Problem 5.5
ch5<-read.table("CH05PR05.txt")
n <- nrow(ch5)
X <- cbind(rep(1,n),ch5[,2])
Y <- (ch5[,1])
X
## [,1] [,2]
## [1,] 1 4
## [2,] 1 1
## [3,] 1 2
## [4,] 1 3
## [5,] 1 3
## [6,] 1 4
Y
## [1] 16 5 10 15 13 22
### Part A
t(Y)%*%Y
## [,1]
## [1,] 1259
### Part B
t(X)%*%X
## [,1] [,2]
## [1,] 6 17
## [2,] 17 55
### Part C
t(X)%*%Y
## [,1]
## [1,] 81
## [2,] 261
Problem 5.13
solve(t(X)%*%X) #X'*X^-1
## [,1] [,2]
## [1,] 1.3414634 -0.4146341
## [2,] -0.4146341 0.1463415
Problem 5.24
Part A.
#1- Vector of Estimated Regression Coefficients
I=diag(n)
b <- solve(t(X)%*%X)%*%t(X)%*%Y
b
## [,1]
## [1,] 0.4390244
## [2,] 4.6097561
#2-Vector of Residuals
res <- Y-(X%*%b)
res
## [,1]
## [1,] -2.87804878
## [2,] -0.04878049
## [3,] 0.34146341
## [4,] 0.73170732
## [5,] -1.26829268
## [6,] 3.12195122
#3-SSR
H <- X%*%solve(t(X)%*%X)%*%t(X)
J=rep(1,n)%*%t(rep(1,n))
SSR <- t(Y)%*%(H-J/n)%*%Y
SSR
## [,1]
## [1,] 145.2073
#4-SSE
SSE <- t(Y)%*%(I-H)%*%Y
SSE
## [,1]
## [1,] 20.29268
#5-Estimated Variance-Covariance Matrix of b
MSE <- SSE/(n-2)
MSE <-as.vector(MSE)
i_xpx<- solve(t(X)%*%X)
MSE*i_xpx
## [,1] [,2]
## [1,] 6.805473 -2.1035098
## [2,] -2.103510 0.7424152
#6-Estimated point estimate of E[Y_h] when X_h = 4
Xh <- matrix(c(1,4),2,1)
t(Xh)%*%b
## [,1]
## [1,] 18.87805
#7-s^2{pred} when X_h = 4
MSE*(1 + t(Xh)%*%(i_xpx)%*%Xh)
## [,1]
## [1,] 6.929209
Part B.
sb0b1 <- (MSE*solve(t(X)%*%X)[1,2])
sb0b1
## [1] -2.10351
sb0_sq <- (MSE*solve(t(X)%*%X))[1,1]
sb0_sq
## [1] 6.805473
sb1 <- sqrt(MSE*solve(t(X)%*%X)[2,2])
sb1
## [1] 0.8616352
Part C.
#Hat Matrix H
H <- X%*%solve(t(X)%*%X)%*%t(X)
H
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.36585366 -0.1463415 0.02439024 0.1951220 0.1951220 0.36585366
## [2,] -0.14634146 0.6585366 0.39024390 0.1219512 0.1219512 -0.14634146
## [3,] 0.02439024 0.3902439 0.26829268 0.1463415 0.1463415 0.02439024
## [4,] 0.19512195 0.1219512 0.14634146 0.1707317 0.1707317 0.19512195
## [5,] 0.19512195 0.1219512 0.14634146 0.1707317 0.1707317 0.19512195
## [6,] 0.36585366 -0.1463415 0.02439024 0.1951220 0.1951220 0.36585366
Part D.
#s_sqr(e)
MSE*(I-H)
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 3.2171327 0.7424152 -0.1237359 -0.9898870 -0.9898870 -1.8560381
## [2,] 0.7424152 1.7323022 -1.9797739 -0.6186794 -0.6186794 0.7424152
## [3,] -0.1237359 -1.9797739 3.7120761 -0.7424152 -0.7424152 -0.1237359
## [4,] -0.9898870 -0.6186794 -0.7424152 4.2070196 -0.8661511 -0.9898870
## [5,] -0.9898870 -0.6186794 -0.7424152 -0.8661511 4.2070196 -0.9898870
## [6,] -1.8560381 0.7424152 -0.1237359 -0.9898870 -0.9898870 3.2171327
Problem 2
Known Values
e_Y1 <- 1
e_Y2 <- 2
e_y3 <- 3
var_Y1 <- 6
var_Y2 <-7
var_Y3 <-8
cov_y1_y2 <- 0
cov_y1_y3 <- -4
cov_y2_y3 <- 5
a <- matrix(c(10, 20, 30),3,1)
A <- matrix(c(1,4,7,2,5,8,3,6,9), 3,3)
Part A.
#Expectation of Y
t(matrix(c(1,2,3),nrow=1))
## [,1]
## [1,] 1
## [2,] 2
## [3,] 3
#Variance Covariance of Y
var <- matrix(c(6,0,-4,0,7,5,-4,5,8), 3,3)
var
## [,1] [,2] [,3]
## [1,] 6 0 -4
## [2,] 0 7 5
## [3,] -4 5 8
Part B.
#Expectation
#E(a'Y) = a'*E(Y)
t(a)%*%matrix(c(1,2,3),3,1)
## [,1]
## [1,] 140
#E(AY) = A*E(Y)
A%*%matrix(c(1,2,3),3,1)
## [,1]
## [1,] 14
## [2,] 32
## [3,] 50
Part C.
#Variance
#Var(a'Y) = a'*Var*a
t(a) %*% var %*% a
## [,1]
## [1,] 14200
#Var(AY) = A*Var*A'
A %*% var %*% t(A)
## [,1] [,2] [,3]
## [1,] 142 301 460
## [2,] 301 667 1033
## [3,] 460 1033 1606