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 bookEconomic Time Series: Modeling and Seasonality
edited by Bell, Holan, and McElroypublished by CRC press, 2012
Model: Seasonal AR(1)
Yt = r Yt-s + et, et is White Noise
Goal: Test H0: r=1
2ˆ /t s t t sY e Yr r
Y1,1 Y1,2 Y1,3 Y1,4 Y1,5 Y1,6 Y1,7 Y1,8 Y1,9 Y1,11 Y1,11 Y1,12=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 Yi,j=Ymonth, year
Yt = r Yt-s + et Yt – Yt-s = (r-1) Yt-s + ei,j Yi,j - Yi,j-1 = (r-1) Yi,j-1 + ei,j
J F M A M J J A S O N D (s=12)Yr. 1Yr. 2 |
Yr. m
21,
11, ,
1 121 1
ˆ: 1 / /m
j
s m
j i
s s s
i ij i i
ji i
j i i
OL Y NS De Yr
22
2 ( 1)( 2) / 3( 1)( 2) / 3( 1) / 2 ( 1)( 1) / 3
0 ( 1~
/,
) / 2/i
i
m m m m m mDm m m
m m mm mN
Yt = r Yt-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:
(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) MSEs2
n.b.: Does not apply to seasonal dummy variables
Previous work:
2, 1
1, 1 ,
21 1 1 1
ˆ: / /m
i
s s s s
i i ij i
j
m
ii j i j i
j
Y e NOL YS Dr r
22
2 0 1/ ( 2) / 3(( 1) / 2
~( 1) / 32) / 3
, 1//
2i
i
mm m m
Nm
mDm
m
2, 1 ,11 1 1
,2 1
, 1 ,( )ˆ: / /m m
i j i ij
i j i i
s s s s
i i ii
j ij YY Y e NS DL YO r r
*****Add seasonal dummy variables:*****
2
2 2
6
6
~( 2)/ (2 4 9)6 9
6( 2)/ 122
0
, 2i
i
m m
mm mD mN
m m
m
m
Known mean 0
0 0,N D
0 0,N D
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)
2 2,
1 2
2 2
/ ( 2)(
ˆ2)
s m
i j ii j
Ns mmD
e NeD
s
2
222
20
0
ˆ 0( 2) 2
ps
NN only for mean casem D m D
w.o.l.o.g. Assume 2 = 1
2, 1 ,11 1 1
,2 1
, 1 ,( )ˆ: / /m m
i j i ij
i j i i
s s s s
i i ii
j ij YY Y e NS DL YO r r
!!!
t statistic, seasonal means model:
Standard error [(X’X)-1(MSE)]1/2 = / ( )MSE sD
2
2/ ( )ˆ
2
sND N
MSE sD NDm D
322 2
0 02
2
1 3( 2)ˆ( , , ) , , ( 1)2 2 3
ˆ2
ps
N mf N D f N DmND
m D
No Mean Seasonal Means
0
0
22
( 2)6
mN
m mD
2ˆ( , , ) 0psf N D
Taylor Series, seasonal means :2
20
0/~
0,
/N NDi
ND Di
V VND V V
ND
20
222 0
00
ˆ, ,
1ˆ( 2)2
sNNs g N D RNN DD
m Dm D
(N0=(m-2)/2<0)
22
1/2
21 3 ( 2) 1ˆ, ,2 2 3
ˆ ps ms g Nm sN
Om
D D
202
2
32
00
0
ˆ(1)
ˆ, ,
1 12 2
1
g N
A s N NND
B s sm
D D
D
C
020
2
32
00
1 12
ˆ(1 12
) DB s
mCDN
NsA s N D
2
2
22 4 9
90
16 ( 2)
ˆ, , ( 2)6 6( 2)1
( 2)
6
6( 2)22
12
i i
mm
m mCov N D mm
m
m m
m
mm
m
m
0
230
2 8(1)3 2
D mAN m
220
1 4(1)2
BN m
020
23 2
D mCN m
3 3 212 13
21 23 3
31 32 3
1
2
3
2
1
0
02
64 168 108 )1 14 2 80 1.(1)
5
V Vm m m
V Vm mV V
D B V BV A
NC
VA
C
Approximate variance of in seasonal means case
3 2
3
64 168 108 ){ } 0.8
80 1.5 m
m m mVar
m
COMPARISONNo Mean Model Seasonal Means Model
2 2{ }3 ( 1)
Es
mm m
3 2
{ }2 3s m
Em
{ } 1Var
3 2
3
64 168 108 ){ }
80 1.5
m m mVar
m
Calculation “recipe” for Seasonal Means Model
(1) Regress Yt-Yt-s on seasonal dummies and Yt-s. Get = t test for Yt-s
(2) Compute
(3) Compute
(4) Compare to N(0,1) to get p-value.
3 2
{ }2 3s m
Em
( )
EZ
Var
Alternative approach: Expand around (N, D) only, run large (1/2 million) simulation fixup for small m.
Result for variance:
2
3 3/2
0.8777 464 48 .4705 1 2.2215 0.08531.
108
80 1.5 5 1.51.5m msm
m m m
m
Similar empirical adjustments to mean:
1 0.2118 0.05050.37041.5
3 22 3s mm m ss
Compare limit (sinfinity) variance formulas: Taylor 3 variable versus Taylor 2 variable with and without adjustments
Unadjusted (N,D) only1 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 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
Histogram Stats (reps = 1,000,000)
2 2
4 6 0.34 0.65 2.59 2.51 1.39 1.296 6 0.26 0.35 3.06 2.99 1.31 1.2512 6 0.18 0.15 4.16 4.12 1.23 1.2024 6 0.13 0.07 5.77 5.74 1.19 1.1752 6 0.09 0.03 8.40 8.38 1.17 1.14
formula formulas m skew kurt
2 2
4 6 0.34 0.65 2.59 2.51 1.39 1.294 8 0.19 0.33 2.60 2.54 1.16 1.114 10 0.11 0.22 2.60 2.56 1.05 1.034 20 0.02 0.08 2.61 2.59 0.90 0.894 100 0.10 0.04 2.62 2.62 0.80 0.79
formula formulas m skew kurt
4 1000 0.12 0.04 2.62 2.62 0.78 0.76
(formulas from book)
Higher order models (seasonal multiplicative form)
(1 )( )( ( )) ( ( ))
st t
t t
B Y ZB Z e AR pr
(1) (under H0:r=1) Regress Dt = Yt-Yt-s on p lags of D AR(p) and residual rt
(2) Filter Y using AR(p) model for D. Ft = filtered Yt (3) Regress rt on Ft-s & p lags of D (t test on Ft-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.
( ( ))(1 )( )st tB B Y e r
Example 1: Oil Imports (from book, s=12, m=36)
Levels
First Differences& Seasonal Means
Parameter Variable DF Estimate t Value Pr > |t|
Intercept 1 -163.62422 -0.20 0.8442filter12 1 -0.81594 -16.19 <.0001 D1 1 0.01853 0.48 0.6349D2 1 -0.00511 -0.11 0.9128D3 1 0.00016148 0.00 0.9974D4 1 0.02891 0.58 0.5617D5 1 0.02369 0.48 0.6344D6 1 0.00623 0.13 0.8998D7 1 -0.01025 -0.22 0.8268D8 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 | | | +--------------------------------------------------------------+
Ft-12
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
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 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
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 | | | +--------------------------------------------------------------+
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 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
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 || |+----------------------------------------------------------+
