Global Warming Signals in Brazil and its Macro-Regions - Trend Analysis of Distributional Characteristics Characteristics

8 The aim of this research is to study global warming signals across Brazil. This 9 investigation uses approximately 60 years of daily temperature data set and applies a 10 recent trend test proposed by Rivas and Gonzalo (2020) which analyses not only the 11 average but also diﬀerent distributional characteristics. Besides, the test provides 12 robust results for both I(0) and I(1) processes. We found signiﬁcant trends in almost 13 all characteristics in the analysis of the whole country. The mean and the maximum 14 are increasing over time and the dispersion measures indicate decreasing trends. For 15 the region analysis, we found out that, apart from the South, which does not appear 16 to be drastically aﬀected by global warming, the other regions present clear signs 17 of global warming.

1 Introduction 20 Climate Change issues, especially those related to Global Warming, are one of the 21 most discussed matters nowadays. Increasing temperature and extreme climate events, 22 regardless of their anthropogenic or natural origin, are perhaps the most challenging 23 problem in modern society. Given the importance of this matter and the consequences 24 it may bring, it is rather essential some basic characteristics inherent of the increasing 25 process of climate variables. 26 To study Climate Change we must deeply understand four issues questions, which form in its distributional characteristics. 37 Many studies have been carried out to analyze the possible trends in temperature 38 data as well as other climate variables over the years. While most of them have explored 39 the temperature for the whole globe or for Europe, relatively fewer have been focused on 40 Brazilian recorded temperature. Besides, considering the extension of Brazilian territory, 41 and the heterogeneous characteristic of the regions, it is important to analyze each one 42 of them to be sure that the possible effect of climate change is homogeneous throughout 43 the country. Therefore, our objective with this paper is to try to understand the first 44 question by analyzing the characteristics of temperature in Brazil. 45 The rest of the paper is organized as follows. In Section 2, we define Global Warming 46 and the trends needed to investigate Global Warming, as well as the Rivas  The tricky part of this paper is to determine a suitable definition of a trend. We   3 Global Warming Signals in Brazil and in its Macro-Regions

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Definition 3 (Stochastic trend (White and Granger, 2011)) Let X t be a stochas-73 tic process determined as a function trend, X t has a (weak) increasing trend in kth central moment.

Let
trend, X t has a (weak) increasing trend in kth absolute central moment.
(weak) increasing trend in quantile p. 85 Rivas and Gonzalo (2020) argue that from Definition 3, we can consider the stochastic 86 trend the way it is defined by the econometrics literature as a deterministic trend in the 87 second moment of the distribution. This is where the whole approach of the Rivas and 88 Gonzalo (2020) trend test begins since they develop a method able to detect deterministic 89 trends, and apply this method to different distributional characteristics. In order to fulfill some requirements of the trend test, we will consider that temperature 92 is a functional stochastic process X that must satisfy certain regularity conditions (the 93 state densities and distribution) in order for the quantiles to be estimated consistently.
For the purpose of this paper, we use a linear trend to represent the function h(t) in 98 Equation (1). Then we can write (1) as Rivas and Gonzalo (2020) highlights three important remarks referring to this defini-100 tion: (i) the estimation of (1) has to be understood as the linear least squared approx-101 imation of an unknown function h(t); (ii) the parâmeter β is the plim ofβ OLS ; (iii) in 102 practice, in order to test β = 0, it is recommended to use a robust HAC version of t β=0 .

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Therefore, in the empirical application, we estimate Equation (2) via OLS and implement 104 the HAC version of t β=0 as proposed by Newey and West (1986). 105 We also need the Definition 4 to satisfy the summability and Strength requirements 106 so the β have the right properties. The nexts two definitions guarantee that they are 107 fulfilled.

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Definition 5 (Order of Summability) A trend h(t) is summable of order θ, or (S(θ)) 109 if there exists a slowly-varying function L(T ), such that As we consider a linear function h(t) = t (then h(t) > 0) for the trend, we can rewrite 111 the Equation (3) as: therefore, θ = 1.

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Definition 6 (Trend Strength) A trend function h(t) is said to be stronger than an- Global Warming Signals in Brazil and in its Macro-Regions

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Definitions 5 and 6 allow us to proceed with the RG trend test in a given characteristic . First though, we have to call two propositions presented by Rivas and 117 Gonzalo (2020).

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Proposition In the OLS regression we have The OLS estimatorβ satisfies To proceed the behavioral analysis of the t-statistic t β = 0, we assume that the function Then, the t-statistic diverges at the rates In our case, h(t) = t, so the summability parameters are θ = 1 and γ = 1. Then, in 127 Equation (7),β = Op(1); but, t β=0 diverges as T → ∞. (2020) say that in this case, T 1 2β = Op(1) and t β = 0 diverges as T → ∞. Therefore, RG 131 trend test can detect the stochastic trend generated also by I(1) processes. is impossible to make any trustable forecast or risk analysis because of the confidence 142 interval is too high, which means we lost confidence in our predictions.

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Our approach has no intention to make such assumptions, as we aim to verify the 144 existence of climate change and not its implications in other variables. Even though, 145 testing for stationarity has an important role in this analysis. As we mentioned before, 146 stationarity can be altered by climate change, so we proceed with a stationarity test in 147 order to make an initial guess about the existence of climate change in Brazil. 148 We proceed with the augmented Dickey-Fuller (ADF) test for unit root proposed by 149 Dickey and Fuller (1979) and adapted by Said and Dickey (1984) where y is the temperature characteristic at time t, N is the number of observations, ρ is 155 the parameter to be estimated and ϵ is the error assumed to be white noise.If |ρ| ≥ 1, y t is nonstationary and the variance of y t tends to infinity with time (explodes). If |ρ| < 1, 157 then y t is trend stationary. The hypothesis of trend stationarity can be evaluated by 158 testing whether the absolute value of ρ is strictly less than one. In ADF test the null 159 hypothesis H 0 : ρ = 1 is tested against the one-sided alternative hypothesis.The standard 160 DF test is carried out by estimating (10) after subtracting y t−1 in both sides. Then it can 161 be written as: where δ = ρ − 1. The null and alternative hypotheses are H 0 : δ = 0 and H 1 : δ ≤ 0. The 164 maximum likelihood estimator of δ is denoted byδ and calculated as follows: The statistic for testing the null hypothesis that ρ = 1 in (10), which is equivalent to 166 δ = 0 in (11b), is based on the usual OLS (Ordinary Least Squares) t-test as: whereσδ is the usual OLS standard error defined as: where S e denotes the standard deviation of the OLS estimate of the residuals in the 169 regression model and it can be calculated as: The ADF approach differs from the standard DF test by allowing for general AR(p, q) models with unknown orders. In this case, the test equation is given by: The procedure follows the same steps as the standard DF presented above and p is  Next, we apply the trend test to the distributional characteristics of Brazilian tempera-204 ture. Table 2 reports the OLS trend coefficients and a HAC t β = 0 (significance level) and

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The negative coefficients for the standard deviation indicate that the country is facing 217 temperatures closer to the mean as the years go by. Also, as the mean temperature is 218 increasing, we can say that the temperature is less volatile around an increasingly average.

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Furthermore, the trend coefficients of the lower quantiles presented higher magnitude than 220 the upper ones, indicating that the temperature is decreasing its dispersion.

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In order to highlight the importance of the quantiles analysis, we performed auxiliary 222 tests to verify whether the dispersion between the quantiles are decreasing or not. We also estimate the Wald test to check for equality in trend coefficients.  between lower and middle quantiles presents no statistical significance.

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13 Global Warming Signals in Brazil and in its Macro-Regions

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As discussed in the introduction, Brazil is a country with continental proportions, 240 therefore, its climate is not homogeneous in all regions. Temperature discrepancies vary 241 considerably from the north (where a warm climate prevails throughout the year) to the 242 south (which has four well-defined seasons reaching quite low temperatures, especially 243 during the winter). In order to contemplate this heterogeneity, in the following section, 244 we conduct the unit root and trend test for each region.     The analysis for the whole country shows a significant and positive trend for the mean, and all quantiles seem to be increasing over time, which reveals that temperatures are 308 increasing systematically over the years. In other words, the closer to the equator line, 309 the more likely it is to feel the signs of Global Warming.

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The most affected regions by Global Warming effects seem to be the Southeast and The data used to support the findings of this study were taken from the Brazilian