Transcript Review of Unit Root Testing

Review of Unit Root Testing
D. A. Dickey
North Carolina State
University
Nonstationary Forecast
Stationary Forecast
”Trend Stationary” Forecast
Nonstationary Forecast
Autoregressive Model
AR(1)
Yt - m = r (Yt-1-m) + et
Yt = m (1- r) + rYt-1 + et
DYt = m (1- r) + (r-1)Yt-1 + et
DYt = (r-1)(Yt-1 - m) + et
where DYt is Yt -Yt-1
AR(p)
Yt - m = a1(Yt-1-m) + a2(Yt-2-m) + ...+ ap(Yt-1-m) + et
AR(1) Stationary  |r| < 1
– OLS Regression Estimators – Stationary case
– Mann and Wald (1940’s) : For |r| < 1
n
n
t =2
t =2
2
ˆ
r m =  (Yt - Y )(Yt -1 - Y ) /  (Yt -1 - Y )
n
n
t =2
t =2
n ( rˆ m - r ) = n -1/ 2  (Yt -1 - Y )et / n -1  (Yt -1 - Y ) 2
2
p
1 n

 (Yt -1 - Y ) 2 
= Var{Y }
2
n t =2
1- r
More exciting algebra coming up ……
AR(1) Stationary  |r| < 1
– OLS Regression Estimators – Stationary case
1 n
4
Var{
 (Yt -1 - Y )et } 
2
t
=
2
1- r
n
L
1 n
4
 (Yt -1 - Y )et  N (0,
)
2
1- r
n t =2
L
Slutzky :
n ( rˆ m - r )  N (0,1 - r 2 )
(1)Same limit if sample mean replaced by m
(2) AR(p)  Multivariate Normal Limits 
|r| < 1
Yt-m = r(Yt-1-m) + et = r(r(Yt-2-m) + et-1) + et = ... =
et + ret-1 +r2et-2 + … +rk-1 et-k+1+ rk (Yt-k-m)
Yt=m +
(converges for |r| < 1)
Var{Yt } = 2/(1-r2)
r=1
But if r=1, then Yt = Yt-1 + et, a random walk.
Yt = Y0 + et + et-1 + et-2 + … + e1
Var{Yt - Y0} = t2
E{Yt} = E{Y0}
.
|r| < 1
AR(1)
E{Yt} = m
Var{Yt } is constant
Forecast of Yt+L converges to m (exponentially fast)
Forecast error variance is bounded
AR(1)
r=1
Yt = Yt-1 + et
E{Yt} = E{Y0}
Var{Yt} grows without bound
Forecast not mean reverting
E = MC2
r=?
Nonstationary (r=1) cases:
Case 1: m known (=0)
Regression Estimators (Yt on Yt-1 noint )
n
n
( rˆ - 1) =
Y
t =2
n
e
/n
2
t -1
/n2
t -1 t
Y
t =2
r=1  Nonstationary
n
n( rˆ - 1) =
Y
e /n
t -1 t
t =2
n
L
2
2
Y
/
n
 t -1
 " DF "
t =2
n
t statistic =
Y
e
t -1 t
t =2
n
L

s 2  Yt -21
t =2
Recall stationary results:
L
n ( rˆ - r )  N (0,1 - r ),
2
L
t  N (0,1)
Note: all results independent of  2
Where
H0:r=1are
?
my
clothes?
H1:|r|<1
DF Distribution ??
Numerator:
e1
e2
e3
:
en
 2 n 2
Yn -  et 

n
1
1 2
2
t =1


Y
e
/(
n

)
=

(  - 1)

t -1 t
2
2  n
 2
t =2




en
e2
e12
e1e2
e22
e1e3 … e1en
e2e3 … e2en
e32 … e3en
:
en2
Y1e2
Y2e3
:
e3
…
e1
… Yn-1en
Denominator
4
2
2
2
2
Y
=
e
+
(
e
+
e
)
+
(
e
+
e
+
e
)
t 1 1 2
1
2
3
t =2
= ( e1 e2
 3 2 1 e1 

 
e3 )  2 2 1 e2  = ( z1
 1 1 1 e 

 3 
For n
Observations:
 n - 1 n - 2 n - 3 ...

 n - 2 n - 2 n - 3 ...
An =  n - 3 n - 3 n - 3 ...

:
:
\
 :
 1
1
1
1

1  1 -1 0
 
1  -1 2 -1
1 =  0 -1 2
 
:  :
:
:
1  0 0 0
 ? 0 0  z1 

 
z2 z3 )  0 ? 0  z2 
 0 0 ?  z 

 3 
( Z ~ N (0,1))
... 0 

... 0 
... 0 

\ :
... 2 
-1
(eigenvalues are reciprocals of each other)
Eigenvalues :
 in2 =
1 2  (n - i ) 
sec 

4
 2n - 1 
Results:
eTAne =
lim
n
n-2
n
2
2
2

Z

 in in
Z ~ N (0,1)
i =1
eTAne
Graph of
i,502 and limit :
2
 2(-1)  2 2
=
 Zi 
i =1  (2i - 1) 

i +1
Z ~ N (0,1)
Histograms for n=50:
-1.96
-8.1
Theory 1: Donsker’s Theorem (pg. 68, 137 Billingsley)
{et} an iid(0,2)
sequence 
(n=100)
Sn =
e1+e2+ …+en
X(t,n) = S[nt]/(n1/2)=Sn normalized
Theory 1: Donsker’s Theorem (pg. 137 Billingsley)
Donsker: X(t,n) converges in law to W(z), a “Wiener Process”
plots of X(t,n) versus z= t/n for n=20, 100, 2000
20 realizations of
X(t,100) vs. z=t/n
Theory 2: Continuous mapping theorem
(Billingsley pg. 72)
L
h( ) a continuous functional => h( X(t,n) )
 h(W(t))
L
For our estimators, Yn / n   W (1)
 (Y /
n
and
t -1
n( rˆ - 1) =
t
1
so……
0
n
1 2
2
(Yn / n -  et2 / n 2 )
2
t =1
 (Y /
n
1 2
(W (1) - 1)
  2
2
W
 (t )dt
0
L
n (1/ n)   W 2 (t )dt
t -1
1
)
2
t
)
2
n (1/ n)
1 2
(W (1) - 1)
L
2
 1
2
W
 (t )dt
0
Distribution is …. ???????
Extension 1: Add a mean (intercept)
Yt - m = r (Yt -1 - m ) + et
def
Yt - m - (Yt -1 - m ) = Yt = ( r - 1)(Yt -1 - m ) + et
New quadratic forms.
New distributions
Yt = Y0 + e1 + e2 +
^
rm , m
+ et
n -1
Y = Y0 + e1 +
e2 +
n
Yt - Y
1
+ et
n
Estimator independent of Y0
Extension 2: Add linear trend
Yt - (a +  t ) = r (Yt -1 - (a +  (t - 1)) + et
Yt =  + ( r - 1)(Yt -1 - (a +  (t - 1)) + et
and under H 0
Yt =  + et = " drift " + et
Y1 = [Y0 +  ] + e1
Y2 = [Y1 +  ] + e2 = [Y0 + 2 ] + e1 + e2
Regress  Yt
on 1, t, Yt-1 annihilates Y0 , t
New quadratic forms.
New distributions
^
r , 
The 6 Distributions
-8.1
coefficient
n(rj-1)
-14.1
-21.8
0
t test

f(t) =
--1.95
1.96
-2.93
0
mean
-3.50
trend
 percentiles, n=50
pr<
0.01
0.025 0.05
0.10
0.50
0.90
0.95
0.975 0.99
---
-2.62 -2.25 -1.95 -1.61 -0.49 0.91
1.31
1.66
2.08
1
-3.59 -3.32 -2.93 -2.60 -1.55 -0.41 -0.04 0.28
0.66
f(t)
(1,t)
-4.16 -3.80 -3.50 -3.18 -2.16 -1.19 -0.87 -0.58 -0.24
 percentiles, limit
pr<
0.01
0.025 0.05
0.10
0.50
0.90
0.95
0.975 0.99
---
-2.58 -2.23 -1.95 -1.62 -0.51 0.89
1.28
1.62
2.01
1
-3.42 -3.12 -2.86 -2.57 -1.57 -0.44 -0.08 0.23
0.60
f(t)
(1,t)
-3.96 -3.67 -3.41 -3.13 -2.18 -1.25 -0.94 -0.66 -0.32
Higher Order Models
stationary:
Yt - m = 1.3(Yt -1 - m ) - .4 (Yt - 2 - m ) + et
Yt = - 0.1(Yt -1 - m ) + .4 ( Yt -1 ) + et
m 2 -1.3m + 0.4 = (m - .5)(m - .8) = 0
“characteristic eqn.”
roots 0.5, 0.8 ( < 1)
note: (1-.5)(1-.8) = -0.1
nonstationary
Yt - m = 1.3(Yt -1 - m ) - .3(Yt - 2 - m ) + et
Yt = - 0.0 (Yt -1 - m ) + .3( Yt -1 ) + et ,
m2 -1.3m + 0.3 = (m - .3)(m - 1)
" unit root !"
Yt = .3( Yt -1 ) + et
Higher Order Models- General AR(2)
roots: (m - a )( m -  ) = m2 - ( a +  )m + a
AR(2): ( Yt - m ) = ( a +  ) ( Yt-1 - m ) - a ( Yt-2 - m ) + et
Yt - m = (a +  ) (Yt -1 - m ) - a (Yt - 2 - m ) + et
Yt = - (1 - a -  + a ) (Yt -1 - m ) + a ( Yt -1 ) + et
(1 - a -  + a ) = (1 - a )(1 -  )
nonstationary
(0 if unit root)
Yt = - (1 - a -  + a ) (Yt -1 - m ) + a ( Yt -1 ) + et
t test same as AR(1).
Coefficient requires
modification
t test  N(0,1) !!
Tests
These coefficients  normal!
|
|
Regress:
Yt
on (1, t)
Yt-1 Yt -1 , Yt - 2 ,
, Yt - p +1
( “ADF” test )
r-1
augmenting affects limit distn.
()  “
does not affect “
“
Silver example:
Nonstationary Forecast
Stationary Forecast
Is AR(2) sufficient ? test vs. AR(5).
proc reg; model D = Y1 D1-D4; test D2=0, D3=0, D4=0;
Source df Coeff.
t
Intercept 1 121.03 3.09
Yt-1
1 -0.188 -3.07
Yt-1-Yt-2 1 0.639 4.59
Yt-2-Yt-3 1 0.050 0.30
Yt-3-Yt-4 1 0.000 0.00
Yt-4-Yt-5 1 0.263 1.72
F413 = 1152 / 871 = 1.32
Pr>|t|
0.0035
X
0.0038
0.0001
0.7691
0.9985
0.0924
Pr>F = 0.2803
Fit AR(2) and do unit root test
Method 1: OLS output and tabled critical value (-2.86)
proc reg; model D = Y1 D1;
Source df Coeff. t
Pr>|t|
Intercept 1 75.581 2.762 0.0082 X
Yt-1
1 -0.117 -2.776 0.0038 X
Yt-1-Yt-2 1 0.671 6.211 0.0001 
Method 2: OLS output and tabled critical values
proc arima; identify var=silver stationarity = (dickey=(1));
Augmented Dickey-Fuller Unit Root Tests
Type
Zero Mean
Single Mean
Trend
Lags
1
1
1
t
-0.2803
-2.7757
-2.6294
Prob<t
0.5800
0.0689 
0.2697
?
First part ACF
IACF
PACF
Full data ACF
IACF
PACF
Differences
Levels
Amazon.com Stock ln(Closing Price)
Levels
Augmented Dickey-Fuller Unit Root Tests
Type
Zero Mean
Single Mean
Trend
Lags
2
2
2
Tau
1.85
-0.90
-2.83
Pr < Tau
0.9849
0.7882
0.1866
Differences
Augmented Dickey-Fuller Unit Root Tests
Type
Zero Mean
Single Mean
Trend
Lags
Tau
Pr<Tau
1
1
1
-14.90
-15.15
-15.14
<.0001
<.0001
<.0001
Are differences white noise (p=q=0) ?
Autocorrelation Check for White Noise
To
Lag
ChiSquare
DF
Pr >
ChiSq
6
12
18
24
3.22
6.24
9.77
12.28
6
12
18
24
0.7803
0.9037
0.9391
0.9766
-------------Autocorrelations------------0.047 0.021 0.046 -0.036 -0.004 0.014
-0.062 -0.032 -0.024 0.006 0.004 0.019
0.042 0.015 -0.042 0.023 0.020 0.046
-0.010 -0.005 -0.035 -0.045 0.008 -0.035
Differences
Levels
Amazon.com Stock Volume
Augmented Dickey-Fuller Unit Root Tests
Type
Lags
Zero Mean
Single Mean
Trend
4
4
4
Tau
Pr < Tau
0.07
-2.05
-5.76
0.7063
0.2638
<.0001
Maximum Likelihood Estimation
Parameter Estimate
MU
-71.81516
MA1,1
0.26125
AR1,1
0.63705
AR1,2
0.22655
NUM1
0.0061294
To
Lag
ChiSquare
DF
Pr >
ChiSq
6
12
18
24
30
36
42
48
0.59
9.41
11.10
17.10
21.86
28.58
35.53
37.13
3
9
15
21
27
33
39
45
0.8978
0.4003
0.7456
0.7052
0.7444
0.6869
0.6291
0.7916
t Value
-8.83
4.53
14.35
4.32
10.56
Approx
Pr > |t|
<.0001
<.0001
<.0001
<.0001
<.0001
Lag
0
2
1
2
0
Variable
volume
volume
volume
volume
date
-------------Autocorrelations-------------0.009 -0.002 -0.015
-0.042 0.002 0.068
-0.042 0.006 0.013
0.064 -0.043 0.029
0.003 0.022 -0.068
-0.020 0.015 0.093
0.070 0.038 -0.052
0.026 -0.021 0.018
-0.023
-0.075
-0.014
-0.045
0.010
0.033
0.033
0.002
-0.008 -0.016
0.026 0.065
-0.017 0.027
-0.034 0.035
0.014 0.058
-0.041 -0.015
-0.044 0.023
0.004 0.037
Differences
Levels
Amazon.com Spread = ln(High/Low)
Augmented Dickey-Fuller Unit Root Tests
Type
Lags
Zero Mean
Single Mean
Trend
4
4
4
Tau
Pr<Tau
-2.37
-6.27
-6.75
0.0174
<.0001
<.0001
Maximum Likelihood Estimation
Parm
Estimate
MU
-0.48745
MA1,1
0.42869
AR1,1
0.38296
AR1,2
0.42306
NUM1 0.00004021
To
Lag
6
12
18
24
30
36
42
48
ChiSquare
2.87
3.83
7.62
15.96
19.01
22.38
25.39
30.90
DF
3
9
15
21
27
33
39
45
Pr >
ChiSq
0.4114
0.9221
0.9381
0.7721
0.8695
0.9187
0.9546
0.9459
t Value
-1.57
5.57
8.85
5.97
1.82
Approx
Pr>|t|
0.1159
<.0001
<.0001
<.0001
0.0690
Lag
0
2
1
2
0
Variable
spread
spread
spread
spread
date
-------------Autocorrelations-------------0.004 0.021 0.025 -0.039 0.014 -0.053
0.000 0.016 0.013 -0.000 0.008 0.037
-0.038 -0.062 0.010 -0.032 -0.004 0.027
-0.006 0.008 -0.076 -0.085 0.045 0.022
0.008 0.043 0.013 -0.018 -0.007 0.057
0.004 0.027 0.041 -0.030 0.014 -0.052
0.043 0.042 0.019 0.003 0.034 -0.016
0.015 -0.054 -0.061 -0.049 -0.004 -0.021
S.E. Said: Use AR(k) model even if MA
terms in true model.
N. Fountis: Vector Process with One Unit
Root
D. Lee: Double Unit Root Effect
M. Chang: Overdifference Checks
G. Gonzalez-Farias: Exact MLE
K. Shin: Multivariate Exact MLE
T. Lee: Seasonal Exact MLE
Y. Akdi, B. Evans – Periodograms of Unit
Root Processes
H. Kim: Panel Data tests
S. Huang: Nonlinear AR processes
S. Huh: Intervals: Order Statistics
S. Kim: Intervals: Level Adjustment &
Robustness
J. Zhang: Long Period Seasonal.
Q. Zhang: Comparing Seasonal
Cointegration Methods.