Tarea_2

Librerías necesarias

library(reshape)

## Warning: package 'reshape' was built under R version 4.4.3

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following object is masked from 'package:reshape':
## 
##     rename

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(ggplot2)
library(plotly)

## Warning: package 'plotly' was built under R version 4.4.3

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:reshape':
## 
##     rename

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

library(moments)
library(car)

## Warning: package 'car' was built under R version 4.4.3

## Loading required package: carData

## Warning: package 'carData' was built under R version 4.4.3

## 
## Attaching package: 'car'

## The following object is masked from 'package:dplyr':
## 
##     recode

library(viridisLite)

1. Simule los siguientes procesos:

Inciso a. yi = 5 + 3 ∗ xi + ui donde ui ∼ Ν(μ = 0, σ = 5), xi posee una distribución unitaria con soporte positivo de su elección y cov(ui, xi) = 0.

# Fijamos una semilla
set.seed(223)
n  <- 4532           # últimos num | matricula

x  <- runif(n, 0, 1)  # distribución >= 0
u  <- rnorm(n, mean = 0, sd = 5) # error 
y  <- 5 + 3 * x + u   # variable dependiente

# Comprobación de la varianza
cov(u, x)

## [1] -0.01708463

Pasamos a ajustar la regresión y a revisar tanto el intercepto como los demás estimadores

modelo <- lm(y ~ x)
summary(modelo)

## 
## Call:
## lm(formula = y ~ x)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -18.1547  -3.4358   0.1112   3.3718  15.9968 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   5.0761     0.1510   33.62   <2e-16 ***
## x             2.7871     0.2615   10.66   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.988 on 4530 degrees of freedom
## Multiple R-squared:  0.02446,    Adjusted R-squared:  0.02424 
## F-statistic: 113.6 on 1 and 4530 DF,  p-value: < 2.2e-16

El intercepto mantiene el valor esperado de 5 (y cuando x= 0). Sin embargo en la pendiente se estimo un valor de 2.88.

Visualizaciones a.

# combinación de plots 
par(mfrow = c(2, 3),   
    las   = 1,         
    mar   = c(4, 4, 2, 1
              ))


plot(modelo, las=1, col="#817", which=1)          
plot(modelo, las=1, col='#929', which=2)          
plot(modelo, las=1, col='#948', which=3)          
plot(x, modelo$residuals, col='#945',
     xlab="x", ylab="Residuals",
     main="x vs Residuals")                      
plot(x, modelo$residuals / sd(modelo$residuals),
     col='#901', xlab="x",
     ylab="Standardized Residuals",
     main="Scale-Location")                      

Inciso b

set.seed(231)

n1 <- 5555
x1  <- runif(n1, 0, 1) # distribucion exp positiva

# crear vector para la varianza
sigma <- numeric(n1)
for(i in 1:floor(n1/3)) {
  sigma[1] <- 2 +(4*x1)[1]
} 

for(i in (floor(n1/3)+1):floor(2*n1/3)) {
  sigma[i] <- 0.5
}

for(i in (floor(2*n1/3)+1):n1) {
  sigma[i] <- 0.9 * sigma[i-1] + 2
}
   
# desviación estandar
sd_i <- sqrt(sigma)

# 
u1 <- rnorm(n1, mean = 0, sd = sd_i)
y1 <- 5 + 3*x1 + u1

# Comprobación de la varianza
cov(u1, x1)

## [1] 0.01028037

modelo1 <- lm(y1 ~ x1)
summary(modelo1)

## 
## Call:
## lm(formula = y1 ~ x1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.9066  -0.4203   0.0401   0.4193  14.5516 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  4.89659    0.06957   70.38   <2e-16 ***
## x1           3.12205    0.11950   26.13   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.585 on 5553 degrees of freedom
## Multiple R-squared:  0.1095, Adjusted R-squared:  0.1093 
## F-statistic: 682.6 on 1 and 5553 DF,  p-value: < 2.2e-16

Inciso c

# Fijamos una semilla
set.seed(203)

Inciso a

Inciso a

Inciso a

Inciso a

Inciso a

Inciso a