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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

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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


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 ……


  • AR(1) Stationary  |r| < 1

    • OLS Regression Estimators – Stationary case

  • 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


Histograms for n=50:

-1.96

-8.1


Extension 1: Add a mean (intercept)

New quadratic forms.

New distributions

Estimator independent of Y0


Extension 2: Add linear trend

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

Regress Yt

New quadratic forms.

New 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


Full data ACF IACF PACF


Amazon.com Stock ln(Closing Price)

Levels

Differences

Demo:

Rho_3.sas


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


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


Amazon.com Stock Volume

Levels

Differences


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)

Levels

Differences


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


  • 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


  • 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


  • 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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