STAT GU4205/GR5205 (Section 004) Linear Regression Models

HW4 by Dongrui Liu, Uni:dl3390

5.25

Part (a) (1)X’X =

x = c(1,0,2,0,3,1,0,1,2,0)
y = c(16,9,17,12,22,13,8,15,19,11)
X = matrix(x,length(x),2)
X[,1]=rep(1,10)
Y = matrix(y,length(y),1)
XtXinv = solve(t(X)%*%X)
XtXinv

##      [,1] [,2]
## [1,]  0.2 -0.1
## [2,] -0.1  0.1

2. b =

b = XtXinv%*%t(X)%*%Y
b

##      [,1]
## [1,] 10.2
## [2,]  4.0

3. e

e = Y-X%*%b
e

##       [,1]
##  [1,]  1.8
##  [2,] -1.2
##  [3,] -1.2
##  [4,]  1.8
##  [5,] -0.2
##  [6,] -1.2
##  [7,] -2.2
##  [8,]  0.8
##  [9,]  0.8
## [10,]  0.8

4. H

H = X%*%XtXinv%*%t(X)
H

##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
##  [1,]  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.1   0.1
##  [2,]  0.1  0.2  0.0  0.2 -0.1  0.1  0.2  0.1  0.0   0.2
##  [3,]  0.1  0.0  0.2  0.0  0.3  0.1  0.0  0.1  0.2   0.0
##  [4,]  0.1  0.2  0.0  0.2 -0.1  0.1  0.2  0.1  0.0   0.2
##  [5,]  0.1 -0.1  0.3 -0.1  0.5  0.1 -0.1  0.1  0.3  -0.1
##  [6,]  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.1   0.1
##  [7,]  0.1  0.2  0.0  0.2 -0.1  0.1  0.2  0.1  0.0   0.2
##  [8,]  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.1   0.1
##  [9,]  0.1  0.0  0.2  0.0  0.3  0.1  0.0  0.1  0.2   0.0
## [10,]  0.1  0.2  0.0  0.2 -0.1  0.1  0.2  0.1  0.0   0.2

5. SSE

SSE = t(e)%*%e
SSE[1,1]

## [1] 17.6

6. \(s^2\)(b)

n = length(y)
MSE = SSE/(n-2)
s2b = MSE[1,1]*XtXinv
s2b

##       [,1]  [,2]
## [1,]  0.44 -0.22
## [2,] -0.22  0.22

7. Yˆh when Xh = 2,

xh = c(1,2)
Yh = xh%*%b
Yh[1,1]

## [1] 18.2

8. s2{Yˆh } when Xh = 2.

xh = c(1, 2)
s2Yh = MSE[1,1]*t(xh)%*%XtXinv%*%t(t(xh))
s2Yh

##      [,1]
## [1,] 0.44

Part(b) (1) s2{b1}

s2b[2,2]

## [1] 0.22

2. s{b0, b1}

s2b[1,2]

## [1] -0.22

3. s{b0}

sqrt(s2b[1,1])

## [1] 0.663325

part (c) \[ SSR = Y'\bigg[H - \bigg(\frac{1}{n}\bigg)J\bigg]Y \] \(Y'\)

Yt=t(Y)
Yt

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,]   16    9   17   12   22   13    8   15   19    11

\(A = H - \bigg(\frac{1}{n}\bigg)J\)

A = H-(1/n)*matrix(rep(1,n),n,n)
A

##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
##  [1,]    0  0.0  0.0  0.0  0.0    0  0.0    0  0.0   0.0
##  [2,]    0  0.1 -0.1  0.1 -0.2    0  0.1    0 -0.1   0.1
##  [3,]    0 -0.1  0.1 -0.1  0.2    0 -0.1    0  0.1  -0.1
##  [4,]    0  0.1 -0.1  0.1 -0.2    0  0.1    0 -0.1   0.1
##  [5,]    0 -0.2  0.2 -0.2  0.4    0 -0.2    0  0.2  -0.2
##  [6,]    0  0.0  0.0  0.0  0.0    0  0.0    0  0.0   0.0
##  [7,]    0  0.1 -0.1  0.1 -0.2    0  0.1    0 -0.1   0.1
##  [8,]    0  0.0  0.0  0.0  0.0    0  0.0    0  0.0   0.0
##  [9,]    0 -0.1  0.1 -0.1  0.2    0 -0.1    0  0.1  -0.1
## [10,]    0  0.1 -0.1  0.1 -0.2    0  0.1    0 -0.1   0.1

And the result will be:

SSR = Yt%*%(H-(1/n)*matrix(rep(1,n),n,n))%*%Y
SSR

##      [,1]
## [1,]  160