Relación entre el PIB y el coeficente de GINI

Para este trabajo tomaremos datos desde 1990 hasta 2021, la idea es probar si el crecimiento económico génera que haya más igualdad.

Cargamos el documento

#install.packages("readxl")#
#install.packages("tidyverse")#
#install.packages("tseries")
#install.packages("car")
#install.packages("timsac")
# Instalar el paquete "urca" (solo si aún no está instalado)
#install.packages("urca")
#install.packages("readxl")

# Cargar el paquete "urca"
library(readxl)
library(urca)
library(lubridate)
## 
## Attaching package: 'lubridate'
## The following objects are masked from 'package:base':
## 
##     date, intersect, setdiff, union
library(tseries)
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library(lubridate)
library(tidyverse)
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ✔ forcats 1.0.0     ✔ stringr 1.5.0
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library(car)
## Loading required package: carData
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## Attaching package: 'car'
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## The following object is masked from 'package:dplyr':
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library(foreign)
library(timsac)

data <- read_excel("C:\\Users\\david\\Desktop\\UNAL\\Octavo Semestre\\Eco II\\ginipib.xlsx")

Ahora, vamos a ajustar la variable tiempo

data$Año <- NULL
# Cambiar los nombres de las columnas en un marco de datos data
colnames(data) <- c("gini", "pib")
gini.ts = ts(data$gini, start = 1990, frequency = 1)
pib.ts = ts(data$pib, start = 1990, frequency = 1)
data$Año <- NULL
class(data$pib)
## [1] "numeric"
class(data$gini)
## [1] "numeric"
#Convierto PIB a variable en Log
data <- data %>%
  mutate(lpib = log(pib.ts))
#hagamos lo mismo para lpib
lpib.ts = ts(data$lpib, start = 1990, frequency = 1)
plot(lpib.ts, main="Tendencia")

#Agrupemos ambos en un solo dataframe
datos=cbind(lpib.ts,gini.ts)
datos
## Time Series:
## Start = 1990 
## End = 2021 
## Frequency = 1 
##       lpib.ts gini.ts
## 1990 25.51482   0.523
## 1991 25.53464   0.513
## 1992 25.57429   0.515
## 1993 25.62674   0.538
## 1994 25.68326   0.541
## 1995 25.73398   0.562
## 1996 25.75433   0.569
## 1997 25.78806   0.567
## 1998 25.79374   0.538
## 1999 25.75079   0.569
## 2000 25.77962   0.569
## 2001 25.79626   0.587
## 2002 25.82099   0.587
## 2003 25.85942   0.575
## 2004 25.91138   0.560
## 2005 25.95854   0.536
## 2006 26.02355   0.550
## 2007 26.08876   0.539
## 2008 26.12106   0.553
## 2009 26.13240   0.543
## 2010 26.17636   0.546
## 2011 26.24353   0.535
## 2012 26.28191   0.526
## 2013 26.33198   0.526
## 2014 26.37599   0.526
## 2015 26.40512   0.510
## 2016 26.42578   0.506
## 2017 26.43928   0.497
## 2018 26.46460   0.504
## 2019 26.49597   0.513
## 2020 26.42068   0.535
## 2021 26.52519   0.515

siguiente, vamos a ver el autocorrelograma

autocorrelograma_pib <- acf(data$pib)

autocorrelograma_lpib <- acf(data$lpib)

autocorrelograma_gini <- acf(data$gini)

Pasamos al autocorrelograma parcial

autocorrelograma_parcial_pib <- pacf(data$pib)

autocorrelograma_parcial_lpib <- pacf(data$lpib)

autocorrelograma_parcial_gini <- pacf(data$gini)

plot(gini.ts)

plot(pib.ts)

Ahora, miremos estadística describtiva

summary(data)
##       gini             pib                 lpib      
##  Min.   :0.4970   Min.   :1.205e+11   Min.   :25.51  
##  1st Qu.:0.5210   1st Qu.:1.560e+11   1st Qu.:25.77  
##  Median :0.5380   Median :1.941e+11   Median :25.99  
##  Mean   :0.5398   Mean   :2.112e+11   Mean   :26.03  
##  3rd Qu.:0.5605   3rd Qu.:2.759e+11   3rd Qu.:26.34  
##  Max.   :0.5870   Max.   :3.309e+11   Max.   :26.53
str(data)
## tibble [32 × 3] (S3: tbl_df/tbl/data.frame)
##  $ gini: num [1:32] 0.523 0.513 0.515 0.538 0.541 0.562 0.569 0.567 0.538 0.569 ...
##  $ pib : num [1:32] 1.20e+11 1.23e+11 1.28e+11 1.35e+11 1.43e+11 ...
##  $ lpib: Time-Series [1:32] from 1990 to 2021: 25.5 25.5 25.6 25.6 25.7 ...
summary(gini.ts)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.4970  0.5210  0.5380  0.5398  0.5605  0.5870
summary(pib.ts)
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 1.205e+11 1.560e+11 1.941e+11 2.112e+11 2.759e+11 3.309e+11

Analizado lo anterior, comenzaremos por una prueba de estacionariedad para ambas variables, para ellos nos remitimos a un Test Dickey-Fuller aumentado

dfu_gini <- ur.df(gini.ts, type = "trend", lags = 1)
print(summary(dfu_gini))
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression trend 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.028021 -0.007229 -0.000387  0.007365  0.024475 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  0.1796294  0.0665947   2.697   0.0121 *
## z.lag.1     -0.3074726  0.1179579  -2.607   0.0149 *
## tt          -0.0007969  0.0003225  -2.471   0.0204 *
## z.diff.lag  -0.0516674  0.1791724  -0.288   0.7754  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01332 on 26 degrees of freedom
## Multiple R-squared:  0.267,  Adjusted R-squared:  0.1825 
## F-statistic: 3.157 on 3 and 26 DF,  p-value: 0.04157
## 
## 
## Value of test-statistic is: -2.6066 2.9041 4.3548 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau3 -4.15 -3.50 -3.18
## phi2  7.02  5.13  4.31
## phi3  9.31  6.73  5.61

Ahora, hacemos lo mismo para el PIB

dfu_pib <- ur.df(pib.ts, type = "trend", lags = 1)
print(summary(dfu_gini))
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression trend 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.028021 -0.007229 -0.000387  0.007365  0.024475 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  0.1796294  0.0665947   2.697   0.0121 *
## z.lag.1     -0.3074726  0.1179579  -2.607   0.0149 *
## tt          -0.0007969  0.0003225  -2.471   0.0204 *
## z.diff.lag  -0.0516674  0.1791724  -0.288   0.7754  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01332 on 26 degrees of freedom
## Multiple R-squared:  0.267,  Adjusted R-squared:  0.1825 
## F-statistic: 3.157 on 3 and 26 DF,  p-value: 0.04157
## 
## 
## Value of test-statistic is: -2.6066 2.9041 4.3548 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau3 -4.15 -3.50 -3.18
## phi2  7.02  5.13  4.31
## phi3  9.31  6.73  5.61

Miremos que tal se comportan los residuos, una forma de observar estacionariedad

modelo_ver_estacionariedad=lm(gini.ts ~ pib.ts)
summary(modelo_ver_estacionariedad)
## 
## Call:
## lm(formula = gini.ts ~ pib.ts)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.045585 -0.012977  0.001351  0.014131  0.037097 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.848e-01  1.201e-02  48.705  < 2e-16 ***
## pib.ts      -2.131e-13  5.423e-14  -3.929 0.000464 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.02042 on 30 degrees of freedom
## Multiple R-squared:  0.3397, Adjusted R-squared:  0.3177 
## F-statistic: 15.44 on 1 and 30 DF,  p-value: 0.0004642
residuales_mv=modelo_ver_estacionariedad$residuals
summary(residuales_mv)
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -0.045585 -0.012977  0.001351  0.000000  0.014131  0.037097
residualPlot(modelo_ver_estacionariedad)

y=ur.df(residuales_mv)
summary(y)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression none 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.023150 -0.005587  0.002027  0.010266  0.025479 
## 
## Coefficients:
##            Estimate Std. Error t value Pr(>|t|)   
## z.lag.1    -0.35349    0.12359  -2.860  0.00792 **
## z.diff.lag  0.01541    0.17335   0.089  0.92981   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01254 on 28 degrees of freedom
## Multiple R-squared:  0.2384, Adjusted R-squared:  0.184 
## F-statistic: 4.382 on 2 and 28 DF,  p-value: 0.0221
## 
## 
## Value of test-statistic is: -2.8601 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau1 -2.62 -1.95 -1.61

En definitiva ambas series de tiempo son estacionarias, lo cual es un buen sintoma para seguir con el proceso, no se pierde información veamos la cointegración

X=ur.df(residuales_mv, type="trend", selectlags = "AIC" )
summary(X)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression trend 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0277274 -0.0065285  0.0008746  0.0075240  0.0211262 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  0.0066706  0.0050023   1.333   0.1939  
## z.lag.1     -0.3444974  0.1244848  -2.767   0.0103 *
## tt          -0.0002838  0.0002675  -1.061   0.2985  
## z.diff.lag  -0.0228160  0.1760920  -0.130   0.8979  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01257 on 26 degrees of freedom
## Multiple R-squared:   0.28,  Adjusted R-squared:  0.1969 
## F-statistic:  3.37 on 3 and 26 DF,  p-value: 0.0336
## 
## 
## Value of test-statistic is: -2.7674 3.3277 4.6949 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau3 -4.15 -3.50 -3.18
## phi2  7.02  5.13  4.31
## phi3  9.31  6.73  5.61
X2=ur.df(residuales_mv, type="drift", selectlags = "AIC" )
summary(X2)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression drift 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0250825 -0.0075625  0.0000228  0.0085729  0.0230146 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  0.0019652  0.0023182   0.848  0.40405   
## z.lag.1     -0.3564103  0.1242647  -2.868  0.00792 **
## z.diff.lag  -0.0006034  0.1752481  -0.003  0.99728   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0126 on 27 degrees of freedom
## Multiple R-squared:  0.2488, Adjusted R-squared:  0.1932 
## F-statistic: 4.472 on 2 and 27 DF,  p-value: 0.02102
## 
## 
## Value of test-statistic is: -2.8682 4.4084 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau2 -3.58 -2.93 -2.60
## phi1  7.06  4.86  3.94
X3=ur.df(residuales_mv, type="none", selectlags = "AIC" )
summary(X3)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression none 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.023150 -0.005587  0.002027  0.010266  0.025479 
## 
## Coefficients:
##            Estimate Std. Error t value Pr(>|t|)   
## z.lag.1    -0.35349    0.12359  -2.860  0.00792 **
## z.diff.lag  0.01541    0.17335   0.089  0.92981   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01254 on 28 degrees of freedom
## Multiple R-squared:  0.2384, Adjusted R-squared:  0.184 
## F-statistic: 4.382 on 2 and 28 DF,  p-value: 0.0221
## 
## 
## Value of test-statistic is: -2.8601 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau1 -2.62 -1.95 -1.61