Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Central England Monthly Temperature Series, 1772— 2016

642 views

Published on

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

Published in: Economy & Finance
  • Be the first to comment

  • Be the first to like this

The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Central England Monthly Temperature Series, 1772— 2016

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 10. Harvey and Mills (2003), Four seasons: Spring
  11. 11. Harvey and Mills (2003), Four seasons: Summer
  12. 12. Harvey and Mills (2003), Four seasons: Autumn
  13. 13. Harvey and Mills (2003), Four seasons: Winter
  14. 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. 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. 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. 17. Vogelsang and Franses (2005)
  18. 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. 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. 20. Proietti and Hillebrand (2017): Testing stationarity
  21. 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. 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. 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. 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. 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. 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. 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. 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. 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. 30. The shifting mean The shifting mean equals EySk+s = φ 1 (1)δs ( Sk + s SK ).
  31. 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. 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. 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. 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. 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. 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. 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. 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. 39. Back to the time series The monthly CET series, 1772–2016 1800 1850 1900 1950 2000 05101520
  40. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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’).

×