Seasonal Unit Root Tests in Long Periodicity Cases

D. A. DickeyYing Zhang

Natural gas-a colorless, odorless, gaseous hydrocarbon-may be stored in a number of different ways. It is most commonly held in inventory underground under pressure in three types of facilities. These are: (1) depleted reservoirs in oil and/or gas fields, (2) aquifers, and (3) salt cavern formations. (Natural gas is also stored in liquid form in above-ground tanks).

1. Regression with Time Series Errors Y(t) = a + bt + seasonal effects + Z(t), Z(t) a stationary time series

Seasonal effects: Sinusoids,Seasonal dummy variables

2. Dynamic Seasonal ModelsY(t) = Y(t-d) + e(t) copy of last season Y(t) = Y(t-d) + e(t) – e(t-d) EWMA of past seasons

Y(t) = Y(t-1) + [Y(t-d)-Y(t-d-1)] + Z(t) Z(t) = e(t) PROC CUT&PASTE; Z(t) = e(t) – e(t-1) – e(t-d) + e(t-d-1)

“airline”

Z(t) = (1-B)(1-Bd) e(t)

Y(t) = Y(t-1) + [Y(t-d)-Y(t-d-1)] + e(t)

Y(t) = 10 + t + 8X3 – 8X5 -5X8 – 5X9 – 5X10+e(t)

Summary:

1.Both models can give same predictions for pure trend + seasonal functions.

2.For data, lag model looks back 1 year and ignores (or discounts) others. Good for slowly changing seasonality.

3.For data, dummy variable model weights all years equally. Good for very regular seasonality. 4. Differences in forecast errors too!

Weekly natural gas data – unit root forecast

Weekly natural gas data – seasonal dummy variable forecast

A general seasonal model:

Yt –f(t) = Yt-d –f(t-d)) + et

f(t) = deterministic components

H0:

Under H0, period d functions annihilated.

f periodic Yt –Yt-d = (Yt-d –f(t-d)) + et

=1 Yt –Yt-d = et

ˆ( )m d 1

1

1 2 2( 1) 1 ( 1) ( 1

1) 1

1 1

((1/ ) / ([ ]) )d m m

d i s d i s d is i

d

ss

i

m Y de m Yd

Y1=e1 (Y1,1)

Y2=e2

(Y1,2)Y3=e3

(Y1,3)Y4=e4

(Y1,4)

Y5=e5+e1

(Y2,1)Y6=e6+e2

(Y2,2)Y7=e7+e3

(Y2,3)Y8=e8+e4

(Y2,4)

Use double subscripts: Quarterly (d=4) example, m years, f(t)=0

12 2

0( )sD W t dt

2 ( ) ( )

( 1)

sN W

f

W t d t

i

4t t tY Y e

Numerator is

Denominator is

Known unit root facts (2=1):

(1) Moments (d=1 case or individual terms)

E{Ns} = 0, E{Ds} = (m-1)/(2m)1/2 Var{Ns} = (m-1)/(2m)1/2 Var{Ds} = (m-1)(m2-m+1)/(3m3)1/3 Cov{Ns, Ds} = (m-1)(m-2)/(3m2) 1/3

(2) Studentized statistic asymptotically equivalent to (numerator sum) / (denominator sum)1/2

1

/d

ss

d NN d

1

/d

ss

dD D

Basic idea is simple: Large d numerator approximately normalLarge d denominator converges to E{denominator}

( 1) 11

1 ( )1

1(1/ ) (0 )/ ,

2[ ]

d m

d i s d i si

D

s

Y ed Nm

2 2( 1) 1

1

1

1

[ ] /1

2

m

d i si

dP

s

d Y m

0ˆ( ) (0,1 2) ( )Dratio m d N H

:

/ (0,1)D

t statistic

numerator denominator N

Nice proof Grandpa!

As you see, I’m very excited.

( ,1, 1)d N Y

( ,2, 1)d N Y

( ,4, 1)d N Y

d=4 and N(0,1)

-1.645 0 1.645

CDFs

(SAS)

-2.386 0 2.386

d=4 md1/2(-1) and N(0,2)

CDFs

Improving the Normal Approximation:

JASA paper (Dickey, Hasza, Fuller, 1984 ) gives limitdistribution for studentized statistic (d=12)

5th %ile = -1.8095th %ile = 1.5250th %ile: -0.14 (Note: (1.52-1.80)/2 = -0.14 !!)

Difference: 1.52+1.80 = 3.32, 2(1.645) = 3.29 (close !!) Suggestion: shift by median

CLT limit distribution median is 0.

Median as function of seasonality d: 1. Get medians for d=2, 4,12 from DHF 2. Plot median vs. d-1/2 (d=2,4,12,limit)

1/ d

Median as function of seasonality d:

Regress median on d-1/2

Slope very close to ½, Intercept very close to 0.

Median Shifts and Tau Percentiles.

d med -1/(2 ) p01 p025 p05 p10

d 2 -0.35 -0.35355 -2.67990 -2.31352 -1.99841 -1.63510

4 -0.24 -0.25000 -2.57635 -2.20996 -1.89485 -1.53155

12 -0.14 -0.14434 -2.47069 -2.10430 -1.78919 -1.42589

inf 0.00 0 -2.32685 -1.96046 -1.64535 -1.28205

2 2N

D

:

1 numerator terms, mean ~ ( ) ( 0, 1/ 2 )

2 denominator terms, mean ~ ?(1/ 2,1/ 2 )

( , ) 1/ (3 )

i

i

d

Taylor

d dd

d

Cov N D d

N

D

/ 0 2 0 2 ( 1/ 2) + remainderdN D dN dN D

{ / } 0 0 2 cov( , ) = ( 2 / 3 )

0

.

could

4714 /

use 1 2 ) /(

E dN D d

d

N D d d

Simulation Evidence

• m= 100, various d values• 2 sets of 40,000 t statistics at each (m,d)• e.g. d=365 and m=100, (daily data 100 years)

– 36500x40000 = 1.46 billion generated data points.– SAS: only 10 minutes run time !– Overlay percentiles (adjusted t) on N(0,1)– Duplicates almost exactly the same.

Simulation Evidence - Detrending

• m= 20, d =4, 6, 12, 24, 52, 96, 168, 365• 96 quarter hours/day, 168 hours/week• Detrending:

– None– Constant, linear, quadratic– Period d sinusoids (fundamental & harmonic)

• Sets of 20,000 t statistics at each (m,d).

20 years of weekly data, 20,000 simulated series TAU

20 years of weekly data, 20,000 simulated series TAU

20 years of weekly data, 20,000 simulated series TAU

20 years of weekly data, 20,000 simulated series TAU

20 years of weekly data, 20,000 simulated series TAU

20 years of weekly data, 20,000 simulated series TAU

Standard tau percentiles for various adjustments

Three replicates (of 20,000) per d value

Conclusions:

Spread between percentiles about constant (and close to N(0,1) spread)

Medians smooth function of 1/sqrt(d)

Degree of detrending matters

Cubic smoothing regression plotted with raw %iles.

4,

1/ 0.5

d

d

Claim: As d infinity, Tau N(0,1) for all of these forms of detrendingSeasonal random walk Z, data Y. Y = X + Z Detrend by OLS:

1 1( ( ' ) ') ( ( ' ) ')R PY I X X X X Y I X X X X Z Seasonal Random Walk has d “channels” of m values

Denominator is sum of d quadratic formsWithout detrending each has eigenvalues

1

2 24 sin ( )2(2 1)

O mm

1( ' ) 'X X X Xcan be written as

'T T'T T I

k = rank of X matrixMiddle matrix is diagonal.Projection =>

k diagonal entries 1 rest 0Denominator quadratic form contains

'T T

1' ( ' ) ' ' 'Z X X X X Z Z T T Z

k times maximum eigenvalue = O(km2)Upper probability bound on unnormalized quadratic form.Normalization is m2d so k/d0 suffices for

no limit effect of detrending.Same for numerator, estimator, tau statistic.

2(1/ 3 / 2) /k dBased on Taylor series (for large m) adjustment is

for regression adjustments with k columns selected from intercept and Fourier sines and cosines.

Your talk seems better now Grandpa!

Focus on Medians:

4,

1/ 0.5

d

d

Focus on Medians:

Allowing for augmenting terms, as in seasonal multiplicative model, follows the same proof as in DHF.

Natural gas data: Procedure

(1) Compute residuals (trend + harmonics)(2) AR(2) fit to span 52 differences of residuals(3) Filter with AR(2)

Ft = filtered seriesWt = span 52 differences Ft – Ft-52

(4) Regress Wt on Ft-52 Wt-1 Wt-2

The REG Procedure

Dependent Variable: Diff

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 3 718362 239454 231.53 <.0001

Error 679 702233 1034.21632

Corrected Total 682 1420595

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 -0.68125 1.23111 -0.55 0.5802

L52FY 1 -0.99746 0.03800 -26.25 <.0001

Diff1 1 0.01417 0.00777 1.82 0.0686

Diff2 1 -0.01152 0.00730 -1.58 0.1151

Follow up:

Lag 52 coefficient near -1 suggests 52-1 near -1

Perhaps no lag correlation in the presence of sinusoids

Fit ARIMAX model as a check (AR(2), no seasonal lag): Standard ApproxParameter Estimate Error t Value Pr > |t| Lag Variable

MU 727.58194 684.44164 1.06 0.2878 0 total AR1,1 1.37442 0.03379 40.67 <.0001 1 total AR1,2 -0.38964 0.03381 -11.53 <.0001 2 total NUM1 0.09520 0.04525 2.10 0.0354 0 date NUM2 -883.25146 23.18237 -38.10 <.0001 0 s1 NUM3 240.92573 23.05715 10.45 <.0001 0 c1 NUM4 -133.27021 11.51098 -11.58 <.0001 0 s2 NUM5 122.42419 11.53277 10.62 <.0001 0 c2

Lack of fit? Box-Ljung test on residuals

Autocorrelation Check of Residuals

To Chi- Pr >Lag Square DF ChiSq -------------Autocorrelations------------ 6 1.40 4 0.8449 0.008 -0.012 0.001 -0.000 -0.023 0.033 12 18.66 10 0.0448 -0.086 0.034 0.089 -0.009 0.017 0.077 18 23.67 16 0.0970 0.022 0.002 0.025 0.012 0.047 0.055 24 26.61 22 0.2263 -0.014 -0.037 0.022 -0.027 -0.028 -0.017 30 29.61 28 0.3821 0.010 0.036 0.042 -0.012 -0.021 0.012 36 33.03 34 0.5150 0.001 0.030 -0.027 -0.031 0.042 -0.010 42 46.84 40 0.2122 -0.026 -0.081 -0.035 -0.034 0.078 -0.042 48 51.65 46 0.2625 0.011 0.042 -0.044 -0.027 0.036 0.014 54 65.50 52 0.0989 -0.055 0.037 -0.024 -0.008 0.085 -0.070 60 75.05 58 0.0654 -0.096 0.023 -0.027 -0.002 -0.029 0.022 66 80.14 64 0.0838 -0.006 -0.035 -0.053 -0.030 -0.035 -0.009 72 85.28 70 0.1033 -0.060 -0.017 0.034 0.032 -0.007 0.011 78 87.52 76 0.1724 -0.034 -0.012 -0.026 -0.004 -0.027 -0.001 84 91.06 82 0.2312 0.018 -0.029 -0.011 -0.050 0.010 0.017 90 96.17 88 0.2586 0.000 -0.030 -0.048 0.049 0.006 -0.018 96 107.69 94 0.1582 -0.011 -0.053 0.006 -0.020 -0.066 -0.075102 117.16 100 0.1158 0.082 -0.059 -0.013 0.018 0.016 -0.003108 137.48 106 0.0215 -0.021 -0.058 0.044 0.021 -0.067 -0.112

Lag 104, 52

AR(2) characteristic polynomial m2 - 1.37442 m + 0.38964 (m=1/B)

QUESTIONS?

D. A. Dickey

D.A.D.

QUESTIONS?

D. A. Dickey

D.A.D.

OK, we’re outa here!

Following up – No adjustment add 1.0 /(2d1/2) = (1/2 + 0(2/3) )/d1/2

Polynomial add 2.3/(2d1/2) ≈ (1/2 + 1(2/3) )/d1/2

Sine (fund.) add 5.0 /(2d1/2) ≈ (1/2 + 3(2/3) )/d1/2

+ harmonic add 7.6 /(2d1/2) ≈ (1/2 + 5(2/3) )/d1/2

Sine + linear about the same as sine Generated 3 sets of pctles (20,000 reps) for both models Sorted on d and 5th percentile Result: percentiles interspersed (see below) Moral: Use same adjustments for sine, sine + linear.

-----------------20,000 reps per line------------ d=52 -----------------------

trend t_1 t_2_5 t_5 t_10 t_25 t_50 t_75 t_90 t_95 t_97_5 t_99 n

harmonic -2.95 -2.58 -2.24 -1.86 -1.23 -0.53 0.16 0.78 1.14 1.48 1.85 1040Harmonic -2.93 -2.54 -2.23 -1.84 -1.22 -0.54 0.15 0.78 1.15 1.46 1.86 1040harmonic -2.94 -2.55 -2.21 -1.85 -1.21 -0.52 0.17 0.77 1.16 1.50 1.88 1040

sine wave -2.75 -2.36 -2.03 -1.66 -1.04 -0.35 0.34 0.95 1.34 1.65 2.03 1040sine wave -2.73 -2.34 -2.03 -1.65 -1.03 -0.34 0.34 0.96 1.34 1.67 2.05 1040lin&sine -2.73 -2.35 -2.03 -1.66 -1.03 -0.34 0.34 0.97 1.34 1.66 2.01 1040sine wave -2.69 -2.35 -2.01 -1.64 -1.03 -0.34 0.34 0.95 1.31 1.65 2.03 1040lin&sine -2.71 -2.33 -1.98 -1.62 -1.01 -0.33 0.35 0.98 1.35 1.65 2.04 1040 lin&sine -2.71 -2.33 -1.98 -1.62 -1.01 -0.33 0.35 0.98 1.35 1.65 2.04 1040

mean -2.49 -2.15 -1.83 -1.47 -0.84 -0.17 0.52 1.13 1.48 1.80 2.16 1040mean -2.52 -2.16 -1.83 -1.46 -0.84 -0.16 0.53 1.16 1.52 1.84 2.21 1040linear -2.51 -2.13 -1.82 -1.45 -0.81 -0.15 0.54 1.16 1.51 1.82 2.18 1040quadratic -2.49 -2.13 -1.82 -1.45 -0.84 -0.15 0.52 1.12 1.48 1.80 2.22 1040 linear -2.53 -2.14 -1.81 -1.45 -0.83 -0.15 0.54 1.15 1.53 1.87 2.22 1040quadratic -2.53 -2.12 -1.80 -1.42 -0.82 -0.14 0.53 1.13 1.49 1.83 2.19 1040mean -2.44 -2.09 -1.79 -1.44 -0.83 -0.16 0.52 1.13 1.50 1.84 2.25 1040quadratic -2.50 -2.10 -1.78 -1.43 -0.84 -0.15 0.52 1.14 1.51 1.83 2.18 1040linear -2.52 -2.10 -1.78 -1.42 -0.83 -0.15 0.53 1.16 1.52 1.85 2.22 1040

none -2.38 -2.05 -1.73 -1.36 -0.75 -0.07 0.62 1.23 1.60 1.90 2.25 1040 none -2.46 -2.07 -1.73 -1.36 -0.75 -0.07 0.61 1.22 1.61 1.93 2.31 1040 none -2.43 -2.04 -1.73 -1.37 -0.75 -0.07 0.62 1.23 1.59 1.90 2.27 1040