review of unit root testing d. a. dickey north carolina state university
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Review of Unit Root TestingReview of Unit Root Testing
D. A. DickeyD. A. DickeyNorth Carolina State North Carolina State
UniversityUniversity
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Nonstationary Forecast
Stationary Forecast
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”Trend Stationary” Forecast
Nonstationary Forecast
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Autoregressive ModelAutoregressive Model AR(1) AR(1)
Yt Yt-1et
Yt Yt-1et
Yt Yt-1
et
Yt Yt-1 et
where Yt is Yt Yt-1
AR(p) AR(p)
Yt Yt-1Yt-2pYt-1et
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AR(1) Stationary AR(1) Stationary | |– OLS Regression Estimators – Stationary caseOLS Regression Estimators – Stationary case– Mann and Wald (1940’s) : For |Mann and Wald (1940’s) : For |
21 1
2 2
1/ 2 1 21 1
2 2
22
1 22
ˆ ( )( ) / ( )
ˆ( ) ( ) / ( )
1( ) { }
1
n n
t t tt t
n n
t t tt t
pn
tt
Y Y Y Y Y Y
n n Y Y e n Y Y
Y Y Var Yn
More exciting algebra coming up ……
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AR(1) Stationary AR(1) Stationary | |– OLS Regression Estimators – Stationary caseOLS Regression Estimators – Stationary case
4
1 22
4
1 22
2
1{ ( ) }
1
1( ) (0, )
1
ˆ: ( ) (0,1 )
n
t tt
n L
t tt
L
Var Y Y en
Y Y e Nn
Slutzky n N
(1)Same limit if sample mean replaced by AR(p) Multivariate Normal Limits
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||
YYttYYt-1t-1eett YYt-2t-2eet-1t-1eett
eetteet-1t-1 eet-2t-2 … … k-1k-1eet-k+1t-k+1kkYYt-t-
kk
YYttconverges for converges for
Var{YVar{Ytt } }
ButBut if if , then Y, then Ytt YYt-1t-1 e ett, a , a random walkrandom walk. .
YYtt YY00 e et t e et-1 t-1 e et-2 t-2 … … e e11
VarVarYYtt YY00 t t
YYttYY00
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AR(1) AR(1) ||
E{YE{Ytt} }
Var{YVar{Ytt } is constant } is constant
Forecast of YForecast of Yt+Lt+L converges to converges to (exponentially fast) (exponentially fast)
Forecast error variance is boundedForecast error variance is bounded
YYtt YYt-1t-1 e ett
YYttYY00
VarVarYYttgrows without boundgrows without bound
Forecast Forecast notnot mean reverting mean reverting
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E = MC2
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Nonstationary cases:
Case 1: known (=0)
Regression Estimators (Yt on Yt-1 noint )
12
21
2
ˆ( 1)
n
t tt
n
tt
Y e
Y
n
/n
/n2
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12
2 21
2
/ˆ( 1) " "
/
L
n
t ttn
tt
Y e nn DF
Y n
12
2 21
2
L
n
t tt
n
tt
Y et statistic
s Y
2ˆ( ) (0,1 ), (0,1)L L
n N t N
Nonstationary
Recall stationary results:
Note: all results independent of
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Where are my clothes?H0: H1:
?
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DF Distribution ??
Numerator: 2 2
2 211 2
2
1 1/( ) ( 1)
2 2
n
n tnt
t tt
Y eY e n
n
e1 e2 e3 … en
e1 e12 e1e2 e1e3 … e1en
e2 e22 e2e3 … e2en
e3 e32 … e3en
: :en en
2
Y2e3Y1e2 Yn-1en…
:
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Denominator
42 2 2 2
1 1 2 1 2 32
( ) ( )tt
Y e e e e e e
1 1
1 2 3 2 1 2 3 2
3 3
3 2 1 ? 0 0
2 2 1 0 ? 0
1 1 1 0 0 ?
( ~ (0,1))
e z
e e e e z z z z
e z
Z N
For n Observations:
11 2 3 ... 1 1 1 0 ... 0
2 2 3 ... 1 1 2 1 ... 0
3 3 3 ... 1 0 1 2 ... 0
: : : \ : : : : \ :
1 1 1 1 1 0 0 0 ... 2
n
n n n
n n n
A n n n
(eigenvalues are reciprocals of each other)
2
2
:
1 ( )sec
4 2 1
in
Eigenvalues
n i
n
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Results:
Graph of
and limit :
eTAne = 2 2 2
1
~ (0,1)n
in i ni
Z Z N
n-2 eTAne = limn
212 2
1
2( 1)~ (0,1)
(2 1)
i
ii
Z Z Ni
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Histograms for n=50:
-8.1
-1.96
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Theory 1: Donsker’s Theorem (pg. 68, 137 Billingsley)
{et} an iid(0,)
sequence
Sn = e1+e2+ …+en
X(t,n) = S[nt]/(n1/2)=Sn normalized
(n=100)
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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
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Theory 2: Continuous mapping theorem (Billingsley pg. 72)
h( ) a continuous functional => h( X(t,n) ) h(W(t)) L
For our estimators, / (1)L
nY n W
and 2 1
2
1 0
/ (1/ ) ( )n L
tt
Y n n W t dt
so……
2 2 2 2 2
12 1
2
1 0
2
12
0
1 1( / / ) ( (1) 1)2 2ˆ( 1)
( )/ (1/ )
1( (1) 1)
2
( )
n
n t Lt
n
tt
Y n e n Wn
W t dtY n n
W
W t dt
Distribution is …. ???????
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Extension 1: Add a mean (intercept)Extension 1: Add a mean (intercept)
1
1 1
( )
( ) ( 1)( )
t t t
def
t t t t t
Y Y e
Y Y Y Y e
^
, New quadratic forms.New distributions
Estimator independent of Y0
0 1 2
0 1 2
1 1t t
t
t
Y Y e e e
nY Y e e e
n nY Y
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Extension 2: Add linear trendExtension 2: Add linear trend
1
1
0
( ) ( ( ( 1))
( 1)( ( ( 1))
" "
t t t
t t t
t t t
Y t Y t e
Y Y t e
and under H
Y e drift e
^
,
New quadratic forms.New distributions
Regress Yt on 1, t, Yt-1 annihilates Y0 , t
1 0 1
2 1 2 0 1 2
[ ]
[ ] [ 2 ]
Y Y e
Y Y e Y e e
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The 6 DistributionsThe 6 Distributions
coefficient n(j-1)
t test
f(t) = 0 mean trend
- 1.96
0
-1.95
-8.1-14.1 -21.8
-2.93 -3.50
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pr< 0.01 0.025 0.05 0.10 0.50 0.90 0.95 0.975 0.99
f(t)
--- -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
(1,t) -4.16 -3.80 -3.50 -3.18 -2.16 -1.19 -0.87 -0.58 -0.24
percentiles, n=50
pr< 0.01 0.025 0.05 0.10 0.50 0.90 0.95 0.975 0.99
f(t)
--- -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
(1,t) -3.96 -3.67 -3.41 -3.13 -2.18 -1.25 -0.94 -0.66 -0.32
percentiles, limit
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Higher Order Models
1 2
1 1
2
1.3( ) .4( )
0.1( ) .4( )
1.3 0.4 ( .5)( .8) 0
t t t t
t t t t
Y Y Y e
Y Y Y e
m m m m
“characteristic eqn.”roots 0.5, 0.8 ( < 1)
1 2
1 1 1
2
1.3( ) .3( )
0.0( ) .3( ) , .3( )
1.3 0.3 ( .3)( 1)
" !"
t t t t
t t t t t t t
Y Y Y e
Y Y Y e Y Y e
m m m m
unit root
note: (1-.5)(1-.8) = -0.1
stationary:
nonstationary
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Higher Order Models- General AR(2)
1 2
1 1
( ) ( ) ( )
(1 ) ( ) ( )
(1 ) (1 )(1 )
t t t t
t t t t
Y Y Y e
Y Y Y e
roots: (m )( m ) = m2 m AR(2): ( Yt ) =( Yt-1 ) ( Yt-2 ) + et
nonstationary
1 1(1 ) ( ) ( )t t t tY Y Y e
(0 if unit root)
t test same as AR(1).Coefficient requiresmodification
t test N(0,1) !!
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Tests
Regress:
tY 1 2 1, , ,t t t pY Y Y on (1, t) Yt-1
( “ADF” test )
-1 ( )
augmenting affects limit distn.
“ does not affect “ “
These coefficients normal!| |
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Nonstationary Forecast
Stationary Forecast
Silver example:Silver example:
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Is AR(2) sufficient ? test vs. AR(5).Is AR(2) sufficient ? test vs. AR(5).proc reg; model D = Y1 D1-D4;proc reg; model D = Y1 D1-D4; test D2=0, D3=0, D4=0;test D2=0, D3=0, D4=0;
Source df Coeff. t Pr>|t|Source df Coeff. t Pr>|t|Intercept 1 Intercept 1 121.03 3.09 0.0035121.03 3.09 0.0035
YYt-1t-1 1 1 -0.188 -3.07 0.0038-0.188 -3.07 0.0038
YYt-1t-1-Y-Yt-2t-2 1 0.639 4.59 0.0001 1 0.639 4.59 0.0001
YYt-2t-2-Y-Yt-3t-3 1 0.050 0.30 1 0.050 0.30 0.76910.7691
YYt-3t-3-Y-Yt-4t-4 1 0.000 0.00 1 0.000 0.00 0.99850.9985
YYt-4t-4-Y-Yt-5t-5 1 0.263 1.72 1 0.263 1.72 0.09240.0924
FF41413 3 = 1152 / 871 = 1.32 Pr>F = 0.2803 = 1152 / 871 = 1.32 Pr>F = 0.2803
X
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Fit AR(2) and do unit root testMethod 1: OLS output and tabled critical value (-2.86)proc reg; model D = Y1 D1;
Source df Coeff. t Pr>|t|Source df Coeff. t Pr>|t|Intercept 1 Intercept 1 75.581 2.762 0.0082 X75.581 2.762 0.0082 X
YYt-1t-1 1 1 -0.117 -0.117 -2.776-2.776 0.0038 X 0.0038 X
YYt-1t-1-Y-Yt-2t-2 1 0.671 6.211 0.0001 1 0.671 6.211 0.0001
Method 2: OLS output and tabled critical valuesproc arima; identify var=silver stationarity = (dickey=(1));
Augmented Dickey-Fuller Unit Root Tests
Type Lags t Prob<t Zero Mean 1 -0.2803 0.5800 Single Mean 1 -2.7757 0.0689 Trend 1 -2.6294 0.2697
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?
First part ACF IACF PACF
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Full data ACF IACF PACF
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Amazon.com Stock ln(Closing Price) L
evel
sD
iffe
ren
ces
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Augmented Dickey-Fuller Unit Root Tests
Type Lags Tau Pr < Tau
Zero Mean 2 1.85 0.9849 Single Mean 2 -0.90 0.7882 Trend 2 -2.83 0.1866
Levels
Differences
Augmented Dickey-Fuller Unit Root Tests
Type Lags Tau Pr<Tau
Zero Mean 1 -14.90 <.0001 Single Mean 1 -15.15 <.0001 Trend 1 -15.14 <.0001
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Autocorrelation Check for White Noise
To Chi- Pr > Lag Square DF ChiSq -------------Autocorrelations-------------
6 3.22 6 0.7803 0.047 0.021 0.046 -0.036 -0.004 0.014 12 6.24 12 0.9037 -0.062 -0.032 -0.024 0.006 0.004 0.019 18 9.77 18 0.9391 0.042 0.015 -0.042 0.023 0.020 0.046 24 12.28 24 0.9766 -0.010 -0.005 -0.035 -0.045 0.008 -0.035
Are differences white noise (p=q=0) ?
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Amazon.com Stock Volume L
evel
sD
iffe
ren
ces
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Augmented Dickey-Fuller Unit Root Tests
Type Lags Tau Pr < Tau
Zero Mean 4 0.07 0.7063 Single Mean 4 -2.05 0.2638 Trend 4 -5.76 <.0001
Maximum Likelihood Estimation
Approx Parameter Estimate t Value Pr > |t| Lag Variable MU -71.81516 -8.83 <.0001 0 volume MA1,1 0.26125 4.53 <.0001 2 volume AR1,1 0.63705 14.35 <.0001 1 volume AR1,2 0.22655 4.32 <.0001 2 volume NUM1 0.0061294 10.56 <.0001 0 date
To Chi- Pr >Lag Square DF ChiSq -------------Autocorrelations-------------
6 0.59 3 0.8978 -0.009 -0.002 -0.015 -0.023 -0.008 -0.016 12 9.41 9 0.4003 -0.042 0.002 0.068 -0.075 0.026 0.065 18 11.10 15 0.7456 -0.042 0.006 0.013 -0.014 -0.017 0.027 24 17.10 21 0.7052 0.064 -0.043 0.029 -0.045 -0.034 0.035 30 21.86 27 0.7444 0.003 0.022 -0.068 0.010 0.014 0.058 36 28.58 33 0.6869 -0.020 0.015 0.093 0.033 -0.041 -0.015 42 35.53 39 0.6291 0.070 0.038 -0.052 0.033 -0.044 0.023 48 37.13 45 0.7916 0.026 -0.021 0.018 0.002 0.004 0.037
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Amazon.com Spread = ln(High/Low)L
evel
sD
iffe
ren
ces
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Augmented Dickey-Fuller Unit Root Tests
Type Lags Tau Pr<Tau
Zero Mean 4 -2.37 0.0174 Single Mean 4 -6.27 <.0001 Trend 4 -6.75 <.0001
Maximum Likelihood Estimation
Approx Parm Estimate t Value Pr>|t| Lag Variable MU -0.48745 -1.57 0.1159 0 spread MA1,1 0.42869 5.57 <.0001 2 spread AR1,1 0.38296 8.85 <.0001 1 spread AR1,2 0.42306 5.97 <.0001 2 spread NUM1 0.00004021 1.82 0.0690 0 date
To Chi- Pr > Lag Square DF ChiSq -------------Autocorrelations------------- 6 2.87 3 0.4114 -0.004 0.021 0.025 -0.039 0.014 -0.053 12 3.83 9 0.9221 0.000 0.016 0.013 -0.000 0.008 0.037 18 7.62 15 0.9381 -0.038 -0.062 0.010 -0.032 -0.004 0.027 24 15.96 21 0.7721 -0.006 0.008 -0.076 -0.085 0.045 0.022 30 19.01 27 0.8695 0.008 0.043 0.013 -0.018 -0.007 0.057 36 22.38 33 0.9187 0.004 0.027 0.041 -0.030 0.014 -0.052 42 25.39 39 0.9546 0.043 0.042 0.019 0.003 0.034 -0.016 48 30.90 45 0.9459 0.015 -0.054 -0.061 -0.049 -0.004 -0.021
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S.E. Said: Use AR(k) model even if MA S.E. Said: Use AR(k) model even if MA terms in true model.terms in true model.
N. Fountis: Vector Process with One Unit N. Fountis: Vector Process with One Unit Root Root
D. Lee: Double Unit Root EffectD. Lee: Double Unit Root Effect
M. Chang: Overdifference ChecksM. Chang: Overdifference Checks
G. Gonzalez-Farias: Exact MLEG. Gonzalez-Farias: Exact MLE
K. Shin: Multivariate Exact MLE K. Shin: Multivariate Exact MLE
T. Lee: Seasonal Exact MLET. Lee: Seasonal Exact MLE
Y. Akdi, B. Evans – Periodograms of Unit Y. Akdi, B. Evans – Periodograms of Unit Root ProcessesRoot Processes
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H. Kim: Panel Data testsH. Kim: Panel Data tests
S. Huang: Nonlinear AR processesS. Huang: Nonlinear AR processes
S. Huh: Intervals: Order StatisticsS. Huh: Intervals: Order Statistics
S. Kim: Intervals: Level Adjustment & S. Kim: Intervals: Level Adjustment & RobustnessRobustness
J. Zhang: Long Period Seasonal. J. Zhang: Long Period Seasonal.
Q. Zhang: Comparing Seasonal Q. Zhang: Comparing Seasonal Cointegration Methods.Cointegration Methods.