Rob J Hyndman
Functional time series
with applications in demography
4. Connections, extensions and applications
Outline
1 Yield curves
2 Electricity prices
3 Dynamic updating with partially observed
functions
4 Functional ARH models
5 References
Functional time series with applications in demography
4. Connections, extensions and applications
2
Example: Yield curves
10
5
Yield
15
US Treasury Bonds
0
50
100
150
Functional time series with applications in demography
200
250
300
4. Connections, extensions and applications
350
3
Example: Yield curves
US Treasury bonds
8
Jan 1989
6
Jan 1991
Jan
Jan 1995
1993
Jan 1987
Jan 1997
Jan 2001
Jan 1999
4
Bid−ask midpoint average
10
Jan 1985
0
100
Functional time series with applications in demography
200
300
4. Connections, extensions and applications
400
3
Example: Yield curves
9
10
11
Jan 1985
Jan 1980
8
Jan 1990
Jan 1975
Jan
Jan 1970
1995
7
Bid−ask midpoint average
12
US Treasury bonds
6
Jan 2000
0
100
Functional time series with applications in demography
200
300
4. Connections, extensions and applications
400
3
Model for yield curves
PC3
0.15
PC2
50
150
250
350
0.05
0
50
150
250
350
0
50
150
250
Term
Term
Term
B1
B2
B3
4
350
−8
−50
−10
−4
0
0
0
5
50
2
100
10 15
0
−0.05
0.060
−0.20
0.064
−0.10
0.068
0.00
0.072
PC1
1970
1980
1990
2000
1970
1980
Month
Functional time series with applications in demography
1990
Month
2000
1970
1980
1990
2000
Month
4. Connections, extensions and applications
4
Forecasts of coefficients
ARIMA forecasts (with first differencing). 80%
prediction intervals shown in yellow.
Forecasts from ARIMA(4,1,0)
Forecasts from ARIMA(1,1,2)
1970
1980
1990
2000
0
1970
1980
Month
Functional time series with applications in demography
1990
Month
2000
−8 −6 −4 −2
B3
B2
−100
−10
−50
0
0
B1
10
2
50
4
20
100
Forecasts from ARIMA(0,1,3)
1970
1980
1990
2000
Month
4. Connections, extensions and applications
5
2 year forecasts for yield curves
4
3
2
1
Bid−ask midpoint average
5
Yield curve forecasts: 2003−2004
0
50
100
150
200
250
300
350
Term
Functional time series with applications in demography
4. Connections, extensions and applications
6
Nelson-Siegel models
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel model
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t e−λt x + rt (x)
Well-behaved at long maturities.
λt is usually fixed and pre-specified.
Useful for describing yield curves, and for estimating
yield for unobserved x.
Coefficients highly collinear making interpretation
and forecasting difficult.
Functional time series with applications in demography
4. Connections, extensions and applications
7
Nelson-Siegel models
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel model
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t e−λt x + rt (x)
Well-behaved at long maturities.
λt is usually fixed and pre-specified.
Useful for describing yield curves, and for estimating
yield for unobserved x.
Coefficients highly collinear making interpretation
and forecasting difficult.
Functional time series with applications in demography
4. Connections, extensions and applications
7
Nelson-Siegel models
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel model
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t e−λt x + rt (x)
Well-behaved at long maturities.
λt is usually fixed and pre-specified.
Useful for describing yield curves, and for estimating
yield for unobserved x.
Coefficients highly collinear making interpretation
and forecasting difficult.
Functional time series with applications in demography
4. Connections, extensions and applications
7
Nelson-Siegel models
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel model
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t e−λt x + rt (x)
Well-behaved at long maturities.
λt is usually fixed and pre-specified.
Useful for describing yield curves, and for estimating
yield for unobserved x.
Coefficients highly collinear making interpretation
and forecasting difficult.
Functional time series with applications in demography
4. Connections, extensions and applications
7
Nelson-Siegel models
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel model
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t e−λt x + rt (x)
Well-behaved at long maturities.
λt is usually fixed and pre-specified.
Useful for describing yield curves, and for estimating
yield for unobserved x.
Coefficients highly collinear making interpretation
and forecasting difficult.
Functional time series with applications in demography
4. Connections, extensions and applications
7
Nelson-Siegel models
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel model
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t e−λt x + rt (x)
Well-behaved at long maturities.
λt is usually fixed and pre-specified.
Useful for describing yield curves, and for estimating
yield for unobserved x.
Coefficients highly collinear making interpretation
and forecasting difficult.
Functional time series with applications in demography
4. Connections, extensions and applications
7
Diebold-Li model
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel reparameterization
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t
1−e−λt x
λt x
− e−λt x
+ rt (x)
This parameterization has lower collinearity than the
original.
Provides interpretable coefficients (level, slope,
curvature)
All coefficients usually forecast with univariate AR(1)
models.
Functional time series with applications in demography
4. Connections, extensions and applications
8
Diebold-Li model
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel reparameterization
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t
1−e−λt x
λt x
− e−λt x
+ rt (x)
This parameterization has lower collinearity than the
original.
Provides interpretable coefficients (level, slope,
curvature)
All coefficients usually forecast with univariate AR(1)
models.
Functional time series with applications in demography
4. Connections, extensions and applications
8
Diebold-Li model
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel reparameterization
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t
1−e−λt x
λt x
− e−λt x
+ rt (x)
This parameterization has lower collinearity than the
original.
Provides interpretable coefficients (level, slope,
curvature)
All coefficients usually forecast with univariate AR(1)
models.
Functional time series with applications in demography
4. Connections, extensions and applications
8
Diebold-Li model
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel reparameterization
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t
1−e−λt x
λt x
− e−λt x
+ rt (x)
This parameterization has lower collinearity than the
original.
Provides interpretable coefficients (level, slope,
curvature)
All coefficients usually forecast with univariate AR(1)
models.
Functional time series with applications in demography
4. Connections, extensions and applications
8
Diebold-Li model
Functional time series model
yt,x = µ(x) +
K
X
βt,k φk (x) + rt (x) + εt,x
k =1
Nelson-Siegel reparameterization
yt,x = β1,t + β2,t
1−e−λt x
λt x
+ β3,t
1−e−λt x
λt x
− e−λt x
+ rt (x)
This parameterization has lower collinearity than the
original.
Provides interpretable coefficients (level, slope,
curvature)
All coefficients usually forecast with univariate AR(1)
models.
Functional time series with applications in demography
4. Connections, extensions and applications
8
Diebold-Li model
Functional time series with applications in demography
4. Connections, extensions and applications
9
Diebold-Li model
x=
Functional time series with applications in demography
4. Connections, extensions and applications
10
Diebold-Li model
Functional time series with applications in demography
4. Connections, extensions and applications
11
Diebold-Li model
Functional time series with applications in demography
4. Connections, extensions and applications
11
Bowsher-Meeks model
Functional time series model
yt,x = st (x) + σt (x)εt,x ,
st (x) = µ(x) +
K
X
βt,k φk (x) + rt (x)
k =1
Bowsher-Meeks model
y t ,x =
J
X
γt,j ξj (x) + σt (x)εt,x
j=1
{ξj (x)} are spline terms;
∆γt+1 = A(Bγt − µ) + Ψ∆γt + et is a cointegrated VAR
with et ∼ N(0, Ω).
Functional time series with applications in demography
4. Connections, extensions and applications
12
Bowsher-Meeks model
Functional time series model
yt,x = st (x) + σt (x)εt,x ,
st (x) = µ(x) +
K
X
βt,k φk (x) + rt (x)
k =1
Bowsher-Meeks model
y t ,x =
J
X
γt,j ξj (x) + σt (x)εt,x
j=1
{ξj (x)} are spline terms;
∆γt+1 = A(Bγt − µ) + Ψ∆γt + et is a cointegrated VAR
with et ∼ N(0, Ω).
Functional time series with applications in demography
4. Connections, extensions and applications
12
Bowsher-Meeks model
Functional time series model
yt,x = st (x) + σt (x)εt,x ,
st (x) = µ(x) +
K
X
βt,k φk (x) + rt (x)
k =1
Bowsher-Meeks model
y t ,x =
J
X
γt,j ξj (x) + σt (x)εt,x
j=1
{ξj (x)} are spline terms;
∆γt+1 = A(Bγt − µ) + Ψ∆γt + et is a cointegrated VAR
with et ∼ N(0, Ω).
Functional time series with applications in demography
4. Connections, extensions and applications
12
Bowsher-Meeks model
Functional time series model
yt,x = st (x) + σt (x)εt,x ,
st (x) = µ(x) +
K
X
βt,k φk (x) + rt (x)
k =1
Bowsher-Meeks model
y t ,x =
J
X
γt,j ξj (x) + σt (x)εt,x
j=1
{ξj (x)} are spline terms;
∆γt+1 = A(Bγt − µ) + Ψ∆γt + et is a cointegrated VAR
with et ∼ N(0, Ω).
Functional time series with applications in demography
4. Connections, extensions and applications
12
Hays-Shen-Huang model
Functional time series model
yt,x = st (x) + σt (x)εt,x ,
st (x) = µ(x) +
K
X
βt,k φk (x) + rt (x)
k =1
Functional dynamic factor model
st (x) =
K
X
βt,k φk (x) + rt (x)
k =1
p
β t ,k − µ k =
X
ψr ,k (βt−r ,k − µk ) + vt,k
r =1
Equations estimated simultaneously using penalized
likelihood (analogous to PCA but taking account of
autocorrelations).
Functional time series with applications in demography
4. Connections, extensions and applications
13
Hays-Shen-Huang model
Functional time series model
yt,x = st (x) + σt (x)εt,x ,
st (x) = µ(x) +
K
X
βt,k φk (x) + rt (x)
k =1
Functional dynamic factor model
st (x) =
K
X
βt,k φk (x) + rt (x)
k =1
p
β t ,k − µ k =
X
ψr ,k (βt−r ,k − µk ) + vt,k
r =1
Equations estimated simultaneously using penalized
likelihood (analogous to PCA but taking account of
autocorrelations).
Functional time series with applications in demography
4. Connections, extensions and applications
13
Hays-Shen-Huang model
Functional time series model
yt,x = st (x) + σt (x)εt,x ,
st (x) = µ(x) +
K
X
βt,k φk (x) + rt (x)
k =1
Functional dynamic factor model
st (x) =
K
X
βt,k φk (x) + rt (x)
k =1
p
β t ,k − µ k =
X
ψr ,k (βt−r ,k − µk ) + vt,k
r =1
Equations estimated simultaneously using penalized
likelihood (analogous to PCA but taking account of
autocorrelations).
Functional time series with applications in demography
4. Connections, extensions and applications
13
Hays-Shen-Huang model
Functional time series with applications in demography
4. Connections, extensions and applications
14
Compare FPCA
PC3
0.15
PC2
50
150
250
350
0.05
0
50
150
250
350
0
50
150
250
Term
Term
Term
B1
B2
B3
4
350
−8
−50
−10
−4
0
0
0
5
50
2
100
10 15
0
−0.05
0.060
−0.20
0.064
−0.10
0.068
0.00
0.072
PC1
1970
1980
1990
2000
1970
1980
Month
Functional time series with applications in demography
1990
Month
2000
1970
1980
1990
2000
Month
4. Connections, extensions and applications
15
Outline
1 Yield curves
2 Electricity prices
3 Dynamic updating with partially observed
functions
4 Functional ARH models
5 References
Functional time series with applications in demography
4. Connections, extensions and applications
16
Electricity prices
Price
Time
of day
Functional time series with applications in demography
Days
4. Connections, extensions and applications
17
Electricity prices
Price set on the
previous day, so
each curve known
a day ahead.
Price is a smooth
function of time of
day, but varies
from day to day.
Price
So a functional
time series model
is appropriate.
Time
of day
Functional time series with applications in demography
Days
4. Connections, extensions and applications
17
Electricity prices
Price set on the
previous day, so
each curve known
a day ahead.
Price is a smooth
function of time of
day, but varies
from day to day.
Price
So a functional
time series model
is appropriate.
Time
of day
Functional time series with applications in demography
Days
4. Connections, extensions and applications
17
Electricity prices
Price set on the
previous day, so
each curve known
a day ahead.
Price is a smooth
function of time of
day, but varies
from day to day.
Price
So a functional
time series model
is appropriate.
Time
of day
Functional time series with applications in demography
Days
4. Connections, extensions and applications
17
Electricity prices
Functional time series models allows for
multiple seasonality: time of day, day of week
and time of year.
Time of day is handled by the functions while
day of week and time of year are handled via
the PC scores.
Different time of day patterns (e.g., weekdays
and weekends) can be handled via different
PCs.
Functional time series with applications in demography
4. Connections, extensions and applications
18
Electricity prices
Functional time series models allows for
multiple seasonality: time of day, day of week
and time of year.
Time of day is handled by the functions while
day of week and time of year are handled via
the PC scores.
Different time of day patterns (e.g., weekdays
and weekends) can be handled via different
PCs.
Functional time series with applications in demography
4. Connections, extensions and applications
18
Electricity prices
Functional time series models allows for
multiple seasonality: time of day, day of week
and time of year.
Time of day is handled by the functions while
day of week and time of year are handled via
the PC scores.
Different time of day patterns (e.g., weekdays
and weekends) can be handled via different
PCs.
Functional time series with applications in demography
4. Connections, extensions and applications
18
Electricity prices
Variation
explained
4%
74%
3%
10%
Eigenfunctions
Functional time series with applications in demography
4. Connections, extensions and applications
19
Outline
1 Yield curves
2 Electricity prices
3 Dynamic updating with partially observed
functions
4 Functional ARH models
5 References
Functional time series with applications in demography
4. Connections, extensions and applications
20
Dynamic updating
We split a seasonal univariate time series into sections of
one year.
yt (x) = observed value in year t and season x.
Functional time series with applications in demography
4. Connections, extensions and applications
21
Dynamic updating
We split a seasonal univariate time series into sections of
one year.
26
24
22
20
Sea surface temperature
28
yt (x) = observed value in year t and season x.
1950
1960
1970
Functional time series with applications in demography
1980
1990
2000
4. Connections, extensions and applications
2010
21
Dynamic updating
We split a seasonal univariate time series into sections of
one year.
26
24
22
20
Sea surface temperature
28
yt (x) = observed value in year t and season x.
2
4
Functional time series with applications in demography
6
8
10
12
4. Connections, extensions and applications
21
26
24
22
20
Sea surface temperature
28
Sea surface temperatures
2
4
6
8
10
12
Month
Functional time series with applications in demography
4. Connections, extensions and applications
22
28
Sea surface temperatures
26
an indicator of El Niño
22
24
å Some outliers identified and
removed
å Problem: how to forecast using a
partially observed year?
20
Sea surface temperature
å Monthly SST: 1950–2008, used as
2
4
6
8
10
12
Month
Functional time series with applications in demography
4. Connections, extensions and applications
22
28
Sea surface temperatures
26
an indicator of El Niño
22
24
å Some outliers identified and
removed
å Problem: how to forecast using a
partially observed year?
20
Sea surface temperature
å Monthly SST: 1950–2008, used as
2
4
6
8
10
12
Month
Functional time series with applications in demography
4. Connections, extensions and applications
22
28
Sea surface temperatures
26
an indicator of El Niño
22
24
å Some outliers identified and
removed
å Problem: how to forecast using a
partially observed year?
20
Sea surface temperature
å Monthly SST: 1950–2008, used as
2
4
6
8
10
12
Month
Functional time series with applications in demography
4. Connections, extensions and applications
22
Dynamic updating: solution 1
p
Season
1
Redefine the year
0
n
n+1
Years
Functional time series with applications in demography
4. Connections, extensions and applications
23
Dynamic updating: solution 1
p
m
Season
1
Redefine the year
0
n
n+1
Years
Functional time series with applications in demography
4. Connections, extensions and applications
23
p
m
Season
1 p
m−1
Dynamic updating: solution 1
0
n
n+1
n+2
Years
Functional time series with applications in demography
4. Connections, extensions and applications
23
p
m
Season
1 p
m−1
Dynamic updating: solution 1
0
n
n+1
n+2
Years
Functional time series with applications in demography
4. Connections, extensions and applications
23
p
m
Season
1 p
m−1
Dynamic updating: solution 1
0
n
n+1
n+2
Years
Functional time series with applications in demography
4. Connections, extensions and applications
23
p
m
Season
1 p
m−1
Dynamic updating: solution 1
0
n
n+1
n+2
Years
Functional time series with applications in demography
4. Connections, extensions and applications
23
Dynamic updating: solution 2
p
m
Season
1
OLS regression
For each month, regress the month against the
partial principal components.
0
n
n+1
Years
Functional time series with applications in demography
4. Connections, extensions and applications
24
Dynamic updating: solution 2
p
m
Season
1
OLS regression
For each month, regress the month against the
partial principal components.
0
n
n+1
Years
Functional time series with applications in demography
4. Connections, extensions and applications
24
Dynamic updating: solution 3
p
m
Season
1
Ridge regression
For each month, regress the month against the
partial principal components.
0
n
n+1
Years
Functional time series with applications in demography
4. Connections, extensions and applications
25
Dynamic updating
Mean squared error
Computed on a rolling forecast origin
Univariate methods
Dynamic updating
MP
RW
SARIMA
FTS BM
OLS RR
0.69 1.45 0.98
0.74 0.69 1.04 0.48
MP = mean predictor (mean of prior data)
RW = random walk
SARIMA = seasonal ARIMA model
FTS = functional time series
BM = block moved
OLS = ordinary least squares
RR = ridge regression
Functional time series with applications in demography
4. Connections, extensions and applications
26
Outline
1 Yield curves
2 Electricity prices
3 Dynamic updating with partially observed
functions
4 Functional ARH models
5 References
Functional time series with applications in demography
4. Connections, extensions and applications
27
Functional ARH model
Autoregressive Hilbertian process of order 1
ARH(1) model Z
ft (x) − µ(x) =
[ft−1 (x) − µ(x)]θ(x, y) dy + et (x)
= θ[ft−1 (x) − µ(x)] + et (x)
RR
θ2 (x, y) dx dy < 1 for stationarity.
Bosq (2000) and Horváth & Kokoszka (2012)
provide basic theory.
Estimating θ(x, y) is very difficult to do well.
Functional time series with applications in demography
4. Connections, extensions and applications
28
Functional ARH model
Autoregressive Hilbertian process of order 1
ARH(1) model Z
ft (x) − µ(x) =
[ft−1 (x) − µ(x)]θ(x, y) dy + et (x)
= θ[ft−1 (x) − µ(x)] + et (x)
RR
θ2 (x, y) dx dy < 1 for stationarity.
Bosq (2000) and Horváth & Kokoszka (2012)
provide basic theory.
Estimating θ(x, y) is very difficult to do well.
Functional time series with applications in demography
4. Connections, extensions and applications
28
Functional ARH model
Autoregressive Hilbertian process of order 1
ARH(1) model Z
ft (x) − µ(x) =
[ft−1 (x) − µ(x)]θ(x, y) dy + et (x)
= θ[ft−1 (x) − µ(x)] + et (x)
RR
θ2 (x, y) dx dy < 1 for stationarity.
Bosq (2000) and Horváth & Kokoszka (2012)
provide basic theory.
Estimating θ(x, y) is very difficult to do well.
Functional time series with applications in demography
4. Connections, extensions and applications
28
Functional ARH model
ARH(p) model
ft (x) − µ(x) = θ1 [ft−1 (x) − µ(x)] +
· · · + θp [ft−p (x) − µ(x)] + et (x)
θj () are linear functions on a Hilbert space
et (x) is H white noise
FPCA decompostion
ft (x) = µ(x) +
∞
X
βt,k φk (x)
k =1
å Equivalence: βt is a VAR(p) process if ft (p) is
an ARH(p).
Functional time series with applications in demography
4. Connections, extensions and applications
29
Functional ARH model
ARH(p) model
ft (x) − µ(x) = θ1 [ft−1 (x) − µ(x)] +
· · · + θp [ft−p (x) − µ(x)] + et (x)
θj () are linear functions on a Hilbert space
et (x) is H white noise
FPCA decompostion
ft (x) = µ(x) +
∞
X
βt,k φk (x)
k =1
å Equivalence: βt is a VAR(p) process if ft (p) is
an ARH(p).
Functional time series with applications in demography
4. Connections, extensions and applications
29
Functional ARH model
ARH(p) model
ft (x) − µ(x) = θ1 [ft−1 (x) − µ(x)] +
· · · + θp [ft−p (x) − µ(x)] + et (x)
θj () are linear functions on a Hilbert space
et (x) is H white noise
FPCA decompostion
ft (x) = µ(x) +
∞
X
βt,k φk (x)
k =1
å Equivalence: βt is a VAR(p) process if ft (p) is
an ARH(p).
Functional time series with applications in demography
4. Connections, extensions and applications
29
Functional ARH model
ARH(p) model
ft (x) − µ(x) = θ1 [ft−1 (x) − µ(x)] +
· · · + θp [ft−p (x) − µ(x)] + et (x)
θj () are linear functions on a Hilbert space
et (x) is H white noise
FPCA decompostion
ft (x) = µ(x) +
∞
X
βt,k φk (x)
k =1
å Equivalence: βt is a VAR(p) process if ft (p) is
an ARH(p).
Functional time series with applications in demography
4. Connections, extensions and applications
29
Functional ARH model
ARH(p) model
ft (x) − µ(x) = θ1 [ft−1 (x) − µ(x)] +
· · · + θp [ft−p (x) − µ(x)] + et (x)
θj () are linear functions on a Hilbert space
et (x) is H white noise
FPCA decompostion
ft (x) = µ(x) +
∞
X
βt,k φk (x)
k =1
å Equivalence: βt is a VAR(p) process if ft (p) is
an ARH(p).
Functional time series with applications in demography
4. Connections, extensions and applications
29
Functional ARH model
I have never seen a convincing application of
ARH models to real data.
But the tools are nearly ready for real work.
The FPCA approach allows much more
sophisticated time series dynamics with
different processes for different coefficients.
The FPCA approach is not based on a stochastic
model, and does not necessarily give consistent
estimates (e.g., when {ft (x)} is not stationary).
Functional time series with applications in demography
4. Connections, extensions and applications
30
Functional ARH model
I have never seen a convincing application of
ARH models to real data.
But the tools are nearly ready for real work.
The FPCA approach allows much more
sophisticated time series dynamics with
different processes for different coefficients.
The FPCA approach is not based on a stochastic
model, and does not necessarily give consistent
estimates (e.g., when {ft (x)} is not stationary).
Functional time series with applications in demography
4. Connections, extensions and applications
30
Functional ARH model
I have never seen a convincing application of
ARH models to real data.
But the tools are nearly ready for real work.
The FPCA approach allows much more
sophisticated time series dynamics with
different processes for different coefficients.
The FPCA approach is not based on a stochastic
model, and does not necessarily give consistent
estimates (e.g., when {ft (x)} is not stationary).
Functional time series with applications in demography
4. Connections, extensions and applications
30
Functional ARH model
I have never seen a convincing application of
ARH models to real data.
But the tools are nearly ready for real work.
The FPCA approach allows much more
sophisticated time series dynamics with
different processes for different coefficients.
The FPCA approach is not based on a stochastic
model, and does not necessarily give consistent
estimates (e.g., when {ft (x)} is not stationary).
Functional time series with applications in demography
4. Connections, extensions and applications
30
Outline
1 Yield curves
2 Electricity prices
3 Dynamic updating with partially observed
functions
4 Functional ARH models
5 References
Functional time series with applications in demography
4. Connections, extensions and applications
31
Selected references
Nelson & Siegel (1987) Parsimonious modeling of yield curves.
J Business.
Diebold & Li (2006) Forecasting the term structure of
government bond yields. J Econometrics.
Bowsher & Meeks (2008) The dynamics of economic functions:
modeling and forecasting the yield curve. JASA.
Hays, Shen & Huang (2012) Functional dynamic factor models
with application to yield curve forecasting. Ann Appl Stat.
Sen & Klüppelberg (2012) Time series of functional data. Tech
report, TU Munich.
Shang & Hyndman (2011) Nonparametric time series
forecasting with dynamic updating. Mathematics and
Computers in Simulation.
Bosq (2000) Linear processes in function spaces. Springer.
Horváth & Kokoszka (2012) Inference for functional data with
applications. Springer. Springer
Functional time series with applications in demography
4. Connections, extensions and applications
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