# Review of Unit Root Testing D. A. Dickey North Carolina State University.

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Review of Unit Root Testing D. A. Dickey North Carolina State University

Nonstationary Forecast Stationary Forecast

Trend Stationary Forecast Nonstationary Forecast

Autoregressive Model AR(1) AR(1) Y t Y t-1 e t where Y t is Y t Y t-1 AR(p) Y t Y t-1 Y t-2 p Y t-1 e t

AR(1) Stationary | –OLS Regression Estimators – Stationary case –Mann and Wald (1940s) : For | More exciting algebra coming up ……

AR(1) Stationary | –OLS Regression Estimators – Stationary case (1)Same limit if sample mean replaced by AR(p) Multivariate Normal Limits

| | Y t Y t-1 e t Y t-2 e t-1 e t e t e t-1 e t-2 … k-1 e t-k+1 k Y t-k Y t converges for Var{Y t } Var{Y t } But if, then Y t Y t-1 e t, a random walk. Y t Y 0 e t e t-1 e t-2 … e 1 Var Y t Y 0 t Var Y t Y 0 t Y t Y 0 Y t Y 0

AR(1) | E{Y t } E{Y t } Var{Y t } is constant Var{Y t } is constant Forecast of Y t+L converges to (exponentially fast) Forecast error variance is bounded Y t Y t-1 e t Y t Y 0 Y t Y 0 Var Y t grows without bound Forecast not mean reverting

E = MC 2

Nonstationary cases: Case 1: known (=0) Regression Estimators (Y t on Y t-1 noint ) n /n /n 2

Nonstationary Recall stationary results: Note: all results independent of

Where are my clothes? H 0 : H 1 : ?

DF Distribution ?? Numerator: e 1 e 2 e 3 … e n e 1 e 1 2 e 1 e 2 e 1 e 3 … e 1 e n e 2 e 2 2 e 2 e 3 … e 2 e n e 3 e 3 2 … e 3 e n : : e n e n 2 Y2e3Y2e3 Y1e2Y1e2 Y n-1 e n … :

Denominator For n Observations: (eigenvalues are reciprocals of each other)

Results: Graph of and limit : e T A n e = n -2 e T A n e =

Histograms for n=50: -8.1 -1.96

Theory 1: Donskers Theorem (pg. 68, 137 Billingsley) {e t } an iid(0, ) sequence S n = e 1 +e 2 + …+e n X(t,n) = S [nt] /(n 1/2 )=S n normalized (n=100)

Theory 1: Donskers 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)) For our estimators, and so…… Distribution is …. ???????

Extension 1: Add a mean (intercept) New quadratic forms. New distributions Estimator independent of Y 0

Extension 2: Add linear trend New quadratic forms. New distributions Regress Y t on 1, t, Y t-1 annihilates Y 0, t

The 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< 0.010.0250.050.100.500.900.950.9750.99 f(t) ----2.62-2.25-1.95-1.61-0.490.911.311.662.08 1-3.59-3.32-2.93-2.60-1.55-0.41-0.040.280.66 (1,t)-4.16-3.80-3.50-3.18-2.16-1.19-0.87-0.58-0.24 percentiles, n=50 pr< 0.010.0250.050.100.500.900.950.9750.99 f(t) ----2.58-2.23-1.95-1.62-0.510.891.281.622.01 1-3.42-3.12-2.86-2.57-1.57-0.44-0.080.230.60 (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 characteristic eqn. roots 0.5, 0.8 ( < 1) note: (1-.5)(1-.8) = -0.1 stationary: nonstationary

Higher Order Models- General AR(2) roots: (m )( m ) = m 2 m AR(2): ( Y t ) = ( Y t-1 ) ( Y t-2 ) + e t nonstationary (0 if unit root) t test same as AR(1). Coefficient requires modification t test N(0,1) !!

Tests Regress: on (1, t)Y t-1 ( ADF test ) ( ) augmenting affects limit distn. does not affect These coefficients normal! |

Nonstationary Forecast Stationary Forecast Silver example:

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 Y t-1 1 -0.188 -3.07 0.0038 Y t-1 -Y t-2 1 0.639 4.59 0.0001 Y t-2 -Y t-3 1 0.050 0.30 0.7691 Y t-3 -Y t-4 1 0.000 0.00 0.9985 Y t-4 -Y t-5 1 0.263 1.72 0.0924 F 41 3 = 1152 / 871 = 1.32 Pr>F = 0.2803 F 41 3 = 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 Y t-1 1 -0.117 -2.776 0.0038 X Y t-1 -Y t-2 1 0.671 6.211 0.0001 Y t-1 -Y t-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 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/5/1596002/slides/slide_29.jpg", "name": "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.", "description": "t Pr>|t| Intercept 1 75.581 2.762 0.0082 X Y t-1 1 -0.117 -2.776 0.0038 X Y t-1 -Y t-2 1 0.671 6.211 0.0001 Y t-1 -Y t-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

? First part ACF IACF PACF

Full data ACF IACF PACF

Amazon.com Stock ln(Closing Price) Levels Differences

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 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/5/1596002/slides/slide_33.jpg", "name": "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

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 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|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 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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