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Outline

Title
Abstract
Figures
Introduction
Material and Methods
Description of the Data
Data Analysis
Results and Discussion
Conclusions
References

Study of Tests for Trend in Time Series

Profile image of Thelma SáfadiThelma Sáfadi

2021, REVISTA BRASILEIRA DE BIOMETRIA

https://doi.org/10.28951/RBB.V39I2.471
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Abstract

The time series methodology is an important tool when using data over time. The time series can be composed of the components trend (Tt), seasonality (St) and the random error (at). The aim of this study was to evaluate the tests used to analyze the trend component, which were: Pettitt, Run, Mann-Kendall, Cox-Stuart and the unit root tests (Dickey-Fuller, Dickey-Fuller Augmented and Zivot and Andrews), given that there is a discrepancy between the test results found in the literature. The four series analyzed were the maximum temperature in the Lavras city, MG, Brazil, the unemployment rate in the Metropolitan Region of S~ao Paulo (RMSP), the Broad Consumer Price Index (IPCA) and the nominal Gross Domestic Product (GDP) of Brazil. It was found that the unit root tests showed similar results in relation to the presence of the stochastic trend for all series. Furthermore, the turning point of the Pettitt test diverged from all the structural breaks found through the Zivot and Andrews ...

Figures (14)
Rev. Bras. Biom., Lavras, v.39, n.2, p.311-333, 2021 - doi: 10.28951/rbb.v39i2.471 Thus, to obtain the value of the test statistic (7), one should start by estimating one of the equations: 19 or 20 or 21, by the OLS method. Subsequently, the estimated coefficient of Z;_1 is divided in each case by its standard error to obtain the statistic 7 as follows
Rev. Bras. Biom., Lavras, v.39, n.2, p.311-333, 2021 - doi: 10.28951/rbb.v39i2.471 Thus, to obtain the value of the test statistic (7), one should start by estimating one of the equations: 19 or 20 or 21, by the OLS method. Subsequently, the estimated coefficient of Z;_1 is divided in each case by its standard error to obtain the statistic 7 as follows
Table 1 - Results of the trend tests for the temperature series The Mann-Kendall test, indicated according to Goossens and Berger (1986, apud Back (2001)) when it comes to climate series, rejected the presence of the trend component along with the unit root and Cox-Stuart tests. Thus, diverging from the Run test. In addition, according to Morettin and Toloi (2018), the Run and Cox-Stuart tests should be used with caution.
Table 1 - Results of the trend tests for the temperature series The Mann-Kendall test, indicated according to Goossens and Berger (1986, apud Back (2001)) when it comes to climate series, rejected the presence of the trend component along with the unit root and Cox-Stuart tests. Thus, diverging from the Run test. In addition, according to Morettin and Toloi (2018), the Run and Cox-Stuart tests should be used with caution.
Figure 2 presents the maximum temperature series with the point of change by means of the Pettitt test and the time series of the T statistic of the ZA test for the detection of structural break.
Figure 2 presents the maximum temperature series with the point of change by means of the Pettitt test and the time series of the T statistic of the ZA test for the detection of structural break.
Table 2 - Results of the trend tests for the unemployment series Based on results from Table 2, it was found through unit root tests, ADF and Zivot and Andrews, that the series showed a stochastic trend and the series is integrated of order one, and a difference is required for it to become stationary.
Table 2 - Results of the trend tests for the unemployment series Based on results from Table 2, it was found through unit root tests, ADF and Zivot and Andrews, that the series showed a stochastic trend and the series is integrated of order one, and a difference is required for it to become stationary.
The second series analyzed was the unemployment rate in the Metropolitan Region of Sao Paulo. The unemployment rate and its autocorrelation function are shown in Figure 3.
The second series analyzed was the unemployment rate in the Metropolitan Region of Sao Paulo. The unemployment rate and its autocorrelation function are shown in Figure 3.
Thus, knowing that the unemployment series presented the trend component through the tests studied, there are the points of change resulting from the Pettitt test (FIGURE 4 (a)) and ZA test (FIGURE 4 (b)).
Thus, knowing that the unemployment series presented the trend component through the tests studied, there are the points of change resulting from the Pettitt test (FIGURE 4 (a)) and ZA test (FIGURE 4 (b)).
Table 3 - Results of the trend tests for the inflation series Like the temperature series, the Broad Consumer Price Index series rejected the presence of a unit root through the ADF test and Zivot and Andrews test, that is, the series did not show a stochastic trend. In addition, the ADF test did not show a significant value (p-value > 0.05) in relation to the estimation of the deterministic trend. Based on the Figure 5 (a), it was observed that the values of the series increase and decrease over time. In addition, the analyzed range of the IPCA series is related to the era of the real plan, which was implemented in Brazil in February 1994. Also analyzing the ACF of the inflation series, the observations slowly decline to zero, which characterizes a non-stationary series (FIGURE 5 (b)). For the inflation series, the tests differed in relation to the presence or not of the trend component, as can be seen in Table 3.
Table 3 - Results of the trend tests for the inflation series Like the temperature series, the Broad Consumer Price Index series rejected the presence of a unit root through the ADF test and Zivot and Andrews test, that is, the series did not show a stochastic trend. In addition, the ADF test did not show a significant value (p-value > 0.05) in relation to the estimation of the deterministic trend. Based on the Figure 5 (a), it was observed that the values of the series increase and decrease over time. In addition, the analyzed range of the IPCA series is related to the era of the real plan, which was implemented in Brazil in February 1994. Also analyzing the ACF of the inflation series, the observations slowly decline to zero, which characterizes a non-stationary series (FIGURE 5 (b)). For the inflation series, the tests differed in relation to the presence or not of the trend component, as can be seen in Table 3.
The last analyzed series refers to the data of the Brazilian nominal Gross Domestic Product (GDP). Figure 7 presented the series and its autocorrelation function (ACF). It was observed through Figure 7 (a) that the series presented a non-stationary behavior, since the values presented a increase and decrease over time. In addition, unlike climatic series, macroeconomic series have effects of policies, of economic crisis, so it was observed some peaks in the series due to these factors. For example,
The last analyzed series refers to the data of the Brazilian nominal Gross Domestic Product (GDP). Figure 7 presented the series and its autocorrelation function (ACF). It was observed through Figure 7 (a) that the series presented a non-stationary behavior, since the values presented a increase and decrease over time. In addition, unlike climatic series, macroeconomic series have effects of policies, of economic crisis, so it was observed some peaks in the series due to these factors. For example,
Table 4 - Results of the trend tests for the GDP series It is noted that the GDP series showed a stochastic trend through the ADF test and Zivot and Andrews test. However, the Mann-Kendall and Cox-Stuart tests rejected the hypothesis of the presence of the trend component, diverging from the other tests.
Table 4 - Results of the trend tests for the GDP series It is noted that the GDP series showed a stochastic trend through the ADF test and Zivot and Andrews test. However, the Mann-Kendall and Cox-Stuart tests rejected the hypothesis of the presence of the trend component, diverging from the other tests.
Despite the tests diverging in relation to the presence of the trend component, on the other hand, the structural breaks found through the Pettitt test and Zivot and Andrews test for the GDP series were equal. This result can be observed in Figure 8.
Despite the tests diverging in relation to the presence of the trend component, on the other hand, the structural breaks found through the Pettitt test and Zivot and Andrews test for the GDP series were equal. This result can be observed in Figure 8.
Table 5 - Summary of trend tests The Pettitt test is not present in Table 5, since the purpose of the test is to show the point of change and whether that point is significant or not. All points of change found through the Pettitt test were significant at 5%.
Table 5 - Summary of trend tests The Pettitt test is not present in Table 5, since the purpose of the test is to show the point of change and whether that point is significant or not. All points of change found through the Pettitt test were significant at 5%.

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Andreu Sansó

The Stata Journal: Promoting communications on statistics and Stata, 2016

The article extends the previous Hylleberg, Engle, Granger, and Yoo seasonal unit-root test commands (Baum and Sperling, 2001, https://ideas.repec.org/c/boc/bocode/s416502.html ; Depalo, 2009, Stata Journal 9: 422–438), which allow for the use of both quarterly and monthly data. It is also possible to choose between ordinary least-squares and generalized least-squares detrending (Rodrigues and Taylor, 2007, Journal of Econometrics 141: 548–573) procedures to deal with the deterministic part of the process. The command allows for the use of the sequential method proposed by Hall (1994, Journal of Business and Economic Statistics 12: 461–470) and Ng and Perron (1995, Journal of the American Statistical Association 90: 268–281), the adaptation of the modified Akaike information criteria to the case of seasonal unit-root tests (del Barrio Castro, Osborn, and Taylor, 2016, Econometric Reviews 35: 122–168) as well as the inclusion of Akaike information and Bayesian information criteria to...

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