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:

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, 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

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 ) ( ) 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 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

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

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.