Review of Unit Root TestingReview of Unit Root Testing

D. A. DickeyD. A. DickeyNorth Carolina State North Carolina State

UniversityUniversity

Nonstationary Forecast

Stationary Forecast

”Trend Stationary” Forecast

Nonstationary Forecast

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

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 ……

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

||

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

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

E = MC2

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

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

Where are my clothes?H0: H1:

?

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…

:

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

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

Histograms for n=50:

-8.1

-1.96

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)

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)

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 …. ???????

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

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

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

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

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

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) !!

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!| |

Nonstationary Forecast

Stationary Forecast

Silver example:Silver example:

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

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

?

First part ACF IACF PACF

Full data ACF IACF PACF

Amazon.com Stock ln(Closing Price) L

evel

sD

iffe

ren

ces

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

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) ?

Amazon.com Stock Volume L

evel

sD

iffe

ren

ces

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

Amazon.com Spread = ln(High/Low)L

evel

sD

iffe

ren

ces

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

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

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.