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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

Review of Unit Root Testing

D. A. Dickey

North Carolina State University

(Previously presented at Purdue Econ Dept.)


Review of unit root testing

Nonstationary Forecast

Stationary Forecast


Review of unit root testing

”Trend Stationary” Forecast

Nonstationary Forecast


Review of unit root testing

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


Review of unit root testing

  • AR(1) Stationary  |r| < 1

    • OLS Regression Estimators – Stationary case

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

More exciting algebra coming up ……


Review of unit root testing

  • AR(1) Stationary  |r| < 1

    • OLS Regression Estimators – Stationary case

  • Same limit if sample mean replaced by m

  • (2) AR(p)  Multivariate Normal Limits


Review of unit root testing

  • |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}


Review of unit root testing

  • 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


Review of unit root testing

E = MC2

r = ?


Review of unit root testing

Nonstationary (r=1) cases:

Case 1: m known (=0)

Regression Estimators (Yt on Yt-1noint )

/n

n

/n2


Review of unit root testing

r=1  Nonstationary

Recall stationary results:

Note: all results independent of s 2


Review of unit root testing

Where are my clothes?

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

?


Review of unit root testing

DF Distribution ??

Numerator:

e1 e2 e3 … en

e1 e12e1e2 e1e3 … e1en

e2 e22e2e3 … e2en

e3 e32 … e3en

: :

en en2

:

Y1e2

Y2e3

Yn-1en


Review of unit root testing

Denominator

For n

Observations:

(eigenvalues are reciprocals of each other)


Review of unit root testing

Results:

eTAne =

n-2eTAne =

Graph of

gi,502and limit :

SAS program:

Simulate_Tau.sas


Review of unit root testing

Histograms for n=50:

-1.96

-8.1


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


Review of unit root testing

t percentiles, n=50

t percentiles, limit


Review of unit root testing

Higher Order Models

stationary:

“characteristic eqn.”

roots 0.5, 0.8( < 1)

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

nonstationary


Review of unit root testing

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) !!


Review of unit root testing

Tests

These coefficients  normal!

|   |

Regress:

on (1, t)

Yt-1

( “ADF” test )

r-1

( t )

  • augmenting affects limit distn.

  • “ does not affect “ “


Review of unit root testing

Silver example:

Nonstationary Forecast

Stationary Forecast

Demo:

Rho_2.sas


Review of unit root testing

  • 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


Review of unit root testing

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


Review of unit root testing

?

First part ACF IACF PACF


Review of unit root testing

Full data ACF IACF PACF


Review of unit root testing

Amazon.com Stock ln(Closing Price)

Levels

Differences

Demo:

Rho_3.sas


Review of unit root testing

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


Review of unit root testing

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


Review of unit root testing

Amazon.com Stock Volume

Levels

Differences


Review of unit root testing

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


Review of unit root testing

Amazon.com Spread = ln(High/Low)

Levels

Differences


Review of unit root testing

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


Review of unit root testing

  • 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


Review of unit root testing

  • 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


Review of unit root testing

  • 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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