A Review on some Advanced Models in Time Series Forecasting

A Review on some Advanced Models in Time Series Forecasting by Ankush Dey 002011701085 Bachelor in Production Engineering 3rd Year 1st Semester Jadavpur University under the guidance of Dr. Shankar Chakraborty Professor Department of Production Engineering Jadavpur University.

Acknowledgement The success and final outcome of this report on term paper required a lot of guidance and assistance and I feel extremely privileged to have received this immense amount of support from the conception of this term paper up until its successful completion. All that I have done is only due to such supervision and assistance and through this section I wish to express my utmost gratitude to everyone involved. I respect and thank Prof. Dr. Shankar Chakraborty, for providing me an opportunity to do the term paper work in the field of “Time Series Forecasting” and giving me all the support and guidance, which helped me complete the project duly. I am extremely thankful to him for taking time out of his busy schedule to assist me. I am also grateful to the rest of the faculty and staff of the Department of Production Engineering, for providing me with the necessary resources, opportunities and support. Finally, I would like to express my heartfelt thanks to my family and friends, for their constant support, understanding and encouragement throughout the time it took to complete this term paper. Their unwavering belief in my abilities and their willingness to listen and offer support have been a source of strength and inspiration throughout the process. Ankush Dey B. Prod. E - Ⅲ 1.

Contents 1. Introduction 4-5 1.1 Time Series 4 1.2 Time Series Forecasting 4 1.3 Importance of Time Series Forecasting in various fields 4 1.4 Variations in Time Series Data -- 5 2. Traditional Forecasting Model 5-8 2.1 Introduction - 5 2.2 Different kinds of Traditional Models - 6-7 2.2.1 Naive Model -- 6 2.2.2 Moving Average(MA) - 6 2.2.3 Autoregressive (AR) - 6 2.2.4 Exponential smoothing -- 6 2.2.5 Autoregressive Integrated Moving Average (ARIMA) - 6-7 2.2.6 Regression Analysis 7 2.3 Limitations of Traditional Models 7-8 3. Some Advanced Models in Time Series Forecasting -- 8-31 3.1 Vector Auto-Regression(VAR) - 8-17 3.1.1 Description - 8 3.1.2 Mathematical Expression - 8-9 3.1.3 Uses in various fields 9 3.1.4 Steps involved in applying VAR -- 10-16 2.

3.1.5 Advantages and Limitations - 17 3.2 Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) - 17-25 3.2.1 Description 17 3.2.2 Mathematical Expression - 18-19 3.2.3 Uses in various fields 19 3.2.4 Steps involved in applying VAR -- 20-24 3.2.5 Advantages and Limitations - 24-25 3.3 Holt’s Linear Trend Exponential Smoothing Model 25-31 3.2.1 Description -- 25 3.2.2 Mathematical Expression - 25-26 3.2.3 Uses in various fields -- 26-27 3.2.4 Steps involved in applying VAR - 27-30 3.2.5 Advantages and Limitations 30-31 4. Conclusion -- 31 5. References - 32-33 3.

1. Introduction 1.1 Time Series: A time series is a set of data points which is collected with its time of occurrence. It is a sequence of data points that are used to capture patterns in the underlying process. Time series data can be used to track changes and patterns over time, such as sales, weather, stock prices, and so on. 1.2 Time Series Forecasting: Time series forecasting is the process of using historical data to make predictions about future events. It is a statistical technique that uses past values of a time series to forecast future values. It is widely used in fields such as economics, finance, and operations research, to predict future trends, patterns and behaviors. Time series forecasting is a powerful tool for understanding and predicting future trends and patterns in data, and can be used to make important business decisions. 1.3 Importance of Time Series Forecasting in various fields: Time series forecasting can be used in any field where there is a need to predict future trends and patterns based on historical data. It is a valuable tool for making informed decisions, managing resources and planning for the future. ● Economic forecasting: Time series forecasting is used in economics to predict future trends in GDP, inflation, unemployment, and other key economic indicators. ● Finance: Time series forecasting is used in finance to predict stock prices, exchange rates, and other financial market trends. ● Sales forecasting: Time series forecasting is used in business and retail to predict future sales and plan inventory levels. ● Weather forecasting: Time series forecasting is used in meteorology to predict future weather patterns and plan for severe weather events. ● Energy demand forecasting: Time series forecasting is used in the energy industry to predict future demand for electricity and plan for power generation and distribution. ● Transportation and logistics: Time series forecasting is used in transportation and logistics to predict future demand for goods and services, and plan for transportation and delivery. ● Healthcare: Time series forecasting is used in healthcare to predict demand for medical services, plan for staffing and resource allocation, and forecast disease outbreaks. ● Manufacturing: Time series forecasting is used in manufacturing to predict demand for products and plan for production schedules. ● Social media: Time series forecasting is used in social media to predict future user engagement and plan for content creation and distribution. ● Sports: Time series forecasting is used in sports to predict future performance of teams and players, and make decisions about team management and player recruitment. 4.

1.4 Variations in Time Series Data: Variations in time series data refer to patterns or changes in the values of the time series over time. Analyzing the variations in time series data are crucial for making accurate predictions through the process of time series forecasting.There are four main types of variations or trends that can be observed in time series data: Secular, Seasonal, Cyclical, and Irregular variations. 1. Secular trend: A secular trend, also known as a long-term trend, is a gradual change in the value of a time series over a period of several years or decades. It reflects long-term changes in the underlying process generating the time series. For example, A company's sales revenue increasing over the years. 2. Seasonal variations: A seasonal variation is a pattern in a time series that is caused by regular, repeating events. It reflects short-term changes that occur with a predictable frequency, such as daily, weekly, monthly or yearly. For example, the demand for air-conditioners increases during summers and decreases during winters is a seasonal variation. 3. Cyclical variations: A cyclical variation in time series data often consists of four phases: prosperity, decline, depression, and recovery. Prosperity is characterized by high economic growth and low unemployment. Decline is characterized by a slowing of economic growth and an increase in unemployment. Depression is characterized by a severe downturn in economic activity, with high unemployment and low economic growth. Recovery is characterized by a return to economic growth and a decrease in unemployment. These phases are often referred to as the business cycle. 4. Irregular variations: An irregular variation is a pattern in a time series that does not fit any of the other categories. It reflects random and unpredictable variations that cannot be explained by any underlying process. For example, random variations in stock prices is an irregular variation. 2. Traditional Forecasting Models 2.1 Introduction: Traditional time series forecasting models are a set of methods used to make predictions about future values based on historical time series data and these models have been widely used for many years. These are based on statistical techniques such as moving averages, exponential smoothing, regression etc. They are commonly used for short-term forecasting and are useful for understanding and predicting patterns and trends in time series data. They are relatively simple to understand and implement, making them a popular choice for businesses and organizations looking to make predictions about future events. 5.

2.2 Different kinds of Traditional Models: 2.2.1 Naive Method: The Naive method is the simplest form of time series forecasting, which uses the last observation as the forecast for the next period. It is based on the assumption that the future will be the same as the recent past. This method is useful when there are no clear patterns in the data. 2.2.2 Moving Average(MA): The Moving average method is a simple time series forecasting method that uses the average of the past observations as the forecast for the next period. It is useful for identifying and removing short-term fluctuations from the time series data. It also uses past forecast errors as predictors to forecast future values. It is based on the assumption that the future value of a time series is a function of its past forecast errors. 2.2.3 Autoregressive (AR): The Autoregressive method is a time series forecasting method that uses past values of the time series as predictors to forecast future values. It is based on the assumption that the future value of a time series is a function of its past values. 2.2.4 Exponential smoothing: The Exponential smoothing method is a time series forecasting method that uses a weighted average of past observations, where more recent observations are given a higher weight. It is useful for identifying and removing short-term fluctuations and long-term trends from the time series data. 2.2.5 Autoregressive Integrated Moving Average (ARIMA): The Autoregressive Integrated Moving Average (ARIMA) method is a time series forecasting method that uses a combination of autoregressive(AR) and moving average(MA) models to make predictions. It combines the strengths of both AR and MA models and can handle time series data with both trends and seasonality. The basic mathematical formulation for an ARIMA(p,d,q) model is as follows: ?? = ? + φ1??−1 + φ2??−2 +.... + φ???−? + Θ1??−1 + Θ2??−2 +.... + Θ???−? + ?? Where, ● is the value of the time series at time t ?? ● c is a constant ● are the autoregressive coefficients for lagged values of the time series φ1, φ2, .... , φ? 6.

● are the moving average coefficients for the residual errors Θ1, Θ2, ...., Θ? ● is the residual error at time t ?? ● d is the number of differences applied to the time series ● p is the number of autoregressive terms ● q is the number of moving average terms The goal of the ARIMA model is to determine the values of p, d, and q that best fit the time series data, and use this model to make predictions about future values. The model is estimated using maximum likelihood or least squares method. It's worth noting that, in this model, d term is used to make the time series stationary by taking the difference between consecutive values in the time series. ARIMA model is a powerful method that can be used to analyze and forecast time series data with both trends and seasonality. It can also be used to analyze and forecast time series data with no patterns such as white noise. 2.2.6 Regression analysis: Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It is used in traditional forecasting models when there are one or more independent variables that affect the time series data. 2.3 Limitations of Traditional Models: The limitations of traditional forecasting models are as follows 1. Limited to linear relationships: Many traditional forecasting models assume a linear relationship between the independent and dependent variables, which may not be the case in many real-world situations. 2. Limited to univariate or multivariate time series: Traditional models are typically designed for univariate or multivariate time series data, and may not be well-suited for other types of data, such as cross-sectional data or panel data. 3. Limited to stationary time series: Many traditional models assume that the time series data is stationary, which means that the statistical properties of the data (mean, variance, etc.) do not change over time. However, many real-world time series are non-stationary, and traditional models may not be able to handle this type of data. 4. Limited to a specific set of patterns: Many traditional models are designed to handle specific types of patterns, such as trend or seasonality, and may not be able to handle other types of patterns or outliers in the data. 5. Limited in handling missing values: Many traditional models may not handle missing values well, which is a common problem in real-world data. 7.

These limitations of traditional models have led to the development of advanced forecasting methods such as VAR, ARFIMA etc, which are capable of handling non-linear relationships, multiple types of data, non-stationary time series and handling missing values, these models can also identify patterns and outliers in the data that traditional models may not be able to detect and also making more accurate predictions. 3. Some Advanced Forecasting Models: 3.1 Vector Auto-Regression (VAR) : 3.1.1 Description: Vector Autoregression (VAR) is a statistical model used for multivariate time series forecasting. In other words, it is used to forecast multiple variables that are possibly correlated with each other, at the same time. It is an extension of the univariate autoregression (AR) model, in which multiple time series are used as inputs to the model. VAR models assume that the current value of each variable is a linear combination of past values of all the variables in the system. The model estimates the coefficients of the linear relationship between the past values of all the variables and the current value of each variable. VAR models are particularly useful when there are multiple variables with complex relationships and interactions, such as in economic forecasting, where interest rates, inflation, GDP, and other variables are all interrelated. The VAR model can capture these relationships and provide more accurate forecasts than models that only consider one variable at a time. It is important to note that, VAR models are only appropriate for stationary time series, if the data is non-stationary, it should be made stationary by applying differencing or a transformation. 3.1.2 Mathematical Expression: The basic mathematical formulation for a VAR(p) model is as follows ?? = ? + ?1??−1 + ?2??−2 +.... + ????−? + ?? Where, ● is the vector of the value of the time series of all the variables at time t ?? ● is a vector of constants ? ● ?1, ?2, .... , ??are the coefficients of the linear relationship between the past or lagged values of all the variables and the current value of each variable 8.

● are the vectors of past or lagged values of all the variables in the ??−1, ??−2, .... , ??−? system ● is a vector of residual errors where ~ N(0,Σ) ?? ?? ● is the number of lags included in the model ? The equation can be represented in matrix form as ?? = ? + ? ??−1+ ?? Where, ● is the matrix of the value of the time series of all the variables at time t ?? ● is a matrix of constants ? ● is the matrix of past or lagged values of all the variables in the system ??−1 ● is a matrix of coefficients ? ● is a matrix of error terms ?? To estimate the coefficients of the VAR model, one can use the method of ordinary least squares (OLS) or maximum likelihood estimation (MLE). Once the coefficients are estimated, the VAR model can be used to forecast future values of the variables by plugging in the estimated coefficients and the past values of the variables. 3.1.3 Uses in various fields: Vector Autoregression (VAR) models are used in a variety of fields to analyze and forecast multivariate time series data. Some examples of where VAR models are used include: ● Economics: VAR models are used in macroeconomics to analyze the relationships between different economic variables, such as GDP, inflation, and interest rates. ● Finance: VAR models are used in finance to analyze the relationships between different financial variables, such as stock prices, exchange rates, and interest rates. ● Weather forecasting: VAR models are used in meteorology to analyze the relationships between different weather variables, such as temperature, precipitation, and wind speed. ● Marketing: VAR models are used in marketing to analyze the relationships between different marketing variables, such as sales, advertising, and customer satisfaction. ● Ecology: VAR models are used in ecology to analyze the relationships between different environmental variables, such as temperature, precipitation, and species populations. ● Political Science: VAR models are used in political science to analyze the relationships between different political variables, such as voter turnout, public opinion, and election outcomes. In summary, VAR models can be used in a variety of fields to analyze the relationships between different variables and make predictions about future values, specifically in fields where multiple time series data is available. 9.

3.1.4 Steps Involved in applying VAR: Step 1 : Checking for Stationarity of the data: First step in the process of using a Vector Autoregression (VAR) model is to check the stationarity of the time series data. Stationarity of a time series means that the statistical properties of the series such as mean, variance, and autocorrelation are constant over time. VAR models are only appropriate for stationary time series, so it is important to check for stationarity before using the model. There are several ways to check for stationarity, but one common method is to use the Augmented Dickey-Fuller (ADF) test. The ADF test is a statistical test that tests the null hypothesis that a time series is non-stationary. The hypothesis taken in the ADF test is: ?0 = ?ℎ? ?????? ℎ?? ? ???? ???? (??? − ??????????) ?1 = ?ℎ? ?????? ?????'? ℎ??? ? ???? ???? (??????????) Now the test statistic related to the test is compared to a critical value from the critical values in the ADF test table.If the test statistic is less than the critical value, the null hypothesis is rejected, and the time series is considered to be stationary. If the test statistic is greater than the critical value, the null hypothesis is not rejected and the time series is considered to be non-stationary. It is important to note that, ADF test assumes that the error term is normally distributed. It is also worth mentioning that there are other tests to check for stationarity such as the KPSS test or PP test. Step 2: Conversion from Non-Stationary to Stationary data: If a time series is non-stationary, it must be made stationary before using a Vector Autoregression (VAR) model. There are several ways to make a non-stationary time series stationary, including differencing and transformation. 1. Differencing: One of the most common ways to make a non-stationary time series stationary is by differencing. The idea behind differencing is to remove the trend and seasonality from the time series. The most common form of differencing is first-order differencing, which involves subtracting the value of the time series at a previous time point from the current value. The mathematical formula for first-order differencing is: Δ?? = ?? − ??−1 Where, ● is the value of the time series at time t ?? ● is the value of the time series at the previous time point ??−1 10.

Second-order differencing can also be used, it involves subtracting the first-order differenced series from the original series. The mathematical formula for second-order differencing is: Δ 2?? = Δ?? − Δ??−1 Where, ● is the first order differencing value at time t Δ?? ● is the first order differencing value at time t-1 Δ??−1 Higher-order differencing can also be used, but it's less common. It's important to note that differencing can be used to remove trends or seasonality, but it also removes the level information in the original data 2. Transformation: Another way to make a non-stationary time series stationary is by using a transformation, such as log or Box-Cox transformation. The Log transformation is applied to the time series by taking the natural logarithm of each value in the series. The mathematical formulation of the log transformation is: (??) ??? = ???(??) Where, ● is the value of time series after transformation (??) ??? ● is the value of the time series at time t ?? The Box-Cox transformation is a power transformation that can be used to stabilize the variance of a time series. The Box-Cox transformation is applied to the time series by raising each value in the series to a power of λ. The mathematical formulation of the Box-Cox transformation is: (??) ??? = ?? λ−1 λ Where, ● is the value of time series after transformation (??) ??? ● is the value of the time series at time t ?? ● λ is a parameter that is chosen to minimize the variance of the transformed series. It's worth noting that, while differencing and transformation are the most common ways to make a non-stationary time series stationary, they are not the only methods. Other techniques include seasonal decomposition, trend removal and more advanced methods like Kalman filters. The best method to use depends on the nature of the data and the specific problem we are trying to solve. 11.

Step 3: Selection of Lag Order(p) of the model: The third step in the process of using a Vector Autoregression (VAR) model is to select the lag order of the model. The lag order of the model is the number of past observations to use as predictors. In other words, it is the number of lags of the variables that are included in the model. A higher lag order will include more past observations as predictors, which can improve the model's accuracy but also increase the complexity of the model. There are several ways to select the lag order of a VAR model, one of the most common methods is to use information criteria such as Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC). 1. Akaike Information Criterion (AIC): AIC is a measure of the relative quality of a statistical model. It balances the goodness of fit of the model with the complexity of the model. The mathematical formulation of AIC is ??? = 2? − 2??(?) Where, ● k = Number of parameters in the model ● L = Maximized value of the likelihood function A lower AIC value indicates a better model. 2. Bayesian Information Criterion (BIC): BIC is similar to AIC, but it gives more weight to the number of parameters in the model, making it more conservative than AIC. The mathematical formulation of BIC is ??? = ???(?) − 2??(?) Where, ● k = Number of parameters in the model ● n = Number of observations ● L = Maximized value of the likelihood function A lower BIC value indicates a better model. Both AIC and BIC are used to compare different models with different lag orders and select the one with the lowest value of AIC or BIC. It is important to note that the selection of the lag order is a trade-off between the model's accuracy and complexity, and it depends on the specific problem. It's worth noting that, while these criteria are widely used in practice, they are not the only methods to select the lag order and other methods such as Hannan-Quinn criterion, Schwarz criterion and Final prediction error also exist and can be used to select the best lag order. 12.

Step 4: Stacking of Variables: The fourth step in the process of using a Vector Autoregression (VAR) model is to create a matrix of observations and a matrix of lagged observations. This step is also known as "Stacking" the variables. The matrix of observations contains the current values of all the variables and the matrix of ?? lagged observations contains the past values of all the variables. The number of lags used in ??−1 the matrix of lagged observations is determined by the lag order selected in step 3. For example, if we have 2 variables (x and y) and we selected a lag order of 2, the matrix of observations would be: ?? ?? = [ ?? ?? ] And the matrix of lagged observations would be: ??−1 ?? = [ ??−1 ??−1 ] ??−2 ??−2 Once the matrices of observations and lagged observations are created, they can be used to estimate the coefficients of the VAR model. It's worth noting that the order of the matrix of lagged observations is determined by the number of variables and the lag order. More specifically, the order of the matrix of lagged observations is (number of variables x lag order). Step 5: Estimation of the Parameters of the model: This step consists of the estimation of the coefficients of the model. The coefficients of the VAR model are the parameters that describe the linear relationship between the past values of all the variables and the current value of each variable. There are two main methods to estimate the coefficients of a VAR model: the method of ordinary least squares (OLS) and maximum likelihood estimation (MLE). 1. Method of Ordinary Least Squares (OLS): OLS is used to estimate the coefficients of the linear relationship between the past values of all the variables and the current value of each variable. The OLS method estimates the coefficients by minimizing the sum of squared differences between the observed values of the variables and the values predicted by the model. The difference between the observed values of the variables( and the values predicted by the ??) model( is given by(ignoring the constant term c) ???−1) 13.

?? = ?? − ???−1 The OLS estimates of the coefficients can be obtained by: ) ? = (??−1 ? ??−1) −1 (??−1 ? ?? Where, ● is the matrix of the estimated coefficients ? ● is the matrix of the lagged values we stacked before ??−1 ● is the matrix of the observed values we stacked before ?? 2. Maximum likelihood estimation (MLE): MLE is used to estimate the coefficients of the linear relationship between the past values of all the variables and the current value of each variable. The MLE method finds the values of the coefficients that maximize the likelihood function. The likelihood function is a measure of the probability of observing the data, given the coefficients and the error term distribution. The mathematical formulation of MLE for VAR is as follows: ?(?, Σ) = 2Π − ?? 2 |Σ| − ? 2 ???(− 1 2 (Σ) −1(?? − ???−1) ?(?? − ???−1)) Where, ● is the likelihood function ?(?, Σ) ● n is the number of observations ● k is the number of variables in the model ● is the matrix of current values of all the variables ?? ● is the matrix of coefficients ? ● is the matrix of past values of all the variables ??−1 ● Σ is the covariance matrix of the error terms ● |Σ| denotes the determinant of the matrix Σ The goal of MLE is to find the values of A and Σ that maximize the likelihood function. This is done by taking the derivative of the likelihood function with respect to A and Σ and setting them equal to zero. Once the derivatives are set to zero, the resulting equations can be solved for the values of A and Σ that maximize the likelihood function. 14.

It is worth noting that MLE assumes that the errors are normally distributed with zero mean and variance Σ, which is unknown and should be estimated as well. Once the coefficients are estimated, the VAR model can be used to forecast future values of the variables by plugging in the estimated coefficients and the past values of the variables. Step 6: Forecasting using the model: This step consists of forecasting future values using past data. Once the parameters of the model have been estimated, it can be used to make predictions about future values of the variables in the model. The VAR model can be used to make point forecasts, which are predictions for specific future values of the variables, or to make interval forecasts, which are predictions for a range of possible future values of the variables. It's worth noting that, to make accurate predictions, it's important to use the most recent data available and to take into account any known future events that may affect the variables in the model. Also, it's important to remember that all forecasts are uncertain and that the future may differ from the predictions made by the model. Step 7: Diagnosis and Evaluation of the model : After the estimation of the parameters of the model and forecasting, this step involves checking the assumptions of the model, assessing the goodness of fit, and checking for any potential problems such as autocorrelation, heteroskedasticity, and multicollinearity. 1. Checking the assumptions of the model: The VAR model makes several assumptions such as linearity of the model, normally distributed errors with zero mean and constant variance, and independence of the errors. It's important to check if these assumptions are met by analyzing the residuals of the model. ● Linearity of the model: The linearity of the model can be checked by plotting the residuals against the predicted values. If a pattern is observed, then the linearity assumption is violated. ● Normality of errors: The normality of errors can be checked by creating a normal probability plot of the residuals, or by performing a normality test such as the Jarque-Bera test. ● Constant variance of errors: The constant variance of errors can be checked by creating a plot of residuals against time or against predicted values, if a pattern is observed, then the assumption of constant variance is violated. 2. Assessing the goodness of fit: The goodness of fit of the model is evaluated by looking at the R-squared value. Additionally, the root mean squared error (RMSE) and mean absolute error (MAE) can also be used to evaluate the goodness of fit. ● R-squared value: The R-squared value is a measure of how well the model fits the data.It is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). it ranges from 0 to 1, a higher value indicates a better fit. 15.

● Root Mean Squared Error (RMSE): The RMSE is a measure of the difference between the predicted values and the actual values. It is calculated as the square root of the mean squared error. ● Mean Absolute Error (MAE): The MAE is a measure of the difference between the predicted values and the actual values. It is calculated as the mean of the absolute errors. 3. Checking for Autocorrelation: Autocorrelation is a problem that occurs when the residuals of the model are correlated. It can be checked using the Durbin-Watson statistic or the Ljung-Box test. 4. Checking for Heteroskedasticity: Heteroskedasticity is a problem that occurs when the variance of the residuals is not constant. It can be checked using the Breusch-Pagan test or the White test. 5. Checking for Multicollinearity: Multicollinearity is a problem that occurs when the independent variables in the model are highly correlated. It can be checked using the variance inflation factor (VIF) or the condition index. Step 8: Iteration: Repeat steps 3-7 for different lag orders and select the best model with the lowest AIC or BIC. In this step, the modeler will try different lag orders for the VAR model and compare the results using model selection criteria such as AIC (Akaike Information Criterion) or BIC (Bayesian Information Criterion). These criteria are used to compare models with different numbers of parameters and to select the model that best balances model fit. The model with the lowest AIC or BIC is considered to be the best model. Step 9: Interpretation of results and Conclusion: The final step for using a Vector Autoregression (VAR) model is to interpret the results and draw conclusions. This step involves summarizing the findings, making recommendations based on the results of the model etc ● Summarize the findings: Summarize the main results of the model, including the coefficients of the relationships between the variables, forecasted results, the goodness of fit measures and any test statistics that were calculated. ● Interpret the results: Interpret the results in the context of the research question and the data. Explain the meaning of the coefficients and test statistics, and how they relate to the research question. ● Make recommendations: Based on the results of the model, make recommendations for further research or for actions that should be taken. ● Discuss the limitations: Discuss the limitations of the model, including any assumptions that were made and any potential sources of error. To conclude, building a VAR model involves a series of these important steps that help to create a valid and reliable model. 16.

3.1.5 Advantages and Limitations: ● Advantages: 1. Flexibility: VAR models can be used to analyze multiple time series, allowing for the modeling of complex relationships among variables. 2. Causality: VAR models can be used to infer causality relationships among variables, by identifying which variables are leading and which are lagging. 3. Simplicity: VAR models are relatively simple to estimate and interpret, compared to other multivariate models. 4. Forecasting of multiple time series: VAR models can be used for forecasting multiple time series, providing predictions for multiple variables at once. ● Limitations: 1. Assumptions: VAR models make strong assumptions about the stationarity, error terms etc, violations of these assumptions can lead to biased or inefficient estimates. 2. Limited interpretability: VAR models can be difficult to interpret in terms of the underlying economic or causal mechanisms. 3. Limited explanatory power: VAR models can struggle to capture non-linear or time-varying relationships among variables. 4. Limited ability to handle exogenous variables: VAR models do not easily accommodate exogenous variables, such as economic policy changes or external shocks. 3.2 Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) : 3.2.1 Description: ARFIMA (AutoRegressive Fractionally Integrated Moving Average) is a statistical model that is used for time series forecasting. It is an extension of the traditional ARIMA (AutoRegressive Integrated Moving Average) model, which is used for stationary time series. The ARFIMA model can be represented as ARFIMA(p,d,q) with a difference parameter d, representing the degree of differencing needed to make the series stationary, p is the order of autoregression, q is the order of moving average.The difference parameter d is a fraction here(also known as fractional differencing), between -0.5 and 0.5, and it allows the model to capture the long-term dependencies in the data. The ARFIMA model is commonly used in financial time series, economics, and other fields where non-stationary time series are present. Additionally, the model can also be used in combination with exogenous variables and it can be used for multivariate time series. 17.

3.2.2 Mathematical Expression: The mathematical representation of an ARFIMA (AutoRegressive Fractionally Integrated Moving Average) model is a combination of the autoregressive (AR) model and the moving average (MA) model, with the added capability of fractional differencing. An ARFIMA model is represented mathematically as: (1 − φ1? − φ2? 2 −.... − φ?? ?)(1 − ?) ??? = (1 + Θ1? + Θ2? 2 +.... + Θ?? ?)?? Where, ● B is the Backshift Operator ● is the value of time series at time t ?? ● d is the order of fractional differencing ? ∈ (− 0. 5, 0. 5) ● p is the order of autoregression ● q is the order of moving average ● ?? is the white noise term at time t which represents the random disturbances or errors in the model, and ~ N(0,Σ) ?? ● The term represents the autoregressive part of the (1 − φ1? − φ2? 2 −.... − φ?? ?) model, which describes the dependence of the current value of the time series on its past values. ● The term represents the fractional differencing part of the model, which is used (1 − ?) ? to capture long-term dependencies in the data. ● The term represents the moving average part of the (1 + Θ1? + Θ2? 2 +.... + Θ?? ?) model, which describes the dependence of the current value of the time series on the past errors or innovations. It's worth noting that this mathematical representation of the ARFIMA model allows it to capture long-term dependencies in the data, as well as the short-term dependencies captured by the AR and MA models. Backshift Operator(B): B is the backshift operator, it is a mathematical operator used in time series analysis to shift the values of a time series back in time. It is used in the mathematical representation of ARIMA (AutoRegressive Integrated Moving Average) and ARFIMA (AutoRegressive Fractionally Integrated Moving Average) models. In the context of time series analysis, the backshift operator is defined as, ? ??? = ??−? 18.

Where, ● is the backshift operator ? ● is the value of time series at time t ?? ● is the number of time periods shifted back ? ● is the value of the time series k period back ??−? The backshift operator is used to express the past values of a time series in terms of the current value. This operator can also be used for denoting differencing and fractional differencing. The differencing can be given by the formula, Δ ??? = (1 − ?) ??? Where, ● is the d th order differencing of Δ ??? ?? ● B is the Backshift Operator ● d is the order of differencing It is worth noting that, after the checking of the stationarity of the data, if the data comes out to be non-stationary then this differencing(fractional) will be applied to make the data stationary and then the model can be estimated using the coefficients and the stationary series. 3.2.3 Uses in various fields: Auto-Regressive Fractionally Integrated Moving Average(ARFIMA) models have a wide range of applications in various fields, some of the most common uses are: ● Economics: ARFIMA models are commonly used in economics to model time series data such as GDP, inflation, or unemployment. ● Finance: ARFIMA models are also used in finance to model time series data such as stock prices, exchange rates, or interest rates. ● Weather and Climate: ARFIMA models are also used in the field of weather and climate to model time series data such as temperature, precipitation, or wind speed. ● Engineering: ARFIMA models are used in engineering to model time series data in fields such as power systems, communication systems, and control systems. ● Medicine: ARFIMA models are used in medical research to model time series data such as heart rate, blood pressure, or patient outcomes. The ability of ARFIMA models to capture long-term dependencies in the data makes them well-suited for different sectors of human interest.In summary, ARFIMA models have a wide range of applications in various fields such as Economics, Finance, Weather and Climate, Engineering, Medicine and many others, They are used to model time series data that exhibit long-term memory, long-term trends or cyclical patterns. 19.

3.2.4 Steps Involved in applying ARFIMA: Step 1: Checking for Stationarity of the data: First step in the process of using an Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) model is to check the stationarity of the time series data. Stationarity of a time series means that the statistical properties of the series such as mean, variance, and autocorrelation are constant over time. ARFIMA models are only appropriate for stationary time series, so it is important to check for stationarity before using the model. There are several statistical tests that can be used for this purpose, such as the Augmented Dickey-Fuller (ADF) test or the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. The ADF test is a statistical test that tests the null hypothesis that a time series is non-stationary. The KPSS test checks the null hypothesis that there is no unit root in the time series. A significantly less p-value in the test indicates that the series is non-stationary. It is important to note that these tests assume that the error term is normally distributed with mean zero and constant variance. Step 2: Selection of Order of differencing(d), Order of Auto-regression(p) and Order of Moving Average(q): In this step selection of Order of differencing(d), Order of Auto-regression(p) and Order of Moving Average(q) is done. There are various estimators for selecting the order of differencing(d), order of auto-regression(p) and order of moving average(q) in the ARFIMA model. Some of them are as follows: 1. Periodogram Estimator: The Periodogram Estimator is a method used to estimate the parameters of the ARFIMA model, specifically the order of differencing (d) and the order of the autoregressive (p) and moving average (q) terms. The periodogram estimator is based on the periodogram, which is a graphical representation of the power spectral density of a time series. The periodogram is used to identify any dominant frequencies or patterns in the time series, which can be used to estimate the parameters of the ARFIMA model. The periodogram estimator begins by plotting the periodogram of the time series, and then looking for patterns or peaks in the periodogram. The presence of a dominant peak in the periodogram suggests the presence of a seasonal pattern in the data, and the order of differencing (d) is estimated based on the location of the peak. Once the order of differencing (d) is estimated, the periodogram estimator can be used to estimate the order of the autoregressive (p) and moving average (q) terms. This is typically done by looking for patterns or peaks in the periodogram that correspond to the autoregressive and moving average terms of the model. 20.

2. Robinson Estimator: The Robinson Estimator is a method used to estimate the parameters of an ARFIMA model, specifically the order of differencing (d) and the order of the autoregressive (p) and moving average (q) terms. The Robinson Estimator is based on the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots of a time series. The ACF and PACF plots are used to identify patterns in the time series, which can be used to estimate the parameters of the ARFIMA model. The Robinson Estimator begins by plotting the ACF and PACF of the time series, and then looking for patterns or significant lags in the plots. The order of differencing (d) is estimated based on the decay of the ACF and PACF plots. The order of autoregression (p) is the number of lags for which the PACF plot is significantly different from zero, and the order of moving average (q) is the number of lags for which the ACF plot is significantly different from zero.. One of the advantages of the Robinson Estimator is its ability to handle non-stationary time series data, unlike other methods like the periodogram estimator which assumes the data to be stationary. 3. Whittle Estimator: The Whittle Estimator is a method used to estimate the parameters of an ARFIMA model, specifically the order of differencing (d) and the order of the autoregressive (p) and moving average (q) terms. The Whittle Estimator is based on the Maximum Likelihood Estimation (MLE) method and uses the power spectrum of the time series to estimate the parameters of the ARFIMA model. The power spectrum is a representation of the distribution of power in a time series, as a function of frequency. The Whittle Estimator begins by plotting the power spectrum of the time series, and then looking for patterns or peaks in the spectrum. The order of differencing (d) is estimated based on the location of the peak in the spectrum, and the order of the autoregressive (p) and moving average (q) terms are estimated based on the shape of the spectrum. One of the advantages of the Whittle Estimator is its ability to provide a frequency-domain interpretation of the ARFIMA model, which can be useful in identifying patterns and trends in the time series data. It's worth noting that, identifying the order of autoregression (p) and moving average (q) is an important step in the ARFIMA model because these parameters control the amount of autoregression and moving average in the model and they play an important role in determining the model's performance and forecasting accuracy. Step 3: Fractional Differencing of the data: A key feature of the ARFIMA model is that it allows the use of fractional differencing, this allows the model to capture the long-term dependencies in the data. The fractional differencing part of the model is denoted by which is obtained by (1 − ?) ? 21.

(1 − ?) ? = ?=0 ∞ ∑ Γ(? − ?) ? ? Γ(−?)Γ(? + 1) Where, ● B is the Backshift operator ● d is the order of differencing ? ∈ (− 0. 5, 0. 5) ● is the gamma function Γ In summary, this method allows the ARFIMA model to capture long-term dependencies in the data by allowing the model to consider past observations with a weight that decreases gradually as the lag increases, this means that the more distant past observations have less weight than more recent observations, which allows the model to capture long-term dependencies in the data. Step 4: Estimation of the Parameters of the model: The fourth step in the ARFIMA model is the estimation of the parameters of the model. The parameters to be estimated in the ARFIMA model are the autoregressive coefficients ( ), the φ? moving average coefficients ( ), and the fractional differencing order (d)(which is already Θ? estimated before) The two most common and widely used estimation methods for the ARFIMA model are Maximum Likelihood Estimation (MLE) and Bayesian Inference. 1. Maximum Likelihood Estimation (MLE): The MLE method is a statistical method that is used to estimate the parameters of the ARFIMA model. It involves finding the values of the parameters that maximize the likelihood of the observed data given the model. The MLE method is widely used in ARFIMA modeling due to its flexibility and robustness. 2. Bayesian Inference: The Bayesian inference method is another widely used method for estimating the parameters of the ARFIMA model. It involves using prior information about the parameters to update the probability distribution of the parameters given the observed data. This method estimates the parameters of the ARFIMA model by sampling from the posterior probability distribution using techniques such as Markov Chain Monte Carlo (MCMC) methods. There are other methods like Least Squares Estimation (LSE), Yule-Walker method, Hannan-Rissanen method etc for estimation of the parameters as well. It is worth noting that estimation of the parameters of the model is a very important step in applying ARFIMA model as the quality of the forecasted values depend upon it. Step 5: Forecasting using the model: This step consists of forecasting future values using past data after the model is selected. Once the parameters of the model have been estimated, it can be used to make predictions about future values of the variables in the model. 22.

This can be done by putting estimated parameter values in the main equation of the model and using past data to forecast the future data. It's worth noting that, to make accurate predictions, it's important to use the most recent data available and to take into account any known future events that may affect the variables in the model. Also, it's important to remember that all forecasts are uncertain and that the future may differ from the predictions made by the model Step 6: Diagnosis and Evaluation of the model: Diagnosis and evaluation of ARFIMA models is an important step in the model building process, as it helps to ensure that the model is a good fit for the data and that it can be used for forecasting and other applications. 1. Residual Analysis: After fitting the ARFIMA model to the data, the residuals of the model are analyzed to check for any patterns or outliers that may indicate a poor fit of the model. The residuals should be normally distributed with zero mean and constant variance. 2. Assessing the goodness of fit: The goodness of fit of the model is evaluated by the Root Mean Squared Error (RMSE), Mean Squared Error (MSE) and Mean Absolute Error (MAE) etc. ● Mean Absolute Error (MAE) and Mean Squared Error (MSE): These are measures of the accuracy of the forecast generated by the model. Lower values of MAE and MSE indicate a better fit of the model to the data. MAE (Mean Absolute Error) is calculated by taking the absolute value of the difference between the forecasted values and the actual values, and then averaging the differences over a set of observations. MSE (Mean Squared Error) is calculated by squaring the differences between the forecasted values and the actual values, averaging the squared differences over a set of observations. ● Root Mean Squared Error (RMSE): The RMSE is a measure of the difference between the predicted values and the actual values. It is calculated as the square root of the mean squared error. 3. Checking for Autocorrelation: Autocorrelation is a problem that occurs when the residuals of the model are correlated.The Ljung-Box test is a statistical test used to check for the presence of autocorrelation in the residuals of the model. 4. Backtesting: Backtesting is a method that involves using historical data to test the performance of the model. The model is used to generate forecasts for historical data, and the accuracy of the forecasts is compared to the actual data. Step 7: Iteration: Repeat steps 2-6 for different differencing orders, autoregressive orders and moving average order and select the best model with the lowest AIC or BIC. In this step, the modeler will try differencing orders, autoregressive orders and moving average order for the ARFIMA model and compare the results using model selection criteria such as AIC (Akaike Information Criterion) or BIC (Bayesian Information Criterion). These criteria are used to compare models with different 23.

numbers of parameters and to select the model that best balances model fit. The model with the lowest AIC or BIC is considered to be the best model. Step 8: Interpretation of results and Conclusion: The final step for using an Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) model is to interpret the results and draw conclusions. This step involves summarizing the findings, making recommendations based on the results of the model etc ● Summarize the findings: Summarize the main results of the model, including the parameter coefficients, forecasted results, the goodness of fit measures and any test statistics that were calculated. ● Interpret the results: Interpret the results in the context of the research question and the data. Explain the meaning of the coefficients and test statistics, and how they relate to the research question. ● Make recommendations: Based on the results of the model, make recommendations for further research or for actions that should be taken. ● Discuss the limitations: Discuss the limitations of the model, including any assumptions that were made and any potential sources of error. To conclude, building an ARFIMA model involves a series of these important steps that help to create a valid and reliable model. 3.2.5 Advantages and Limitations: ● Advantages: 1. Handling of non-stationarity: ARFIMA models are particularly useful for modeling non-stationary time series, which are common in many real-world applications. 2. Handling of long-term dependencies: ARFIMA models can handle long-term dependencies in the data, which is important for modeling many types of time series data. 3. Better forecasting performance: ARFIMA models have been shown to have better forecasting performance than traditional time series models such as ARIMA models and exponential smoothing models. 4. High interpretability: The parameters of ARFIMA models have a clear interpretation in terms of the underlying data-generating process, making it easy to understand the model and its predictions. ● Limitations: 1. Complexity: ARFIMA models can be complex and require a high level of expertise to implement, particularly when estimating the model parameters. 24.

2. Difficulty in model selection: Selecting the appropriate order of differencing (d), autoregressive order (p), and moving average order (q) can be difficult, and different methods may produce different results. 3. Assumptions: Like any statistical model, ARFIMA models make assumptions like normality of error terms etc if these assumptions are not met, the model may not be appropriate. 4. Limited to univariate time series: ARFIMA models can only be used for univariate time series analysis, which limits its applicability in some cases where multivariate time series are needed. 3.3 Holt’s Linear Trend Exponential Smoothing Model: 3.3.1 Description: Holt's Linear Trend Exponential Smoothing model is a type of time series forecasting model that is an extension of simple exponential smoothing. It is used to capture the increasing or decreasing nature of the data, and it is particularly useful for modeling time series with a linear trend. The model is simple and easy to understand and it can be used to forecast data in various fields such as sales, inventory, and economic time series data. The model is based on the idea of estimating the level and trend components of the time series separately. The level component represents the average value of the time series, and the trend component represents the increasing or decreasing nature of the time series. One of the main advantages of Holt's Linear Trend model is its simplicity and ease of use. The model requires only two smoothing parameters to be estimated, which can be done using methods such as MSE, MAE or Grid Search. The model is also suitable for modeling time series with a linear trend and it is robust against small fluctuations. However, it is not suitable for time series with high volatility and irregular fluctuations and it assumes that the time series is linear and that no seasonality exists. 3.3.2 Mathematical Expression: The level in a time series data is explained by the average value of time series for a specific time period. The trend in a time series is explained by the average increase or decrease of the values over a period of time. Now, The simple exponential smoothing equation is given by, ??+1 = α?? + (1 − α)?? Where, ● is the forecasted value at time step ??+1 ? + 1 ● is the smoothing factor α 25.

● is the the most recent observed value of the time series ?? ● is the previous forecast ?? As stated earlier, this model has no trend or seasonal component and leads to a flat forecast (all forecasts will be the same and equal to the most recently observed value). Therefore, in 1957 Charles Holt extended this model to include a trend component , . ?? The Holt’s Linear Trend Exponential Smoothing equation is given by, ??+1 = α?? + (1 − α)(?? + ??) ?? = β(?? − ??−1) + (1 − β)??−1 Where, ● is the forecasted value at time step ??+1 ? + 1 ● is the the most recent observed value of the time series ?? ● and are the previous forecasts at time step t and ?? ??−1 ? − 1 ● is the forecasted trend component at time step ?? ? ● ??−1 is the value of trend at at time step ? − 1 ● α is the smoothing factor, 0 ≤ α ≤ 1 ● β is the trend smoothing factor, 0 ≤ β ≤ 1 It is worth noting that this model is particularly useful for modeling time series with a linear trend, but it can't handle seasonality. 3.3.3 Uses in various fields: Holt's Linear Trend model is used to forecast time series data with a linear trend. It is particularly useful in fields such as: ● Sales forecasting: The model can be used to forecast future sales for a business, based on past sales data. This can help businesses plan for inventory and production needs, as well as make strategic decisions about marketing and pricing. ● Agriculture: The model can be used to predict yield of different agricultural products, vegetables etc based on historical data. These can help farmers and people related to the business in this sector for determining storage size, cost etc. ● Inventory forecasting: The model can be used to forecast the demand for a product, based on past demand data. This can help businesses plan for inventory needs and optimize their supply chain operations. 26.

● Finance: The model can be used to forecast stock prices, currency exchange rates, and other financial indicators, based on historical data. This can help investors and traders make investment decisions. These are the various fields in which Holt’s Linear Trend Exponential Smoothing model can be used. 3.3.4 Steps involved in applying Holt’s Linear Trend Exponential Smoothing: Step 1: Data Preparation: In Holt's Linear Trend model, data preparation is the process of cleaning, transforming, and organizing the data before it is used to fit the model. This may include tasks such as removing missing or duplicate values, converting data into a suitable format, and ensuring that the data is in the correct time-series order. Data preparation for Holt's Linear Trend model may also include: ● Determining the appropriate frequency of the data (e.g. daily, monthly, quarterly) ● Identifying and dealing with outliers or anomalies in the data ● Checking for seasonality and adjusting for it if necessary as this model can’t handle data containing seasonality. ● Verifying that the data is consistent and does not contain any errors or inconsistencies ● Normalizing or scaling the data if necessary It's also worth noting that Holt's Linear Trend model is a type of exponential smoothing model, which is used for predicting future values of a time series. Therefore, it is important that the data is prepared in such a way that it accurately reflects the underlying trends and patterns of the time series. This often requires careful examination of the data and may involve some trial and error to determine the best approach. Step 2: Estimation of Initial conditions: Since all equations for the Holt-Winters method are recurrence relations, we need to supply a set of initial values to these estimating equations to get the forecasting started.Specifically, we need to set the values of and . ?0 ?0 1. Estimation of : can be estimated by just simply using the first observed value in the ?0 ?0 time series data or taking a simple average of some initial values in time series data. ?0 = 1/?(?0 + ?1 +... + ??) Where, ● is the initial forecasted value ?0 ● are initial initial observed value in the time series data ?0, ?1, .... , ?? ? + 1 27.

2. Estimation of : can be estimated by simply taking the initial trend value or taking the ?0 ?0 simple average of some initial trend values calculated from the time series data. ?0 = 1/? Where, ● is the initial forecasted trend value ?0 ● are initial initial observed value in the time series data ?0, ?1, .... , ?? ? + 1 It's worth noting that the choice of initial values can have a big impact on the final results, and it may be necessary to try different values to find the best values for the parameters. Step 3: Estimate the Smoothing Parameters: Estimation of smoothing parameters in Holt's Linear Trend model refers to the process of finding the optimal values for the smoothing coefficients (α and β) that minimize the difference between the predicted and actual values of the time series. This can be done by intuition, grid search or various optimization models. 1. By Intuition: By using intuition, one can set the initial value of smoothing coefficients according to the degree of uncertainty they expect in their data. Commonly used values are between 0.1 and 0.3, but it can be adjusted based on the data and the problem at hand. The intuition behind this approach is that, if you expect a lot of uncertainty in your data, we would want to give more weight to the current and recent values when making predictions. In this case, we should set a higher value for the smoothing coefficient (alpha or beta) which would result in the model giving more weight to the current and recent values. On the other hand, if you expect less uncertainty in your data, we would want to give more weight to the historical values when making predictions. In this case, we should set a lower value for the smoothing coefficient, which would result in the model giving reasonable weight to the historical values. It's important to note that the intuition method for initialization of smoothing coefficients is based on the assumptions of the user and may not always be the best approach. It's a good starting point but it's always good to validate the results using other methods such as grid search or optimization algorithms. 2. Grid Search: Grid search is a method used in estimating the parameters of Holt's linear trend exponential smoothing model, which is a type of time series forecasting method that combines exponential smoothing with a linear trend component. In Grid search, a set of possible parameter values for the model are defined and a search is conducted over the entire set of possible combinations of parameters. The algorithm will 28.

evaluate each combination and select the best-performing set of parameters based on a chosen evaluation metric such as Mean Squared Error (MSE) or Mean Absolute Error (MAE). For example, Holt's method requires two parameters: the smoothing factor (α) and the trend factor (β). A grid search could be used to test a range of values for α and β, such as 0.1 to 0.9 in increments of 0.1. The algorithm would then evaluate all possible combinations of α and β values, such as (0.1, 0.1), (0.1, 0.2), (0.1, 0.3), and so on, and select the combination that results in the best performance according to the chosen evaluation metric. Grid search can be computationally expensive, especially for large sets of possible parameter values, but it can be an effective way to find the best set of parameters for the Holt's method model. Step 4: Forecasting using the Model: Once the model is fit to the data, it can be used to make predictions for future values of the time series. This is done by using the estimated parameters and the historical data to calculate the forecasted values. It's worth noting that, to make accurate predictions, it's important to use the most recent data available and to take into account any known future events that may affect the variables in the model. Also, it's important to remember that all forecasts are uncertain and that the future may differ from the predictions made by the model. Step 5: Updating the model: This step involves updating the level and trend components with the new forecast and observed value. And then repeat the previous process again. Step 6: Diagnosis and Evaluation of the model: Diagnosis and evaluation are important steps in Holt's Linear Trend model that help to ensure that the model is fit to the data, and that the predictions made by the model are accurate. The steps involved in this process include: 1. Residual analysis: This involves plotting the residuals (the difference between the predicted and actual values) and analyzing them to ensure that they are randomly distributed and have a constant variance. If the residuals show a pattern, it can indicate that the model is not a good fit for the data and needs to be refined. 2. Assessing the goodness of fit: Various statistical measures like Mean Absolute Error (MAE), Mean Squared Error (MSE) and Root Mean Squared Error (RMSE) can be used to evaluate the accuracy of the model. These measures provide a quantitative way to compare the model's predictions to the actual values. 3. Plotting the forecasts: By plotting the forecasts against the actual values, one can visually assess how well the model is performing. A good model should have forecasts that closely match the actual values. 29.

4. Model comparison: Comparing the model to other models or to a naive forecast can help to determine if the model is a good fit for the data and if it is better than the alternatives. It’s worth nothing that these evaluation steps are crucial for ensuring that the model is fit to the data and making accurate predictions. However, it's also important to keep in mind that no model is perfect, and there will always be some level of uncertainty involved in the predictions. The goal is to find a model that performs well and has a good balance of accuracy and simplicity. Step 7: Interpretation of results and Conclusion: Interpreting the results and drawing conclusions is the final step in the Holt's Linear Trend model, which involves using the model's predictions and the diagnostic and evaluation steps to understand what the model is telling us about the data. ● Interpreting the model's predictions: The predictions made by the model can be used to understand the underlying patterns and trends in the data. For example, if the model is predicting an upward trend, it may indicate that the time series is increasing over time. ● Identifying important factors: By analyzing the model's parameters, one can identify the factors that have the most impact on the predictions. For example, if the trend parameter is large, it may indicate that the time series is increasing at a fast rate. ● Assessing the model's performance: By evaluating the model's accuracy measures and comparing it to other models, one can assess how well the model is performing. If the model has a high accuracy, it is likely to be a good fit for the data. ● Drawing conclusions: The results of the model can be used to draw conclusions about the underlying patterns and trends in the data. These conclusions can be used to make predictions about future values of the time series and to support decision making. ● Discuss the limitations: Discuss the limitations of the model, including any assumptions that were made and any potential sources of error. It's worth noting that, The results of the Holt's Linear Trend model are only as good as the data and assumptions that go into the model. Therefore, it is important to critically evaluate the results, and to consider the limitations and assumptions of the model when interpreting the results and drawing conclusions. 3.3.5 Advantages and Limitations: ● Advantages: 1. Simple to implement: Holt's Linear Trend model is relatively simple to implement and can be easily understood by both technical and non-technical audiences. 2. Flexible: Holt's Linear Trend model can be used for a wide range of time series forecasting problems, including data with linear trends. 3. Accurate: Holt's Linear Trend model has been found to be accurate in many time series forecasting problems, particularly when the data has a linear trend. 4. Good performance on large datasets: Holt's Linear Trend model is able to capture the trend of the series with a large number of data points. 30.

● Limitations: 1. Limited to linear trends: The model is based on the assumption that the time series has a linear trend, which may not be the case for all data. 2. Limited to non-seasonal data: Holt's exponential smoothing method shows good forecasting performance in the absence of seasonal or cyclical variations. In other words,it does not work with the data which show the seasonal or cyclical pattern. To overcome this problem Holt-Winters Exponential Smoothing is developed, where a seasonal component is added in the calculation. 3. Sensitive to outliers: Holt's Linear Trend model can be sensitive to outliers, which can lead to inaccurate predictions if not handled properly. 4. Limited in handling non-stationary data: Holt's Linear Trend model assumes that the data is stationary, so if the data is non-stationary, the model may not be able to capture the underlying patterns and trends in the data. 4. Conclusion: In conclusion, time series forecasting is a crucial task in many industries, and there are a variety of models available for forecasting time series data. This term paper reviewed some advanced models, including VAR, ARFIMA and Holt's Linear Trend model. Each model has its own advantages and limitations, and the choice of model depends on the characteristics of the data and the complexity of the problem. VAR and ARFIMA models are powerful methods for time series forecasting, particularly when the data has a complex pattern of seasonality and trend. These models are able to capture the underlying patterns and trends in the data, and can be used to make accurate predictions. The VAR model is also useful for multivariate time series and also for long memory processes. However, they require a high level of expertise to implement, and can be computationally expensive. On the other hand, Holt's Linear Trend model is a simple and widely used method for time series forecasting, which is easy to implement and understand. It is particularly useful for data with a linear trend, and can be used for a wide range of time series forecasting problems. However, it is limited in handling non-stationary data and non-constant seasonality. It is worth noting that no model is perfect, and the results will always be subject to some level of uncertainty. Therefore, it is crucial to carefully evaluate the results of the models, and to consider the limitations and assumptions of the models when interpreting the results and drawing conclusions. Additionally, it is also important to update the models as new data becomes available to ensure that the predictions remain accurate over time. 31.

References ● Baillie, RT., Kongcharoen, C., Kapetanios, G.(2012) “Prediction from ARFIMA models: Comparisons between MLE and semiparametric estimation procedures” In International Journal of Forecasting 28(pp. 46-53) [DOI] ● Liu, K. ,Chen, Y ., Zhang, X.(2017, June 17) “An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs” [DOI] ● Sekar, P(2010) “Diagnostic checking of time series models''. In Indian Journal of Science and Technology (vol: 3, no: 9, pp: 1026-1031) [DOI] ● Y.K. Tse, V.V. Anh, Q. Tieng (2002)”Maximum likelihood estimation of the fractional differencing parameter in an ARFIMA model using wavelets” . In Mathematics and Computers in Simulation 59 (pp. 153–161) [DOI] ● Vance, L Martin., Nigel P. Wilkins (1999) “Indirect estimation of ARFIMA and VARFIMA models” In Journal of Econometrics 9(pp. 149-175) [DOI] ● “Time Series Forecasting: De�inition, Applications, and Examples” https://www.tableau.com/learn/articles/time-series-forecasting ● Nau, R. “The mathematical structure of ARIMA models” https://people.duke.edu/~rnau/Mathematical_structure_of_ARIMA_models--Robert _Nau.pdf ● Prabhakaran, S.(2019, Nov 2) “Augmented Dickey Fuller Test (ADF Test) – Must Read Guide” https://www.machinelearningplus.com/time-series/augmented-dickey-fuller-test/ ● “OLS in Matrix Form” https://web.stanford.edu/~mrosenfe/soc_meth_proj3/matrix_OLS_NYU_notes.pdf ● “Coef�icient of determination” https://en.wikipedia.org/wiki/Coef�icient_of_determination#:~:text=R2%20is%20 a%20measure,predictions%20perfectly%20�it%20the%20data. ● Sangarshanan, (2018, Oct 3) “Time series Forecasting — ARIMA models” https://towardsdatascience.com/time-series-forecasting-arima-models-7f221e9eee 06#:~:text=D%20%3D%20In%20an%20ARIMA%20model,variance%20are%20co nstant%20over%20time. ● Howell, E.(2022, Dec 9) “Time Series Forecasting with Simple Exponential Smoothing” https://towardsdatascience.com/forecasting-with-simple-exponential-smoothing-d d8f8470a14c ● Howell, E(2022, Dec 13) “Time Series Forecasting with Holt’s Linear Trend Exponential Smoothing” https://towardsdatascience.com/forecasting-with-holts-linear-trend-exponential-s moothing-af2aa4590c18 ● “Holt’s Linear Trend” https://www.real-statistics.com/time-series-analysis/basic-time-series-forecasting /holt-linear-trend/ ● Baum, CF. (2013) “ARIMA and ARFIMA models” http://fmwww.bc.edu/EC-C/S2013/823/EC823.S2013.nn08.slides.pdf 32.

● Reisen, V., Abraham, B., Lopes ,S. “Estimation of Parameters in ARFIMA Processes: A Simulation Study” http://mat.ufrgs.br/~slopes/artigos/ver�inal.pdf ● Corduas, M. “Preliminary Estimation of ARFIMA Models” ● “Vector autoregression” https://en.wikipedia.org/wiki/Vector_autoregression ● Prabhakaran, S.(2019) “Vector Autoregression (VAR) – Comprehensive Guide with Examples in Python” https://www.machinelearningplus.com/time-series/vector-autoregression-examples-pytho n/ ● “What Is Time-Series Forecasting?” https://www.timescale.com/blog/what-is-time-series-forecasting/#:~:text=In%20the%20si mplest%20terms%2C%20time,specific%20point%20in%20the%20future. 33.