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Review of Unit Root Testing: Nonstationary vs. Stationary Forecast

This review provides an in-depth analysis of unit root testing in time series forecasting, focusing on stationarity criteria and statistical techniques. Topics include autoregressive models, OLS regression estimators, AR(p) models, nonstationary cases, and higher-order models. Special emphasis is placed on the AR(1) model's properties, including forecasts, growth patterns, and mean reversion. Additionally, advanced concepts like nonstationary forecast vs. stationary forecast and model comparisons are discussed.

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Review of Unit Root Testing: Nonstationary vs. Stationary Forecast

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  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. 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

  5. AR(1) Stationary  |r| < 1 • OLS Regression Estimators – Stationary case • Mann and Wald (1940’s) : For |r| < 1 More exciting algebra coming up ……

  6. AR(1) Stationary  |r| < 1 • OLS Regression Estimators – Stationary case • Same limit if sample mean replaced by m • (2) AR(p)  Multivariate Normal Limits

  7. |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}

  8. 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

  9. E = MC2 r = ?

  10. Nonstationary (r=1) cases: Case 1: m known (=0) Regression Estimators (Yt on Yt-1noint ) /n n /n2

  11. r=1  Nonstationary Recall stationary results: Note: all results independent of s 2

  12. Where are my clothes? H0:r=1 H1:|r|<1 ?

  13. DF Distribution ?? Numerator: e1 e2 e3 … en e1 e12e1e2 e1e3 … e1en e2 e22e2e3 … e2en e3 e32 … e3en : : en en2 : Y1e2 Y2e3 … Yn-1en

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

  15. Results: eTAne = n-2eTAne = Graph of gi,502and limit : SAS program: Simulate_Tau.sas

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

  17. Extension 1: Add a mean (intercept) New quadratic forms. New distributions Estimator independent of Y0

  18. Extension 2: Add linear trend on 1, t, Yt-1 annihilates Y0 , bt Regress Yt New quadratic forms. New distributions

  19. 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

  20. t percentiles, n=50 t percentiles, limit

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

  22. 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) !!

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

  24. Silver example: Nonstationary Forecast Stationary Forecast 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 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

  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 • 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

  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. 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

  31. 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

  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 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

  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.

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