Modelo de Regresión Múltiple

1. Ingresamos los datos

y<-c(1.5,0.8,1.2,1.4,0.2,0.8,0.6,1.3,0.4,0.6)
x1<-c(8.4,3.3,5.8,10,4.7,7.7,4.5,8.6,5.9,6.3)
x2<-c(7.7,4.5,8.4,7.8,2.4,4.8,2.5,3.4,2,4.1)

2. Generamos vector de unos

uno<-seq(1,1,length.out =10)
uno

##  [1] 1 1 1 1 1 1 1 1 1 1

3. Construimos las matrices X e Y

X<-matrix(c(uno,x1,x2),nrow=10)
X

##       [,1] [,2] [,3]
##  [1,]    1  8.4  7.7
##  [2,]    1  3.3  4.5
##  [3,]    1  5.8  8.4
##  [4,]    1 10.0  7.8
##  [5,]    1  4.7  2.4
##  [6,]    1  7.7  4.8
##  [7,]    1  4.5  2.5
##  [8,]    1  8.6  3.4
##  [9,]    1  5.9  2.0
## [10,]    1  6.3  4.1

Y<-matrix(c(y),nrow=10)
Y

##       [,1]
##  [1,]  1.5
##  [2,]  0.8
##  [3,]  1.2
##  [4,]  1.4
##  [5,]  0.2
##  [6,]  0.8
##  [7,]  0.6
##  [8,]  1.3
##  [9,]  0.4
## [10,]  0.6

4. Calculamos el vector B

La ecuacion se obtiene mediante \(\hat \beta=(X'X)^{-1}X'y\)

\[ \hat \beta=(X'X)^{-1}X'y \]

Determinamos:

\[ (X'X) \]

XtX<-t(X)%*%X
XtX

##      [,1]   [,2]   [,3]
## [1,] 10.0  65.20  47.60
## [2,] 65.2 465.18 332.61
## [3,] 47.6 332.61 278.36

Obtenemos

\[ (X'X)^{-1} \]

XtX.inv<-solve(XtX)
XtX.inv

##             [,1]        [,2]        [,3]
## [1,]  1.19364308 -0.14664660 -0.02888807
## [2,] -0.14664660  0.03277723 -0.01408844
## [3,] -0.02888807 -0.01408844  0.02536653

Calculamos

\[ X'y \]

XY<-t(X)%*%Y
XY

##       [,1]
## [1,]  8.80
## [2,] 63.32
## [3,] 49.65

Finalmente obtenemos los coeficientes B

\[ \beta=(X'X)^{-1}X'y \]

B<-solve(t(X)%*%X)%*%(t(X)%*%Y)
B

##             [,1]
## [1,] -0.21589662
## [2,]  0.08547343
## [3,]  0.11315334

Modelo de Regresión

\[ Y_{i} = -0.2159 + 0.0855X_{1}+0.1132X_{2} \]

Suma de Cuadrados

\[ Syy = y'y - n \bar y^2 \]

Syy<-t(Y)%*%Y-nrow(Y)*mean(Y)^2
Syy

##       [,1]
## [1,] 1.796

\[ SSE=y'y - \hat \beta'X'y \]

SSE=t(Y)%*%Y - t(B)%*%XY
SSE

##           [,1]
## [1,] 0.4096497

\[ SSR= \hat \beta'X'y \]

SSR<-t(B)%*%XY-nrow(Y)*mean(Y)^2
SSR

##         [,1]
## [1,] 1.38635

Grados de Libertad

gl1=(ncol(X)-1)
gl2=(nrow(Y)-ncol(X))

MSE<-SSE/gl2
MSE

##            [,1]
## [1,] 0.05852139

MSR<-SSR/gl1
MSR

##           [,1]
## [1,] 0.6931751

5. PRUEBAS INDIVIDUALES

5.1. Para B1

\[ H_o: \beta_1=0 \\ \] \[ H_1: \beta_1 \neq 0 \]

Función de Prueba

\[ t=\frac{\hat \beta_1}{D.S(\hat \beta_1)} \]

V.B1<-MSE*XtX.inv[2,2]
t<-B[2]/sqrt(V.B1)
t

##          [,1]
## [1,] 1.951586

v.p.B1<-2*pt(t,gl2,lower.tail=FALSE)
v.p.B1

##            [,1]
## [1,] 0.09195252

El valor p es mayor que el 5%. Entonces la hipótesis nula no se rechaza. La variable \(X_1\) no está asociada con la variable \(Y\).

5.2. Para B2

\[ H_o: \beta_2=0 \] \[ H_1: \beta_2 \neq 0 \]