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Review of Unit Root Testing. D. A. Dickey North Carolina State University (Previously presented at Purdue Econ Dept.). Nonstationary Forecast. Stationary Forecast. ”Trend Stationary” Forecast. Nonstationary Forecast. Y t - m = r ( Y t-1 -m) + e t Y t = m (1- r) + r Y t-1 + e t

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review of unit root testing

Review of Unit Root Testing

D. A. Dickey

North Carolina State University

(Previously presented at Purdue Econ Dept.)

slide2

Nonstationary Forecast

Stationary Forecast

slide3

”Trend Stationary” Forecast

Nonstationary Forecast

slide4

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

whereDYt is Yt-Yt-1

  • Autoregressive Model
  • AR(1)
  • AR(p)

Yt -m = a1(Yt-1-m) + a2(Yt-2-m) + ...+ ap(Yt-1-m) + et

slide5
AR(1) Stationary  |r| < 1
    • OLS Regression Estimators – Stationary case
    • Mann and Wald (1940’s) : For |r| < 1

More exciting algebra coming up ……

slide6
AR(1) Stationary  |r| < 1
    • OLS Regression Estimators – Stationary case
  • Same limit if sample mean replaced by m
  • (2) AR(p)  Multivariate Normal Limits
slide7
|r| < 1
  • Yt-m = r(Yt-1-m) + et=r(r(Yt-2-m)+ et-1) + et= ... = et + ret-1+r2et-2+ … +rk-1et-k+1+ rk (Yt-k-m) .
  • Yt=m + (converges for |r| < 1)
  • Var{Yt } = s2/(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}= ts2
  • E{Yt} = E{Y0}
slide8
AR(1) |r| < 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
slide9

E = MC2

r = ?

slide10

Nonstationary (r=1) cases:

Case 1: m known (=0)

Regression Estimators (Yt on Yt-1noint )

/n

n

/n2

slide11

r=1  Nonstationary

Recall stationary results:

Note: all results independent of s 2

slide12

Where are my clothes?

H0:r=1 H1:|r|<1

?

slide13

DF Distribution ??

Numerator:

e1 e2 e3 … en

e1 e12e1e2 e1e3 … e1en

e2 e22e2e3 … e2en

e3 e32 … e3en

: :

en en2

:

Y1e2

Y2e3

Yn-1en

slide14

Denominator

For n

Observations:

(eigenvalues are reciprocals of each other)

slide15

Results:

eTAne =

n-2eTAne =

Graph of

gi,502and limit :

SAS program:

Simulate_Tau.sas

extension 1 add a mean intercept
Extension 1: Add a mean (intercept)

New quadratic forms.

New distributions

Estimator independent of Y0

extension 2 add linear trend
Extension 2: Add linear trend

on 1, t, Yt-1 annihilates Y0 , bt

Regress Yt

New quadratic forms.

New distributions

the 6 distributions
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

slide20

t percentiles, n=50

t percentiles, limit

slide21

Higher Order Models

stationary:

“characteristic eqn.”

roots 0.5, 0.8( < 1)

note: (1-.5)(1-.8) = -0.1

nonstationary

slide22

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) !!

slide23

Tests

These coefficients  normal!

|   |

Regress:

on (1, t)

Yt-1

( “ADF” test )

r-1

( t )

  • augmenting affects limit distn.
  • “ does not affect “ “
slide24

Silver example:

Nonstationary Forecast

Stationary Forecast

Demo:

Rho_2.sas

slide25
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 Pr>|t|

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 Pr>F = 0.2803

X

slide26

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 Lags t Prob<t

Zero Mean 1 -0.2803 0.5800

Single Mean 1 -2.77570.0689 

Trend 1 -2.6294 0.2697

slide27

?

First part ACF IACF PACF

slide29

Amazon.com Stock ln(Closing Price)

Levels

Differences

Demo:

Rho_3.sas

slide30

Levels

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

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

slide31

Are differences white noise (p=q=0) ?

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

slide32

Amazon.com Stock Volume

Levels

Differences

slide33

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

slide35

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

slide36

Cointegration

    • Two nonstationary time series Yt and Xt with linear combination aYt+bXt stationary
    • Example: spread = log(high)-log(low)
    • a=1, b=-1
    • Unit root test shows stationary.
  • More demos:

Harley.sas

Brewers.sas

slide37
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
slide38
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.
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