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

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

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:

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

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  !!!

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

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

Approximate variance of  in seasonal means case

COMPARISON No Mean Model Seasonal Means Model

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.

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:

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:

Notes: Graphs use sample means (both expansions give same mean approximation) 3 variable Taylor variance closer to simulated statistics’ variance than is empirical adjusted if no s adjustment used. With the finite s part in the empirically adjusted formula, that formula gives 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

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

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,

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

Parameter Variable DF Estimate t Value Pr > |t| Intercept filter <.0001  D D D D D D D D | Formulas from Economic Time Series Modeling and Seasonality | | pg. 398 (Bell, Holan McElroy eds.) | | | | s = 12 m = 36 | | Tau = Mean = variance = | | | | Tau ~ N( ,0.8377) | | | | Z=( ( ))/sqrt(0.8377) | | | | Pr{Z < } = | | | F t-12 

Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag Variable Shift MU amt 0 AR1, amt 0 AR2, < amt 0 AR2, < amt 0 AR2, amt 0 AR2, amt 0 AR2, amt 0 AR2, amt 0 AR2, amt 0 AR2, amt 0 NUM month1 0 NUM < month2 0 NUM month3 0 NUM month4 0 NUM month5 0 NUM month6 0 NUM month7 0 NUM month8 0 NUM month9 0 NUM month10 0 NUM month11 0 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations

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

Log Passengers (1,12) with lags at 1, 12, 23 Standard Approx Parameter Estimate Error t Value Pr > |t| Lag MU AR1, < AR1, < AR1, < Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations

Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept filter  D D D | Formulas from Economic Time Series Modeling and Seasonality | | pg. 398 (Bell, Holan McElroy eds.) | | | | s = 12 m = 12 | | Tau = Mean = variance = | | | | Tau ~ N( ,0.9550) | | | | Z = (-3.05-( ))/sqrt(0.9550) | | Pr{Z < } = | | |

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

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 AR1, < Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations

Natural Gas Example – OLS Regression Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept filter <.0001  D | | s = 52 m = 18 | | Tau = Mean = variance = | | | Tau ~ N( ,0.8877) | | | Z = ( ( ))/sqrt(0.8877) = | | | Pr{Z < } = | |