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Normally Distributed Seasonal Unit Root Tests D. A. Dickey North Carolina State University

Normally Distributed Seasonal Unit Root Tests D. A. Dickey North Carolina State University Note: this presentation is based on the paper “Normally Distributed Seasonal Unit Root Tests” authored by D. A. Dickey in the book Economic Time Series: Modeling and Seasonality

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Normally Distributed Seasonal Unit Root Tests D. A. Dickey North Carolina State University

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  1. Normally Distributed Seasonal Unit Root Tests D. A. Dickey North Carolina State University Note: this presentation is based on the paper “Normally Distributed Seasonal Unit Root Tests” authored by D. A. Dickey in the book Economic Time Series: Modeling and Seasonality edited by Bell, Holan, and McElroy published by CRC press, 2012

  2. Model: Seasonal AR(1) Yt = rYt-s + et, et is White Noise Goal: Test H0: r=1 Yt Yi,j=Ymonth, year J F M A M J J A S O N D (s=12) Yr. 1 Yr. 2 | Yr. m Y1,1 Y1,2 Y1,3 Y1,4 Y1,5 Y1,6 Y1,7 Y1,8 Y1,9 Y1,11 Y1,11Y1,12=Y1,s Y2,1 Y2,2 Y2,3 Y2,4 Y2,5 Y2,6 Y2,7 Y2,8 Y2,9 Y2,21 Y2,11 Y2,12=Y2,s | Ym,1 Ym,2 Ym,3 Ym,4 Ym,5 Ym,6 Ym,7 Ym,8 Ym,9 Ym,21 Ym,11 Ym,12=Ym,s Yt = rYt-s + et Yt – Yt-s = (r-1)Yt-s + ei,j Yi,j -Yi,j-1 = (r-1) Yi,j-1 + ei,j

  3. Previous work: Yt = rYt-s + et Yi,j = r Yi,j-1 + ei,j Yi,j -Yi,j-1 = (r-1) Yi,j-1 + ei,j • Dickey & Zhang (2011, J. Korean Stat. Soc.) • Under H0: • S large  CLT t stat NORMAL (0,1) (O(s-1/2) mean adjustment helpful ) • Known O(s-1/2) adjustments to mean (same) for k periodic regressors added (k<<s) • MSEs2 • n.b.: Does not apply to seasonal dummy variables

  4. Known mean 0 *****Add seasonal dummy variables:*****

  5. w.o.l.o.g. Assume s2 = 1 Notation: E{Ni}=N0 E{Di}=D0 (different for mean 0 versus seasonal means) MSE=Mean Square Error = (Total SSq – Model SSq)/df MSE in seasonal means case is (regressing deseasonalized differences on deseasonalized lag levels)  !!!

  6. Standard error [(X’X)-1(MSE)]1/2 = t statistic, seasonal means model: No Mean Seasonal Means

  7. Taylor Series, seasonal means : (N0=-(m-2)/2<0)

  8. Approximate variance of t in seasonal means case

  9. COMPARISON No Mean Model Seasonal Means Model

  10. Calculation “recipe” for Seasonal Means Model Regress Yt-Yt-s on seasonal dummies and Yt-s. Get t = t test for Yt-s Compute Compute Compare to N(0,1) to get p-value.

  11. Alternative approach: Expand around (N, D) only, run large (1/2 million) simulation  fixup for small m. Result for variance: Similar empirical adjustments to mean:

  12. Compare limit (sinfinity) variance formulas: Taylor 3 variable versus Taylor 2 variable with and without adjustments 1 million replicates s=12, m=6 (10 seconds run time) Reference normal variance from empirical adjustment from 3 variable Taylor: Unadjusted (N,D) only

  13. Notes: Graphs use sample means (both expansions give same mean approximation) 3 variable Taylor variance 1.1556 closer to simulated statistics’ variance 1.2310 than is empirical adjusted if no s adjustment used. With the finite s part in the empirically adjusted formula, that formula gives 1.2024 The choice m=6 gave the biggest vertical gap between the limit (s) variance formulas. In previous graph. NEXT: Sequence with m=6, s =4, 6, 12, 24 THEN: Sequence with s=4, m=6, 8, 10, 20, 100

  14. Histogram Stats (reps = 1,000,000) (formulas from book)

  15. Higher order models (seasonal multiplicative form) Suggested estimation (Dickey, Hasza, Fuller (1984)) (under H0:r=1) RegressDt = Yt-Yt-s on p lags of D  AR(p) and residual rt FilterY using AR(p) model for D. Ft= filtered Yt Regressrt onFt-s& p lags of D (t test on Ft-s is t) Dickey, D. A., D. P. Hasza, and W. A. Fuller (1984). “Testing for Unit Roots in Seasonal Time Series”, Journal of the American Statistical Association,79, 355-367.

  16. Example 1: Oil Imports (from book, s=12, m=36) Levels First Differences & Seasonal Means

  17. Parameter Variable DF Estimate t Value Pr > |t| Intercept 1 -163.62422 -0.20 0.8442 filter12 1 -0.81594 -16.19 <.0001  D1 1 0.01853 0.48 0.6349 D2 1 -0.00511 -0.11 0.9128 D3 1 0.00016148 0.00 0.9974 D4 1 0.02891 0.58 0.5617 D5 1 0.02369 0.48 0.6344 D6 1 0.00623 0.13 0.8998 D7 1 -0.01025 -0.22 0.8268 D8 1 -0.04440 -1.13 0.2604 Ft-12 +--------------------------------------------------------------+ | Formulas from Economic Time Series Modeling and Seasonality | | pg. 398 (Bell, Holan McElroy eds.) | | | | s = 12 m = 36 | | Tau = -16.19 Mean = -4.2118 variance = 0.8377 | | | | Tau ~ N(-4.2118,0.8377) | | | | Z=(-16.19-(-4.2118))/sqrt(0.8377) | | | | Pr{Z <-13.09 } = 0.0000 | | | +--------------------------------------------------------------+

  18. Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag Variable Shift MU 515.88809 4154.0 0.12 0.9012 0 amt 0 AR1,10.17664 0.05113 3.450.000612amt 0 AR2,1 -0.67807 0.04966 -13.65 <.0001 1 amt 0 AR2,2 -0.40409 0.05916 -6.83 <.0001 2 amt 0 AR2,3 -0.19310 0.06194 -3.12 0.0018 3 amt 0 AR2,4 -0.23710 0.06232 -3.80 0.0001 4 amt 0 AR2,5 -0.20400 0.06233 -3.27 0.0011 5 amt 0 AR2,6 -0.17344 0.06259 -2.77 0.0056 6 amt 0 AR2,7 -0.18688 0.06021 -3.10 0.0019 7 amt 0 AR2,8 -0.07802 0.05122 -1.52 0.1277 8 amt 0 NUM1 6522.6 7137.8 0.91 0.3608 0 month1 0 NUM2 -25856.9 5920.4 -4.37 <.0001 0 month2 0 NUM3 16307.1 5505.8 2.96 0.0031 0 month3 0 NUM4 6196.3 6060.5 1.02 0.3066 0 month4 0 NUM5 5578.6 6051.0 0.92 0.3566 0 month5 0 NUM6 2777.3 5744.2 0.48 0.6287 0 month6 0 NUM7 3490.6 6036.5 0.58 0.5631 0 month7 0 NUM8 3538.0 6079.1 0.58 0.5606 0 month8 0 NUM9 -13966.8 5499.4 -2.54 0.0111 0 month9 0 NUM10 5738.4 5910.3 0.97 0.3316 0 month10 0 NUM11 -11305.0 7131.6 -1.59 0.1129 0 month11 0 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq --------------------Autocorrelations-------------------- 6 . 0 . 0.001 -0.003 -0.007 0.004 -0.001 -0.010 12 0.94 3 0.8169 0.016 0.006 -0.009 -0.009 0.038 0.002 18 7.50 9 0.5851 -0.048 0.076 0.016 -0.054 -0.052 -0.029 24 12.23 15 0.6615 0.017 0.059 -0.048 -0.022 0.059 -0.019 30 15.66 21 0.7883 0.014 0.022 0.023 -0.002 0.079 0.000 36 24.97 27 0.5761 -0.002 0.061 -0.007 0.097 0.018 0.080 42 31.62 33 0.5358 -0.019 0.024 -0.033 -0.074 -0.075 -0.030 48 40.27 39 0.4140 0.002 -0.008 -0.097 0.051 0.024 0.072

  19. Example 2: Airline Series from Box & Jenkins Original Scale Logarithmic Scale

  20. Log Passengers (1,12) with lags at 1, 12, 23 Standard Approx Parameter Estimate Error t Value Pr > |t| Lag MU 0.0002871 0.0022107 0.13 0.8967 0 AR1,1 -0.28601 0.06862 -4.17 <.0001 1 AR1,2 -0.43072 0.07154 -6.02 <.0001 12 AR1,3 0.30157 0.07270 4.15 <.0001 23 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq --------------Autocorrelations---------------- 6 6.22 3 0.1016 -0.050 -0.066 -0.096 -0.105 0.112 0.075 12 10.23 9 0.3319 -0.009 -0.045 0.137 -0.044 0.037 -0.063 18 15.87 15 0.3909 -0.110 0.022 0.063 -0.094 0.100 0.045 24 24.25 21 0.2809 -0.154 0.002 0.007 -0.014 0.015 -0.169

  21. Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 -0.00095084 0.00309 -0.31 0.7587 filter12 1 -0.42340 0.13897 -3.05 0.0029 D1 1 0.11925 0.08262 1.44 0.1517 D12 1 0.24740 0.14225 1.74 0.0847 D23 1 0.09924 0.07566 1.31 0.1923 +--------------------------------------------------------------+ | Formulas from Economic Time Series Modeling and Seasonality | | pg. 398 (Bell, Holan McElroy eds.) | | | | s = 12 m = 12 | | Tau = -3.05 Mean = -4.1404 variance = 0.9550 | | | | Tau ~ N(-4.1404,0.9550) | | | | Z = (-3.05-(-4.1404))/sqrt(0.9550) | | Pr{Z <1.1158 } = 0.8677 | | | +--------------------------------------------------------------+

  22. Example 3: Weekly Natural Gas Supplies (Energy Information Agency) November April

  23. Lag 1 model fits well for Natural Gas Series First and Span 52 Differences Conditional Least Squares Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag MU 0.23692 2.51525 0.09 0.9250 0 AR1,1 0.39305 0.03101 12.68 <.0001 1 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq ------------------Autocorrelations----------------- 6 1.61 5 0.8997 0.007 -0.026 0.029 -0.013 -0.006 -0.003 12 12.55 11 0.3237 -0.051 0.000 0.034 0.042 0.008 0.081 18 18.20 17 0.3764 0.018 0.059 0.020 0.027 0.005 0.036 24 19.60 23 0.6656 0.003 -0.033 -0.009 -0.013 -0.010 -0.011 30 20.56 29 0.8747 0.013 0.008 0.014 0.008 -0.019 0.012 36 25.35 35 0.8847 0.023 0.016 -0.011 -0.051 0.011 -0.040 42 34.76 41 0.7432 -0.031 -0.082 -0.007 -0.042 0.009 -0.024 48 40.70 47 0.7295 0.034 0.054 -0.030 0.005 0.036 0.008

  24. Natural Gas Example – OLS Regression Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 -0.27635 1.05426 -0.26 0.7933 filter52 1 -1.04170 0.03343 -31.16 <.0001  D1 1 0.00313 0.02138 0.15 0.8838 +----------------------------------------------------------+ | | | s = 52 m = 18 | | Tau = -31.16 Mean = -8.6969 variance = 0.8877 | | | | Tau ~ N(-8.6969,0.8877) | | | | Z = (-31.16 - (-8.6969))/sqrt(0.8877) = -23.84 | | | | Pr{Z <-23.84 } = 0.0000 | | | +----------------------------------------------------------+

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