Question

In the simple linear regression with zero-constant item for (xi
, yi) where i = 1, 2, · · · , n,
Yi = βxi + i
where {i}
n
i=1 are i.i.d. N(0, σ2
).
(a) Derive the normal equation that the LS estimator, βˆ, satisfies.
(b) Show that the LS estimator of β is given by
βˆ =
Pn
i=1
P
xiYi
n
i=1 x
2
i
.
(c) Show that
E(βˆ) = β, V ar(βˆ) = σ
2
Pn
i=1 x
2
i
, and βˆ ∼ N

β, σ
2
Pn
i=1 x
2
i

.
(d) Let ˆi = Yi − βxˆ
i
. Show that cov( ˆi
, βˆ) = 0, and ˆi and βˆ are independent. Let
σˆ
2 =
1
n − 1
Xn
i=1
(Yi − βxˆ
i)
2 =
1
n − 1
Xn
i=1
ˆi
2
.
Show that ˆσ
2 and βˆ
0 are independent.
(e) It is given that
(n − 1)ˆσ
2
σ
2
∼ χ
2
n−1
.
Show that
βˆ − β
p
σˆ
2/(
Pn
i=1 x
2
i
)
S ∼ tn−1
f) Show that the 100(1 − α)% prediction interval of Y0 taken at x = x0 is given by
βxˆ
0 ± tn−1,α/2 σˆ
s
1 +
x
2
P
0
n
i=1 x
2
i
,
where tn−1,α/2 is the upper α/2 percentile of a t distribution with (n-1) degree of freedom

Answer 1

yi = ßxite: i = 1.2, (a) We minimize Δ , n 7t 7t 7L 7L 7t _--2 Σ(yi _ βχί)x,-0 Normal equation : dB b) From normal equation we get B 0%11 12 2 . sinceyis are independent 2 7L Since β = Σ lìyì, lì = -n-Σ, is a linear function of yi u,here yi independently distributed normal variables so

Var(yi Χίο" Where, ei-Vi-8xī (n - 1)022 7l_ a 2 andare independent (from (d) and then rN