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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.)


Nonstationary Forecast

Stationary Forecast


”Trend Stationary” Forecast

Nonstationary Forecast


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


  • AR(1) Stationary  |r| < 1

    • OLS Regression Estimators – Stationary case

    • Mann and Wald (1940’s) : For |r| < 1

More exciting algebra coming up ……


  • 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-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}


  • 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


E = MC2

r = ?


Nonstationary (r=1) cases:

Case 1: m known (=0)

Regression Estimators (Yt on Yt-1noint )

/n

n

/n2


r=1  Nonstationary

Recall stationary results:

Note: all results independent of s 2


Where are my clothes?

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

?


DF Distribution ??

Numerator:

e1 e2 e3 … en

e1 e12e1e2 e1e3 … e1en

e2 e22e2e3 … e2en

e3 e32 … e3en

: :

en en2

:

Y1e2

Y2e3

Yn-1en


Denominator

For n

Observations:

(eigenvalues are reciprocals of each other)


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


t percentiles, n=50

t percentiles, limit


Higher Order Models

stationary:

“characteristic eqn.”

roots 0.5, 0.8( < 1)

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

nonstationary


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) !!


Tests

These coefficients  normal!

|   |

Regress:

on (1, t)

Yt-1

( “ADF” test )

r-1

( t )

  • augmenting affects limit distn.

  • “ does not affect “ “


Silver example:

Nonstationary Forecast

Stationary Forecast

Demo:

Rho_2.sas


  • 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


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


?

First part ACF IACF PACF



Amazon.com Stock ln(Closing Price) PACF

Levels

Differences

Demo:

Rho_3.sas


Levels PACF

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


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

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


Amazon.com Stock Volume PACF

Levels

Differences


Augmented Dickey-Fuller Unit Root Tests PACF

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) PACF

Levels

Differences


Augmented Dickey-Fuller Unit Root Tests PACF

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


  • Cointegration PACF

    • 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


  • S.E. Said: Use AR(k) model even if MA terms in true model. PACF

  • 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 PACF

  • 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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