Transcript Determining Seasonality: A Comparison of Diagnostics from X-12-ARIMA Demetra Lytras Roxanne Feldpausch William Bell Disclaimer This report is released to inform interested parties of ongoing research and to.

Determining Seasonality: A
Comparison of Diagnostics from
X-12-ARIMA
Demetra Lytras
Roxanne Feldpausch
William Bell
Disclaimer
This report is released to inform
interested parties of ongoing research
and to encourage discussion of work in
progress. Any views expressed on
statistical, methodological, technical, or
operational issues are those of the
authors and not necessarily those of the
U.S. Census Bureau.
Purpose
To evaluate the properties of
diagnostics available in X-12-ARIMA for
detecting seasonality (to determine if a
series is a candidate for seasonal
adjustment)
Overview
•
•
•
•
•
Definition of diagnostics
Nonseasonal series
Fixed seasonal effects models
Airline models
Conclusions
Tests for Seasonality
• D8 F test for stable seasonality
• M7
• Spectrum of the differenced
transformed original series
• c2 test for fixed seasonal effects
– Extension to Fmb test (not in X-12-ARIMA)
Notation
Assume monthly data, changes for
quarterly data are clear
D8 F Test Assuming Stable
Seasonality
H0: m1=m2=…=m12
H1: mp  mq for at least one pair (p,q)
where
m1,…,m12 are the monthly means of the
seasonal-irregular (SI) component (the
detrended series)
D8 F Assumptions
• The SI ratios are independently
distributed as N(mi,2)
• Problems: Estimated SI ratios are
actually dependent and heteroscedastic
(higher variance near the ends)
• Traditionally attempted solution: use 7
as critical value
M7
M7 

1 7
2 Fs

3 Fm
Fs

where
Fs = D8 F statistic for stable seasonality
Fm = D8 F statistic for moving seasonality
Note: M7 < 1, series is seasonal
Spectrum of the Differenced
Transformed Original Series
• To determine seasonality, look for
peaks at seasonal frequencies 1/12,
2/12, 3/12, and 4/12
• A peak of six or more “stars” is
considered seasonal where one star is
1/52nd of the spectral range in decibels
• Used default start, estimated spectrum
based on last 8 years of data
c2 Test for Fixed Seasonal Effects
• Fit a regARIMA model with
– Fixed seasonal effects
– Nonseasonal ARIMA model
• Use results of the c2 test for fixed
seasonal effect regression coefficients
Fixed Seasonal Effects
Regressors
M 1,t
 1 in January

  1 in Decem ber,...,
 0 otherwise

M 11,t
 1 in Novem ber

  1 in Decem ber
 0 otherwise

c2 Test for Fixed Seasonal Effects
1 ˆ
ˆ
ˆ
cˆ   '[Var( ) ]
2
where ˆ is the vector of fixed seasonal
effect regression parameters
2
c
Compare to 11 (.05)  19.7
c2 Test for Fixed Seasonal Effects
Assumptions
• c2 distribution (under H0) holds exactly
only if the ARMA parameters and the
innovation variance 2 are known
• Problem: Need to estimate the parameters
• Attempted solution: use model-based F test
to correct for the estimation of 2
– Still need to estimate ARMA parameters
Estimates of 2
1
2
ˆ
ˆ
ML :
 
at

nd t
1
2
2
~
ˆ
" Unbiased":  
at

nd r t
where
aˆt  residual from fit t edmodel
2
n  number of observations
d  order of differencing
r  number of regression paramet ers
Extension to Model-based F test
for Fixed Seasonal Effects
2
ˆ
c nd r
Fmb 

~ Fk ,nd r
k
nd
where
2
ˆ
c is the chi-squared statistic from X-12-ARIMA
k is the number of fixed seasonal effects
regressors (k=11 for monthly data)
r is the total number of regression variables
Methods
• Simulate nonseasonal series to
determine significance levels of the
diagnostics
• Simulate seasonal series to determine
the power of the diagnostics
Methods - Nonseasonal Series
Simulated 10,000 monthly series with a
length of 20 years for each of the
following models
ARIMA (0 1 0)
ARIMA (0 1 1), with  = 0.3, 0.5, and 0.8
ARIMA (1 1 0), with  = 0.3, 0.5, and 0.8
X-12-ARIMA Settings
• Model: Correct ARIMA model
(estimated parameters) + seasonal
regressors
• Forecasts: 2 years
• Adjustment Type: additive
Percent of Nonseasonal Series Detected as
Seasonal
Model
(0 1 1)
(1 1 0)
(0 1 0)

0.3
0.5
0.8
0.3
0.5
0.8
D8F
0.1
0.0
0.0
6.3
11.2
16.8
1.2
M7
0.9
0.2
0.0
8.2
10.6
10.9
3.1
c2
Fmb
Spect
Peaks 5% test 5% test
18.9
7.9
5.4
14.4
7.6
5.4
8.9
7.7
5.3
21.3
7.7
5.5
17.9
7.9
5.4
10.4
7.8
5.3
23.5
7.5
5.1
Critical Values of the Diagnostics
for a 5% Test
Spect
Model  D8 F M7 Peaks
Original cutoff
7.00 1.00
6.00
(0 1 1) 0.3 3.337 1.22
12.7
0.5 2.712 1.35
10.8
0.8 2.145 1.51
7.9
(1 1 0) 0.3 7.542 0.92
13.7
0.5 9.736 0.86
11.6
0.8 12.15 0.85
8.1
(0 1 0)
4.854 1.07
13.9
Methods – Seasonal Series
• Simulated series with
– Fixed seasonal effects
– Airline series
• Applied seasonality diagnostics
– D8 F, M7, Spectrum – used size adjusted
critical value
– Fmb
Fixed Seasonal Effect Series
Added fixed seasonal effects based on
two real series to the nonseasonal
simulated series
Seasonal Factors
Fixed Seasonal Effects
• Simulated 1,000 series from each of
the following 36 models
– Two sets of base seasonal factors
– Rescaling of base seasonal factors: small,
medium, and large (compared to the
irregular)
– Six (0 1 1) and (1 1 0) nonseasonal
models
Small Seasonal Variation: Percent of Fixed
Seasonal Effect Series Detected as Seasonal
Using Size Adjusted Critical Values
Model 
(0 1 1) 0.3
0.5
0.8
(1 1 0) 0.3
0.5
0.8
D8 F
87.2
94.8
96.5
36.7
24.6
15.6
Fmb
Spectrum
M7
Peaks
5% Test
85.6
20.2
84.3
94.8
22.7
92.3
95.9
25.8
95.7
33.4
10.9
61.3
24.0
12.2
58.6
16.3
12.3
69.9
Medium Seasonal Variation: Percent of Fixed
Seasonal Effect Series Detected as Seasonal
Using Size Adjusted Critical Values
Model 
(0 1 1) 0.3
0.5
0.8
(1 1 0) 0.3
0.5
0.8
D8 F
99.5
100.0
100.0
67.7
43.6
30.0
Fmb
Spectrum
M7
Peaks
5% Test
99.0
36.3
99.2
100.0
43.6
99.9
100.0
49.4
100.0
64.9
18.8
93.3
41.5
20.4
91.4
29.6
21.7
96.1
Methods – Airline Series
•
Simulated 1,000 series from each of the
following 48 models
–
–
–
–
Seasonal  = 0.6 and 0.9
Nonseasonal  = 0.3 and 0.8
Length of 10 and 20 years
Starting values:
i.
ii.
–
Zeros
One of two sets of values based on real series
Innovation variance: 1 and a smaller number
Results – Airline Series
• Fmb test found 99 - 100% of the series
seasonal
• M7, D8 F and spectrum peaks found
89.2-100% of the series seasonal
Conclusions - D8 F, M7 and
Spectrum Peaks
• Significance levels vary greatly
depending on the model
• Power is equal or lower than that of the
Fmb test for fixed seasonal effects
Conclusions – c2 and Fmb Tests for
Fixed Seasonal Effects
• The c2 test was slightly oversized
• This is corrected by Fmb, whose
significance levels are consistently
close to the stated level of the test
• Fmb has higher power than the M7, D8
F and spectrum peaks for most models
Contact Information
[email protected]
[email protected]
[email protected]