Review of Unit Root Testing

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Review of Unit Root Testing

• D. A. Dickey
• North Carolina State University

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Nonstationary Forecast
Stationary Forecast
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Trend Stationary Forecast
Nonstationary Forecast
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• 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

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• AR(1) Stationary ? r lt 1
• OLS Regression Estimators Stationary case
• Mann and Wald (1940s) For r lt 1

More exciting algebra coming up
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• AR(1) Stationary ? r lt 1
• OLS Regression Estimators Stationary case

• Same limit if sample mean replaced by m
• (2) AR(p) ? Multivariate Normal Limits ?

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• r lt 1
• Yt-m r(Yt-1-m) et r(r(Yt-2-m) et-1) et
  ... et ret-1 r2et-2 rk-1
  et-k1 rk (Yt-k-m) .
• Ytm (converges for r lt 1)
  
• VarYt s2/(1-r2)
• r 1
• But if r1, then Yt Yt-1 et, a random walk.
  
• Yt Y0 et et-1 et-2 e1
• VarYt - Y0 ts2
• EYt EY0

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• AR(1) r lt 1
• EYt m
• VarYt is constant
• Forecast of YtL converges to m (exponentially
  fast)
• Forecast error variance is bounded
• AR(1) r 1
• Yt Yt-1 et
• EYt EY0
• VarYt grows without bound
• Forecast not mean reverting

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E MC2
r ?
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Nonstationary (r1) cases
Case 1 m known (0)
Regression Estimators (Yt on Yt-1 noint )
/n
n
/n2
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r1 ? Nonstationary
Recall stationary results
Note all results independent of s 2
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Where are my clothes?
H0r1 H1rlt1
?
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DF Distribution ?? Numerator
e1 e2 e3
en e1 e12 e1e2 e1e3
e1en e2 e22 e2e3
e2en e3
e32 e3en
en
en2


Y2e3
Y1e2
Yn-1en

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Denominator
For n Observations
(eigenvalues are reciprocals of each other)
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Results
eTAne
n-2 eTAne
Graph of gi,502 and limit
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Histograms for n50
-1.96
-8.1
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Theory 1 Donskers Theorem (pg. 68, 137
Billingsley)
et an iid(0,s2) sequence ?
(n100)
Sn e1e2 en?
X(t,n) Snt/(n1/2s)Sn normalized
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Theory 1 Donskers 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 n20, 100, 2000

• 20 realizations of
• X(t,100) vs. zt/n

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Theory 2 Continuous mapping theorem
(Billingsley pg. 72)
h( ) a continuous functional gt h( X(t,n) )
h(W(t))
For our estimators,
and
so
Distribution is . ???????
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Extension 1 Add a mean (intercept)
New quadratic forms. New distributions
Estimator independent of Y0
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Extension 2 Add linear trend
on 1, t, Yt-1 annihilates Y0 , bt
Regress Yt
New quadratic forms. New distributions
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The 6 Distributions
coefficient n(rj-1)
-8.1
-14.1
-21.8
0
t test t
- 1.96
-1.95
-2.93
-3.50
f(t) 0
mean trend
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t percentiles, n50
prltt 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
t percentiles, limit
prltt 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
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Higher Order Models
stationary
characteristic eqn. roots 0.5, 0.8 ( lt 1)
note (1-.5)(1-.8) -0.1
nonstationary
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Higher Order Models- General AR(2)
roots (m - a )( m - b ) m2 - ( a b )m ab
AR(2) ( Yt - m ) ( a b ) ( Yt-1 - m ) - ab
( Yt-2 - m ) et
(0 if unit root)
nonstationary
t test same as AR(1). Coefficient
requires modification
t test ? N(0,1) !!
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Tests
These coefficients ? normal! ?
?
Regress
on (1, t)
Yt-1
( ADF test )
r-1 ( t )

• augmenting affects limit distn.
• does not affect

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Silver example
Nonstationary Forecast
Stationary Forecast
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• Is AR(2) sufficient ? test vs. AR(5).
• proc reg model D Y1 D1-D4 test D20, D30,
  D40
• Source df Coeff. t Prgtt
• Intercept 1 121.03 3.09 0.0035
• Yt-1 1 -0.188 -3.07 0.0038
• Yt-1-Yt-2 1 0.639 4.59 0.0001
• Yt-2-Yt-3 1 0.050 0.30 0.7691
• Yt-3-Yt-4 1 0.000 0.00 0.9985
• Yt-4-Yt-5 1 0.263 1.72 0.0924
• F413 1152 / 871 1.32 PrgtF 0.2803

X
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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
Prgtt 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 varsilver
stationarity (dickey(1))
Augmented Dickey-Fuller Unit Root Tests Type
Lags t Probltt 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)
Levels
Differences
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Levels
Augmented Dickey-Fuller Unit Root Tests
Type Lags Tau Pr lt Tau Zero
Mean 2 1.85 0.9849 Single Mean
2 -0.90 0.7882 Trend
2 -2.83 0.1866
Differences
Augmented Dickey-Fuller Unit Root Tests
Type Lags Tau PrltTau Zero
Mean 1 -14.90 lt.0001 Single Mean
1 -15.15 lt.0001 Trend 1
-15.14 lt.0001
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Are differences white noise (pq0) ?
Autocorrelation Check for White
Noise To Chi- Pr gt 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
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Amazon.com Stock Volume
Levels
Differences
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Augmented Dickey-Fuller Unit Root
Tests Type Lags Tau
Pr lt Tau Zero Mean 4
0.07 0.7063 Single Mean 4
-2.05 0.2638 Trend
4 -5.76 lt.0001
Maximum Likelihood Estimation
Approx Parameter Estimate
t Value Pr gt t Lag Variable MU
-71.81516 -8.83 lt.0001 0 volume
MA1,1 0.26125 4.53 lt.0001
2 volume AR1,1 0.63705 14.35
lt.0001 1 volume AR1,2 0.22655
4.32 lt.0001 2 volume NUM1
0.0061294 10.56 lt.0001 0 date
To Chi- Pr gt 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)
Levels
Differences
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Augmented Dickey-Fuller Unit Root
Tests Type Lags Tau
PrltTau Zero Mean 4 -2.37
0.0174 Single Mean 4 -6.27
lt.0001 Trend 4 -6.75
lt.0001 Maximum Likelihood
Estimation Approx
Parm Estimate t Value Prgtt Lag
Variable MU -0.48745 -1.57 0.1159
0 spread MA1,1 0.42869 5.57 lt.0001
2 spread AR1,1 0.38296 8.85
lt.0001 1 spread AR1,2 0.42306 5.97
lt.0001 2 spread NUM1 0.00004021
1.82 0.0690 0 date
To Chi- Pr gt 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 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

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• 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.