{"id":548,"date":"2023-10-18T18:22:17","date_gmt":"2023-10-18T18:22:17","guid":{"rendered":"https:\/\/blogs.mathworks.com\/finance\/?p=548"},"modified":"2024-03-12T15:02:02","modified_gmt":"2024-03-12T15:02:02","slug":"time-series-analysis-of-trends-in-global-carbon-emissions-from-fossil-fuels","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/finance\/2023\/10\/18\/time-series-analysis-of-trends-in-global-carbon-emissions-from-fossil-fuels\/","title":{"rendered":"Time Series Analysis of Trends in Global Carbon Emissions from Fossil Fuels"},"content":{"rendered":"<div class=\"rtcContent\">\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\"><em>The following post is from Hang Qian, Software Developer on the Econometrics Toolbox Team.<\/em><\/div>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">Global carbon emissions have increased dramatically since 1901. However, the annual growth rates were time-varying in the past century and slowed down in recent years, possibly due to international efforts on climate change under the Kyoto Protocol and Paris Agreement, and exogenous shocks by the pandemic.<\/div>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">This demo analyzes time-varying local trends of carbon emissions from fossil fuels and builds a multivariate dynamic factor model for carbon emissions from coal, gas, and oil.<\/div>\n<h2 style=\"margin: 3px 10px 5px 4px;padding: 0px;line-height: 20px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 20px;font-weight: bold;text-align: left\">Retrieve Data from Haver Analytics\u00ae<\/h2>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">The <a href=\"https:\/\/www.mathworks.com\/products\/datafeed.html\">Datafeed Toolbox&#x2122;<\/a> enables you to connect to <a href=\"https:\/\/www.haver.com\/\">Haver Analytics<\/a> databases to retrieve their economic data for analysis in <a href=\"https:\/\/www.mathworks.com\/products\/matlab.html\">MATLAB\u00ae<\/a>.<\/div>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">Connect to the Haver Analytics Environmental, Social, and Governance Indicators (ESG) database by specifying its database name file located on your local machine, for example, <span style=\"font-family: monospace\">C:\\DLX\\DATA\\ESG.dat<\/span>.<\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">h = haver(<span style=\"color: #a709f5\">&#8220;C:\\DLX\\DATA\\ESG.dat&#8221;<\/span>);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px 0px 4px 4px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">h.DataReturnFormat = <span style=\"color: #a709f5\">&#8220;timetable&#8221;<\/span>;<\/div>\n<\/div>\n<\/div>\n<div style=\"margin: 10px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">The ESG database contains yearly measurements of carbon emissions from burning fossil fuels from 1901, among other series.<\/div>\n<ul style=\"margin: 10px 0px 20px;padding-left: 0px;font-family: Helvetica, Arial, sans-serif;font-size: 14px\">\n<li style=\"margin-left: 56px;line-height: 21px;min-height: 0px;text-align: left\"><span style=\"font-family: monospace\">C001BB<\/span> \u2014 Total yearly carbon emissions from burning fossil fuels (billions of metric tons of CO2)<\/li>\n<li style=\"margin-left: 56px;line-height: 21px;min-height: 0px;text-align: left\"><span style=\"font-family: monospace\">C001BBTP<\/span> \u2014 Carbon-emission contribution from coal<\/li>\n<li style=\"margin-left: 56px;line-height: 21px;min-height: 0px;text-align: left\"><span style=\"font-family: monospace\">C001BBTG<\/span> \u2014 Carbon-emission contribution from gas<\/li>\n<li style=\"margin-left: 56px;line-height: 21px;min-height: 0px;text-align: left\"><span style=\"font-family: monospace\">C001BBTO<\/span> \u2014 Carbon-emission contribution from oil<\/li>\n<\/ul>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px;padding: 6px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">CO2Emissions = fetch(h,[<span style=\"color: #a709f5\">&#8220;C001BB&#8221; &#8220;C001BBTP&#8221; &#8220;C001BBTO&#8221; &#8220;C001BBTG&#8221;<\/span>]);<\/div>\n<\/div>\n<\/div>\n<div style=\"margin: 10px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\"><span style=\"font-family: monospace\">CO2Emissions<\/span> is a 4-column cell array containing timetable objects with the carbon-emission measurements for each retrieved series.<\/div>\n<h2 style=\"margin: 3px 10px 5px 4px;padding: 0px;line-height: 20px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 20px;font-weight: bold;text-align: left\">Visualize the Time Series<\/h2>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">Specify variables for each time series.<\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">dates = CO2Emissions{1}.Time;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">fossil = CO2Emissions{1}.GlobalCarbonEmissionsFromFossilFuelCombustion_Industry_Bil_Metr;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">coal = CO2Emissions{2}.GlobalCarbonEmissionsFromCoal_Bil_MetricTons_;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">oil = CO2Emissions{3}.GlobalCarbonEmissionsFromOil_Bil_MetricTons_;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px 0px 4px 4px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">gas = CO2Emissions{4}.GlobalCarbonEmissionsFromGas_Bil_MetricTons_;<\/div>\n<\/div>\n<\/div>\n<div style=\"margin: 10px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">Visually inspect the variables to determine their dynamic properties.<\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">figure<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">tl = tiledlayout(4,1);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">nexttile<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">plot(dates,coal)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">title(<span style=\"color: #a709f5\">&#8220;Coal&#8221;<\/span>)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">nexttile<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">plot(dates,oil)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">title(<span style=\"color: #a709f5\">&#8220;Gas&#8221;<\/span>)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">nexttile<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">plot(dates,gas)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">title(<span style=\"color: #a709f5\">&#8220;Oil&#8221;<\/span>)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">nexttile<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">plot(dates,fossil)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">title(<span style=\"color: #a709f5\">&#8220;Fossil&#8221;<\/span>)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">xlabel(tl,<span style=\"color: #a709f5\">&#8220;Year&#8221;<\/span>);<\/div>\n<\/div>\n<div class=\"inlineWrapper outputs\">\n<div style=\"border-radius: 0px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">ylabel(tl,<span style=\"color: #a709f5\">&#8220;Carbon Emissions (bil. metric tons)&#8221;<\/span>);<\/div>\n<\/div>\n<div style=\"padding: 10px 0px 6px 17px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px;line-height: 17.234px\">\n<div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsFigure\" data-testid=\"output_0\">\n<div class=\"figureElement eoOutputContent\"><img decoding=\"async\" class=\"figureImage figureContainingNode\" style=\"max-width: 560px;width: 100%;padding-bottom: 0px\" src=\"http:\/\/blogs.mathworks.com\/finance\/files\/2023\/10\/appLocalTrend.mlx-10-18-23_1.png\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"margin: 10px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">The time series plots indicate that carbon emissions have an upward trend, but a deterministic linear trend is inadequate for time series modeling.<\/div>\n<h2 style=\"margin: 3px 10px 5px 4px;padding: 0px;line-height: 20px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 20px;font-weight: bold;text-align: left\">Estimate Local-Trend <a href=\"https:\/\/www.mathworks.com\/help\/econ\/state-space-models.html?s_tid=CRUX_lftnav\">State-Space Model<\/a><\/h2>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"alignnone wp-image-587\" src=\"http:\/\/blogs.mathworks.com\/finance\/files\/2023\/10\/EstimateLocalTrendSSM.png\" alt=\"\" width=\"1110\" height=\"273\" \/><\/p>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">The local function <span style=\"font-family: monospace\">paramMap<\/span> is a parameter mapping function that defines the local trend model of a single variable in state-space form. Create a state-space model by passing the parameter mapping <span style=\"font-family: monospace\">paramMap<\/span> to <a href=\"https:\/\/www.mathworks.com\/help\/econ\/ssm.html\"><span style=\"font-family: monospace\">ssm<\/span><\/a> and estimate the unknown parameters by maximum likelihood using <span style=\"font-family: monospace\">estimate<\/span>.<\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">Mdl = ssm(@paramMap);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">y = fossil;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">params0 = [0.1; 0.1; 0.5; 0.01];<\/div>\n<\/div>\n<div class=\"inlineWrapper outputs\">\n<div style=\"border-radius: 0px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">EstMdl = estimate(Mdl,y,params0,Univariate=true,lb=zeros(4,1));<\/div>\n<div style=\"padding: 10px 0px 6px 17px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px;line-height: 17.234px\">\n<div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsTextElement\" style=\"width: 1168px;font-style: normal;font-size: 12px\" data-testid=\"output_1\">\n<div class=\"textElement eoOutputContent\" style=\"max-height: 261px;font-style: normal;font-size: 12px\" data-previous-available-width=\"1131\" data-previous-scroll-height=\"224\" data-hashorizontaloverflow=\"false\">Method: Maximum likelihood (fmincon)<br \/>\nSample size: 121<br \/>\nLogarithmic likelihood: 73.4142<br \/>\nAkaike info criterion: -138.828<br \/>\nBayesian info criterion: -127.645<br \/>\n| Coeff Std Err t Stat Prob<br \/>\n&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-<br \/>\nc(1) | 0.03895 0.00487 7.99895 0<br \/>\nc(2) | 0.08024 0.00389 20.62245 0<br \/>\nc(3) | 0.51501 0.98526 0.52271 0.60118<br \/>\nc(4) | 0.02905 0.55409 0.05243 0.95818<br \/>\n|<br \/>\n| Final State Std Dev t Stat Prob<br \/>\nx(1) | 10.00236 0.06387 156.60423 0<br \/>\nx(2) | 9.96258 0.04528 220.01330 0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"margin: 10px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">Based on the fitted model, the Kalman filter and smoother provide the state estimation. The first state variable corresponds to the local trend, and the second variable is the lagged variable.<\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">States = smooth(EstMdl,y);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px 0px 4px 4px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">localTrend = States(:,1);<\/div>\n<\/div>\n<\/div>\n<h2 style=\"margin: 3px 10px 5px 4px;padding: 0px;line-height: 20px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 20px;font-weight: bold;text-align: left\">Forecast Local-Trend State-Space Model<\/h2>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">The estimated local trend facilitates the out-of-sample forecast. Without new observations, assume that the current state of the growth momentum passes on to the forecasting periods. The Kalman filter provides the forecasted carbon emission levels.<\/div>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">Forecast the local trend model into a horizon of length <span style=\"font-family: monospace\">n<\/span>. Compute approximate 95% forecast intervals. Plot the forecasts.<\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">n = 10;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">[yf,yfMSE] = forecast(EstMdl,n,y);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">yfLB = yf &#8211; 2*sqrt(yfMSE);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">yfUB = yf + 2*sqrt(yfMSE);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">xf = datetime((year(dates(end))+1:year(dates(end))+n)&#8217;,1 ,1);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\"><\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">figure<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">plot(dates,y,<span style=\"color: #a709f5\">&#8220;-b&#8221;<\/span>, <span style=\"color: #0e00ff\">&#8230;<\/span><\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">\u00a0\u00a0\u00a0\u00a0dates,localTrend,<span style=\"color: #a709f5\">&#8220;&#8211;m&#8221;<\/span>, <span style=\"color: #0e00ff\">&#8230;<\/span><\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">\u00a0\u00a0\u00a0\u00a0xf,yf,<span style=\"color: #a709f5\">&#8220;.b&#8221;<\/span>, <span style=\"color: #0e00ff\">&#8230;<\/span><\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">\u00a0\u00a0\u00a0\u00a0xf,yfLB,<span style=\"color: #a709f5\">&#8220;&#8211;r&#8221;<\/span>, <span style=\"color: #0e00ff\">&#8230;<\/span><\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">\u00a0\u00a0\u00a0\u00a0xf,yfUB,<span style=\"color: #a709f5\">&#8220;&#8211;r&#8221;<\/span>);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">xlabel(<span style=\"color: #a709f5\">&#8220;Year&#8221;<\/span>)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">ylabel(<span style=\"color: #a709f5\">&#8220;Carbon Emissions (bil. metric tons)&#8221;<\/span>)<\/div>\n<\/div>\n<div class=\"inlineWrapper outputs\">\n<div style=\"border-radius: 0px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">title(<span style=\"color: #a709f5\">&#8220;In-Sample Fit and Out-of-Sample Forecast&#8221;<\/span>)<\/div>\n<\/div>\n<div style=\"padding: 10px 0px 6px 17px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px;line-height: 17.234px\">\n<div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsFigure\" data-testid=\"output_2\">\n<div class=\"figureElement eoOutputContent\"><img decoding=\"async\" class=\"figureImage figureContainingNode\" style=\"max-width: 560px;width: 100%;padding-bottom: 0px\" src=\"http:\/\/blogs.mathworks.com\/finance\/files\/2023\/10\/appLocalTrend.mlx-10-18-23_2.png\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2 style=\"margin: 3px 10px 5px 4px;padding: 0px;line-height: 20px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 20px;font-weight: bold;text-align: left\">Dummy Variables for Intervention Effects<\/h2>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">As countries are working towards carbon neutrality, it is of interest to consider the effects of the Kyoto Protocol and the Paris Agreement on carbon emissions. Also, exogenous shocks by the pandemic slowed down the world economy, which resulted in the reduction of greenhouse gas emissions. Create dummy variables that indicate interventions on climate.<\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">Kyoto = year(dates) &gt; 1997;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">Paris = year(dates) &gt; 2015;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">Covid = year(dates) &gt;= 2020;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">Dummy = [Kyoto Paris Covid];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px 0px 4px 4px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">nvar = size(Dummy,2);<\/div>\n<\/div>\n<\/div>\n<div style=\"margin: 10px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">The local function <span style=\"font-family: monospace\">paramMapDummy<\/span> is a parameter mapping that includes the exogenous predictors in the observation equation of the local-trend state-space model. Create a state-space model for the local-trend model with intervention dummy variables, and then fit the model to the total carbon emissions variable <span style=\"font-family: monospace\">fossil<\/span>.<\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">MdlDummy = ssm(@(params)paramMapDummy(params,fossil,Dummy));<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">params0Dummy = [params0; zeros(nvar,1)];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">[~,estParams] = estimate(MdlDummy,fossil,params0Dummy, <span style=\"color: #0e00ff\">&#8230;<\/span><\/div>\n<\/div>\n<div class=\"inlineWrapper outputs\">\n<div style=\"border-radius: 0px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">\u00a0\u00a0\u00a0\u00a0Univariate=true,lb=[zeros(4,1); -Inf(nvar,1)]);<\/div>\n<div style=\"padding: 10px 0px 6px 17px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px;line-height: 17.234px\">\n<div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsTextElement\" style=\"width: 1168px;font-style: normal;font-size: 12px\" data-testid=\"output_3\">\n<div class=\"textElement eoOutputContent\" style=\"max-height: 279px;font-style: normal;font-size: 12px\" data-previous-available-width=\"1131\" data-previous-scroll-height=\"268\" data-hashorizontaloverflow=\"false\">Method: Maximum likelihood (fmincon)<br \/>\nSample size: 121<br \/>\nLogarithmic likelihood: 89.3099<br \/>\nAkaike info criterion: -164.62<br \/>\nBayesian info criterion: -145.049<br \/>\n| Coeff Std Err t Stat Prob<br \/>\n&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-<br \/>\nc(1) | 0.05716 0.00594 9.62038 0<br \/>\nc(2) | 0.05491 0.00369 14.87828 0<br \/>\nc(3) | 0.52095 0.75995 0.68550 0.49303<br \/>\nc(4) | 0.02591 0.84557 0.03064 0.97555<br \/>\nc(5) | -0.17017 0.32139 -0.52948 0.59648<br \/>\nc(6) | -0.13397 0.15493 -0.86474 0.38719<br \/>\nc(7) | -0.73923 0.10720 -6.89565 0<br \/>\n|<br \/>\n| Final State Std Dev t Stat Prob<br \/>\nx(1) | 11.09709 0.04838 229.38155 0<br \/>\nx(2) | 10.74625 0.03542 303.37964 0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"margin: 10px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">The last three parameters correspond to the intervention effects. Empirical results confirm that carbon emissions have been reduced under the Kyoto Protocol and Paris Agreement. The pandemic also has a significant negative effect on greenhouse gas emissions.<\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper outputs\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">disp(estParams(5:7)&#8217;)<\/div>\n<div style=\"padding: 10px 0px 6px 17px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px;line-height: 17.234px\">\n<div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsTextElement\" style=\"width: 1168px;font-style: normal;font-size: 12px\" data-testid=\"output_4\">\n<div class=\"textElement eoOutputContent\" style=\"max-height: 261px;font-style: normal;font-size: 12px\" data-previous-available-width=\"1131\" data-previous-scroll-height=\"17\" data-hashorizontaloverflow=\"false\">-0.1702 -0.1340 -0.7392<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2 style=\"margin: 3px 10px 5px 4px;padding: 0px;line-height: 20px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 20px;font-weight: bold;text-align: left\">Common Local Trend for Three Series<\/h2>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">Assume that a common stochastic trend drives global carbon emissions from coal, gas, and oil. This is a dynamic factor model, in which the common local trend serves as an unobserved factor underlying three observed time series. To prepare for the multivariate extension of the local trend model, reparametrize the local trend as<\/div>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone wp-image-590\" src=\"http:\/\/blogs.mathworks.com\/finance\/files\/2023\/10\/CommonLocalTrend.png\" alt=\"\" width=\"936\" height=\"277\" \/><\/div>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">Y = [coal oil gas];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">MdlFactor = ssm(@(params)paramMapFactor(params,Y));<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">params0Factor = [0.1*ones(3,1); 0.1*ones(3,1); 0.5; 0.01];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">estimate(MdlFactor,Y,params0Factor,Univariate=true, <span style=\"color: #0e00ff\">&#8230;<\/span><\/div>\n<\/div>\n<div class=\"inlineWrapper outputs\">\n<div style=\"border-radius: 0px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">\u00a0\u00a0\u00a0\u00a0lb=zeros(size(params0Factor)));<\/div>\n<div style=\"padding: 10px 0px 6px 17px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px;line-height: 17.234px\">\n<div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsTextElement\" style=\"width: 1168px;font-style: normal;font-size: 12px\" data-testid=\"output_5\">\n<div class=\"textElement eoOutputContent\" style=\"max-height: 294px;font-style: normal;font-size: 12px\" data-previous-available-width=\"1131\" data-previous-scroll-height=\"283\" data-hashorizontaloverflow=\"false\">Method: Maximum likelihood (fmincon)<br \/>\nSample size: 121<br \/>\nLogarithmic likelihood: 271.547<br \/>\nAkaike info criterion: -527.094<br \/>\nBayesian info criterion: -504.728<br \/>\n| Coeff Std Err t Stat Prob<br \/>\n&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;<br \/>\nc(1) | 0.00499 0.00076 6.53820 0<br \/>\nc(2) | 0.00687 0.00103 6.66628 0<br \/>\nc(3) | 0.00335 0.00050 6.68231 0<br \/>\nc(4) | 0.16843 0.01224 13.75993 0<br \/>\nc(5) | 0.45911 0.03079 14.90903 0<br \/>\nc(6) | 0.01370 0.00061 22.48582 0<br \/>\nc(7) | 0.73514 0.06504 11.30249 0<br \/>\nc(8) | 0.00205 0.00172 1.19137 0.23351<br \/>\n|<br \/>\n| Final State Std Dev t Stat Prob<br \/>\nx(1) | 641.47037 2.89157 221.84121 0<br \/>\nx(2) | 627.97142 2.14629 292.58522 0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2 style=\"margin: 20px 10px 5px 4px;padding: 0px;line-height: 20px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 20px;font-weight: bold;text-align: left\">Conclusion<\/h2>\n<div style=\"margin: 2px 10px 9px 4px;padding: 0px;line-height: 21px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 14px;font-weight: 400;text-align: left\">A feature of the state-space model is to incorporate the unobserved components like local levels and trends in the state variables. As a nonstationary model, the random-walk local trend captures the time-varying trends that provide a good fit for the global carbon emission data.<\/div>\n<h2 style=\"margin: 3px 10px 5px 4px;padding: 0px;line-height: 20px;min-height: 0px;font-family: Helvetica, Arial, sans-serif;font-style: normal;font-size: 20px;font-weight: bold;text-align: left\">Local Functions<\/h2>\n<div style=\"background-color: #f5f5f5;margin: 10px 0 10px 0\">\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 4px 4px 0px 0px;padding: 6px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\"><span style=\"color: #0e00ff\">function <\/span>[A,B,C,D,mean0,Cov0] = paramMap(params)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">sigmaU = params(1);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">sigmaE = params(2);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">intercept = params(3);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">slope = params(4);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">A = [2 -1;1 0];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">B = [sigmaU;0];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">C = [1 0];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">D = sigmaE;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">mean0 = [intercept;intercept-slope];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">Cov0 = zeros(2);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\"><span style=\"color: #0e00ff\">end<\/span><\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\"><\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\"><span style=\"color: #0e00ff\">function <\/span>[A,B,C,D,mean0,Cov0,stateType,yDeflate] = paramMapDummy(params,y,Dummy)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">sigmaU = params(1);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">sigmaE = params(2);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">intercept = params(3);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">slope = params(4);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">dummyCoeff = params(5:end);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">A = [2 -1;1 0];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">B = [sigmaU;0];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">C = [1 0];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">D = sigmaE;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">mean0 = [intercept;intercept-slope];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">Cov0 = zeros(2);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">stateType = [2;2];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">yDeflate = y &#8211; Dummy*dummyCoeff;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\"><span style=\"color: #0e00ff\">end<\/span><\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\"><\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\"><span style=\"color: #0e00ff\">function <\/span>[A,B,C,D,mean0,Cov0,stateType,YDeflate] = paramMapFactor(params,Y)<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">sigmaU = params(1:3);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">sigmaE = params(4:6);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">intercept = params(7);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">slope = params(8);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">A = [2 -1;1 0];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">B = [1;0];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">C = [sigmaU, zeros(3,1)];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">D = diag(sigmaE);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">mean0 = [0;0];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">Cov0 = zeros(2);<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">stateType = [2;2];<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">linear = [intercept;0;0] + [slope;0;0] .* (1:size(Y,1));<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px;padding: 0px 45px 0px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\">YDeflate = Y &#8211; linear&#8217;;<\/div>\n<\/div>\n<div class=\"inlineWrapper\">\n<div style=\"border-radius: 0px 0px 4px 4px;padding: 0px 45px 4px 13px;line-height: 18.004px;min-height: 0px;font-family: Menlo, Monaco, Consolas, 'Courier New', monospace;font-size: 14px\"><span style=\"color: #0e00ff\">end<\/span><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p><script type=\"text\/javascript\">\n{ let css = '\/* Styling that is common to warnings and errors is in diagnosticOutput.css *\/.embeddedOutputsErrorElement {    min-height: 18px;    max-height: 550px;} .embeddedOutputsErrorElement 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However, the annual growth rates were&#8230; <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/finance\/2023\/10\/18\/time-series-analysis-of-trends-in-global-carbon-emissions-from-fossil-fuels\/\">read more >><\/a><\/p>\n","protected":false},"author":204,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[40,16,4],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/posts\/548"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/users\/204"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/comments?post=548"}],"version-history":[{"count":7,"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/posts\/548\/revisions"}],"predecessor-version":[{"id":596,"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/posts\/548\/revisions\/596"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/media?parent=548"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/categories?post=548"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/finance\/wp-json\/wp\/v2\/tags?post=548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}