(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 1
Unit Root Tests
Test for Stationarity
• Correlogram (Graphical)
• Unit Root Tests
–Dickey-Fuller –Phillip-Perron
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 3
Dickey-Fuller’s UR Test (1)
Assumptions Y t ~AR(1) Hypothesis
H 0 : Y t ~ non-stationary H 1 : Y t ~ stationary
Dickey-Fuller’s UR Test(2)
CASE I
Y t ~AR(1) w/o drift and trend Testing Model
1 1
t t t
Y = φ Y − + ε
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 5
Dickey-Fuller’s UR Test(3)
Testing Model
1 1 1
1 1
1
0 1
1 1
where
( 1)
1 H : 1 or 0 H : 1 or 0
t t t t
t t t
Y Y Y
Y Y
φ ε
γ ε
γ φ
φ γ
φ γ
− −
−
− = − +
∆ = +
= −
≥ ≥
< < Necessary condition
Dickey-Fuller’s UR Test(4)
Steps
1) run OLS of ∆ Y t on Y t-1 2) test for γ <0
DF have shown by simulation that
τ does not follow a t-dist or a normal
ˆ ~ ??
( ) ˆ se τ γ
= γ
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 7
Dickey-Fuller’s UR Test(5)
Ref: Table 4.1 page 223 (Enders), Table D.7 (Gujarati)
τ
-1.645 -1.95
z
.05
0
Dickey-Fuller’s UR Test(6)
CASE II
Y t ~AR(1) w/ drift but w/o trend Testing Model
0 1 1
0 1 1
t t t
t t t
Y a Y
Y a Y
φ ε
γ ε
−
−
= + +
∆ = + +
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 9
Dickey-Fuller’s UR Test(7)
Steps
1) run OLS of ∆ Y t on constant Y t-1 2) test for γ <0
DF have shown by simulation that
τ µ does not follow a t-dist or a normal
ˆ ~ ??
( ) ˆ
µ se τ γ
= γ
Dickey-Fuller’s UR Test(8)
Ref: Table 4.1 page 223
τ
-1.645 -1.95
z
.05
0 -2.89
τ µ
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 11
Dickey-Fuller’s UR Test(9)
CASE III
Y t ~AR(1) w/ drift and trend Testing Model
0 1 1 2
0 1 1 2
t t t
t t t
Y a Y a t
Y a Y a t
φ ε
γ ε
−
−
= + + +
∆ = + + +
Dickey-Fuller’s UR Test(10)
Steps
1) run OLS of ∆ Y t on constant t Y t-1 2) test for γ <0
DF have shown by simulation that
τ τ does not follow a t-dist or a normal
ˆ ~ ??
( ) ˆ
τ se τ γ
= γ
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 13
Dickey-Fuller’s UR Test(11)
Ref: Table 4.1 page 223
τ
-1.645 -1.95
z
.05
0 -2.89
τ µ
-3.45 τ τ
Dickey-Fuller’s UR Test(12)
Note that
Drift and time trend will make it more difficult to reject H 0 .
If H 0 is rejected when both are
included, it will be rejected in the other cases.
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 15
Dickey-Fuller’s UR Test(12)
Simple Rule
First, include both drift and trend in the testing model. Do τ τ test.
If reject, Y ~ stationary
Otherwise, remove trend and do τ µ test.
If reject, Y ~ stationary. Otherwise, remove drift and do τ test.
If reject, Y ~ stationary. Otherwise, Y is non-stationary
Dickey-Fuller’s UR Test(12)
Full-scheme testing rule
See Figure 4.7 on page 257
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 17
Augmented Dickey-Fuller (1)
Assumptions Y t ~AR(p) Testing Model
1 1 2 2 1 1
1 ( 1 )
t t t p t p
p t p p t p t p t
Y Y Y Y
Y Y Y
φ φ φ
φ φ ε
− − − − +
− + − + −
= + + +
+ − − +
L
Augmented Dickey-Fuller (2)
1 1 2 2 1 2
1 2 1
1
1 1 2 2 1 2
1 2 1
( )
(- )( )
( )
( )
(- ) ( )
t t t p p t p
p p t p t p
p t p t
t t p p t p
p p t p p t p t
Y Y Y Y
Y Y
Y
Y Y Y
Y Y
φ φ φ φ
φ φ
φ ε
φ φ φ φ
φ φ φ ε
− − − − +
− − + − +
− +
− − − − +
− − + − +
= + + + +
+ − −
+ − ∆ +
= + + + +
+ − ∆ + − ∆ +
L
L M
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 19
Augmented Dickey-Fuller (3)
1 2 2 1
2 2 1
2 1 3
1 2
1
( 1)
+( )
(- - )
(- )
( )
t p t
p t
p p p t p
p p t p
p t p t
Y Y
Y Y Y
Y
φ φ φ φ
φ φ φ
φ φ φ φ φ
φ ε
−
−
− − − +
− − +
− +
∆ = + + + + −
− − − − ∆ +
+ − ∆
+ − ∆
+ − ∆ + L
L L
p-1 Augmented terms
Hypothesis
Fortunately, we can use the same τ τ ,
τ µ and τ tests as DF unit root tests
0 1
1 2 2
where
H : 0 H : 0
p 1 γ
γ
γ φ φ φ φ
≥
<
= + + + + L −
Augmented Dickey-Fuller (4)
