/Supuesto.jpg)
background-size: cover
background-position: middle
## Verificación de supuestos

---
# Supuesto 1: Media cero
.pull-left[
<center>
<img src="https://latex.codecogs.com/svg.latex?\Large&space;m(\hat{a})=\frac{\sum_{t=t'}^{T}\hat a_T}{T-d-p}
\\
\\
\hat\sigma_a=\sqrt\frac{\sum_{t=t'}^{T}(\hat a_t-m\hat a_t))^2}{T-d-q-p}" 
title="Media cero"/
 width="500" 
 height="500"> 
</center>
]

.pull-right[

```r
A.hat <- rnorm(100,0,1) #para cuestion del ejemplo 
#SACAR LOS AT CON EL MODELO ARIMA 
N <- 100
d <- 1
p <- 5
q <- 3

m_A.hat <- sum(A.hat)/(N-d-p)
Sigma2aHat <- sqrt( (sum(A.hat-m_A.hat)^2)/(N-d-p-q) )

# Ho: median equal to 0 
# Ha: Violation of assume median equal to 0

Value_proof_median0 <- abs((sqrt(N-d-p)*m_A.hat)/(Sigma2aHat))

if(Value_proof_median0<2){ 
 print("There is no evidence to REJECT median = 0")} else{
 print("There evidence to reject median=0") }
```

```
## [1] "There evidence to reject median=0"
```
]

---
<h1 align="left"> Supuesto 2: Varianza Constante-Inspección visual </h1>

---
# Supuesto 3: Mutuamente Independientes

```r
A.hatK <- rnorm(1000,0.49,2) # Here we evaluate the differentiate series versus non differentiate series.

rho_Hat.a <- (sum(A.hat*A.hatK))/(sum(A.hat)^2)

if(abs(rho_Hat.a)>=(1/(sqrt(N-d-p)))){
  print("The autocorrelation is ??? 0, so the 'at' are mutually independent")
} else {print("The autocorrelation is = 0, so the 'at' are not mutually independent")}
```

```
## [1] "The autocorrelation is ??? 0, so the 'at' are mutually independent"
```

---
class: scrollable-slide
# Supuesto 4: Normalidad

.pull-left[
<center>
<img src="https://latex.codecogs.com/svg.latex?\Large&space;JB=N(\frac{S^2}6+\frac{K^2}{24})\\
si \quad JB \leq X^2--> Acepto \quad H0\\
si \quad JB > X^2--> Rechazo \quad H0\\"title="Jarque-Bera"/
 width="500" 
 height="400">

</center>
]
.pull-right[
.scrollable[

```r
N <- NROW(A.hat) # Here put the manu's values
SIGMA2 <- sum(A.hat^2)/N

M2 <-(((t(A.hat))%*%A.hat)/N)

SIGMA <- sqrt(SIGMA2)
SIGMA
```

```
## [1] 1.005997
```

```r
# TERCER MOMENTO
M3 <-sum(A.hat^3)/N 
M3
```

```
## [1] -0.4398876
```

```r
#CUARTO MOMENTO
M4 <- sum(A.hat^4)/N
M4
```

```
## [1] 3.027681
```

```r
#SIMETRIA
SIMETRIA <- M3/(SIGMA^3)
SIMETRIA
```

```
## [1] -0.4320673
```

```r
#EXCESO DE CURTOSIS
EXCESODECURTOSIS <- ((M4-3*(SIGMA^4))/(SIGMA^4))
EXCESODECURTOSIS
```

```
## [1] -0.04387384
```

```r
#3 CONSTRUCCION DEL ESTADISTICO JARQUE BERA
JARQUEBERA <- N*(((SIMETRIA^2)/6)+((EXCESODECURTOSIS^2)/24))
JARQUEBERA
```

```
## [1] 3.119389
```

```r
#PVALUE
PVALUEJARQUEBERA <- 1-pchisq(JARQUEBERA,2) #JARQUEBERA SE SITRIBUYE CON 2 GRADOS DE LIBERTAD
PVALUEJARQUEBERA
```

```
## [1] 0.2102003
```

```r
#NIVEL DE SIGNIFICANCIA DEL 0.05 PVALUE SE CONTRASTA CON EL NIVEL DE SIGNIFICANCIA 
```

Como tenemos el pvalue mayor al nivel de significancia NO rechazamos la `\(Ho\)` de que los datos vengan de una distribcion normal `\(\sim N(\mu,\sigma)\)`

```r
library(tseries)
JBTESTCOMANDOVERIFICACION <- jarque.bera.test(A.hat)
JBTESTCOMANDOVERIFICACION
```

```
## 
##  Jarque Bera Test
## 
## data:  A.hat
## X-squared = 1.3473, df = 2, p-value = 0.5098
```

```r
VALORCRITICO <- qchisq(1-0.05,2) 
VALORCRITICO
```

```
## [1] 5.991465
```

```r
#ESTADISTICO DE PRUEBA SE CONTARTA CON EL VALOR CRITICO
#como el estadistico de prueba jarque bera es menor al valor critico no rechazamos H0
#SI EXISTE NORMALIDAD

shapiro.test(A.hat)
```

```
## 
##  Shapiro-Wilk normality test
## 
## data:  A.hat
## W = 0.9893, p-value = 0.6086
```

```r
qqnorm(A.hat)
```

![](Slides_files/figure-html/CON COMANDO DIRECTO-1.png)
]
]

---
class: inverse center middle
background-image: url(Immagini/Conclusiones.jpg)
background-size: cover
background-position: middle
# Conclusiones

---
# Conlusiones
Del desarrollo de este ejercicio podemos concluir que:

* La serie de IPP por oferta interna es muy sensible a _schocks_ de oferta externa
* Tomar un año base diferente a 2014 puede presentar valores nominales diferentes
* Se debe pulir el desarrollo de la metodología de Box Jetkins para obtener resultados mejores