Changli He, Jian Kang and Timo Teräsvirta Tianjin University of Finance and Economics CREATES, Aarhus University C.A.S.E., Humboldt-Universität zu Berlin Seminar at Eesti pank 21 September 2017

1. The Shifting Seasonal Mean Autoregressive
Model and Seasonality in the Central England
Monthly Temperature Series, 1772— 2016
Changli He , Jian Kang and Timo Teräsvirta†‡
Tianjin University of Finance and Economics
†CREATES, Aarhus University
‡C.A.S.E., Humboldt-Universität zu Berlin
Seminar at Eesti pank
21 September 2017

2. Introduction
Documentation of the time series:
I Manley (1974) provides monthly mean temperatures for
central England for the years 1659–1973.
I Parker, Legg and Folland (1992) present a new, revised, series
for the years 1772–1991.
I In this analysis, the series also begins from 1772 as in Proietti
and Hillebrand (2017), and is extended to 2016. The source:
http://www.metoffice.gov.uk/hadobs/hadcet/data/
download.html.

3. Introduction
Several researchers have studied and modelled this seasonal time
series using various time series techniques.
I Harvey and Mills (2003) use local cubic polynomial trends and
low-pass …lters for the quarterly series, 1723–2000.
I Vogelsang and Franses (2005) …t dynamic autoregressive
models with a deterministic trend to the 12 monthly series,
1659–2000.
I Proietti and Hillebrand (2017) consider stationarity of the 12
monthly series and …t a structural time series model to them,
1772–2014. (Inspiration for our work comes from theirs.)

4. Introduction
Several researchers have studied and modelled this seasonal time
series using various time series techniques.
I Harvey and Mills (2003) use local cubic polynomial trends and
low-pass …lters for the quarterly series, 1723–2000.
I Vogelsang and Franses (2005) …t dynamic autoregressive
models with a deterministic trend to the 12 monthly series,
1659–2000.
I Proietti and Hillebrand (2017) consider stationarity of the 12
monthly series and …t a structural time series model to them,
1772–2014. (Inspiration for our work comes from theirs.)
Nonlinear seasonal time series models:
I Seasonal STAR (SEASTAR) model: Franses and de Bruin
(2000).
I For modelling changing seasonality there exists the Seasonal
TV-STAR model: van Dijk, Strikholm and Teräsvirta (2003).

5. Introduction
Object of interest: the monthly CET series 1772–2016 and changes
in its seasonal pattern over time:
1800 1850 1900 1950 2000
05101520

6. Harvey and Mills (2003)
Use annual and quarterly series 1723–1999
I Nonparametric methods
I Local polynomial approach
I Filtering (…lter out high-frequency components)
We shall take a look at the local polynomial approach.

7. Harvey and Mills (2003)
Polynomial regression
The model:
yt = β0 + β1(t t0) + β2(t t0)2
+ β3(t t0)3
+ εt (1)
where
I εt iid(0, σ2).
The model (1) is a normal third-order polynomial model (yt
explained by the …rst three powers of (t t0).
Estimation by least squares (assuming T observations):
Minimise
T
∑
t=1
fyt β0 β1(t t0) β2(t t0)2
β3(t t0)3
g2
w.r.t. β0, β1, β2, β3.

8. Harvey and Mills (2003)
Local cubic polynomial regression
Minimise
T
∑
t=1
fyt β0 β1(t t0) β2(t t0)2
β3(t t0)3
g2
K(
t t0
h
)
w.r.t. β0, β1, β2, β3 at t0, where
I the kernel function (in this case the Gaussian kernel) equals
K(
t t0
h
) = (2πh2
) 1/2
expf (
t t0
h
)2
g.
I The parameter h is called bandwidth.
I Estimation separately for each t0.

9. Harvey and Mills (2003)
Seasonal models (four seasons):
I Spring: March, April, May,
I Summer: June, July, August
I Autumn: September, October, November
I Winter: December, January, February

10. Harvey and Mills (2003), Four seasons: Spring

11. Harvey and Mills (2003), Four seasons: Summer

12. Harvey and Mills (2003), Four seasons: Autumn

13. Harvey and Mills (2003), Four seasons: Winter

14. Harvey and Mills (2003): Conclusions
I ’Although trend temperatures have been rising during the
twentieth century, there is no evidence, apart from the spring,
that the latter half of this century has seen any acceleration in
trend temperature.’
I ’Given the uncertainties in the estimates near the boundaries,
then, apart from the spring, there is little evidence to suggest
that trend temperatures at the end of the twentieth century
were higher than in the early part of the eighteenth century.’

15. Notation
I S is the length of the seasonal cycle (for example a year,
S = 12)
I Symbol j represents the jth unit (for example a month,
j = 1, ..., 12) within the cycle
I Symbol k indicates the (k + 1)st cycle (for example the
(k + 1)st year), and K is the total number of cycles (for
example years).
I The general tth observation becomes Sk + j),
k = 0, 1, ..., K 1. (Example: the …rst 12 months together
form the …rst year.)
I The aforementioned authors did not use this notation in their
work.

16. Vogelsang and Franses (2005)
These authors model the series beginning 1659, the period is
1659–2000
Approach: Form 12 separate monthly series and build a model for
each of them.
The model for month s:
ySk+s = µs + βs (Sk + s) + δs I(Sk + s t0 > 0)
+ γs (Sk + s t0)I(Sk + s t0 > 0) +
p
∑
i=1
csi ySk+s i + εSk+s
for k = 0, 1, ..., K 1, where I(A) is an indicator variable:
I(A) = 1 if A is valid, zero otherwise.
I Critical values for testing βs = 0: 5% level: 5.222, 1% level:
8.100.

17. Vogelsang and Franses (2005)

18. Proietti and Hillebrand (2017): Testing stationarity
The authors consider the period 1772–2014.
Object of interest in this presentation: are the monthly series,
formed as in Vogelsang and Franses (2005), stationary?
The structural time series model for month s:
ySk+s = µSk+s + εSk+s
µSk+s = µS(k 1)+s + ηsk+s
where Eε2
Sk+s = σ2
εs and Eη2
Sk+s = σ2
ηs .
I The null hypothesis, H0: σ2
ηs = 0.

19. Proietti and Hillebrand (2017): Testing stationarity
The locally best invariant test statistic (Nyblom and Mäkeläinen,
1983) has the form
ξsµ =
1
bσ2
εs
K 1
∑
J=0
f
J
∑
k=0
(ySk+s ys )g2
where
I bσ2
εs = (1/K) ∑K 1
k=0 (ySk+s ys )2.
I The asymptotic null distribution of ξsµ is the …rst-level
Cramér-von Mises distribution, the 5% critical value equals
0.461.

20. Proietti and Hillebrand (2017): Testing stationarity

21. Proietti and Hillebrand (2017): Modelling
The monthly structural time series model for month s:
ySk+s = µSk+s + ψSk+s
µSk+s = βs (Sk + s) + θs δSk+s
δSk+s = δSk+s 1 + ηSk+s
ψSk+s = φs ψSk+s 1 + ζSk+s
where, for k = 0, 1, ..., K,
I ηSk+s iidN (0, 1) and ζSk+s iidN (0, σ2
ζs ).

22. Proietti and Hillebrand (2017): Modelling
Summary of the results (from the paper):
(a) The estimated slopes β are higher in the winter months
(January and December, in particular) and November. In contrast,
the drift is not signi…cantly di¤erent from 0 for June and July.
(b) The estimated average drift implies a rise in temperatures
amounting to 0.37oC over a century.

23. Proietti and Hillebrand (2017): Modelling
Summary of the results (from the paper):
(a) The estimated slopes β are higher in the winter months
(January and December, in particular) and November. In contrast,
the drift is not signi…cantly di¤erent from 0 for June and July.
(b) The estimated average drift implies a rise in temperatures
amounting to 0.37oC over a century.
(c) The loadings θ on the global stochastic trend are higher (in
absolute value) for April and May, and August, September and
October. The loadings are close to 0 for January and December.

24. Proietti and Hillebrand (2017): Modelling
Summary of the results (from the paper):
(a) The estimated slopes β are higher in the winter months
(January and December, in particular) and November. In contrast,
the drift is not signi…cantly di¤erent from 0 for June and July.
(b) The estimated average drift implies a rise in temperatures
amounting to 0.37oC over a century.
(c) The loadings θ on the global stochastic trend are higher (in
absolute value) for April and May, and August, September and
October. The loadings are close to 0 for January and December.
(d) The transitory component ψSk+s has higher variability in the
winter months: both the variance and the periodic autoregressive
coe¢ cients are higher. The periodic autoregressive coe¢ cients are
also higher in the summer months.

25. SSM-AR model
The Seasonal Shifting-Mean Autoregressive (SSM-AR) model is
de…ned as follows:
ySk+s =
S
∑
j=1
δj (
Sk + j
SK
)D
(j)
Sk+s +
p
∑
i=1
φi ySk+s i + εSk+s (2)
where
I D
(j)
Sk+s = 1 when j = s, zero otherwise,
I εSk+s WN(0, σ2),
I roots of 1 ∑
p
i=1 φi zi lie outside the unit circle, and
I for notational simplicity, the number of observations equals
SK.
Setting S = 1 yields the Shifting Mean Autoregressive (SM-AR)
model by González and Teräsvirta (2008).

26. SSM-AR model
The jth time-varying coe¢ cient δj (Sk+j
SK ) equals
δj (
Sk + j
SK
) = δj0 +
qj
∑
i=1
δji gj (
Sk + j
SK
; γji , cji ) (3)
j = 1, ..., S, where
gj (
Sk + j
SK
; γji , cji ) = (1 + expf γji (
Sk + j
SK
cji )g) 1
(4)
or
gj (
Sk + j
SK
; γji , c1ji , c2ij ) = (1 + expf γji (
Sk + j
SK
c1ji )
(
Sk + j
SK
c2ji )g) 1
(5)
In both (4) and (5), γji > 0 for all i, j.

27. SSM-AR model
Simpli…ed notation: the model (2) at time Sk + s may be written
as
ySk+s = δs (
Sk + s
SK
) +
p
∑
i=1
φi ySk+s i + εSk+s . (6)

28. SSM-AR model
Simpli…ed notation: the model (2) at time Sk + s may be written
as
ySk+s = δs (
Sk + s
SK
) +
p
∑
i=1
φi ySk+s i + εSk+s . (6)
The SSM-AR model di¤ers from the SEASTAR model in three
respects.
I All units have a di¤erent transition function.
I The transition variable is rescaled time.
I The autoregressive part is linear (that can be generalised if
needed).

29. The shifting mean
Let L be the lag operator: Lxt = xt 1, and write
φ(L) = 1
p
∑
i=1
φi Li
The model (2) for j = s can be written as follows:
ySk+s = φ 1
(L)fδs (
Sk + s
SK
)D
(s)
Sk+s + εSk+s g
where δs (x) = 0 for x < 0.

30. The shifting mean
The shifting mean equals
EySk+s = φ 1
(1)δs (
Sk + s
SK
).

31. Log-likelihood
The quasi log-likelihood function (SK observations) of the model
is de…ned as follows:
LSK (θ, ε) = c
SK
2
ln σ2 1
2SK
K 1
∑
k=0
S
∑
j=1
ε2
Sk+j
σ2
= c
SK
2
ln σ2 1
2σ2K
K 1
∑
k=0
1
S
S
∑
j=1
(ySk+j
δj (
Sk + j
SK
)D
(j)
Sk+s
p
∑
i=1
φi ySk+s i )2
. (7)

32. Asymptotic properties
Triangular array asymptotics required (time is rescaled:
0 < (Sk + j)/SK 1, SK is the number of observations).
Assumptions for proving consistency and asymptotic normality of
maximum likelihood estimators:
A1 In the transition function gs (Sk+s
SK ; γsi , csi ), γsi > 0,
i = 1, ..., qs ; cs1 < ... < csqs . This implies
gs (
Sk + s
SK
; γsi , csi ) 6= gs (
Sk + s
SK
; γsj , csj )
for i 6= j. In addition, δsi 6= 0, i = 1, ..., qs .

33. Asymptotic properties
A2 Parameter space Θ is compact,
A3 The density is positive (bounded away from zero) for all θ 2 Θ.
A4 The errors εSk+s iidN (0, σ2).
A5 The roots of 1 ∑
p
i=1 φi zi = 0 lie outside the unit circle.
If A1 is relaxed such that csj = cs,j+1 for some j, then
gs (Sk+s
SK ; γsj , csj ) 6= gs (Sk+s
SK ; γs,j+1, cs,j+1) requires γsj 6= γs,j+1.
Normality of errors may be relaxed by assuming that εSk+s
iid(0, σ2).
To prove asymptotic normality, add
A6: The true parameter θ0
is an interior point of Θ.

34. Testing constancy of seasonal coe¢ cients
The …rst step in building SSM-AR models is to test stability of the
coe¢ cients of dummy variables.
The null model is the standard autoregressive model with (only)
seasonally varying means:
ySk+s =
S
∑
j=1
δj0D
(j)
Sk+s +
p
∑
i=1
φi ySk+s i + εSk+s (8)
whereas the alternative is the SSM-AR model (2).
Identi…cation problem: If H0: γj = 0, j = 1, ..., S, is our null
hypothesis, then δj1 and cj1, or cj1 and cj2, are unidenti…ed
nuisance parameters when this hypothesis holds.

35. Testing constancy of seasonal coe¢ cients
Solution: Follow Luukkonen et al. (1988) by expanding
(1 + expf γj (Sk+j
SK cj )g) 1 into a Taylor series around γj = 0
and reparameterise (4) (or (5)) accordingly.
Assuming δj1 6= 0, j = 1, ..., S, this gives the following third-order
polynomial expression
δj (
Sk + j
SK
) = αj0 + αj1
Sk + j
SK
+ αj2(
Sk + j
SK
)2
+ αj3(
Sk + j
SK
)3
+R3,Sk+j
where R3,Sk+j is the remainder.

36. Testing constancy of seasonal coe¢ cients
The resulting auxiliary SSM-AR model has the following form:
ySk+s =
S
∑
j=1
fαj0 + αj1
Sk + j
SK
+ αj2(
Sk + j
SK
)2
+αj3(
Sk + j
SK
)3
gD
(j)
Sk+s +
p
∑
i=1
φi ySk+s i + εSk+s (9)
where εSk+s = εSk+s + R3,Sk+s .
The new null hypothesis equals H0
0: α1 = ... = αS = 0, where
αj = (αj1, αj2, αj3)0 for j = 1, ..., S.
Under H0, the LM-type test statistic has an asymptotic
χ2-distribution with 3S degrees of freedom. (This is proved in the
paper.)

37. Testing constancy of seasonal coe¢ cients
I In many applications, including the present one, it is both
useful and informative to test constancy for each season
separately, assuming that the other seasonal dummies have
constant coe¢ cients.

38. Testing constancy of seasonal coe¢ cients
I In many applications, including the present one, it is both
useful and informative to test constancy for each season
separately, assuming that the other seasonal dummies have
constant coe¢ cients.
I Can also select, season by season, the number of transitions in
the shifting seasonal means. This is done by sequential
testing.

39. Back to the time series
The monthly CET series, 1772–2016
1800 1850 1900 1950 2000
05101520

40. Back to the time series
Monthly averages of the CET series, 1772–2016
-505101520
J an F eb M ar A pr M a y J un J ul A ug S ep O c t N o v D ec

41. Back to the time series
Empirical monthly marginal densities of the CET series, 1772–2016
-505101520
J an F eb Mar Apr Ma y J un J ul A ug Sep Oct No v Dec

42. Testing constancy
1 2 3
Jan 0.0000 0.0000 0.0000
Feb 0.3012 0.4010 0.2876
Mar 0.0005 0.0002 0.0007
Apr 0.6509 0.3874 0.5350
May 0.9336 0.3759 0.5777
Jun 0.3490 0.4045 0.5933
Jul 0.0717 0.0201 0.0476
Aug 0.1193 0.0157 0.0346
Sep 0.0174 0.0184 0.0461
Oct 0.0001 0.0000 0.0000
Nov 0.0000 0.0000 0.0001
Dec 0.0009 0.0030 0.0082
p-values of tests of constancy by month, based on the …rst, second
and third order Taylor approximation of the transition function

43. Estimation results
1800 1900 2000
2.02.53.03.5
Jan
1800 1900 2000
6.06.57.0
Mar
1800 1900 2000
16.817.217.6
J ul
1800 1900 2000
15.616.016.4
A ug
1800 1900 2000
12.212.613.013.4
Sep
1800 1900 2000
8.08.59.0
Oct
1800 1900 2000
4.04.55.05.5
N o v
1800 1900 2000
2.22.63.03.4
D ec
Estimated shifting means by month, 1772–2016

44. Estimation results
Month 1772-2016 1772-1899 1900-2016
Jan 1.67 1.67 0.00
Feb 0.00 0.00 0.00
Mar 1.56 0.15 1.40
Apr 0.00 0.00 0.00
May 0.00 0.00 0.00
Jun 0.00 0.00 0.00
Jul 1.07 0.00 1.07
Aug 1.02 0.00 1.02
Sep 1.33 0.11 1.22
Oct 1.57 0.021 1.54
Nov 1.97 0.69 1.28
Dec 1.31 1.31 0.00
Estimated changes in the seasonal means, 1772–2016, 1772–1899,
and 1900–2016

45. Misspeci…cation tests
I Testing the null hypothesis of no error autocorrelation
Max lag p-value
1 0.53
2 0.23
3 0.33
6 0.24
12 0.27
p-values of the Godfrey-Breusch test of no error autocorrelation
modi…ed to apply to the (nonlinear) SSM-AR model

46. Misspeci…cation tests
I Testing one transition against two
1 2 3
Jan 0.2568 0.1782 0.1630
Mar 0.4744 0.4928 0.6553
Jul 0.8577 0.6911 0.4481
Aug 0.7190 0.2325 0.3103
Sep 0.7076 0.8175 0.8626
Oct 0.6851 0.8584 0.5736
Nov 0.4441 0.7313 0.8803
Dec 0.2672 0.1328 0.0947
p-values of tests of one transition against two by month, based on
the …rst, second and third order Taylor approximation of the
transition function

47. Misspeci…cation tests
I Testing constancy of the error variance by month
The alternative: the error variance of month s is
σ2
Sk+s = σ2
s + ψs (1 + expf γs (
Sk + s
SK
cs )g) 1
where
I σ2
s > 0, σ2
s + ψs > 0, s = 1, ..., S.
I Test month by month.
I The null hypothesis for month s is γs = 0 (identi…cation
problem).

48. Misspeci…cation tests
-6-4-20246
J an F eb Mar Apr Ma y J un J ul A ug Sep Oct No v Dec
Empirical densities of residuals by month

49. Misspeci…cation tests
1 2 3
1 0.00 0.00 0.00
2 0.00 0.00 0.00
3 0.94 0.85 0.59
4 0.20 0.11 0.11
5 0.12 0.12 0.10
6 0.00 0.00 0.00
7 0.12 0.12 0.11
8 0.00 0.00 0.00
9 0.00 0.00 0.00
10 0.51 0.50 0.46
11 1.00 0.91 0.68
12 0.00 0.00 0.00
p-values of the test of constant error variance by month using the
…rst, second and third order approximations to the alternative

50. Estimation of the error variance
To be done for the six months for which constancy is rejected.
I Can be carried out one month at a time (orthogonality).
I For an analogous estimation problem, see for example Amado
and Teräsvirta (2013) or Silvennoinen and Teräsvirta (2016).

51. Final remarks
The results concern only a single temperature series and cannot be
generalised.
Three groups:
I ’19th century warming’: Nov, Dec, Jan
I ’20th century warming’: Mar, Jul, Aug, Sep, Oct, Nov
I ’No warming’: Feb, Apr, May, Jun
Next step: Consider several series and see whether there are
similarities between them (’co-shifting’).