E703: Advanced Econometrics I
Exercise Questions - Serial Correlation
Please attemp all questions
1. If {yt } is the covariance-stationary solution to the AR(1) equation
yt = c + φyt−1 + εt
or yt − φyt−1 = c + εt
or (1 − φL)yt = c + εt ,
where {εt } is white noise with |φ| < 1, what is the least square (linear)
projetion?
b ∗ (yt |1, yt−1 )
E
b ∗ [yt |xt ] = x0t β ∗ , where
Hint: The least square (liner) projection of y on x is E
0
β ∗ = E[xt xt ]−1 E[xt yt ] (see Hayashi, pp. 139-140).
b ∗ (yt |1, yt−1 ) =
E(yt−1 εt ) = 0. Do the projection coefficients depend on t? Is E
E(yt |yt−1 )? Hint: E(yt−1 εt ) = 0 but does it mean E(εt |yt−1 ) = 0? What is
Ê ∗ (yt |1, yt−1 , yt−2 )? Now suppose that {yt } is covariance-stationary solution
to the AR(1) equation when |φ| > 1. Is it true that E(yt−1 εt ) = 0? [Answer:
b ∗ (yt |1, yt−1 ) = c + φyt−1 as was the case when |φ| < 1?
No.] Does E
2. Suppose that the money supply process has the form mt = m+ρmt−1 +εt ,
where m is constant and 0 < ρ < 1.
a. Show that it is possible to express mt+n in terms of the known value
mt and the sequence {εt+1 , εt+2 , . . . , εt+n }.
b. Suppose that all values of εt+i for i > 0 have mean value of zero (i.e.
E[εt+i |mt ] = 0 ∀i > 0). Explain how you could use your result in part
(a) to forecast the money supply n periods into the future.
3. (Proof of Proposition 6.1, Hayashi, p. 367) A well-known result about
mean square convergence is
i. zn converges in mean square if and only if E[(zm − zn )2 ] → 0 as m, n →
∞.
ii. If xn →m.s. x and zn →m.s. z, then
lim E(xn ) = E(x) and
n→∞
lim E(xn zn ) = E(xz).
n→∞
Take this result for granted in answering the following.
P
a. Prove that yt = µ + ∞
j=0 ψj εt−j converges in mean square as n → ∞
under the hypothesis of Proposition 6.1. Hint: Let
yt,n = µ +
n
X
ψj εt−j .
j=0
What needs to be proved is that {yt,n } convergence in mean square
as n → ∞ for each t. Given (i) above, it is sufficient to prove that
(assuming m > n without loss of generality)
#
" m
m
X
X
2
2
ψj2 → 0 as m, n → ∞.
E (
ψj εt−j ) = σ
j=n+1
j=n+1
Use the fact that an absolutely summable sequence is square summable,
i.e.,
∞
∞
X
X
|ψj | < ∞ ⇒
ψj2 < ∞,
j=0
j=0
and the fact that a sequence of real numbers {αn } converges (to a finite
limit) if and only if αn is a Cauchy sequence:
αn → α ⇔ |αm − αn | → 0 as m, n → ∞.
Set αn =
Pn
j=0
ψj2 .
b. Prove that E(yt ) = µ. Hint: Let yt,n = µ +
was shown in (a) that yt,n →m.s. yt .
Pn
j=0
ψj εt−j as in (a). It
c. (Proof of part(b) of Proposition 6.1) Show that
E[(yt − µ)(yt−j − µ)] = lim E[(yt,n − µ)(yt−j,n − µ)].
n→∞
Have we shown that {yt } is covariance-stationary? [Answer:Yes.]
d. Prove part (c) of Propostion 6.1, taking the follwoing facts from calculus
for granted.
(a) If
j } is absolutely summable, then {aj } is summable (i.e. −∞ <
P{a
∞
j=0 aj < ∞) and
∞
∞
X
X
aj ≤
|aj | .
j=0
j=0
(b) Consider P
a sequence with two subscripts, {ajk }(j,P
k = 0, 1, 2, . . .).
∞
Suppose ∞
|a
|
<
∞
for
each
k
and
let
s
≡
jk
k
j=0
j=0 |ajk |. Suppose also that {sk } is summable. Then
!
!
!
∞
∞
∞
∞
∞
∞
X
X
X
X
X
X
ajk < ∞ and
ajk =
ajk < ∞.
j=0
j=0
k=0
k=0
k=0
j=0
Hint: Derive
∞
X
j=0
|γj | ≤ σ 2
∞
∞
X
X
j=0
k=0
!
|ψj+k | · |ψk |
< ∞.