Global Carbon di Oxide Emissions in Hamilton Filter Model

The paper examined the cyclical trends, seasonal variation and seasonal adjustment of global CO2 emission from 1970 to 2018 through the application of Hamilton regression filter model. ARIMA (4,0,0) forecasting model for 2030 has been added with the Hamilton filter model and observed that the new model is stable, stationary and significant in which volatility is being minimised and the heteroscedasticity problem is totally disappeared. Keywords— CO2 emission, Hamilton filter, seasonal adjustment, cyclical trend, ARIMA forecasting. JEL Classification codes-C13,C22,C50,O13,O40,Q54


INTRODUCTION
A series contains both the cyclical and seasonal behaviours. A seasonal behaviour is very strictly regular, meaning there is a precise amount of time between the peaks and troughs of the data. Cyclical behaviour on the other hand can drift over time because the time between periods isn't precise. A series with strong seasonality will show clear peaks in the partial auto-correlation function as well as the autocorrelation function, whereas a cyclical series will only have the strong peaks in the auto-correlation function. Patterns of peaks and troughs occur in both cyclical and seasonal analysis wherein seasonal trends the period between successive peaks (or troughs) is constant. But in cyclical trends the distance between successive peaks is irregular.
The decomposition of macroeconomic time series with trend and cyclical components is crucial to many macroeconomic concepts such as potential output, natural interest rate, share prices and inflation rate and so on. Econometric analysis requires filtering techniques which cater data sequences having short and strong trends. Filter can be used to decompose a time series into trend, seasonal and irregular components.
H.P.Filter (1997) has been used as a low pass smoothing filter in a numerous macroeconomic investigation setting smoothing parameter to certain arbitrary conventional values. It can decompose trend and cycles of the time series data to produce new time series such as potential GDP and output gap that are useful in macroeconomic discussion and in some recent public debate. It also used ARMA forecasting technique that deliver the required forecasted values that are needed in H.P. algorithm. The cyclical component obtained by H.P.Filter is unpredictable which indicates that we cannot do any investigation into the cyclical component because they are independent.
Many business cycles analysis, volatility of stock price indices and unemployment trends and cyclical patterns have been being studied through Baxter and King (1999) and Christiano and Fitzgerald (2003) models of decomposition of trends and cycles through filter till now. Therefore, the author endeavours to examine the patterns of trends and seasonal and cyclical fluctuations of global CO2 emission during 1970-2018 by applying the methodology of Hamilton Filter model(2018).The paper also finds the projected rise of global CO2 emission in 2050 and examined the forecasting ARIMA(p,d,q) model incorporating in the Hamilton regression filter model up to 2030. Lindsey(2020) reported that during 1750-2019,the atmospheric concentration of CO2 in ppm has been steadily increasing but the CO2 emission has been catapulting along with small cycles.Author mentioned that CO2 concentration was never exceeded over 300ppm until preindustrial revolution, but in 1958 it reached to 315 and in 2015 it accelerated to 400ppm.If global energy demand continues to grow,the CO2 emission is projected to exceed 900ppm by this century(2100).

III. METHODOLOGY AND SOURCE OF DATA
Semi-log regression model is applied to find linear trend which is given by, Where xi= variable to be estimated, a and b are constant,t =time,ui=random error and i=1,2,……..n.
Then Cochrane and Orcutt(1949) model was applied to find the projection of global CO2 emission for 2030.
vt+h=yt+h -yt is the difference i.e. how the series changes over h periods. For h=8, the filter 1-L h wipes out any cycle with frequencies exactly one year and thus taking out both long run trend as well as any strictly seasonal components.
It also applies random walk: yt=yt-1-εt where d=1 and ωt h =εt+h+εt+h-1+………….+εt+1 Regression filter reduces to a difference filter when applied to a random walk. Hamilton  vt=α+β0ivt-i+εt+βiεt-i+ѐt where vt=Hamilton regression filter residual, α is constant, β0i andβiare the coefficients of AR and MA. AR terms and MA terms are Vt-iand εt-irespectively where i=1,2………..n ,andѐt= random error/residuals. For convergence criteria, β0i andβishould be less than one and the values of AR and MA roots must be less than one for stability and stationary condition.
Data of global CO2 emission in Giga Ton from 1970 to 2018 were collected from the World Bank.

IV. OBSERVATIONS FROM THE MODELS
Global CO2 emission has been catapulting at the rate of 1.74% per annum significantly during 1970-2018 which is estimated by the trend line given below. In 2018,the global CO2 emission was 36.10gigaton which is projected to rise 63.503Gigaton in 2050 which is depicted in Figure 1 below. This non-linear trend is significant at 5% level having autocorrelation problem.
The global CO2 emission has the property of having five structural breaks in its path from 1970 to 2018 in 1977,1987,1996,2003 and 2010 all of which are significant at 5% level and all showed upward rising. It is depicted in Figure 2 below. The Hamilton regression filter is found as a good fit with high R 2 and significant F values with low AIC and SC respectively and all the coefficients are less than one.The residual vt is calculated as This equation can be treated as Hamilton filter model of global CO2 emission. Figure 3 where fitted and actual lines are shown as upward rising and the residual consists of 7 peaks and troughs respectively whose cyclical trend is diverging away from equilibrium with rising volatility.  Hamilton filter residual of global CO2 emission during 1970-2018 has been decomposed into trend, cycle, seasonal variation, remainder, seasonally adjusted cycle respectively in a panel of diagrams in Figure 4 below. In panel1,the Hamilton filter residual is shown as a cyclical pattern having seven peaks and troughs. In panel 2,the smooth cyclical trend is plotted which minimises cycles showing two peaks and troughs where amplitude is widening. In panel 3,the seasonal variation is shown where its patterns are v shaped. In panel 4,the remainder is presented which is fluctuating and in panel 5,the seasonally adjusted cycle is visible which looks as like as residual line. .00
In Figure 7,the forecast ARIMA(4,0,0) is depicted neatly where the forecast value for 2030 is (0.044088) in comparison with the value of (-0.0950) at 2018 and forecast line is moving upward up to 2027 and then it is declining although its amplitudes were increasing up till 2010. The heteroscedasticity test for residual of ARIMA(4,0,0) by applying ARCH(4) revealed that nR 2 =3.0166 because n=34,R 2 =0.088872 which is accepted by Chi-square(4) test whose probability is found as 0.55 ( where F=0.7058 whose probability is 0.59) after adjusting the sample during 1985-2018 which implied that no heteroscedasticity is accepted at null hypothesis.

V. LIMITATIONS AND SCOPE FOR FUTURE STUDIES
The paper did not compare the observations of decompositions of Hamilton regression filter model with the Hodrick-Prescott Filter model.If the quarterly or monthly data of co2 emission were taken then the observations from Hamilton filter will be more perfect and analytic. The details of autocorrelation and partial autocorrelation functions are not shown step by step of all lags.Application of Hamilton filter through ARIMA(p,d,q) forecast model may produce many debates. The clarifications of explaining amplitudes were not included in this analysis.Even,the explanation of volatility through appropriate GARCH model is also absent here.Therefore,there is enough scope of future research in the context of this paper. 2/global/scientific-projections/projections-for-carbondioxide).By 2100,cumulative global CO2 emission will exceed 1 trillion tons of carbons (3.67 trillion of CO2) threshold which according to IPCC will raise the earth's surface temperature to 2°C above the pre-industrial minimum and trigger dangerous interference. (http://euanmearns.com/global-co2-emissions-forecast-to-2100/). Paris Climate Agreement targets to limit temperature rise with in 1.5°C that might increase global GDP by 60-65% by 2030.The agreement claims to reduce CO2 emission by 20% along with 20% increase in renewable energy and energy efficiency respectively. As a whole, GHG emissions intensity must be reduced by 33-35% below 2005 level within 2030. So, to limit global temperature rise to 1.5°C,emissions must be below 25gigatons by 2030.As on 2020,February,194 states with EU agreed the Paris Agreement where 188 countries plus EU represent more than 87% of GHGs. China had committed to cut GHGs by 26-28% by 2025 and USA assured to cut through Clear Power Plan. India is also committed to reduce the emissions intensity of its GDP by 33 to 35 percent by 2030 from 2005 levels. India's emission intensity has reduced by 21% over the period 2005-2014. By 2030, India's emission intensity is projected to be even lower-in the range of 35 to 50 percent. (https://www.nrdc.org/experts/anjali-jaiswal/roadparis-indias-advancement-its-climate-pled).

VII. CONCLUSIONS
The paper concludes that the global CO2 emission has been catapulting at the rate of 1.74% per annum significantly during 1970-2018 and its projected value of 2050 is 63.503gigaton in comparison to 36.10 gigaton in 2018.Hamilton regression filter showed that global CO2 is cyclical in patterns having 7 peaks and troughs but the smooth cyclical trend consists of two distinct peaks and troughs which are widening in the amplitude and the seasonal variation is v shaped when it is decomposed by STL method. The seasonal variation is confirmed by the patterns of ACF and PACF whose Q stat are significant at 5% level. Automatic forecasting ARIMA(4,0,0) model for 2030 is incorporated with Hamilton filter and the result is appeared as convergent, stable and stationary with minimum volatility. This model is free from heteroscedasticity problem.