part iii seats
DESCRIPTION
DemetraTRANSCRIPT
October 2008
NOTES ON PROGRAMS
TRAMO AND SEATS
SEATS PART
Signal Extracion in ARIMA Times Series
Agustín Maravall
Bank of Spain
2
In the remote past, unobserved components where estimated
using
Deterministic Models
xt = pt + st + ut
For example:
pt = a +bt linear trend
st = Σβi dit dummies ( t1d = 1 for January, 0 otherwise;…)
or, equivalenty, sine-cos functions
st = ΣAj cos (ωjt + Bj)
ut = white noise: niid (0, Vu)
pt st ut ut
3
− concept of deterministic: if we know the "true" parameters of the
model, the variable can be forecast with no error (ex.: tp or ts
above)
− concept of white noise : [ ]T
1ta is w.n. iff (a1….aT) ∼ niid (0, Va)
MOST COMMON OBJECTIVE OF SEASONAL ADJUSTMENT:
Better understanding of underlying (still, short-term) evolution of the series. In so far as highly transitory noise can also distort the picture, it is often helpful to look at: Trend-Cycle estimation (i.e. the SHORT-TERM TREND)
Trend and SA series
0
200000
400000
600000
800000
1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
stochastic trend stochastic SA series
4
"Short term" analysis ≡ at the most two-year horizon
When main interest is to remove:
− Seasonal noise, − (Highly transitory) irregular noise, so as to read data better in short-term policy, the remaining signal may well contain variation for cyclical frequencies. In this case, trend → "trend-cycle"
Stochastic Trend - cycle
80
85
90
95
100
105
110
115
1 10 20 30 40 50 60 70 80 90 100 110 120 130 140
stochastic trend
5
Gradual realization that seasonality evolves in time ("moving seasonality")
[An obvious and basic example: the weather,
one of the main causes of seasonality]
↓ MOVING AVERAGE METHODS
1) Fixed ("band-pass") filters
Some limitations:
* Spurious results
* Can overadjust
Can underadjust
......................
6
Spectrum of white noise
0 π
Spectrum of Seasonal Component in w.n.
0 π
Squared gain of X11 (default) filter
0 π
7
2) An alternative approach:
Use simple stochastic models to capture structure of series. (ARIMA models)
Derive optimal filter
(Signal Extraction)
THIS IS OUR APPROACH
The method permits us to jointly solve many problems of applied interest.
In the most general case:
A series contaminated by outliers, affected by regression variables,
subject to deterministic effects (TD, EE, Intervention variable, ...)
has been cleaned by TRAMO ("preadjustment").
Then the preadjusted or "linearized" series (the output of the
ARIMA model) is decomposed into components by SEATS.
8
IPI: Original Series
20
40
60
80
100
120
140
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155
IPI: Preadjustment Factors
20
40
60
80
100
120
140
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155
IPI: linear series
20
40
60
80
100
120
140
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155
9
Use of TRAMO as a PREADJUSTMENT program
Observed series
TRAMO ↓ (Interpolated series) ↓
↓ ↓ Stochastic Part:
"Linearized" Series
(output of ARIMA)
Deterministic Part:
Regression Effects
↓ SEATS AO (Outlier) ↓ TC (Outlier) Trend LS (Outlier) + Seasonal Trading Day/Leap Year + Easter Effect (Transitory) Holidays + Intervention Variables Irregular Regression Variables (can be assigned to any component) ↓ ↓ Final components
10
BASIC IDEA BEHIND THE FILTERS IN SEATS
a) NONSEASONAL SERIES b) PURELY SEASONAL
SERIES
x t = white noise S x t = w t (stationary MA)
filter for SA series: filter for SA series:
1 = ) F B, ( υ 0 = ) F B, ( υ Conclusion: SERIES WITH DIFFERENT STOCHASTIC STRUCTURES REQUIRE DIFFERENT FILTERS What SEATS does: To taylor the filter to the structure of the series (in some optimal way)
Spectrum of non seasonal series
0 π 0 π/2 π
Spectrum of seasonal series
11
The decomposition can be multiplicative: irregular x )transitory(x seasonal x cycletrend_ t =Χ in which case:
- trend-cycle gives level - others expressed as factors
(in this case, usually multiplied by 100. Thus st = 103.7 implies that the seasonal effect for month t is an increase of 3.7 percent for that month series’ value.)
or additive: irregular + ) transitory + ( seasonal + cycletrend_ = x t . Since Χ= tt log x makes multiplicative → additive (in logs), we discuss additive decomposition.
12
The decomposition is of the type: zt = pt + st + (ct) + ut pt : trend st : seasonal ct : transitory ut : irregular or
zt = nt + st
nt : seasonally adjusted series nt = pt + (ct) + ut
Assumption:
COMPONENTS ARE ORTHOGONAL
(what causes seasonal fluctuations -weather, holidays, ...- has little to do with what causes the long-term evolution of the series -productivity, technology, ... )
13
SEATS allows for the sum of the components to respect the stochastic structure of the observed series. This stochastic structure is captured with an ARIMA model. Given the ARIMA model for the observed data:
) roots unit includes ( polynomial AR" Full " : ) B (
a ) B ( = z ) B ( tt
φ
θφ
or:
a ) B ( ) B (
= z tt φθ
SEATS decomposes zt in the following manner: 1) Factorize the AR polynomial (B) φ as in: (B) x (B) x (B) = (B) csp φφφφ
where: : (B)pφ trend roots
: (B)sφ seasonal roots : (B)cφ "transitory" roots (roots are assigned according to their associated frequency) Assumption: Two different components cannot share the
same AR root. Strictly speaking, the assumption is only needed for UNIT AR
roots. But it simplifies exposition.
14
2) Express zt as:
, u + a ) B ( ) B (
+ a ) B ( ) B (
+ a ) B (
) B ( = a
) B ( ) B (
tct
c
cst
s
spt
p
pt φ
θφθ
φθ
φθ
with ut white noise. Hence, model for components are: a ) B ( =p ) B ( ptptp θφ
a ) B ( = s ) B ( st sts θφ a ) B ( = c ) B ( ct ctc θφ ut = white noise If the spectra of all components are nonnegative, the decomposition is called ADMISSIBLE
15
Example: Let the model be tt12 a)B(x)B4.1( θ=∇∇− . Then, =∇∇−=φ 12)B4.1()B(
S)B4.1( 2∇−= We know that * )B4.1( − generates stationary (highly transitory) behavior,
* ∇ (and 2∇ ) generates trends, * S generates seasonality. Thus the allocation of roots will be
,)B4.1()B(
S)B(
)B(
c
s
2p
−=φ
=φ
∇=φ
16
and the series tx is decomposed as in the “Stochastic Partial Fraction Expansion”
tttt
tctc
sts
pt2
p
tt
ucsp
uaB4.1
)B(a
S
)B(a
)B(
a)B(
)B(x
+++=
+−
θ+
θ+
∇
θ=
=φθ
=
where
,a)B(c)B4.1(
,a)B(sS
,a)B(p
ctct
stst
ptpt2
θ=−
θ=
θ=∇
and tu is white noise. All components are mutually orthogonal. Notice that components also follow ARIMA-type models and can be interpreted.
17
SEASONAL COMPONENT For a deterministic seasonal component, the sum over a year period of the component should be zero, st+ st-1 + … + st-11 = 0 (monthly data) or (1 + B + B2 + …+ B11) st = 0 . In short, if S = 1 + B + B2 + … + B11,
S st = 0
For "moving" or stochastic seasonality, this condition cannot hold for every t. (Precisely because component is moving.) But, in any case, the annual sum of st should, on average, be zero, and should not depart too much from it. Thus we may say S st = ast , where ast is w.n. , with E ast = 0 , Var (ast) = Vs relatively small; this yields a stochastic component (Harvey-Todd, 1983).
18
More generally, for the seasonal component, often: B + + B + B + 1 = S = ) B ( 1s-2
s Kφ where s = # of observations / year. Hence, a model for the seasonal of the type , w = s S tt where wt is a stationary ARMA model with: * zero mean * small variance, implies "annual aggregation of the seasonal component will on average be
zero, and will not depart too much from it”.
A Comment on Stationary Seasonal AR Roots
Assume the ARIMA model for the observed series contains the
seasonal AR factor
)B1( ssφ+ .
19
* When ks <φ ,
k = a preassigned (moderate) value (in SEATS: parameter
RMOD = .5 by default),
then factor is assigned to the transitory component.
(A small correlation whose effect disappears, in practice, after
one or two years cannot be properly called “seasonality”.)
* When ks >φ , (very rarely encountered)
the factor )B1( ssφ+ is associated with a stationary 2-year
period. It is thus assigned to the transitory component.
* When ks −<φ ,
the following identity is used
[ similar to )BB1()B1(B1 1ss −+++−=− K ] .
Let sφ denote now )( sφ− . Then,
)BBB1()B1(B1 1s1s22ss
−−ϕ++ϕ+ϕ+ϕ−=φ− K ,
where
[ ] s/1sφ=ϕ
(Ex.: →=φ 7.4 915.=ϕ
→=φ 7.12 987.=ϕ ) .
20
Then,
- the root )B1( ϕ− is assigned to the trend-cycle component.
- the roots of the polynomial
1s1s22s BBB1)B( −−ϕ++ϕ+ϕ+=ϕ K
are assigned to the seasonal component.
Thus the model for the seasonal component will in general be of the type
ststd
s a)B(sS)B( s θ=ϕ , (most often with 0=ϕ and 1d s = ), with sta a zero mean, small variance w.n. The model will be balanced (i.e.: total AR order = total MA order).
21
TREND Analogously, we may start with a deterministic trend, say pt = a + bt We know that ∇ pt = b, or ∇2 pt = 0 We cannot expect a "moving" trend to exactly satisfy the above conditions at every t. Instead, we require that departures from those conditions should, on average, cancel out, and that they should not be too large. This yields as a possible specification: ∇ pt = b + apt with E apt = 0 Var (apt) = Vp relatively small This stochastic trend specification is the well-known "random walk + drift" model. Alternatively, we could use as stochastic specification ∇2 pt = apt
22
with E apt = 0
Var (apt) = Vp relatively small
This is the so-called "second-order random walk" model. Notice that the 2 stochastic models are different:
∇ pt = b + apt implies a random shock in the slope of the trend
∇2 pt = apt implies a random shock in the change of the slope
of the trend
More generally, the specification of the stochastic trend will be of the type tt
d wpp =∇ where wt is a -zero mean -stationary ARMA process.
23
The model for the trend component can be expressed, in general, as ptpt
d
p a)B(p)B( p θ=∇φ ,
with (Maravall, 1993)
- φp (B) stationary (for example, (1--.8B)),
- d = (0), 1, 2, (3),
- θp(B) of low order,
- Var (apt) = a small fraction of Va.
The model will also be balanced.
24
In essence: SEATS finds admissible models for the components a ) B ( = p ) B ( ptptp θφ
a ) B ( = s ) B ( ststs θφ a ) B ( = c ) B ( ctctc θφ .n . w= ut such that u + c + s + p = x ttttt (Sum of component models ≡ ARIMA for observed series) At time t = T, SEATS PROVIDES - FOR t = 1,…, T, T+1,…T+FH ( FH = Forecast Horizon ) the decomposition:
u + c + s + p = x T|t T|t T|t T|t t
) x forecast ARIMAitsby replaced is x T > twhen ( |Ttt →→
where (for ex.) ) x x | s ( E = estimator MMSE = s T1t T|t K .
SEATS also provides standard error of estimators and forecasts.
25
ALLOCATION OF AR ROOTS Ex.: Quartely data Pseudospectrum:
Trend Roots
Unit AR roots at 0 = ω ( i.e., root B=1 in AR polynomial
0)1( =φ→ ).
Also:
Stationary roots for 0 =ω if large enough modulus.
Ex.: (1 - .8B) in AR polynomial.
"Large enough" = above the value of parameter RMOD
Spectrum ( Quaterly Series )
π/2 π 0
26
Seasonal roots
Seasonal frequencies:
ππ ,
2 (once-and twice-a year frequencies)
Roots at
[ ] - ,
2
εππ∈ω
ε±π∈ω
will be treated as seasonal ( : ε controlled by EPSPHI) Transitory * AR factors of the type ( 1 - .4B ) or ( 1 + .4B ) (i.e. roots for 0 = ω or πω = with small moduli, as
determined by RMOD)
* AR roots for
επε∈ω - 2
, + 0 (range of "cyclical frequency")
(i.e. between trend and first harmonic) * AR roots for "intraseasonal" frequencies * when Q > P : In this case, the SEATS decomposition yields
a pure MA ( Q - P ) component (hence transitory). Notice that, when Q > P, a transitory component will appear even when there is no AR factor allocated to it. Irregular
Always white noise
(Convenient for testing)
27
The TRANSITORY COMPONENT is always stationary, and hence
its effect is, by construction, transitory.
It will typically capture short-lived, fairly erratic behavior that is not
white noise, sometimes associated with ackward frequencies.
Its separate presence is justified by two considerations:
a) The variation it contains should not contaminate the trend or
seasonal components. Its removal permits to obtain smoother,
more stable trends or seasonals.
b) From the testing and diagnostics point of view, it is desirable to
preserve a purely white-noise irregular, computed as a residual.
However, in the final decomposition, it may be convenient to
combine the transitory and irregular components into a single
"transitory-irregular" component.
28
ALLOCATION OF AR ROOTS
Example: Quarterly data
εεεε εεεε εεεε
freq .
0 ππππ/2 ππππ
range of cyclical
frequencies
range of intraseasonal
frequencies
TO TRANSITORY TO TRANSITORY
TO TREND TO SEASONAL TO SEASONAL
29
An example:
Model for monthly observed series:
tt1232 a ) B ( = x ) B .512 + B .624 - B .78 - 1 ( θ∇∇
Regular AR polynomial factorizes as:
) B .64 + B 1.58 - 1 ( ) B .8 + 1 ( = B .512 + B .624 - B .78 - 1 232
Root of πω⇒ = ) B .8 + 1 ( 2/2 =ωπ=τ⇒ months
hence seasonal root
( 6 times-a-year frequency )
Roots of ⇒ ) B .64 + B 1.58 - 1 ( 2
complex root with modulus 8.r 2 =φ= ;
frequency ω (in radians) = = ) r 2 / ( arcos 1φ
rads .16 = ;
period months 40 2
= =ωπ
.
Thus complex root is associated with a 31/3 year
stationary cycle
⇒ to transitory component
30
Roots of ⇒∇ B - 1 = trend
Roots of ∇12 = B - 1 = 12
= ) B + + B + 1 ( ) B - 1 ( = 11K
S = ∇
⇒∇ trend
⇒S seasonal
* Grouping the roots, the series would be decomposed into:
- trend: a ) B ( = p pt pt2 θ∇
- seasonal: a ) B ( = s S ) B .8 + 1 ( st st θ
- transitory: a ) B ( = c ) B .64 + B 1.58 - 1 ( ctct2 θ
- irregular: .n . w= ut
* The AR polynomials of the models for the components are
determined.
31
DECOMPOSITION FOR THE “DEFAULT” (AIRLINE) MODEL
t121t12 a ) B + 1 ( ) B + 1 ( = x θθ∇∇
models for the components are of the type: )S( 212 ∇=∇∇
TREND-CYCLE
a ) B ( = p ptpt2 θ∇ , (A)
with order [ )B(pθ ] = 2 .
SEASONAL
a ) B ( = s ) B + + B + 1 ( stst11 θK (B)
with order [ )B(sθ ] = s - 1
(there is no transitory component)
IRREGULAR
ut ~ white noise (C)
33
SOME EXAMPLES OF MODEL SPECIFICATION (Monthly series)
A : Basic Structural Model (Harvey-Todd , 1983); ARIMA specifications.
B : ARIMA-Model-Based decomposition of Airline model (Default model TRAMO-SEATS)
C : ARIMA-Model-Based interpretation of X11 (Cleveland, 1975)
A B C
Trend Component a ) B + 1 ( = p t p t2 α∇ a ) B + 1 ( ) B + 1 ( = p t p t
2 α∇ a ) B .32 - B .30 + B .26 + 1 ( = p t p 32
t 2 ∇
Seasonal Component
a = s ) B ( S t s t a ) B ( = s ) B ( S t s st θ
11 order of ) B ( sθ
st12
t a )B .26 + 1 ( = s ) B ( S
Irregular Component
.n.w .n.w .n.w
Overall Series
a ) B ( = x t t 12 θ∇∇ ; 13 order of ) B ( θ parameters 3
a ) B + (1 B) + (1 = x t12
121t12 θθ∇∇ ; 13 order of ) B ( θ
parameters 2
a ) B ( = x t t 12 θ∇∇ ; 14 order of ) B ( θ parameters 0
See Maravall, 1985.
34
Model for SEASONALLY ADJUSTED SERIES can be obtained
by aggregation
nt = pt + ct + ut ,
For ex., for default model, since
) 2 2, (IMA .n . w+ ) 2 2, (IMA → ,
nt ~ IMA (2,2) , say
∇2 nt = θ2 (B) ant .
Typically one obtains:
θ2 (B) ≈ (1 - .9 B) (1 + α B) ,
with α of moderate size.
If (1 - .9B) cancels one ∇, the model becomes
∇ nt = ( 1 + α B) ant + k ,
with α small.
35
Hence model for SA series often is not far from the popular
"random walk + drift" model.
Remark:
Also we could aggregate the transitory and the irregular to yield a
stationary (transitory- irregular) component
vt = ct + ut
If ct is ARMA (pc, qc), then
vt is ARMA (pv, qv) with
pv = pc
qv = max (pc, qc)
However, a word of caution:
Transitory + Irreg. = Stationary deviations from SA and
detrended series
But trend is "short-term" trend
(i.e., a trend for short-term analysis)
and may contain variation for cyclical frequencies. More properly
called “trend-cycle”.
36
Ex: Quarterly data:
Thus "transitory-irregular component" in SEATS is not meant to
be interpreted as the economic "business cycle".
Note: The trend-cycle of SEATS can be decomposed in a second
run of SEATS into a "long-term trend" plus a "business cycle"
component (Kaiser and Maravall, 2001).
Trend for quaterly data
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0π/30 π/4 π/2 π
37
IDENTIFICATION PROBLEM Example:
They both yield identical aggregate. Alternatively, if tp is invertible, we can remove some noise and add it to the irregular:
old trend = new trend + w.n.
Old trend
0 π
New irregular (w.n.)
0 π
New trend
0 π
New Components
0 π
Old Components
0 π
38
In gral: Can exchange noise among invertible components. Hence: * UNDERIDENTIFICATION problem: - There are ∞ models that yield the same aggregate - They only differ in the relative allocation of white noise to the
components.
SEATS follows solution of Pierce, Box-Hillmer - Tiao, and Burman: THROW ALL WHITE NOISE TO THE (WHITE-NOISE)
IRREGULAR COMPONENT
(⇒ MAXIMIZE THE VARIANCE OF IRREGULAR)
39
CANONICAL SOLUTION
: canonical component : any other admissible decomposition
Trend-Cycle
0 π
Seasonal Component
0 π
Irregular Component
0 π
Canonical Decomposition
0 π
40
PROPERTIES OF CANONICAL DECOMPOSITION - Maximizes ) u ( Var t
- Makes other components noninvertible:
• they display a spectral zero;
• they contain a unit MA root.
Ex. Seasonal:
) B - 1 ( factor contains ) B ( 0 = ) 0 ( g ss θ⇒
Canonical Seasonal Component
0 π
41
Two important properties of the canonical decomposition (Hillmer-Tiao) (1) Let p't be the trend-cycle component in any admissible
decomposition.
It can always be decomposed as
e + p = p ttt′ , (B) where: - ⊥ are e , p tt -pt is canonical trend-cycle -et is w.n. with 0 Ve ≥ . ( p can be replaced by s or c ) Hence: For an observed ARIMA model, the canonical
decomposition provides the "cleanest" signal.
42
(2) Canonical decomposition minimizes
Var (p-innovations) in components (except for ut ) Since the p-innov. is the source of the stochastic behavior of
component,
min. Var (p-innov.) ⇒ most stable components
(compatible with observed ARIMA)
Notice that:
- if there is an admissible decomposition,
there is a canonical decomposition.
- Given any admissible decomposition, the
canonical one can be obtained trivially.
43
Remark:
Sometimes, observed ARIMA model does not accept an
admissible decomposition.
Ex: Airline (default) model
a ) B + 1 ( ) B + 1 ( = x t 12
12 1t12 θθ∇∇
for 0 > ) ( > 112 θθθ , (a case seldom found),
the spectrum of ut becomes negative
SEATS modifies the model until a reasonable decomposable
approximation is found.
44
ESTIMATION OF THE COMPONENTS In brief: Assume, first, an ∞ realization. ] x x x [ = t- ∞∞Χ KK MMSE estimator of st: ) x= x F ; B = F ( jtt
j1
+−
= ) |s( E = s tt Χ
x ) F + B ( + = tjj
j1=j
0
υυ ∑
∞
+ ) x + x ( + ) x + x ( + = 2t- 2t+ 21t- t+1 10 Kυυυ x ) F B, ( = tυ
) F B, ( ≡υ Wiener-Kolmogorov filter. - Convergent;
- Symmetric and centered;
- Adapts to the series;
45
WK FILTER Easy algorithm to obtain it: Assume we wish to estimate a signal, given by the model: φs (B) st = θs (B) ast ast ~ w.n (0, Vs) in series given by model: φ (B) xt = θ (B) at at ~ w.n (0, Va) as in xt = st + rt , rt = "rest" [Notice: φ (B) = φs (B) φr (B)] Write:
st = Ψs (B) ast ; )B()B(
)B(s
ss φ
θ=Ψ ;
xt = Ψ (B) at ; )B()B(
)B(φθ=Ψ ;
46
Then, for a doubly ∞ realization, the MMSE estimator of st is
given by the WK filter
tx
)F,B(filterWK
)F()B(
)F(s
)B(s
aV
sV
ts
444 3444 21ν=−
ΨΨ
ΨΨ= .
Thus, in order to estimate the signal, once the ARIMA model for
xt has been identified, only the model for the signal is needed.
(The other components can be ignored).
[ Note: if series is stationary, WK filter is equal to
)x(ACGF)s(ACGF
)F,B(t
t=ν ]
47
Expressing the Ψ - polynomials as functions of the θ - and φ -
polynomials, after cancelation of roots, one obtains:
)F(
)F()F()B(
)B()B(VV
)F,B( rsrs
a
s
θφθ
θφθ=ν ,
Hence, the filter is
- Symmetric
- Centered
- Convergent (invertibility of θ (B))
- Bounded
or:
WK filter to estimate st is ≡ ACGF of the ARMA model [ ] ;a)B()B(y)B( ytrst φθ=θ
=
a
syt V
V,0wna ,
a stationary model.
48
WK Seasonal Component
-0,05
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
1 12 24 36 48 60
WK Trend-cycle
-0,05
0
0,05
0,1
0,15
0,2
0,25
1 12 24 36 48 60
WK SA series
-0,4
-0,2
0
0,2
0,4
0,6
0,8
1 12 24 36 48 60
WK Irregular Component
-0,2
-0,1
0
0,1
0,2
0,3
0,4
0,5
0,6
1 12 24 36 48 60
49
Example: Estimate trend-cycle ∇2 pt = θp (B) apt in series with model ∇∇4 xt = θ(B) at , as in xt = pt + rt . We have:
2
pp
)B()B(
∇θ
=Ψ
S)B()B(
)B(2
4 ∇θ=
∇∇θ=Ψ
and
)F(
S)F(
)B(
S)B(k)F,B( pp
pp θθ
θθ
=ν
)FFF1S( 32 +++= The filter )F,B(pν is the ACF of the model
][ ytpt aS)B(y)B( θ=θ , ayt ~ w.n (0, kp)
Invertibility of )B(θ guarantees that the filter )F,B(ν is convergent in B and F.
50
CONVERGENCE of the filter implies that it can always be
truncated. Thus, for large enough series, the estimator tt x)F,B(s ν= can be assumed for the middle years of the sample. For ex., if data spans 20 years, for most series the full filter can be
assumed for the central 10 - 14 years. This estimator =ts FINAL or HISTORICAL ESTIMATOR We look next at its structure.
51
FINAL OR HISTORICAL ESTIMATOR We have: tst x)F,B(s ν= , (1) from which one obtains:
[ ] )(g)(G)(g x2
ss ωω=ω where
[ ]2ss )(~)(G ων=ω
Squared Gain of filter it determines which frequencies will contribute to the signal (that is, it filters the spectrum of the series by frecuencies).
52
Squared Gain of Filters: SA series
0
0,2
0,4
0,6
0,8
1
1,2
Squared Gain of Filters: Trend-cycle
0
0,2
0,4
0,6
0,8
1
1,2
Squared Gain of Filers: Seasonal Component
0,00
0,20
0,40
0,60
0,80
1,00
1,20
Squared Gain of Filters: Irregular Component
0,00
0,20
0,40
0,60
0,80
1,00
1,20
53
TESTING
PRESENCE / ABSENCE OF SEASONALITY;
DETERMINISTIC / STOCHASTIC SEASONALITY.
Absence or presence of seasonality:
Determined in AMI.
However:
Given that concept of seasonality somewhat implies NS, AMI in
TRAMO-SEATS is slightly biased towards seasonal
differencing.
Thus, on occasion, when a model of the type
[ ] t12
t12d a)B98.1()B(x)B( −θ=∇∇φ (D)
is obtained, it may be because of seasonal overdifferencing of
the model
µ+θ=∇φ ttd a)B(x)B( ,
a model that has no seasonality.
It can also be the result of the presence of deterministic
seasonality
∑ β+µ+θ=∇φ=
11
1iititt
d da)B(x)B( , (E)
where itd is a monthly seasonal dummy.
54
In both cases, superconsistency of 12θ will yield a value close
to -1.
To distinguish between the two cases, a simple F-test (easily
performed in TRAMO) yields good results.
(More on this issue latter.)
However ,
in the case in which there is highly stable seasonality in the
series, the stochastic specification (D) is maintained, instead of
the dummy-variable specification. Both are very close, and the
starting values lost in (D) are compensated by the 12 additional
parameters in (E) (µ plus 11 dummies). Yet (D) implicitly
allows the µ and β parameters in (E) to evolve –if need be–
very slowly, and the stable stochastic specification is likely to
outperform the dummy-seasonal specification.
Thus, no special treatment for stable (deterministic) seasonality
is needed.
It will be picked up well with the multiplicative structure
t12
t12 a)B99.1()(x)( −=∇ .
55
TESTING FOR UNDER/OVER ADJUSTMENT Underestimation of seasonality ⇒ Excess Variance in SA series Overestimation of seasonality ⇒ Variance of SA series is too small. In SEATS, the following comparison is performed. The variance of the stationary transformation of the SA series
and of the seasonal component are obtained for
- the theoretical value of the optimal estimator: )s(V t - the empirical value obtained for actual estimator: )s(V t (Bartlet’s approximation for )V(SD yields
2
1m
1j
2j0s 21
T2)V(SD
∑ ρ+γ==
).
56
Then: )s(V)s(V:H tt0 = . When V > (significantly) V ⇒ overestimation of seasonality; when
V < (significantly) V ⇒ underestimation of seasonality.
For ex.:
)010.SD(100.V
067.V
s
s
==
=
⇒ “EVIDENCE OF OVERESTIMATION OF SEASONALITY”.
57
ANOTHER REPRESENTATION OF INTEREST :
THE ESTIMATOR AS A FILTER APPLIED TO THE
INNOVATIONS IN THE OBSERVED SERIES
tst x)F,B(s ν=
using xt = ta)B()B(
φθ
⇒
[ ] tstst a)F,B(a)B()B(
)F,B(s ξ=
φθν=
)F,B(sξ = “PSIE-weights” (easy to obtain: Maravall, 1994) +ξ+ξ++ξ+=ξ 01
jjs BB)F,B( KK
applies to prior and concurrent innovations KK +ξ++ξ+ −−
jj1 FF
applies to "future" innovations (posterior to t) ξj = contribution of at-j to ts ξ-j = contribution of at+j to ts
58
Note ξξξξs (B,F) is: asymmetric non-convergent in B (unless series is stationary) convergent in F (always)
PSIE-WEIGHTS: SA series
-0,6
-0,4
-0,2
0
0,20,4
0,6
0,8
1
1,2
-24-21-18-15-12-9-6-303691215182124
PSIE-WEIGHTS: Trend-cycle
-0,2
0
0,2
0,4
0,6
0,8
1
1,2
-24-21-18-15-12-9-6-303691215182124
PSIE-WEIGHTS: Seasonal Component
-0,30
-0,20
-0,10
0,00
0,10
0,20
0,30
0,40
0,50
-24-21-18-15-12-9-6-303691215182124
59
As we shall see later, the filter is important to analyse revisions and convergence of the estimator, as well as SE of preliminary estimators . Example: ∇2 pt =θp (B) apt : SIGNAL ∇∇4 xt =θ (B) at : SERIES Then: tpt x)F,B(p ν= ,
ppp
p k)F(
S)F(
)B(
S)B()F,B(
θθ
θθ
=ν ,
where kp = Vp / Va , S = 1 + F + F2 + F3 . Thus
=∇θν= t2pt a
S)B(
)F,B(p
t
p
2
p
p a)F(
S)F()B(k
θθ
∇θ
=
part in B part in F Notice:
't
ppt
2 a)F(
S)F()B(p
θ
θθ=∇
with a2p
't Vk)a(Var = . Somewhat different from model for
SIGNAL above.
60
Previous remark brings a point of general interest: MODEL FOR COMPONENT versus MODEL FOR ESTIMATOR
We have
MODEL FOR SERIES
tt a)B(x)B( θ=φ (“observed”)
MODEL FOR COMPONENT (two components)
[ ]) B ( ) B ( ) B ( , nsx nsttt φφ=φ+=
a ) B ( = s) B ( stss t θφ
MMSE estimator for ts (doubtly infinite realization):
tnsns
st x)F(
)F()F(
)B(
)B()B(ks
θ
φθ
θ
φθ=
it is found:
MODEL FOR ESTIMATOR
't
nssts a
)F(
)F()F()B(s)B(
θ
φθθ=φ
)aka( ts't =
61
Comparison of the model for the component with that of the
estimator shows the effects induced by the estimation filter.
Stationary
transformation
Stationary model
Part in B Part in F
Component
Estimator
=φ ts s)B(
=φ ts s)B(
)B(sθ
)B(sθ )F(
)F()F( ns
θ
φθ
Component and estimator share
− the stationarity-inducing transformation (in particular,
the differencing)
− the (stationary) part in B
Difference: estimator includes a part in F (reflecting the 2-sided
character of the filter). This part is a convergent polynomial in F.
Component and estimator will have different ACF and spectrum.
The different model structures of component and estimator have
some implications of applied relevance.
62
One implication
It is (close to) standard practice to build models on seasonally
adjusted data.
This is based on the belief that, by removing seasonality, model
dimensions can be reduced.
This belief is wrong. Example: DEFAULT MODEL a ) B + 1 ( ) B + 1 ( = x t
12121t 12 θθ∇∇
Decomposes into:
θ
θ∇
a ) B ( =s S
a ) B ( = n
st st
nt nt2
The model for the estimator of the SA series has ACF of model:
= n ) B + 1 ( ) B + 1 ( t
212121 ∇θθ
a S ) B ( ) B ( = t nn θθ , an ARIMA (13,2,15) model.
63
Set, for instance, .4- = 1θ .6- = 12θ . The MA expansions (or ψ -weights) of the stationary transformation of - the original series - the seasonally adjusted series, are the following:
64
LAG ORIGINAL SERIES SA-SERIES (ESTIM.)
0 1 1.00 1 - 0.4 -1.33 2 - 0.38 3 - - 4 - - 5 - - 6 - - 7 - - 8 - - 9 - -
10 - - 11 - - 12 - 0.6 -0.40 13 0.24 0.53 14 - -0.15 15 - - 16 - 0.37 17 - 0.15 18 - 0.06 19 - 0.02 20 - 0.01 21 - - 22 - - 23 - - 24 - -0.24 25 - 0.32 26 - -0.09 27 - - 28 - - 29 - - 30 - - 31 - - 32 - - 33 - - 34 - - 35 - - 36 - -0.14 37 - 0.19 38 - -0.05
Hence: * Model for SA series: MORE COMPLEX
* No reduction in dimension if SA series is used.
65
This is a reason to avoid modelling SA series ↓ they will have coefficients for seasonal and large lags.
A second implication: Broadly, the difference between theoretical component and
estimator is the following
Component: a ) B ( = s ) B ( t s sts θφ
Estimator: a ) F ( ) B ( = s ) B ( tssts αθφ
Difference: )F(
)F()F() F ( ns
s θ
φθ=α
When )F(sθ or )F(nφ contain unit roots, given that these
roots will appear in the MA part of the estimator, the estimator will
not be invertible.
When tn (what is removed in order to obtain ts ) is NON-
STATIONARY → )B(nφ will contain unit roots.
66
Example: Default (Airline) model.
Component models:
a ) B ( = n nt2t2 θ∇
a ) B ( = s S st11t θ
Thus S)B(s =φ .
S ≡ 11 unit root.
Therefore, tn will be NI because of these unit roots.
Recall: the presence of seasonality (in general) ⇒ unit AR
roots in model for seasonal.
67
Consequence:
In gral, for tn
* No convergent AR representation (nor VAR representation)
exists.
AVOID USING AR MODELS TO MODEL SA SERIES
68
General result:
n + m = x t t t
m t ⇒ is Noninvertible if
tn is Nonstationary (Maravall, 1995)
Hence in a standard
trend + seasonal + irregular
decomposition, with NS trend and NS seasonality, all three :
u and , s , p t t t will be NI.
Noninvertibility of the estimators
(and hence previous implications)
is a fairly general property of SA and detrending methods
(including X11)
69
Another important applied result:
We saw ) ( g ] ) ( G [ = ) ( g x
2uu ωωω ,
where u denotes now any of the components, and
) ( g
) ( g = ) ( G
x
uu ω
ωω ≡ Gain of filter .
Thus
) ( g ) ( g
) ( g = ) ( g u
x
uu ω
ω
ωω
Since 1 ) ( g
) ( g
x
u ≤
ω
ω,
for all components:
spectrum of component ≥ spectrum of estimator.
In particular, for the stationary transformation, When some other component is present, the estimator will
always underestimate the stochastic variance of the component
(bias towards “stability”).
)estimator(Var.)comp(Var ≥ .
70
The loss of variance counterpart is the appearance of
crosscovariances between components’ estimators. (As shall be
seen later, these crosscovariances can also be modelled.)
In a particular application, to see if the empirical estimates agree
with the model, their variances and ACF should be compared to
those of the model for the estimator, not to those of the model for
the component.
71
ACF OF TREND: Theoretical Component
-0,60
-0,50
-0,40
-0,30
-0,20
-0,10
0,00
0,10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
ACF OF TREND: Theoretical MMSE Estimator
-0,4
-0,3-0,2
-0,1
00,1
0,20,30,4
0,5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
ACF OF TREND: Empirical Estimate
-0,4-0,3-0,2-0,1
00,10,20,30,40,50,6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
72
JOINT DISTRIBUTION OF THE ESTIMATORS
From the models for the estimators, the variances, ACFs,
spectra, and so on, can be obtained for their stationary
transformations (ST).
We can further obtain the
(THEORETICAL) CROSS-COVARIANCE FUNCTION
between any pair of estimators (ST).
Therefore:
- Given our Normality assumption, the joint distribution of the (ST
of the) estimators is known;
- it is fully determined by the “observed” ARIMA;
- this knowledge permits us to devise simple tests having to do
with issues related to the decomposition obtained in a particular
application.
73
The properties of the estimators have been derived for the case
of an ∞ realization.
Since WK-filter is convergent (in B and in F), in practice it could
be approx. with a finite (2-sided) filter.
Estimators that are obtained with the full filter:
FINAL (OR HISTORICAL) ESTIMATOR
The time it takes for the filter to converge depends on:
- stochastic structure of the series,
- stochastic structure of the component.
Ex. Monthly series with seasonality.
For many, 3 years of revisions are enough.
( Most are completed in 5 years )
Assuming 3 years, if series has 180 observations,
for the central 108 the component estimator could be
considered final.
74
FINITE SERIES ; PRELIMINARY ESTIMATION
Observed series: [x1, x2, …, xT]
Consider WK filter to estimate SA series ( tn )
To obtain T|t 1
n there is no problem: filter has converged to that
for final estimator.
WK SA series
-0,2
0
0,2
0,4
0,6
0,8
1
-k 0 k
Final Estimator and End-Points Problem
-0,5
0
0,5
1
1,5
2
2,5
3
3,5
4
t1 t2 T
75
However, the filter cannot be used to obtain T|t 2n because
convergence of the filter requires future observations, not yet
available.
(same problem near the beginning of the series)
way to proceed (Preliminary Estimators)
1) EXTEND SERIES WITH FORECASTS AND
BACKCASTS (ARIMA ONES)
2) APPLY FILTER TO EXTENDED SERIES
(Cleveland and Tiao, JASA, 1976)
In this way, a PRELIMINARY ESTIMATOR is obtained.
Preliminary Estimator
-0,5
0
0,5
1
1,5
2
2,5
3
3,5
4
T forecasts
76
As new observations become available, it will be revised, until the
final estimator is obtained.
The most important preliminary estimator: T|Tn
Forecasts of n are obtained in the same way as preliminary
estimators, simply by extending the series further.
Concurrent Estimator
-0,5
0
0,5
1
1,5
2
2,5
3
3,5
4
T
Forecast
-0,5
0
0,5
1
1,5
2
2,5
3
3,5
4
77
In SEATS, for monthly data:
24-months-ahead forecasts are computed for the series and
components.
In general, if MQ is the number of observations per year, the
number of forecasts computed is max (2MQ, 8).
FINAL ESTIMATOR:
=ν= tst x)F,B(s
= … + ν1 xt-1 + ν0 xt + ν1 xt+1 + …
PRELIMINARY ESTIMATOR:
Obtained by replacing observations not yet available with
forecasts.
Let
T|ts : Estimator of st when last observation is xT
78
For example:
=−1t|ts … + ν1 xt-1 + ν0 1t|tx − + ν1 1t|1tx −+ + …
↑ ↑
1 p.a.f. 2 p.a.f.
obtained as
ARIMA forecasts
In summary, for finite realization:
[x1 , x2, …, xT]
Preliminary Estimator:
x ) F B, ( = s eT|t sT|t υ
= xe
T|t [backcasts, observations, forecasts]
i.e., series extended with forecasts and backcasts
No need for long extensions:
Burman-Wilson algorithm: only a few forecasts and backcasts
are needed. (Typically, about 2 years)
Note: Preliminary estimator will imply an asymmetric filter, and
will be subject thus to a phase effect.
79
As new observations become available:
s T|t → s 1T+|t → ... s kT+|t → ...
the estimator of s t is revised.
As ∞→ k , (in practice, “large enough”)
→s kT+|t s t (the "final" or "historical" estimator)
(In practice,
st ≡ Historical or Final Estimator is valid for
central years of the series)
80
STRUCTURE OF THE SA SERIES AVAILABLE AT TIME T:
For a particular realization ] x , , x , x [ T21 K ,
what we have is a sequence of estimators:
... s T|100T- ... s |TjT- ... s T|T
s T|100T- ≡ FINAL EST. = s 100T-
s |TjT- ≡ PRELIMINARY EST. (j covering a few years)
s T|T ≡ CONCURRENT EST.
Each one of these estimators is the output of a different model.
(Each j ⇒ a different model.)
Therefore, SA series is a mixture of realizations with different
underlying models.
Thus: SA series available at some point in time is nonlinear
(≅ time-varying parameters model).
* Another reason to avoid using SA in modeling.
81
PRELIMINARY ESTIMATORS AND REVISIONS
Preliminary estimators and revisions are implied by the use of
TWO-SIDED FILTER
KK + x + x + x + x + = s 2t+ 2t+1 1t01t- 1t υυυυ
Two-sided filter is
- necessary to avoid phase effects;
- implied by MMSE ("optimal") estimation in model-based
approach.
Starting with concurrent estimator:
Observations: ] x , , x [ t1 K
KK +x + x + x + x + = s t | 3t+ 3t | 2t+ 2t | 1t+ 1t0t|t υυυυ
1 p.a. forecast 2 p.a. forecast
Notice that:
Given that all forecasts are l.c. of jtx − (j = 0, 1, 2, …), t|ts is
implicitely obtained with a one-sided filter.
82
New observation )x ( 1t+ arrives.
New revised estimator (1-period revision)
KK + x +x + x + x + = s 1t+ | 3t+ 31t+ | 2t+ 21t+ 1t01t+ | t υυυυ
new observation
updated forecast
Likewise, when 2tx + becomes available, the 2-period revision of the concurrent estimator will be given by
KK + x +x +x +x + x + = s 2t|3t32t21t1t01t12t+ | t ++++− υυυυυ
and so on.
Of course, to revise series is always disturbing and an
inconvenience.
But it is due to the fact that knowledge of the future helps us to
understand the present ( a very basic fact of life! ).
To suppress revisions is
- to ignore relevant information,
- to distort our measurements.
83
Revisions:
1-period revision:
.)n.w(a
)xx()xx(ssr
1t1
t|2t1t|2t2t|1t1t1t|t1t|t)1(
t|t
=ξ=
=+−ν+−ν=−=
+
++++++ K
( 1ξ = a constant) .
2-period revision:
)1(MA
aa
)xx(
)xx()xx(ssr
2t21t1
t|3t2t|3t3
t|2t2t2t|1t1t1t|t2t|t)2(
t|t
=
ξ+ξ=
=+−ν+
+−ν+−ν=−=
++
+++
+++++
K
…………………………………..
k-period revision:
)1k(MA
aassr ktk1t1t|tkt|t)k(
t|t
−=
=ξ++ξ=−= +++ K
84
For the full revision in the concurrent estimator :
s - s = r t|t tt + ) x - x ( = t|1t+ 1t+ 1υ
K + ) x - x ( + t | 2t+ 2t+ 2υ
K + (2) e + (1) e = t2t1 υυ
where:
) j ( et : j-th-period-ahead forecast error of the series
[ 1tt a)1(e +=
1t12tt aa)2(e ++ ψ+=
…………………………… ].
85
Hence
∑∞
=υ=
1jtjt )j(er
depends on:
- forecast errors
- weights of the WK filter
Thus:
- interest in "small" forecast errors ( X11 → X11 ARIMA)
- but revision still depends on the s j ′υ ,
WHICH DEPEND, in turn, ON THE STOCHASTIC
STRUCTURE OF THE SERIES
( i.e., the ARIMA model ).
For some series, the revision can be large;
for other series, they may be small.
Also, for some series the revision will last long;
for others it will disappear fast.
86
THUS, FOR A GIVEN SERIES, THERE IS AN
APPROPRIATE AMOUNT OF REVISION.
THE REVISION SHOULD NOT BE LARGER THAN THAT,
NOR SHOULD IT BE SMALLER.
Two features of the revision process are of relevance:
- the size of the revision
- the duration of the revision process.
Often one finds there is a trade-off between them.
87
ERROR IN THE ESTIMATOR OF A COMPONENT
In the context of Seasonal Adjustment, concern with the error
made when measuring seasonality has been periodically
expressed (Bach et al. 1976; Moore et al. 1981; Bank of
England 1992). This need is especially left for key variables that
are (explicitly or implicitly) being subject to some type of
targeting (e.g., a monetary aggregate or a consumer price
index). In these cases, intrayear monitoring and policy reaction
is based on the SA series (e.g., see Maravall 1988).
We are concerned with the precision of the
* concurrent estimator and forecasts
* first revisions
* final estimator
* some rates of growth.
The associated MSE are straightforward to obtain.
From the previous discussion, it is clear that the error will be different for
a) final estimator, ts
b) preliminary estimator, K,2,1k;s kt|t ±±=+ ,
(which also includes forecasts).
88
Total estimation error in the estimator T|ts
T|ttT|t ss −=ε ,
it can be expressed as the sum of the two errors,
)ss()ss( T|ttttT|t −+−=ε ,
where the first error
ttt sse −=
is the error in the final estimator , and the second error
T|ttT|t ssr −=
denotes the revision in the estimator T|ts .
* te and T|tr are orthogonal (Pierce 1979)
thus, for example, )r(V)e(V)(V T|ttT|t +=ε .
89
REVISION ERROR: Size and Convergence
Express, as before, component as filter of innovations in series:
x ) F B, ( = s tst υ
a ) B ( ) B (
) F B, ( = ts φθ
υ ,
or
a ) F B, ( = s tst ξ
* Divergent in B
* Convergent in F
Write:
a ) F ( + a ) B ( = s 1t+ +
st -
st ξξ
The filter F)B, ( sξ can be easily computed (Maravall, Journal of
Forecasting, 94).
90
For a concurrent estimator:
a ) B ( t-
sξ : Effect of starting conditions and present and
past innovations in series.
a ) F ( 1t++
sξ : Effect of future innovations.
Taking conditional expectations at time t ,
a ) B ( = s t-
st|t ξ
the revision is given by rt = s - s t|t t ; or
,
a zero-mean convergent one-sided (stationary) MA.
a ) F ( = r 1t+ +
st ξ
91
HISTORICAL (OR FINAL) ESTIMATION ERROR
Because of its stochastic nature, the historical estimator s t contains an estimation error
)ss(e ttt −=
* "unobservable"
* finite variance
* can derive distribution.
In particular, te has ACF and spectrum of the model (Pierce,
80)
[ ] etnst a)B()B(e)B( θθ=θ ,
eta ∼ wn
a
ns
V
VV,0
92
ERROR ANALYSIS: SOME APPLICATIONS
A. From knowledge of the models for the different types of
estimation errors, one can build standard
TESTS FOR THE SIGNIFICANCE OF SEASONALITY
in a particular application, such as, for example,
0s:H 0 = ,
where s is a vector of estimators with known covariance matrix.
Notice that it may be possible to detect significant seasonality
with the final estimator, yet the forecasts of the seasonal
component for the next year may be worthless.
B. Proper intrayear monitoring of the economy is greatly
facilitated.
For example: Assume an increase in unemployment of 10000
persons in last month, as measured with (the concurrent
estimator of) the SA series. We can easily test for whether this
increase is significantly different from zero.
Naturally, economic policies based on some (explicit or implicit)
annual (or biannual, …) targeting, where short-term control is
typically based on the SA series, the variance of the estimation
error of seasonality can be used to build confidence intervals
around the estimated SA series . In this way, the question “are
we growing too much?” (or “too little”) can be answered in a
more rigorous manner.
93
Historical Estimation Error: never known.
The best we can do: Historical Estimator.
Hence, from applied point of view, perhaps CI should only
consider the Revision Error.
(“How far can I be from my eventual best measurement?”)
C. USE IN MODEL ESPECIFICATION
The possibility of deriving properties of the components can be
of help in the choice of the proper model.
It is often the case that several ARIMA specifications seem
about equally acceptable from the fitting and out-of-sample
forecasting criteria. In these cases, one can look at the
decompositions implied by the “sample equivalent” models, and
select the one that offers the most appealing decomposition.
Some important criteria that can be used in the comparison are
the following:
• Stability of the components
One may wish to remove a seasonal component as stable as
possible. Thus one may seek the decomposition with
min [ ])a(Var st .
• Precission of the estimator
(better detection of seasonality…)
• SMALLER REVISIONS
(and fast convergence) .
(examples in Bank of Spain web site).
94
AMB used as “fixed filter”: SEATS by default (RSA = 0). A remark on the DEFAULT MODEL "Airline Model" (Box-Jenkins, 1970)
µθθ∇∇ + a ) B +1 ( ) B +1 ( = x t12
121t12 The annual difference of the monthly growth (rate-of-growth if in
logs) is a stationary process, with constant variance
* Parameters have "structural” interpretation θ1 → stability of trend θ12 → stability of seasonal ( values close to -1 produce stable components ) σa → overall unpredictability * Often found * Displays very well-behaved filters * Can encompass many models (in a fairly robust way)
95
Ex.: - deterministic trend: 9.9- 1 →θ - deterministic seasonal: 9.9- 12 →θ - even white noise ! θ1( and )9.9- 12 →θ • No need for the dilema: Deterministic vs. Stochastic. • Overdifferencing does little damage. • Good idempotency properties. • Good for pretesting.