This document summarizes a study that aimed to forecast demand for petroleum products in Ghana using time series models. The study analyzed annual demand data from 2000-2010 using nested conditional mean (ARMA) and conditional variance (GARCH, GJR, EGARCH) models. It proposed and studied a regression-based forecast filtering simulation to potentially improve forecast results. Key findings included that ARMA(2,2) models best forecasted mean demand, while GARCH(1,1)/GJR(1,1)/EGARCH(1,1) models best forecasted demand variance. The study compared forecast accuracy of these nested models to random walk and regression models using error and inequality metrics.
Developing model for fuel consumption optimization in aviationAlexander Decker
This document discusses developing a model for optimizing fuel consumption in the aviation industry. It begins with an introduction on the importance of fuel optimization given rising costs and limited resources.
The paper then reviews literature identifying three key dimensions that impact fuel consumption: technology/design, operations/performance, and alternative fuels. 96 variables across these dimensions are identified from previous research.
The document describes evaluating these variables through an expert survey. Factor analysis is then used to refine the variables down to the 23 most influential. These 23 variables form the basis of a neural network model developed to optimize fuel consumption for a specific aircraft, which is tested to validate the model.
Regressions allow development of compressor cost estimation models print th...zhenhuarui
This article presents 10 regression models to estimate costs of different components for pipeline compressor stations with varying capacities in different US regions. The models show large cost differences between regions, with the Western region having the highest costs. The models also indicate that all compressor station cost components have economies of scale, with unit costs decreasing as capacity increases. Limitations of the models include uneven data distribution and missing variables.
This document summarizes an analysis of energy and exergy utilization in India's transportation sector from 2005-2011. It finds that the overall energy efficiency of the sector is between 21.3-30.03% when considering various modes of transport including aviation, pipelines, roadways and railways. When compared to other countries, the transportation sector in India is the least efficient. The results are intended to help develop energy policies to improve efficiency and achieve energy security goals.
The document discusses oil prices and whether speculators are to blame for price increases. It notes that from 2007 to 2008, oil prices rose from $P1 to $P2 per barrel as investors entered the market. However, for the higher price to balance the market, a stock build equal to the shaded area (A-B) was required to account for increased demand D1. Therefore, if stock levels did not increase accordingly, the demand growth itself likely contributed more to higher prices than speculative investing activities.
1) Oil prices have decreased over 50% in recent months due to higher supply from US shale oil production and lower demand expectations.
2) Lower prices are good for consumers in the short-term through reduced energy bills, but could lead to less investment and job losses in the oil industry if prices remain low.
3) The level of oil production depends on both the price level and what policies are adopted regarding climate change, as limits to global warming could render some fossil fuel reserves economically unrecoverable.
In this paper we attempt to review the models, process, qualitative and quantitative methods of forecasting. We also review the needs and reasons for forecasting and what methods and approaches are employed for forecasting, requirements for forecasting, what are the shortcomings and business implications of forecasting.
This document discusses demand estimation and forecasting. It notes that demand estimation involves understanding the relationship between demand and its determinants, quantifying the nature of demand, and developing a demand function. The key aspects of demand estimation are identifying dependent and independent variables, developing a mathematical model, collecting primary and secondary data, estimating model parameters, and making estimates based on the model. However, the model cannot be exact due to qualitative consumer behavior. Demand forecasting estimates future demand values based on past data for purposes like production planning, materials purchasing, sales targeting, and financial planning. While useful, demand forecasting has uncertainties since demand depends on many variables and consumer psychology.
This document discusses demand estimation through regression analysis. It explains that regression analysis is used to model the relationship between a dependent variable (like quantity demanded) and independent variables (like price, income, etc.). By minimizing the errors between actual data points and the estimated regression line, regression analysis provides the "line of best fit" for estimating demand relationships. The document outlines different marketing research approaches used to collect demand data, including consumer surveys and market experiments. It also discusses the identification problem in directly observing demand from price-quantity data due to shifting supply curves.
Developing model for fuel consumption optimization in aviationAlexander Decker
This document discusses developing a model for optimizing fuel consumption in the aviation industry. It begins with an introduction on the importance of fuel optimization given rising costs and limited resources.
The paper then reviews literature identifying three key dimensions that impact fuel consumption: technology/design, operations/performance, and alternative fuels. 96 variables across these dimensions are identified from previous research.
The document describes evaluating these variables through an expert survey. Factor analysis is then used to refine the variables down to the 23 most influential. These 23 variables form the basis of a neural network model developed to optimize fuel consumption for a specific aircraft, which is tested to validate the model.
Regressions allow development of compressor cost estimation models print th...zhenhuarui
This article presents 10 regression models to estimate costs of different components for pipeline compressor stations with varying capacities in different US regions. The models show large cost differences between regions, with the Western region having the highest costs. The models also indicate that all compressor station cost components have economies of scale, with unit costs decreasing as capacity increases. Limitations of the models include uneven data distribution and missing variables.
This document summarizes an analysis of energy and exergy utilization in India's transportation sector from 2005-2011. It finds that the overall energy efficiency of the sector is between 21.3-30.03% when considering various modes of transport including aviation, pipelines, roadways and railways. When compared to other countries, the transportation sector in India is the least efficient. The results are intended to help develop energy policies to improve efficiency and achieve energy security goals.
The document discusses oil prices and whether speculators are to blame for price increases. It notes that from 2007 to 2008, oil prices rose from $P1 to $P2 per barrel as investors entered the market. However, for the higher price to balance the market, a stock build equal to the shaded area (A-B) was required to account for increased demand D1. Therefore, if stock levels did not increase accordingly, the demand growth itself likely contributed more to higher prices than speculative investing activities.
1) Oil prices have decreased over 50% in recent months due to higher supply from US shale oil production and lower demand expectations.
2) Lower prices are good for consumers in the short-term through reduced energy bills, but could lead to less investment and job losses in the oil industry if prices remain low.
3) The level of oil production depends on both the price level and what policies are adopted regarding climate change, as limits to global warming could render some fossil fuel reserves economically unrecoverable.
In this paper we attempt to review the models, process, qualitative and quantitative methods of forecasting. We also review the needs and reasons for forecasting and what methods and approaches are employed for forecasting, requirements for forecasting, what are the shortcomings and business implications of forecasting.
This document discusses demand estimation and forecasting. It notes that demand estimation involves understanding the relationship between demand and its determinants, quantifying the nature of demand, and developing a demand function. The key aspects of demand estimation are identifying dependent and independent variables, developing a mathematical model, collecting primary and secondary data, estimating model parameters, and making estimates based on the model. However, the model cannot be exact due to qualitative consumer behavior. Demand forecasting estimates future demand values based on past data for purposes like production planning, materials purchasing, sales targeting, and financial planning. While useful, demand forecasting has uncertainties since demand depends on many variables and consumer psychology.
This document discusses demand estimation through regression analysis. It explains that regression analysis is used to model the relationship between a dependent variable (like quantity demanded) and independent variables (like price, income, etc.). By minimizing the errors between actual data points and the estimated regression line, regression analysis provides the "line of best fit" for estimating demand relationships. The document outlines different marketing research approaches used to collect demand data, including consumer surveys and market experiments. It also discusses the identification problem in directly observing demand from price-quantity data due to shifting supply curves.
This document discusses various methods for classifying and forecasting demand. It categorizes demand based on whether goods are for consumers or producers, whether they are perishable or durable, and whether demand is derived, autonomous, for a firm or industry, or for total markets versus market segments. It then discusses demand forecasting and different quantitative and qualitative techniques for forecasting, including expert opinion methods, complete/sample consumer enumeration surveys, sales force opinion surveys, and consumer end use surveys. Each technique is described along with its advantages and disadvantages.
This document discusses various techniques for demand forecasting. It outlines factors that affect demand forecasting such as the time horizon and level of forecasting (macro, industry, or firm). Short-term forecasts are used for production scheduling and inventory management, while long-term forecasts are for strategic planning of new projects. Demand is determined by factors like income, price, demographics, and product characteristics. The document also describes qualitative methods like expert opinion and quantitative methods like time series analysis and regression for forecasting demand.
The document summarizes key concepts about forecasting from the 8th edition of the textbook "Operations Management" by William J. Stevenson. It discusses definitions of forecasting, the importance and uses of forecasts in various business functions. Methods of forecasting include qualitative judgmental forecasts, quantitative time series analysis, and associative models using explanatory variables. Specific forecasting techniques covered include naive forecasts, moving averages, exponential smoothing, trend analysis, and regression. The document also addresses evaluating forecast accuracy and controlling forecasts.
The document discusses various quantitative forecasting techniques including time series methods like moving averages and exponential smoothing. It provides examples of how to calculate 3-period moving averages and exponential smoothing forecasts using sample sales data. Exponential smoothing places more weight on recent observations compared to moving averages. The smoothing constant determines how quickly older data is discounted.
Interventions required to meet business objectives - from Forecasting Methods,
Forecast Accuracy / Error Reduction,
Integrate – Sales Forecast / Production to undertaking a CPFR
The document describes modeling and forecasting of tur production in India using the ARIMA model. Time series data of tur production from 1950-1951 to 2014-2015 was analyzed using time series methods. The autocorrelation and partial autocorrelation functions were calculated and the Box-Jenkins ARIMA methodology was used. The ARIMA(1,1,1) model was found to be appropriate based on diagnostic checking. Forecasts of tur production from 2015-2016 to 2024-2025 were then calculated using the selected ARIMA(1,1,1) model. The forecasts could help policymakers plan for future requirements of tur seed, imports, and exports.
IRJET - Crude Oil Price Forecasting using ARIMA ModelIRJET Journal
1) The document examines using an ARIMA model to forecast crude oil prices based on WTI crude oil price data from 1987 to 2020.
2) It finds the WTI oil price data is non-stationary and takes the log of the prices and removes trends and seasonality to make the data stationary.
3) It then identifies the best fitting ARIMA model as ARIMA(0,1,4) and applies it to forecast future oil prices, finding a mean squared error of 1.606 on test data.
This document summarizes a study that models crude oil prices using a Lévy process. The study finds that a MA(8) model best fits the time series properties of oil price returns. However, there is also evidence of GARCH effects. Therefore, the best overall model is a GARCH(1,1) with errors modeled by a Johnson SU distribution. This hybrid Lévy-GARCH process captures the temporal, spectral and distributional properties of the crude oil price data set.
Measuring the volatility in ghana’s gross domestic product (gdp) rate using t...Alexander Decker
This document summarizes a study that analyzed volatility in Ghana's GDP growth rate using GARCH models. The study found that GDP volatility exhibited characteristics like clustering and leverage effects. A GARCH(1,1) model provided a reasonably good fit to quarterly GDP data. Volatility and leverage effects were found to have significantly increased. The best fitting models for GDP volatility were ARIMA(1,1,1)(0,0,1)12 and ARIMA(1,1,2)(0,0,1)12 models.
APPLICATION OF COMPUTER FOR ANALYZING WORLD CO2 EMISSIONIJCSEA Journal
Global climate change due to CO2 emissions is an issue of international concern that primarily attributed
to fossil fuels. In this study, Genetic Algorithm (GA) is used for analyzing world CO2 emission based on the
global energy consumption. Linear and non-linear forms of equations were developed to forecast CO2
emission using Genetic Algorithm (GA) based on the global oil, natural gas, coal, and primary energy
consumption figures. The related data between 1980 and 2010 were used, partly for installing the models
(finding candidates of best weighting factors for each model (1980-2003)) and partly for testing the models
(2004–2010). Global CO2 emission is forecasted up to year 2030.
This document discusses forecasting gasoline prices in the United States using an ARIMA model. It provides background on gasoline, including its consumption and retail prices. The objective is to understand price volatility due to supply and demand constraints. Data on US gasoline prices from 1993-2014 is obtained from the EIA. After checking for stationarity and transforming the data, an ARIMA(1,1,3) model is identified as best. This model reveals gasoline prices are significantly related to past prices and unobserved factors. The validated model is used to forecast future gasoline prices.
This document summarizes a study that aimed to identify the best linear time series models to forecast paddy production in Batticaloa District, Sri Lanka. The study analyzed time series data on paddy production from 1980-2013 using various trend and time series models like exponential smoothing, Holt-Winters' method, and ARIMA. The Holt-Winters' method was found to be the best model based on the lowest Mean Absolute Percentage Error and residual analysis. The model forecasted paddy production values of 158695 tons for 2013/14 Maha season, 105481 tons for 2014 Yala season, and 213964 tons for 2014/15 Maha season.
Cascade networks model to predict the crude oil prices in IraqIJECEIAES
Oil prices are inherently volatile, and they used to suffer from many fluctuations and changes. Therefore, oil prices prediction is the subject of many studies in the field, some researchers concentrated on the key factors that could influence the prediction accuracy, while the others focused on designing models that forecast the prices with high accuracy. To help the institutions and companies to hedge against any sudden changes and develop right decisions that support the global economy, in this project the concept of cascade networks model to predict the crude oil prices has been adopted, that can be considered relatively as new initiative in the field. The model is used to predict the Iraqi oil prices since as its commonly known that the economy in Iraq is totally depend on oil. Therefore, it is vital to develop a better perception about the crude oil price dynamics because its volatility can cause a sudden economic crisis.
Efficient estimation of natural gas compressibility factor usingAbelardo Contreras
This document presents a new method for estimating natural gas compressibility factor (Z-factor) using least square support vector machine (LSSVM) modeling. The LSSVM model is developed and tested using a database of over 2,200 samples of sour and sweet gas compositions. The model predicts Z-factor as a function of gas composition, molecular weight, pressure, and temperature. Statistical analysis shows the LSSVM model outperforms existing empirical correlations with an average absolute relative error of 0.19% and correlation coefficient of 0.999. The accurate prediction of Z-factor is important for natural gas engineering calculations.
This document presents models developed to predict rates of different types of accidents (total, fatal, injury, damage) on a rural road in Saudi Arabia based on geometric design elements and traffic volume. Statistical analyses were conducted using accident and road data over 5 years. Multiple linear regression models were developed relating accident rates to average curvature, average gradient, number of horizontal/vertical curves, and average annual daily traffic. The models showed acceptable correlation and were found to be statistically significant, indicating relationships between accident rates and road/traffic characteristics. The developed models can be used for short-term accident prediction and identification of safety-influencing factors.
GWO-based estimation of input-output parameters of thermal power plantsTELKOMNIKA JOURNAL
This document presents a study that uses the Grey Wolf Optimizer (GWO) method to estimate the input-output parameters of the fuel cost curve for thermal power plants.
The fuel cost curve represents the relationship between a plant's fuel costs and power output, and needs to be periodically re-estimated due to temperature and aging effects. Accurately estimating the curve's parameters is important for economic dispatch calculations.
The study formulates parameter estimation as an optimization problem to minimize errors between actual and estimated fuel costs. It applies GWO to find the parameters for different fuel cost curve models using test data from three power plants. Simulation results show GWO provides better parameter estimates than other estimation methods.
Short-Term Load Forecasting Using ARIMA Model For Karnataka State Electrical ...IJERDJOURNAL
ABSTRACT: Short-term load forecasting is a key issue for reliable and economic operation of power systems. This paper aims to develop short-term electric load forecasting ARIMA Model for Karnataka Electrical Load pattern based on Stochastic Time Series Analysis. The logical and organised procedures for model development using Autocorrelation Function and Partial Autocorrelation Function make ARIMA Model particularly attractive. The methodology involves Initial Model Development Phase, Parameter Estimation Phase and Forecasting Phase. To confirm the effectiveness, the proposed model is developed and tested using the historical data of Karnataka Electrical Load pattern (2016). The forecasting error of ARIMA Model is computed and results have shown favourable forecasting accuracy.
Model for Evaluating CO2 Emissions and the Projection of the Transport Sector IJECEIAES
This document presents a system dynamics model to analyze carbon emissions from Bogota's transport sector under different policy scenarios from 2005 to 2050. The model considers population growth, private and public vehicle fleet growth, passenger trips, and resulting CO2 emissions. Four scenarios combine urban development and environmental policies. Scenario 1, with balanced development and environmental policies, shows the lowest emissions growth. Scenario 2, focused on development without environmental policies, shows the highest emissions and fleet growth. Implementing balanced and coordinated urban and environmental policies can effectively reduce transport sector emissions.
A Hybrid Model of MEMD and PSO-LSSVR for Steel Price ForecastingDr. Amarjeet Singh
Herein, we propose a novel hybrid method for forecasting steel prices by modeling nonlinearity and time variations together to enhance forecasting adaptability. The multivariate empirical mode decomposition (MEMD)–ensemble-EMD (EEMD) approach was employed for preprocessing to separate the nonlinear and time variation components of a hot-rolled coil (HRC) price return series, and a particle swarm optimization (PSO)-based least squares support vector regression (LSSVR) approach and a generalized autoregressive conditional heteroskedasticity (GARCH) model were applied to capture the nonlinear and time variation characteristics of steel returns, respectively. The empirical results revealed that compared with the traditional models, the proposed hybrid method yields superior forecasting performance for HRC returns. The evidence also suggested that in capturing the price dynamics of HRC during the COVID-19 pandemic period, the asymmetric GARCH model with MEMD–LSSVR outperformed not only standard GARCH models but also the EEMD-LSSVR models. The proposed MEMD–LSSVR–GARCH model for steel price forecasting provides a useful decision support tool for steelmakers and consumers to evaluate steel price trends.
Forecasting Crude Oil Prices by using Deep Learning Based ModelIRJET Journal
This document discusses using deep learning models to forecast crude oil prices. It proposes a new hybrid model that uses deep learning techniques like LSTM, CNN, and RNNs. The model is trained on West Texas Intermediate crude oil market data and shows improved accuracy in price predictions compared to other methods. The document also reviews several other studies applying machine learning and deep learning approaches to crude oil price and energy market forecasting.
This document discusses various methods for classifying and forecasting demand. It categorizes demand based on whether goods are for consumers or producers, whether they are perishable or durable, and whether demand is derived, autonomous, for a firm or industry, or for total markets versus market segments. It then discusses demand forecasting and different quantitative and qualitative techniques for forecasting, including expert opinion methods, complete/sample consumer enumeration surveys, sales force opinion surveys, and consumer end use surveys. Each technique is described along with its advantages and disadvantages.
This document discusses various techniques for demand forecasting. It outlines factors that affect demand forecasting such as the time horizon and level of forecasting (macro, industry, or firm). Short-term forecasts are used for production scheduling and inventory management, while long-term forecasts are for strategic planning of new projects. Demand is determined by factors like income, price, demographics, and product characteristics. The document also describes qualitative methods like expert opinion and quantitative methods like time series analysis and regression for forecasting demand.
The document summarizes key concepts about forecasting from the 8th edition of the textbook "Operations Management" by William J. Stevenson. It discusses definitions of forecasting, the importance and uses of forecasts in various business functions. Methods of forecasting include qualitative judgmental forecasts, quantitative time series analysis, and associative models using explanatory variables. Specific forecasting techniques covered include naive forecasts, moving averages, exponential smoothing, trend analysis, and regression. The document also addresses evaluating forecast accuracy and controlling forecasts.
The document discusses various quantitative forecasting techniques including time series methods like moving averages and exponential smoothing. It provides examples of how to calculate 3-period moving averages and exponential smoothing forecasts using sample sales data. Exponential smoothing places more weight on recent observations compared to moving averages. The smoothing constant determines how quickly older data is discounted.
Interventions required to meet business objectives - from Forecasting Methods,
Forecast Accuracy / Error Reduction,
Integrate – Sales Forecast / Production to undertaking a CPFR
The document describes modeling and forecasting of tur production in India using the ARIMA model. Time series data of tur production from 1950-1951 to 2014-2015 was analyzed using time series methods. The autocorrelation and partial autocorrelation functions were calculated and the Box-Jenkins ARIMA methodology was used. The ARIMA(1,1,1) model was found to be appropriate based on diagnostic checking. Forecasts of tur production from 2015-2016 to 2024-2025 were then calculated using the selected ARIMA(1,1,1) model. The forecasts could help policymakers plan for future requirements of tur seed, imports, and exports.
IRJET - Crude Oil Price Forecasting using ARIMA ModelIRJET Journal
1) The document examines using an ARIMA model to forecast crude oil prices based on WTI crude oil price data from 1987 to 2020.
2) It finds the WTI oil price data is non-stationary and takes the log of the prices and removes trends and seasonality to make the data stationary.
3) It then identifies the best fitting ARIMA model as ARIMA(0,1,4) and applies it to forecast future oil prices, finding a mean squared error of 1.606 on test data.
This document summarizes a study that models crude oil prices using a Lévy process. The study finds that a MA(8) model best fits the time series properties of oil price returns. However, there is also evidence of GARCH effects. Therefore, the best overall model is a GARCH(1,1) with errors modeled by a Johnson SU distribution. This hybrid Lévy-GARCH process captures the temporal, spectral and distributional properties of the crude oil price data set.
Measuring the volatility in ghana’s gross domestic product (gdp) rate using t...Alexander Decker
This document summarizes a study that analyzed volatility in Ghana's GDP growth rate using GARCH models. The study found that GDP volatility exhibited characteristics like clustering and leverage effects. A GARCH(1,1) model provided a reasonably good fit to quarterly GDP data. Volatility and leverage effects were found to have significantly increased. The best fitting models for GDP volatility were ARIMA(1,1,1)(0,0,1)12 and ARIMA(1,1,2)(0,0,1)12 models.
APPLICATION OF COMPUTER FOR ANALYZING WORLD CO2 EMISSIONIJCSEA Journal
Global climate change due to CO2 emissions is an issue of international concern that primarily attributed
to fossil fuels. In this study, Genetic Algorithm (GA) is used for analyzing world CO2 emission based on the
global energy consumption. Linear and non-linear forms of equations were developed to forecast CO2
emission using Genetic Algorithm (GA) based on the global oil, natural gas, coal, and primary energy
consumption figures. The related data between 1980 and 2010 were used, partly for installing the models
(finding candidates of best weighting factors for each model (1980-2003)) and partly for testing the models
(2004–2010). Global CO2 emission is forecasted up to year 2030.
This document discusses forecasting gasoline prices in the United States using an ARIMA model. It provides background on gasoline, including its consumption and retail prices. The objective is to understand price volatility due to supply and demand constraints. Data on US gasoline prices from 1993-2014 is obtained from the EIA. After checking for stationarity and transforming the data, an ARIMA(1,1,3) model is identified as best. This model reveals gasoline prices are significantly related to past prices and unobserved factors. The validated model is used to forecast future gasoline prices.
This document summarizes a study that aimed to identify the best linear time series models to forecast paddy production in Batticaloa District, Sri Lanka. The study analyzed time series data on paddy production from 1980-2013 using various trend and time series models like exponential smoothing, Holt-Winters' method, and ARIMA. The Holt-Winters' method was found to be the best model based on the lowest Mean Absolute Percentage Error and residual analysis. The model forecasted paddy production values of 158695 tons for 2013/14 Maha season, 105481 tons for 2014 Yala season, and 213964 tons for 2014/15 Maha season.
Cascade networks model to predict the crude oil prices in IraqIJECEIAES
Oil prices are inherently volatile, and they used to suffer from many fluctuations and changes. Therefore, oil prices prediction is the subject of many studies in the field, some researchers concentrated on the key factors that could influence the prediction accuracy, while the others focused on designing models that forecast the prices with high accuracy. To help the institutions and companies to hedge against any sudden changes and develop right decisions that support the global economy, in this project the concept of cascade networks model to predict the crude oil prices has been adopted, that can be considered relatively as new initiative in the field. The model is used to predict the Iraqi oil prices since as its commonly known that the economy in Iraq is totally depend on oil. Therefore, it is vital to develop a better perception about the crude oil price dynamics because its volatility can cause a sudden economic crisis.
Efficient estimation of natural gas compressibility factor usingAbelardo Contreras
This document presents a new method for estimating natural gas compressibility factor (Z-factor) using least square support vector machine (LSSVM) modeling. The LSSVM model is developed and tested using a database of over 2,200 samples of sour and sweet gas compositions. The model predicts Z-factor as a function of gas composition, molecular weight, pressure, and temperature. Statistical analysis shows the LSSVM model outperforms existing empirical correlations with an average absolute relative error of 0.19% and correlation coefficient of 0.999. The accurate prediction of Z-factor is important for natural gas engineering calculations.
This document presents models developed to predict rates of different types of accidents (total, fatal, injury, damage) on a rural road in Saudi Arabia based on geometric design elements and traffic volume. Statistical analyses were conducted using accident and road data over 5 years. Multiple linear regression models were developed relating accident rates to average curvature, average gradient, number of horizontal/vertical curves, and average annual daily traffic. The models showed acceptable correlation and were found to be statistically significant, indicating relationships between accident rates and road/traffic characteristics. The developed models can be used for short-term accident prediction and identification of safety-influencing factors.
GWO-based estimation of input-output parameters of thermal power plantsTELKOMNIKA JOURNAL
This document presents a study that uses the Grey Wolf Optimizer (GWO) method to estimate the input-output parameters of the fuel cost curve for thermal power plants.
The fuel cost curve represents the relationship between a plant's fuel costs and power output, and needs to be periodically re-estimated due to temperature and aging effects. Accurately estimating the curve's parameters is important for economic dispatch calculations.
The study formulates parameter estimation as an optimization problem to minimize errors between actual and estimated fuel costs. It applies GWO to find the parameters for different fuel cost curve models using test data from three power plants. Simulation results show GWO provides better parameter estimates than other estimation methods.
Short-Term Load Forecasting Using ARIMA Model For Karnataka State Electrical ...IJERDJOURNAL
ABSTRACT: Short-term load forecasting is a key issue for reliable and economic operation of power systems. This paper aims to develop short-term electric load forecasting ARIMA Model for Karnataka Electrical Load pattern based on Stochastic Time Series Analysis. The logical and organised procedures for model development using Autocorrelation Function and Partial Autocorrelation Function make ARIMA Model particularly attractive. The methodology involves Initial Model Development Phase, Parameter Estimation Phase and Forecasting Phase. To confirm the effectiveness, the proposed model is developed and tested using the historical data of Karnataka Electrical Load pattern (2016). The forecasting error of ARIMA Model is computed and results have shown favourable forecasting accuracy.
Model for Evaluating CO2 Emissions and the Projection of the Transport Sector IJECEIAES
This document presents a system dynamics model to analyze carbon emissions from Bogota's transport sector under different policy scenarios from 2005 to 2050. The model considers population growth, private and public vehicle fleet growth, passenger trips, and resulting CO2 emissions. Four scenarios combine urban development and environmental policies. Scenario 1, with balanced development and environmental policies, shows the lowest emissions growth. Scenario 2, focused on development without environmental policies, shows the highest emissions and fleet growth. Implementing balanced and coordinated urban and environmental policies can effectively reduce transport sector emissions.
A Hybrid Model of MEMD and PSO-LSSVR for Steel Price ForecastingDr. Amarjeet Singh
Herein, we propose a novel hybrid method for forecasting steel prices by modeling nonlinearity and time variations together to enhance forecasting adaptability. The multivariate empirical mode decomposition (MEMD)–ensemble-EMD (EEMD) approach was employed for preprocessing to separate the nonlinear and time variation components of a hot-rolled coil (HRC) price return series, and a particle swarm optimization (PSO)-based least squares support vector regression (LSSVR) approach and a generalized autoregressive conditional heteroskedasticity (GARCH) model were applied to capture the nonlinear and time variation characteristics of steel returns, respectively. The empirical results revealed that compared with the traditional models, the proposed hybrid method yields superior forecasting performance for HRC returns. The evidence also suggested that in capturing the price dynamics of HRC during the COVID-19 pandemic period, the asymmetric GARCH model with MEMD–LSSVR outperformed not only standard GARCH models but also the EEMD-LSSVR models. The proposed MEMD–LSSVR–GARCH model for steel price forecasting provides a useful decision support tool for steelmakers and consumers to evaluate steel price trends.
Forecasting Crude Oil Prices by using Deep Learning Based ModelIRJET Journal
This document discusses using deep learning models to forecast crude oil prices. It proposes a new hybrid model that uses deep learning techniques like LSTM, CNN, and RNNs. The model is trained on West Texas Intermediate crude oil market data and shows improved accuracy in price predictions compared to other methods. The document also reviews several other studies applying machine learning and deep learning approaches to crude oil price and energy market forecasting.
This document presents a time series model for the exchange rate between the Euro (EUR) and the Egyptian Pound (EGP) using a GARCH model. The author analyzes the time series data of the exchange rate for 2008 and finds that it exhibits volatility clustering where large changes tend to follow large changes. An ARCH or GARCH model is needed to capture the changing conditional variances over time. The author estimates several GARCH models and selects the GARCH(1,2) model based on statistical significance of coefficients and AIC values. Diagnostic tests show that the GARCH(1,2) model adequately captures the heteroskedasticity in the data. The fitted model is then used to predict future exchange rates
Tried Analysis and Forecast for Aviation Accident from a selective period of time and concluded my results through trend analysis, Mann Kendall abrupt and observed ACF and PACF of stationary series and analysis for the model accuracy. also used ARIMA model to forecast the time series of number of fatalities of civil aviation accidents.
Does price shock in electricity sector correct the consumptionAlexander Decker
This document summarizes a study that examines whether price shocks in Iran's electricity sector can correct consumption patterns. The authors estimate an econometric model using time series data from 1973-2007. They find that electricity demand has a price elasticity of less than 1, indicating it is a necessary good. As such, shocking prices is not an efficient way to rectify consumption patterns, since demand does not respond much to price changes for necessary goods. They also find little substitution between electricity and gas. The authors conclude that abrupt price changes alone will not effectively change consumption patterns in Iran's electricity sector due to the inelastic nature of demand and lack of substitutes.
A two-dimensional mathematical, model is developed to simulate the flow regime,
of the upper part of Dibdibba Formation. The proposed, conceptual model, which is
advocated to simulate the flow regime of aquifer is fixed for one layer, i.e. the activity
of the deeper aquifer is negligible. The model is calibrated using, trial and error
method. According to the calibration process, the hydraulic characteristics of the
upper aquifer has been identified the hydraulic conductivity in the study area ranged
(60-200) m/day while the specific, yield ranges, between, (0.08- 0.45).In this research,
the obtaining of the optimum management of groundwater flow by linked simulationoptimization
model. MODFLOW packages are used to simulate the flow in the system
of groundwater. This model is completed with an optimization model which is
depending on the Genetic Algorithm (GA) and Tabu Search (TS). Two management
cases (fixed well location and flexible well location with the moving, well option)
were considered by executing the model with adopting calibratedparameters. In the,
first case the objective function is converged to a maximum value of (3.35E+5 m3/day)
by using GA, while this function is closed to 4.00E+5 m3/day by using TS. The
objective function in second case converges to the maximum value (7.64E+05m3/day)
and (8.25E+05m3/day) when using GA and TS respectively. The choice option for the
optimal location of the wells in the second case leads to an increase of 106%
Similar to Forecasting demand for petroleum products in ghana using time series models (20)
Abnormalities of hormones and inflammatory cytokines in women affected with p...Alexander Decker
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Forecasting demand for petroleum products in ghana using time series models
1. Journal of Economics and Sustainable Development
ISSN 2222-1700 (Paper) ISSN 2222-2855 (Online)
Vol.4, No.17, 2013
www.iiste.org
Forecasting Demand for Petroleum Products in Ghana using Time
Series Models
1.
2.
3.
Godfred Kwame Abledu(PhD)1*, Boakye Agyemang2,and Semevor Reubin3
School of Applied Science and Technology, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana
Statistics Department, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana
Applied Mathematics Department, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana
* E-mail of the corresponding author: godfredabledu@gmail.com
Abstract
The objective of this study was to forecast and analyse the demand for petroleum products in Ghana using
annual data from 2000-2010. It focused on studying the feasibility forecast using nested conditional mean
(ARMA) and conditional variance (GARCH, GJR, EGARCH) family of models under such volatile market
conditions. A regression based forecast filtering simulation was proposed and studied for any improvements in
the forecast results.
Keywords: time Series models, regression model, forecast filtering, petroleum products, stationarity of time
Series data.
1.
Introduction
The demand for petroleum has increased in the last decade all over the world including the United
States, Middle Eastern nations, and other Asian nations, which has contributed in the high prices. The demand
for petroleum products in India has been increasing at a rate higher than the increase of domestic availability
(Banapurmath, et. al., 2011). At the same time, there is continuous pressure on emission control through
periodically tightened regulations particularly in metropolitan cities. Over the period 1980-2008, the price of
crude oil had fluctuated significantly, with a mean, minimum and maximum values of $ 32.31 (bbl), $ 12.72
(bbl) and $ 140 (bbl) respectively(WAMA(2008). The above statistics, in addition to a standard deviation of
17.08 over the sample period show that the prices of crude have always been characterized with severe
instability. Monthly fluctuations have in fact been more severe than these annual trends, with the price of crude
oil reaching $140 (bbl) in July 2008. Such instability in the prices of crude oil is bound to cause macroeconomic
distortions, especially in net-oil importing countries, like some ECOWAS countries (WAMA, 2008).
Ghana’s demand for crude oil and refined petroleum products has also been growing over the past
decade, and, the country’s demand for oil increased dramatically surprising many energy analysts. This growth
has been driven by socio-economic and technical factors that have influenced each category of final energy use.
These changing petroleum requirements are closely related to its high rates of growth in economic output and
personal incomes. The growth in incomes and the accompanying changes in petroleum demand are themselves
driven by an ongoing population shift from rural to urban areas. That growing urban population is demanding
new vehicles and new roads, raising the demand for energy in the transportation sector. The growth in output in
the industrial sector is driving the high demand for petrochemical feed stocks, including naphtha-based
petrochemicals, which are similar in composition to motor gasoline. Fluid catalytic cracking of heavy ends to
high-value liquid fuels is a common unit operation in oil refineries (Khan, et al., 2011). In this process, the heavy
feedstock that contains sulfur is cracked to light products
At the core of the development of every nation is petroleum. Currently, petroleum is among the most
important natural resources. Every nation uses petroleum products such as gasoline, jet fuel, liquefied petroleum
gas and diesel to run cars, trucks, aircraft and other vehicles. There is therefore the need to build stocks to meet
the seasonal demands. In the long term, blending non-petroleum additives into petroleum products such as
ethanol or other oxygenating agents into gasoline will be necessary to reduce crude oil demand. Efficient
refining capacity is a requirement to meet the demand of the nations. The past few years have witnessed an
increased impetus toward renewable energy to replace fossil fuels that has been driven both by environmental
and national security concerns (Hensel, 2011).
Many researchers analyzing the demand for petroleum products have looked at the aggregate
consumption of petroleum. Sa'ad(2009), used annual time series data for the period 1973 to 2007 in two
econometric techniques namely the structural time series model (STSM) and unrestricted error correction model
(UECM) to estimate demand for petroleum products. The results from both models revealed that the demand for
petroleum products is price inelastic. The robust optimization methodology is applied to deal with uncertainties
in the prices of saleable products, operating costs, product demand, and product yield in the context of refinery
operational planning (Leiras, et al , 2010 and Munim, et al.,2010
129
2. Journal of Economics and Sustainable Development
ISSN 2222-1700 (Paper) ISSN 2222-2855 (Online)
Vol.4, No.17, 2013
www.iiste.org
MATERIALS AND METHOD
Conditional mean models were used to forecast mean while the conditional variance models were used
for forecasting variance or volatility in the demand for oil. In this study, the nesting of these two models was
used first to forecast the conditional mean and then the conditional variance was estimated to get the value of
forecast demands for oil(Shrivastava, 2009). After analysis of data for quarter 1 (Q1r) and quarter 2 (Q2r),
ARMA(2,2) models had been found most appropriate for forecasting mean, hence ARMA(2,2) and
(GARCH(1,1)/GJR(1,1) /EGARCH(1,1)) were used for forecasting oil demand for 2012 and 2013.
The conditional mean and variance models have been viewed from a linear filtering perspective, and
then the application of the iterated conditional expectations to the recursive equations was conducted, one
forecast period at a time (Dhar, et. al. , 2009). For example for forecasting demand for the second quarter (Q2),
demand data were taken of quarter 1 (Q1) and forecast using above defined method has been done. Using
observed quarter 2(Q2) demand as dependent variable and (ARMA/GARCH) forecast as independent variable
regression model was obtained. Calculation of quarter 3 demand (Q3r) using observed data of Q2 was obtained
using nested models and the regression model obtained in the previous step was used to filter quarter 3 (Q3)
demand forecast.
Conditional variance models (Shrivastava, 2009), unlike the traditional or extreme value estimators,
incorporate time varying characteristics of second moment/volatility explicitly. The following models fall into
the category of conditional volatility models(Hull, 2006):
i.
ARCH (m) Model (Auto Regressive Conditional Heteroscedasticity)
ii.
EWMA Model (Exponentially Weighted Moving Average Model)
iii.
GARCH(a,b)Model(Generalized Autoregressive Conditional Heteroscedasticity).
iv.
EGARCH Model.
The stationarity was tested using ADF test with and without drift and trend, the AR(p) was determined
using PACF and MA(q) was determined using ACF. The number of lagged terms to be included in the model
was identified based on the minimum value of AIC and SBC criteria. The ARMA model was tested for ARCH
effects using the ARCH LM test and the measures of performance were calculated for the static and dynamic
forecasts made for the out-sample period. The in-sample data constituting 80% was used for estimating the
coefficients of the parameters and 20% the out-of- sample data was forecasted.
The forecasted results from random walk model, ARMA, ARMA-GARCH, ARMA-EGARCH models
using static and dynamic forecasting were compared based on the predictive power using the three forecasting
accuracy measures: Root Mean Square Error, Mean Absolute Error and Thiel Inequality Coefficient. Theil’s U
statistic was rescaled and decomposed into 3 proportions of inequality – bias, variance and covariance – such
that bias + variance + covariance = 1 and these measures were also calculated.
1.1. Autoregressive Moving Average( ARMA )Models
The Autoregressive Moving Average (ARMA) Models have been used by many researcher for
Z
forecasting(Shrivastava, et, al. , 2010; Abu and Behrooz, 2011) . Given a time series of data t , the ARMA
model is a tool for understanding and, perhaps, predicting future values in this series. The model consists of two
parts, an autoregressive (AR) part and a moving average (MA) part. The model is usually then referred to as the
ARMA (a, b) model where “a” is the order of the autoregressive part and “b” is the order of the moving average
part. The notation ARMA (a, b) refers to the model with “a” autoregressive terms and “b” moving-average
terms. This model contains the AR(a) and MA(b) models. A time series
a
Zt
follows an ARMA (1, 1) model if it
b
Z1 = k + ωt + ∑ β i Z t −i + ∑ α iωt −i
ω
i −1
i =1
, where { t } is a white noise series. The above equation
satisfies
implies that the forecast value is depended on the past value and previous shocks.
The notation MA(b) which refers to the moving average model of order b is written as
b
Z t = k + ωt + ∑ α iωt −i
i =1
and the notation AR(a) which refers to the autoregressive model of order a, is written
a
Z1 = k + ωt + ∑ β i Z t −i
as
i −1
where the
α1 ,...,αb
are the parameters of the model, µ is the expectation of Zt
ω ,...,ω
t −b are again, white noise error terms.
(often assumed to equal to 0), and the t
The Autoregressive Moving Average model(ARMA) is a method which can be used when the time
series variable is related to past values of itself. By regressing Zt on some combination of its past values, we are
130
3. Journal of Economics and Sustainable Development
ISSN 2222-1700 (Paper) ISSN 2222-2855 (Online)
Vol.4, No.17, 2013
www.iiste.org
able to derive a forecasting equation. We expect the autoregressive technique to perform reasonably well for a
time series that:
1.
Is not extremely volatile and does not contain extreme amounts of random
movement.
2.
Requires “q” short-term or medium- term forecast, that is less than two years
The fact that the autoregressive procedure does not perform well on a time series is not a serious
disadvantage. Practically all forecasting techniques perform poorly in this situation. Suppose we want to predict
the values of Zt using the previous observation, we use the expanded equation:
where t takes the values = 3, 4, 5, …. The values
b0 b1
, ,
and
Z t = b0 + b1Z t −1 + b2 Z t − 2 ,
b2 are the least squares regression estimates
Zt −1
obtained from any multiple linear regression. There are two predictor variables here, the lagged variables,
and
Zt − 2
. The above equation is a second order autoregressive equation because it uses the two lagged terms. In
^
th
general, the a – order autoregressive equation is given as :
Z t = b0 + b1Z t −1 + b2 Z t − 2 + ... + ba Z t − a
, and
t = a + 1, a + 2,...
The assumption underlying the ARMA model is that the future value of a variable is a linear function of
past observations and random errors. In this model it is possible to find an adequate description of data set. This
method consists of four steps:
1. Model identification,
2. Parameter estimation,
3. Diagnostic checking and
4. Forecasting.
In the identification step, it can be seen that a model generated from an ARMA process may contain some
autocorrelation properties, so there will be some potential models that can fit the data set but the best fitted
model is selected according to AIC information criteria. Stationarity is a necessary condition in building an
ARMA model used for forecasting, so data transformation is often required to make the time series to be
stationary. In this study, the unit root test, known as the Dickey and Fuller test (Shrivastava, et, al., 2010;
Gujarati, 2006; Abu and Behrooz, 2011 ), is used to test the stationarity of the time series.
Based on the result obtained, the data set is stationary at first difference even with the existence of
structural break. Once a tentative model is obtained, estimation of the model parameters is applicable. The
parameters are estimated such that an overall measure of errors is minimized. The third step is diagnostic
checking for model adequacy. Autocorrelation and also serial correlation of the residuals are used to test the
goodness of fit of the tentatively obtained model to the original data. When the final model is approved then it
will be used for prediction of future values of the oil demand.
1.2. The ARCH/GARCH Models
The first model that provides a systematic framework for volatility modeling is the ARCH model of Engle
(Gujarati, 2006). The basic idea of the ARCH model is that the shock
uncorrected but dependent; also the dependence of
lagged
values.
2
t
2
1 t−1
Specifically,
an
ARCH
(m)
2
m t−m (Gujarati, 2006), where
model
assumes
that
at = σtεt
α0 > 0 and αi ≥ 0
for
must satisfy some regularity condition to ensure that the unconditional variance of
ε
,
{ε t } is a sequence of independent and identically
distributed (i.i.d.) random variables with mean zero and variance 1,
αi
of an asset return is serially
αt can be described by a simple quadratic function of its
σ =α0 +α a +.....+α a
coefficient
αt
i > 0 . The
at
is finite. In
practices, t is often assumed to follow the standard normal or a standardized student t distribution or a
generalized error distribution.
GARCH models are used as a successful treatment to the financial data which often demonstrate timepersistence, volatility clustering and deviation from the normal distribution. Among the earliest models is Engel
(1982) linear ARCH model, which captures the time varying
feature of the conditional variance. Bollerslev (1986) develops Generalized ARCH (GARCH) model, allowing
for persistency of the conditional variance and more efficient testing. Engle and
131
4. Journal of Economics and Sustainable Development
ISSN 2222-1700 (Paper) ISSN 2222-2855 (Online)
Vol.4, No.17, 2013
www.iiste.org
Bollerslev (1986) invent the Integrated GARCH (IGARCH) model that provides consistent estimation under the
unit root condition. Engle, Lilien, and Robins (1987) design the ARCH-inMean (ARCH-M) model to allow for time varying conditional mean. Nelson’s (1990a & b) Exponential
GARCH (EGARCH) model allows asymmetric effects and negative coefficients in
the conditional variance function.
The leveraged GARCH (LGARCH) model documented in Glosten, Jagannathan and Runkle (1993)
takes into account the asymmetric effects of shocks from different directions. Since their introduction by Engle
(1982), Autoregressive Conditional Heteroskedastic (ARCH) models and their extension by Bollerslev (1986) to
generalised ARCH (GARCH) processes, GARCH models have been used widely by practitioners. At a first
glance, their structure may seem simple, but their mathematical treatment has turned out to be quite complex.
Although the ARCH is simple, it often requires many parameters to adequately describe the volatility process of
an asset return some alternative models must be sought. Shrivastava, et al. (2010) and Hull(2006) proposed a
useful extension known as the generalized ARCH (GARCH) model. An important feature of GARCH-type
models is that the unconditional volatility σ depends on the entire sample, while the conditional volatilities
are determined by model parameters and recent return observations.
σt
(ε t ) t ∈ ℤ be a sequence of independent and identically distributed (i.i.d.) random variables, and
p ∈ ℕ = {1, 2, 3,..., )
p ∈ ℕ o = ℕ ∪ {0}
α0 > 0 , α1 ,..., α p −1 ≥ 0 , α p > 0 ,
let
and
. Further, let
β1 ,..., β q −1 ≥ 0
β >0
( X t ) t ∈ℤ with volatility
and q
be non-negative parameters. A GARCH(p, q) process
(σ ) t ∈ℤ is then a solution to the equations:
process is t
Let
X t = σ tε t , t ∈ℤ
(1)
p
q
i =1
j =1
σ t2 = α t + ∑α i X t2−1 + ∑ β jσ t2−1
t ∈ℤ
,
(2)
(σ ) t ∈ℤ
(ε t ) t ∈ ℤ
t
where the process
is non-negative. The sequence
is referred to as the driving noise
sequence. GARCH (p, 0) processes are called ARCH (p) processes. The case of a GARCH (0, q) process is
excluded since in that case, the volatility equation (2) decouples from the observed process and the driving noise
sequence.
It is a desirable property that
is measurable with respect to
σ
σt
algebra generated by (ε t − h ) h ∈ ℕ . If this condition holds, we shall call the
GARCH (p, q) process causal. Then
generated by
(εt − h)h∈ℕ0 .
should depend only on the past innovations (ε t − h ) h ∈ ℕ , that is, it
Also,
( Xt ) is measurable with respect to σ
σt
is
independent of
algebra
(εt + h)h∈ℕ0 ,
and
σ (εt − h : h ∈ℕ0 ) ,
Xt
is independent of
σ (εt + h : h ∈ℕ ) , for fixed t. The requirement that all the coefficients α1,...,αp and β1,..., β q
2
σ
are non-
2
negative ensures that σ is non-negative, so that t can indeed be defined as the square root of σ .
Equation(1) is the mean equation and is specified as an AR(p) process. Equation (2) is the conditional
variance equation and it is specified as the GARCH(1, 1) process. Conditional variance models (Shrivastava,
2009), unlike the traditional or extreme value estimators, incorporate time varying characteristics of second
moment/volatility explicitly. By successively substituting for the lagged conditional variance into equation(2),
the following expression is obtained:
ht =
α0
∞
+ α1 ∑ i =1 βi −1ε t2−i
1− β
(3)
An ordinary sample variance would give each of the past squares an equal weight rather than declining
weights. Thus the GARCH variance is like a sample variance but it emphasizes the most recent observations.
Since
ht
is the one period ahead forecast variance based on past data, it is called the conditional variance. The
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squared residual is given by:
vt = ε t2 − ht
(4)
Equation (4) is by definition unpredictable based on the past. Substituting equation (4) into equation(2)
yields an alternative expression as follows:
ε t2 = ω + (α1 + β )ε t2−1 + vt − β vt −1
(5)
{a }
From the structure of the model, it is seen that large past squared shocks
m
2
t −i i =1
σ
2
t
a
imply a large
a
for the innovation t . Consequently, t tends to assume a large value (in modulus).
conditional variance
This means that, under the ARCH framework, large shock tend to be followed by another shock; because a large
variance does not necessarily produce a large realization. It only says that the probability of obtaining a large
variate is greater than that of a smaller variance. To understand the ARCH models, it pays to carefully study
ARCH (I) model
at = σtεt
,
σt2 =α0 +α1at2−1, where α0 > 0 and α
≥0
I
. The unconditional mean of
at
remains zero
because
E ( at ) = E E ( at / Ft −1 ) = E σ t E ( ε t ) = 0
The
conditional
variance
if
at
can
be
var ( at ) = E at2 = E E at2 / Ft −1 = E α 0 + α1at2−1 = α 0 + α1 E at2−1 .
E ( at2−1 ) = 0 var ( at ) = var ( at −1 ) = E ( at2−1 )
at
( )
(
)
(
)
( )
is a stationary process with
Because
var ( at ) = α 0 + α1 var ( at )
0 ≤ α1 ≤ 1
and
α
var ( at ) = 0 .
1 − α1 Since
(6)
as
obtained
the variance of
. In some applications, we need higher order moments of
at
at
. Therefore, we have
must be positive, we require
to exist and, hence,
α1 must also satisfy
some additional constraints. For instance, to satisfy its tail behavior, we require that the fourth moment of
finite. Under the normality assumption, we have
2
2
E at4 / Ft −1 = 3 E at2 / Ft −1 = 3 α 0 + α1at2−1
(Brockwell and Davis, 1996).
2
2
E at4 = E E at4 / Ft −1 = 3E α 0 + α1at2−1 = 3E α 0 + 2α 0α1at2−1 + α12 at4−1
Therefore ,
at
(
If
)
(
)
(
( )
(
)
)
at
(
)
(
)
is fourth – order stationary with
m4 = E ( at4 )
, then we have
m4 = 3 α + 2α 0α1 var ( at ) + α1m4
α
2
= 3α 0 1 + 2 1 + 3α12 m4
1 − α1
2
0
Consequently
m4 =
2
3α 0 (1 + α1 )
(1 − α1 ) (1 − 3α12 )
.
This result has two important implications: since the fourth moment of
must also satisfy the condition
1 − 3α13 > 0;
that is,
0 ≤ α12 ≤ 1
133
at
is positive, we see that
α1
3 ; and the unconditional Kurtosis of at is
is
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( )
E at4
var ( at )
2
=3
α 02 (1 + α1 )
(1 − α1 ) (1 − 3α12 )
Thus, the excess of
at
×
(1 − α1 )
α 02
2
1 − α12
=3
1 − 3α12>3
is positive and the tail distribution of
(7)
at
is heavier than that of a normal
a
distribution. In other words, the shock t of a conditional Gaussian ARCH (I) model is more likely than
Gaussian white noise series to produce “outcome”. This is in agreement with the empirical finding that “outliers”
appear more often in asset returns than that implied by an iid sequence of normal random variates. These
properties continue to hold for general ARCH models, but the formula becomes more complicated for higher
order ARCH models.
The condition
αi ≥ 0 in σ t2 = α 0 + α1at2−1 + .... + α m at2− m can be related. It is a condition to ensure
σ2
that the conditional variance t is positive for all t. The model has some weakness: it assume that positive and
negative shocks have the same effects on volatility because it depends on the square of the previous shocks. In
practices it is well known that price of a financial asset responds differently to positive and negative shocks.
The ARCH model is rather restrictive. For instance,
α12 of an ARCH (I) model must be in the interval
0 1
3 if the series has a finite fourth moment. The constraint becomes complicated for higher order ARCH
models. In limits, the ability of ARCH models with Gaussian innovations is to capture excess kurtosis. The
ARCH model does not provide any new insight for understanding the sources of variation of a financial time
series. It merely provides a mechanical way to describe the behavior of the conditional variation. It gives no
indications of what causes such behavior to occur. ARCH models are likely to over predict the volatility because
they respond slowly to large isolated shocks to the return series(Brockwell and Davis, 1996).
1.3. The EGARCH Model
This model is used to allow for symmetric effects between positive and negative asset returns. An EGARCH
(m, s) model can be written as (Dhar, et. Al. , 2009).
at = σtεt
wher
α0
( )
In σ
,
2
t
1 + β1B + ... + βs−1Bs−1
= α0
g ( εt −1 )
1 − α1B....αm Bm
is a constant, B is the back-shift (or lag) operator such that
1 + β1B + β s − B + ...
(8)
Bg ( ε t ) = g ( ε t −1 )
are polynomials with zeros outside the unit circle and have no common factors. By outside
the unit circle, we mean that absolute values of the zeros are greater than 1. Here, it is understood that
i > m and
βj = 0
and
αi = 0
for
for j > s. The latter constraint on αi and βi implies that the unconditional variance αt is finite,
σ2
ε
whereas its conditional variance t evolves over time, and t is often assumed to be a standard normal
standardized student-t distribution or generalized error distribution:
m
m
i =1
i =1
σ t2 = α 0 + ∑ α iα t2−1 + ∑ β jσ t2− j
(9)
reduces to a pure ARCH (m) model if S=0.
The αi and βi are referred to as ARCH and GARCH parameters respectively. The unconditional mean of
In(σ t2 )
is
α 0 . It uses logged conditional variance to relax the positiveness constraint of model coefficients. The
g (ε t ) enables the model to respond asymmetrically to the positive and negative lagged values of α . The
t
θ ≠ 0 . Since negative shocks tend to have larger impacts, we expect θ to be negative.
model is nonlinear if
use of
For higher order EGARCH model, the nonlinearity becomes much more complicated. This model can be used to
obtain multistep ahead volatility forecasts.
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1.4. Fitting the Parameters of the Model
Once a model is selected and data are collected, it is the job of the researcher to estimate the parameters
of the Model. These are values that best fit the historical data. It is hypothesized that the resulting model will
provide a prediction of the future observation. It is also hypothesized that all values in a given sample are equal.
The time series model includes one or more parameters. We identify the estimated values with a hat.
⌢
β is denoted β . The procedure also provide estimates of the standard
For instance, the estimated value of
σ
deviation of the noise, ε .
1.5. Forecasting from The Model
The main purpose of modeling a time series is to make forecasts which are then used directly for making
decisions. In this analysis, we let the current time be T, and assume that the demand for periods 1 through T are
known. We now want to forecast the demand for the period (T+ ς ). The unknown demand is the random
x
ς
variable X(T+ ς )., and its realization is
(T+ ).. Our forecast for the realization is
1.6. Measuring the Accuracy of the Model
The forecast error is the difference between the realization and the forecast. Thus
eς
x
xT + ς
.
x
T +ς
ς
=
.
(T+ )...Assuming the model is correct, then we have
eς
=
E [ X T +ς ] + ε ς −
(10)
xT +ς
(11)
We investigate the probability distribution of the error by computing its mean and variance. One
desirable characteristics of the forecast
xT +ς
is that it is unbiased. For an unbiased estimate, the expected value
of the forecast is the same as the expected value of the time series. Because
zero, an unbiased forecast implies
E[ε ς ]
2
is assumed to have a mean of
. The fact that the noise in independent from one period to the next
period means that the variance of the error is:
σ ε (ς ) = σ E (ς ) + σ
εt
Var[ε t ] = Var{E[ X T +ς ] − xT +ς } + Var[εT +ς ]
and
2
2
.
(12)
2.
3. DATA ANALYSIS AND RESULTS
3.1. Data and method of analysis
The data for the study was obtained from Tema Oil refinery. The AFC and the PACF of the time series
are shown in Figure 1. The PACF shows a single spike at the first lag and the ACF shows a tapering pattern. The
φ
positive, geometrically decaying pattern of the AFC, coupled with single significant coefficient 11 strongly
suggest an AR(1){=ARMA(1,0)} process.
The time series plot(Figure 2) of the standardized residuals mostly indicates that there is no trend in the
residuals, no outliers, and in general, no changing variance across time. The ACF of the residuals shows no
significant autocorrelations, an indication of a good result. The Q-Q plot is a normal probability plot. It doesn’t
look too bad, so the assumption of normally distributed residuals looks okay. The bottom plot gives p-values for
the Ljung-Box-Pierce statistics for each lag up to 20. These statistics consider the accumulated residual
autocorrelation from lag 1 up to and including the lag on the horizontal axis. The dashed blue line is at 0.05. All
p-values are above it indicating that this is a good result
The time series data ranged from January 2000 until December 2012. The coefficient of variation (V) was
used to measure the index of instability of the time series data. The coefficient of variation (V) is defined as:
V=
σ
Y
(13)
where σ is the standard deviation and
Y=
1 n
∑ Yt
n t =1
(14)
is the mean of petroleum prices changes.
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A completely stable data has V = 1, but unstable data are characterized by a V>1 (Telesca et al., 2008).
Regression analysis was used to test whether trends and seasonal factors exist in the time series data. The
existence of linear trend factors was tested through this regression equation:
2
Y = β0 + β1T + ε ε ∼ WN (0, σ )
,
(15)
Stationarity is tested using unit root test. The stochastic time series of interest is Zt. Taking the first
∆Z = A + A + A Z + µ
∆Z
t
1
2
3 t −1
t is the first difference, t is the trend
where
difference we have the following:
variable taking on the values from 1, 2, 3, …, n. and Zt-1 is the one period lagged value of the variable Z. The
null hypothesis is that A3, the coefficient of Zt-1 is zero. That is to say that the underlying time series is
nonstationary. This is called the unit root hypothesis. We proceed to show that a3, the estimated value of A3 is
zero. The unit root test is used since we have already assumed that the time series is nonstationary. The tau test
whose critical values are tabulated by the creators on the basis on Monte Carlo simulation are used(Gujarati,
2006 ). The rule for testing the hypothesis is that if the computed t(tau) value of the estimated A3 is greater (in
absolute value) than the critical Dickey Fuller(DF) tau values, we reject the root hypothesis, that is, we conclude
that the said time series is stationary. On the other hand, if the computed tau value is smaller (in absolute values)
than the critical DF tau values, we do not reject the unit hypothesis. In that case, the time series is nonstationary.
Data in Table 1 describe the nested ARMA(2,2) and GARCH(1,1) models. The forecasts have closest
mean with respect to observed mean while EGARCH(1,1) model has shown maximum correlation with the
observed Q2 returns. In all these cases Q1 returns data is used to calculate GARCH family model’s parameters.
3.2. Empirical Results
Eight model selection criteria as suggested by Ramanathan (2002) were used to choose the best
forecasting models among ARIMA and GARCH models, while the best time series methods for forecasting
demand for petroleum products was chosen based on the values of four criteria, namely RMSE, MAE, MAPE
and U-statistics (Table 2). Finally, the selected model was used to perform short-term forecasting for the next
twelve months for demand for petroleum products starting from January 2013 until December 2013.
The results showed that the coefficient of variation (V) of the time series data was 1.312 (V>1).
Because the V value was closed to 1, it was concluded that the time series data was stable (Telesca et al., 2008).
The results of the regression analysis had shown that positive linear trend factor existed in the time series data
but seasonal factor was not. Referring to the Augmented Dickey-Fuller tests results, the time series data of the
study was not stationary. But after the first order of differencing was carried out, the time series data became
stationary.
The double exponential smoothing method was used as the regression result had shown that positive
linear trend factor exists in the time series data. Double exponential smoothing models consisted with two
parameters which were symbolized as α for the mean and β for the trend. The best model of the double
exponential smoothing was selected based on the lowest value of MSE (Mean Square Error) from the
combination of α and β with condition 0<α, β<1.
The result showed that the combination α = 0.9 and β = 0.1 was the best forecasting model of double
exponential smoothing method (Table 3). The double exponential smoothing model was written in equation
F
= a + bh = 4764.2345 + h *(−32.3465)
form, from Table 4, as T + k
All models which fulfilled the criteria of p+q≤5 have been considered and compared in this study.
There were twenty ARIMA (p, d, q) models which fulfilled the criteria(Table 5). The parameters of the models
were estimated with the least square method. Parameters which were not significant at 5% confidence level were
dropped from the model. Using the eight model selection criteria suggested by Ramanathan (2002), the ARIMA
(3, 1, 2) model was selected as the best model among the other ARIMA models. However, the parameters of AR
(1) and MA (1) were found not significant and thus dropped from the model.
Identification and estimation of GARCH (p, q) models in this study were done by following the four
steps that were ARCH effect checking, estimation, model checking and forecasting. Four GARCH (p,q) models
were selected and compared, namely GARCH (1, 1), GARCH (1, 2), GARCH (2, 1) and GARCH (2, 2). Using
the eight model selection criteria suggested by Ramanathan (2002), the GARCH (1, 1) model was selected as the
best model among the other three GARCH models(Table 6). ARCH effect which was tested by using a
regression analysis existed in the ARIMA (3, 1, 2) model. What this meant was that the ARIMA (3, 1, 2) model
could be mixed with the best GARCH model (i.e., GARCH(1, 1)). Four model selection criteria were used to
select the best forecasting model from the four different types of time series methods. Based on the results of the
ex-post forecasting (starting from January until December 2013), the ARIMA (3, 1, 2)/GARCH (1, 1) model was
the best short-term forecasting model for the demand for petroleum products (Table 9).
A linear relationship between Q2r and GARCH family forecast for different combinations was also
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obtained. R-sq values in table of models gave the percentage of variations which the regression was able to
explain. It was clear that relationship (7) best explains the variations in the actual returns and forecasted returns,
while relationship (5) was the second best in explaining the variations while relationship (3) was the third best in
Table 8.
To determine whether there is significant difference for the mean demand and the standard deviation
values of the observed and predicted data for each month, a z-test (for means) and F-test (for standard
deviations) were applied (Haan, 1977; Devore and Peck, 1993). Since monthly mean values from observed and
predicted data is between z-critical table values (± 1.96 for 2 tailed at the 5% significance level), the data support
the claim that there is no difference between the mean values of observed and predicted data. Similarly, monthly
standard deviation values from observed and predicted data were smaller than F- critical table values at the 5%
significance level. Furthermore, these results show that the predicted data preserve the basic statistical properties
of the observed series.
The coefficient of correlation (R), which measures the strength of the association between 2 variables,
and the significance level (Rsig) related to the R of regression shows that there is a statistically significant linear
relationship between the observed and predicted data. On the other hand, the coefficient of determination (Rsquare), which is interpreted as the proportionate reduction of total variation associated with the use of the
predictor variable (the observed data in this study), and adjusted R-square measure, which presents the sample
response of the population for each regression, were close to one. In addition, the results (F-value and FSig)
concerning tests applied for determining whether the estimated regression functions adequately fit the data
emphasize that the association between the observed and predicted monthly data sequences is linear. Based on
these results, it is concluded that the selected best ARIMA model for each station can make accurate estimates.
CONCLUSION
Seven multiple regression relationships for different combination of nested ARMA / GARCH were
used to filter their Q3 demand forecast. Filtered result analysis shows improvements in the correlation coefficient
of the forecast demands and observed Q3 demands. Correlation coefficient is positive in some simulations,
which were always negative with GARCH family model’s forecast. Regression filtered results follow market
trend better, while other descriptive parameters like variance, skewness and kurtosis become more comparable to
actual Q3 demand. Therefore the proposed simulation framework under given observations to some extent has
improved nested conditional mean and variance models forecast of Q3 forecast for petroleum products under
such market conditions of 2013. However, it is not generally possible to get a definite relationship between
observed and forecasted result.
This study also investigated four different types of univariate time series methods, namely exponential
smoothing, ARIMA, GARCH and the mixed ARIMA/GARCH. The results showed that the mixed
ARIMA/GARCH model outperformed the exponential smoothing, ARIMA and GARCH for forecasting the
demand for petroleum products.
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Tables
Table 1: Descriptive Statistics of ARMA/GARCH forecast and Regression filtered forecast
Variables
Q3r
Mean
-0.0049
StDev
0.0398
Variance
0.0016
Minimum
-0.1160
Maximum
0.0790
Range
0.1950
Skewness
-0.0600
Kurtosis
-0.0500
Q3g2211f
-0.0067
0.0178
0.0003
-0.0056
0.0065
0.0121
1.0100
5.7100
Q3j2211f
0.0088
0.0707
0.0050
-0.0088
0.0110
0.0198
0.0400
-1.5100
Q3e2211f
-0.0072
0.0141
0.0002
0.0637
-0.0358
-0.0996
0.1111
-0.0767
Q3g2211rf
-0.0010
0.0283
0.0008
0.1173
-0.0698
-0.1872
0.0044
-0.8067
Q3j2211rf
0.0001
0.0566
0.0032
0.1709
-0.1038
-0.2748
0.0122
-1.5367
Q3e2211rf
-0.0112
0.1131
0.0128
0.2245
-0.1378
-0.3624
0.0005
-2.2667
Q3gjrf
0.0201
0.2262
0.0512
0.2781
-0.1718
-0.4500
0.0013
-2.9967
Q3jerf
-0.0313
0.4524
0.2047
0.3317
-0.2058
-0.5376
0.0134
-3.7267
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Table 2: Criteria for Assessing Forecast Accuracy
Double
Exponential
Smoothing
Criteria
Formula
RSME
GARCH(1, 1)
ARIMA(3, 1, 2)
/ GARCH(1, 1)
414.4506
ESS
N
ARIMA(3, 1, )
193 .3087
158.8801
155.5007
⌢
1
∑ Yt − Yt
n t =1
⌢
1 n Yt − Yt
× 100%
∑
n t =1 Yt
392.6509
1348.9835
122.8083
126.7645
6.8052
2.8309
2.4178
2.5528
RSME
⌢ 1 n
1
Y 2 + ∑Y 2
∑ n t =1
n t =1
0.0237
0.0192
0.0161
0.0157
n
MAE
MAPE
n
U- Statistics
Table 3: Error Sum of Square (ESS) according to α and β values
α
β
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
19345
81246
46341
31941
24941
21943
121426
31244
451427
0.2
18311
78242
47398
34742
23946
20846
110478
30142
410672
0.3
14567
61965
35943
33943
22647
20143
101495
30245
401421
0.4
13425
56349
26332
22946
21548
20959
100433
30456
400231
0.5
12833
45247
22344
21948
21041
22932
221409
32537
421502
Table 4: Output of the double exponential smoothing model
Parameters
Values
α
0.9800
β
0.1000
Sum of squared residuals
187364
Root mean squared error
234.3465
Mean
4764.2345
Trend
-32.3465
Table 5: Estimation of ARIMA (3, 1, 2)
Variables
Coefficient
Standard error
Z-statistic
p-value
Constant
19.2319
28.2345
0.6934
0.4562
AR(2)
-0.7843
0.0453
-18.4653
0.0001*
AR(3)
0.1287
0.0321
2.8675
0.0023*
MA(2)
0.9128
0.0234
20.3465
0.0002*
MA(3)
0.3465
0.0458
28.9874
0.0134*
*significant at 0.005 level
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ISSN 2222-1700 (Paper) ISSN 2222-2855 (Online)
Vol.4, No.17, 2013
Figure 1: AFC and PACF Plots
Figure 2: Time Series Plots of Residuals
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