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1
H.W. (unitroot)
1. Consider a Difference Stationary Model X
t
 X
t1
=
β
+
ε
t
, where
ε
t
is a stationary ARMA term
and X
0
=
α
.
a. Rewrite X
t
as a function of
α
,
β
and
ε
i
.
b. Verify that the variance of X
t
is tdependent and
ε
t
has a permanent effect on X
t
.
c. Show that the variance of the forecast error X
t+h
 E(X
t+h
│
I
t
) increases as h increases.
d. Show that we cannot obtain a stationary series by removing a trend term from X
t
.
e. Which data transformation should be used to achieve stationarity?
Ans.
Consider the difference stationary model
X
t
 X
t1
=
β
+
ε
t
,
where
ε
t
is a stationary ARMA term. To simplify the discussion, let us assume
ε
t
is
iid~(0,
σ
2
) and the initial value of X
t
is given by X
0
=
α
, then
X
t
=
α
+
β
t +
1
t
ii
ε
=
Σ
.1
The intercept
α
depends on the initial condition X
0
. The variance of X
t
is given by
V(X
t
) = t
σ
2
.
That is, the variance of X
t
is tdependent. As
X
t+h
=
α
+
β
(t+h) +
1
th
+
=
Σ
2,
the effect of
ε
t
shows up in X
t+h
for h > 0. thus,
ε
t
has an permanent impact on {X
t
}.
The forecast error is
X
t+h
 E(X
t+h
│
I
t
) =
1
it
i
+
=+
Σ
3
and its variance is
V[X
t+h
 E(X
t+h
│
I
t
)] = h
σ
2
.
Thus, the prediction error is unbound as the forecast horizon increases.
When we remove the constant and the trend from the data, we have
X
t
=
1
t
=
Σ
4,
which again is nonstationary (WHY?).
To achieve stationarity, we have to first difference the data; that is Y
t
≡
X
t
 X
t1
is
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stationary.
2. Consider a Trend stationary model X
t
=
α
+
β
t +
ε
t
, where
ε
t
is a stationary ARMA term.
a. Verify that the variance of X
t
is t invariant and the effect of
ε
t
on X
t
dissipates
asymptotically.
b. Show that the variance of the forecast error X
t+h
 E(X
t+h
│
I
t
) is (asymptotically) constant.
c. Show that differencing the data removes the trend and, at the same time, introduces a
moving average unit root.
d. Which data transformation should be used to achieve stationarity?
Ans.
Consider the trend stationary model
X
t
=
α
+
β
t +
ε
t
where
ε
t
is a stationary ARMA term. To simplify the discussion, let us assume
ε
t
~ iid(0,
σ
2
).
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 Winter '09
 Fairlie

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