1. Review of Unit Root Testing D. A. Dickey North Carolina State University (Previously presented at Purdue Econ Dept.)

2. Nonstationary Forecast Stationary Forecast

3. ”Trend Stationary” Forecast Nonstationary Forecast

4. 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

5. AR(1) Stationary  |  –OLS Regression Estimators – Stationary case –Mann and Wald (1940’s) : For |  More exciting algebra coming up ……

6. AR(1) Stationary  |  –OLS Regression Estimators – Stationary case (1)Same limit if sample mean replaced by   AR(p)  Multivariate Normal Limits

7. |  |  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 

8. 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

9. E = MC 2 

10. Nonstationary  cases: Case 1:  known (=0) Regression Estimators (Y t on Y t-1 noint ) n /n /n 2

11.   Nonstationary Recall stationary results: Note: all results independent of  

12. Where are my clothes? H 0 :  H 1 :  ?

13. 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 … :

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

15. Results: Graph of     and limit : e T A n e = n -2 e T A n e = SAS program: Simulate_Tau.sas

16. Histograms for n=50: -8.1 -1.96

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

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

19. 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

20. 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

21. Higher Order Models “characteristic eqn.” roots 0.5, 0.8 ( < 1) note: (1-.5)(1-.8) = -0.1 stationary: nonstationary

22. 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) !!

23. Tests Regress: on (1, t)Y t-1 ( “ADF” test )  -1 (  )  augmenting affects limit distn.  “ does not affect “ “ These coefficients  normal! |   |

24. Nonstationary Forecast Stationary Forecast Silver example: Demo: Rho_2.sas

25. 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

26. 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<t Zero Mean 1 -0.2803 0.5800 Single Mean 1 -2.7757 0.0689 Trend 1 -2.6294 0.2697

27. ? First part ACF IACF PACF

28. Full data ACF IACF PACF

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

30. 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<Tau Zero Mean 1 -14.90 <.0001 Single Mean 1 -15.15 <.0001 Trend 1 -15.14 <.0001

31. 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) ?

32. Amazon.com Stock Volume Levels Differences

33. 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

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

35. 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

36. 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

37. 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

38. 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.