Elsevier

Econometrics and Statistics

Volume 12, October 2019, Pages 1-24
Econometrics and Statistics

The shifting seasonal mean autoregressive model and seasonality in the Central England monthly temperature series, 1772–2016

https://doi.org/10.1016/j.ecosta.2019.05.005Get rights and content

Abstract

A new autoregressive model with seasonal dummy variables in which coefficients of seasonal dummies vary smoothly and deterministically over time is introduced. The error variance of the model is seasonally heteroskedastic and multiplicatively decomposed as in ARCH models. This variance is also allowed to be smoothly and deterministically time-varying. Under regularity conditions, consistency and asymptotic normality of the maximum likelihood estimators of parameters of this model is proved. The purpose of the model is to find out how the average monthly temperatures in the well-known central England temperature series have been varying during the period of more than 240 years. The main result is that warming has occurred but that there are notable differences between months. In particular, no warming is found for February, April, May and June.

Introduction

As a monthly temperature time series, the central England temperature (CET) series is quite unique because of its length. It extends over three and a half centuries and thus provides an opportunity to consider possible changes in the climate on a ‘micro-level’. The series was originally compiled by Manley (1974) and covered the years 1659–1973; for a revised and extended series see Parker et al. (1992). Several authors have studied properties of the series for various time periods. Harvey and Mills (2003) aggregated the series from 1723 to 1999 to the annual and seasonal level and considered deterministic trends using both local cubic trends and low-pass filters. Their conclusion was that within this period, no warming trend can be discerned. The main reason for this is that they found a rather strong downward movement in both the annual and seasonal series until around 1775, and the upward movement beginning thereafter only brought the temperatures to the level where they were in the beginning of the period. Vogelsang and Franses (2005) analysed the whole CET series from 1659 to 2000 using an autoregressive model augmented by a linear trend and a possible trend-break. Their conclusion was that there is a positive linear trend for months from October to April. This excludes the summer months. Recently, Proietti and Hillebrand (2017) used a structural time series model, separating the series from 1772 to 2013 into permanent and transitory components. The permanent component contained both a deterministic (linear) and a stochastic trend. They found that the deterministic trend is strongest for November, December and January, whereas the stochastic trend has the highest coefficients for April and May, and again for August, September and October.

This paper has two important purposes. The first one is to develop a seasonal time series model which can adequately describe changes in seasonality over time in situations in which no single both observable and quantifiable cause for the change can be identified. The idea is to generalise the standard autoregressive model with seasonal dummy variables to the situation in which the seasonal pattern of the time series may not remain constant over time. The second purpose is to apply the model, called the Seasonal Shifting Mean Autoregressive (SSM-AR) model, to quantify potential warming in the monthly CET series. This implies a more detailed scrutiny of seasonality in this series than what is reported in hitherto published papers. The months in which warming, if any, has occurred will be found and its strength for each month estimated.

The plan of the paper is as follows. The SSM-AR model is introduced in Section 2. Its properties, such as the log-likelihood, score and the information matrix are presented in Section 3 and the Hessian in Section 4. Asymptotic theory for maximum likelihood estimators of parameters of the model is considered in Section 5. Specification and testing of the SSM-AR model is the topic of Section 6. Application to the CET series is described in Section 7. Section 8 contains discussion and final remarks. Proofs, estimated equations and some additional material can be found in Appendices.

Section snippets

The model

The number of nonlinear seasonal time series models is not large, but a few examples exist. Franses and de Bruin (2000) introduced a seasonal smooth transition autoregressive (SEASTAR) model and fitted it to seasonally unadjusted unemployment series. The purpose of the study was to study the effects of seasonal adjustments on the properties of these series. Ajmi et al. (2008) generalised the model to the case where the variable to be explained is fractionally integrated and fitted the model to

Log-likelihood, score and the information matrix

Assuming independent errors with mean zero and time-varying variance σSk+s2, the quasi log-likelihood function (SK observations) of the model for the seasonal unit s is defined as follows:LSK(θ,ɛ)=const12k=0K1s=1SlnσSk+s212k=0K1s=1SεSk+s2σSk+s2,where, from (1), εSk+s=ySk+sδs(Sk+jSK)i=1pϕiySk+si, and σSk+s2 is defined by (6).

For notational simplicity it is in this section assumed that qj=1 in (2) and rj=1 in (6) and that the transition function is defined by (3). Generalisations are

The Hessian

If the errors are not assumed normal, the Hessian matrix of (12) is needed for statistical inference. It is block diagonal as the mean and the variance components do not have common parameters. The next two lemmas specify the nonzero blocks of this matrix.

Lemma 7

The average Hessian matrix for the mean component of the log-likelihood (12) equalsHSKM=[HθSKMHθϕSKMHϕSKM],where HθSKM= diag(Hθ1KM,,HθSKM) withHθsKM=1Kk=0K11σSk+s2{εSk+sθsεSk+sθs+εSk+s2εSk+sθsθs}.In (19),2δs(Sk+sSK)θsθs=[2δs(S

Zig-zag algorithm

It follows from Lemmas 1 and 2 that the log-likelihood function is continuous. In addition, the mean and the variance components of the model do not have common parameters. Maximisation of the log-likelihood can be carried out by splitting each iteration into two components as Sargan (1964) suggested. The parameters in the mean part are estimated first, and parameters in the error variance thereafter, conditionally on the estimates of the mean parameters. Next the mean parameters are

General

As is clear from Section 2, before an SSM-AR model can be estimated its form has to be specified. The number of transitions has to be determined from the data because typically there is little or no theory available to help the model builder to make the correct decision. Overestimating this number leads to difficulties because a model containing too many transitions either in the mean or the variance or both is not identified. The parameters of such a model cannot be estimated consistently.

The series

The central England temperature series is one of the longest existing monthly temperature series. Because of its length and frequency it offers a possibility to study effects of climate change on temperatures by season. It is updated and available at http://www.metoffice.gov.uk/hadobs/hadcet/data/download.html. As mentioned in the Introduction, the series has for varying subperiods been analysed by several researchers. Following Proietti and Hillebrand (2017), we disregard the earliest

Final remarks

In this work we develop a flexible nonlinear model capable of describing changes in the seasonal pattern of a time series over time. It is applied to the CET series using the same time period as Proietti and Hillebrand (2017) did but extended to the end of 2016. The main result is that with one exception, temperatures for the months from July to March have increased and that the warming has been strongest in the winter months. The results also show that there are differences in the timing of

Acknowledgments

The second author acknowledges support by the Tianjin Natural Science Foundation (17JCYBJC43300). Part of the work for this paper was carried out when the third author was visiting Tianjin University of Finance and Economics, whose kind hospitality is gratefully acknowledged. Material from this paper has been presented at the 2017 Conference on Econometric Models of Climate Change, Nuffield College, Oxford, September 2017, the workshop ‘Advances in financial and time series econometrics’,

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