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

Autoregressive Model AR(1) AR(1) Y t   Y t-1  e t Y t   Y t-1  e t  Y t   Y t-1  e t  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 (1940’s) : 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    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  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 = SAS program: Simulate_Tau.sas

Histograms for n=50:

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

pr<  f(t) (1,t)  percentiles, n=50 pr<  f(t) (1,t)  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 )  -1 (  )  augmenting affects limit distn.  “ does not affect “ “ These coefficients  normal! |   |

Nonstationary Forecast Stationary Forecast Silver example: 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 Y t Y t-1 -Y t Y t-2 -Y t Y t-3 -Y t Y t-4 -Y t F 41 3 = 1152 / 871 = 1.32 Pr>F = F 41 3 = 1152 / 871 = 1.32 Pr>F = 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 X Y t X Y t-1 -Y t Y t-1 -Y t 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 Single Mean Trend

? First part ACF IACF PACF

Full data ACF IACF PACF

Amazon.com Stock ln(Closing Price) Levels Differences Demo: Rho_3.sas

Augmented Dickey-Fuller Unit Root Tests Type Lags Tau Pr < Tau Zero Mean Single Mean Trend Levels Differences Augmented Dickey-Fuller Unit Root Tests Type Lags Tau Pr<Tau Zero Mean <.0001 Single Mean <.0001 Trend <.0001

Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF ChiSq Autocorrelations 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 Single Mean Trend <.0001 Maximum Likelihood Estimation Approx Parameter Estimate t Value Pr > |t| Lag Variable MU < volume MA1, < volume AR1, < volume AR1, < volume NUM < date To Chi- Pr > Lag Square DF ChiSq Autocorrelations

Amazon.com Spread = ln(High/Low) Levels Differences

Augmented Dickey-Fuller Unit Root Tests Type Lags Tau Pr<Tau Zero Mean Single Mean <.0001 Trend <.0001 Maximum Likelihood Estimation Approx Parm Estimate t Value Pr>|t| Lag Variable MU spread MA1, < spread AR1, < spread AR1, < spread NUM date To Chi- Pr > Lag Square DF ChiSq Autocorrelations

Cointegration –Two nonstationary time series Y t and X t with linear combination aY t +bX t stationary –Example: spread = log(high)-log(low) –a=1, b=-1 –Unit root test shows stationary. More demos: Harley.sasBrewers.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.