# Persistence and Time Trends in the Temperatures in Spain

Authors: Luis A. Gil-Alana
Journal: Advances in Meteorology
 (2009)

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## Abstract

This paper deals with the analysis of the temperatures in several locations in Spain during the last 50 years. We focus on the degree of persistence of the series, measured through a fractional differencing parameter. This is crucial to properly estimate the parameters of the time trend coefficients in order to determine the degree of warming in the area. The results indicate that all series are fractionally integrated with orders of integration ranging between 0 and 0.5. Moreover, the time trend coefficients are all positive though they are  statistically insignificant, which is in contrast with the results based on nonfractional integration.


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## Body

## 1. Introduction

Modelling climatological time series is an issue that still remains
controversial. Although the evidence supporting the existence of climate change
is now well established throughout the world, the statistical methods employed
to determine the precise magnitude of the change are an unresolved topic. The standard way of modelling temperatures is
to assume a linear function of time of the form(1)yt=β0+β1t+xt,t=1,2,…, where  yt  is the observed temperature at period of time t,  and
xt is the deviation term that  is supposed to be relatively stable across
time. The parameter β1 measures the average change in yt per time period. Thus, in the context of temperature time series, long-run
warming may be occurring if  β1 is positive, in which case there
is an increasing trend in temperature. On the other hand, it is a well-known fact that temperatures are highly time-dependent.
Mathematically, there exist different ways of modelling that behaviour and a
key issue here are
to determine if the deviation term is then stationary or not. The most standard
approach is to assume that  xt in (1) is stationary I(0). Rigorously
speaking, an integration
of order 0 process (denoted by I(0)) is defined as a covariance stationary
process with a spectral density function that is positive and finite at the
zero frequency. In plain words, it means that the process must be relatively
stable across time, and it includes a wide number of mathematical models, like
the white noise, stationary autoregressive (AR), moving averages (MAs), stationary
ARMA, and so forth. Of all these models, the classic autoregressive AR (1) model (2)xt=ρxt−1+εt,t=1,2,…, where  |ρ|<1, and εt is white noise that has been widely employed in the climatological community (e.g.,
[1]) because of its relation with the stochastic the first-order
differential equation [2]. If the detrended series xt is not I(0), the classic alternative is the unit root, also called integration
of order 1 (I(1)) and described as (3)xt=xt−1+vt,t=1,2,…, and first differences are then adopted to achieve the I(0) error vt.
(Note that in this context  vt may be white noise
but also a weakly autocorrelated (AR(1)) process.) These
two approaches have been widely employed in the climatological community to
describe climate change and the warming effect in temperatures throughout the
world. Thus, for example, Bloomfield and Nychka [3] and Woodward and Gray
[4] among others assume that the error term is I(0), while unit roots or I(1) models have been employed in
temperature time series in Woodward and Gray [5], Stern and Kaufmann [6],
Kaufmann and Stern [7], and Kaufmann et al. [8].However, theI(0) and the I(1) models described above
are merely two particular cases of a much more general class of processes
called I(d), where d (the number of differences required to get I(0)) may not necessarily be an integer value
(usually 1) but a real value between 0 and 1 or even above 1. In such a case, the
process is said to be fractionally integrated or integrated of order d (denoted
by I(d)). That is, xt is said to be I(d) if (4)(1−L)dxt=vt,t=1,2,…, where L  is the lag operator (i.e., Lxt=xt−1), and vt is I(0). Note
that the polynomial on the left-hand-side in (4) can be expressed in terms of
its binomial expansion, such that, for all real d, (5)(1−L)d=∑j=0∞ψjLj=∑j=0∞(dj)(−1)jLj=1−dL+d(d−1)2L2−⋯, and (4) can be
written as (6)xt=dxt−1−d(d−1)2xt−2+⋯+vt.  Thus, d is a parameter that indicates the degree of temporal dependence
between the data, and the higher the value of d is, the higher is the level of
association (or correlation) between the observations. Also, note that if d is an
integer value, xt depends only on a finite number of previous
values, while if it is a noninteger value, it depends on all its past history. Comprehensive
surveys of fractionally integrated models can be found in Robinson  [9, 10], Beran [11], Baillie [12], Doukhan et al. [13], and
Gil-Alana and Hualde [14], and examples
of this type of model in meteorological time series data are the papers of
Bloomfield [15], Smith et al. [16], Lewis and Ray [17], Pethkar and Selvam
[18], Koscielny-Bunde et al. [19], Pelletier and
Turcotte [20], Percival et al. [21], Maraun et al. [22], and Gil-Alana [23, 24]. Most of these
authors coincide that the order of integration in regional, global (and even
for specific locations) temperature time series is positive though smaller than
0.5.In
this paper, we
focus on the Spanish case. The reason for the choice is that among the European
countries, Spain is one of the most highly affected by climate change.
According to a recent report from the European Environmental Agency (EEA, 2004),
Spain and Portugal will be most affected within the European Union by climate
change. In this paper, it is also mentioned that averaged temperature in the
last 100 years has risen slightly more in Spain than the rest of the EU.Spain climate is determined by its
world position, on the south western edge of Eurasia and just 13.4 kilometers from Africa at
its southernmost point, with an ocean to the west and a sea to the east, and by
its continental land mass and high mountainous terrain, producing a mosaic of
climates, one of the most varied in Europe. In a recent paper, Brunet et al.
[25] examined long-term (1850–2003) Spanish
temperatures by employing a new daily adjusted record of daily mean temperature
collected by the European Community (EC-) funded project EMULATE. Their results
indicate a highly significant warming over the entire period, with a pronounce
increase in temperature since 1973 [26]. (Other papers dealing
with temperatures in Spain are Jones and Moberg [27] and Sigró et al. [28].)The
outline of the article is as follows. In Section2, we present the statistical
model. Section 3 describes the data and presents the test results, while
Section 4 contains some concluding comments and extensions.


## 2. The Statistical Framework

Throughout this
paper, we consider a model given by (1) and (4), that is, (7)yt=β0+β1t+xt,(1−L)dxt=vt,t=1,2,…,   and given the seasonal (monthly) structure of the series under
analysis, we suppose that the disturbance term, vt, follows a
seasonal AR(1) process of the form (8)vt=ρvt−12+εt. In other words, the value of the time series at time t depends not
on the previous value (as is the case in the nonseasonal AR(1) model described
in (3)) but on the value at the same month in the previous year plus an error
term. Thus, we assume that all the seasonal dependences are captured through the seasonal AR
polynomial. Alternatively, we could have employed seasonal dummy variables.
However, in doing so in the empirical work carried out in the following
section, the coefficients were found to be statistically insignificant in the
majority of the cases.We estimate the
above model using a Whittle function in the frequency domain along with a
procedure proposed by Robinson [9] that permits us to test any real value
for the differencing parameter d. (The Whittle function is an approximation to the likelihood function
that is more suitable for applications in many cases [29].) That
is, we test the null hypothesis (9)Ho:d=do in (7) and (8), where do can be any real value. This is a
very general specification since it allows us to consider many interesting cases.
Thus, if we cannot reject the null hypothesis (9) with do=0,  we
support the trend stationary representation advocated by many authors,
including Brunet et al. [25] work for the Spanish case. On the other hand, if
do=1 cannot be rejected, we support the unit root or I(1) representation.
However, the differencing parameter d can also be a value between 0 and 1, or
even above 1.


## 3. Data and Results

The data employed in the article correspond to the daily maximum (Tmax) and minimum (Tmin) temperatures (in 0.1°C) in six locations in Spain obtained from the European Climate Assessment & Dataset (ECA&D), collected by Tank et al. [30]. (Mean temperature data were also considered and the results, though not reported, were completely in line with those presented here for the maximum and minimum temperatures.) The specific locations are Badajoz (Talavera): LAT: +38:53:00, LON: −06:48:15; Madrid: LAT: +40:24:40, LON: −03:39:19; Malaga: LAT: +36:40:00, LON: −04:28:43; San Sebastian-Donostia: LAT: +43:18:24, LON: −02:01:38; Valencia: LAT: +39:28:48, LON: −00:21:08; Zaragoza: LAT: +41:39:43, LON: −00:59:31. We focus on these stations since they are those with the longest available data without periods of missing observations. The starting dates are 27/10/1936 for San Sebastian-Donostia; 01/01/1950 for Madrid and Valencia; 01/01/1955 for Badajoz; 24/10/1959 for Zaragoza, and 01/08/1980 for Malaga. All the series end on 31/07/2008. The dataset was build based on available data at ECA&D, and the starting dates were chosen in such a way that there was no more than one consecutive observation missing in the sample. For such observation, the arithmetic mean of the previous and the following observations was adopted. The daily Tmax temperatures are displayed in Figure 1, and we observe a clear seasonal pattern in all of them that is more visible when we compute the monthly mean Tmax temperatures in Figure 2. Figure 3 displays the annual means of Tmax, noting a slight trend in the temperatures in all cases. This implies that a deterministic trend may also be present in the monthly and daily data though it may be obscured by the presence of the seasonal structures. Though not reported, similar plots were obtained in case of the minimum temperatures.Daily maximum temperatures (Tmax) time series (in 0.1°C).
(a)(b)(c)(d)(e)(f)Monthly mean maximum temperatures (Tmax) time series (in 0.1°C).
(a)(b)(c)(d)(e)(f)Annual mean maximum temperatures (Tmax) time series (in 0.1°C).
(a)(b)(c)(d)(e)(f)In this section, we
focus on the monthly mean temperatures and estimate the model given by (7) and
(8). Table 1  focuses on Tmax, and it displays the estimates of the
intercepts and the time trend coefficients, the fractional differencing
parameters (with the 95% confidence band), and the seasonal AR coefficients.
The first thing we observe in this table is that the orders of integration are
in all cases constrained between 0 and 1, and the two integer orders of
differentiation (i.e., do=0 and do=1) are decisively
rejected in favour of fractional integration. We see that the highest estimates
correspond to Badajoz, with do=0.478, followed by Madrid (0.380)
and Valencia and Zaragoza (0.267 and 0.255, resp.), while Malaga
(do=0.166) and San
Sebastian-Donostia (do=0.118) present the lowest degrees of integration. In general, the degree of
seasonal dependence is high in all cases, ranging from 0.8430 (San Sebastian-Donostia)
to 0.9506 (Malaga).
If we focus on the deterministic terms, we first notice that the intercepts are
statistically significant in all cases, while the time trends are all
insignificantly different from zero. Thus, according to this specification, the
evidence of warming may be questionable though the coefficients are positive in
all cases: the highest values correspond to Badajoz (0.065), Madrid (0.051),
and Zaragoza (0.046) implying that temperatures have increased about 0.0065,
0.0051, and 0.0046 degrees Celsius per month, respectively. These values
represent an annual increase of about 0.078°C, 0.061°C, and 0.055°C, respectively,
for each of the three locations.Table 1
Estimates of the
time trend coefficients, the fractional differencing parameters, and the
seasonal AR(1) coefficients. Series:Tmax. (t-values in parenthesis. In brackets the 95%
confidence bands for the orders of integration.)
LocationIntercept (β0)Time Tr. (β1)Order diff. (d)Seas. AR (ρ)BADAJOZ204.8720.065360.4780.9116(6.089)(0.594)[0.324,  0.663]MADRID173.3420.051080.3800.9270(6.506)(0.748)[0.254,  0.546]MALAGA229.3420.022990.1660.9506(22.949)(0.465)[0.064,  0.300]SAN SEBASTIAN156.7870.010780.1180.8430(29.486)(1.042)[0.049,  0.201]VALENCIA211.4270.029210.2670.9186(18.271)(1.058)[0.173,  0.389]ZARAGOZA190.3830.046180.2550.9348(10.657)(0.905)[0.151,  0.394]Table2 is similar
to Table 1 but referring to the minimum temperatures (Tmin). The
same conclusions as in Table 1 are obtained here. Thus, fractional degrees of
differentation are observed in all cases, with values ranging from 0.155 (San
Sebastian-Donostia) to 0.384 (Badajoz); the intercepts
are all statistically significant, while the time trends, though positive, are
statistically insignificant. The highest coefficients are those corresponding
to Malaga (0.05101) and Valencia
(0.05046), while the lowest
value is obtained at San Sebastian-Donostia with β1=0.01268.Table 2
Estimates of the
time trend coefficients, the fractional differencing parameters and the
seasonal AR(1) coefficients. Series:Tmin. t-values in parenthesis. In brackets the 95%
confidence bands for the orders of integration.)
LocationIntercept (β0)Time Tr. (β1)Order diff. (d)Seas.  AR (ρ)BADAJOZ93.75920.021520.3840.8432(5.229)(0.427)[0.279,  0.518]MADRID84.7360.027790.3360.9123(4.927)(0.656)[0.234,  0.466]MALAGA128.8770.051010.1990.9401(11.953)(0.958)[0.110,  0.312]SAN SEBASTIAN97.4720.012680.1550.8613(17.197)(1.156)[0.071,  0.261]VALENCIA111.6690.050460.3620.9122(6.582)(1.182)[0.247,  0.519]ZARAGOZA81.6530.043340.2540.9288(6.248)(1.161)[0.156,  0.381]In Tables3 and 4,
we present the results, respectively, for Tmax and Tmin,
following the standard practice of assuming that the error term in (7) is I(0)
stationary. In other words, we impose a priori d=0 in (7) and though this
hypothesis was strongly rejected in Tables 1 and 2, it is interesting to check
how the time trend coefficients change in this case. The most noticeable fact
observed here is that the time trend coefficients, though smaller in magnitude
than in Tables 1 and 2, they are statistically significant in the majority of
the cases: Badajoz, Madrid, Valencia, and Zaragoza in case of Tmax,
and in all the locations except Badajoz in case of Tmin. The
seasonal structure is again highly dependent in all cases.Table 3
Estimates of the
time trend coefficients and the seasonal AR(1) coefficients assuming that the
process isI(0), i.e., d=0. Series: Tmax. t-values in parenthesis.
In bold, significant time trend coefficients.)
LocationIntercept (β0)Time Tr. (β1)Seas. AR (ρ)BADAJOZ224.1190.026750.9529(38.275)(1.698)MADRID182.2130.031340.9562(31.400)(2.194)MALAGA228.3800.024460.9590(41.844)(0.871)SAN SEBASTIAN157.2180.009800.8629(51.411)(1.594)VALENCIA213.1560.025380.9424(57.146)(2.765)ZARAGOZA193.9820.036240.9535(30.683)(1.938)Table 4
Estimates of the
time trend coefficients and the seasonal AR(1) coefficients assuming that the
process isI(0), i.e., d=0. Series: Tmin. t-values in
parenthesis. In bold, significant time trend coefficients.)
LocationIntercept (β0)Time Tr. (β1)Seas. AR (ρ)BADAJOZ94.2390.017320.8961(23.736)(1.621)MADRID89.3520.019460.9430(20.632)(1.826)MALAGA127.3810.053930.9511(24.101)(1.984)SAN SEBASTIAN98.2660.010610.8838(35.387)(1.901)VALENCIA120.0050.034220.9446(30.348)(3.516)ZARAGOZA83.6180.037700.9488(17.905)(2.730)Table5 reports the
annual increases in Tmax  according to the time trend coefficients
obtained in Tables 1 and 3, that is, with d estimated from the data and with d=0 a priori. We see that the values are much higher in the cases of Badajoz, Madrid,
Valencia, and Zaragoza if d is estimated rather than imposed. The same happens
with Tmin (see Table 6), observing higher values if d is estimated
rather than imposed in all locations with the exception of Malaga.Table 5
Annual increase in the temperatures (in 0.1°C). Series: Tmax.
LocationEstimateddd=0BADAJOZ0.78430.3210MADRID0.61290.3761MALAGA0.27580.2935SAN SEBASTIAN0.12930.1176VALENCIA0.35050.3045ZARAGOZA0.55410.4348Table 6
Annual increase in the temperatures (in 0.1°C). Series: Tmin.
LocationEstimateddd=0BADAJOZ0.25820.2078MADRID0.33340.2335MALAGA0.20410.6471SAN SEBASTIAN0.15210.1273VALENCIA0.60550.4106ZARAGOZA0.52000.4524In the final part
of this paper, we wonder if the differences in the results across locations may
be due to the different sample sizes employed in each case. Thus, in Tables7
and 8, we report the same estimates as in Tables 5 and 6 (i.e., annual
increases in 0.1°C)
but using the same sample size in all cases, with the starting date in August
1980. The values are now similar in the two cases, especially for Tmax (Table 7) though slightly smaller when using the estimated values of the d’s.
In case of Tmax, the highest warming effects correspond to the cases
of San Sebastian-Donostia (with an
increase of about 0.048°C per year), Valencia (0.041°C),
and Zaragoza (0.035°C),
while for Tmin, the highest increases are in San Sebastian-Donostia (0.049°C), Valencia (0.039°C), and Badajoz (0.036°C).Table 7
Annual increase in the temperatures (in 0.1°C) using the same sample
size. Series: Tmax.
LocationEstimateddd=0BADAJOZ0.21930.2520MADRID0.24790.2675MALAGA0.27580.2935SAN SEBASTIAN0.48500.4865VALENCIA0.41240.4673ZARAGOZA0.35070.3558Table 8
Annual increase in the temperatures (in 0.1°C) using the same sample
size. Series: Tmin.
LocationEstimateddd=0BADAJOZ0.36030.4067MADRID0.16230.2276MALAGA0.20410.6471SAN SEBASTIAN0.49590.5288VALENCIA0.39330.4864ZARAGOZA0.31030.3176These results are partially
consistent with those obtained in Brunet et al. [25]. They used a daily
adjusted dataset composed of the 22 longest Spanish temperature records, and,
on average, they found an annual increase of about 0.054°C for the time period
1973–2003 employing a
trend stationary representation based on I(0) errors. (Note, however, that they do not directly work with
observable data but with reconstructed data generated in the framework of the
EC EMULATE project.) This is in line with our results displayed in the
right columns in Tables 7 and 8 though our values are slightly smaller than
theirs, which may be explained by the lack of autocorrelation in Brunet et al.
[25] work.


## 4. Concluding Comments

In this paper, we have examined the monthly mean maximum and minimum
temperature (Tmax and Tmin) time series in six Spanish
locations, analysing the degree of dependence of the series in order to obtain
estimates of the time trend coefficients in a much more precise way than when
ignoring the time dependence across the data.The results
indicate that all series are fractionally integrated, the orders of integration
ranging between 0.166 and 0.478
in case ofTmax, and between 0.155 and 0.384 in case of  Tmin.
As expected, a strong degree of seasonal dependence is observed in all cases.
With respect to the time trend coefficients, the values are positive though they
are statistically insignificant, which is in contrast with the results based on
I(0) disturbances. The fact that the time trend coefficients are found to be
insignificant in our work does not necessarily rule out the hypothesis of warming
in the temperatures since this can be a consequence of the sample size used or
even misspecification of the model. Note that the model presented here, though
more general than other specifications based on purely deterministic trend
models, it may still be a simplistic approximation to the true data process.This paper can be extended in
several directions. Thus, for example, the seasonal structure may be examined
in a much more elaborated way, allowing for strong dependence at the seasonal
frequencies. This can be easily implemented by replacing the polynomial(1−L)d  in (4) by one of the form (1−L12)d. The possibility
of segmented trends in these data is another issue that will be examined in
future papers.



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