Math - Time Series Analysis - James D Hamilton - 1994 Pdf

Math - Time Series Analysis - James D Hamilton - 1994 Pdf

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Many books on the subject fall into two categories: classic texts with the basic theories and fundamentals of time series analysis, and revised editions of academic textbooks with real-world examples and exercises. We picked an array that covers the initial introduction to references and guides along with your time series analysis self-study.

This book is a basic introduction to time series and the open-source software R, and is intended for readers who have little to no R knowledge. It gives step-by-step instructions for getting started with time series analysis and how to use R to make it all happen. Each module features practical applications and data to test the analysis. The co-author Paul Cowpertwait also features the data sets on a companion website.

The book gives a good overview of time series analysis without being overwhelming. It covers the basics, including methods, forecasting models, systems, and ARIMA probability models that include studying seasonality. It also includes examples and practical advice and comes with a free online appendix.

Time series analysis is a complex subject, and even these books barely scratch the surface of its uses and evolution. In order to utilize the analysis to its fullest, you have to stay current with new trends and theories, as well as continue to deepen your understanding. To learn more about theories and read real customer stories, check out our time series analysis resources page.

The last decade has brought dramatic changes in the way that researchers analyze economic and financial time series. This book synthesizes these recent advances and makes them accessible to first-year graduate students. James Hamilton provides the first adequate text-book treatments of important innovations such as vector autoregressions, generalized method of moments, the economic and statistical consequences of unit roots, time-varying variances, and nonlinear time series models. In addition, he presents basic tools for analyzing dynamic systems (including linear representations, autocovariance generating functions, spectral analysis, and the Kalman filter) in a way that integrates economic theory with the practical difficulties of analyzing and interpreting real-world data. Time Series Analysis fills an important need for a textbook that integrates economic theory, econometrics, and new results. The book is intended to provide students and researchers with a self-contained survey of time series analysis. It starts from first principles and should be readily accessible to any beginning graduate student, while it is also intended to serve as a reference book for researchers.

Time-Critical Decision Makingfor Business AdministrationPara mis visitantes del mundo de habla hispana, este sitio se encuentra disponible en español en:Sitio Espejo para América Latina Sitio en los E.E.U.U.Realization of the fact that "Time is Money" in business activities, the dynamic decision technologies presented here, have been a necessary tool for applying to a wide range of managerial decisions successfully where time and money are directly related. In making strategic decisions under uncertainty, we all make forecasts. We may not think that we are forecasting, but our choices will be directed by our anticipation of results of our actions or inactions.Indecision and delays are the parents of failure. This site is intended to help managers and administrators do a better job of anticipating, and hence a better job of managing uncertainty, by using effective forecasting and other predictive techniques.Professor Hossein Arsham To search the site, try Edit | Find in page [Ctrl + f]. Enter a word or phrase in the dialogue box, e.g. "cash flow" or "capital cycle" If the first appearance of the word/phrase is not what you are looking for, try Find Next. MENU Chapter 1: Time-Critical Decision Modeling and Analysis Chapter 2: Causal Modeling and ForecastingChapter 3: Smoothing TechniquesChapter 4: Box-Jenkins MethodologyChapter 5: Filtering TechniquesChapter 6: A Summary of Special ModelsChapter 7: Modeling Financial and Economics Time SeriesChapter 8: Cost/Benefit AnalysisChapter 9: Marketing and Modeling Advertising CampaignChapter 10: Economic Order and Production Quantity Models for Inventory ManagementChapter 11: Modeling Financial Economics DecisionsChapter 12: Learning and the Learning CurveChapter 13: Economics and Financial Ratios and Price IndicesChapter 14: JavaScript E-labs Learning ObjectsCompanion Sites:Business StatisticsExcel For Statistical Data Analysis Topics in Statistical Data AnalysisComputers and Computational StatisticsQuestionnaire Design and Surveys SamplingProbabilistic ModelingSystems SimulationProbability and Statistics ResourcesSuccess Science Leadership Decision Making Linear Programming (LP) and Goal-Seeking StrategyLinear Optimization Solvers to Download Artificial-variable Free LP Solution Algorithms Integer Optimization and the Network Models Tools for LP Modeling ValidationThe Classical Simplex MethodZero-Sum Games with ApplicationsComputer-assisted Learning Concepts and TechniquesLinear Algebra and LP ConnectionsFrom Linear to Nonlinear Optimization with Business Applications Construction of the Sensitivity Region for LP Models Zero Sagas in Four DimensionsBusiness Keywords and Phrases Collection of JavaScript E-labs Learning ObjectsCompendium of Web Site Review Chapter 1: Time-Critical Decision Modeling and Analysis IntroductionEffective Modeling for Good Decision-MakingBalancing Success in BusinessModeling for Forecasting Stationary Time SeriesStatistics for Correlated Data Chapter 2: Causal Modeling and ForecastingIntroduction and SummaryModeling the Causal Time SeriesHow to Do Forecasting by Regression AnalysisPredictions by RegressionPlanning, Development, and Maintenance of a Linear ModelTrend AnalysisModeling Seasonality and TrendTrend Removal and Cyclical AnalysisDecomposition Analysis Chapter 3: Smoothing TechniquesIntroductionMoving Averages and Weighted Moving AveragesMoving Averages with TrendsExponential Smoothing TechniquesExponenentially Weighted Moving AverageHolt's Linear Exponential Smoothing TechniqueThe Holt-Winters' Forecasting TechniqueForecasting by the Z-Chart Concluding Remarks Chapter 4: Box-Jenkins MethodologyBox-Jenkins MethodologyAutoregressive Models Chapter 5: Filtering TechniquesAdaptive FilteringHodrick-Prescott FilterKalman Filter Chapter 6: A Summary of Special Modeling TechniquesNeural NetworkModeling and Simulation Probabilistic ModelsEvent History AnalysisPredicting Market ResponsePrediction Interval for a Random VariableCensus II Method of Seasonal AnalysisDelphi AnalysisSystem Dynamics ModelingTransfer Functions MethodologyTesting for and Estimation of Multiple Structural ChangesCombination of ForecastsMeasuring for Accuracy Chapter 7: Modeling Financial and Economics Time SeriesIntroductionModeling Financial Time Series and EconometricsEconometrics and Time Series ModelsSimultaneous EquationsFurther Readings Chapter 8: Cost/Benefit AnalysisThe Best Age to Replace EquipmentPareto AnalysisEconomic QuantityChapter 9: Marketing and Modeling Advertising CampaignMarketing and Modeling Advertising CampaignSelling ModelsBuying ModelsThe Advertising Pulsing PolicyInternet AdvertisingPredicting Online Purchasing BehaviorConcluding RemarksFurther Readings Chapter 10: Economic Order and Production Quantity Models for Inventory ManagementIntroductionEconomic Order and Production Quantity for Inventory ControlOptimal Order Quantity DiscountsFinite Planning Horizon Inventory Inventory Control with Uncertain DemandManaging and Controlling Inventory Chapter 11: Modeling Financial Economics Decisions Markov ChainsLeontief's Input-Output ModelRisk as a Measuring Tool and Decision CriterionBreak-even and Cost AnalysesModeling the Bidding ProcessProduct’s Life Cycle Analysis and ForecastingChapter 12: Learning and The Learning CurveIntroductionPsychology of LearningModeling the Learning CurveAn ApplicationTime-Critical Decision Modeling and Analysis The ability to model and perform decision modeling and analysis is an essential feature of many real-world applications ranging from emergency medical treatment in intensive care units to military command and control systems. Existing formalisms and methods of inference have not been effective in real-time applications where tradeoffs between decision quality and computational tractability are essential. In practice, an effective approach to time-critical dynamic decision modeling should provide explicit support for the modeling of temporal processes and for dealing with time-critical situations. One of the most essential elements of being a high-performing manager is the ability to lead effectively one's own life, then to model those leadership skills for employees in the organization. This site comprehensively covers theory and practice of most topics in forecasting and economics. I believe such a comprehensive approach is necessary to fully understand the subject. A central objective of the site is to unify the various forms of business topics to link them closely to each other and to the supporting fields of statistics and economics. Nevertheless, the topics and coverage do reflect choices about what is important to understand for business decision making.Almost all managerial decisions are based on forecasts. Every decision becomes operational at some point in the future, so it should be based on forecasts of future conditions. Forecasts are needed throughout an organization -- and they should certainly not be produced by an isolated group of forecasters. Neither is forecasting ever "finished". Forecasts are needed continually, and as time moves on, the impact of the forecasts on actual performance is measured; original forecasts are updated; and decisions are modified, and so on.For example, many inventory systems cater for uncertain demand. The inventoryparameters in these systems require estimates of the demand and forecasterror distributions. The two stages of these systems, forecasting andinventory control, are often examined independently. Most studies tend to lookat demand forecasting as if this were an end in itself, or at stockcontrol models as if there were no preceding stages of computation.Nevertheless, it is important to understand the interaction between demandforecasting and inventory control since this influences the performance ofthe inventory system. This integrated process is shown in the following figure: The decision-maker uses forecasting models to assist him or her in decision-making process. The decision-making often uses the modeling process to investigate the impact of different courses of action retrospectively; that is, "as if" the decision has already been made under a course of action. That is why the sequence of steps in the modeling process, in the above figure must be considered in reverse order. For example, the output (which is the result of the action) must be considered first. It is helpful to break the components of decision making into three groups: Uncontrollable, Controllable, and Resources (that defines the problem situation). As indicated in the above activity chart, the decision-making process has the following components: Performance measure (or indicator, or objective): Measuring business performance is the top priority for managers. Management by objective works if you know the objectives. Unfortunately, most business managers do not know explicitly what it is. The development of effective performance measures is seen as increasingly important in almost all organizations. However, the challenges of achieving this in the public and for non-profit sectors are arguably considerable. Performance measure provides the desirable level of outcome, i.e., objective of your decision. Objective is important in identifying the forecasting activity. The following table provides a few examples of performance measures for different levels of management:LevelPerformanceMeasureStrategic Return of Investment, Growth,and InnovationsTactical Cost, Quantity, andCustomer satisfactionOperational Target setting, and Conformance with standardClearly, if you are seeking to improve a system's performance, an operational view is really what you are after. Such a view gets at how a forecasting system really works; for example, by what correlation its past output behaviors have generated. It is essential to understand how a forecast system currently is working if you want to change how it will work in the future. Forecasting activity is an iterative process. It starts with effective and efficient planning and ends in compensation of other forecasts for their performanceWhat is a System? Systems are formed with parts put together in a particular manner in order to pursue an objective. The relationship between the parts determines what the system does and how it functions as a whole. Therefore, the relationships in a system are often more important than the individual parts. In general, systems that are building blocks for other systems are called subsystemsThe Dynamics of a System: A system that does not change is a static system. Many of the business systems are dynamic systems, which mean their states change over time. We refer to the way a system changes over time as the system's behavior. And when the system's development follows a typical pattern, we say the system has a behavior pattern. Whether a system is static or dynamic depends on which time horizon you choose and on which variables you concentrate. The time horizon is the time period within which you study the system. The variables are changeable values on the system. Resources: Resources are the constant elements that do not change during the time horizon of the forecast. Resources are the factors that define the decision problem. Strategic decisions usually have longer time horizons than both the Tactical and the Operational decisions. Forecasts: Forecasts input come from the decision maker's environment. Uncontrollable inputs must be forecasted or predicted. Decisions: Decisions inputs ate the known collection of all possible courses of action you might take. Interaction: Interactions among the above decision components are the logical, mathematical functions representing the cause-and-effect relationships among inputs, resources, forecasts, and the outcome. Interactions are the most important type of relationship involved in the decision-making process. When the outcome of a decision depends on the course of action, we change one or more aspects of the problematic situation with the intention of bringing about a desirable change in some other aspect of it. We succeed if we have knowledge about the interaction among the components of the problem.There may have also sets of constraints which apply to each of these components. Therefore, they do not need to be treated separately. Actions: Action is the ultimate decision and is the best course of strategy to achieve the desirable goal.Decision-making involves the selection of a course of action (means) in pursue of the decision maker's objective (ends). The way that our course of action affects the outcome of a decision depends on how the forecasts and other inputs are interrelated and how they relate to the outcome. Controlling the Decision Problem/Opportunity: Few problems in life, once solved, stay that way. Changing conditions tend to un-solve problems that were previously solved, and their solutions create new problems. One must identify and anticipate these new problems. Remember: If you cannot control it, then measure it in order to forecast or predict it.Forecasting is a prediction of what will occur in the future, and it is an uncertain process. Because of the uncertainty, the accuracy of a forecast is as important as the outcome predicted by the forecast. This site presents a general overview of business forecasting techniques as classified in the following figure:Progressive Approach to Modeling: Modeling for decision making involves two distinct parties, one is the decision-maker and the other is the model-builder known as the analyst. The analyst is to assist the decision-maker in his/her decision-making process. Therefore, the analyst must be equipped with more than a set of analytical methods.Integrating External Risks and Uncertainties: The mechanisms of thought are often distributed over brain, body and world. At the heart of this view is the fact that where the causal contribution of certain internal elements and the causal contribution of certain external elements are equal in governing behavior, there is no good reason to count the internal elements as proper parts of a cognitive system while denying that status to the external elements.In improving the decision process, it is critical issue to translating environmental information into the process and action. Climate can no longer be taken for granted:Societies are becoming increasingly interdependent.The climate system is changing.Losses associated with climatic hazards are rising.These facts must be purposeful taken into account in adaptation to climate conditions and management of climate-related risks.The decision process is a platform for both the modeler and the decision maker to engage with human-made climate change. This includes ontological,ethical, and historical aspects of climate change, as well as relevant questions such as:Doesclimate change shed light on the foundational dynamics of realitystructures?Does it indicate a looming bankruptcy of traditional conceptions of human-nature interplays?Does it indicate the need for utilizing nonwestern approaches, and if so, how?Does the imperative of sustainable development entail a new groundwork for decision maker?How will human-made climate change affect academic modelers -- and how can they contribute positively to the global science and policy of climate change?Quantitative Decision Making: Schools of Business and Management are flourishing with more and more students taking up degree program at all level. In particular there is a growing market for conversion courses such as MSc in Business or Management and post experience courses such as MBAs. In general, a strong mathematical background is not a pre-requisite for admission to these programs. Perceptions of the content frequently focus on well-understood functional areas such as Marketing, Human Resources, Accounting, Strategy, and Production and Operations. A Quantitative Decision Making, such as this course is an unfamiliar concept and often considered as too hard and too mathematical. There is clearly an important role this course can play in contributing to a well-rounded Business Management degree program specialized, for example in finance.Specialists in model building are often tempted to study a problem, and then go off in isolation to develop an elaborate mathematical model for use by the manager (i.e., the decision-maker). Unfortunately the manager may not understand this model and may either use it blindly or reject it entirely. The specialist may believe that the manager is too ignorant and unsophisticated to appreciate the model, while the manager may believe that the specialist lives in a dream world of unrealistic assumptions and irrelevant mathematical language.Such miscommunication can be avoided if the manager works with the specialist to develop first a simple model that provides a crude but understandable analysis. After the manager has built up confidence in this model, additional detail and sophistication can be added, perhaps progressively only a bit at atime. This process requires an investment of time on the part of the manager and sincere interest on the part of the specialist in solving the manager's real problem, rather than in creating and trying to explain sophisticated models. This progressive model building is often referred to as the bootstrapping approach and is the most important factor in determining successful implementation of a decision model. Moreover the bootstrapping approach simplifies the otherwise difficult task of model validation and verification processes. The time series analysis has three goals: forecasting (also called predicting), modeling, and characterization. What would be the logical order in which to tackle these three goals such that one task leads to and /or and justifies the other tasks? Clearly, it depends on what the prime objective is. Sometimes you wish to model in order to get better forecasts. Then the order is obvious. Sometimes, you just want to understand and explain what is going on. Then modeling is again the key, though out-of-sample forecasting may be used to test any model. Often modeling and forecasting proceed in an iterative way and there is no 'logical order' in the broadest sense. You may model to get forecasts, which enable better control, but iteration is again likely to be present and there are sometimes special approaches to control problems.Outliers: One cannot nor should not study time series data without being sensitive to outliers. Outliers can be one-time outliers or seasonal pulses or a sequential set of outliers with nearly the same magnitude and direction (level shift) or local time trends. A pulse is a difference of a step while a step is a difference of a time trend. In order to assess or declare "an unusual value" one must develop "the expected or usual value". Time series techniques extended for outlier detection, i.e. intervention variables like pulses, seasonal pulses, level shifts and local time trends can be useful in "data cleansing" or pre-filtering of observations.Further Readings:Borovkov K., Elements of Stochastic Modeling, World Scientific Publishing, 2003.Christoffersen P., Elements of Financial Risk Management, Academic Press, 2003.Holton G., Value-at-Risk: Theory and Practice, Academic Press, 2003.Effective Modeling for Good Decision-Making What is a model? A Model is an external and explicit representation of a part of reality, as it is seen by individuals who wish to use this model to understand, change, manage and control that part of reality."Why are so many models designed and so few used?" is a question often discussed within the Quantitative Modeling (QM) community. The formulation of the question seems simple, but the concepts and theories that must be mobilized to give it an answer are far more sophisticated. Would there be a selection process from "many models designed" to "few models used" and, if so, which particular properties do the "happy few" have? This site first analyzes the various definitions of "models" presented in the QM literature and proposes a synthesis of the functions a model can handle. Then, the concept of "implementation" is defined, and we progressively shift from a traditional "design then implementation" standpoint to a more general theory of a model design/implementation, seen as a cross-construction process between the model and the organization in which it is implemented. Consequently, the organization is considered not as a simple context, but as an active component in the design of models. This leads logically to six models of model implementation: the technocratic model, the political model, the managerial model, the self-learning model, the conquest model and the experimental model.Succeeding in Implementing a Model: In order that an analyst succeeds in implementing a model that could be both valid and legitimate, here are some guidelines:Be ready to work in close co-operation with the strategic stakeholders in order to acquire a sound understanding of the organizational context. In addition, the QM should constantly try to discern the kernel of organizational values from its more contingent part.The QM should attempt to strike a balance between the level of model sophistication/complexity and the competence level of stakeholders. The model must be adapted both to the task at hand and to the cognitive capacity of the stakeholders.The QM should attempt to become familiar with the various preferences prevailing in the organization. This is important since the interpretation and the use of the model will vary according to the dominant preferences of the various organizational actors.The QM should make sure that the possible instrumental uses of the model are well documented and that the strategic stakeholders of the decision making process are quite knowledgeable about and comfortable with the contents and the working of the model. The QM should be prepared to modify or develop a new version of the model, or even a completely new model, if needed, that allows an adequate exploration of heretofore unforeseen problem formulation and solution alternatives.The QM should make sure that the model developed provides a buffer or leaves room for the stakeholders to adjust and readjust themselves to the situation created by the use of the model and The QM should be aware of the pre-conceived ideas and concepts of the stakeholders regarding problem definition and likely solutions; many decisions in this respect might have been taken implicitly long before they become explicit.In model-based decision-making, we are particularly interested in the idea that a model is designed with a view to action.Descriptive and prescriptive models: A descriptive model is often a function of figuration, abstraction based on reality. However, a prescriptive model is moving from reality to a model a function of development plan, means of action, moving from model to the reality.One must distinguishes between descriptive and prescriptive models in the perspective of a traditional analytical distinction between knowledge and action. The prescriptive models are in fact the furthest points in a chain cognitive, predictive, and decision making. Why modeling? The purpose of models is to aid in designing solutions. They are to assist understanding the problem and to aid deliberation and choice by allowing us to evaluate the consequence of our action before implementing them.The principle of bounded rationality assumes that the decision maker is able to optimize but only within the limits of his/her representation of the decision problem. Such a requirement is fully compatible with many results in the psychology of memory: an expert uses strategies compiled in the long-term memory and solves a decision problem with the help of his/her short-term working memory. Problem solving is decision making that may involves heuristics such as satisfaction principle, and availability. It often, involves global evaluations of alternatives that could be supported by the short-term working memory and that should be compatible with various kinds of attractiveness scales. Decision-making might be viewed as the achievement of a more or less complex information process and anchored in the search for a dominance structure: the Decision Maker updates his/her representation of the problem with the goal of finding a case where one alternative dominant all the others for example; in a mathematical approach based on dynamic systems under three principles:Parsimony: the decision maker uses a small amount of information.Reliability: the processed information is relevant enough to justify -- personally or socially -- decision outcomes. Decidability: the processed information may change from one decision to another. Cognitive science provides us with the insight that a cognitive system, in general, is an association of a physical working device that is environment sensitive through perception and action, with a mind generating mental activities designed as operations, representations, categorizations and/or programs leading to efficient problem-solving strategies. Mental activities act on the environment, which itself acts again on the system by way of perceptions produced by representations. Designing and implementing human-centered systems for planning, control, decision and reasoning require studying the operational domains of a cognitive system in three dimensions:An environmental dimension, where first, actions performed by a cognitive system may be observed by way of changes in the environment; and second, communication is an observable mode of exchange between different cognitive systems.An internal dimension, where mental activities; i.e., memorization and information processing generate changes in the internal states of the system. These activities are, however, influenced by partial factorizations through the environment, such as planning, deciding, and reasoning. An autonomous dimension where learning and knowledge acquisition enhance mental activities by leading to the notions of self- reflexivity and consciousness.Validation and Verification: As part of the calibration process of a model, the modeler must validate and verified the model. The term validation is applied to those processes, which seek to determine whether or not a model is correct with respect to the "real" system. More prosaically, validation is concerned with the question "Are we building the right system?" Verification, on the other hand, seeks to answer the question "Are we building the system right?"Balancing Success in BusinessWithout metrics, management can be a nebulous, if not impossible, exercise. How can we tell if we have met our goals if we do not know what our goals are? How do we know if our business strategies are effective if they have not been well defined? For example, one needs a methodology for measuring success and setting goals from financial and operational viewpoints. With those measures, any business can manage its strategic vision and adjust it for any change. Setting a performance measure is a multi-perspective at least from financial, customer, innovation, learning, and internal business viewpoints processes. The financial perspective provides a view of how the shareholders see the company; i.e. the company's bottom-line. The customer perspective provides a view of how the customers see the company. While the financial perspective deals with the projected value of the company, the innovation and learning perspective sets measures that help the company compete in a changing business environment. The focus for this innovation is in the formation of new or the improvement of existing products and processes. The internal business process perspective provides a view of what the company must excel at to be competitive. The focus of this perspective then is the translation of customer-based measures into measures reflecting the company's internal operations.Each of the above four perspectives must be considered with respect to four parameters: Goals: What do we need to achieve to become successful?Measures: What parameters will we use to know if we are successful? Targets: What quantitative value will we use to determine success of the measure?Initiatives: What will we do to meet our goals?Clearly, it is not enough to produce an instrument to document and monitor success. Without proper implementation and leadership, creating a performance measure will remain only an exercise as opposed to a system to manage change.Further Readings:Calabro L. On balance, Chief Financial Officer Magazine, February 01, 2001. Almost 10 years after developing the balanced scorecard, authors Robert Kaplan and David Norton share what they've learned.Craven B., and S. Islam, Optimization in Economics and Finance, Springer , 2005.Kaplan R., and D. Norton, The balanced scorecard: Measures that drive performance, Harvard Business Review, 71, 1992. Modeling for Forecasting:Accuracy and Validation AssessmentsForecasting is a necessary input to planning, whether in business, or government. Often, forecasts are generated subjectively and at great cost by group discussion, even when relatively simple quantitative methods can perform just as well or, at very least; provide an informed input to such discussions.Data Gathering for Verification of Model: Data gathering is often considered "expensive". Indeed, technology "softens" the mind, in that we become reliant on devices; however, reliable data are needed to verify a quantitative model. Mathematical models, no matter how elegant, sometimes escape the appreciation of the decision-maker. In other words, some people think algebraically; others see geometrically. When the data are complex or multidimensional, there is the more reason for working with equations, though appealing to the intellect has a more down-to-earth undertone: beauty is in the eye of the other beholder - not you; yourself.The following flowchart highlights the systematic development of the modeling and forecasting phases: Modeling for Forecasting Click on the image to enlarge it and THEN print it The above modeling process is useful to: understand the underlying mechanism generating the time series. This includes describing and explaining any variations, seasonallity, trend, etc. predict the future under "business as usual" condition.control the system, which is to perform the "what-if" scenarios.Statistical Forecasting: The selection and implementation of the proper forecast methodology has always been an important planning and control issue for most firms and agencies. Often, the financial well-being of the entire operation rely on the accuracy of the forecast since such information will likely be used to make interrelated budgetary and operative decisions in areas of personnel management, purchasing, marketing and advertising, capital financing, etc. For example, any significant over-or-under sales forecast error may cause the firm to be overly burdened with excess inventory carrying costs or else create lost sales revenue through unanticipated item shortages. When demand is fairly stable, e.g., unchanging or else growing or declining at a known constant rate, making an accurate forecast is less difficult. If, on the other hand, the firm has historically experienced an up-and-down sales pattern, then the complexity of the forecasting task is compounded.There are two main approaches to forecasting. Either the estimate of future value is based on an analysis of factors which are believed to influence future values, i.e., the explanatory method, or else the prediction is based on an inferred study of past general data behavior over time, i.e., the extrapolation method. For example, the belief that the sale of doll clothing will increase from current levels because of a recent advertising blitz rather than proximity to Christmas illustrates the difference between the two philosophies. It is possible that both approaches will lead to the creation of accurate and useful forecasts, but it must be remembered that, even for a modest degree of desired accuracy, the former method is often more difficult to implement and validate than the latter approach. Autocorrelation: Autocorrelation is the serial correlation of equally spaced time series between its members one or more lags apart. Alternative terms are the lagged correlation, and persistence. Unlike the statistical data which are random samples allowing us to perform statistical analysis, the time series are strongly autocorrelated, making it possible to predict and forecast. Three tools for assessing the autocorrelation of a time series are the time series plot, the lagged scatterplot, and at least the first and second order autocorrelation values.Standard Error for a Stationary Time-Series: The sample mean for a time-series, has standard error not equal to S / n ½, but S[(1-r) / (n-nr)] ½, where S is the sample standard deviation, n is the length of the time-series, and r is its first order correlation.Performance Measures and Control Chart for Examine Forecasting Errors:Beside the Standard Error there are other performance measures. The following are some of the widely used performance measures: Performance Measures for Forecasting Click on the image to enlarge it and THEN print itIf the forecast error is stable, then the distribution of it is approximately normal. With this in mind, we can plot and then analyze the on the control charts to see if they might be a need torevise the forecasting method being used. To do this, if we divide a normal distribution into zones, with each zone one standard deviation wide, then one obtains the approximate percentage we expect to find in each zone from a stable process.Modeling for Forecasting with Accuracy and Validation Assessments:Control limits could be one-standard-error, or two-standard-error, and any point beyond these limits (i.e., outside of the error control limit) is an indication the need to revise the forecasting process, as shown below: A Zone on a Control Chart for Controlling Forecasting Errors Click on the image to enlarge it and THEN print it The plotted forecast errors on this chart, not only should remain with the control limits, they should not show any obvious pattern, collectively.Since validation is used for the purpose of establishing a model’s credibility it is important that the method used for the validation is, itself, credible. Features of time series, which might be revealed by examining its graph, with the forecasted values, and the residuals behavior, condition forecasting modeling.An effective approach to modeling forecasting validation is to hold out a specific number of data points for estimation validation (i.e., estimation period), and a specific number of data points for forecasting accuracy (i.e., validation period). The data, which are not held out, are used to estimate the parameters of the model, the model is then tested on data in the validation period, if the results are satisfactory, and forecasts are then generated beyond the end of the estimation and validation periods. As an illustrative example, the following graph depicts the above process on a set of data with trend component only: Estimation Period, Validation Period, and the Forecasts Click on the image to enlarge it and THEN print it In general, the data in the estimation period are used to help select the model and to estimate its parameters. Forecasts into the future are "real" forecasts that are made for time periods beyond the end of the available data.The data in the validation period are held out during parameter estimation. One might also withhold these values during the forecasting analysis after model selection, and then one-step-ahead forecasts are made.A good model should have small error measures in both the estimation and validation periods, compared to other models, and its validation period statistics should be similar to its own estimation period statistics.Holding data out for validation purposes is probably the single most important diagnostic test of a model: it gives the best indication of the accuracy that can be expected when forecasting the future. It is a rule-of-thumb that one should hold out at least 20% of data for validation purposes.You may like using the Time Series' Statistics JavaScript for computing some of the essential statistics needed for a preliminary investigation of your time series.Stationary Time SeriesStationarity has always played a major role in time series analysis. To perform forecasting, most techniques required stationarity conditions. Therefore, we need to establish some conditions, e.g. time series must be a first and second order stationary process.First Order Stationary: A time series is a first order stationary if expected value of X(t) remains the same for all t.For example in economic time series, a process is first order stationary when we remove any kinds of trend by some mechanisms such as differencing.Second Order Stationary: A time series is a second order stationary if it is first order stationary and covariance between X(t) and X(s) is function of length (t-s) only.Again, in economic time series, a process is second order stationary when we stabilize also its variance by some kind of transformations, such as taking square root. You may like using Test for Stationary Time Series JavaScript.Statistics for Correlated DataWe concern ourselves with n realizations that are related to time, that is having n correlated observations; the estimate of the mean is given by 2b1af7f3a8