BAYESIAN UNIVARIATE MODELING
OF SEASONAL TIME SERIES:
CANOVA PRIOR
Enrique M. Quilis
Macroeconomic Research Department Ministry of Economy and Finance. Spain.
CONTENTS
Standard model: unit roots.
Litterman prior: non-seasonal unit root.
Canova prior: seasonal unit roots and zero
frequency unit root.
frequency unit root.
UNIT ROOTS: STANDARD MODEL
6 7 8 9 10
1 2 3 4 5
G
a
UNIT ROOTS: STANDARD MODEL
Unit root at w=0
frequency period = ∝.
Stochastic trend.
)
B
1
(
−
=
∇
))
B
(
U
(
4
=
∇
∇
∇∇
Stochastic trend.
Represented by the
Litterman prior (random walk + drift).
)
B
1
(
−
UNIT ROOTS:
∇
=
(
1
−
B
)
6 7 8 9 10
1 2 3 4 5
G
a
UNIT ROOTS: LITTERMAN PRIOR
)
B
1
(
−
=
∇
Mean centered around a random walk with
drift:
t
t
u
z
)
B
1
LITTERMAN PRIOR: Mean
µ
=
=
φ
0
*
=
=
φ
=
φ
else
0
1
h
1
LITTERMAN PRIOR: Variance
µ
=
θ
σ
=
µ φ)
(
)
(
v
)
V
(
diag
2
∀
σ
θ
θ
=
φ
φh
)
)
(
g
(
)
(
LITTERMAN PRIOR: Variance
Temporal decay
θ
≤
θ
≤
=
1
h
1
0
1
g
∞
<
θ
≤
=
θ
−
1
h
0
h
g
LITTERMAN PRIOR: Variance
Temporal decay
0.6 0.7 0.8 0.9 1 Geometric 0.1 0.5 0.9 0.6 0.7 0.8 0.9 1 Harmonic 0.1 0.5 0.91 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 Lag
LITTERMAN PRIOR: Variance
0.6 0.8 1 1.2
Prior on phi parameters: th1=0.1 th2=0.5
1 1.5 2
Prior on phi parameters: th1=1 th2=0.5
0 1 2 3 4 5 6 7 -0.2
0 0.2 0.4 0.6
0 1 2 3 4 5 6 7 -1
UNIT ROOTS: CANOVA PRIOR
)
B
1
(
−
=
∇
)
B
(
U
4
=
∇
∇
)
B
B
B
1
(
)
B
(
U
=
+
+
2+
3Unit root at w=0
frequency period = ∝.
Stochastic trend. Represented by stochastic constraints
)
B
1
(
−
=
∇
Seasonal unit roots at
w=π/2 and w=π
frequencies period = 4q and 2q.
Stochastic seasonality. Represented by
)
B
B
B
1
(
)
B
(
SEASONAL UNIT ROOTS: CANOVA PRIOR
)
B
1
)(
B
1
(
)
B
B
B
1
(
)
B
(
U
=
+
+
2+
3=
+
2+
Unit root at w=π/2
frequency period = 4q=1y.
Stochastic annual
seasonality.
Unit root at w=π
frequency period = 2q=0.5y.
Stochastic semiannual
SEASONAL UNIT ROOTS:
(
1
+
B
2)
6 7 8 9 10
0 1 2
0 1 2 3 4 5
Cycle per year
G
a
SEASONAL UNIT ROOTS:
(
1
+
B
)
6 7 8 9 10
1 2 3 4 5
G
a
UNIT ROOTS: CANOVA PRIOR
6 7 8 9 10
0 1 2
0 1 2 3 4 5
Cycle per year
G
a
BASIC MODEL: AR(p)
s
p
u
z
z
z
z
t 1 t 1 1 t 2 p t p t≥
+
φ
+
+
φ
+
φ
+
µ
=
− −L
−BASIC MODEL: AR(p): Spectrum
π
φ
−
=
∑
=
2
v
)
w
cos(
1
1
)
w
(
f
p a1 h
h
Unit roots at critical frequencies wc={0,π/2, π} mean
an infinity peak at them: f(wc)=∝.
Hence, the denominator should be zero the AR
BASIC MODEL: AR(p): Constraints.
}
,
2
/
,
0
{
w
1
)
w
cos(
cp
1 h
h
c
φ
=
=
π
π
∑
=
Unit roots at critical frequencies wc={0,π/2, π} mean
an infinity peak at them: f(wc)=∝.
Hence, the denominator should be zero linear
combinations of the AR parameters should be constraint to add up to one.
BASIC MODEL: AR(p): Constraints.
0 2 4 6 8 10 12
-1 0 1
w=0
1
w=pi/2
0 2 4 6 8 10 12
-1 0
0 2 4 6 8 10 12
-1 0 1
BASIC MODEL: AR(p): Constraints.
We consider this prior information in a
stochastic (i.e., inexact or approximate) way,
adding a disturbance to the constraints.
The disturbance reflects:
Approximation errors due to the selection of a
finite order AR polynomial.
finite order AR polynomial.
It is not generally known at which frequency a
peak will appear.
BASIC MODEL: AR(p): Constraints.
+
=
φ
)
Q
,
0
(
N
~
e
e
r
R
The prior considers anhyperparameter that represents the global
tightness of the prior and a set of variances to take
θ
=
π πv
0
0
0
v
0
0
0
v
Q
/20 c
a set of variances to take into account the
uncertainties concerning the peak at each of the unit root frequencies:
trend, annual seasonality and semiannual
FULL CANOVA PRIOR
We have two prior structures:
Litterman prior: representation of a unit root at
the zero frequency (stochastic trend) by means of
a prior centered around a random walk with drift.
Partial Canova prior: unit roots at critical
frequencies (trend, annual seasonality and
frequencies (trend, annual seasonality and
semiannual seasonality) are introduced using
stochastic constraints on the AR parameters.
We combine both sources of prior information
FULL CANOVA PRIOR: Hyperparameters
We have two set of hyperparameters:
Litterman prior: global tightness, decay, and
tightness on intercept.
Partial Canova prior: global tightness of the prior
FULL CANOVA PRIOR
(
V
R
'
Q
R
) (
V
R
'
Q
r
)
~
1 1 1 −1 ∗ −1φ −
− −
φ
+
φ
+
=
φ
(
1 1)
1~
V
R
'
Q
R
~
− − −φ
φ
=
+
Σ
φ=
(
V
φ+
R
'
Q
R
)
Σ
)
~
,
~
(
N
~
~
C
φ
Σ
APPLICATION
Box-Jenkins Airline data.
Quarterly totals: s=4.
Log-transformed.
AIRLINE DATA: Logs
6.8 7 7.2 7.4 7.6
Q1-48 Q1-50 Q1-52 Q1-54 Q1-56 Q1-58 Q1-60 Q1-62
AIRLINE DATA: Logperiodogram
6 8 10 12
Periodogram
lo
g
(p
o
w
e
r)
0 2 4
lo
g
(p
o
w
e
FULL CANOVA PRIOR (Canova, table 1)
p --> 12
Canova
Global
Tightness 0 π/2 π
4.00 0.90 2.00 3.00
Litterman
Global
BAR ESTIMATION
0 2 4 6 8 10 12
-2 0 2
AR estimation
2
PRIOR
0 2 4 6 8 10 12
-2 0
0 1
AR ESTIMATION
5 6 7 8
GAIN FUNCTION
0 0.5 1
0 1 2 3 4
BAR ESTIMATION
5 6 7 8
GAIN FUNCTION
REFERENCES
CANOVA, F. (1992) "An alternative approach
to modeling and forecasting seasonal time
series",
Journal of Business and Economic
Statistics
, vol. 10, n. 1, p. 97-108.
