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) Y t = r Y t-s + e t, e t is White Noise Goal: Test H 0 :  =1 Y 1,1 Y 1,2 Y 1,3 Y 1,4 Y 1,5 Y 1,6 Y 1,7 Y 1,8 Y 1,9 Y 1,11 Y 1,11 Y 1,12 =Y 1,s Y 2,1 Y 2,2 Y 2,3 Y 2,4 Y 2,5 Y 2,6 Y 2,7 Y 2,8 Y 2,9 Y 2,21 Y 2,11 Y 2,12 =Y 2,s | Y m,1 Y m,2 Y m,3 Y m,4 Y m,5 Y m,6 Y m,7 Y m,8 Y m,9 Y m,21 Y m,11 Y m,12 =Y m,s Y t  Y i,j =Y month, year Y t = r Y t-s + e t  Y t – Y t-s = ( r-1) Y t-s + e i,j  Y i,j - Y i,j-1 = ( r-1) Y i,j-1 + e i,j J F M A M J J A S O N D (s=12) Yr. 1 Yr. 2 | Yr. m

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

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

5. Notation: E{N i }=N 0 E{D i }=D 0 (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) w.o.l.o.g. Assume  2 = 1  !!!

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

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

8. Approximate variance of  in seasonal means case

9. COMPARISON No Mean Model Seasonal Means Model

10. Calculation “recipe” for Seasonal Means Model (1)Regress Y t -Y t-s on seasonal dummies and Y t-s. Get  = t test for Y t-s (2)Compute (3)Compute (4)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 Unadjusted (N,D) only 1 million replicates s=12, m=6 (10 seconds run time) Reference normal variance from empirical adjustment from 3 variable Taylor:

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

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

25. Higher order models (seasonal multiplicative form) (1)(under H 0 :  =1) Regress D t = Y t -Y t-s on p lags of D  AR(p) and residual r t (2)Filter Y using AR(p) model for D. F t = filtered Y t (3)Regress r t on F t-s & p lags of D (t test on F t-s is  Suggested estimation (Dickey, Hasza, Fuller (1984)) 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.

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

27. 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 +--------------------------------------------------------------+ | 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 | | | +--------------------------------------------------------------+ F t-12 

29. 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 AR1,1 0.17664 0.05113 3.45 0.0006 12 amt 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

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

31. 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

32. 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 | | | +--------------------------------------------------------------+

33. Example 3: Weekly Natural Gas Supplies (Energy Information Agency) Weekly Lower 48 States Natural Gas Working Underground Storage (Billion Cubic Feet) April November

34. 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

35. 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 | | +----------------------------------------------------------+