Actividad 1.3

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library(tidyverse)
library(forecast)
library(quantmod)
library("AER")
library(MASS)
library(tseries)
library(car)
library(lmtest)
library(urca)
library ("aTSA")

n <- 240
set.seed(180188)
u <- rnorm(n)
v <- rnorm(n)

X <- numeric(n)
Y <- numeric(n)

for (t in 2:n){
  X[t] <- X[t-1] + u[t]
  Y[t] <- Y[t-1] + v[t]
}

modelo <- lm(Y ~ X)
summary(modelo)

## 
## Call:
## lm(formula = Y ~ X)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.0492 -3.3070 -0.0299  2.5518 14.4081 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.03357    0.28586   17.61   <2e-16 ***
## X            1.29574    0.08664   14.96   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.365 on 238 degrees of freedom
## Multiple R-squared:  0.4845, Adjusted R-squared:  0.4823 
## F-statistic: 223.7 on 1 and 238 DF,  p-value: < 2.2e-16

adf.test(X)

## Augmented Dickey-Fuller Test 
## alternative: stationary 
##  
## Type 1: no drift no trend 
##      lag   ADF p.value
## [1,]   0 -1.80  0.0734
## [2,]   1 -1.68  0.0903
## [3,]   2 -1.72  0.0849
## [4,]   3 -1.79  0.0748
## [5,]   4 -1.33  0.2020
## Type 2: with drift no trend 
##      lag   ADF p.value
## [1,]   0 -1.74   0.431
## [2,]   1 -1.60   0.485
## [3,]   2 -1.62   0.476
## [4,]   3 -1.69   0.448
## [5,]   4 -1.23   0.616
## Type 3: with drift and trend 
##      lag   ADF p.value
## [1,]   0 -3.47  0.0459
## [2,]   1 -3.18  0.0911
## [3,]   2 -3.15  0.0962
## [4,]   3 -3.32  0.0675
## [5,]   4 -2.98  0.1643
## ---- 
## Note: in fact, p.value = 0.01 means p.value <= 0.01

adf.test(Y)

## Augmented Dickey-Fuller Test 
## alternative: stationary 
##  
## Type 1: no drift no trend 
##      lag    ADF p.value
## [1,]   0 -0.608   0.461
## [2,]   1 -0.659   0.443
## [3,]   2 -0.526   0.491
## [4,]   3 -0.580   0.471
## [5,]   4 -0.630   0.454
## Type 2: with drift no trend 
##      lag   ADF p.value
## [1,]   0 -1.20   0.628
## [2,]   1 -1.22   0.618
## [3,]   2 -1.06   0.674
## [4,]   3 -1.17   0.637
## [5,]   4 -1.29   0.595
## Type 3: with drift and trend 
##      lag   ADF p.value
## [1,]   0 -2.18   0.498
## [2,]   1 -2.32   0.442
## [3,]   2 -2.23   0.478
## [4,]   3 -2.21   0.485
## [5,]   4 -2.17   0.504
## ---- 
## Note: in fact, p.value = 0.01 means p.value <= 0.01

acf(X)

acf(Y)

coint.test(Y, X)

## Response: Y 
## Input: X 
## Number of inputs: 1 
## Model: y ~ X + 1 
## ------------------------------- 
## Engle-Granger Cointegration Test 
## alternative: cointegrated 
## 
## Type 1: no trend 
##     lag      EG p.value 
##    4.00   -2.57    0.10 
## ----- 
##  Type 2: linear trend 
##     lag      EG p.value 
##   4.000   0.296   0.100 
## ----- 
##  Type 3: quadratic trend 
##     lag      EG p.value 
##   4.000   0.228   0.100 
## ----------- 
## Note: p.value = 0.01 means p.value <= 0.01 
##     : p.value = 0.10 means p.value >= 0.10

Aunque la regresión lineal entre Y y X muestra coeficientes estadísticamente significativos y un R2 relativamente alto, esto no implica una relación económica real. Las pruebas ADF y los correlogramas indican que ambas series son no estacionarias (presentan raíz unitaria), lo que sugiere que la regresión puede ser espuria o “sin sentido”.

#La prueba de cointegración de Engle-Granger arroja valores p altos, por lo que no se rechaza la hipótesis nula de no cointegración. En consecuencia, no hay evidencia de un equilibrio de largo plazo entre las variables. Por lo tanto, se concluye que se trata de una regresión espuria y no de un fenómeno de cointegración.

data("USMacroSWM", package = "AER")
head(USMacroSWM)

##          production oil   cpi expenditure
## Jan 1947      17.04  NA 21.48          NA
## Feb 1947      17.14  NA 21.62          NA
## Mar 1947      17.24  NA 22.00          NA
## Apr 1947      17.10  NA 22.00          NA
## May 1947      17.17  NA 21.95          NA
## Jun 1947      17.17  NA 22.08          NA

TS = ts(USMacroSWM[-(1:12),1:2], start=1948, freq=12)
head(TS)

##          production  oil
## Jan 1948      17.87 0.06
## Feb 1948      17.91 0.00
## Mar 1948      17.71 0.00
## Apr 1948      17.74 0.00
## May 1948      18.04 0.00
## Jun 1948      18.28 0.00

plot(TS[,1], main="Producción")

plot(TS[,2], main="Petróleo")

adf.test(TS[,1])

## Augmented Dickey-Fuller Test 
## alternative: stationary 
##  
## Type 1: no drift no trend 
##      lag  ADF p.value
## [1,]   0 8.79    0.99
## [2,]   1 5.91    0.99
## [3,]   2 4.67    0.99
## [4,]   3 4.07    0.99
## [5,]   4 3.97    0.99
## [6,]   5 4.02    0.99
## [7,]   6 3.85    0.99
## Type 2: with drift no trend 
##      lag  ADF p.value
## [1,]   0 2.53    0.99
## [2,]   1 1.69    0.99
## [3,]   2 1.25    0.99
## [4,]   3 1.06    0.99
## [5,]   4 1.08    0.99
## [6,]   5 1.14    0.99
## [7,]   6 1.07    0.99
## Type 3: with drift and trend 
##      lag    ADF p.value
## [1,]   0 -0.752   0.966
## [2,]   1 -1.221   0.903
## [3,]   2 -1.545   0.770
## [4,]   3 -1.774   0.673
## [5,]   4 -1.833   0.648
## [6,]   5 -1.804   0.660
## [7,]   6 -1.849   0.641
## ---- 
## Note: in fact, p.value = 0.01 means p.value <= 0.01

adf.test(TS[,2])

## Augmented Dickey-Fuller Test 
## alternative: stationary 
##  
## Type 1: no drift no trend 
##      lag    ADF p.value
## [1,]   0 -15.97    0.01
## [2,]   1 -12.17    0.01
## [3,]   2 -10.97    0.01
## [4,]   3  -9.34    0.01
## [5,]   4  -7.67    0.01
## [6,]   5  -7.18    0.01
## [7,]   6  -6.45    0.01
## Type 2: with drift no trend 
##      lag    ADF p.value
## [1,]   0 -16.83    0.01
## [2,]   1 -13.00    0.01
## [3,]   2 -11.89    0.01
## [4,]   3 -10.24    0.01
## [5,]   4  -8.52    0.01
## [6,]   5  -8.05    0.01
## [7,]   6  -7.30    0.01
## Type 3: with drift and trend 
##      lag    ADF p.value
## [1,]   0 -17.22    0.01
## [2,]   1 -13.38    0.01
## [3,]   2 -12.30    0.01
## [4,]   3 -10.66    0.01
## [5,]   4  -8.92    0.01
## [6,]   5  -8.47    0.01
## [7,]   6  -7.72    0.01
## ---- 
## Note: in fact, p.value = 0.01 means p.value <= 0.01

modelo2 <- lm(TS[,1] ~ TS[,2])
summary(modelo2)

## 
## Call:
## lm(formula = TS[, 1] ~ TS[, 2])
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -51.018 -26.163  -2.519  19.464  60.811 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   57.099      1.127   50.68  < 2e-16 ***
## TS[, 2]      196.485     38.007    5.17 3.08e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 28.32 on 682 degrees of freedom
## Multiple R-squared:  0.03771,    Adjusted R-squared:  0.0363 
## F-statistic: 26.73 on 1 and 682 DF,  p-value: 3.085e-07

coint.test(TS[,1], TS[,2])

## Response: TS[, 1] 
## Input: TS[, 2] 
## Number of inputs: 1 
## Model: y ~ X + 1 
## ------------------------------- 
## Engle-Granger Cointegration Test 
## alternative: cointegrated 
## 
## Type 1: no trend 
##     lag      EG p.value 
##   6.000  -0.544   0.100 
## ----- 
##  Type 2: linear trend 
##     lag      EG p.value 
##    6.00    2.26    0.10 
## ----- 
##  Type 3: quadratic trend 
##     lag      EG p.value 
##     6.0    -1.2     0.1 
## ----------- 
## Note: p.value = 0.01 means p.value <= 0.01 
##     : p.value = 0.10 means p.value >= 0.10

La serie de producción presenta una clara tendencia creciente a lo largo del tiempo, lo que indica que no es estacionaria. En cambio, la serie de petróleo muestra alta volatilidad con picos frecuentes, sugiriendo fluctuaciones intensas sin un patrón estable. Ambas características son típicas de series macroeconómicas no estacionarias.

Los resultados muestran valores p muy altos para la producción, por lo que no se rechaza la hipótesis de raíz unitaria; es decir, la serie no es estacionaria. Para el petróleo, algunos resultados indican estacionariedad bajo ciertas especificaciones, pero en general también presenta comportamiento inestable.

La regresión entre producción y petróleo arroja coeficientes estadísticamente significativos, pero un R2 muy bajo, lo que indica que el petróleo explica muy poco de la variación en la producción. Esto sugiere una relación débil en términos económicos.

#Los valores p obtenidos son altos, por lo que no se rechaza la hipótesis nula de no cointegración. Esto implica que no existe evidencia de una relación de equilibrio de largo plazo entre ambas variables.